Mathematical Discourse: Language, Symbolism and Visual Images

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Mathematical Discourse Language, Symbolism and Visual Images

Kay L. O'Halloran

continuum LONDON

•

NEW YORK

Continuum The Tower Building, 11 York Road, London SE1 7NX

15 East 26th Street, New York, NY 10010

© Kay L. O'Halloran 2005 All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0-8264-6857-8 (hardback) Library of Congress Cataloging-in-Publication Data A catalogue record for this book is available from the Library of Congress Typeset by RefineCatch Ltd, Bungay, Suffolk Printed and bound in Great Britain by Cromwell Press Ltd, Trowbridge, Wilts

Contents Acknowledgements

viii

Copyright Permission Acknowledgements

ix

1

Mathematics as a Multisemiotic Discourse 1.1 The Creation of Order 1.2 Halliday's Social Semiotic Approach 1.3 Mathematics as Multisemiotic 1.4 Implications of a Multisemiotic View 1.5 Tracing the Semiotics of Mathematics 1.6 Systemic Functional Research in Multimodality

1 1 6 10 13 17 19

2

Evolution of the Semiotics of Mathematics 2.1 Historical Development of Mathematical Discourse 2.2 Early Printed Mathematics Books 2.3 Mathematics in the Early Renaissance 2.4 Beginnings of Modern Mathematics: Descartes and Newton 2.5 Descartes' Philosophy and Semiotic Representations 2.6 A New World Order

22 22 24 33 38 46 57

3

Systemic Functional Linguistics (SFL) and Mathematical Language 3.1 The Systemic Functional Model of Language 3.2 Interpersonal Meaning in Mathematics 3.3 Mathematics and the Language of Experience 3.4 The Construction of Logical Meaning 3.5 The Textual Organization of Language 3.6 Grammatical Metaphor and Mathematical Language 3.7 Language, Context and Ideology

60 60 67 75 78 81 83 88

4

The Grammar of Mathematical Symbolism 4.1 Mathematical Symbolism 4.2 Language-Based Approach to Mathematical Symbolism 4.3 SF Framework for Mathematical Symbolism 4.4 Contraction and Expansion of Experiential Meaning 4.5 Contraction of Interpersonal Meaning 4.6 A Resource for Logical Reasoning 4.7 Specification of Textual Meaning 4.8 Discourse, Grammar and Display 4.9 Concluding Comments

94 94 96 97 103 114 118 121 125 128

VI 5

CONTENTS The Grammar of Mathematical Visual Images

129

5.1 5.2 5.3 5.4 5.5 5.6

129 133 139 142 145

The Role of Visualization in Mathematics SF Framework for Mathematical Visual Images Interpersonally Orientating the Viewer Visual Construction of Experiential Meaning Reasoning through Mathematical Visual Images Compositional Meaning and Conventionalized Styles of Organization 5.7 Computer Graphics and the New Image of Mathematics 6

7

146 148

Intersemiosis: Meaning Across Language, Visual Images and Symbolism

158

6.1 6.2 6.3 6.4 6.5 6.6

158 163 171 177 179 184

The Semantic Circuit in Mathematics Intersemiosis: Mechanisms, Systems and Semantics Analysing Intersemiosis in Mathematical Texts Intersemiotic Re-Contexualization in Newton's Writings Semiotic Metaphor and Metaphorical Expansions of Meaning Reconceptualizing Grammatical Metaphor

Mathematical Constructions of Reality

189

7.1 Multisemiotic Analysis of a Contemporary Mathematics Problem 7.2 Educational Implications of a Multisemiotic Approach to Mathematics 7.3 Pedagogical Discourse in Mathematics Classrooms 7.4 The Nature and Use of Mathematical Constructions

189 199 205 208

References

211

Index

223

In memory of my father, Jim O'Halloran. For my brother Greg and his family.

Acknowledgements

This study of mathematical discourse is based on Michael Halliday's systemic functional model of language and Jim Martin's extensive contributions to systemic theory. Michael O'Toole's application of systemic functional theory to displayed art provides the inspiration for the models for mathematical symbolism and visual image presented here. Jay Lemke pioneered the application of systemic functional theory to science and mathematics as multisemiotic discourses. This work would not be possible without these founding contributions. My special thanks to the director and librarians from the John Rylands University Library of Manchester (JRULM) for making so readily available the mathematics manuscripts in the Mathematical Printed Collection. I thank Linda Thompson (Director of the Language and Literacy Studies Research Group, Faculty of Education, University of Manchester) for supporting my visit to JRULM. My special thanks to Philip J. Davis (Emeritus Professor, Applied Mathematics Division, Brown University) for his interest in this project. Our lively correspondence has contributed to the contents of this book. I thank Michael O'Toole and Frances Christie for their friendship and support, and I thank my past and present friends and colleagues at the National University of Singapore - most notably Joe Foley, Chris Stroud, Linda Thompson, Desmond Allison, Ed McDonald, Umberto Ansaldo and Lisa Lim.

Copyright Permission Acknowledgements

The author is grateful to the following organizations for the right to reproduce the images which appear in this book. Every effort has been made to contact copyright holders of material produced in this book. The publishers apologize for any omissions and will be pleased to rectify them at the earliest opportunity. Chapter 1 Plate 1.1(1)

Plate 1.3(1)

Photographs from Beevor (2002: Chapter 24) The photographs are reprinted with kind permission from: Photograph 43: Bildarchiv PreuBischer Kulturbesitz, Berlin Photograph 44: Ullstein Bild, Berlin Photograph 45: Jiirgen Stumpff/Bildarchiv PreuBischer Kulturbesitz, Berlin Language, Visual Images and Symbolism (Kockelkoren et al, 2003: 173) Reprinted with kind permission from Elsevier

Chapter 2 Plate 2.2(1) The Treviso Arithmetic (reproduced from Swetz, 1987: 140) Reprinted by permission from Open Court Publishing Company, a division of Carus Publishing Company, Peru, IL from Capitalism and Arithmetic by F. Swetz, © 1987 by Open Court Publishing Company The following have been reprinted by courtesy of the director and librarian, the John Rylands University Library of Manchester: Plate 2.2(2) Plate 2.2(3)

The Hindu-Arabic system versus counters and lines (Reisch, 1535: 267) Printing counter and line calculations (Reisch, 1535: 326)

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COPYRIGHT PERMISSION ACKNOWLEDGEMENTS

Plate 2.2(4a) Euclid's Elements: Venice 1482, Erhard Ratdolt (ThomasStanford, 1926: Illustration II) Plate 2.2(4b) Euclid's Elements: Venice 1505, J. Tacuinus (ThomasStanford, 1926: Illustration IV) Plate 2.2(6) QuadraturaParaboles (Archimedes, 1615: 437) Plate 2.3(2) Hitting a target (Tartaglia, 1546: 7) Plate 2.3(3) Arithmetic calculations to hit a target (Tartaglia, 1546: 106) Plate 2.3(4) Positioning a target (Galileo, 1638: 67) Plate 2.4(la) Removing the human body (Descartes, 1682: 111) Plate 2.4(2a) Movement in space and time: the stone (Descartes, 1682: 217) Plate 2.4(2b) Movement in space and time: the model (Descartes, 1682:

217) Plate 2.4(4a) Plate 2.4(4b) Plate 2.4(5a) Plate 2.4(5b) Plate 2.5(1)

Context, circles and lines (Descartes, 1682: 226) Circles and lines (Descartes, 1682: 228) Descartes' semiotic compass (1683: 54) (Book Two) Drawing the curves (Descartes, 1683: 20) (Book Two) Illustration from Newton's (1729) The Mathematical Principles of Natural Philosophy

My thanks to the Bodleian Library, Oxford for access to the microfilm of the following manuscript held by the British Library, London: Plate 2.4(8a) Newton's (1736: 80-81) Method of Fluxions and Infinite Series Plate 2.4(8b) Newton's (1736: 100) Method of Fluxions and Infinite Series The following have been reprinted by courtesy of Dover Publications: Plate Plate Plate Plate Plate

2.2(5) 2.4(lb) 2.4(6a) 2.4(6b) 2.4(7)

Translation of Euclid (reproduced from Euclid, 1956: 283) Removing the human body: Newton (1952: 9) Descartes' description of curves (1954: 234) Descartes' use of symbolism (1954: 186) Newton's algebraic notes on Euclid (reproduced from Cajori, 1993: 209) Latin edition (1655) of Barrow's Euclid (Taken from Isaac Newton: A Memorial Volume [ed. WJ. Greenstreet: London, 1927], p. 168)

Chapter 4 Plate 4.3(1)

Mathematical Symbolic Text (Stewart, 1999: 139) From Calculus: Combined Single and Multivariable 4th edition by Stewart. © 1999. Reprinted with permission of Brooks/ Cole, a division of Thompson Learning: www.thompsonrights.com. Fax: 800 730-2215

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The following are reprinted with permission from Elsevier: Plate 4.5(1) Plate 4.7(1)

Mathematical Symbolic Text (Wei and Winter, 2003: 159) Textual Organization of Mathematical Symbolism (Clerc, 2003: 117)

Chapter 5 Plate 5.2(1)

Plate 5.7(1)

Interpretation of the Derivative as the Slope of a Tangent (Stewart, 1999: 130) From Calculus: Combined Single and Multivariable 4th edition by Stewart. © 1999. Reprinted with permission of Brooks/ Cole, a division of Thompson Learning: www.thompsonrights.com. Fax: 800 730-2215 Evolving Images of Computer Graphics (a) Figure 5 Stills from a computer-made movie: wrapping a rectangle to form a torus (Courtesy T. Banchoff and C. M. Strauss) (Davis, 1974: 126) Reprinted by kind permission of T. Banchoff and C. M. Strauss through Philip J. Davis (Emeritus Professor, Applied Mathematics Division, Brown University Providence, RI, USA) (b) MATLAB graphics, circa 1985 (courtesy of Philip J. Davis) Reprinted with kind permission Philip J. Davis (Emeritus Professor, Applied Mathematics Division, Brown University Providence, RI, USA) (d) Graphical and Diagrammatic Display of Patterns (Berge et al., 2003: 194) Reprinted with kind permission from Elsevier

Chapter 6 Plate 6.31

Plate 6.32

Newton's (1736: 46) Procedure for Drawing Tangents Reprinted from microfilm in the Bodleian Library, Oxford. Reproduced with permission from the British Library (London) which holds the original manuscript. The Derivate as the Instantaneous Rate of Change (Stewart, 1999: 132) From Calculus: Combined Single and Multivariable 4th edition by Stewart. © 1999. Reprinted with permission of Brooks/ Cole, a division of Thompson Learning: www.thompsonrights.com. Fax: 800 730-2215

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Chapter 7 Plate 7.1 (1)

Mathematics Example 2.24 (Burgmeier et al, 1990: 76-77) From Burgmeier, J. W., Boisen, M. B. and Larsen, M. D. (1990) Brief Calculus with Applications. New York: McGraw-Hill. © 1990. Reprinted with kind permission from The McGraw-Hill Publishing Company, New York

1 Mathematics as a Multisemiotic Discourse

1.1 The Creation of Order Success is right. What does not succeed is wrong. It was, for example, wrong to persecute the Jews before the war since that set the Anglo-Americans against Germany. It would have been right to postpone the anti-Jewish campaign and begin it after Germany had won the war. It was wrong to bomb England in 1940. If they had refrained, Great Britain, so they believe, would have joined Hitler in the war against Russia. It was wrong to treat Russian and Polish [prisoners of war] like cattle since now they will treat Germans in the same way. It was wrong to declare war against the USA and Russia because they were together stronger than Germany.

In this extract from Berlin: The Downfall 1945, Beevor (2002:429) summarizes the views of over three hundred pro-Nazi generals after Germany's defeat in the Second World War, based on a report of interviews by the Supreme Headquarters Allied Expeditionary Force in Europe (SHAEF). The German generals are seen to possess a view of events; one they envisaged would have worked towards victory rather than defeat. Their guiding principle, as expressed by Beevor (2002: 429), is 'Success is right. What does not succeed is wrong.' Many millions participated in the enactment of those views, and the familiar question arises as to how this could be possible. How could so many people be persuaded to take part in the events which unfolded during the course of the Second World War? There have been a variety of responses to this question. Goldhagen (1996), for instance, suggests that most of the ordinary Germans involved in the holocaust were 'willing executioners' who actually believed in the events that took place. No doubt a variety of means were used incrementally over a long period of time in order to mobilize the population in the war effort. In the past century, such massive mobilizations have not been confined to Germany. Weitz (2003), for example, documents the unprecedented programmes of genocide which have taken place in the twentieth century, including Stalin's Soviet Union, Cambodia under the Khmer Rouge and the former Yugoslavia. In these cases and many others, significant portions of the population take part in the war effort. But how can so many people be convinced of the necessity of such programmes, the impact of which lasts for generations?

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In attempting to answer such a question, it is worthwhile to consider a simple reformulation of the German generals' guiding principles 'Success is right. What does not succeed is wrong' (Beevor, 2002: 429). That is, if the phrase for the Nazi party is inserted, the statement becomes 'Success [for the Nazi party] is right. What does not succeed [for the Nazi party] is wrong.' Such a reformulation introduces in unequivocal terms the basis upon which the guiding principles are constructed. The simple inclusion of the beneficiary 'the Nazi party' makes clear the premise underlying the linguistic statement, and the specific interests which are being served. Such an inclusion also provides room for argumentation and negation, whereas the finality accompanying the original cliched statement 'Success is right' is much more difficult to counteract. In a similar manner, the import of linguistic choices may be seen in George W. Bush's statement to the world after 11 September 2001 attacks on the United States: 'Either you are with us, or you are with the terrorists' (CNN.com/US 20 September 2001). Expressed in simple terms of a relational set of circumstances, the dichotomy is based on pro-American interests ('with us') versus anti-American interests ('against us'). Such a simple division of the world into two opposing sets of relations leaves few options for a negotiated peace settlement along other possible lines of interest. Language functions in this way to structure the world largely in terms of categories, the nature of which depends upon the choices which are made. The value of using language and other systems of meaning to create a world view conducive to the war effort was well recognized in Nazi Germany. These strategies included the use of the media for news reports and documentaries (involving language, visual images and music), political speeches and rallies (for example, language, visual images, embodied action, music, and architectural features of the platform and seating arrangements), and particular styles of dress and the distinctive salute of the Nazi party (for example, the uniforms, insignias, actions and gestures). These strategies have direct parallels in existence today, where choices from the different resources combine to create particular meanings to the exclusion of others. However, the contexts which give rise to the ordering of reality are not confined to those which are specifically designed for mass consumption in the form of 'propaganda' programmes. Order is maintained, negotiated and challenged in every situation which involves choices from language, visual images, gesture, styles of dress and so forth. Page (2001: 10) comments: 'There is a privilege in being raised in a time of peace. A luxury that your life is not under immediate threat. War becomes something labelled as heroic, often patriotic, nationalistic. There is a cause, it is just and right, and it somehow excuses all the pain and all the loss.' The use of language and other sign systems for the structuring of thought and reality in the ways described by Page is the subject of this study. This approach is not intended to downplay strategies of physical and mental coercion and abuse. However, violence commences somewhere, and in many cases, for ordinary citizens at least, the starting point is the ordering

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of reality along certain lines through semiosis; that is, acts of meaning through choices from language and other sign systems. The major aim of this study is to introduce a theory and approach for examining the nature and impact of semiosis in contexts which span the supposedly inane to the discourses of immense influence, which include the subject matter of this investigation; namely, mathematics and science. War is chosen as the topic to introduce this approach. The role of language for structuring thought and reality is well recognized today within a wide range of disciplines which include sociolinguistics, critical discourse theory, communication studies, psychology and sociology (for example, Berger and Luckmann, 1991; Bourdieu, 1991; Fairclough, 1989; Gumperz, 1982; Halliday, 1978; Herman and Chomsky, 1988; Vygotsky, 1986). In addition, the functions of visual images are increasingly taken into account (for example, Barthes, 1972; Lynch and Woolgar, 1990; Mirzoeff, 1998; van Leeuwen andjewitt, 2001). This is especially important in the electronic age where the ease with which pictorial representations may be reproduced is expanding. Beevor (2002), for example, includes visual images in the form of black and white photographs and maps to depict the advance of the Red Army and the final collapse of the Third Reich. Berlin: The Downfall 1945 is a text or discourse constructed through choices from the English language, photographs and maps. These choices work together to create Beevor's account of the horror of the final months of the Second World War in Germany. In what follows, the types of meanings afforded by Beevor's (2002) photographs are investigated and compared to meanings which are made using language. Photographs 43-45 displayed in Plate 1.1(1) appear in Chapter 24 in Beevor (2002: 354—369). These photographs appear among a group of inserted photographs which are numbered 30-49. As seen in Plate 1.1(1), Photograph 43 is a picture of a German teenage conscript at the end of the war, Photograph 44 shows a Russian female medical assistant attending to a wounded Russian soldier, and the official signing of the final surrender by General Stumpff, Field Marshal Keitel and Admiral von Friedburg in May 1945 is shown in Photograph 45. In Beevor (2002), these pictures are preceded by photographs of Russians engaged in street fighting in Berlin, scenes outside the Reich Chancellery, convoys of Russian-controlled armed forces, German soldiers surrendering in Berlin, Russian soldiers washing and civilians cooking in the streets of Berlin, victory celebrations between delegates from the Red Army and the US Army, and German civilians escaping across the Elbe River to American territory. Immediately following Plate 1.1(1), there are further photographs of soldiers in the streets of Berlin, the Russian victory parade, and a full-page photograph of Red Army officials visiting the battleground inside the Reichstag. The photographs displayed in Plate 1.1(1) have contextual meaning within this sequence of photographs. Beevor's (2002) linguistic account of the fall of Berlin similarly unfolds as a staged text consisting of sentences, paragraphs, pages

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43

45

Plate 1.1(1) Photographs from Beevor (2002: Chapter 24) and chapters which have contextual meaning within the sequence of the narrative. However, there are differences in the types of meaning afforded by Beevor's linguistic and photographic account of the fall of Berlin. These

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differences relate to the meaning potential of language and visual images. This point is developed below. From existing photographs of the fighting and aftermath in Berlin in 1945, a selection of photographs has been chosen to be included in Beevor (2002). In turn, each photograph in the sequence represents a set of choices made by the photographer, which, in the case of war, most likely happen more by chance rather than design. The photographer captures an instance of time according to the camera angle, the camera distance, the perspective and light conditions, for example. Certain scenes are frozen within the frame, and within those frames human figures are engaged in some form of action in a setting. Further to this, the photographs are developed and reproduced under certain conditions which include choices in terms of paper quality, darkroom techniques, and the possibility for various forms of editing, including cropping and erasure. Putting aside the materiality of the medium and the production process, following O'Toole's (1994) framework for the analysis of paintings, each photograph represents choices at the rank of the whole frame or the Work (in terms of the setting, actions and circumstance), the Episodes in each frame (the activities which are captured), the Figures (the individual people and other objects) and their Members (in terms of body parts and parts of the objects). The impact of these choices in the photographs displayed in Plate 1.1(1) merit close attention. The settings, physical actions, gestures, facial expressions and the nature of the averted gazes of the human figures in the photographs are juxtaposed in what is a grotesque opposition between the devastation faced by those involved in the fighting (Photographs 43-44) and the well-fed and well-attired defiance of those taking part in the official surrender (Photograph 45). This opposition is marked at each rank of the Work, Episode, Figure and Member. For example, the contrast between the physical and emotional state of the soldiers, the medical attendant and the German generals becomes evident in a glance. The quality, style and condition of their respective uniforms at the rank of Figure and Member are similarly diametrically opposed. Compositionally, even the grainy quality of the street scene where the Russian medical assistant attends to the injuries suffered by a soldier (Photograph 44) is placed in stark opposition to the smooth textual quality of the photograph of the official German surrender (Photograph 45). The situational contexts, actions, experiences and the emotional and physical states of the participants in the fall of Berlin according to circumstance, nationality, age, gender and position are thus constructed by the photographs. Even if Beevor had the space to describe these dimensions, the meanings of these black and white photographs are impossible to exactly reproduce in narrative form. A linguistic description cannot make the same meanings as Photographs 43-45. The scenes, the interplay of Episodes, the actions and events, the mood of the Figures realized through their embodied actions and appearance cannot be captured using words.

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In the same manner, 'Success [for the Nazi party] is right. What does not succeed [for the Nazi party] is wrong' cannot be captured pictorially. Different resources such as language and visual images have different potentials to create meaning. In simplest terms, language tends to order the world in terms of categorical-type distinctions, while visual images such as photographs create order in a manner which to varying degrees accords with our dynamic perceptual experience of the world. The two types of meanings afforded by language and visual images combine in Beevor's account of the fall of Berlin and the collapse of the Third Reich. The semantic realm explored in this study is not war, rather it is the world offered by mathematics, the discourse which underlies the scientific view of the world. This world came into being largely through the development and refinement of a new sign system, namely mathematical symbolism, which was designed to function in co-operation with language and specialized forms of visual images. The mathematical and the scientific ways of ordering the world permeate our everyday existence, and thus the aim of this study is to understand the nature and the implications of such a view. Before moving to the field of mathematics, Michael Halliday's social-semiotic approach which informs this study is introduced. 1.2 Halliday's Social Semiotic Approach

We impose order on the world, and that order is expressed semiotically through choices from a variety of sign systems. These semiotic resources, or sign systems, include language, paintings and other forms of visual images, music, embodied systems of meaning such as gesture, action and stance, and three-dimensional man-made items and objects such as clothes, sculptures and buildings. A culture may be understood as typical configurations of choices from a variety of semiotic resources. The lecture, the pop song, the political speech, the news report and the textbook are to a large extent predictable configurations of semiotic choices. In a general sense, this understanding of semiotics pertains to 'the specificity of human semiosis' (Cobley, 2001: 260) where 'Semiosis is the name given to the action of signs. Semiotics might therefore be understood as the study of semiosis or even as a "metasemiosis", producing "signs about signs" '(Cobley, 2001: 259). As Cobley (2001: 259) claims, 'Behind this simple definition [of semiotics] lies a universe of complexity.' Noth (1990) describes the diversity in theoretical and applied approaches to study of semiotics and Chandler (2002: 207) sees semiotics as 'a relatively loosely defined critical practice rather than a unified, fullyfledged analytical method or theory'. There are many schools and branches of theoretical and applied semiotics, with various definitions and meanings. Noth (1990), for instance, categorizes semiotics as being concerned with the study of language and language-based codes, text (for example, rhetoric and stylistics, poetry, theatre and drama, narrative, myth, ideology and theology), non-verbal communication, aesthetics and visual

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communication. Noth (1990: 5-6) provides alternative subdivisions which include the semiotics of culture, multimedia communication, popular culture, anthropology, ethnosemiotics, and other topics such as psychosemiotics, socio-semiotics and semiotic sociology, together with the semiotics of disciplines such as mathematics, psychiatry, history and so forth. Michael Halliday's (1978, 1994, 2004) social-semiotic theory of language known as Systemic Functional Linguistics (SFL) is located within the theoretical realm of what Noth (1990: 6) terms 'socio-semiotics'. Halliday is concerned with the social interpretation of the meaning of language, and this view is extended to include other semiotic resources such as the maps and photographs found in Beevor (2002) and the mathematical symbolism and diagrams found in the discourse of mathematics. While the basic tenets of the Hallidayan approach to language are introduced below, more comprehensive accounts may be found elsewhere (for example, Bloor and Bloor, 1995; Eggins, 1994; Martin, 1992; Martin and Rose, 2003; Thompson, 1996). Halliday (1978, 1994) sees language as a tool, where the means through which language is used to achieve the desired results are located within the grammar. The grammar is theorized according to the functions language is required to serve. Halliday (1994) identifies the 'metafunctions of language' as (i) the experiential - the construction of our experience of the world, (ii) the logical - the construction of logical relations in that world, (iii) the interpersonal - the enactment of social relations, and (iv) the textual - the means for organizing the message. The grammatical systems through which these four metafunctions of language are realized are described in Chapter 3. From the Hallidayan perspective, meaning is thus made through choices from the metafunctionally based grammatical systems. The meaning of a choice (the sign or the syntagm) is understood in relation to the other possible choices within the system networks (the paradigmatic options). Halliday uses the term 'social semiotic' to explain that the meanings of the signs (the semiotic choices) depend on the context of use (the social). The meanings arising from choices from the system networks are negotiated within the social and cultural context in which those choices are made. For example, a linguistic statement such as 'Success is right' does not exist as an abstract independent entity. Rather, the statement means within a context of use, in this case in Beevor's (2002) account of the fall of Berlin. In the same fashion, contexts are established semiotically. For example, the fall of Berlin is constructed by Beevor (2002) and other historians through choices from the semiotic resources of language, maps and photographs. Similarly, the academic lecture is a typical configuration of semiotic choices from the resources of language, visual images, dress, gesture, objects, architecture, seating, lighting and so forth. The configuration of the academic lecture is recognizable by members of a culture, even though the form varies according to discipline and institution.

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In order to communicate, members of a culture, or groups within that culture, must possess some sense of shared contextual meaning. Being part of a culture means learning, using and experimenting with the meaning potential of the semiotic systems to create, maintain and negotiate the reality which is socially constructed. Semiotic activity is also used for acts of resistance, which may materialize, for example, in the form of email messages or websites where 'standard' linguistic practices are subverted from the point of view of grammar, lexical choice, text colour and graphics. The dynamic nature of the electronic medium is such that the distinction between the spoken and written modes becomes increasingly blurred with the variations in genre configurations, language choices and graphical representations. However, these new practices eventually become in themselves standardized in much the same way that video texts in the music industry become predictable. The resistance which some discourses initially appear to offer (for example, in the music and film industry, sport and the internet) typically become absorbed into mainstream culture, often in the form of re-packaged commercial products. The contextual values attached to different choices or combinations of choices from semiotic resources are socially and culturally determined. Members of a culture recognize and maintain or resist those values. Companies such as McDonalds, Nike and Coca Cola, for example, invest large amounts of money in advertising to ensure that their brands and accompanying icons maintain 'the right' social value among the other products on offer. In this way they seek to create and maintain a market for groups of consumers. In one study, Cheong (2004) found that apart from the interpersonally salient component of an advertisement designed to attract the attention of the reader (in many cases a visual image), the only obligatory item in a print advertisement is the company logo. Presumably if the logo was missing, the intertextual relations with other texts in the advertising campaign would ensure that the brand is easily identifiable. Advertising as such means creating an image so that the product or service is viewed as desirable by groups of members of a community. Buying the product thus means acquiring the social and cultural connotative value of that product (Barthes, 1972, 1974). Human life is negotiated through semiotic exchange within the realms of situational and cultural contexts. Certain combinations of selections function more prominently to structure reality to the exclusion of others. Studies in Systemic Functional Linguistics (SFL) attempt to document and account for the typical linguistic patterns in different types of social interaction or genres; for example, casual conversation (Eggins and Slade, 1997), service encounters (Ventola, 1987), pedagogical discourse (Christie, 1999; Christie and Martin, 1997; O'Halloran, 2000, forthcoming b; Unsworth, 2000) and scientific writing (Martin, 1993b; Martin and Veel, 1998). Other studies of language look at typical patterns along contextual parameters such as gender (for example, Tannen, 1995) and sexuality (for example, Cameron and Kulick, 2003). Forensic linguistics, on the other

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hand, is concerned with identifying typical language patterns of the individual (for example, Coulthard, 1993). Bourdieu's (1991) notion of symbolic and cultural capital of the 'habitus', which is the set of acquired dispositions of an individual or group of individuals, may be conceptualized as semiotic capital; that is, the ability to construct, interpret and reconstruct the world in contextually specific ways. However, the ability to make appropriate meanings in a range of contexts through the use of semiotic resources is unevenly distributed across sections of any community or culture. The reason for this unequal distribution of semiotic capital is related to the educational, economic, social and cultural background of individuals and groups within any community. For example, Bernstein (1977,1990) identifies the disadvantages students from lower social class backgrounds face in participating in the linguistic practices rewarded in educational institutions. In a sense, being 'educated' means being able to participate in certain types of 'valued' semiotic exchange; for instance, the discourses of medicine, science, business, law, music and art. Certain groups within a society, typically those with wealth and connections, are relatively well placed within the semantic domains which are rewarded (usually by members of that same group). Other groups to varying degrees are marginalized. Increasingly the market-driven practices adopted in schools and universities, such as making entrance dependent on money rather than merit, function to reinforce these divisions of inclusion and exclusion. Participation in everyday discourse includes semiotic exchange in terms of performative action; that is, selections in the form of gesture, stance, proxemics and dress. Whether delivering a conference paper or giving a political speech, the speaker needs to talk the talk (using appropriate linguistic and phonological choices), walk the walk (in terms of non-verbal behaviour and action), and increasingly look the look (in terms of clothing, hairstyle, make-up, body size, body shape, height, and skin and hair colour, for example) according to the parameters established as desirable in that culture. More generally men and women are urged to identify their 'unique selling point' (USP), be it the talk, the walk or the look. Increasingly acts of meaning inscribed on and through the human body (for example, physical appearance which is increasingly the product of medical procedures and other forms of practices involving drugs, chemicals and so forth) often outweigh the import of other acts of meaning (the talk). In the electronic medium, the performative action and physical creation of identity becomes a textual act. One is no longer constrained by semiosis emanating from the body and the immediate material context. Multiple identities can be established according to the limits of the electronic medium and platforms that are offered, and the user's ability to make use of different semiotic resources, including language, visual images, music and so forth. Semiotic capital comes into play in new ways through computer technology. The social-semiotic construction of reality (Berger and Luckmann, 1991)

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is determined as much by what is included as to what is excluded. As seen in the example of the German generals' 'guiding principles', (i) there are limits to what options are selected, and (ii) there are limits to what can be selected from the existing systems. In the first case, semiotic selections function as meaning through choice, and so some options (for example, 'success') are chosen to the exclusion of others (for example, 'justice' or 'freedom'), while other possible options are left out (for example, 'for the Nazi party'). In the second case, although systems are dynamic and constantly changing with each contextual instantiation, there are nonetheless at any one time a limited number of options available. We are contained within particular semantic domains according to the limitations of the systems which are available. These systems, however, constantly evolve so that meaning making is a dynamic practice in which change is possible. Realms of meaning do not exist until they become semiotic choices; for example, the concepts of women's rights, gender and Freud's (for example, 1952, 1954) concept of psychoanalysis are comparatively new linguistic choices. Although perhaps pre-existing as disparate practices, the introduction of these options in language led to radically new ways of conceptualizing women, women's roles and what has become the inner psychosexual self. Similarly, the scientific revolution in the seventeenth century introduced radically new ways of conceptualizing the physical world. The basis for this scientific re-ordering of reality was the development of mathematics which offered new resources in the form of the symbolism and visual display. These semiotic resources combine in significant ways with language to create a new world order. The nature of that order is investigated in this study. 1.3 Mathematics as Multisemiotic

Mathematics and science are considered as 'multisemiotic' constructions; that is, discourses formed through choices from the functional sign systems of language, mathematical symbolism and visual display. These discourses are commonly constituted as written texts, although mathematical and scientific practices are not confined to these forms of semiotic activity. There are many different 'multimodal' genres constituting mathematical and scientific practices; for example, lectures, conference papers, software programs and laboratory investigations. In addition to the written mode, these types of semiotic activity involve spoken discourse, physical action and gesture in environments, which include digital media and day-to-day three-dimensional material reality. The major line of enquiry in this study, however, is directed towards multisemiosis in printed discourses of mathematics, largely because modern mathematical symbolism is a semiotic resource which developed in written format. In order to develop theoretical frameworks for mathematical symbolism and visual display, the print medium has been chosen for investigation. In addition, the effects of computer technology on the nature of mathematical discourse are also

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considered in this study. With the exception of the systemic functional (SF) approach to mathematics (Lemke, 1998b; O'Halloran, 1996, 1999b, 2000, 2003a, 2003b, forthcoming a; Veel, 1999), few studies exist in the field of the semiotics of mathematics (for example, Anderson et al, 2003; Rotman, 1987,1988,1993,2000). Mathematical discourse involves language, mathematical symbolism and visual images as displayed in Plate 1.3(1), a page reproduced from Physica D, a journal for research in dynamical systems theory. Plate 1.3(1) contains equations (11), (12) and (13), which are mathematical symbolic statements spatially separated from the main body of the linguistic text. Symbolic statements and elements are also embedded within the linguistic text. For example, symbolic elements function as elements within the linguistic statements in the text located between equations (11) and (13). In addition, there are visual images in the form of mathematical graphs in the three panels labelled Fig. 2 in Plate 1.3(1). Mathematical written discourse may also contain tables which are forms of textual organization where the reader may access information quickly and efficiently (Baldry, 2000a; Lemke, 1998b). As seen in Plate 1.3(1), mathematical printed texts are typically organized in very specific ways which simultaneously permit segregation and integration of the three semiotic resources. An SF approach to mathematics as social-multisemiotic discourse means that each of the three semiotic resources - language, visual images and mathematical symbolism - is perceived to be organized according to unique discourse and grammatical systems through which meaning is realized. That is, each semiotic resource is considered to be a functional sign system which is organized grammatically. Mathematical texts such as those displayed in Plate 1.3(1) represent specific semiotic choices from the available grammatical systems in each of the three resources. As seen in the graphs and linguistic and symbolic components of the mathematics text in Plate 1.3(1), choices from the three semiotic resources function integratively. That is, the linguistic text and the graphs contain symbolic elements and the symbolic text contains linguistic elements. This feature of mathematical discourse means that the grammars of each resource must be considered in relation to each other. The similarities and differences in the organizing principles of the three semiotic resources are considered intra-semiotically in terms of the grammars and functions of each resource. In addition, mathematical discourse is considered inter-semiotically; that is, in terms of the meaning which arises from the relations and shifts between the three semiotic resources. Royce (1998a, 1998b, 1999) refers to intersemiotic semantic relations between linguistic and visual components of a text as 'intersemiotic complementarity', and ledema (2003: 30) calls the process of semiotic shift as 'resemioticization', which he defines as 'the analytical means for . . . tracing how semiotics are translated from one into another as social processes unfold'. In mathematics, intersemiotic shifts take place on a macro-scale across stretches of text, and they also take place on a micro-scale within stretches

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MATHEMATICAL DISCOURSE J. Kockelkoren et al./Physica D 174 (2003) 16X-175

of the wavelength Xc of the patterns is at criticality about 13% off from the theoretical value; however, we are not interested here in the absolute value, but in the relative variation of X C /X. The difficulty of comparing theory and experiment on the variation of the wavelength is that the only theoretically sharply defined quantity is the wavelength

173

sufficiently far behind the front, A.^, and that one has to go beyond the lowest order Ginzburg-Landau treatment to be able to study the pattern wavelength left behind. For example, if we use a Swift-Hohenberg equation for a system with critical wavenumber kc and bare correlation length £o.

(11) then a node counting argument [4,6] yields for the asymptotic wavelength A.as far behind the front [6]:

(12) In the Rayleigh-Benard experiments, kc « 2.75/d, where d is the cell height; the theoretical value is £o = 0.385rf, so if our conjecture that the value is some 15% larger is correct, we get £Q ^ 0.4<W. This then gives

(13) As we stressed already above Xas is the wavelength far behind the front; for a propagating pulled front, there is another important quantity which one can calculate analytically, the local wavelength A* measured in the leading edge of the front. For the Swift-Hohenberg

Fig. 2. Top panel: shadowgraph trace of a propagating front in the experiments of FS for f = 0.012 [16]. The time difference between successive traces is 0.42fv, where f v is the vertical diffusion time in the experiments, and the distances are measured in units d (the cell height) (from [9]). Middle panel: similar data obtained from numerical integration of the Swift-Hohenberg equation also at e = 0.012 starting with a localized initial condition. The time difference between successive traces corresponds to 0.42/v. Bottom panel: velocity versus time in the experiment, as obtained by interpolating the maxima of the traces in the top panel, as explained in the text. The dashed line shows the analytical result (8) and the dotted curve the result of the amplitude equation simulation of Fig. 1 with nln" = 1.2. Note that the curves are not fitted, only the absolute scale is affected by adjusting £o

Plate 1.3(1) Language, visual images and symbolism (Kockelkoren et al, 2003: 173) of text. The potential of intersemiotic processes to produce metaphorical construals is formulated through the notion of 'semiotic metaphor'. Through close examination of the meaning realized within and across the three semiotic resources, the functions and the semantic realm of

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mathematics as a discourse are tentatively formulated in this study. It must be stressed that this is not an account of the entire field of mathematics. Rather it is an account of the semiotic processes and the discourse and grammatical strategies through which mathematics operates to structure the world. From this position, the semantic realm with which mathematics is concerned may be appreciated. This is in part achieved through a comparison of the functions of mathematics with those of language. However, mathematics evolved as a discourse capable of creating a world view which extends beyond that possible using linguistic resources alone. The result of that re-ordering in what is viewed as the scientific revolution is also considered in this study. The implications of viewing mathematical discourse as a multisemiotic construction are considered below. 1.4 Implications of a Multisemiotic View The multisemiotic approach, where language, visual images and mathematical symbolism are considered as semiotic resources (O'Halloran, 1996), originally stems from O'Toole's (1994, 1995, 1999) extensions of Halliday's (1978, 1994) SF approach to displayed art, and Lemke's (1998b, 2000, 2003) early work in mathematical and scientific discourse. The SF approach to mathematics is welcomed by Rotman (2000: 42) who explains that such an approach offers 'a linguistic/semiotic framework well grounded in natural language that . . . [is] abstract enough to include the making of meaning in mathematics'. Halliday's (1994) Systemic Functional Grammar (SFG) includes documentation of the metafunctionally based systems which are the grammatical resources through which meaning is made. Halliday's account of the abstract language systems includes statements of how these choices are realized in text. SFG is essentially a 'natural' grammar as it explains how language is organized to fulfil the metafunctions of language: the experiential, logical, interpersonal and textual. Halliday's (1994) model of language described in Chapter 3 provides the basis for the Systemic Functional Grammars (SFG) presented for mathematical symbolism and visual images in Chapters 4 and 5 respectively. These grammars and a framework with systems for intersemiosis are used for discourse analyses of mathematical texts in Chapters 6 and 7. The discussion includes an account of the educational implications of a multisemiotic view of mathematics and the nature of pedagogical discourses in mathematics classrooms. The SFGs for mathematical symbolism and visual images are inspired by O'Toole's (1994, 1995) systemic frameworks for the analysis of semiosis in paintings, architecture and sculpture. O'Toole (1994) demonstrates how the SF frameworks may be used so that the viewer can learn to engage directly with instantiations of displayed art rather than depending on the 'knowledge' handed down by art historians and other accredited experts. Bourdieu (1989) further explains that aesthetics and art appreciation are discourses which function covertly to maintain existing social class

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distinctions. In this view, 'taste' is a social and cultural product through which group and individual identities are indexed and, as with all symbolic investments, different values are placed on those indices. Needless to say, the highest values are accorded to those who constitute the powerful in society. the main reason for this close [semiotic] engagement with the details before our eyes is that it enables everyone to sharpen their perceptions and join in the discussion as soon as they begin to recognize the systems at work in the painting. And everyone can say something new and insightful about the work in front of them. Art history, on the other hand, requires a long apprenticeship . . . before they are expected to be able to contribute any new information to a discussion of the work in question. And what kind of information might this be? . . . Don't they in fact 'mystify' the painting and make us feel we have nothing to contribute? . . . the result is to build an insurmountable wall around this precious property. (O'Toole, 1994: 171)

Following O'Toole's (1994) example, rather than producing a discursive commentary about the nature of mathematics and its intellectual achievements, the intention behind the SF approach in this study is active engagement with mathematical text in order to understand the strategies through which the presented reality is structured, the content of that reality and the nature of the social relations which are subsequently established. The result is an appreciation and understanding of the functions of mathematical discourse and the strategies through which this is achieved. This is essentially a new approach to mathematics for practising mathematicians, and teachers and students of mathematics. This approach also offers insights for outsiders who typically possess a limited understanding and knowledge of mathematics. The implications of an SF approach to mathematics as a multisemiotic discourse are outlined below in relation to the key ideas and formulations developed in this study. These ideas are revisited in Chapter 7 after the theory and approach have been developed in Chapters 2-6. Mathematical and Scientific Language

The view of mathematics as multisemiotic has implications about the ways mathematical and scientific language are understood. Traditionally, the nature of scientific language has been viewed in isolation rather than as a semiotic resource which has been shaped through the use of mathematical symbolism and visual display. Scientific language developed in certain ways as a response to the functions which were fulfilled symbolically and visually. On a more global scale, our entire linguistic repertoire has been shaped by the use of other semiotic resources, with the result that many of our contemporary linguistic constructions are metaphorical in nature. For example, certain views become common sense under the guise of metaphorical labels such as 'economic rationalism', 'entrepreneurship' and

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'freedom and democracy'. Despite their grammatical instantiation as nouns, these are not concrete or material objects. On the contrary, rather complex and dynamic sets of practices are subsumed under such labels. An understanding of the functions of mathematical symbolism, visual images and other semiotic resources permits a re-evaluation of the role of language in constructing such a naturalized view of the world. As with the vested interest behind the guiding principle, 'Success is right', metaphorical terms need to be critically understood in a historical and contextual manner in order to appreciate the premises behind their construction. The Grammar of Mathematical Symbolism

An SF framework for mathematical symbolism is presented so that the grammatical strategies through which meaning is encoded symbolically can be documented. This is significant because the grammatical strategies for organizing meaning in symbolic statements differ from those found in language. While members of a culture are capable of using language as a functional resource in various ways, typically the use of mathematical symbolism is restricted to certain groups. One reason for this limited access is that the grammar of mathematical symbolism is not generally well understood. It is important to demonstrate how mathematical meaning is organized, and how the unique grammatical strategies specifically developed in mathematical symbolism so that this semiotic could be used for the solution of mathematics problems. The underlying premise is that mathematical symbolism developed as a semiotic resource with a grammar which had the capacity to solve problems in a manner that is not possible with other semiotic resources. The SFG of mathematical symbolism presented in Chapter 4 explains how this functionality is achieved. Grammar of Visual Images in Mathematics

Visual images in mathematics are specialized types of visual representation, most typically in the form of abstract graphs, statistical graphs and diagrams. The systemic functional framework for abstract graphs is used to explain how the systems are organized to make very specific meanings which provide a link between the linguistic description of a problem and the symbolic solution. Once again, the functions fulfilled by mathematical visual images are different to those achieved linguistically and symbolically. The systems through which the functions of abstract graphs are achieved are discussed in Chapter 5. This discussion includes insights into the changing roles of visual images in mathematics due to the impact of computer technology. Visualization is undergoing a rapid resurgence due to the increasing sophistication of computer graphics which display numerical solutions generated by the computer. The new ways of manipulating and viewing data through computers are discussed in Chapter 5.

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Intrasemiosis and Intersemiosis

While the three semiotic resources in mathematics fulfil individual functions which are not replicable across the other resources (Lemke, 1998b, 2003; O'Halloran, 1996), the success of mathematics depends on utilizing and combining the unique meaning potentials of language, symbolism and visual display in such a way that the semantic expansion is greater than the sum of meanings derived from each of the three resources. Lemke (1998b) refers to this expansion of meaning as the multiplicative aspect of multisemiosis. Mathematical discourse thus depends on intrasemiotic activity, or semiosis through choices from the grammatical systems within each resource, and intersemiotic activity, or semiosis through grammatical systems which function across the three resources. Intersemiosis involves reconstrual of particular elements in a second or third resource through intersemiotic shifts or 'code-switching'. Intrasemiosis, or meaning within one semiotic resource, is important because the types of meaning made by each semiotic are fundamentally different. Intersemiosis, however, is equally important because not only is the new meaning potential of another resource accessed, but also metaphorical expressions can arise with such shifts. This important process, which may arise in any multisemiotic discourse, is developed in this study through the notion of semiotic metaphor. The functions of mathematics are therefore achieved through intrasemiosis and intersemiosis; that is, meaning through each semiotic resource, and meaning across the three semiotic resources where metaphor plays an important role in the expansion of meaning. Intersemiotic Mechanisms, Systems and Semiotic Metaphor

Intersemiotic mechanisms provide a description of the ways in which intersemiosis takes place across language, visual images and mathematical symbolism. The intersemiotic mechanisms take place through metafunctionally based systems which are documented in Chapter 6. Semiotic metaphor refers to the phenomenon of metaphorical construals which arise from such shifts across semiotic resources. This process means that expansions in meaning can occur when a functional element is reconstrued in a different resource. For instance, an action realized through a verb in language (for example, 'measuring') may be reconstrued as an entity in a second semiotic resource (for example, a visual line segment or a symbolical distance). Such reconstruals permit expansions of meaning on a scale which is not possible within a single semiotic resource. As explained in Chapter 6, one of the key elements in the success of mathematics is the metaphorical reformulation of elements across the three semiotic resources. Mathematics Education

The view of mathematics as a multisemiotic discourse is significant in a pedagogical context as often teachers and students do not seem to be aware

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of the grammatical systems for mathematical symbolism and visual display, and the types of metaphorical construals which take place in mathematics texts and in the classroom. The ways in which a social-semiotic perspective can inform mathematics teaching and learning are described in Chapter 7. This discussion is based on the functions of language, visual images and the symbolism, their respective grammatical systems and the nature of the intersemiotic activity. Chapter 7 includes a discussion of the nature of pedagogical discourse in mathematics. 1.5 Tracing the Semiotics of Mathematics

In order to introduce the types of meaning found in modern mathematics, a historical perspective is adopted in Chapter 2 to examine the semiotic unfolding of mathematics from the period of the early Renaissance to modern contemporary mathematics. The nature of the projects of early modern mathematics, as exemplified by Descartes and Newton, is seen to lead to the creation of a mathematical and scientific reality which is located within a limited semantic domain. However, at the same time, the semantic expansions afforded by the visual images and mathematical symbolism permitted expansions in the form of scientific description, prediction and prescription. Contemporary thought in mathematics, for example, chaos and dynamical systems theory, also reveals the changes in mathematical theorizations of reality. Significantly, the mathematical practices advocated by Descartes and Newton have been re-inscripted into new contexts in contemporary times. The beginnings of modern mathematics and science developed in what was originally conceived as a transcendental realm which necessitated the existence of God, as seen in the discussion of Cartesian and Newtonian philosophy in Chapter 2. The re-inscription of the supposedly 'value free' discourses of mathematics and science as universal truth into new realms of human endeavour such as the social sciences, education, business, economics and politics is questioned from the relatively fresh perspective of the socio-semiotics of mathematics in Chapter 7. This discussion also contributes to an appreciation of the metaphorical nature of our semiotic constructions and the limitations of the contexts in which mathematics may be usefully applied. Mathematics is thus first viewed in a historical context so the functions for which mathematics was originally designed and the context of that development may be appreciated. From this point, the new contexts in which mathematics is re-inscribed are critically examined. Although mathematics has expanded into new fields, the semiotic resources nonetheless essentially remain linguistic, visual and symbolic. Computation is considered a symbolic undertaking which is instantiated in an electronic medium. An understanding of the scientific view of the world made possible through mathematics is an overriding theme because such a view is vital for an understanding of contemporary Western culture which

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materializes as a technological project shaped by the discourse of mathematics and science. Looking back, the rationalist project of the eighteenth century and the consequent mathematical, scientific and technological achievements of the modern period appeared to hold much promise for the world. As Horkheimer and Ardorno (1972) claim, the much-touted aim of progress was the improvement of the human condition accompanied by freedom, equality and justice. In retrospect, however, such progress seems to have been made for the advantage of the relatively privileged few. In addition to providing the infrastructure for unequally distributed goods and services such as healthcare and education, advances in mathematical and scientific knowledge appear to have primarily provided the means for technological development which is directed and controlled by military, business and political interests. As Davis (2000: 291) claims: 'Through advanced science and technology, warfare utilizes many mathematical ideas and techniques. The creation of vast numbers of new mathematical theories over the past fifty years was due in considerable measure to the pressures and the financial support of the military.' The self-evident deliverables of the scientific project were underscored in the aftermath of the Second World War and, in a more recent case, the US-led war in Iraq in 2003 where the destructive power of military technological innovation was widely televised. As Horkheimer (1972: 3) claims: Tn the most general sense of progressive thought, the Enlightenment has always aimed at liberating men from fear and establishing their sovereignty. Yet the fully enlightened earth radiates disaster triumphant.' Today the extent to which the military, business and the political institutions can be differentiated as separate functioning bodies becomes increasingly difficult to ascertain. One could include universities on the list of institutions which increasingly function pragmatically along the lines of business-orientated commercial interests. The soundness of reason depends on the explicit or implicit premises upon which that reasoning is based. The view that mathematical and scientific reasoning is constructed to order the world along certain principles which change is not new (for example, Derrida, 1978; Foucault, 1970,1972; Kuhn, 1970). However, the approach adopted in this study is to understand the systems and strategies through which that ordering takes place. In this way, the functions of these discourses may be understood, and through such awareness we can understand our own positions and explore possibilities other than those directly offered. This is an exploration of the world view offered by mathematics and science, a view which dominates our everyday thinking. It is also a critique of that world view which is so often misunderstood as universal truth. The path is developed through an excursion through early printed mathematical texts to understand the context behind modern mathematics. SFGs are used to critically interpret the nature of meanings made in contemporary mathematics. Through an understanding of the discourse,

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we may start to count the gains and costs of the mathematical and scientific view of the world. In the view of Davis (2000: 291): The mathematical spirit both solves problems and creates other problems. What is the mathematical spirit? It is the spirit of abstraction, of objectification, of generalization, of rational or 'logical' deduction, of universal quantization, of computational recipes. It claims universality and indubitability. I have the conviction . . . that this spirit is now . . . pushing us too hard, pushing us to the edge of dehumanization.

The ways in which 'mathematics is pushing us too hard' are investigated through an understanding of mathematics as a multisemiotic resource. Only then can we begin to appreciate the ways in which this discourse and scientific order function to shape our view of ourselves, and our relations to others and the world around us. 1.6 Systemic Functional Research in Multimodality

This study of mathematics represents part of a growing movement in SFL (see ledema, 2003) where language is conceptualized as one resource which functions alongside other semiotic resources. This research field is commonly called 'multimodality', or the study of 'multimodal discourse' (for example, Baldry, 2000b; Baldry and Thibault, forthcoming a; Kress, 2000, 2003; Kress et al., 2001; Kress and van Leeuwen, 1996, 2001; Levine and Scollon, 2004; O'Halloran, 2004a; Unsworth, 2001; Ventola et al., forthcoming). Apart from the research in mathematics (Lemke, 2003; O'Halloran, 1996, 1999b, 2003b, forthcoming a), studies have been completed in a wide range of fields including science (Baldry, 2000a; Kress et al, 2001; Lemke, 1998b, 2000, 2002), biology (Guo, 2004b; Thibault, 2001), multiliteracy (Lemke, 1998a; Unsworth, 2001), film and television (ledema, 2001; O'Halloran, 2004b; Thibault, 2000), music (Callaghan and McDonald, 2002), museum exhibitions (Pang, 2004), shopping displays (Ravelli, 2000), TESOL (Royce, 2002), hypertext and the electronic medium (Jewitt, 2002; Kok, 2004; Lemke, 2002) and advertising (for example, Cheong, 2004). Research in the field of multimodality also includes the development by Anthony Baldry et al (Baldry, 2004, forthcoming; Baldry and Thibault, 2001, forthcoming a, forthcoming b) of an on-line multimodal concordancer, the Multimodal Corpus Authoring (MCA) system, which is web-based software for the analysis of phase and transitions in dynamic texts such as television advertisements, film and web pages. There have been attempts to construct grammatical frameworks for different semiotic resources (see Kress and van Leeuwen, 1996, 2002; O'Halloran, 2004a). However, with the exception of Thibault's (2001) approach to the theory and practice of multimodal transcription and Baldry and Thibault's notion of phase for the analysis of dynamic texts (Baldry, 2004, forthcoming; Baldry and Thibault, 2001, forthcoming b), few comprehensive theoretical and practical approaches have been developed in the field of multimodality. Consequently, a meta-language for

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an overarching model for the theory and practice of multimodal discourse analysis remains at a preliminary stage. Partly as a consequence of this lack of a meta-theory, there exist problems of terminology in studies of multimodality, as noted by ledema (2003: 50). For example, there is confusion over the use of the terms 'mode' versus 'semiotic', and, consequently, 'multimodal' versus 'multisemiotic'. Given that this field represents a relatively new area of research, this is to be expected as the much needed frameworks undergo development. As an example of mixed terminology, Kress and van Leeuwen (2001: 21-22) define 'mode' as the 'semiotic resources which allow the simultaneous realization of discourses and types of (inter) action. Designs then use these resources, combining semiotic modes, and selecting from the options which they make available according to the interests of a particular communication situation.' From this position, Kress and van Leeuwen (2001: 22) see Narrative, for example, as a mode. In this study, however, the term 'semiotic' is used to refer to semiotic resources such as language, visual images and mathematical symbolism. These semiotic resources have unique grammatical systems through which they are organized. Any discourse that involves more than one semiotic resource is therefore termed 'multisemiotic' rather than 'multimodal'. The use of the term 'multimodal' is explained below. The term 'mode' in SFL, following Halliday and Hasan (1985), typically means the role language is playing (spoken or written) in an interaction. This sense, adopted in this study, is concerned with the nature of the action of semiosis; that is, whether it is auditory, visual or tactile, for example. It follows that different semiotic resources are constrained in terms of possible modes through which the semiotic activity can take place. For example, language may be instantiated orally or visually, but visual images are instantiated through the visual mode in different media such as print, electronic media and three-dimensional space. On the other hand, Kress and van Leeuwen (2001: 22) use the term 'medium' to refer to the '[m]aterial resources used in the production of semiotic products and events, including both the tools and the materials used (for example, the musical instrument and air; the chisel and the block of wood. They are usually specially produced for this purpose, not only in culture (ink, paint, cameras, computers), but also in nature.' In order to maintain existing systemic terminology, in this study the term mode is used to refer to the channel (auditory, visual or tactile, for example) through which semiotic activity takes place, medium for the material resources of the channel, and genre for text types such as the Narrative (which is realized through language in the spoken or written form). The term multisemiotic is used for texts which are constructed from more than one semiotic resource and multimodality is used for discourses which involve more than one mode of semiosis. A radio play featuring speech, music and diegetic sound is therefore multisemiotic rather than multimodal as it involves multiple semiotic resources realized through the auditory mode of

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sound through the medium of the radio. However, a website which contains written linguistic text and a music video clip is multisemiotic (involving language, visual images, music) and multimodal (visual and auditory). The practices adopted here do not attempt to solve the problems of mixed usages of terminology, rather they seek to clarify the use of the terms adopted in this study. In this respect, mathematics is referred to as multisemiotic as it consists of three semiotic resources, language, visual images and mathematical symbolism. Mathematics is considered to be primarily a written discourse produced in printed and electronic media. There are also multimodal genres in the field of mathematics, such as the academic lecture, which involves spoken discourse and other semiotic resources. The multimodal nature of mathematical pedagogical discourse is discussed in Chapter 7.

2 Evolution of the Semiotics of Mathematics

2.1 Historical Development of Mathematical Discourse

A historical view of the changing nature of multisemiosis in mathematical discourse from the early Renaissance to the present is a useful method for introducing in general terms the development of the semantic realm of mathematics (O'Halloran, 2003b, forthcoming a). Such an examination of visual images, symbolism and language in mathematical texts demonstrates how particular dimensions of meaning are incorporated to the exclusion of others. This excursion includes a discussion of the first known printed mathematics book, the Treviso Arithmetic 1478, and an examination of early mathematical and scientific printed texts from the sixteenth to the eighteenth centuries. In particular, Descartes' shift of emphasis from perception to what he called 'the intellect' and Newton's reformulation of nature in mathematical terms are investigated. Descartes (1596-1650) and Newton (1642-1727) are seen to provide important points of departure in the seventeenth century for what was to become the contemporary mathematical and scientific project. In the first case, Descartes successfully used mathematical symbolism to describe and differentiate between curves. It appears that this success with symbolic and visual semiotic tools was incorporated into an approach upon which Descartes could base his philosophical method, a method aimed at securing 'true knowledge'. This method involved dispensing with the 'secondary' qualities of matter, such as colour, odour and taste perceived by the bodily senses, and accepting only 'primary' qualities which could be dealt with through 'the mind'. Newton developed Descartes' mathematical semiotic tools to provide a symbolic description of physical reality. In doing so, Newton in fact re-admitted sense experience to the philosophical and scientific realm in such a way that made the invisible (for example, force and attraction) visible through mathematicized symbolic description (Barry, 1996). The scientific age and the use of experimentation and technology began in earnest with Newton. In this movement, much effort was expended in developing mathematical symbolism as a semiotic resource with a grammar that could directly interact with the grammars of graphs,

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diagrams and language. In this chapter, some necessarily fragmented glimpses of these early events are traced in order to introduce the nature of meanings found in contemporary mathematics. To make this undertaking more accessible, the historical investigation into the semiotic realm of mathematics and science takes the form of a discussion of the illustrations and diagrams which appear in early printed mathematical texts. In addition, observations concerning the different forms of symbolism in these texts are made. As Cajori (1993) explains, the history of the development of mathematical symbolism is complex and involves rivalry among mathematicians. Cajori's (1993) detailed account of the history of mathematical notation reveals that the majority of forms of mathematical symbolism became obsolete with only a few forms surviving to the present day. These developments are not included in the account presented here, nor is it possible to include a discussion of the origins of algebra documented by mathematical historians (for example, Klein, 1968). The broader view presented in this study is that algebra developed in three stages (see Joseph, 1991; Swetz, 1987). First there existed rhetorical algebra which involved linguistic descriptions and solutions to problems. The second stage was syncopated algebra where quantities and operations which were used frequently were symbolized. The last stage of the development was symbolic algebra where the mathematical symbolism developed as a semiotic resource in its own right. Rather than providing a complete description of the three stages, the changing nature of multisemiosis in mathematics as the symbolism developed is explored. This discussion of the history of mathematics differs from most accounts in that the view is essentially semiotic. In other words, out of the possible options within the different sign systems for language, visual images and symbolism, it may be seen that only certain ranges of choice were incorporated in mathematics during different time periods. The shifting nature of those choices becomes evident as the printed classical mathematical texts of antiquity and early practical arithmetic books were replaced with new forms of semiosis in the mathematics of the early Renaissance. Descartes and Newton reformulated the mathematical realm in what marks the beginning of modern science during the seventeenth century. However, today that mathematical realm functions in different contexts from that which supported the original mathematical formulations. The implications of this re-contextualization of mathematical and scientific formulations in fields which include the arts, social sciences and humanities are addressed in Chapter 7. The reasons for the changes in the nature of mathematics are traced to the cultural, intellectual and economic climate of the different time periods, the functions which the mathematics was designed to serve and the available technology. This remains true today where economic, commercial, political and private interests combine with advances in computer technology to determine the type of mathematics developed and the nature of scientific projects which are undertaken. As Wilder (1986) claims,

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mathematics is a cultural practice, and so, like other forms of discourse, it is politically motivated. Consequently, following Koestler (1959) and later philosophers of science such as Kuhn (1970), the development of mathematics and science has not been orderly: 'The progress of science [and mathematics] is generally regarded as a clean, rational advance along a straight descending line; in fact it has followed a zigzag course, more bewildering than the evolution of political thought' (Koestler, 1959: 11). In the following discussion, views of the changing nature of the semiotics of mathematics are oudined in relation to the cultural and situational contexts which gave rise to those discourses. In explaining the effectiveness of mathematics, Hamming (1980) claims that what is seen is what is looked for, that the kind of mathematics used is selected from a range of possible choices and that in this process very few problems are answered. Following this line of argument, mathematics is seen to deal with a limited semantic field in limited ways, but in doing so has the potential to solve problems which would be impossible to solve using other semiotic resources. Seen in this light, the breakthrough which led to the scientific revolution was a new way of conceptualizing the world using new forms of semiosis. This is basically the position developed in the following discussion of the evolution of modern mathematics. 2.2 Early Printed Mathematics Books

The first known printed Western mathematical book is the Treviso Arithmetic 1478. Partially translated from Italian into English by David Eugene Smith in the 1920s, the first complete translation appears in Frank Swetz's (1987) Capitalism and Arithmetic. The author of the original manuscript is unknown and the title arises from the date and place of publication, the Italian town of Treviso. The book is concerned with practical arithmetic for calculations in trade and commerce. Swetz (1987) explains that the content is typical of the earliest known mathematics books in Europe. The majority of the books were written by masters in reckoning schools and guilds which flourished in Italy during the fifteenth century. These institutions were popular places to learn the mathematics necessary for the expanding merchant trade. As seen in Plate 2.2(1), the Hindu-Arabic system is used for the calculations in the Treviso Arithmetic, and there are drawings to demonstrate how the calculations are performed. Although Swetz (1987) is an English translation of an Italian text, it can be seen that the original text is constructed semiotically through the use of language, numerical symbols and particular forms of drawings. The style is rhetorical algebra where unknown quantities are realized as entities such as 'profit' rather than symbolic quantities such as 'P'. According to Swetz (1987), in the late fifteenth century some Italian writers were starting to use syncopated algebra with forms of abbreviations for recurring terms and mathematical operations.

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Thus the problem is solved, and the answer is that there falls to Piero as profit 138 ducats, 21 grossi, n pizoli and Plate 2.2(1) The Treviso Arithmetic (reproduced from Swetz, 1987: 140) At the time of the Treviso Arithmetic, there were controversies over the best method for performing arithmetical calculations as pictured in Plate 2.2(2). The controversy concerned the abacists, who manually used counters and ruled lines to perform the calculations and recorded the result in Roman numerals, and the algorists, who used the Hindu-Arabic numeration system and algorithms to calculate and record. Computation at this time centred around prestigious counting tables, and the proposed shift to algorists' Hindu-Arabic system represented a threat to those who had vested interests in maintaining the tables. Despite the obvious benefits of the new system, the shift to the Hindu-Arabic numeration system, first introduced in Europe as early as AD 1000 was slow because of the resistance exerted by those who controlled the tables. By the fifteenth century Italy, however, was ahead of other European countries in using the

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Plate 2.2(2) The Hindu-Arabic system versus counters and lines (Reisch, 1535: 267) Hindu-Arabic system as the means for performing arithmetic calculations. The Treviso Arithmetic demonstrates how the nature of mathematics is influenced by cultural and economic concerns of the time (in this case, merchant trade and commerce) and pressure from special interest groups (for example, those supporting the Hindu-Arabic number system). In addition, technology plays an important role as seen below in the discussion of the

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impact of the printing press on mathematics. More generally, the nature of the development of mathematics is a convergence of these factors. The use of printing press technology explains the increased popularity of the Hindu-Arabic system which lent itself to this type of reproduction (for example, Dantzig, 1954; Eisenstein, 1979; Swetz, 1987). The calculations completed through lines and counters on the counting tables were clumsy to reproduce and required special printing techniques. As may be seen in Plate 2.2(3), the diagrams on the left-hand side (which include small pictures of hands) demonstrate how the calculations are performed using the lines and counters. As mathematical texts increasingly appeared in print, this form of representation could not compete with the more efficient semiotic form of the Hindu-Arabic system of computation. The expansion of the use of the Hindu-Arabic system is significant for two reasons. First, as commercial mathematics increasingly became semiotic instantiations in the written mode, the algorithms for the calculations became more widely disseminated and commercial arithmetic moved from the hands and counters of the few to a wide audience. Second, Swetz (1987: 32) explains that the increased focus, attention and recording of the mathematical techniques in the Hindu-Arabic practical arithmetical texts in effect paved the way for the development of symbolic algebra. The printed text permitted close examination and development of arithmetical algorithms, and the standardization of mathematical procedures, techniques and symbols which led to the range of mathematical notations documented by Cajori (1993). Under the economic and intellectual impetus of this time, not only were mathematical techniques being more widely learned but they were, in many cases, new techniques based on the use of Hindu-Arabic numerals and their accompanying algorithms. From this period onward, computation involving numbers would be more easily executed and efficiently recorded. The visual stimulus of a mathematical process written out allowed for a re-examination and questioning of the process; patterns could be noted and mathematical structure discerned. Printing also forced a standardization of mathematical terms, symbols, and concepts. The way was now opened for even greater computational advances and the movement from a rhetorical algebra to a symbolic one. (Swetz, 1987: 284)

Swetz (1987) explains that at the time the Treviso Arithmetic was published, the Classicist mathematicians demanded a printed edition of Euclid. However, it appears that this priority was placed second to arithmetic where 'Practical necessity was the motivating force in this printing decision' (Swetz, 1987: 25). 'Perhaps the typographical problems inherent in setting type for geometrical figures were responsible for the delay, but more likely it was due to the economic and intellectual demands of the marketplace' (ibid.). In Thomas-Stanford's (1926: 3) view, the early Venetian printing presses published few mathematics books due to the problems of printing the diagrams. Two examples of early printed editions of Euclid's Elements are displayed in Plates 2.2(4a-b). Thomas-Stanford (1926: 3) states that the

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Plate 2.2(3) Printing counter and line calculations (Reisch, 1535: 326) first printed edition, which appeared in 1482, was 'an epoch-making' book in many respects: 'It was [one of] the first attempt[s] - and a highly successful one - to produce a long mathematical book illustrated by diagrams.' In this version of Euclid displayed in Plate 2.2(4a), the running linguistic text in the style of rhetorical algebra is ornately decorated and the diagrams are neatly offset to one side. Plate 2.2(4b) shows the richness of the border patterns and text which appears in the early editions of Euclid. In the

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Plate 2.2(4a) Euclid's Elements: Venice 1482, Erhard Ratdolt (ThomasStanford, 1926: Illustration II) original version of Plate 2.2(4b) reproduced in Thomas-Stanford (1926: Illustration IV), parts of the text and the decoration are coloured red. Thomas-Stanford (1926: 4) observes that' [it] would almost seem that at Venice especially the printers sought by a refinement of ornamentation to relieve the austerity of the subject-matter', the nature of which may be appreciated from the modern translation of Euclid which is displayed in Plate 2.2(5). The mathematics appears as objective statements which are accompanied by perfect geometrical shapes. Apart from the statement, 'I say that. . .' in Euclid's discourse, the author is absent. In Euclid's writings

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Plate 2.2(4b) Euclid's Elements: Venice 1505, J. Tacuinus (ThomasStanford, 1926: Illustration IV) displayed in Plates 2.2(4a-b) and 2.2(5), there is also a noticeable lack of symbolism in the text, which appears only in the form of a, b, c and d and A, B, C and D to refer to the points, sides, angles and triangles in the

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PROPOSITION 18. In any triangle the greater side subtends the greater angle. For let ABC be a triangle having the side AC greater than AB; I say that the angle ABC is also greater than the angle BCA. For, since AC is greater than AB, let AD be made equal toAB [i. 3], and let BD be joined. Then, since the angle ADB is an exterior angle of the triangle BCD, it is greater than the interior and opposite angle DCB. [i. 16] But the angle ADB is equal

to the angle ABD, since the side AB is equal to AD; therefore the angle ABD is also greater than the angle ACB; therefore the angle ABC is much greater than the angle ACS. Therefore etc. Q. E. D. Plate 2.2(5) Translation of Euclid (reproduced from Euclid, 1956: 283) mathematical diagrams. As we shall soon see, Newton rewrote Euclid's geometry in symbolic form. Needless to say, the quality of the production of the mathematics printed texts was not always consistent. For example, as displayed in Plate 2.2(6), the translated version of Archimedes (1615: 437) printed in Paris contains mathematical diagrams which are crude especially with respect to line width and horizontal alignment on the page. In addition, typesetting lines separate the Greek and Latin versions of the text, and the headings and margins. Presumably early printing presses possessed to different degrees the technology, expertise and finance to produce printed mathematical books. While the absence of mathematical symbolism in Archimedes' (287-212 BC) rhetorical-style text is not surprising, the textual layout includes spatial separation of the text and diagrams, which is a feature of contemporary mathematical texts. Febvre and Martin (1976: 259) claim that 'printing does not seem to have played much part in developing scientific theory at the start'. This view is based on an observation that some influential works in arithmetic and

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Plate 2.2(6) Quadratura Paraboles (Archimedes, 1615: 437)

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algebra, such as Nicolas Chuquet's (1484) The Triparty, remained in manuscript form in the late fifteenth century. However, early printing presses must have taken time to become established and to be economically viable, the printers must have made careful choices as to what they published. Given the economic demands of the time and the pressure from certain established circles, it is not surprising that the first books in the late fifteenth century were concerned with practical arithmetic and classical texts rather than new works in algebra. When the printed texts on algebra did appear, they were influential. Whitrow (1988), for example, states that the key to the mathematical revolution in the sixteenth century was the beginnings of the development of algebra, and the first book on the subject was Luca Pacioli's Summa de Arithmetica (1494). Whitrow (1988: 267) notes: '[this book] was extremely influential, presumably because it was printed'. The printing of mathematical texts had an immense impact on the mathematics that was subsequently developed, for as Eisenstein (1979: 467) claims '[cjounting on one's fingers or even using an abacus did not encourage the invention of Cartesian coordinates'. Eisenstein (1979) further explains that Newton mastered the classical works of the ancients and contemporary mathematicians such as Descartes from the books he obtained from libraries and book fairs. Newton was self-taught, and this differed greatly from previous practices where learning took place in an oral tradition which involved the elder masters. Likewise, Leibniz had read most of the important mathematical texts of his time before he was twenty years of age (Smith, 1951). Mathematics became widely accessible, and in some sense standardized, through the medium of the printing press. Before moving beyond the times of the Treviso Arithmetic, it is important to note that commercial arithmetic was not the only concern of this time simply for the reason that commerce and trade do not only involve counting. As Swetz (1987: 25) explains, the reckoning masters were the forerunners of applied mathematicians, and their concerns spanned commercial arithmetic to land surveying, construction of calendars, and cask gauging. As trade and colonialization expanded, there was a need to refine navigational techniques and increase military strength. As becomes evident in Section 2.3, mathematical and scientific descriptions at the beginning of the Renaissance included the study of warfare. At this time, mathematics became a recognized profession which was freed from the mysticism of the Middle Ages, and it developed into a discipline that ranked alongside or above other more established fields of study. In this climate, studies in mathematics expanded rapidly into new ways of thinking about old ideas, and new ways of thinking about new ideas. 2.3 Mathematics in the Early Renaissance

The new movements in sixteenth-century Europe were fuelled through the decline of feudalism and the growth of cities and towns in which the

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wealth created through manufacture, commerce and trade meant the introduction of a new power base. In the climate of relative social stability in the West, a re-examination of ideas occurred in what is generally termed the Renaissance. Swetz (1987: 5) explains that 'Intellectual humanism was patronised by capitalism and secularism, which broadened man's horizons of inquiry and innovation.' The arts flourished and the nature of mathematical and scientific thinking also changed. The nature of the change in mathematics is explored below through the examination of several printed mathematical texts of that time. Niccolo Fontana, known as Niccolo Tartaglia, was a pupil of the Italian reckoning schools and later became a prominent mathematician in his own right. Tartaglia wrote La Nova Scientia (The New Science) in 1537, and the frontispiece to this book, displayed in Plate 2.3(1), contains a picture of a tower with different academic fields represented by human figures. The new place of mathematics in what marks the beginning of the Renaissance is made clear in this scene. After being admitted by Euclid and passing by the two firing cannons with their attendants, the two female figures standing by Tartaglia to greet the visitor are labelled Arithmetic and Geometry. The other female figures are labelled Astronomy, Music, Poetry, and Astrology. The figures of Plato and Aristotle stand at the entrance to the second level and the female figure of Philosophy is located at the top level. The banner that Plato holds reads 'hue geometriae expers ingrediatur' or 'Nobody enters who is not expert of mathematics' (translated by David Pingree in Davis, 2000: 293). As Davis (2000: 293) explains, 'What we have here is the hierarchy of knowledge as set out by St. Thomas Aquinas . . . that lacks its topmost thomist level: theology!' The message is that those seeking wisdom must know mathematics, and it appears that an integral part of that knowledge is somehow associated with cannons, a topic which is further investigated below. In the new spirit of the early Renaissance, mathematical discourse began to appear in a very different form from the earlier classics which were still in place as the authorative texts. The circumstantial context of the mathematical problem was often made explicit, and, in addition, the human realm was depicted. For instance, the concept of volume is illustrated through spears which pierce the body of a naked man standing on a mound of earth in Reisch (1535: 424). This conception of volume differs dramatically from that found in Euclid, for example. Tartaglia's (1546: 7) research into the trajectory of cannonballs displayed in Plate 2.3(2) clearly shows the circumstantial context of the problem including the target which is to be hit. The accompanying mathematical symbolic text includes extended arithmetical calculations as seen in Plate 2.3(3). The arithmetic is difficult to read as it is embedded within the linguistic text, and the notation does not include the shorthand forms which are found today; for example, 1020 for 100,000,000,000,000,000,000. This number appears as '100000000000000000000' in Tartaglia (1546). Apart from Tartaglia, military concerns in the form of hitting targets are reflected in other studies;

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Plate 2.3(1) Tartaglia's (1537) La Nova Scientia Frontispiece for example, Galileo (1638: 67) who attempts to calculate the angle of elevation of a building on a hill, this time from different positions on the ground as seen in Plate 2.3(4). Many mathematical texts in the early Renaissance appear to involve human figures participating in some form of physical or perceptual activity

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Plate 2.3(2) Hitting a target (Tartaglia, 1546: 7) where the circumstantial context of the exercise is included. For example, the context surrounding Tartaglia's (1537) concern with predicting the path of a cannonball is explicit in Plate 2.3(5). A man is engaged in firing a cannon to hit a target which appears to be the building on the other side of the river or lake. There is a human actor engaged in a material activity which is presented as the problem to be solved. The problem is approached through geometrical constructions involving lines and a circle (which was later proved to be incorrect through the work of Galileo). While Tartaglia later regretted his work on cannon fire as a contribution to the art of warfare (see Davis, 2000: 293), these texts were nonetheless explicit as to the purpose of the mathematical exercise. Contemporary textbooks in mathematics also have images where the context of the problem is visualized; for example, introductory exercises pose problems in order to introduce the mathematical theory which is to be developed, and practice exercises apply that theory. However, reasons for the theory such as warfare are not typically depicted, at least in the public eye. Rather, the mathematical theory is developed in the abstract, and retrospectively shown to have applications, or alternatively, a suitable example of a problem is posed so that the reader can appreciate the usefulness of the mathematics which is subsequently developed. Mathematical theory is, however, often presented non-contextually in contemporary discourse. However, in the

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Plate 2.3(3) Arithmetic calculations to hit a target (Tartaglia, 1546: 106)

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Plate 2.3(4) Positioning a target (Galileo, 1638: 67) Renaissance texts viewed here, the realm of human activity is an integral part of the mathematics texts. The semiotic visual rendition of the mathematics problem permits a re-organization of perceptual reality. For example, Tartaglia's (1537) drawing of the men firing the cannon in Plate 2.3(5) demonstrates how the visualization of the problem allows new objects or entities to be introduced semiotically. That is, the line segments, circles and arcs and the resultant triangles are entities which only exist in the semiotic construal of the material context of the problem. Significantly, these new entities become the focus of attention for Descartes who attempts to construct different curves and in doing so discovers that they may be described algebraically. From this point, Descartes discards the human realm of sense perception. In what follows, an examination of Descartes' mathematical and philosophical writings reveals that mathematical symbolism developed as a semiotic system to form 'a semantic circuit' with the visual images and language. That is, the visual images of the curves, the symbolic description of those curves and the use of language function hand-in-hand to create a new version of reality. 2.4 Beginnings of Modern Mathematics: Descartes and Newton

There was a shift in the nature of the semiotic construction of mathematical problems in the seventeenth to eighteenth centuries where the circles, curves and line segments increasingly became the major focus of attention rather than the depiction of the material context of the problem that featured so prominently in the early Renaissance texts. For example, the human body gradually disappears, or alternatively is replaced by a part of

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Plate 2.3(5) Predicting the path of cannon fire (Tartaglia, 1537)

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the body, as we may see in the case of Descartes' (1682: 111) eyes in Plate 2.4(la) and Newton's drawings of the path of light in Plate 2.4(lb). While the mental process of perception is still construed, the body part of 'the eye' now acts as the sensor. In these new diagrams, the focus shifts away from the human actor and the context of the problem to the semiotic entities of lines, curves and triangles. With the decline of the human agent, the line segments, circles and arcs and their accompanying spatial and temporal relations take centre stage. In Newton's diagram in Plate 2.4(lb),

Plate 2.4(la) Removing the human body: Descartes (1682: 111)

Plate 2.4(lb) Removing the human body: Newton (1952: 9)

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for example, the dynamic process of the refraction of light is constructed through intersecting line segments through a lens. The drawing of the eye contextualizes the scene, but this eye possesses secondary importance compared to the lens and the path of light depicted by the line segments. This secondary importance is signalled by the size and position of the eye compared to size, centrality and labelling of the line segments and the lens. The human figure disappears in visual representations of material actions as well as acts of perception. For example, in Descartes' (1985c: 259) drawing in Plate 2.4(2a), a hand rather than a complete human figure is drawn swinging a stone from point A to point F. The situational context of the problem is absent and the stone is not swung for any conceivable purpose other than to trace the movement of the stone. Descartes labels the path A, B and F at different points which adds a temporal dimension to the visual semiotic representation. Descartes is concerned with spatial and temporal dimensions of the path in the visual semiotic construction of the problem. Although the path of the cannonball is drawn in Plate 2.3(5), Tartaglia does not attempt to mark so explicitly the unfolding temporal dynamics at particular points of time, but rather the more dynamic aspect

Plate 2.4(2a) Movement in space and time: the stone (Descartes, 1682: 217)

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of his drawing appears in the firing of the cannon. Descartes' attention, however, lies with the path of the stone at different times which are explicitly drawn, labelled and marked spatially with line segments. Descartes draws a model of a compass to trace the movement of the stone in Plate 2.4(2b). This drawing depicts a material compass that swings on a pivot at point E. As we shall soon see, the semiotic compass plays a major role in the development of Descartes' geometry. Newton's (1953: 31) construction of the path of two swinging pendulums

Plate 2.4(2b) Movement in space and time: the model (Descartes, 1682: 217)

Plate 2.4(3) Movement in space and time: the pendulum (Newton, 1953: 31)

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is displayed in Plate 2.4(3). In this diagram the human figure and the context have completely disappeared, and the material everyday object such as a stone is replaced with a pendulum, which is a piece of scientific equipment like the lens in Plate 2.4(lb). Newton marks the relative position of the pendulums in more detail than Descartes' path of the stone in Plate 2.4(2a). One can see the development of the semiotic construction of the prediction of the path of objects as a continuous mapping of spatial and temporal dimensions. The shift from the realm of the material, everyday world of human action and perception to de-contextualized visual images in the beginnings of modern science is apparent in Descartes' drawings in Plates 2.4(4a-b). These illustrations display the path of a ball through water. In Plate 2.4(4a), a man is shown hitting the ball downwards into what looks like a lake or a pond. We have, in a manner similar to Tartaglia's drawings, features of the context of the situation which include a complete human figure involved in some material action in a setting. The path of the ball is constructed as a series of line segments in relation to a circle. However, in Plate 2.4(4b) Descartes shifts his attention to the line segments and curves. Once again, the human figure and the context are eliminated and the major participants are the new semiotic entities of line segments and circles which are situated in specific relations to each other. The appearance of de-contextualized visual images where the major processes are spatial, temporal and relational with entities in the form of

Plate 2.4(4a) Context, circles and lines (Descartes, 1682: 226)

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Plate 2.4(4b) Circles and lines (Descartes, 1682: 228) the line segments, circles and curves requires explanation. One reason is the growing significance of the role of the mathematical symbolism. Descartes discovered that curves could be described using mathematical symbolism, and therefore he moved (a) from the semiotic construction of the material context (b) to the semiotic construction of the lines and curves (c) which were described symbolically (d) to solve the problem. Newton and other mathematicians used this path to lay the foundations of modern science. This path is explored in the remainder of this chapter. Descartes was interested in constructing curves using a semiotically grounded compass which was conceived from the material compass and ruler used by the Greeks. From the material actions depicted in Plate 2.4(5a), Descartes devised a method of semiotically constructing curves based on proportionality as displayed in Plate 2.4(5b). For Descartes, 'This new [semiotic] instrument does not have to be physically applied; it is enough to be able to visualise it and use it as a computing device. In other words, pen and paper is all that is required, since the nature of the curve is revealed in its tracing' (Shea, 1991: 45). The shifts in the nature of the semiotic construals of Descartes' geometry seem to occur in stages. In the initial stages, the semiotic rendition included drawing the material action of tracing the curve as displayed in Plate 2.4(5a). However, this material drawing of the curve (which includes the actions of the hand) developed into a semiotic rendition of the material compass to trace curves as displayed in Figure 2.4(5b). Descartes' main concern was the proportional relations which he mapped visually and spatially as curves using his semiotic compass. Descartes discovered that mathematical symbolism could be used to differentiate between the curves he constructed. Although this was a major

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Plate 2.4(5a) Descartes' semiotic compass (1683: 54) (Book Two)

Plate 2.4(5b) Drawing the curves (Descartes, 1683: 20) (Book Two) breakthrough, Descartes' interest remained in the construction of the curves and he did not realize, as later mathematicians such Newton and Leibniz did, that the symbolism provided a complete description of the curves rather than a means for construction. While the method amounted to an algebraization of ruler-and-compass constructions (Davis and Hersh, 1986), Descartes nonetheless 'simplified algebraic notation and

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set geometry on a new course by his discovery that algebraic equations were useful not only in classifying geometrical curves, but in actually devising the simplest possible construction' (Shea, 1991: 67). Davis and Hersh (1986: 5) comment: 'In its current form, Cartesian geometry is due as much to Descartes' own contemporaries and successors as to himself.' Despite this, Descartes' increasing reliance on the symbolism is evident in his geometry as displayed in Plates 2.4(6a-b). The symbolism features as an integral part of Descartes' geometry, one that now depends on language, visual images and mathematical symbolism. The significance of Descartes' algebraic descriptions for curves cannot be underestimated because this is the point from which modern mathematics and science developed as an integrated multisemiotic discourse in which a central role was assigned to the symbolism. For example, Newton's reliance on mathematical symbolic descriptions is seen in Plates 2.4(7) and 2.4(8a-b). Newton rewrites Euclid's geometry in algebraic terms in Plate 2.4(7), and in printed versions of Newton's work in Plates 2.4(8a-b) the reliance on algebraic symbolism as a method for reasoning is evident. Newton proceeds symbolically step-by-step in Plate 2.4(8a), and efficiently organizes these symbolic descriptions into table format in Plate 2.4(8b). Newton's semiotic is the symbolism which functions in conjunction with language and the mathematical graphs. The implications and circumstances surrounding Descartes' move to the symbolic are worthy of further investigation, not only because this provided Newton with the tools to rewrite nature, but also because it appears that the newly de-contextualized and algebraicized geometry provided Descartes with the foundations for his influential philosophy. Descartes' method is concerned with establishing what is 'true knowledge' through the path of intelligibility rather than sensory experience. This method appears to operate within the boundaries of Descartes' new form of symbolic semiosis, which offers much more at the price of admitting substantially less. The programme of objectivity and truth in the mathematical descriptions inherited from Descartes exists today. 2.5 Descartes' Philosophy and Semiotic Representations

While Descartes did not fully utilize the potential of the mathematical symbolic descriptions, he certainly appreciated the power of this form of semiosis. Descartes repeatedly insists, for instance, that language is inadequate for his purpose of achieving certainty of knowledge beyond the commonsense kind. Descartes' distrust of the linguistic semiotic is openly expressed in the Second Meditation in 'Meditations on First Philosophy' (Descartes, 1952, 1985b) where he attempts to describe what can be known with certainty through a discussion of a ball of wax. Descartes explains that knowledge achieved through the senses (for example, colour, flavour, smell, shape and size) is unreliable because these properties change as the wax is heated. Descartes concludes that mental facilities alone permit

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*to

LA GEOMETRIE.

Plate 2.4(6a) Descartes' description of curves (1954: 234)

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Plate 2.4(6b) Descartes' use of symbolism (1954: 186)

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Plate 2.4(7) Newton's algebraic notes on Euclid

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80

The Method ..of FLUXIONS,

llli To determine -a Conic SeSiimt, at any.Point of which, the Curvntitre and Pofitian of the Tangent, (inrefpeftofthe AxhJ) may be like to the Curvature and' Pofition of the. tangent, at a Point aj/igiid of any other Curve. "21. The ufe of which Problem is. this* that inftead of Ellipfes of the fecond kind, whole Properties- of refradling Light are explain'd by DCS Cartes in his Geometry, Conic Sections may be fubftituted, which Hull perform the fame | thing, very nearly, as to their Reirasftions. And the fame may be underftooci of other Curves.

P R O B. VII. 20 find as .many Curves as you pleafe^ wbofe Areas may be exhibited by finite. Equations.

EXAMPLES.

Plate 2.4 (8a) Newton's (1736: 80-81) Method of Fluxions and Infinite Series

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10. And thus from the Areas, however they may be feign'd, you may always determine the Ordinates to which they belong. P R O B.

VIII.

70 find as many Curves as you pleafe^ wbofe Areas Jhall have a relation to the Area of any given Curve> ajjlgnable by finite Equations. i. Let FDH be a given Curve, and GEI the Curve required, and conceive their Ordinates DB and EC to move, at right Angles upon

their Abfcifles or Bafes AB and AC. Then the Increments or Fluxions o( the Areas which they defcribe, will be as thofe Ordinates drawn

M Plate 2.4(8a) - cont

into

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fhe Method < ? / F L U X I O N S ,

Plate 2.4(8b) Newton's (1736: 100) Method of Fluxions and Infinite Series

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examination of the reliable essence of matter which he conceptualizes as motion and extension in the form of length, breadth and depth. Two translations where Descartes explicitly criticizes the use of language are reproduced below: the first is an English translation of Descartes' original 1642 Latin text and the second is a translation of the 1647 French version of that Latin text. But as I reach this conclusion I am amazed at how [weak and] prone to error my mind is. For although I am thinking about these matters within myself, silently and without speaking, nonetheless the actual words bring me up short, and I am almost tricked by ordinary ways of talking. (Descartes, 1985b: 21) I am indeed amazed when I consider how weak my mind is and how prone to error. For although I can, dispensing with words, [directly] apprehend all this in myself, none the less words have a hampering hold upon me, and the accepted usages of ordinary speech tend to mislead me. (Descartes, 1952: 209)

The two translations express Descartes' criticism of the use of language to describe knowledge which he claims is certain. His intent is clear: 'But aiming as I do at knowledge superior to the common, I should be ashamed to draw grounds for doubt from the forms and terms of ordinary speech' (ibid.: 210). The semiotic Descartes installs as the one most appropriate for his purposes is mathematical symbolism. In this process, the accompanying geometrical curves are only to be used as an aid to thought. Descartes describes his symbolic expressions which are to replace the linguistic descriptions: Whatever, therefore, is to be regarded as an item . . . we shall designate by a unique sign, which can be freely chosen. For convenience sake, we employ the letters, a, b and c, etc., to express magnitudes already known, and A, B, C, etc. for unknown magnitudes. To them we shall often prefix the signs 1, 2, 3, etc., to indicate their numerical quantity, and shall also append them to indicate the number of relations which are to be recognised in them. Thus if I write 2a\ that will be as if I should write the double of the magnitude signified by the letter a, which contains three relations. By this device not only do we obtain a great economy in words, but also, what is more important, we present the terms of the difficulty so plain and unencumbered that, while omitting nothing which is needed, there is also nothing superfluous, nothing which engages our mental powers to no purpose . . . Rule XVI (ibid.: 101)

Descartes (1952: 101-102) claims that linguistic descriptions such as 'the square' or 'the cube' are confusing, and that they should be abandoned. The first is entitled the root, the second the square, the third the cube, the fourth the biquadratic, etc. These terms have, I confess, long misled me. For, after the line and square, nothing it seemed to me allowed of being more clearly exhibited to the imagination than the cube and other shapes; and with their aid I solved not a few difficulties. But at last after many trials I came to realise that by this way of conceiving things I had discovered nothing which I could not have learnt much more easily and

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distinctly without it, and that all such denominations should be entirely abandoned, as being likely to cause confusion in our thinking.

The underlying reason for Descartes' claim that the linguistic descriptions cause 'confusion in our thinking' is that the linguistic version means something quite different to what is generally considered to be the symbolic equivalent. As explained in the discussion of language and photographs in Chapter 1, different semiotic resources have the potential to mean different things. In Descartes' example, the linguistic term 'the cube' is semantically a fixed entity through its grammatical instantiation as a noun. It is an object, a thing, a participant or entity. On the other hand, the symbolic ax1 is not a fixed object, rather it is a complex of interactive participants which combine through the process of multiplication as seen in the expanded form ax xx x x x. The symbolic expression is not a stable fixed entity like the linguistic nominal group, rather it is a dynamic complex which may be reconfigured in different ways. This type of symbolic expression offers countless alternatives to describe different curves; for example x3, 2X3, Sx3 and so forth. Given that Descartes' aim is to construct and differentiate between curves, it is not surprising that the symbolic descriptions are preferred. Descartes' (1985a: 9-78, 111-151) 'Rules for the Direction of the Mind' and 'Discourse on the Method for Rightly Conducting One's Reason and Seeking the Truth in the Sciences' explain his method for securing true knowledge. Basically the method entails finding the simplest parts which are known to be true, and ordering and enumerating these parts to understand the more complex whole. The results should be checked to make sure that there are no errors. Shea (1991: 131) explains more fully the method:' (a) nothing is to be assented to unless evidently known to be true; (b) every subject-matter is to be divided into the smallest possible parts, and each dealt with separately; (c) each part is to be considered in the right order, the simplest first; and (d) no part is to be omitted in reviewing the whole'. This is Descartes' method. Construct the problem out of the simplest elements possible, and rearrange those elements to solve the problem of the more complex. The symbolic descriptions fulfil Descartes' criterion for the right philosophical method upon which to proceed to secure true knowledge because the method appears to be based upon his success in algebraicized geometry. The relationship between Descartes' philosophical method and his algebraicized geometry is quite apparent. First, the (algebraic) elements are broken down into their simplest components to understand the more complex (symbolic) configuration. Second, the (symbolic) statements do not include any superfluous information which may function as a distraction. Third, the (symbolic) expressions may be checked to ensure there is no error. Descartes' procedures in geometry match his method for securing true knowledge. There are several important implications of Descartes' mathematics and philosophy which shaped the course of modern mathematics and science.

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Descartes' philosophy rested on mathematical formulations and procedures which revolutionized the nature of semiosis through the prominence accorded to the symbolic and the secondary position accorded to the linguistic. Language was considered inadequate for the knowledge to which Descartes aspired. Language belonged to the common-sense world of perceptions and sensory input which was deemed unreliable. The famous Cartesian mind and body duality was attached to the semiotic forms which were used to represent each realm. Symbolic and specialized mathematical forms of visual semiosis were located in the realm of the mind which was, according to Descartes, the proper site for securing knowledge. Language belonged to the realm of the everyday and to the body and its sensory apparatus. This remarkable shift in emphasis resulted in a sharp dichotomy between the different forms of semiosis, and this included a difference in the values attached to each. The differing values accorded to the sciences and the arts and social sciences continue today. The focus of concern in mathematics became curves or patterns which were exactly describable through the symbolism. The types of processes in these visual representations are spatial and temporal relations, and relative proportional rates of change. The major visual participants are lines, line segments, circles, arcs and curves and geometrical shapes which are the visual representations of the relations. Those relations are described symbolically through mathematical processes such as multiplication, addition, subtraction and division between the symbolic participants. The continuous nature of these relations is depicted graphically and described exactly through the symbolism. Human actors participating in material, affective, perceptive, behavioural or verbal processes in the context of the everyday were removed from the realm of mathematics and science. The new semiotic tools are designed to work within particular semantic fields, and these do not include the human realm of the material, the emotive, and the sensory which were considered superfluous. The human realm was put aside in this major re-evaluation of knowledge. The mystical claims of the Middle Ages were replaced with a new type of knowledge and a new basis for legitimizing truth. In addition, the claims upheld on the basis of mathematical descriptions were backed by experimental evidence. With the material success achieved through the mathematicization of nature, gradually the God which was central to Descartes and Newton's formulations (see Plate 2.5(1)) was removed from modern science. The significance of this re-contextualization in the modern mathematical view of the world is explored in Chapter 7. As Koestler (1959: 11) comments 'all cosmological systems [visions of the universe] reflect the unconscious prejudices, the philosophical or even political biases of their authors; and from physics to physiology, no branch of Science, ancient or modern, can boast freedom from metaphysical bias of one kind or another'. The question remains as to the impact of our adoption of the mathematical in a present-day context which differs from the cosmology within which it developed.

Plate 2.5(1) Illustration from Newton's (1729) The Mathematical Principles of Natural Philosophy

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Mathematical symbolism originated from rhetorical algebra in the form of linguistic descriptions and commands which explained the method through which to proceed. Syncopated algebra saw abbreviations for recurring participants, but basically the linguistic grammar still provided the basis for these discourses. However, through the work of Descartes, Newton and other mathematicians, especially Leibniz who 'made a prolonged study of matters of notation' (Book Two, Cajori, 1993: 180), the symbolism developed as the semiotic tool which was central to the mathematics which subsequently developed. The symbolic grammar was based on economy through the need to describe in the simplest and most condensed form that which also needed to be rearranged to explain the more complex. This meant the development of new systems in the grammar of mathematical symbolism which did not exist in language. There was to be no confusion, no room for error and no superfluous information in the new forms of reasoning provided by the mathematical symbolism. This grammar of mathematical symbolism is the focus of Chapter 4. From this point, Newton started a trend which could only be called a new world order. 2.6 A New World Order The new approach advocated by Descartes proved to be significant because Newton and others created a movement which involved a new representation of the physical world using new semiotic tools. In this movement, matter and perceptual data were re-admitted by Newton, but in a new mathematicized form (Barry, 1996). As Sweet Stayer (1988: 3) claims, while Newton explained the motion of bodies through his calculus and completed research in the fields of optics, tides, thin films and gravitation, The Mathematical Principles of Natural Philosophy was the culmination of his work, and it 'profoundly changed the perspective with which we view the world'. Newton's new semiotic constructions explained the visible world through invisible properties which were made 'real' or 'concrete' through mathematical symbolic description. One key to this success was that the mathematical symbolism, the visual images and language worked together. Descartes discarded sense data and developed a method with a form of semiosis which could describe exactly relations. These entities could be visualized and they could be described exactly in a symbolic form which allowed the rearrangement of those relations to solve problems and conceptualize the more complex phenomena. Newton dispensed with Descartes' position in that he accepted the world of perception, but at the same time he reconstructed that world using Descartes' semiotic tools. Newton used mathematical symbolism to create metaphorical entities which explained the everyday material world. Newton's new mathematical tools permitted exact description of that which was perceived in terms of the properties of matter. The physical world became the object of concern, and with this new engagement, the means for industrialization, colonialization and commerce rapidly increased. The reason for the status of mathematics is

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precisely the goals and objectives which have been fulfilled through this form of semiotic representation. Mathematics and science fulfil functions which have transformed the face of the world and life on earth. Newton instigated a movement which increased control over the physical world because he included experimentation as an integral part of his scientific method. Galileo first established the ideas of experimentation: 'Galileo's work laid the foundations of the modern scientific method which regards the collection of experimental evidence as the essential prelude to the formulation of scientific laws and theories' (Hooper, 1949: 201). One of Newton's major contributions was the use of technology and scientific equipment in a laboratory setting to test empirically his theories. The theories had to fit the empirical evidence, or at least are seen to fit. Science became a matter of description, prediction and prescription within the confines of the practices established by the laboratory and the semiotic tools which permitted those representations. Although matter may have been re-admitted by Newton, there were constraints on how the sensory phenomena could be viewed and described, and those constraints were established through his semiotic tools and the technology of his scientific equipment. As Descartes openly states, all 'superfluous' information was removed, and what remained was what was possible with the symbolism, visual images and language which formed the semantic circuit with which Newton constructed the new world order. New branches of mathematics have developed since Newton, and the idea of an ordered mechanical physical world has been largely abandoned with the development of the notion of chaos and dynamical systems theory. This new view of the world is based on the idea of non-linearity where it is assumed that the behaviour of physical systems is in fact indeterminate; that is, the behaviour of a system cannot be predicted exactly. Davies (1990) explains that the approach advocated by Descartes where systems are broken down into constituent components to understand the complex whole is reasonably successful because most physical systems behave in this linear format up to a certain point. This method of analysis is only partially successful however: 'On the other hand, they [all physical systems] turn out to be nonlinear at some level. When nonlinearity becomes important, it is no longer possible to proceed by analysis, because the whole is now greater than the sum of the parts' (ibid.: 16). When this point is reached, the constraints, boundary conditions and initial conditions of the system must be taken into account if the behaviour of the system is to be predicted with some degree of success. The new mathematics of non-linear dynamical systems theory is made possible through computer technology. As the computing ability increases, together with the potential for highly sophisticated dynamic graphical images, so the nature of the mathematics changes; that is, mathematics and science are intimately linked to the state of the art of computer technology which affords new possibilities in what has literally become a virtual world. Indeed, computer technology is such that visual images are now

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increasingly exploited as a semiotic resource that offers new possibilities for modelling the world. It could be that the superfluous information once discarded by Descartes can now be incorporated in a new more fully inclusive view of the world. In conclusion, mathematics and science offer a particular representation of the world - one that is limited by the semiotic tools and the technology employed in its construction. In order to appreciate the nature of that construction, texts and contexts must be analysed to understand the types of meanings that are made, and the means through which this is achieved. For this reason, the grammatical systems for mathematical symbolism and visual display are presented in Chapters 4—5. After discussing the unique grammars for each resource, the semantic circuit in mathematics involving the linguistic, the visual and the symbolic is discussed in Chapters 6-7. Discourse analyses of mathematical texts demonstrate that intersemiosis across the three resources is critical for the semantic expansions that take place in mathematics. The analyses represent a close engagement with mathematics as a multisemiotic discourse in order to appreciate the potential and the limitations of the meanings which are made.

3 Systemic Functional Linguistics (SFL) and Mathematical Language

3.1 The Systemic Functional Model of Language

The investigation of multisemiosis in mathematics is based on Michael Halliday's (1973, 1978, 1985, 1994) systemic functional (SF) approach to language which has been extended by Jim Martin and others to incorporate discourse systems (Martin, 1992; Martin and Rose, 2003), genre and ideology (for example, Christie, 1999; Christie and Martin, 1997; Hasan, 1996b; Martin, 1997). An outline of systemic functional (SF) theory and the accompanying grammatical and discourse systems of language is provided in order to explain the conceptual apparatus underlying this study of mathematics. The discussion is necessarily technical, but further explanations of SF theory are provided elsewhere (for example, Bloor and Bloor, 1995; Eggins, 1994; Eggins and Slade, 1997; Halliday, 1994; Halliday and Matthiessen, 1999; Martin etal, 1997; Matthiessen, 1995; Thompson, 1996) including a collection of Halliday's writings (Webster, 2002-). The description of Systemic Functional Linguistics (SFL) and the discussion of the nature of mathematical language in this chapter function to contextualize the systemic frameworks for mathematical symbolism and visual display developed in Chapters 4-5. This leads to an investigation of the meaning arising from the integrated use of language, mathematical symbolism and visual display in Chapters 6-7. The description of mathematical and scientific language in this chapter is general (for a more detailed analysis, see Halliday and Martin, 1993; Halliday and Matthiessen, 1999; Martin and Veel, 1998) as the major concern of this study is the extension of SF theory to mathematical symbolism and visual display in order to investigate the multisemiotic nature of mathematical discourse. In what follows, Halliday's SFL model and linguistic systems at the rank of clause and clause complex and Martin's discourse systems at the level of paragraph and text are used to examine the nature of mathematical language. It becomes apparent in this discussion that the study of mathematical and scientific language needs to take into account the meaning arising from the symbolism and visual display. Martin's discourse systems where meaning is made across stretches of text are therefore

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extended in Chapter 6 to include meaning arising from intersemiosis across linguistic, visual and symbolic components of the text. In addition, the description of grammatical metaphor in this chapter is further developed through an examination of semiotic metaphor in Chapters 6-7. In a similar fashion, the discussion of register, genre and ideology is revisited in relation to the multisemiotic nature of the discourse of mathematics in Chapter 7. Halliday 's SF Theory of Language The fundamental assumption behind Halliday's SF social-semiotic theory is that language is a resource for meaning through choice. Halliday (1994) comprehensively documents the grammatical systems through which language is used to achieve different functions. For, as Halliday explains, language has evolved to satisfy human needs and its grammatical organization is therefore functional with respect to those needs. Halliday (ibid.: xiii) states: 'A functional grammar is essentially a "natural" grammar, in the sense that everything in it can be explained, ultimately, by reference to how language is used.' Any instance of written or spoken language does not unfold haphazardly as an abstract artefact as formal linguists would lead us to believe, but rather all texts are constructed in some context of use. The choices in the text's patternings reflect the uses that language is serving in that particular instance. The underlying assumptions of SFL may be contrasted to the position adopted in formal linguistics where language is conceptualized as a system of rules. Descriptions in these traditions show which sentences are acceptable and explanations reveal 'why the line between in and out falls where it does in terms of an innate neurological speech organism' (Martin, 1992: 3). Rather than adopting an individual mentalist perspective, SFL views language as a resource consisting of a network of relationships. Descriptions show 'how these relationships are interrelated' and explanations reveal 'the connections between these relations and the use to which language is put' (ibid.: 3). SFL is thus orientated to choice, 'what speakers might and tend to do', as opposed to restriction, 'what speakers are neurologically required not to do' (ibid.: 3-4). The SFL approach is concerned with the analysis of how language is used to achieve certain goals through the description of lexicogrammatical (that is, lexical and grammatical) and discourse systems, and the analysis of the choices that have been made in any instance of language use. SFL discourse analysis is a critical interpretation of how language choices function to construct a particular view of reality, and the nature of social relations that are enacted in that construction. SFL evolved from Firthian linguistics and consequently is a type of system/structure theory where the key idea is meaning through choice from the available systems. Following Hjelmslev (1961), paradigmatic relations are mapped onto the available options in the system network (the range of

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choices) and syntagmatic relations are mapped on to actual choices (the process which takes the form of a chain of words). The concept of 'realization' relates system and process in that realization statements specify the systems in process and give the structural arrangement of the selected options. Halliday does not privilege either system or process: 'I prefer to think of these [system and process] as a single complex phenomenon: the "system" only exists as potential for the process, and the process is the actualisation of that potential. Since this is a language potential, the "process" takes the form of what we call a text' (Thibault, 1987: 603). The systems in Halliday's grammar of English are organized according to meaning. Halliday separates the two main types of meaning into the 'ideational' or the reflective, and the 'interpersonal' or the socially active. Halliday further separates ideational meaning into two components, the 'experiential' and the 'logical', which are respectively concerned with the construction of experience and logical relations in the world. Halliday also identifies the enabling function of language, the 'textual' component which organizes language choices into coherent message forms. These four types of meaning, the experiential, logical, interpersonal and textual, are called the metafunctions of language as they are manifestations of the general purposes of language: ' (i) to understand the environment (ideational), and (ii) to act on the others in it (interpersonal)' (Halliday, 1994: xiii). The SF approach means that although the grammatical classes such as nouns, verbs, adjectives and so forth still have a place, for example, in the descriptions of grammatical metaphor (Derewianka, 1995; Halliday and Martin, 1993; Martin et al., 1997; Simon-Vandenbergen et al, 2003), the elements of language are described by functional rather than word class labels. Language is conceived as an 'organic configuration of functions' and 'each part is interpreted as functional with respect to the whole' (Halliday, 1994: xiv). There is equal emphasis on the interpretation of the interpersonal metafunction as well as experiential, logical and textual meanings in SFL. As Poynton (1990) explains, the focus on social relations and the expression of personal attitudes and feelings has traditionally been marginalized in the majority of linguistic theories. The focus on system and referential meaning in linguistics, perpetuated with Chomsky's reformulation of Saussure's (1966) langue/parole (language system versus language use) distinction as competence/performance, was accompanied by an explicit emphasis on the cognitive domain (for example, Chomsky, 1965, 2000). Poynton (1990) explains that such cognitively orientated conceptions of language support dichotomies such as objective versus subjective, and reason versus emotion. As Poynton claims, the higher values accorded to objectivity and reason have obvious significance in the development of mathematics and science, and they have also been invoked in areas of social control. These issues are explored through an interpretation of the concepts of reason, objectivity and truth based on the analysis of a mathematics text in Chapter 7.

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Halliday (1994) is largely concerned with lexicogrammar at the ranks of word, word group/phrase, clause and the clause complex. On the other hand, Martin (1992; Martin and Rose, 2003) is concerned with metafunctionally based systems which operate across paragraphs and the whole text. Martin's work follows from Halliday and Hasan's (1976) systemic analysis of textual cohesion where the basic opposition is between structural (grammatical) and non-structural (cohesive) devices. Martin (1992: 1) organizes his divisions stratally 'as an opposition between grammar and semantics (between clause orientated and text orientated resources for meaning)'. Martin thus establishes a separate discourse semantics stratum to complement Halliday's lexicogrammar. Martin's proposals lead to a language plane with two strata,1 discourse semantics and lexicogrammar, and an expression plane which is concerned with phonology and graphology/ typography (Eggins, 1994: 81-82). The resulting SF model of language, which also includes the communication planes of register, genre and ideology, is displayed in Table 3.1 (1). Halliday's Lexicogrammatical Systems and Martin's Discourse Systems

The major systems in Halliday's lexicogrammar and Martin's (1992; Martin and Rose, 2003) metafunctionally based discourse systems are listed in Table 3.1(2). Following systemic conventions, the lexicogrammatical and discourse systems are capitalized. The major lexicogrammatical systems are MOOD for interpersonal meaning, THEME for textual meaning and Table 3.1(1) Language, Expression and Communication Planes IDEOLOGY GENRE REGISTER LANGUAGE CONTENT

Discourse semantics

Paragraph and text Lexicogrammar

Clause complex Clause Word group and phrase Word

EXPRESSION

Phonology Graphology/Typography

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Table 3.1(2) Metafunctional Organization of Halliday's (1994) Lexicogrammatical Systems and Martin's (1992; Martin and Rose, 2003) Discourse Systems Metafunction

Lexicogrammar

Discourse Systems

interpersonal

clause: MOOD; MODALIZATION; MODULATION; POLARITY; TAGGING; VOCATION; ELLIPSIS

NEGOTIATION (exchange rank including SPEECH FUNCTION at the move rank) Structure: Exchange Structure linking moves

word group: PERSON; ATTITUDE (attitudinal modifiers, intensifiers); COMMENT (comment adjuncts); LEXIS (expressive words, stylistic organization of vocabulary) textual

clause: THEME clause and word group: SUBSTITUTION; ELLIPSIS word group: DEIXIS (nominal)

APPRAISAL Structure: surges, flows and falls mapped through word groups, phrases and clauses in text

IDENTIFICATION (phoricity, reference) Structure: reference chains linking participants

logical

clause complex: LOGICO-SEMANTIC RELATIONS and INTERDEPENDENT

CONJUNCTION and CONTINUITY (based on classifications of LOGICO-SEMANTICS RELATIONS and semantic relations respectively) Structure: conjunctive reticula linking messages

experiential

clause: TRANSITIVITY; AGENCY

IDEATION (lexical relations) Structure: lexical strings and nuclear relations linking message parts

word group: TENSE; LEXIS (lexical 'content'); collocation

TRANSITIVITY for experiential meaning at the clause rank, which is the basic unit in which the semantic features are represented. The elements in the clause (word, word group/phrase) are explained by their functions in each of the metafunctionally based systems. At the rank of clause complex or sentence, the systems for logical meaning are LOGICO-SEMANTIC RELATIONS and INTERDEPENDENT. Halliday's systems are discussed in

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relation to the types of selections found in mathematical discourse in Sections 3.2-3.5. These systems are also considered in relation to the grammatical organization of mathematical symbolism in Chapter 4. The realization structures for Halliday's grammatical systems take different forms. Textual meanings arising from the system of THEME and interpersonal meanings through the system of MOOD are described by paradigmatic oppositions realized through syntagmatic structures of the functional categories. Experiential meanings from TRANSITIVITY choices, on the other hand, are represented as clusters of participant/process/ circumstance rather than sequences of functional elements in the clause. In addition, the patterns of realization vary across the metafunctions. Following Halliday, the realizations of the metafunctions in discourse take the forms of particle-like experiential meanings, irregular prosodic swells of interpersonal meanings (where the concept of volume comes into play) and regular periodic wave-like textual meanings. The realization of the logical metafunction at the rank of clause complex through the system of LOGICAL-SEMANTIC RELATIONS is somewhat different from the other grammatical systems as particular elements (for example, the structural conjunctions 'and' and 'or') may be selected more than once in a clause complex, while the other systems have multiple variables which may be selected only once in the clause. Logical meaning is described by the types of INTERDEPENDENCY relations (dependent or independent) and by LOGICO-SEMANTIC relations between clauses (relations of logical expansion or projection of speech and thought). Martin's (1992; Martin and Rose, 2003) discourse systems include NEGOTIATION, APPRAISAL, IDENTIFICATION, CONJUNCTION and CONTINUITY, and IDEATION. The metafunctional organization and structure of the discourse systems are included in Table 3.1(2). NEGOTIATION is orientated towards spoken discourse, but Martin and Rose (2003) also include the system of APPRAISAL for capturing graduations of attitude (affect, judgement, appreciation) and engagement in written (and spoken) discourse. 'Appraisal is concerned with evaluation: the kinds of attitudes that are negotiated in a text, the strength of those feelings involved and the ways in which values are sourced and readers aligned' (ibid.: 22). Martin's concept of discourse systems is useful for the analysis of stretches of text which involve language, visual images and mathematical symbolism. Martin's frameworks, however, need reworking as they are developed for the analysis of linguistic text (that is, intrasemiosis in language), rather than the analysis of meaning within and across different semiotic resources (that is, intrasemiosis in mathematical symbolism and visual images, and intersemiosis across the three semiotic resources). Discourse systems similar to those proposed by Martin are introduced in the SF frameworks for mathematical symbolism and visual images in Chapters 4 and 5, and the framework for intersemiosis in Chapter 6. The analysis of intra- and inter-semiosis in mathematical discourse includes the typography

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of the text at the expression stratum, the importance of which becomes evident in Chapters 4-6. This stratum has not typically been included in SFL analysis. Martin's discourse structures interact systematically with each other and the lexicogrammatical structures (giving rise to incidences of grammatical metaphor, for example) resulting in the 'texture' of a text. The ways in which the discourse systems co-operate with each other to make a text is not as well understood as the nature of interaction across the grammatical and discourse strata (see for example, Halliday, 1994; Hasan, 1984). Martin (1992:392) refers to the systematic interaction between discoursal and grammatical structures as modal responsibility, cohesive harmony and the method of development of the text. While these formulations are not specifically developed in this study, the grammatical density arising from the interactions between language, visual images and the symbolism across different strata becomes apparent in the analyses of the mathematics texts presented in Chapters 6-7. The texture of discourse in this case involves the dense patterns which emerge from the integrated use of language, mathematical symbolism and visual display (O'Halloran, 2000, 2004c). SFL Discourse Analysis

In SFL discourse analysis, clauses are marked by slashes / / . . . / / and the elements within each clause are analysed according to the metafunctionally based grammatical and discourse systems. Elements in the clause are analysed several times, and functional labels are attached according to choices made from each system. Clauses are also classified as major or minor, and complete or ellipsed. In the case of spoken discourse, abandoned clauses may also be tagged. Minor clauses are 'clauses with no mood or transitivity structure, typically functioning as calls, greetings and exclamations' (Halliday, 1994: 63). Eggins (1994: 172) explains that these minor clauses are 'typically brief, but their brevity is not the result of ellipsis'. The classification of clause type is useful for examining interpersonal patterns of domination and deference. Clauses are classified as to whether they contain rankshifted or embedded elements. Using Halliday's (1994: 63) notion of ranks (word, word group/phrase, clause and clause complex), rankshifting is the process whereby a clause or phrase functions at the lower rank of word or word group. That is, embedded clauses (indicated by sets of square brackets [ [ . . . ] ]) and phrases (indicated by the square brackets [ . . . ] ) function within the structure of a word or word group, thus shifting rank from clause and phrase to the lower rank of word/word group.2 Halliday (ibid.: 242) explains that embedded clausal and phrasal elements may function as a Postmodifier in nominal groups; for example, //the job [[I want]] was advertised//; and adverbial groups; for example, //she reacted more strongly [[than they expected]]//. Alternatively the rankshifted element may function as a Head of a nominal group; for example, //[[that so many

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staff are leaving]] is cause for concern//. As may be seen from these examples, rankshifting in language serves the important function of packing information into the clause. The concept of rankshift is particularly significant in the grammar of mathematical symbolism in Chapter 4 where the lexicogrammatical strategies for encoding meaning in the symbolism are seen to be different from those found in mathematical and scientific language. In the following discussion, the major grammatical and discourse systems are explained, and the nature of the selections found in mathematical and scientific language is discussed according to metafunction. The following linguistic extract from Stewart's (1999: 132) textbook Calculus is used to illustrate features of mathematical language: From Equation 3 we recognize this limit as being the derivative off at x,, that is /' (x,). This gives a second interpretation of the derivative: The derivative/' (a) is the instantaneous rate of change of y=f(x) with respect to xwhen x= a. The connection with the first interpretation is that if we sketch the curve y=f(x), then the instantaneous rate of change is the slope of the tangent to this curve at the point where x = a. This means that when the derivative is large (and therefore the curve is steep, as at the Point P in Figure 4), the ^values change rapidly. When the derivative is small, the curve is relatively flat and the ^values change slowly.

The multisemiotic text for this extract is reproduced in Plate 6.3(2) in Chapter 6 where the meanings arising from intersemiosis between the linguistic, symbolic and visual choices in the text are analysed. It becomes apparent in the analysis of Stewart (1999: 132) that mathematical and scientific language must necessarily take into account the visual and symbolic components of the text. 3.2 Interpersonal Meaning in Mathematics

The analysis of interpersonal meaning is concerned with the nature of the social relations which are enacted through linguistic choices from the systems listed in Table 3.1 (2). The description of the system of MOOD is given in Halliday (1994: 71-105). In essence, the MOOD system (where choices are made for Subject, Finite, Mood Adjuncts, Comment Adjuncts, Predicator, Complement and Circumstantial Adjuncts) is related to the SPEECH FUNCTION which is concerned with the giving/demanding information (statements and questions) and goods and services (commands and offers). The SFL analysis is concerned with choice: how do interactants negotiate the exchange of information and goods and services, and what does this reveal about their social relations? A one-to-one relationship between the grammatical classes of MOOD (declarative, interrogative, imperative and exclamative) and the SPEECH FUNCTION (statement, question, command and offer) does not exist, so the co-text and context are taken into consideration in the analysis. However, the congruent or unmarked case is that statements are realized

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through declarative Mood (SubjectAFinite, for example, 'this is . . . ' ) , questions are realized through interrogative Mood (FiniteASubject, for example, 'is this . . .?') and the WH-element (WH, for example, 'why . . .?'), commands through imperative Mood (Predicater, for example, 'solve . . .') and offers through modulated interrogative Mood (modulated FiniteASubject, for example, 'would you . . .'). Incongruent selections result in interpersonal metaphors where there is variation (for whatever reason) in the enactment of the social relations. The system networks for MOOD and SPEECH FUNCTION are displayed in Figure 3.2 (I), 3 together with the Exchange Structure which consists of sequences of moves. SPEECH FUNCTIONS include responses to the statements, questions, commands and offers and Martin's (1992: 66-76) 'dynamic moves' for spoken discourse, which are requests and responses for the tracking moves of Backchannel, Clarification, Check, Confirmation and Challenging moves. The SPEECH FUNCTIONS of Call and Greeting have also been included, together with moves of Reacting in Figure 3.2(1). These classifications frame the types of moves found in written mathematical texts. As displayed in figure 3.2(1), the SPEECH FUNCTION and MOOD systems relate to the Exchange Structure, which is Martin's (1992) discourse system of NEGOTIATION. The Exchange Structure is based on Halliday (1994) where the opposition is between information and goods and services moves. Following Berry (1981), moves are classified as primary knower (Kl) and primary actor (Al) moves, and secondary knower (K2) and secondary actor (A2) moves. Kl is the speaker/writer who has the information which is being negotiated, and Al is the participant who performs the action. The Exchange Structure consists of obligatory moves Kl and Al and three optional moves. These are delaying moves (dKl and dAl), secondary knower (K2) and secondary actor moves (A2). Berry's (1981) moves are developed from Halliday's (1994: 69) SPEECH FUNCTION classifications of 'initiating and responding' to the 'giving and demanding' of goods and services and information. While all the classifications presented in Figure 3.2(1) do not typically occur in written mathematical texts, it is useful to consider the selections afforded in dynamic spoken contexts to situate the type of discourse found in written mathematics. In order to incorporate Ventola's (1987, 1988) 'move complexes', the categories of initiation, request, response and closure moves are supplemented with 'K-Continuation' and 'A-Continuation', and 'K-x' and 'A-x' moves. These move-complexes are based on Halliday's (1994: 220) LOGICO-SEMANTIC RELATIONS of expansion (elaboration, extension and enhancement) and projection (locution and idea) and interdependency relations (see Section 3.4 on logical meaning). A 'continuation' move realizes a continuation of the same SPEECH FUNCTION which was established in the previous move even though the clause selects independently for MOOD. This move is realized through paratactic relations of interdependency between the two clauses. An 'x' move realizes a continuation of

Proposition Dynamic move Followup Reacting

Call Greeting Knowledge

EXCHANGE new

continuation exchange number

dK1 K1 K2 K-lnitiation K-Continuation K-x K-Response K-Request

SPEECH FUNCTION

Action

A-lnitiation A-Continuation A-x A-Response A-Request

declarative WH - interrogative MOOD

YN - interrogative imperative

Proposal Dynamic move Followup Reacting dA1 A1 A2

statement exclamative expletive question acknowledgeme contradiction answer disclaimer clarification challenge backchannel check confirmation

SPEECH FUNCTION

command offer compliance refusal acceptance rejection clarification challenge backchannel check confirmation

Figure 3.2(1) NEGOTIATION (Exchange Structure), SPEECH FUNCTION and MOOD

paralinguistic none

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the same SPEECH FUNCTION in hypotactically related finite and nonfinite clauses. In this way, it is possible to map each clause as an element in the unfolding Exchange Structure. SFL analysis involves analysing choices in the text for mood, speech function and negotiation (exchange structure). While this framework is used to discuss the linguistic selections in Stewart (1999: 132), it is apparent that discourse moves in written mathematics often involve shifts from one semiotic resource to another. Stewart (1999: 132) makes reference to 'Equation 3' and 'Point Pin Figure 4'. In a similar fashion, the command to solve a problem is typically undertaken symbolically. However, the SFL grammatical and discourse frameworks provide the starting point for the development of grammars for the symbolism and visual display and the theorization of intersemiotic shifts and transitions in mathematical discourse. In what follows, the nature of linguistic selections for interpersonal meaning in Stewart (1999: 132) are investigated. The selections from the systems of MOOD, SPEECH FUNCTION and NEGOTIATION function to establish unequal relations between the writer and the reader of the mathematics text. In the following chapters, it becomes apparent that the dominant position of the writer is reinforced across choices for mathematical symbolism and visual display. Similarly, unequal social relations are established between the teacher and the students in the context of the mathematics classroom. While mathematical pedagogical discourse is dominated by the teacher (Veel, 1999), the nature of those social relations in classrooms differs on the basis of gender and social class (O'Halloran, 1996, 2004c). The nature of the linguistic selections which reinforce the position of dominance of the author of the mathematics text is discussed below. Given the monologic format of the written discourse of mathematics, the writer assumes the speech roles. Foremost, the writer is the primary knower (Kl) who gives information in the form of statements through declarative Mood. These statements are typically complete, and so the writer provides detailed information. In the case of mathematics textbooks such as Stewart (1999), the author also assumes the role of secondary knower (K2) who asks the questions and, most typically, provides the answers (Kl). The writer takes the role of the one who commands (A2). In addition, as primary actor (Al), the writer checks that commands have been completed correctly by providing the solutions to problems. Mathematics lends itself to these types of social relations between the writer and the reader as mathematics is a written discourse. The Exchange Structure typically involves long sequences of moves as seen in Stewart (1999: 132). The extended exchanges contribute to the steady interpersonal rhythm of mathematical discourse, with its overarching aim of deriving results through long sequences of logical reasoning. Stewart (1999: 132) attempts to vary the interpersonal nature of the extended exchanges through various strategies, which include the use of 'we' as Subject. However, such selections as 'we' give rise to interpersonal

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metaphors where it is clear that the relations are being manipulated. For example, in the statement 'From Equation 3 we recognize this limit as being the derivative of/at xl that is,/' (xj)', the inclusive 'we' serves interpersonal rather than experiential meaning as the statement may be seen as a metaphorical variant for the command 'Recognize this limit as being the derivative of/at x1} that is,/' (xj)'. Alternatively, the process 'recognize' in the projecting clause may be seen to be metaphorical and unnecessary with respect to the more direct statement 'this limit is the derivative of/at x1; that is/' (xj)'. Although the writer attempts to vary the social relations with the reader through Subject choice and metaphorical expressions, the reader nonetheless remains the receiver of information, and the one whose answers and responses are checked against those provided by the author of the mathematics text. The degrees of probability and obligation associated with the linguistic statements, questions, commands and answers in mathematical discourse are similarly consistent. Halliday's (1994: 354—363) graduations in probability and usuality (MODALIZATION) and inclination, obligation and potentiality (MODULATION) are associated with propositions ('information') and proposals ('goods and services') respectively. The descriptive categories for MODALIZATION and MODULATION and the value and orientation of the selection are displayed in Figure 3.2(2).

probability MODALITY usuality obligation C-MODULATION potentiality inclination O-MODULATION potentiality

VALUE

-maximal high median low objective subjective

ORIENTATION

explicit implicit

Figure 3.2(2) MODALIZATION and MODULATION

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The statements are unmodalized in terms of graduations of probability and usuality in the extract from Stewart (1999: 132). For instance, the absence of modality (realized through the Finite selections such as 'might', 'could', or 'should') functions to make the mathematics statements appear as correct and factual. The lack of modalization is accompanied by maximal obligation in the commands. POLARITY is simply positive ('is') and negative ('is not'). This contributes to the steady interpersonal orientation of a discourse which possesses an unqualified level of certainty. As noted in Chapter 4, probability in mathematical discourse is typically expressed symbolically through relational clauses. For example, probability may be expressed through approximations such as x =» 0.5. The typical absence of selections from Halliday's (1994: 82-83) system of MOOD ADJUNCTS displayed in Figure 3.2(3) also functions to create an aura of factuality. For example, in Stewart (1999: 132) Mood Adjuncts indicating plays with probability (for example, 'possibly', 'perhaps' and 'certainly') are not selected. Instead a certain presumption arises from the unmodalized statements and unmodulated commands, the nature of the processes which are selected (see Section 3.3) and the long implication sequences in the Exchange Structure arising from selections for logical meaning (see Section 3.4). This is not to say that Mood Adjuncts are not selected in mathematical discourse. However, the nature of such adjuncts may replicate the high level of presumption and obviousness found in the pedagogical discourse of mathematics (O'Halloran, 1996, 2004c). The objective, rational and factual stance of mathematics is the product of the nature of the selections for interpersonal meaning as they combine with a limited range of process types and participants (see Section 3.3) with an emphasis towards logical meaning (see Section 3.4). Descartes' removal of the human realm in mathematics and science is apparent in modern mathematics. The types of interpersonal choices from language, mathematical symbolism and visual display function to simultaneously restrict and expand experiential and logical meaning in mathematics. In the context of the mathematics classroom, teachers introduce a variety of interpersonal strategies to maintain solidarity and group cohesiveness and to relieve the interpersonal stance of the subject matter (O'Halloran, 1996, 2004c). However, if one considers high school mathematics texts, books on mathematics and other generic forms of mathematical discourse, the probability usuality MOOD ADJUNCT

presumption inclination time degree

Figure 3.2(3) MOOD ADJUNCTS

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interpersonal stance of mathematical written discourse is largely consistent. This observation explains the inclusion of metaphorical forms of expression and non-generic choices such as cartoons, drawings and photographs which are inserted to disrupt the interpersonal orientation of mathematics texts. Furthermore, the issue of lexical choice for interpersonal meaning in mathematics is addressed below. Halliday (1961: 267) states that 'The grammarian's dream is ... to turn the whole of linguistic form into grammar, hoping to show that lexis can be denned as "most delicate grammar" '(see Hasan, 1996a). Two major areas of interest are lexis specific to the field of mathematics which is considered under experiential meaning (see Section 3.3), and lexical items which are interpersonally marked. The relevant notion is one of 'core vocabulary' where certain lexical items are more central than others in describing experiential or intersubjective reality. Carter describes tests for coreness which involve syntactic and semantic relations, and neutrality. From this perspective, the coreness of lexical items is the extent to which they are 'more tightly integrated than others into the language system; that is, they occupy places in a highly organized network of mostly structurally-semantic and syntactic interrelations' and are 'more discoursally neutral than others, that is, generally they function in pragmatic contexts of language use as unmarked and non-expressive' (Carter, 1998: 36). Expressive linguistic selections orientated towards interpersonal meaning are included in Martin and Rose's (2003: 54) system network for APPRAISAL, which is reproduced in Figure 3.2(4). APPRAISAL is concerned with evaluation: how the text functions to align the reader or speaker with the various propositions or proposals which are put forth. This includes lexical items and cases of amplification, special forms of address and so forth. Further research will see the development of the system of APPRAISAL so that different strategies for positive and negative evaluation monogloss ENGAGEMENT heterogloss

PROJECTION MODALITY CONCESSION

AFFECT APPRAISAL

ATTITUDE

JUDGEMENT APPRECIATION

FORCE GRADUATION FOCUS

Figure 3.2(4) APPRAISAL Systems Reproduced from Martin and Rose (2003: 54)

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may be uncovered. The three main appraisal systems given by Martin and Rose (2003: 54-55) are attitude, amplification (or graduation) and source (or engagement): Attitude comprises affect, judgement and appreciation: our three major regions of feeling. Amplification covers grading, including force and focus; force involves the choice to raise or lower the intensity of gradable items, focus the option of sharpening or softening an experiential boundary. Source covers resources that introduce additional voices into a discourse, via projection, modalization, or concession; the key choice is one voice (monogloss) or more than one voice (heterogloss).

The absence of lexical items orientated towards expressive or evaluative interpersonal meaning is apparent in the extract from Stewart (1999: 132). Lexical choice in mathematics is largely orientated towards experiential and logical meaning rather than interpersonal meaning. This does not mean, however, that the mathematics writer does not make evaluations. Appraisals of what is presented in mathematics exist in different genres in different forms. For example, the authors of research papers in mathematics presumably cast a favourable impression on the results which are established. However, such judgements presumably appear as factual rather than evaluative. For example, Mood Adjunct selections (for example, 'of course' and 'typically') which may combine with an explicit objective orientation towards modality ('it is certain' and 'it appears that') mean that evaluations are made through grammatical choices which do not necessarily include interpersonally expressive lexis (for example, 'that is excellent' or 'that is really ridiculous'). The linguistic strategies for evaluation in mathematics and scientific discourse require further research. In addition, perhaps APPRAISAL may be understood as a meta-system arising from the layering and juxtaposition of functional choices across experiential, interpersonal, textual and logical systems, rather than a discourse system in its own right. Further work is needed, however, to establish how such layers function to orientate the reader. The apparent lack of the need to explicitly evaluate contributes to the view of mathematics as a rational discourse of truth. However, as discussed in Chapter 2, mathematics dispensed with many realms of human activity. As the semiotic which provides the meta-discourse for that which is performed symbolically and visually, language choices in mathematics are circumscribed within certain semantic domains. The limited fields of meaning are considered in the discussion of experiential meaning in Section 3.3 and the packing of that information through grammatical metaphor in Section 3.6. However, the point is that the discourse of mathematics appears as factual and objective truth because of the types of interpersonal choices which are made using language, and the precise organization of those choices in the mathematics text (see Section 3.5 for textual meaning). This orientation is supported by the nature of experiential and logical choices (see Sections 3.3 and 3.4), and by the available options in the system networks for the grammar of visual images and the

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symbolism. This point is explored further in the chapters concerned with the symbolism, visual images and intersemiosis in mathematics. 3.3 Mathematics and the Language of Experience

Following Halliday (1994), experiential meaning at the rank of clause is realized through the system of TRANSITIVITY displayed in Figure 3.3(1). The construction of experience takes the form of choices for process, participants and circumstance. Halliday (1994) includes the ergative interpretation of experience in the form of AGENCY. The associated functional elements are the Medium, Agent, Beneficiary, Range and Circumstance. Halliday's (1994: 166) descriptive categories have been extended to include mathematical 'Operative' processes in mathematical discourse as displayed in Figure 3.3(1). This new process type initially appeared in mathematical symbolism in the form of mathematical processes such as addition, subtraction, multiplication, division, powers and roots, and other mathematical operations. The meanings of these processes in mathematical symbolism do not accord with existing processes categories. The linguistic versions of these process types have thus been categorized as Operative processes. The rationale and justification for the inclusion of Operative processes in the mathematical symbolism is found in Chapter 4. The stages through which mathematics became concerned with particular realms of meaning to the exclusion of others are discussed in Chapter 2. Mathematics dispensed with the human realm, and became concerned with dynamic relations which could be viewed visually and described symbolically. Relations took a visual form, and linguistic descriptions shifted to the symbolic formulations. As language functions as the meta-discourse for these descriptions and visual instantiations, the nature of experiential meaning in mathematics simultaneously expanded to incorporate the new meanings, and contracted to the limited semantic realms with which the visualizations and symbolic descriptions were concerned. The impact on the nature of mathematical language arising from the semantic expansions made possible through the symbolism and visual display may be seen in Stewart (1999: 132). This includes the relatively high incidence of relational processes and the metaphorical nature of the participants. These features of experiential meaning in mathematical language are considered below. The major process type found in mathematical language appears to be the relational process, which Halliday (1998: 193) explains is the favoured process type in science. It appears that as mathematical symbolism became concerned with the description of relations, the same shift occurred within language which was being used to describe and contexualize the visualizations and symbolic descriptions. Halliday (1993a, 1993b, 1998), Halliday and Matthiessen (1999) and Martin (1993a, 1993b) explain that the regrammaticization of experience which takes place through scientific language involves relational processes and entities in the form of grammatical

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Actor Goal Range Recipient Client

Material

Senser Phenomenon

Mental

Sayer Verbiage Receiver

Verbal

Intensive Attributive

Circumstantial

PROCESS

Possessive

Relational

Carrier Attributor Beneficiary Attribute

Intensive

PARTICIPANTS Identifying

Circumstantial Possessive

Existential

Existent Behaver

Behavioural

Behaviour

X1 X2 X3 Xs

Operator Operative

Participant

Extent

duration distance

Location

Manner CIRCUMSTANCE Cause

temporal spatial means quality comparison reason purpose behalf fcomitation

Accompaniment

u

Matter Role

Figure 3.3(1) The TRANSITIVITY System

addition

Identified Identifier

Token Value

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metaphors, in particular, nominalizations. Such metaphorical participants permit experiential meaning to be economically packaged within the nominal group structures which are aligned through relational clauses. The impact of this regrammaticization of experience is discussed in relation to grammatical metaphor in Section 3.6. The relatively high incidence of relational processes and metaphorical participants is found in mathematical language. For example, relational processes realized through 'is' and grammatical metaphors (bold) appear in Stewart (1999: 132): '//The connection with the first interpretation is [ [that if we sketch the curve y =f(x), \ \ then the instantaneous rate of change is the slope of the tangent to this curve at the point [[where x= a ]]]]'. The repacking of experiential content through relational processes and grammatical metaphor is reconsidered in Chapter 6 where the notion of grammatical metaphor is linked to semiotic metaphor. The reasons for the current forms of scientific and mathematical language include the impact of the functions which are fulfilled by mathematical symbolism and visual images. The basis of the discourse system of IDEATION for experiential meaning is lexis (Martin, 1992). The discourse units underlying the lexical items are lexical relations which are concerned with (i) taxonomic relations, (ii) nuclear relations and (iii) activity sequences. The discourse structures realizing lexical relations are called lexical strings which run through the text. In the case of taxonomy, the two types of lexical relations are superordination involving subclassification and composition involving part/whole relations. The types of taxonomic relations are summarized in Eggins (1994: 101-102). In mathematics, the taxonomies for mathematical terms are extended and precise; for example, triangles are defined according to the size of the angles and sides. This serves to order mathematical reality in exact ways, leading to condensation in mathematical texts; for example, the term 'isosceles triangle' incorporates a range of meanings. Mathematical taxonomies, however, are not explored here. The second category of lexical relations involves nuclear relations. 'Nuclear relations reflect the ways in which actions, people, place, things and qualities configure as activities in activity sequences' (Martin, 1992: 309). These relations have previously been handled in SFL under collocation. In the case of mathematics, nuclear relations stretch across linguistic, visual and the symbolic components of the mathematics text. Nuclear relations are realized through configurations of Halliday's functional categories of process, participant and circumstance in the system of TRANSITIVITY, and the corresponding systems in mathematical symbolism and visual display. The model of nuclearity adopted for language and the mathematical symbolism follows Martin (1992: 319). Centre PROCESS = Range: process

Nucleus + MEDIUM + Range: entity

Margin Periphery + AGENT x CIRCUMSTANCE + BENEFICIARY

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The third category of lexical relations is expectancy and implication relations between activities in activity sequences: 'These relations are based on the way in which the nuclear configurations . . . are recurrently sequenced in a given field' (Martin, 1992: 321). The relations in mathematics are realized through conjunctive relations, with implication relations typically involving conditional and consequential type relations. As suggested by the extract from Stewart (1999: 132), given the emphasis on logical meaning and the derivation of results, implication chains involving the semiotic construction of mathematical knowledge through language, symbolism and visual images are extended and complex (O'Halloran, 1996, 2000). Halliday (1978) explains that the need to conceptualize abstract relations in mathematics using linguistic modes of expression causes grammatical problems. Apart from borrowing everyday linguistic terms, mathematical language is technical and often involves complex taxonomies of terms in nominalized forms. Halliday (1993b: 69-85) describes the difficulties in mathematical and scientific language which involve interlocking definitions, technical taxonomies, special expressions, lexical density, syntactical ambiguity, grammatical metaphor and semantic discontinuity. However, these problems cannot be viewed in isolation. Rather the difficulties with mathematical language must be viewed in connection with symbolic and visual descriptions. Further to this, the texture of mathematical discourse (linguistic, visual and symbolic) involves grammatical intricacy (like spoken discourse) and lexical density (like written discourse) which results in grammatical density (O'Halloran, 1996, 2004c). In other words, the language of mathematics is best investigated in relation to functions and grammar of mathematical symbolism and visual display to understand the functions of contemporary linguistic constructions in mathematics. 3.4 The Construction of Logical Meaning

Martin's (1992) discourse systems of CONJUNCTION and CONTINUITY are informed by Halliday's paradigm for clause complex relations in the form of INTERDEPENDENCY and LOGICO-SEMANTIC RELATIONS. Halliday's (1994: 221) description of clause complex relations is based on the system of TAXIS which is common to word, group, phrase and clause complexes alike. Halliday (1994) distinguishes hypotaxis as a dependent modifying relation and parataxis as an independent continuing relation. As illustrated in Figure 3.4(1), clause complexes are classified as paratactic and hypotactic. In addition, cohesive or intersentence logical relations are based on Halliday (1994: 220) and Martin (1992: 179). Halliday's (1994: 219-220) system of LOGICO-SEMANTIC RELATIONS is also concerned with EXPANSION and PROJECTION. The categories of EXPANSION describe the relations whereby a secondary clause expands the primary clause through Elaboration ('='), Extension ('+') and Enhancement ('x'). Secondary clauses realizing Elaboration ('that is' type relations) function to restate, specify, comment on or exemplify the

elaboration

elucidative

opposition clarification

additative

additative alternation

comparative •

similarity contrast

temporal

simultaneous successive

extension expansion enhancement paratactic independent

consequential

cohesive .none

locution projection idea implicit

dependent

hypotactic explicit internal external

Figure 3.4(1) LOGICO-SEMANTIC RELATIONS and INTERDEPENDENCE

purpose condition consequence concession manner

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content of the primary clause. Secondary clauses realizing Extension ('and') add new elements to the primary clause by giving exceptions or offering alternatives. Secondary clauses realizing Enhancement ('so', 'yet', 'then') serve to qualify the primary clause with circumstantial features of time, place, cause or condition. Projection describes the situation whereby the secondary clause is projected through the primary clause as a Locution or Idea. Locution is the realization of the secondary clause as wording (") while Idea realizes the secondary clause as an idea ('). As illustrated in Figure 3.4(1), the type of INTERDEPENDENCE (paratactic or hypotactic) is cross-referenced with the type of LOGICO-SEMANTIC RELATION (expansion or projection). The discourse semantic systems of CONJUNCTION and CONTINUITY are modelled through covariate dependency structures called conjunctive reticula (Martin, 1992). The discourse system of CONTINUITY differs in that items are realized in the Rheme as opposed to textual Theme. Following Martin (1992), these systems are organized by listing the clauses down the page.4 Succeeding moves are shown to be dependent on preceding ones by dependency arrows pointing upwards towards the presumed message. Typically conjunctive relations are anaphoric but in the case of the forward relations, an arrow is placed at both ends of the dependency line. Implicit conjunctions are shown where they could have been made explicit in the discourse. The systems of CONJUNCTION and CONTINUITY may be used to describe logical relations in mathematics (O'Halloran, 2000: 378) which typically involve discourse moves across linguistic, symbolic and visual parts of the text. The step-by-step development of logical reasoning is an important function of symbolic mathematical discourse discussed in Chapter 4. The analysis of logical meaning in mathematics involves long and complex chains of reasoning which favour consequential-type relations (O'Halloran, 1996, 1999b, 2000). Typically these chains of reasoning (at least in the symbolic text) are primarily based on pre-established mathematical results. The significance of logical meaning in mathematical linguistic text is evident in the analysis of the extract from Stewart (1999: 132) displayed in Figure 3.4(2). There are complex nested structures of logical relations realized through structural conjunctions and conjunctive adjuncts, and there are also clause complex relations within rankshifted clause configurations. The analysis also reveals that logical meaning is realized metaphorically in the form of processes. That is, logical meaning is realized through the processes 'gives' and 'means' in the following clauses: 'This gives a second interpretation of the derivative', and 'This means [[that when the derivative is large «(and therefore the curve is steep, as at the Point P in Figure 4)» || the y-values change rapidly]]. Such processes are examples of grammatical metaphor (see Section 3.6) where logical meaning is encoded through process type. However, as evidenced in the short extract from Stewart (1999: 132), logical relations typically stretch across symbolic, visual and linguistic components of the text.

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From Equation 3 we recognize this limit as being the derivative of f at x 1 that is, f'(x1) This gives a second interpretation of the derivative: (that is) The derivative f'(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a. The connection with the first interpretation is [[that if we sketch the curve y = f(x), || then the instantaneous rate of change is the slope of the tangent to this curve at the point [[where x

= a]]]] This means [[that when the derivative is large «(and therefore the curve is steep, as at the Point P in Figure 4)» ||the y-values change rapidly]]. When the derivative is small the curve is relatively flat and the y-values change slowly Stewart[, 1999 #465: 132]

Figure 3.4(2) Logical Relations in Stewart (1999: 132) 3.5 The Textual Organization of Language At the lexicogrammatical stratum, textual meaning is realized as GivenA New through the system of THEME (Halliday, 1994) which is composed of two functional elements: the Theme and Rheme. Following Halliday (1994: 38), ' [t]he Theme is the element which serves as the point of departure for the message; it is that with which the clause is concerned. The remainder of the message, the part in which the Theme is developed, is called . . . the Rheme'. The system network for THEME is given in Figure 3.5(1). The Theme analysis, which is concerned with the organization of New information, permits the development of the text to be tracked at the rank of clause and clause complex. In addition to Theme, Martin and Rose (2003: 175-205) discuss thematic development in terms of phase and the whole text. That is, hyperThemes function to organize information at the rank of paragraph, and macroThemes provide the focus for the text. Thus the Conjunction - structural textual

Conjunctive Adjunct Continative Vocative

THEME interpersonal -

Modal Adjunct Finite

ideational

RHEME

Figure 3.5(1) The System of THEME

topic

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organization of the text is investigated as regular periodic waves of increasing amplitudes at the ranks of clause, clause complex, paragraph and text. The THEME analysis for Stewart (1999: 132) (where Theme selections appear in bold) is given below: //From Equation 3 we recognize// //this limit as being derivative of/at x^// //that is, /'(*!)//

//This gives a second interpretation of the derivative:// //(that is) The derivative/' (a) is the instantaneous rate of change of y—f(x) with respect to x// //when x = a// //The connection with the first interpretation is [ [that if we sketch the curve y = f (x), 11 then the instantaneous rate of change is the slope of the tangent to this curve at the point [ [where x = a] ] ] ] // //This means [ [that when the derivative is large « (and therefore the curve is steep, as at the Point P in Figure 4)» 11 the y-values change rapidly]]// //When the derivative is small// //the curve is relatively flat// //and the y-values change slowly// The analysis demonstrates that the mathematical linguistic text is carefully organized to carry forth the argument. Marked Themes are selected ('From Equation 3', and 'When the derivative is small') to foreground important experiential content. Information is not only packaged into nominal group structures through grammatical metaphor, but clausal rankshift also appears to be a significant method of organizing experiential meaning. In addition, selections such as 'this' link the clause to previously established results. Martin (1992: 416) sees these types of selections as a case of textual grammatical metaphor (see Section 3.6). The linguistic text in Stewart (1999: 132) reflects grammatical intricacy as well as lexical density. More generally, these two types of complexity combine in mathematical discourse to give grammatical density (O'Halloran, 2000, 2004c) as discussed in Chapter 7. The discourse system of IDENTIFICATION is used to track participants where the basic opposition involves phoricity whereby information is recoverable from the text or context. That is, a participant is either newly presented ('addition'), or alternatively the identity of the presumed participant has to be retrieved in some way from the text or context (Halliday, 1994: 312-316). The means of retrieval are described by the types of phora (see the system network in Martin, 1992: 126). This includes 'bridging reference' where the referent has to be inferentially derived from the context rather than by direct reference, and 'multiple reference' which results in ambiguity. There is also 'generic' or 'specific' reference: 'Generic reference is selected when the whole of some experiential class of participants is at stake rather than a specific manifestation of that class' (Martin, 1992: 103).

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These classifications are used to track the participants in mathematical discourse in order to understand how reference functions in mathematics. The linguistic text in Stewart (1999: 132) illustrates that tracking participants in mathematics necessarily involves the linguistic, symbolic and visual components of the text. In addition, reference chains in mathematics are complex as they split and cojoin as mathematical participants are rearranged for the solution to problems and the mathematical relations which are described are visualized (O'Halloran, 1996, 2000). The complexity of tracking participants may be seen in Stewart where 'the derivative' is variously referred to as '/'(xj)' and '/'(a)' and participants are reconfigured in other ways, for example, 'at x^ and 'x= a\ Tracing participant reconfigurations across the three semiotic resources necessarily involves knowledge of the grammars of language, mathematical symbolism and visual display. 3.6 Grammatical Metaphor and Mathematical Language

Grammatical metaphor is an important concept for understanding the nature of scientific language (for example, Chen, 2001; Derewianka, 1995; Halliday, 1994; Halliday and Martin, 1993; Martin, 1992, 1997; Martin and Veel, 1998; O'Halloran, 2003b; Simon-Vandenbergen et al, 2003). This discussion forms the basis in Chapter 6 for the extension of the concept of grammatical metaphor to semiotic metaphor. In this formulation, the notion of grammatical metaphor is extended to take into account the types of meaning expansions which take place intersemiotically in multisemiotic texts. The nature of the systems and lexicogrammatical strategies for encoding meaning in language, visual images and symbolism are the product of the interaction of the three resources. Grammatical metaphor is a 'variation in the expression of a given meaning' which appears in a grammatical form although some lexical variation may occur as well (Halliday, 1994: 342). The typical or unmarked form is referred to as the congruent realization and the other forms which realize some transference of meaning as the metaphorical form. The presence of grammatical metaphor necessitates more than one level of interpretation, the metaphorical (or the transferred meaning) and the congruent. Martin (1993a: 237) states: 'the fact that we have to read the clause on more than one level is critical - the metaphor makes the clause mean what it does'. If, therefore, an expression can be unpacked grammatically to a congruent meaning, it is a case of grammatical metaphor. Halliday's categorization of the types of grammatical metaphor (see Table 1.9 in Martin, 1997: 32) is given in Table 3.6(1). The types of grammatical metaphor are organized metafunctionally according to rank in Table 3.6(1). There exist logical, experiential and interpersonal metaphors at ranks of clause complex, clause and word group. Grammatical metaphor involves the shifts to 'entity', 'quality', 'process' and 'circumstance' from congruent realizations listed in Table 3.6(1).

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Table 3.6(1) Halliday's Grammatical Metaphor (see Table 1.9 in Martin, 1997: 32) RANK AND METAFUNCTION

GRAMMATICAL METAPHOR

Clause complex: LOGICAL relator

entity

Examples: so if because

(nominal group) cause/proof condition reason

relator

quality

Examples: then so

(nominal group) subsequent/follow resulting

relator

process (clause)

Examples: then

follow cause complement

so and

circumstance (clause)

relator Examples: when

in times of/in . . . times under conditions of/under . . . conditions due to

if

therefore Clause: LOGICAL (internal relations) EXPERIENTIAL and INTERPERSONAL process

Examples: event auxiliary - tense - phase - modality

(nominal group) transformation transform will/going to try to can/could/may/will

process Examples: event

entity

prospect attempt possibility, potential, tendency quality

(nominal group) increasing poverty poverty is increasing

( S F L ) A N D MATHEMATICAL LANGUAGE Table 3.6(1) - cont process auxiliary - tense - phase - modality

quality was/used to begin to must/will [always] may

previous initial constant possible/permissable

process Examples: divide

circumstance

circumstance Examples: with to

entity (nominal group)

circumstance

quality (nominal group)

Examples: manner other other

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'h' [on V]

accompaniment destination

[decided] hastily [argued] for a long time cracked on the surface

circumstance (clause) Examples: be about be instead of

hasty decision lengthy [argument] surface [cracks] process (clause) concern replace

Word group: LOGICAL (internal relations), EXPERIENTIAL and INTERPERSONAL quality Examples: unstable entity Examples: the government [decided] the government couldn't decide

entity (nominal group) instability modifier [expansion] (nominal group) the government [decision] [a/the decision] of/by the government the government's [indecision] [the indecision of the government] governmental [indecision]

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Table 3.6(1) - cont Non-entity: LOGICAL (internal relations), EXPERIENTIAL and INTERPERSONAL entity (nominal group) the phenomenon o f . . .

process (clause) . . . occurs/ensues

The new case where a process is realized by circumstance (as illustrated by 'h on r' to mean 'h divided by r') which has appeared in mathematical classroom discourse (O'Halloran, 1996) is added to Halliday's categories in Table 3.6(1). The majority of cases of grammatical metaphor involve the process of nominalization whereby a grammatical class or structure realizing a process, circumstance, quality or conjunction is turned into another grammatical class, that of a nominal group realizing a participant. Following Halliday (1993a, 1993b, 1998) nominalization is conceived as 'the predominant semantic drift of grammatical metaphor in modern English' (Martin, 1992: 406), which has largely resulted from changes in the English language to realize a scientific view of the world. That is, 'a new variety of English' was created 'for a new kind of knowledge' (Halliday, 1993b: 81), one in which the main concern was to establish causal relations. As Halliday explains, the most effective way to construct logical arguments is to establish steps within a single clause, with the two parts 'what was established' and 'what follows from it' reified as two 'things' or participants realized through nominal group structures. These two participants are then connected with a process in a single clause. The strategy of recursive modification of the nominal group is also employed in scientific discourse. These two devices are typical of contemporary written discourse, and as Halliday and Martin (1993: 39) point out, nominalizations may serve important ideological functions because they are less negotiable than the congruent form: 'you can argue with a clause but you can't argue with a nominal group'. Cases of grammatical metaphor may be mapped through the system network as displayed in Figure 3.6(1). In addition to experientially based types of grammatical metaphor, interpersonal metaphors occur in conjunction with the systems of MODALIZATION and MODULATION (see Martin et al, 1997: 70). Following Halliday (1994: 354—363) metaphors of modalization and modulation are realized through the use of modal auxiliaries (modal Finites) with high, median and low values of probability and usuality, and obligation, inclination and potentiality respectively. MODALIZATION and MODULATION vary in orientation with respect to two criteria: first, objectivity and subjectivity; and

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experiential logical interpersonal textual relator

circumstance

circumstance

process

process

quality

quality entity -zero

entity nodifier process entity

Figure 3.6(1) System Network for Grammatical Metaphor second, implicitness and explicitness. A subjective explicit orientation is realized through a projecting clause; for example, 'I know this is correct'. An explicit objective orientation is realized through encoding of the objectivity; for example, 'it is certain this is correct'. Interpersonal metaphors are also realized through incongruence between MOOD and SPEECH FUNCTION selections (Halliday, 1994: 363-367). As explained in Section 3.2, the unmarked MOOD realizations of the SPEECH FUNCTIONS are statement realized by declarative (SubjectAFinite), question by interrogative (FiniteASubject and WH), command by imperative (Predicator), and Offer by modalized interrogative (modalized FiniteASubject). Mathematical discourse includes Modal and Mood metaphors (see Section 3.2). Martin (1992: 416) introduces textual grammatical metaphor which is orientated towards organizing the text as ' "material" social reality'. Martin gives four types of textual metaphor which contribute to this organization of text: (i) 'meta-message relations' as found in Francis' (1985) anaphoric nouns (for example, 'reason', 'example', 'point' and 'factor'); (ii) 'text reference' which identifies facts rather than participants (for example, 'this'); (iii) 'negotiating texture' which can, for example, exploit monologic text as dialogic (for example, 'let me begin by'); and (iv) internal conjunction which orchestrates text organization as opposed to field organization (for example, 'as a final point'). As Martin (1992: 416) points out, rather than being orientated towards logical meaning, these types of textual metaphors may be orientated towards the interpersonal. For example, That point is just silly' (Martin, 1992: 417) is a textual metaphor of the type 'meta-message relation' which is orientated to interpersonal meaning in the form of APPRAISAL. Derewianka (1995: 238) explains that the

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functioning of a nominal element to 'summarise or "distil" a figure or sequence of figures' does not necessarily mean that 'any instance of this type is inherently metaphorical'. According to Derewianka (1995: 238), what needs to be taken into account is a change in the level of generalization and abstraction. Nonetheless, Martin (1992: 395) sees grammatical metaphor as an important strategy for creating texture: 'The resources for weaving chains and strings through different grammatical functions . . . are important ones: but they provide only a very partial picture of the way in which meanings are packaged for grammatical realisation. The real gatekeeper is grammatical metaphor.' As evident in the discussion of experiential meaning in Stewart (1999: 132) in Section 3.3, mathematical discourse involves grammatical metaphor. The analysis of multisemiosis in mathematical and scientific texts enhances our understanding of the role and function of grammatical metaphor. Once the notion of semiotic metaphor is introduced in the form of metaphorical realizations which take place with intersemiotic shifts across semiotic resources, the semantic drift in language where grammatical metaphor developed intrasemiotically in language as a means of re-packaging information becomes understandable in the context of the functions and roles which are fulfilled symbolically and visually. This important point necessitates further discussion of the nature and functions of grammatical metaphor in Chapter 6. As well as grammatical and semiotic metaphors, lexical metaphors (Halliday, 1994: 340-342) may also be examined in mathematics, although this is not undertaken in this study. Lexical metaphors are metaphors in the more classical sense of the term where 'a particular lexeme is said to have a "literal" and a "transferred" meaning'(Derewianka, 1995: 109). In terms of distinguishing grammatical and lexical metaphors, both have 'a semantic category which can be realized congruently or metaphorically' but with grammatical metaphor, 'what is varied is not the lexis but the grammar' (ibid.). Although this field of study is worthy of investigation, the major concern here is shifts in meaning which arise grammatically in mathematics through the interactions between the semiotic resources of language, visual images and mathematical symbolism. 3.7 Language, Context and Ideology SFL views language as a social-semiotic, a system of meanings that construe the reality of a culture. This construction is described metafunctionally: the ideational metafunction construes 'natural reality'; the interpersonal metafunction construes 'intersubjective reality'; and the textual metafunction construes 'semiotic reality' (Halliday, 1978; Matthiessen, 1991). This intrinsic functional organization of language is modelled as interacting with the organization of social context in what Halliday (1985) terms as language's extrinsic functionality. In other words, language is viewed as construing the social context with the net result being the reality of a

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culture. Conversely, the social context impinges upon language use. Martin (1992) describes the relationship between language and social context as one of mutual engendering where instances of language use, collectively called texts, are social processes which are analysed as manifesting the culture they in part largely construct. These SFL formulations are extended to other semiotic resources (for example, Baldry, 2000b; Halliday, 1978; Kress and van Leeuwen, 1996; O'Halloran, 2004a; O'Toole, 1994; Ventola el al., forthcoming) which, in the context of this study, are language, mathematical symbolism and specialized forms of visual display. The analysis of text becomes 'the analysis of semantic choice in context' (Martin, 1992: 404) where context is conceived as consisting of the context of the situation and the context of culture. Context is viewed as a semiotic system manifested in whole or part through language and other semiotic resources. The levels of semiosis articulated by this process of realization are referred to as communication planes. The difference between language, mathematical symbolism and visual display on the one hand, and context on the other, is that the former have their own means of organizing expression (through typography/graphology and so forth) while context depends on other semiotic planes for realization (Ventola, 1987). The context of a text consists of two communication planes: register at the level of context of situation, and genre at the level of the context of culture. Register is constituted by contextual variables of field, tenor and mode which work together to achieve a text's goal. Field is concerned with experiential meaning (what is actually taking place), tenor with interpersonal meaning (the nature of the social relations) and mode with textual meanings (the role language is playing) (Halliday, 1978, 1985). The three register variables of field, tenor and mode can be viewed as working together to achieve a text's goals, 'where goals are defined in terms of the systems of social processes at the level of genre' (Martin, 1992: 502-503). Genre networks are formulated on the basis of similarities and differences between text structures which define text types. A culture consists of particular ways of meaning, which are described through genre, register and the integration of different forms of semiosis. In this study, the focus is directed towards the language and expression plane, rather than register and genre. However, a brief discussion of the register of mathematical language in terms of field, tenor and mode is included below. The written mode of mathematics means that semiosis in the form of language, visual images and the symbolism is constitutive of the mathematics which is developed, rather than contextual meaning arising from the immediate material setting. While mathematics does involve genres other than the written (such as the academic lecture, the conference paper and so forth), in essence modern mathematics developed as a written discourse. The textual organization of linguistic, symbolic and visual components and the compositional arrangement of those selections in mathematics are sophisticated. Mathematical discourse is concerned with particular realms of experiential content according to the field of mathematics (for

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example, elementary mathematics, calculus, pure mathematics and applied mathematics) and the genre. At this stage, the nature of the tenor relations which are established in mathematics is worthy of closer inspection as these relations orientate the reader towards the mathematics which is presented: 'Tenor refers to the negotiation of social relationships among participants' (Martin, 1992: 523). Tenor is the projection of interpersonal meaning realized through discourse semantics and lexicogrammatical systems in the language stratum. Tenor is mediated along the three dimensions of power (which Martin refers to as 'status'), contact and affect (Martin, 1992; Poynton, 1984, 1985, 1990) as displayed in Figure 3.7(1). Status refers to 'the relative position of the interlocutors in a culture's social hierarchy', contact is 'their degree of institutional involvement with each other' and affect includes 'what Halliday (1978) refers to as the "degree of emotional charge" in the relationship between participants' (Martin, 1992: 525). The principle of reciprocity of choice is significant in terms of the realization of status in spoken discourse. Patterns of dominance and deference in which the status of the writer/speaker is reflected take place through the kinds of linguistic choices which are made. Equal status is realized through selections of the same kinds of options for both interlocutors while unequal status is realized through non-reciprocal choices. As Martin (1992: 528) explains, there is 'a symbolic relationship between position in the social hierarchy and various linguistic systems, especially interpersonal ones'. The contact, or degree of involvement, is equivalent to what Hasan (1985) describes as 'social distance', the frequency and range of interaction. The principle of proliferation is used in which a high degree of contact means a wider range of options are available, while a low degree of contact means a smaller range of options. The basic realization principle of affect is amplification in which speakers can vary the 'volume' from normal writing/listening levels. Martin (1992: 529-535) lists features of interaction patterns, discourse semantics, lexis, grammar, and phonology which realize patterns of dominance and deference, involved and uninvolved contact, and dimensions of affect. The lexical and grammatical realizations of these tenor dimensions STATUS reciprocity

TENOR

CONTACT proliferation

equal unequal involved distant positive

AFFECT

marked negative

amplification

Figure 3.7(1) TENOR Dimensions adapted from Martin (1992: 526)

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are represented in a three-dimensional space in Figure 3.7(2). While many of these systems for spoken language do not operate in mathematics written texts (for example, swearing, slang and so forth), the three dimensional space is nonetheless useful as a means of framing the range of options which are available in mathematics. Such a framework is also useful for the analysis of pedagogical discourse in mathematics. The nature of the linguistic choices in written mathematics means that the discourse operates from an uncontested position of dominance. The linguistic choices are not reciprocal, there is minimal affect, and the contact is involved but distant. The nature of interpersonal relations is further discussed in relation to the symbolic and visual components of mathematics text in the following chapters. Martin (1992: 507) explains that ideology may be seen as 'the system of coding orientations constituting a culture'. Incorporating Foucault's (1970, 1972, 1980a, 1984, 1991) formulations of knowledge, power and discourse, SFL analysis is concerned with how texts relate to each other, and how one text relates to all texts that may have been. As texts are interpreted in a multidimensional intertextual semiotic space, this allows the selections which have been made to be effectively placed alongside all other possibilities, thus revealing the ideological positioning of the choices that have been made. Ideology has genre, and hence register and language as its expression plane. The ideological orientation of mathematics is discussed in relation to the concepts of abstraction, contextual independence, reason, objectivity and truth in Chapter 7. The adoption of a multisemiotic perspective of discourse facilitates a holistic understanding of text, context and culture. The inclusion of other forms of semiosis and the study of intrasemiotic and intersemiotic processes enhances the theoretical possibilities afforded by SFL. For example, the SFL framework presented in Table 3.1(1) is extended to incorporate other semiotic resources in the 'Integrative Multisemiotic Model' (IMM) in Lim (2002, 2004). Such a multisemiotic systemic functional model incorporates (i) the grammatical systems for other semiotic resources, (ii) intrasemiosis within the semiotic resources, (iii) intersemiotic mechanisms for meaning across semiotic resources, (iv) systems which operate on the Expression stratum (for example, Colour and Font Style and Size), and (v) the materiality and medium of the text (for example, print versus electronic medium). A multisemiotic approach reveals differences between the functions and systems of semiotic resources across different strata. For example, Lim's (2002: 37) division of metafunctionally based systems shows a separation of metafunctional boundaries with respect to the systems which operate at the grammatical stratum for language. However, the 'system-metafunction fidelity' (Lim, 2002, 2004), or the measure of dedication of a system to one particular metafunction, breaks down on the expression plane. These systems (for example, systems such as Font, Colour and so forth) do not have the clear metafunctional orientations which are found in grammatical systems. Choices from the system of Colour, for example, can function interpersonally to attract attention, textually for cohesive purposes, and

GRAMMAR Residue ellipsis polarity matched attitude concur comment invited vocation respectful person 2nd tagging checking agency: I/medium modalization low " modulation: inclination LEXIS euphemize tempered swearing covert GRAMMAR minor clauses Mood ellipsis Mood contraction vocation ' range of names nick-name

LEXIS attitudinal taboo swearing

POWER - Defer

GRAMMAR exclamative attitude comment minor expressive intensification repetition prosodic nm gp diminutives; mental affection manner degree

GRAMMAR major clauses no ellipsis no contraction no vocation single name full name CONTACT Distant/ uninvolved

AFFECT Positive POWER Dominate AFFECT Negative

CONTACT Intimate/ involved

LEXIS specialized technical slang general words

GRAMMAR exclamative attitude comment minor expressive intensification repetition prosodic nm gp diminutives; mental affection manner degree

LEXIS attitudinal taboo swearing

LEXIS core non-technical standard specific words

GRAMMAR no ellipsis polarity asserted attitude manifested ~ comment presented vocation familiar person 1st tagging invited agency: I/agent modalization high modulation: obligation LEXIS explicit bodily functions swearing overt

Figure 3.7(2) Lexicogrammatical Aspects of the Realization of TENOR adapted from Martin (1992: 529-535)

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experientially for representational meaning. In a similar fashion, semiotic resources have different grammatical systems, and such differences in the meaning potential have implications for the functions which are fulfilled by that resource, as seen in the discussion of the grammar of mathematical symbolism and visual images. The formulation of SF frameworks for mathematical symbolism and visual display and the analysis of intersemiotic processes reveal that the key to the success of mathematical discourse is the ability to create a semantic circuit across the linguistic, symbolic and visual components of the mathematics text through the specialized grammars of each resource. These semantic circuits give rise to metaphorical expressions in the form of semiotic metaphors. The analysis of the mathematical texts in Chapter 7 leads to a further discussion of the ideology and orientation of mathematics. Martin's (1992: 507) focus is situated within the dynamic view of ideology which is 'concerned with the redistribution of power - with semiotic evolution'. These concerns provide the impetus for this study. Notes 1 Martin's (1992: 14-21) arguments for stratification of the content plane include the following limitations of the lexicogrammar: semantic motifs cannot be generalized because of diverse structural realizations; the multiple levels of semantic layers resulting from grammatical metaphor cannot be fully accounted for; generalizations across structural and nonstructural textual relations such as those found in cohesion are not possible; and the semantic stratum is more abstract and the systems are composed of larger units which differ in structure from those found elsewhere. Martin (1992) and Martin and Rose (2003) are concerned with overcoming these limitations and capturing semantic interdependencies in the whole text which are otherwise only partially accounted for by the lexicogrammar. The type of structures are open ended in so far as the issue is not one of constituency, but rather interdependency. 2 Although word groups and phrases occupy the intermediate position on the rank scale, Halliday (1994: 180) distinguishes between the two: 'A PHRASE is different from a group in that, whereas a group is an expansion of a word, a phrase is a contraction of clause.' Halliday's (ibid.: 242) classification of rankshifting thus covers clausal and phrasal elements. 3 In SF system networks, the curly brackets mean 'select from each of the systems' (that is, 'select from this and this'), while the square brackets mean 'select only one of the options' (that is, 'select this or that'). 4 Typically in conjunctive reticula, external relations are modelled on the right-hand side, and the internal relations are modelled down the left with external additive relations positioned in the centre. This allows the conjunctive relations to be separated into those which function in a rhetorical (internal) sense compared to those which function in a more experiential (external) sense.

4 The Grammar of Mathematical Symbolism

4.1 Mathematical Symbolism

The historical perspective covered in Chapter 2 reveals how mathematical symbolism developed as a tool for reasoning through the discovery that curves could be described algebraically and the increasingly important aim of rewriting the physical world in mathematicized form. Mathematical descriptions eventually replaced metaphysical, theological and mechanical explanations of the universe (see for example, Barry, 1996; Kline, 1972, 1980; Wilder, 1981). Today, many fields of human endeavour are written in mathematicized or pseudo-scientific form. The scientific view of the world is not confined to the physical universe; rather it underlies our day-to-day conception of reality. Mathematical discourse succeeds through the interwoven grammars of language, mathematical symbolism and visual images, which means that shifts may be made seamlessly across these three resources. However, each semiotic resource has a particular contribution or function within mathematical discourse. Language is often used to introduce, contextualize and describe the mathematics problem. The next step is typically the visualization of the problem in graphical or diagrammatic form. Finally the problem is solved using mathematical symbolism through a variety of approaches which include the recognition of patterns, the use of analogy, an examination of different cases, working backwards from a solution to arrive at the original data, establishing sub-goals for complex problems, indirect reasoning in the form of proof by contradiction, mathematical induction (if Sk is true, and Sk+1 is true whenever Sk is true, then Sn is true for all n) and mathematical deduction using previously established results (Stewart, 1999: 59-60). The generalized solutions and mathematical models are used for predictive purposes. Before discussing intersemiotic processes which take place across language, the symbolism and visual images in Chapters 6-7, intrasemiosis within mathematical symbolism and visual display is explored in Chapters 4—5 respectively. The unique functions of each resource are discussed through SF frameworks and an investigation of choices from the systems which are found in symbolic and visual

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parts of the mathematical texts. In this way, the grammatical strategies for encoding meaning in each resource may be understood before proceeding to the complex problem of understanding how meaning is made intersemiotically across the three resources. The general nature of meanings afforded by language, mathematical symbolism and visual images is described by Lemke (1998b, 2003). That is, language is seen to be orientated towards making categorical-type distinctions (for example, Bateson, 1972; de Saussure, 1966; Messaris, 1994); that is, typological-type meanings. Mathematical symbolism, on the other hand, is seen to make meanings by degree in the form of continuous descriptions of patterns of co-variation; that is, topological-type meanings. For example, one can observe that the tiger population in an ecosystem is decreasing, and one may even comment: 'there are not many tigers around these days'. The linguistic statement makes a categorical type assessment of the situation regarding tigers: 'there are not' (Existential process with negative polarity) 'many tigers' (Existent) 'around' (circumstance-Location of place) 'these days' (circumstance-Location of time). However, through 'predator-prey' type mathematical models, the relationship between the number of tigers and the number of men, for example, can be specified in order to study the patterns of the interaction between the two species, and to predict the tiger population at any one time, including when they may be expected to become extinct. The mathematical model expresses the relationship between the number of men and the number of tigers as a continuous function over time. For example, if M represents the predator 'man', T represents the prey 'tiger', and t represents time, then such a model would take the form

(Stewart, 1999: 662). In

addition to describing patterns of variation over time, the symbolism has the potential to express the exact relations of parts to a whole. For example, a triangle with base b and height h is related to the area of the triangle A (A) through the symbolic statement As the nature of patterns of variation is not easily discernible from the symbolic statements, the graphs and diagrams are used to give more intuitive understanding of the relationships which are encoded symbolically (Lemke, 1998b). For example, the predator-prey mathematical model for the relationship between the population of tigers and the number of men can be displayed graphically to give a sense of the type of relationship encoded in the mathematical model. Visual images supersede language in terms of the ability to represent continuous spatial relations. However, mathematics visual patterns are often only partial descriptions over a limited domain, and they are limited in terms of their ability to be used for calculations. This shortfall is becoming less marked with the development of the power of computers to display and manipulate visual patterns, a theme which is explored in Chapter 5. The computer is revolutionizing the

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types of mathematics being developed due to the increased facility for performing numerical calculations and displaying the resulting visual patterns through computer graphics. The functions and grammar of mathematical symbolism are examined through the development of an SF framework. This framework is used to explore the nature of interpersonal, experiential, logical and textual meanings afforded by symbolism, and the strategies through which these meanings are encoded. A similar exploration of intrasemiosis in visual images takes place in Chapter 5. From this point, it is possible to examine how language, the symbolism and the visual images combine intersemiotically to create meaning in mathematical discourse. The examination of the grammars of mathematical symbolism and visual display on a separate basis is a somewhat artificial approach as historically these semiotic resources developed together in mathematical discourse. The key to the success of mathematics is that the three grammars function integratively. However, if the process of semiosis is 'frozen' in stages where meaning is made primarily within one resource rather than across the three resources, the contribution of that one resource may be appreciated. This appears to be a necessary preliminary first step to understand how the three semiotic resources function together. The functions and grammar of the mathematical symbolism and visual images are therefore first investigated individually. As will become evident, the grammatical strategies for encoding meaning in mathematical symbolism differ from those found in scientific language. This is not surprising as the symbolism was designed to fulfil different functions, and its grammar evolved accordingly. The nature of scientific language, with its propensity to pack experiential meaning into extended nominal group structures in the form of grammatical metaphors which are configured with relational processes (for example, Halliday and Martin, 1993; Martin and Veel, 1998), is the resultant product of the impact of the use of the symbolism and the visual display in mathematical and scientific discourse. From the discussion of intrasemiosis in mathematical symbolism and visual display, intersemiotic processes and their impact on scientific language are investigated in Chapter 6. 4.2 Language-Based Approach to Mathematical Symbolism

The language-based approach to the SF framework for mathematical symbolism adopted in this study is justified by the fact that the symbolism developed as a semiotic resource which evolved from language. The stages of the development of algebra, for example, have been characterized as rhetorical where instructions were in the form of linguistic commands, syncopated where recurring linguistic elements for participants and processes were symbolized, and symbolic where mathematical symbolism developed as a semiotic resource (see Chapter 2 and Joseph, 1991). The symbolism developed a functionality through new grammatical systems

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which permitted semantic expansions beyond that capable with language, but at the same time it depended upon employing certain linguistic elements and a range of grammatical strategies inherited from language. Furthermore, symbolic statements are typically embedded within linguistic text. Thus, despite the new functionality of mathematical symbolism, it nonetheless requires a surrounding linguistic co-text to contextualize the symbolic descriptions and procedures that take place. The dependence on the linguistic semiotic suggests that the symbolism did not develop a wellrounded functionality, which becomes evident in the discussion of the types of meaning which are possible using mathematical symbolism. The language-based approach permits the semantics of the mathematical symbolism to be understood and contextualized in relation to the types of meaning afforded by the linguistic semiotic. The unique relations between language and mathematical symbolism explain the nature of the mappings that may be made between the two semiotic resources. For example, there exist acceptable wordings in natural language for mathematical symbolic statements, although this is not an exact one-to-one correspondence. Mathematical statements are recoverable from linguistic statements, although in some cases this is problematic because the linguistic construals are metaphorical (see the discussion of semiotic metaphor). The unique relations between language and mathematical symbolism serve to highlight an important difference between these two resources and other semiotic resources such as art, sculpture and architecture where such accurate mappings do not exist. For instance, unlike a mathematical symbolic statement, a painting or a sculpture is not recoverable from any combination of words. After introducing the SF framework for mathematical symbolism, the types of semantic shift in the evolution of the symbolism are discussed in terms of the expansion and contraction of experiential meaning, the narrowing of interpersonal meaning, the development of selected types of logical meaning, and the refinement of textual meaning. These types of semantic shift mean that mathematical symbolism developed as a semiotic resource with a grammar through which meaning is unambiguously encoded in ways which involve maximal economy and condensation. The economical means of encoding meaning in the symbolism permit the easy rearrangement and manipulation of relations so that mathematical models can be constructed and problems can be solved. This perspective is developed in the following discussion of the grammar of mathematical symbolism. A summary of the major points concerning the grammar of mathematical symbolism appears in Section 4.8 4.3 SF Framework for Mathematical Symbolism

The SF model for mathematical symbolism displayed in Table 4.3(1) is based upon Halliday (1994) and Martin's (1992; Martin and Rose, 2003) systemic model for language. The communicative planes of ideology, genre

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Table 4.3(1) SF Model for Mathematical Symbolism MATHEMATICAL SYMBOLISM CONTENT

Discourse Semantics Inter-statemental relations Grammar Statements (or clause complex) Clause (// //) Expressions ([[ ]]) (rankshifted participants of the clause which are the result of mathematical operations) Components (the functional elements in expressions)

DISPLAY

Graphology and Typography

and register are applicable to the multisemiotic mathematical texts considered in Chapters 6-7. In the language plane, the content stratum for mathematical symbolism consists of discourse semantics and grammar strata with the ranks of statement (clause complex), clause, expression and component. The model parallels the discourse stratum and lexicogrammatical ranks of clause complex, clause, word group/phrase and word for language. The 'display plane' for mathematical symbolism corresponds to the 'expression plane' for language in the model. The term 'display plane' is used rather than 'expression plane' because a new grammatical rank of 'expression' is introduced for mathematical symbolism in Table 4.3(1). The need for the inclusion of the rank of expression in the grammar of mathematical symbolism will become apparent in Section 4.4. The SF framework for a grammar for mathematical symbolism is presented in Table 4.3(2). This framework provides a description of the major systems through which mathematical symbolism is organized as a semiotic resource for experiential, logical, interpersonal and textual meaning for the content and display planes. The discourse systems for mathematical symbolism parallel those found in language. However, as discourse moves often span linguistic, symbolic and visual components of the text, Martin's discourse systems are extended in Chapter 6 in the attempt to theorize intersemiosis between the three resources. In the model presented in Table 4.3(2), systems which operate at the level of the display plane are also included. It is recognized that options in the expression of the semiotic choices in the mathematics text (for example, Colour, Font Size and Style) function to create meaning (for example, Kress and van Leeuwen, 2002; Lim, 2004; O'Halloran, 2004a). Traditionally, the expression stratum in language has been under-theorized in SFL where the major concerns have been the language plane and the communicative planes of register, genre and ideology.

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Table 4.3(2) Grammar and Discourse Systems for Mathematical Symbolism DISCOURSE SEMANTICS EXPERIENTIAL

LOGICAL

INTERPERSONAL

TEXTUAL

IDEATION • Activity Sequences consisting of Operative process and participant reconfigurations (progressive steps of simplification and solution) • Nuclear relations (participant and process) • Collocation (symbolic relations and strings through taxonomies, definitions, axioms and theorems)

CONJUNCTION and CONTINUITY (based on EXPANSION) • Sequential placement of statements (explicitly marked when the logical connection is non-sequential) • Extension of TAXIS into long implication sequences

NEGOTIATION Exchange Structure and SPEECH FUNCTION at the move rank • Consists of moves and move-complexes

IDENTIFICATION • Direct Repetition • Referential cohesion (based on definition, operational properties with explicit repetition of reference) • Positional notation (the sequential downward placement of statements and positional placement functional components)

Structure: Exchange Structure linking moves

Structure: conjunctive reticula

Structure: reference chains linking participants

Structure: strings for tracing activity sequence reconfigurations

INTER-STATEMENTAL RELATIONS EXPERIENTIAL

LOGICAL

INTERPERSONAL

TEXTUAL

• Positional notation to indicate continuations of Activity Sequences • Repetition of processes and participants in new configurations

EXPANSION • Conjunctions and cohesive conjunctions • Implicit and explicit conjunctions (external symbolic and linguistic conjunctive devices; internal substitution and operative properties) • Apposition • Parenthesis • Labelling

SPEECH FUNCTION (statements and limited forms of command) • Intricacy of symbolic representation • Abstractness (nature of participants, processes) • Discursive links (using verbal code of main text within the mathematical array) • Labelling

• Positional notation (the sequential downward placement of statements and positional placement functional components) • Dependent clauses (thematic or spatially marked) • Ellipsis (marked by spatial position) • Discursive links (using verbal code of main text within the mathematical array) • Labelling

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Table 4.3(2) - cont STATEMENTS // EXPERIENTIAL

LOGICAL

• Rhetorical 'temporal' TRANSITIVITY conjunctive relations • Processes (Operative, realized through relational and Rule of Order of existential) operations and use of • Participants are brackets rankshifted configurations of Operative processes • Circumstantial features (minor clauses, dependent clauses or fused within participant structure) • Ellipsis of Operative processes • Rule of Order of operations

INTERPERSONAL

TEXTUAL

MOOD (with one symbol for the Finite and Predicator) • MODALITY (consistently high, implicit objective orientation) POLARITY (presence or absence of a slash through the process symbol) • Intricacy (embedded processes) • Abstractness (participants and processes)

• THEME (unmarked choice is Subject of the clause with marked case indicates steps in simplification) • Multiple Theme (textual element spatially placed) • Ellipsis (spatial positioning) • Dependent clauses (thematic or otherwise spatially separated) • Conventional spatial organization • Rule of Order and use of brackets for unfolding of Operative processes

EXPRESSIONS [[ EXPERIENTIAL

LOGICAL

//

]]

INTERPERSONAL

TEXTUAL

• Rhetorical 'temporal' • Intricacy (degree and • Rule of Order and • Operative processes (condensation occurs conjunctive relations explicitness of use of brackets for unfolding of realized through embedding) via high level of conventionalized rankshift within and Operative processes • Degree of between participants) Rule of Order of abstractness (nature of participants and • Degree of rankshift operations and use of brackets processes) indicated by [ [ ] ] • Circumstantial elements (through processes and fused participant structures • Ellipsis of Operative processes • Rule of Order of operations

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Table 4.3(2) - cont COMPONENTS EXPERIENTIAL

LOGICAL

• Conventionalized • Restricted range of combinatory units in the nominal practices group (the absence of DEIXIS, attitudinal and experiential epithets) • Qualifiers (form part of the nominal group without the need for embedding as phrases) • Classifiers • Conventionalized use of specific symbols (numerals, Roman, Greek, Hebrew alphabet)

INTERPERSONAL

TEXTUAL

• Degree of abstractness • Degree of modification

• Function of constituents (spatial, serial position and brackets)

DISPLAY PLANE EXPERIENTIAL

LOGICAL

• Variations in the form • Spatial organization of symbolic text of case, font, scripts and size for special symbols, abbreviations, icons, punctuation, brackets, and combinations of symbols • Use of spatial and positional notation

INTERPERSONAL

TEXTUAL

• Style of production • Spatial arrangement (hand written, of text at each rank computer generated) • Font style and format • Contrasts in font, • Ellipsis of process script and size

The SF framework in Table 4.3(2) provides insights into the ways in which the grammar of mathematical symbolism is organized to fulfil the functions of mathematics, and the ways in which the systems and lexicogrammatical strategies in the symbolism depart from those found in language. Further research is needed in the analysis of mathematical texts, however, in order to fully document the systems, which remain at a preliminary stage of theorization. The framework presented in Table 4.3(2) is best viewed as a first step towards a comprehensive SFG for mathematical symbolism. The metafunctional systems in the SF framework for mathematical symbolism are discussed with reference to the mathematical symbolic text displayed in Plate 4.3(1). This mathematics problem is concerned with

EXAMPLE 4

If f ( x ) = J x - 1 , f i n d t h e d e r i v a t i v e o f / . S t a t e t h e d o m a i n o f / ' .

SOLUTION

Here we r a t i o n a l i / e the numerator

We see that f'(x) exists if x > 1, so the domain of/' is (1, o°). This is smaller than the domain of/, which is [1, °°). Plate 4.3(1) Mathematical Symbolic Text (Stewart, 1999: 139)

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finding the derivative/'(x) of a function/(x) by finding the limit off(x) as x is approached; that is, as h tends to zero. The derivative is the rate of change of the function, which may be interpreted geometrically as the slope of the curve at the point (x,f(x)). The visualization of the derivative in the form of a graph for the geometrical interpretation of the derivative is examined in Chapter 5. 4.4 Contraction and Expansion of Experiential Meaning

Experiential meaning in mathematical symbolism is largely concerned with a semantic field in the form of the description and manipulation of relations. The semantic field of mathematics therefore includes a limited experiential domain compared to language. With the narrowing of the semantic domain, an expansion of meaning took place in mathematics with respect to the description of relations and patterns of variation. The ways in which mathematical symbolism achieves this simultaneous contraction and expansion of the experiential meaning are discussed below. One major innovation in mathematical symbolism is the evolution of a new process type, the Operative process, which takes the form of arithmetic operations and other processes found in the different fields of mathematics. Operative processes initially arose in early numerical systems, which were among the earliest forms of mathematical symbolism. Numerical notation appeared in different cultures arising from practical needs such as recording quantities and marking time intervals for social and economic activities. The nature of the early numerical systems in cultures which include the European, Egyptian, Mesopotamian, Indian, Arabian and Chinese, and independent traditions such as the Mayan in South America, depended upon the functions which were required to be fulfilled and the availability of material resources. Once established, numerical systems circumscribed mathematical activities and new developments in much the same way that grammatical systems in language function to structure reality through the nature of the linguistic choices which are available. As we have seen, the adoption of the Hindu-Arabic numerical system, for example, had a major impact on the development of mathematics in Europe. Symbolic processes in early numerical systems developed from Material processes (Lemke, 1998b; O'Halloran, 1996) which were concerned with counting, adding, multiplying, subtracting, dividing and measuring. However, new mathematical Operative participants and processes began to appear with the development of numerical systems. For example, new participants in the form of very small and very large numbers, which could not materialize in concrete form, arose in the symbolism. Moreover, Operative processes replaced the semantics of Material processes. That is, Operative processes of adding, multiplying, subtracting and dividing symbolic numbers initially paralleled existing Material processes of combining, increasing, decreasing and sharing physical objects. However, the complexity of Material processes undertaken by human participants in a physical

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world had practical limitations and intuitive expectations which did not necessarily extend to the semiotic Operative processes performed using symbolic notation. It became possible to perform complex combinations of Operative processes which were not otherwise feasible or even conceivable, and to obtain results unlike those previously expected. An example of this type of semantic extension occurs in the case of multiplication of fractions where the product is less than the numbers which are multiplied. This contravenes the common-sense understanding where 'to multiply' means 'to increase'. A similar situation arises with the division of fractions where the result is larger than the number which is being divided. The limits of Operative processes within early numerical systems were dependent upon parameters such as the base of the system, the existence of place value, the inclusion of a symbol for zero, a means of separating fractional components and the intricacy and number of symbols. When calculations became complex, material computational devices based on the number systems, such as counting boards, table reckoners and the abacus, were employed. With the development of symbolic algebra, attention turned to generalized descriptions of relations using algebraic methods. The success of these descriptions meant that mathematical symbolism developed as a semiotic resource with grammatical systems which were unique to that resource. These systems developed in accordance with the aim of mathematics: the descriptions of patterns and the means to solve problems relating to those descriptions. This largely involved capturing and rearranging generalized descriptions of relations between variables through Operative processes. With the evolution of mathematical symbolism as a semiotic resource, arithmetic Operative processes were supplemented with processes concerned with powers, roots, complex numbers, limits and other processes found in different branches of mathematics, as seen in the calculus example in Plate 4.3(1) where the limit as h —> 0 is derived for Operative processes are typically performed by human agents on symbolic semiotic participants in the form of numbers and later variable participants, as seen in Plate 4.3(1) where the reader is instructed 'If f(x) = J x — l , find the derivative of/'. In the development of symbolic algebra, the human agent was not included in the mathematical symbolic statements which were more concerned with describing the nature of relations based on established mathematical results, rather than encoding the rhetorical commands which accompanied the solution to the problem. As a result, the human agent tends to be located within the linguistic part of the text which is concerned with the commands (for example, ' [you] find the derivative of/' in Plate 4.3(1)) and statements such as 'We recognize this limit as being the derivative of /at xl} that is,/' (xj' (Stewart, 1999:132). This statement takes the form of the metaphorical projecting clause 'we recognize' for 'this limit is the derivative o f / a t xlt that is, /'(*i)' (see Section 3.2). As we have seen, the human agent also disappeared in math-

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ematical visual images as the concern with lines and curves grew during the seventeenth and eighteenth centuries. At the same time, the notion of agency, where one participant impacts on another, appears to have developed in the symbolism in rather a different fashion from that found in language. For example, in a mathematical function the value of the independent variable x 'impacts' on the value of the dependent variable y in so much as the value of y depends on the value of x. However, the grammatical strategies for encoding such relations take the form of interactions between multiple participants rather than direct impact of one participant on another participant. This idea is developed below through an examination of the way in which Operative processes and participants are configured in mathematical symbolic statements. Operative processes appear to be grammatically different from the linguistic processes documented in Halliday's systems of TRANSITIVITY and the related system of ERGATTVTTY which is concerned with agency. The process types in language are Material, Mental, Behavioural, Verbal, Relational and Existential processes. Halliday (1994: 163) explains that in language there is a key participant, the Medium, which is associated with each process. In the clause 'Jack opened the door', the verb 'opened' is a Material process with 'Jack' as the Actor/Agent who acts on 'the door' which is the Goal/Medium. In this case the Medium is 'the door'. Without 'the door', the action of opening could not have been performed by Jack, the Agent. Halliday (1994: 163) calls the key participant the Medium in the ergative interpretation of the clause: Every process has associated with it one participant that is the key figure in that process: this is the one through which the process is actualized, and without which there would be no process at all. Let us call this element the MEDIUM, since it is the entity through the medium of which the process comes into existence.

Every process in language has an associated Medium, and only in some cases is there an Agent. For example, 'Jack talked' is a Verbal process with the Sayer/Medium 'Jack'. In this example there is no Agent. In the case of 'Jack and Jill walked up the hill', the Medium is 'Jack and Jill', realized as a complex nominal word group and the Range is 'up the hill'. There is no Agent associated with the process of walking up the hill. In a clause such as 'the best idea [ [that Jack and Jill had all day] ] was [ [to walk up the hill'] ], the Token/Identified (Medium) in the Relational process is 'The best idea [ [that Jack and Jill had all day] ]', as evidenced by the probe 'what is the best idea [[that Jack and Jill had all day]]?' The Value/Identifier (Range) is ' [ [to walk up the hill] ]'. In this case, the Medium is the rankshifted clause 'The best idea [[that Jack and Jill had all day]]'. Within this rankshifted clause Jack and Jill' function as the Medium for the process 'had'. The typical nuclear configuration of functional elements for experiential meaning in language has the form: Participant (Medium) + Process ± Participant (Agent) ± Range ± Circumstance/s

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However, Operative processes in mathematical symbolic clauses do not appear to replicate the nature of the experiential meaning in language. While the notion of a Medium and an Agent exists at the rank of clause in mathematical relational processes (for example, realized through '=') in the form of the 'Token (Medium or Agent) = Value (Range or Medium)', the corresponding mathematical participants do not take the correlate form of a word, word group/phrase or a rankshifted clause (with one embedded Medium) as discussed in the case of 'Jack', 'Jack and Jill', and 'the best idea [[that Jack and Jill had all day]]'. Rather, the Medium and other participants in relational symbolic statements are most typically configurations of Operative processes with multiple participants which appear to play equally key roles. For example, the notion of 'a single key participant' in the configuration of Operative processes which constitute the value of the derivative

does not seem to apply. Rather,

there appear to be several key participants, x, h and 1, in the algebraic expression for the derivative/' (x). In a similar fashion, there appear to be multiple key participants which are central to the Operative process configurations in the examples given below. Arithmetic Operations: Exponents: (xyz) n—x"y" zn Factoring Special Polynomials: x* — y2 = (x + y)(x— y) Geometric Formulae: Cosine Law: a? = tf +
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x y . . . y (n times) x z X z x z . . . z ( n times). However, it is not perhaps feasible to ask which is 'the key participant' in these configurations of x, y and z where each variable appears to play an equally significant role. This case is different to 'Jack and Jill' which is a complex nominal group in language, or the case of the rankshifted clause 'the best idea [ [that Jack and Jill had all day] ]' where the Medium in the rankshifted clause is 'Jack and Jill'. The x, y and z appear to interact through the Operative process of multiplication as equally key figures in the configurations which constitute the Medium and Range of the relational process. Similarly, the generalized algebraic form for the addition of two fractions may be seen to consist of the six multiple key participants a,b,c,d,e and fin This is a statement which contains a Relational Identifying process with '=' equating the Value/Identified (Medium) with Token/Identifier (Agent) the probe 'what is the generalized form for

as demonstrated by (Identified). In this

'encoding' clause (Halliday, 1994: 165-167), the Medium Agent

with

However, once again the Medium and the Agent are j

configurations of Operative processes with multiple key processes and participants, in this case division, addition and multiplication with participants a,b,c,d,e and / It is interesting to note that the majority of mathematical clauses for encoding relations seem to appear in the passive form is represented by

rather than

represents

In the case of Operative processes, it appears that the idea of one Agent impacting on one Medium, or one Medium with a Range, is not necessarily appropriate. Rather Operative processes appear as complex rankshifted configurations with participants X t , X2, X3 . . . Xn. This becomes evident if the presence of Operative processes is indicated by the systemic functional convention of square brackets [[]]. In each case, complex rankshifted configurations of Operative processes and participants arise as seen in the

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In order to mark this difference in the grammar of mathematical symbolism, the rank below the clause is called 'the expression' (see Table 4.3(2)) which corresponds to the equivalent rank of word groups/phrases in language. At the rank of the mathematical expression, the potential for agency and the nature of the participants within the nuclear configurations of process/participants need further research in order to classify the participant functions according to the types of Operative processes. This remains an important research goal. At this stage, the typical form of the configurations of expressions in ranking clauses in symbolic mathematics is seen to be: [ [Participant + Processj + Participant + Process2 + Participant. . . Processn]]n (Medium) + Process ± Agent ± Range ± Circumstance/s where the Agent, Range and Circumstance also have the potential to take the form [ [Participant + Process! + Participant + Process2 + Participant. . . ProcessJ]". The potential for rankshifted configurations of Operative processes and participants is one of the key factors in the success of mathematical symbolism because this strategy preserves process/participant structures which may be reconfigured for the solution to problems. This is a significant point in understanding how the grammar of mathematical symbolism is functionally organized to fulfil the goals of mathematics: to order, to model situations, to present patterns, to solve problems and to make predictions. The degree of rankshift found in mathematical symbolism exceeds that found in language. This may be demonstrated through closer inspection of the derivative

in Plate 4.3(1). If the rank-

shifted Operative processes are marked by [[]], we see that the level of embedding is complex.

The degree of rankshift is displayed in Table 4.4(1), where it may be seen that there are six levels of embedding of Operative processes in the mathematical expression for the derivative. At each stage, the process/ participant configuration is preserved so that the expression can be rearranged and simplified. This is an important grammatical strategy in the evolution of mathematical symbolism as the semiotic which is used to solve problems. This is a different grammatical strategy to that found in scientific language, where the potential of the nominal group structures expanded in order to encode experiential meaning. The two different grammatical strategies reflect the different functions of mathematical symbolism and language. This point is further developed in relation to grammatical metaphor in Chapter 6.

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Table 4.4(1) Levels of Rankshift in Mathematical Symbolism Rank

Process

Operative Process/Participant Configuration

The process types in mathematics and their attendant participants are concerned with capturing and reformulating patterns of relations. The processes in mathematical symbolism largely consist of Operative processes, Relational processes in the form of identifying and attributive processes (intensive, circumstantial and possessive) and Existential processes. While the potential of mathematics as a semiotic resource expanded in the field of description of relations, the overall scope of experiential meaning with which mathematics was concerned narrowed. That is, processes involving the human realm of feeling, behaving and talking were largely excluded in the quest to describe patterns. Mathematical symbolic participants became numbers and variables which function as general representations rather than the specific entities or groups realized through lexical choice in language. Circumstance became limited to those types which had applicability in the description of relations. Circumstance in the symbolism is realized through minor clauses, dependent clauses or fused elements within participant group structures in symbolic mathematics, as described below. A summary of the range of experiential meaning in terms of processes, participants and circumstance in symbolic mathematics is displayed in Table 4.4(2). Mathematics became concerned with an

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Table 4.4(2) Restricted Range of Experiential Meaning in Mathematical Symbolism Relational: Identifying

Relational: Operative Attributive

Existential

Token Value/ Identifier

Carrier

Existent

Agent

Token/Identifier

Attributor

Range

Value/Identifier

Attribute

PROCESS

PARTICIPANTS Medium

Participants X], X2 . . . Xp

Operator Participants X], X2 . . . Xp

CIRCUMSTANCE Extent Duration/Distance

Temporal and Spatial

Location: Temporal/Spatial

Time and Place

Contingency:

Condition

Accompaniment

Commitation/Addition

expanded realm of meaning within a restricted experiential field. Further study, however, is needed to fully document the process types, the associated participants, and the types of circumstance which are found in symbolic mathematics. In the effort to efficiently encode meaning in mathematical symbolism in an economical and exact manner, special means have developed for realizing the Operative process/participant configurations in mathematical symbolic statements. These strategies include ellipsis of Operative processes, the Rule of Order for Operative processes, and the use of brackets to re-organize the order of operations. These strategies for efficiently encoding experiential meaning are discussed in relation to textual meaning and the organization of the symbolic statements, and to logical meaning and the temporal unfolding of the Operative processes. Another significant strategy for efficiently encoding experiential meaning in mathematical symbolism is the use of positional and spatial notation. This strategy is discussed below in relation to systems for experiential meaning in the display stratum. Another strategy to efficiendy encode experiential meaning in the symbolism relates to circumstantial elements which appear within participant and process structures, in addition to being separate functional elements in the form of prepositional phrases. For example, temporal Location 'at a time of four seconds' may be represented by the dependent relational statement 'if t = 4', and the circumstance of Extent 'after four seconds' may

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be included in participant structure '5(4)'. Combinations for packing circumstantial information in the symbolic clauses also occur, for example, 'the displacement in the first four seconds is ten metres' may be represented by 5(4)-5(0) = 10. Circumstantial prepositional phrases and adverbial groups also occur in the surrounding linguistic co-text within which mathematical symbolism is embedded. At the rank of discourse semantics in mathematical symbolic text, the system of IDEATION is largely concerned with Activity Sequences in the form of Operative process/participant re-configurations for the solution of mathematics problems. Such Activity Sequences are clearly marked in a step-by-step fashion as seen in the case of the derivation of the derivative /' (x) in Plate 4.3(1). In this example, the steps are organized according to vertically placed lines where participants and process configurations are repeated, substituted, re-organized and simplified according to mathematical definitions (for example, the definition for the derivative of a function), algebraic laws (for example, x2 — f = (x + y) (x — y) which is used to rationalize the numerator), and other established results for algebraic operations. The Activity Sequence consisting of the strings for Operative processes and participants is clearly marked through the textual and spatial organization of the solution to the problem (see textual meaning in Section 4.8). The Rule of Order determines the sequence in which Operative processes are performed within statements and expressions (see logical meaning in Section 4.7). Within the Operative participant structures, there is a restricted range of elements; for example, experiential epithets and choices from the system of DIEXIS do not occur. With the development of the grammar for mathematical symbolism, the conventionalized use of specialized symbols in mathematics took place. At the rank of component, symbols include letters from the Roman alphabet (with upper and lower case letters of varying sizes written with varying fonts and scripts), the Greek alphabet and a range of choices from other alphabets. Other elements include punctuation symbols, brackets, iconic representations, abbreviations and the invention of new symbols as displayed in Table 4.4(3). Cajori (1993) provides evidence to prove that the path to standardized symbolic notation was not smooth. Cajori (1993: 338) explains that' [o]ften the choice of a particular symbol was due to a special configuration of circumstances (large group of pupils, friendships, popularity of a certain book, translation of text) other than those of the intrinsic merit of the symbol'. In reality, the choice of mathematical symbols often depended upon circumstantial, personal and political contingencies rather than merit. This is not so surprising as mathematics is a field of human endeavour. Choices from the systems of the font style, size and format on the display stratum function to realize experiential meaning. For example, symbolic statements such as

appear in italicized form

to mark them as separate elements in the mathematics text. Elements in the

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Table 4.4(3) Examples of Mathematical Symbols Type of Symbol

Example

Meaning

Roman alphabet (upper case)

G

group

Roman alphabet (lower case)

z

complex number

Roman alphabet (lower case, bold)

u

vector

Roman alphabet (upper case, bold)

R

set of real numbers

Greek alphabet

e

angle

Abbreviations

tanA

Punctuation

/(*)

the tangent of A the derivative f (x)

Brackets

3 [(8 + 2) - 2 ]

grouping of terms indicating changed order of operations

Brackets

/<«) _L

the value of the function at a

Iconic New symbols

oo

is perpendicular to infinity

mathematical statements typically appear in standardized fonts according to functional status; for example, different fonts are assigned for functions, variables, text, vector matrices, the Greek alphabet and so forth, with varying sizes according to their function as a subscript (for example, 7 points), sub-subscript (5 points), superscript (7 points), sub-superscript (5 points), symbol (18 points) and sub-symbol (12 points). The different choices from systems in the display stratum may be seen in the expression

The

display of mathematical symbols is standardized through special software applications for mathematical symbolic text. As with all system choices for experiential meaning in mathematical symbolism, the objective is to precisely encode the relations in a condensed format ready for re-organization and solution to the mathematics problem. Mathematical symbolism realizes experiential meaning through spatial and positional notation in a form that is not found in language. This is a significant grammatical strategy for encoding meaning efficiently in mathematics. Simple examples of the use of positional notation are x3, which means x x x x x through the spatial position of the 3 as a superscript, and the process of division which is realized through the spatial arrangement of The use of spatial positioning for experiential meaning is highlighted in the case of matrices where each variable has a value depending upon its

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spatial position in the matrix as seen in the example

Examples of the use of positional notation for experiential meaning in H A B relation to the following configuration G x,f(x) C where x is a variable, F E D f(x) is a function and the spatial positions are marked as A, B, C through to H are given in Table 4.4(4). These examples show that the use of spatial notation extends to Operative processes, participants and expressions of classification and qualification. The contraction of options in the systems for experiential meaning, the use of strategies such as the Rule of Order for Operative processes, ellipsis and the use of spatial notation and special symbols mean that maximal structural condensation can take place in the mathematical symbolic statements. This issue is further developed in relation to textual meaning and the organization of symbolic mathematics. The contraction of options for experiential meaning extends to the other metafunctions, in particular to the realm of interpersonal meaning. The objective and factual Table 4.4(4) Positional Notation Examples Position A

Functional Unit entity

B

Operative process

B

entity

Example

Meaning the mean [of x]

X y

x

/(*) T

B

classifier

M

B

entity

C

classifier/ Operative process

/-' -x

x raised to the power [of y]

the derivative [off] the transpose of matrix x the inverse of function f opposite of x or (-1) times x

C

entity

xl

x factorial

D

deictic

Xo

a specific value of x

E

entity

X

vector x

G

Operative process

.X

multiplied by x

Combination

circumstance Operative process

the definite integral

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appearance of mathematics results from a combination of the restricted selections in the fields of experiential and interpersonal meaning in the mathematical symbolism (which is also reflected in the linguistic parts of the text), the textual strategies of condensation through which meaning is efficiently encoded, and the emphasis directed towards logical meaning. 4.5 Contraction of Interpersonal Meaning

The range of interpersonal meanings in language narrowed as mathematical symbolism became concerned with the description and reformulation of patterns of variation between generalized variables and numerical quantities. The quest for exactness in the encoding of those relations in the simplest manner possible for reconfiguration in the solution to problems meant that all 'superficial' information had to be removed. In combination with the restricted experiential meaning, the rigorous ordering of fewer components of interpersonal meaning in the mathematical symbolism allowed accessible and intelligible conventions to be established. The resultant impact on the nature of interpersonal meaning in mathematical symbolism is explored through a discussion of the symbolic text in Plate 4.3(1). EXAMPLE 4 I off.

f

f

i

n

d t h e derivative off. State t h e domain

SOLUTION

At the rank of clause, a range of SPEECH FUNCTIONS (statement, question, command, offer and so forth) with different values of MODALITY (high, medium, low) are possible using language through the changeable order and selection of different functional units in the Mood structure as described in Chapter 3. For example, 'darling, could you please think how we might get out of this jam' is a metaphorical variation of the command 'think how we can get out of this jam' realized through modulated interrogative Mood ('could you'). The Vocative 'darling' and the colloquial lexis 'jam' add further interpersonal dimensions in the play of the social relations being enacted. Mathematical symbolism, on the other hand, is concerned largely with descriptive statements and a more restricted sense of commands which are realized linguistically through lexical choices such as 'let' or, in the case of Plate 4.3(1), 'find', 'state', and 'we see that' for the command 'see that'. As discussed below, mathematical symbolism does not include lexical choices which are orientated towards interpersonal meaning.

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The Mood structure in relational symbolic clauses such as/(#) = Jx— 1 may be seen to correspond to that found in language; that is, Subject/(%) A Finite/Predicator '='. In mathematical symbolism, the Finite element does not appear as a separate functional element (for example, 'did =', 'was =', or 'could =', 'might =') to realize plays with TENSE (past, present or future) and MODALITY (in the form of probability and usuality). Rather, the Finite element is fused with the process in selections such as '=' for 'equals'. Similarly MOOD ADJUNCTS which function to realize probability (for example, 'possibly' and 'probably'), usuality (for example, 'sometimes' and 'usually') and other semantic domains such as readiness, obligation, time, typicality, obviousness, intensity, degree are typically excluded in mathematical symbolic statements. In mathematics, choices for MODALITY in the form of probability may be realized through symbolic statements for measures of probability; for example, levels of significance: p <0.5 (where the notion of uncertainty is quantified) and different forms of approximations. Mathematics thus narrows the options of language which permit expression of a wide range of probability through the Finite element, the Predicator and MOOD ADJUNCTS. As a further condensatory strategy, positive and negative POLARITY ('is' or 'is not') is realized through one element for the process in mathematical symbolic statements such as

For example, '=' and

'e ' represent positive polarity, and negative polarity is typically indicated by a slash through that same element, for example, V and '£ '. The multiple strategies of condensation, which function to reduce the number of functional elements in a mathematical symbolic clause, mean that a maximal level of certainty is typically associated with mathematical statements and a high degree of obligation is associated with commands. Lexical choices such as 'darling', and expressions of attitude such as 'that is horrific' are not found in the mathematical symbolism. Using the notion of coreness of lexical items (see Chapter 3), mathematical symbolism only consists of items which have a specific meaning in the register of mathematics. Shades of meaning derived from the selection of symbolic elements which occupy a non-central position do not exist in mathematical symbolic statements. Every element is precisely defined in relation to other functional elements where the emphasis is orientated towards experiential and logical meaning. In addition, the symbolic elements appear in generalized forms as variables, special symbols and Operative processes rather than the individuality which is possible through lexical choice. At the level of discourse semantics, the choices from the discourse system of NEGOTIATION largely result in an Exchange Structure where information is provided and commands are issued so that Activity Sequences unfold in a strictly regulated fashion. The mathematical symbolism typically consists of a series of statements where the Exchange Structure is constructed from the position of the primary knower (Kl moves and move-complexes) and the primary Actor (Al moves) (see Chapter 3). In the Exchange

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Structure, there is a strong sense of direction which appears to be nonnegotiable. This strong sense of direction is also realized compositionally through the spatial arrangement of the mathematical symbolic statements which literally 'run' down the page. In addition, choices from the discourse system of APPRAISAL for graduations of evaluation and attitude (affect, judgement, appreciation) (for example, 'I like that', 'that is good' and 'thanks, I appreciate that') are typically absent in mathematical symbolism. The discourse appears as non-evaluative and value-free. The nature of the interpersonal choices in mathematical symbolism results in a discourse which appears as factual and objective, largely as a series of symbolic statements which lack modalization, modulation and choices for affective realms of meaning. The truth-value of the symbolic statements appears as consistently high with an implicit objective orientation; there appears to be no question about symbolic statements such as x = y. The expression of probability takes different forms in mathematics, including probability statements and unmodalized approximations. In addition, the restricted range of process types and the nature of the participants as generalized variables and numerical quantities function to make the text appear as abstract. The symbolic clauses and statements are realized in a tightly organized Exchange Structure. Indeed, mathematics appears as the ultimate truth which is difficult to re-negotiate, question or even interrupt. The factual and objective stance of mathematical symbolism is also communicated through the style of production and contrasts in choices for scripts and fonts from the systems in the display plane. The development of computer software devoted to the expression of mathematical symbolic text functions to ensure that conventionalized styles of typesetting and symbolic representation are standardized. The interpersonal meaning realized through such professionally produced texts, which are extraordinarily intricate and complex due to the high degree of rankshifted expressions and use of specialized symbols, contributes to the image of the text as dominating in terms of the social relations which are established between the writer/producer and reader. Without a knowledge of the grammar of mathematical symbolism, the texts are impenetrable as may be appreciated from Plate 4.5(1). Mathematical symbolic discourse is positioned as authoritative. However, what needs to be made explicit is that this position is created by the nature of the choices that are available from the systems in the language and display planes for mathematical symbolism. Mathematical symbolic texts are concerned with a limited semantic field which appears as factual and objective due to the range of interpersonal and experiential meanings which are admitted. The reason for this is that mathematics is designed to fulfil certain functions which do not include the re-negotiation of social relations or the expression of typically human processes of feeling and emotion. Mathematics is designed to capture, model and predict patterns in the most economical fashion, and this overriding aim has resulted in pre-

THE G R A M M A R OF M A T H E M A T I C A L S Y M B O L I S M In the case ft —»• 0, we calculate at .v = Pf, i = 1

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K:

Lei

(5.7) Then we have

Writing in matrix form, we obtain

where

X is the identity matrix and

(5.8)

Plate 4.5(1) Mathematical Symbolic Text (Wei and Winter, 2003: 159)

determined patterns of experiential, interpersonal, textual and logical meaning. Mathematics is typically assigned a high truth-value, most likely through its success as a tool for science and technology which plays a major role in determining economic, social, cultural and political relations. Mathematicians and scientists, however, are aware of the fallibility and impoverished nature of mathematical descriptions and models. To the

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outsider who cannot engage with the discourse, the myth is harder to counteract. The concept of truth and the cultural value attached to mathematical discourse is explored further in Chapter 7. 4.6 A Resource for Logical Reasoning

In his pursuit of sure knowledge, Descartes desired a tool that could be used to reason with, and that tool was designated to be mathematical symbolism. Mathematical symbolism has been specifically designed as a semiotic resource to describe patterns which can be rearranged for the solution to problems. The encoding experiential meaning in the form of Relational, Operative and Existential processes which preserve process/participant configurations through rankshift, and the narrow range of interpersonal meanings result in a grammar which encodes meaning in the most efficient manner possible. Logical reasoning flows smoothly down the page as seen in the solution to the problem in Plate 4.3(1). In what follows, that reasoning is explored. Mathematical symbolism is concerned with relations of EXPANSION (Halliday, 1994: 328-329) in the form of (i) elaboration, or a re-statement in the form of apposition or clarification; (ii) extension, or additative and variation type relations; and (iii) enhancement, which is predominantly causal-conditional and spatio-temporal type relations. Logical relations of PROJECTION in the form of quotations and reporting of locutions and ideas (ibid.: 220) are not typically found in symbolic mathematics. This represents a narrowing of the options for logical meaning admitted into mathematical symbolism. The 'temporal' conjunctive relations which are concerned with the order in which Operative processes unfold in statements and expressions are realized through the Rule of Order for Operative processes. The conventionalized order is brackets, powers, multiplication/division and addition/subtraction, and this means that Operative processes do not necessarily unfold in a left-to-right fashion. The Rule of Order is a condensatory strategy in that the explicit order with which the mathematical operations are to be performed need not be explicitly marked in mathematical symbolic statements. The order may be changed, however, through the use of differing types of brackets which function to re-organize the standard sequence of operations. For example, the use of brackets in the numerator of

means that the order is changed so that the

Operative processes of addition in (x+ h— 1) and (x— 1) are completed first, rather than the left-to-right sequential processing of the Operative processes of addition and subtraction. At the rank of discourse semantics and inter-statemental relations, CONJUNCTION and CONTINUITY relations are realized through explicit and implicit structural conjunctions which link clauses, and non-structural conjunctive adjuncts which function cohesively across stretches of text (see

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Section 3.4 where Halliday's systems for logical meaning are explained). Some structural conjunctions and conjunctive adjuncts are symbolized in mathematics (for example, .'. and while others occur in linguistic form (for example, 'and', 'or', 'for example', 'also' and so forth). Statements are explicitly marked in cases where the logical connection is non-sequential. For example, statements labelled (1) and (2) may be referred to later as 'from (1) and (2)'. The conceptions upon which logical meaning at the rank of discourse semantics and inter-statemental relations are based are investigated below. Figure 4.6(1) contains the analysis of logical meaning in terms of the conjunctive relations which occur in the solution of the problem displayed in Plate 4.3(1). Conjunctive relations in mathematical symbolism typically function in an internal-rhetorical sense rather than an externalexperiential sense. Following Martin (1992), these types of relations are modelled on the left-hand side as shown in Figure 4.6(1). Relations are modelled through arrows which indicate the dependency structure. Moves are shown to be dependent on preceding ones by dependency arrows pointing upwards, and the arrows point outwards in the cases where the result depends on mathematical definitions, axioms, laws, theorems and other results. In Figure 4.6(1), the implicit conjunctions are causalconditional type relations of the form 'therefore'. On the right-hand side, the basis for the logical reasoning is given. Mathematical symbolic discourse typically involves long implication sequences as seen in Figure 4.6(1). However, it becomes evident from this analysis that mathematical deductive reasoning depends to a large extent on laws, axioms, theorems and established results which are not made explicit in the derivation of the solution to the problem. Tiles (1991) claims that the image of mathematics as dealing solely with deductive reasoning based on syllogisms (that is, deductive inference consisting of two premises and a conclusion which are categorical propositions) is misleading. As seen in Figure 4.6(1), the steps in the Activity Sequence in the solution of the problem involve implicit procedures using established results. Azzouni (1994: 79) characterizes mathematics as 'a collection of algorithmic systems, where any such system, in general, may have terms in it that co-refer with terms in other systems'. It appears that the success of mathematics is dependent on this co-referral of terms based on previously established results which are formalized as definitions, axioms, theorems, laws and so forth. The fallibility of mathematics as a formal system, however, has been demonstrated through contributions by Frege, Hilbert and Russell and Godel. New technological advances are contributing to this process: 'Computers are in the process of dismantling the very image of reason which generated them' (Tiles, 1991: 169). The loss of certainty in mathematics (Kline, 1980) has been accompanied by new mathematical approaches to physical systems; for example, chaos and dynamical systems theory. Perhaps in the pursuit of knowledge (see also Chapter 2), the realms of meaning

f(x) given in Example 4 Definition of derivative

Substitution for f(x)

Rationalize numerator Multiplication Property ofOne(MPOne) Factorization x2-y2=(x+y)(x-y) Distributive Property of Multiplication over Addition (DPM/A) Simplification based on algebraic laws Definition limit Simplication based on algebraic laws

Figure 4.6(1) Logical Meaning (Conjunctive Relations) in Plate 4.3(1)

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which were excluded in mathematical symbolism resulted in an inadequate set of semiotic tools to describe physical phenomena. However, technological innovations such as those accompanying the latest development in computers permit patterns to be mapped computationally and visually in new ways. While mathematical symbolism remains the major semiotic resource for the construction of logical meaning in mathematics, perhaps a new semiotic resource will evolve through the increased power of computers to represent patterns of relations. This may be in the form of the development of a new grammar of visual images, or the construction of a new resource yet to be imagined. As may now be appreciated, mathematics is constrained by the nature of the semiotic resources it has at its disposal. 4.7 Specification of Textual Meaning

The textual organization of mathematical symbolism is sophisticated and highly formalized in order to facilitate the economical encoding of relations which permits immediate engagement with the experiential and logical meaning of the text. The nature of textual meaning is evident in Plate 4.3(1) where, at the rank of discourse semantics, the statements are placed sequentially down the page to permit easy tracking of the nuclear reconfiguration of process/participant structures. The overall spatial arrangement of mathematical symbolic text is generic so that key equations, definitions and solutions are immediately accessible in texts. In addition to spatial sequential organization of the symbolic text, the mathematical statements and clauses are organized syntagmatically in pre-defined forms for Relational and Operative process/participant configurations. Examples of spatial and syntagmatic relations which are typically found in mathematical symbolism are given in Table 4.7(1). The textual organization of the symbolism may be contrasted to experiential meaning in language which is typically represented as clusters of participant/process/circumstantial functions rather than definitive syntagms which have a specific order. That is, alternatives are possible in the configuration of participants, process and circumstance in language; for example, T will sit quietly for ten minutes', 'For ten minutes I will sit quietly', T will quietly sit for ten minutes' and 'Quietly will I sit for ten minutes'. Such re-organization is not a typical feature of mathematical symbolic statements. The tendency towards standardized syntagmatic organization of symbolic statements is evident in Table 4.7(1) and the definition of the derivative

in Plate 4.3(1). The

specificity of textual organization relates to the need to organize the experiential meaning of the mathematical symbolic statement in an exact and simplified format so that problems may be solved. 'To simplify' in mathematics means to arrange mathematical Operative processes and participants in the most simple standardized format using algebraic rules and pre-established mathematical results.

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Table 4.7(1) Textual Organization: Syntagmatic and Spatial Relations 1 Equations and formulae Circ Qualifier

Conj

{x; if

inAABD

Participants 2x + 4 3x AB ABD

Process > > 7i =

Participants 7 y} CD 60°

Process <

Participant 2

2 Simplification of expressions and formulae Process

=

Participant Participant 7(2x-3)-5(l-x) 14x - 21 - 5 + 5x

3 Mathematical statements may be combined Participant -3

Process <

=

Participant x

=

4 Combinations may occur with certain conjunctions if

x + 2y = 7

and

2x + 3y = 4

Spatial positioning in combination with syntagmatic organization plays an important role in the semantics of mathematical symbolism as illustrated in Table 4.7(1). At the clause level of the lexicogrammar, with exceptions (for example, 'if, 'and', and 'or'), conjunctions typically occur on the left- hand side of a mathematical sentence, slightly separated spatially from the mathematical clause as seen in Plate 4.7(1). It should be noted, however, that mathematical texts vary in terms of spatial layout and the extent to which the symbolic text is integrated with language. Many other possibilities occur including the use of language conjunctions and complete integration of mathematical statements within the linguistic text as found in 'If f(x) = J x — l , find the derivative of/' in Plate 4.3(1). Key symbolic statements of special interest, however, are typically spatially separated from the body of the text. The use of spatiality is one key element of mathematical symbolism which differs from the line-wrapped syntagmatic arrangement of linguistic text. Such visual arrangements permit easy engagement with the text. This is necessary as the reading path is not necessarily linear in mathematical and scientific texts which consist of language, visual images and symbolism. The use of spatial arrangement also permits ellipsis on a scale which is not found in language. Lemke (1998b) explains that the table, for example,

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Next,

Next

so that

For the last term,

On the other hand.

Plate 4.7(1) Textual Organization of Mathematical Symbolism (Clerc, 2003: 117)

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carries textual ellipsis to the extreme so that the result is a 'textualized visual display', meaning that the table uses spatial arrangement to compensate for the lack of grammatical constructions. Baldry (2000a) discusses meaning compression and the integration of visual, verbal and symbolic resources in scientific tables. Mathematical symbolic statements function in a similar manner, permitting a visual arrangement for condensatory purposes so that the experiential and logical meaning is immediately accessible. The organization of mathematical symbolism as a message is therefore highly conventionalized in terms of genre, inter-statemental relations, and the statement. Within statements, the notion of the Theme as the point of departure of a message and the Rheme as the part in which the Theme is developed is fundamental to the textual organization of mathematical symbolic clauses. The left-hand side of an equation usually involves the conflation of Subject with Theme. If a Circumstance Adjuncts occurs first, such as 'in A ABD' in Table 4.7(1), this functional element is typically placed further apart from the mathematical equation so that the Subject/ Theme conflation is not disrupted. Given the concern with logical meaning, multiple Themes containing textual elements in the form of conjunctions and conjunctive adjuncts (for example, 'if, 'then', 'therefore') are common in mathematical symbolic text. Marked Themes in the form of a dependent clause (for example, 'if x= 4') also frequently occur. The spatial arrangement of the mathematical symbolic clauses permits ellipsis; for example, /' (x) need not be repeated in the derivation of the derivative in Plate 4.3(1). Within the symbolic statements, the Rule of Order for Operative processes (see Section 4.6 on logical meaning) determines the temporal unfolding of processes/participant configurations, which does not necessarily follow a left-to-right format. Mathematical symbolism possesses its own rules through which mathematical Operative processes unfold. Within mathematical statements, two further textual strategies for economical encoding of meaning are the generic forms of ellipsis established in mathematical symbolism, and the use of spatial position to realize Operative processes (see Section 4.4 on experiential meaning). For example, x for the Operative process of multiplication in expressions is not required in expressions such as xyz (for x x y x z) and (a + b) (c + d) for (a + b) x (c+ d). In addition, the use of spatial position in x3 realizes the Operative process of multiplication for x x x x x. In a similar fashion, spatial notation for division is used in expressions such as

for x ~ y. Another

strategy for economy of expression is the serial placement of the components of an expression which follows conventions which differ from the structure of the nominal group in language. An example of this type of extension occurs within the decimal place value system (for example, 3.1415926) where each place or location in the sequence has a particular value; that is, the decimal system involves place value. At the rank of discourse semantics, the reference chains linking partici-

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pants for the system of IDENTIFICATION are complex as the nuclear configurations of Operative processes and participants are constantly rearranged in the derivation of the solution to a problem. However, the spatial organization is such that these chains are typically organized down the page in a manner where the configurations of Operative processes and participants are standardized. Mathematical symbolic participants are explicitly repeated clause by clause to permit tracking, and the fate of participants is based on definitions, algebraic laws, theorems and so forth. Referential cohesion, the tracking of participants, however, requires a knowledge of the implicit basis upon which participants are rearranged, as discussed in relation to logical meaning. That is, participants may be transformed through axiomatic definitions, derived results, operational properties and theorems. In addition, there may be discursive links to situate the symbolic notation in relation to the main body of the text. Statements are also often labelled for easy reference. Mathematical symbolic texts are textually organized in a way that permits maximum condensation in the instantiation of the texts. Strategies include the spatial or visual arrangement of the text which permits easy tracking of process/participant configurations, standardized sequential placement of elements in the statements and clauses in terms of Theme and Rheme, standardized forms of ellipsis, the Rules of Order for Operative processes and the use of brackets to change that order, and place value at the rank of component. The use of space as a textual means of organizing and displaying mathematical symbolism is an important strategy which, as we have seen, extends to the experiential and logical meaning. Spatial arrangement and positional notation thus developed in the grammar of mathematical symbolism as an important means for fulfilling the functions of this resource. 4.8 Discourse, Grammar and Display

Hand in hand with language and visual images, mathematical symbolism developed as a semiotic resource which had clearly defined functions. These functions include the description of patterns of relations and the reordering of those relations to create models of the physical world, to solve problems and to make predictions. The focus on experiential and logical meaning necessitated the development of a specific grammar in order to fulfil those functions. The discourse, grammatical and display strategies that consequently developed in mathematical symbolism were based on the need for an economy of expression so that meaning could be encoded in a manner which permitted the reconfiguration of process/participant structures. The grammar also needed to be efficient in terms of making use of mathematical definitions, axioms, laws and theorems and other established results. In addition, that grammar interacts with linguistic and visual semiotic resources in the manner described in Chapters 6-7. The discourse, grammatical and display strategies described in this chapter are

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summarized below in terms of two major aims: (1) to encode patterns of relations economically and exactly; and (2) to permit reconfiguration of process/participant structures through the strategy of rankshift. Condensatory Strategies for Exact Encoding of Meaning

Experiential Meaning Mathematical symbolism became concerned with limited fields of experiential meaning in the form of Relational, Operative and Existential processes, the associated participants and restricted forms of circumstance. The removal of experiential realms of meaning considered extraneous (for example, Mental, Behavioural and Verbal processes, their attendant participants and a range of circumstances) aided the expansion of experiential meaning in terms of capturing and rearranging descriptions of patterns of relations through the configurations of Operative process/participants. Activity sequences which document the solution to mathematical problems through the reconfiguration of Operative processes are clearly denned. The development of an economical and precise grammar for encoding experiential meaning permitted the development of mathematical symbolism as a tool for logical reasoning as process/participant structures are easily reconfigured in the steps for deriving the required mathematical result. The grammatical strategies which contribute to the economical encoding of experiential meaning include the use of spatial positioning which results in an economy of expression that is impossible in language. In essence, mathematical symbolism incorporated and built on resources from language (in the form of syntagmatic relations and sequential and serial positioning) and visual images (in the form of spatial arrangement and spatial notation). Within the syntagmatic structures in mathematical statements, symbolic processes do not unfold in a left-to-right fashion because of a variety of strategies, including the Rule of Order for Operative processes and the use of brackets. The order of operations is brackets, powers, and multiplication/division and addition/subtraction. The temporal unfolding of mathematical operations need only be grammatically marked through brackets when the Rule of Order is altered. The grammar of mathematical symbolism permits Operative processes to be ellipsed. This aids condensation in terms of encoding relations in a simplified format. There is a restricted range of elements in the participant structures which take the form of generalized variables, numerical quantities and specialized symbolic expressions for mathematical terms. Qualifiers form part of participant group structures without the need for embedding as phrases. Circumstance is encoded within process and participant structures in the clause in addition to appearing as separate functional elements. The condensatory grammatical strategies in mathematical symbolism include the use of special standardized symbols and place value. At the display stratum, meaning is encoded through Font Style, Format and Size; for example, there are specific formats for functions, variables,

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text, vector matrices, Greek alphabet and so forth. These grammatical strategies contribute to the efficient encoding of relations in mathematical symbolism. Interpersonal Meaning Interpersonal meaning in mathematical symbolism is largely restricted to unmodalized statements and commands. Probability is encoded as probability statements and approximations. Lexis is replaced with variable and numerical participants and other symbolic terms. Functional elements such as the Finite element and Mood Adjuncts are excluded. The Exchange Structure consists of rigid sequences of statements and commands which lack choices of judgement, affect and evaluation. The restricted range of interpersonal meanings in mathematical symbolism contributes to the objective and factual appearance of the discourse. The style of the production of the mathematical symbolic text arising from software applications reinforces the dominating type relations established between the writer/ producer and the reader of the text. Logical Meaning Mathematical symbolism developed as a tool for reasoning in the form of elaboration (clarification and reinstatement), extension (addition and variation) and enhancement (predominantly causal-conditional and spatio-temporal type relations). The extended implication sequences of internal/rhetorical-type relations are most typically based on implicit results such as definitions, laws, axioms, theorems and other established results. The ease at which the grammar of mathematical symbolism permits the reconfigurations of mathematical participants and processes for the descriptions of relations means that logical reasoning flows in the symbolic text. The spatial arrangement of the text enables the conjunctive relations to be apprehended at a glance. At the rank of the statement, the temporal unfolding of Operative processes is determined by the Rule of Order for Operative processes. This means that selections for conjunctive adjuncts for temporal relations need not be made within mathematical statements. Brackets are used to change the conventionalized order of Operative processes. Textual Meaning Mathematical symbolic notation developed specific techniques for organizing experiential and logical meaning at the ranks of discourse, grammar and display strata. These include the use of spatiality in ways which are not found in language. Visual spatial layout includes the organization of the entire mathematical text (genre), the sequential ordering of statements within that text (discourse semantics), the syntagmatic ordering of components within the clauses (grammatical) and the use of positional notation at the rank of expression and component (expressions and components). That is, mathematical statements are sequentially positioned

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vertically and horizontally, while syntagmatic structures unfold horizontally in a manner which depends on the Rule of Order for operations and the use of brackets. Textual organization also functions at the rank of component, for example, place value. Rankshifting of Operative Process/Participant Configurations

Mathematical symbolism and scientific language developed different lexicogrammatical strategies for encoding experiential meaning. Scientific language evolved to expand the meaning potential of nominal group structures through grammatical metaphor, where the relations between these abstract entities were encoded as relational processes (Halliday and Martin, 1993; Martin and Veel, 1998). However, the grammatical strategy which developed in mathematical symbolism is the preservation of Operative process/participant configurations through rankshift. This strategy is essential because it permits the re-organization of the patterns of relations for the solution to problems. In other words, mathematical symbolism succeeds precisely because it preserves the nuclear configurations of Operative process/participant structures in an exact and economical format which may be rearranged for the solution to mathematical problems. Mathematical symbolism succeeds as a tool for logical reasoning because the grammar preserves relations in a dynamic format which can be manipulated to obtain the required result. The mathematical grammatical strategy of rankshift is the platform upon which the grammar is built. 4.9 Concluding Comments This discussion provides a framework for a grammar of mathematical symbolism. As becomes apparent, there is much work to be completed in order to construct a grammar which compares with the comprehensiveness of Michael Halliday's SFG for language and Jim Martin's discourse systems for language. For example, the grammar for Operative processes needs to be further investigated to understand how the notions of the Medium and Agent apply to this process type. In addition, Martin's discourse systems need extending to incorporate shifts across semiotic resources (see Chapter 6). More generally, however, the discussion presented here demonstrates that although mathematical symbolism is impoverished with respect to the meanings afforded by language, it nonetheless is capable of meaning beyond that which is achieved linguistically. An outline of the strategies through which this meaning expansion occurs in mathematical symbolism has been given above. These strategies require further investigation, especially in relation to the new developments in mathematics which are taking place through computer technology. The impact of this technology is considered with respect to visual images in Chapter 5.

5 The Grammar of Mathematical Visual Images

5.1 The Role of Visualization in Mathematics

Visual images are such an important component of our resources for meaning (Kress, 2003; Staley, 2003) that they have been defaced and destroyed in scientific, religious and artistic contexts (see Latour and Weibel, 2002), sometimes for reasons which are not even immediately clear. Visual images in mathematics give an intuitive understanding of the reality constructed through the symbolism and language (for example, Galison, 2002; Lemke, 1998b). Mathematical visual images mirror our perceptual understanding of the world and thus connect and extend common-sense experience to the mathematical symbolic descriptions. However, there have been longstanding tensions among mathematicians over the place of the visual image versus the symbolic in mathematics. Traditionally, the functions of the visual image are seen to be important, but limited compared to those fulfilled by symbolism. In what follows, the tensions and differing perceptions of the roles of mathematical symbolism and visual images are explored. From this point, an SF framework for mathematical visual images is used to analyse the types of metafunctional choices found in abstract mathematical graphs. In the final section, the changing role of mathematical images arising from the use of computer graphics programs to display patterns generated from digitalized data (Colonna, 1994; Davis 1974, 2003) is considered. The struggle concerning the role of the visual image versus the symbolism in mathematics has a long history (Davis, 1974; Galison, 2002; Shin, 1994). Since the time of Descartes, traditionalists have seen mathematics as primarily being the symbolic. From this perspective, the visual image is seen as a heuristic tool rather than a means of establishing a valid proof. A 'proper' theorem takes the form of a statement derived from axioms by a sequence of logical steps, rather than a visual pattern which requires neither verbal statement nor symbolic proof. The opposing perspective is that visual displays are a valid form of establishing results, leading to what has been conceptualized as 'a theorem of the perceived type' (Davis, 1974). Galison (2002: 323) sees the ongoing tension in mathematical circles as an oscillation between two poles: 'We must have images; we cannot have

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images.' In fields 'from geometry to quantum mechanics, from astrophysics to microphysics, the richness of the image and the austerity of the numerical are always falling into each other' (Galison, 2002: 323). The struggle between the image and the symbolic as legitimate forms of semiotic practice in mathematics deserves special attention. Shin (1994) gives two common reasons why mathematicians and scientists claim a limited functionality for visual images: the limitations of visual images in presenting 'knowledge'; and the possible misuse of diagrams, such as making unwarranted assumptions in geometry. In the first instance, the meaning of mathematical visual images is seen to be less precise than the symbolism which allows for logical deduction from a set of established results and formalized procedures for mathematical induction, proof by contradiction, calculation of exact values and so forth. In the second instance, the trends displayed in abstract graphs, for example, are seen to be incomplete descriptions which vary according to how they are displayed. There is also room for error in interpreting visual images, for example, in the case of optical illusions. The symbolism is thus seen to be more powerful (Lemke, 1998b) and less prone to error than visualization. Shin (1994), however, claims that the negative prejudice against the use of diagrams as a means for establishing logical proof is not entirely warranted. Shin (1994) uses C. S. Peirce's system of logical diagrams, called 'existential graphs', to demonstrate that Venn diagrams can be used for valid proof, and that diagrams are not inherently misleading. Mathematicians have in fact always drawn conclusions from visual images, but the results have traditionally been expressed through symbolic means. Davis (1974: 115-116) gives a comprehensive list of reasons why this form of practice developed in mathematics: 1 The tremendous impact of Descartes' Discours de la Methode (1637) by which geometry was reduced to algebra; also the subsequent turnabout wherein the medium (algebra) became the message (algebraic geometry). 2 The collapse, in the early 19th century, of the view, derived largely from limited sense experience, that Euclidean geometry has a priori truth for the universe; that it is the model for physical space. 3 The incompleteness of the logical structure of Euclidean geometry as discovered in the 19th century and as corrected by Hilbert and others . . . 4 The limitations of two or three physical dimensions which form the natural backdrop for visual geometry. 5 The limitations of the visual ground field over which visual geometry is built as opposed to the great generality that is possible abstractly (finite geometries, complex geometries, etc.) when geometry has been algebraicized. 6 The limitations of the eye in its perception of mathematical 'truths' (e.g., the existence of continuous everywhere nondifferentiable functions, optical illusions, suggestive but misleading special cases, etc.).

Davis (1974) explains that mathematics never really recovered from the impact of the Cartesian project which involved the algebraicization of geometry and other fields of mathematics with geometric content. Math-

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ematicians such as Henri Poincare (1854-1912) sought to establish a legitimate place for visual images, which he saw as being the source of intuition and the means for keeping mathematics in contact with the real or 'concrete world', rather than spiraling off into 'the abstract' through the symbolism. 'As far as Poincare was concerned, without intuition the mathematician was like a writer shackled forever in a cell with nothing but grammar' (Galison, 2002: 302). However, Poincare came to realize that the complexity of the solar system defied visualization, and his efforts in effect sparked the study of chaos. But in the quest for a general rule, Poincare had turned to visual images. Galison (2002: 302) summarizes the situation: 'In the sciences of the last century and a half, the pictorial and the logical have stood unstably perched, each forever suspended over the abyss of the other.' From a semiotic perspective, it is possible to see why modern mathematicians prefer to use mathematical symbolism as the means for establishing results. Descartes and later mathematicians favoured algebra as the means for describing curves, and the grammar of mathematical symbolism developed accordingly so that these exact relations could be rearranged and manipulated for the solutions to problems. This is possible because the grammar of mathematical symbolism preserves nuclear configurations of process/participant structures in an economical and exact fashion. These structures are encoded through elaborate forms of rankshift to form multiple levels of embedded process and participant configurations. These configurations are rearranged to establish the required mathematical results. The grammar of mathematical symbolism is thus based on a range of condensatory strategies which facilitate rankshift for the rearrangement of relations. As described in Chapter 4, the condensatory strategies include limited forms of process types, the use of spatial and positional notation, the use of special symbols, the Rule of Order for operations and so forth. The result is a semiotic resource which can be used as a tool for reasoning. The dynamic semiotic resource, the one that can be rearranged and manipulated for the proof of results, was designated to be mathematical symbolism, and the grammar developed accordingly. From Descartes onwards, the emphasis shifted from the construction of curves to symbolic description and manipulation of relations, which were visualized using mathematical graphs, diagrams and other forms of visual display. Mathematical symbolism thus developed a grammar which permitted it to be used for establishing mathematical results, and so the dynamic aspect of the process of mathematicization was allocated to the symbolism. The same effort was not extended to the grammar of visual images, which retained the important but seemingly less significant role of the display of the patterns. In addition, modern mathematics developed as a written discourse which circulated in manuscript and print form. The grammars for mathematical symbolism and visual images were certainly the product of the functions allocated to each resource, but they were also the product of the technology through which the texts could be produced. The advent

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of the printing press was a key factor in the increased popularity of the Hindu-Arabic numerals, their accompanying algorithms, and the eventual development of standardized algebraic procedures. However, the visualization of the patterns traditionally appeared in the form of static graphs and diagrams in printed texts. These visual images could not be manipulated, and they were also time-consuming to produce. Mathematical visual images thus developed as a static representation of the dynamic procedures undertaken through the symbolism, the tool through which the formal reasoning took place. There was a loss of meaning incurred in mathematics through the use of the symbolism. In the process of developing a specific grammar for the functions to which it had been assigned, mathematical symbolism became concerned with a limited semantic domain which not only excluded realms of meaning in language as explained in Chapter 4, but also excluded the realms of meaning afforded by visual semiotic. Mathematical symbolism may be seen as more powerful, but the descriptions of continuity nonetheless arise from categorical distinctions of variation made from a limited range of choices within the symbolism. For example, only certain categories exist for symbolic process types, participants and circumstance. Visual images, on the other hand, are capable of representing graduations of different phenomena (Bateson, 1972; de Saussure, 1966; Lemke, 1998b; Messaris, 1994). 'Words [and at a more delicate level, mathematical symbolism] divide the world into black and white (and in some languages, gray), large and small, strong and weak, good and bad. Images, however, can represent shades of gray, ranges of size, and degrees of those external attributes that viewers use in making inferences . . .' (Messaris, 1994: 121). The loss of the input from the visual semiotic in the symbolized environment of mathematics cannot be underestimated. Davis (1974: 119) explains: The algebraicization of geometry must be regarded as a prosthetic device of great power which maps certain aspects of space into analytical symbols. The blind might be [en]abled to manipulate space through the instrumentality of these symbols, but since one channel of sense experience is denied to the blind, one feels that a corresponding fraction of the mathematical world must be lost to them [and to us] . . . The analytic program [the symbolism], then, is a prosthetic device, acting as a surrogate for the 'real thing' [the visual image].

This situation is rapidly changing, however, with the development of computer graphics which is revolutionizing the role of the visual image in mathematics and science, and, more generally, across most other fields of human activity. Visual images may now be manipulated and synthesized in today's computer environment. This revolution is considered in terms of the increased functionality of the visual image in Section 5.7. However, in order to make sense of what can be achieved visually, a framework for an SF grammar for mathematical visual images is first presented in Section 5.2. The framework, based on O'Toole's (1994, 1995) SF framework for paintings, is extended to account for systems of meaning in mathematical visual

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images. The systems for representational (experiential and logical), interpersonal and compositional meaning are discussed through an analysis of an abstract graph in Sections 5.3-5.6. From this point, the changing nature of visualization in mathematics is considered from the point of view of computer graphics. 5.2 SF Framework for Mathematical Visual Images

Mathematical visual images include abstract and statistical graphs, a range of genres of diagrams and computer-generated graphics. Abstract graphs display the functional relationship between two or more variables in the form of lines, curves and three-dimensional figures. The points are plotted on a set of co-ordinate axes and include only those points which satisfy the given relation. Statistical graphs show the relationship between sets of quantities in the form of bar graphs, column graphs, line graphs, histograms, pie charts, scatter diagrams and so forth. The term 'diagram' is used here in the broadest sense to include pictorial representation of entities and relations such as Venn diagrams, geometrical figures and other figures such as those found in graph theory and topology (Borowski and Borwein, 1989). Computer-generated graphics include traditional forms of abstract and statistical graphs and diagrams, in addition to new forms of dynamic images of graphs which unfold over time. The visual images generated through computer graphics include fractal geometry, views of mathematical models and methods, and other images in applied mathematics, such as graphical representations of diffusion, turbulence and flow, for example (see Colonna, 1994). The evolving range of new genres of mathematical visual images requires investigation and documentation. However, in what follows the focus of attention is the discourse and grammatical systems which constitute the grammar of mathematical visual images. The SF framework is used for the analysis of printed mathematical abstract graphs. The systemic model for visual images in mathematics is presented in Table 5.2(1). The content plane includes a discourse semantics stratum Table 5.2(1) SF Model for Mathematical Visual Images MATHEMATICAL VISUAL IMAGES CONTENT

Discourse Semantics

Inter-Visual Relations Work/Genre Grammar

Episode Figure Parts DISPLAY

Graphics

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which is concerned with Inter-Visual Relations established across a sequence of visual images such as those found in mathematics journal articles, books and websites (see Plate 5.2(1) and Plate 5.7(ld)). Discourse moves across visual, linguistic and symbolic parts of the text are theorized in Chapter 6. Following O'Toole's (1994: 24) ranks for paintings, the entire visual image is the Work and the grammar is concerned with the Episode or the configurations of process/participant and circumstance in the visual image, the Figures which are the participants in the Episodes, and the Parts of the display. The Work is the diagram, graph or computer graphics which result from interactions of Episodes with Figures which are composed of Parts. The sub-division of the mathematical visual image into separate Episodes is not always relevant as there may only be a single Episode and Figure; for example, a drawing of a line. As with language and mathematical symbolism, the systems which operate on the display plane are also considered. Despite the classification of mathematical discourse into three different semiotic resources as suggested by the three SFGs for language, mathematical symbolism and visual display presented in Chapters 3-5, mathematical visual images are typically multisemiotic texts which contain linguistic and symbolic elements in the form of Titles, Labels and Captions. The multisemiotic nature of mathematical visual images are investigated according to metafunction in Sections 5.3-5.6 O'Toole (1994: 24) organizes the systems for paintings according to the representational, modal and compositional metafunctions which correspond to ideational, interpersonal and textual metafunctions in language. The same approach is adopted in Table 5.2(2), which displays the SFG for mathematical visual images. A similar approach for mathematical statistical graphs and biological schematic drawings is adopted by Guo (2004a, 2004b). In Table 5.2(2), the representational metafunction is sub-divided into the experiential and the logical. The metafunctionally organized systems are displayed according to rank and strata: discourse semantics for the Sequence of Graphs/Diagrams/Computer Graphics; grammar with ranks Graphs/Diagrams/Computer Graphics, Episode, Figure, Parts; and the display plane which, following Lim (2002, 2004), is called the 'graphics' plane. The systems for the logical metafunction appear in the discourse and grammar strata at the rank of Graphs/Diagrams/Computer Graphics because logical meaning is seen to primarily arise from sequences of mathematical visual images and interactions of Episodes. The categories for logical meaning are based on Halliday's (1994) system of EXPANSION, that is, elaboration, extension and enhancement. In particular, logical meaning in the form of spatio-temporal relations in visual images is discussed in Section 5.5. The applicability of O'Toole's (1994: 24) framework for displayed art to visual images in mathematics makes sense given the relationships between mathematics and visual art and design from the times of antiquity and the Renaissance to the present. The engagement between mathematics and art includes the mathematicization of the human figure in

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Table 5.2(2) Grammar and Discourse Systems for Mathematical Visual Images DISCOURSE SEMANTICS SEQUENCE OF GRAPHS/ DIAGRAMS/ COMPUTER GRAPHICS

CONCEPTUAL DEVELOPMENT

ENGAGEMENT (Inter-Visual Relations)

THEMATIC DEVELOPMENT

Development of mathematical content through sequences of Episodes, Relations, Figures and Parts

Discourse moves through visual sequence (through repetition and change in Episodes)

Textual organization for tracking participants, processes and relations

LOGICAL RELATIONS EXTENSION in the form of elaboration, extension and enhancement (through multiple TimeFrames)

GRAMMAR Units

REPRESENTATIONAL/ EXPERIENTIAL

INTERPERSONAL

COMPOSITIONAL

GRAPHS/ DIAGRAMS/ COMPUTER GRAPHICS (Genre)

• Display of patterns of relations (as lines, curves and three-dimensional shapes) • Process types: - Relational (graphs of functions) — Transformational x^f^y • Perceptual Reality (for example, geometrical displays) • Mathematical Symbolic Reality (for example, Venn diagrams, data graphs) • Interplay of Episodes • Multiple Time-Frames with Temporal Unfolding through Spatiality • Comparisons of patterns of variation • Circumstance • Dynamic Temporal/Spatial Unfolding (Computer Graphics)

• Metaphorical Narrative • Modality and the Degree of Idealization, Abstraction, Quantification • Accompanying text in the form of Caption, Title and Labelling which are emphasized by Size, Positioning, Underlining and Font • Colour • Line Width, Shading, Line Solidarity, Slope, Arrows • Rhythm • Curvature • Perspective • Framing • Style of Production • Nature of participants • Production process • Intricacy of display • Directionality

• Gestalt: Framing, Horizontals, Verticals and Diagonals • Positioning • Perspective (2D, 3D) • Use of Lines and Curves • Inter-connections established through symbolism and language through Labels • Cohesion (Parallelism, Contrast, Rhythm) • Reference through language, symbolism)

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Table 5.2(2) - cont GRAMMAR Units

EPISODE

FIGURE

PARTS

REPRESENTATIONAL/ EXPERIENTIAL

LOGICAL RELATIONS • Spatial relations (from Interplay of Episodes) • Temporal relations (Multiple Time-Frames) • Spatial/temporal Relations • Types: Elaboration, extension and enhancement • Interplay of Actions or Relations between Figures • Portrayal of process (for example, relation as Curves or Lines)

LOGICAL RELATIONS • Spatial relations (in Episodes) • Spatial/temporal Relations (Multiple Time-Frames) • Types: Elaboration, extension and enhancement • Participants • Circumstantial features

• Title • Axes, Scale, Arrows • Labels • Lines, Curves, Shading, Intersetion Points • Slope

INTERPERSONAL

COMPOSITIONAL

• Prominence of Interplay • Labelling of Interplay (through symbolism, (Size, Colour, Labelling, language) Framing, Prominence, • Portrayal of Process and Position and so forth) Participants (relative • Display of process (Line, Positioning, Size of Curve) Figure and salient features as displayed by Lines, Curves, Colour, Line Width, Shadings)

• Prominence of Figures

(Size, Colour, Labelling, Framing, Prominence, Position and so forth)

• Stylization • Conventionalization

• Labelling of Figure (through symbolism, language) • Stylistic Features (Size, Shape, Dynamics, Colour, and marking of Parts) • Textual markedness (through Labelling, Colour, Size and so forth)

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Table 5.2(2) - cont DISPLAY

GRAPHICS

• Variations in line width, dotted lines and arrows • Variations in font, scripts and size • Colour, Shading, Brightness and Hue

• Variations in line width, • Perspective dotted lines and arrows • Cohesiveness and Contrast (through • Variations in font, scripts Colour, Font, Size and and size so forth) • Colour • Spatial arrangement • Stylization and production (computer generated, drawn and so forth)

terms of proportion, the development and use of perspective, 'derepresentationalized artistic productions' involving geometric constructions and computer-generated and assisted art (for example, Danaher, 2001; Davis, 1994: 167; Emmer, 1993; Field, 1997). Given the interconnectedness between art and mathematics, overlaps between O'Toole's (1994) systems and those proposed in the SF model for mathematical visual images in Table 5.2(2) occur. However, mathematical visual images developed specific grammatical systems which permit the integration of the symbolism and language with the visual images. The nature of these systems is discussed according to metafunction in Sections 5.3-5.6. The SF systems documented in Table 5.2(2) are discussed in relation to the analysis of the abstract mathematical graphs (a) and (b) in Plate 5.2(1). These graphs are the geometrical interpretations of the derivative of a function/' (x) as the slope of a curve. The algebraic form of the derivative is considered in Chapter 4. Abstract graphs are chosen for the analysis as these forms of graphs are central to mathematical descriptions of patterns of variation and, further to this, these graphs employ unique strategies to encode experiential meaning. The application of the framework for images generated by computer graphics (see Plates 5.7(la-d)) is also considered. The investigation of the choices in the grammatical systems for mathematical symbolism in Chapter 4 reveals a contraction and simultaneous expansion of experiential meaning, a narrow range of interpersonal meaning, a refinement of textual meaning and specific forms of logical meaning. In what follows, it will become apparent that these types of meanings are largely replicated in conventionalized forms of mathematical diagrams and graphs. As mathematical symbolism dispensed with different forms of meaning in the quest to encode and rearrange patterns of relations, it appears that a similar trend occurred in mathematical visual representations which were designed to function in collaboration with the symbolic descriptions. The dominant metafunction in the visual display is representational (experiential and logical) meaning, which is aided by the elimination of what is considered to be superfluous contextual information. Compositional styles of the visual images are conventionalized, and

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MATHEMATICAL DISCOURSE "Si Interpretation of the Derivative as the Slope of a Tangent In Section 2.6 we defined the tangent line to the curve y —f(x) at the point P(a,f(a)) to be the line that passes through P and has slope m given by Equation 1. Since, by Definition 2, this is the same as the derivative /'(«), we can now say the following. The tangent line to y — f(x) at (o,/(a)) is the line through (a,/(a)) whose slope is equal to /'(a), the derivative of / at a. Thus, the geometric interpretation of a derivative [as defined by either (2) or (3)] is as shown in Figure 1.

FIGURE 1 Geometric interpretation of the derivative

= slope of tangent at P = slope of curve at P

= slope of tangent at P = slope of curve at P

If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y = f(x) at the point («,/(«)): y-/(a)=/'(fl)U-fl)

Plate 5.2(1) Interpretation of the Derivative as the Slope of a Tangent (Stewart, 1999: 130) interpersonal meaning is contracted and direct. Once the viewer's attention is focused on the relevant parts of the mathematics visual image, the representational meaning is the major function of the mathematical visual display. The experiential meaning is concerned with the Episodes and relevant Parts of the Figures, and logical meaning is largely concerned with spatio-temporal relations. Computer graphics are contributing to the development of new systems of meaning; for example, Colour, Shading, Brightness and Hue. The systems for representational, interpersonal and compositional meanings through computer graphics are included in the grammar and discourse systems in Table 5.2(2). In what follows, the metafunctionally based discourse and grammatical systems for visual images in mathematics are discussed through the analysis of Plate 5.2(1). In the final section, the computer graphics displayed in Plate 5.7(lc-d) are discussed. However, the framework is not complete, and further research is needed to document the systems through which meaning in mathematical images is made. Table 5.2(1) and Guo's (2004a, 2004b) SFGs for mathematical statistical graphs and biological schematic drawings are first steps in this direction.

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5.3 Interpersonally Orientating the Viewer

The choices from the systems for interpersonal function in a visual image determine how the viewer engages with the Work. O'Toole (1994: 7), for example, carefully traces how the viewer becomes involved with Botticelli's Primavera through choices of Rhythm, Gaze, Frame, Light and Perspective: 'The painting has a gentle, undulating rhythm which is in harmony with the graceful gestures and stance of the figures, with the flow of draped clothing, with the placing of fruit, flowers and foliage and with the easily blending colours.' In the painting, Venus engages the viewer 'directly with her eyes' and 'even signals a greeting or benediction' with 'the gesture of her right hand, the tilt of her head, and the poise of her body' (1994: 8). O'Toole discusses how the Compositional forces at work, such as the use of concentric Frames and Colour, also contribute to the viewer's engagement with Venus. Further to this, the position of other Figures demarcates the centrality of Venus. As O'Toole (1994: 11) comments, if the painting is viewed Episode by Episode, there is an interplay of modalities with the 'Rhythm changing from episode to episode as our eye moves from right to left across the canvas'. The result is an exquisite unfolding of Botticelli's masterpiece. Similarly, the viewer's gaze is also directed to certain aspects of mathematical visual images. The result is not a gentle rhythmic engagement through the subtle use of Rhythm, Gaze, Frame, Light and Perspective, however. Rather, the choices for interpersonal meaning function directly so that the viewer immediately engages with the significant aspects of the representational meaning of the graph or diagram. Botticelli's careful choices for individualization of the Figures through Stylization may be contrasted to the uniformity of the mathematical visual images through Conventionalization. This feature is functional in mathematical discourse as it enables experienced viewers to immediately apprehend representational meaning of a display and it also lowers the likelihood of misinterpretation. The interpersonal meaning of traditional mathematical visual images is best examined through an analysis of Plate 5.2(1) using the SF framework provided in Table 5.2(2). The geometrical interpretation of the derivative of the curve y = f(x) as the slope of tangent line at the point P(a,f(a)} is displayed in Plate 5.2(1). There are two graphs for the two equivalent algebraic forms of the derivative: (a)

and

In

what follows, Graph (a) is analysed for interpersonal meaning. The interpersonal meaning arising from the Sequence of Graphs for (a) and (b) is also examined. Interpersonally, at the rank of Work the viewer is presented with a Metaphorical Narrative where the Figure of the Line intersects the Figure of the Curve in Graph (a). The Episode or 'plot' attracts the viewer's attention through the Prominence of the Individual Figures and the Prominence of

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the Interplay; that is, the straight line is 'marked' through the Colour red (not displayed here) and attention is also drawn through the choices for Slope, Line Width and Line Solidarity. The viewer's attention is also drawn to the Curve through the dynamic aspect of the Curvature, particularly the 'flourish' at the tail of the curve positioned around the origin of the axes which is marked with the Label '0'. Closer inspection reveals that the curve is Labelled y—f(x). This Label attracts attention through the curved arrow which is used to point directly to the Curve. The viewer is also drawn to the critical aspect of the Metaphorical Narrative, the point of intersection between the line and the curve which is marked by a dot and Labelled point P in italicized font. The Episode involving the Line and Curve appears as dynamic because the Interplay is framed by the set of horizontal and vertical axes, respectively Labelled x and y. After the initial impact, the Rhythm achieved through the Slope of the Line and the Curvature of the Curve creates a reading path which initially tends from left to right. A series of 'minor' Episodes are marked by the dotted vertical Lines positioned above the values a and a + h on the x axis in Graph (a). These dotted lines intersect the Curve, and a triangle is formed by the horizontal Line Labelled h and the hypotenuse extending from point P. The minor Episodes lack the Prominence of the Interplay of the major Episode as the Colour of the lines is yellow (not displayed here) and the Line Solidarity is dotted rather than solid. In addition, the hypotenuse of the triangle is marked by the Colour blue (not displayed here), which is not prominent because of its Position and close proximity to the Curve. These minor Episodes function as rankshifted instances of Circumstance in Graph (a). The x and y axes provide the visual context for the major and minor Episodes, and attention is drawn to certain parts of the graph through the Title, Labelling and Colour of the Figures and Parts. Apart from this, the context is provided by the symbolic and linguistic Caption, and the surrounding linguistic text. Captions and Titles of Graphs and Diagrams are typically placed in a prominent position and usually are marked through Size and Font. These Labels connect the Episodes, Figures and Parts to the symbolic and linguistic descriptions appearing in the Captions for Graphs (a) and (b). The Style of Production indicates that the graph has been professionally produced through a specialized software application. At the rank of Part, Stylization and Conventionalization are standardized through the means of production. The graph appears to be contextually independent as background information and 'noise' are erased. The graph appears as an abstraction which consists of generalized participants involved in a metaphorically abstract interaction. The Modality and the degree of Idealization, Abstraction and Quantification are high, leading to a maximal truth-value. In other words, the Graph appears to have a high degree of certainty as to the correctness of the representational meaning it encodes. The Modality value is the cultural value assigned to those choices made from the available systems for interpersonal meaning in mathematical graphs.

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At the rank of discourse semantics, the Sequence of Graphs (a) and (b) in Plate 5.2(1) involves repetition. The graphs are identical with the exception of the Labels, which change from a and a + h to a and x, and h and/( a + K) —f(a) to x— aandf(x) -/(«)• In this way, a minor Episode, which appears as Circumstance in Graph (a), is given prominence through Inter-Visual Relations while the nature of interpersonal meaning is reinforced with the exact repetition of the major Episode. The reason for marking the prominence of the Circumstance in Graphs (a) and (b) is explained in Section 5.3, which is concerned with experiential meaning. In statistical graphs and mathematical diagrams, the viewer's attention is similarly drawn to salient parts of the visual display through Captions, Labelling, Size and features such as Colour, Slope, Line Width, Shading and Perspective. The viewer's gaze is also drawn to certain points of the diagram through Perspective. The degree of Abstraction and the level of Intricacy of the figure interacts modally with the Style of Production. In line and bar graphs, the viewer is engaged directly through certain features such as Shading, Patterns and Colours. Parts may be labelled and the Slope may be exaggerated. The 'art' of presenting information lies in making salient particular Parts of the Graph so that the viewer's gaze is directed to those parts of the display. Misleading graphs are those which present information in such a fashion that an unwarranted Prominence is attached to particular dimensions of the display. Prominence is achieved through specific choices for interpersonal, experiential and/or composition meaning. The visual display of information in fields such as advertising, newspaper discourse, politics and so forth is worthy of further investigation. From the analysis of choice from systems for interpersonal meaning in Plate 5.2(1), three features of mathematical visual images become evident. First, the viewer is explicitly directed to the relevant parts of the visual display through the nature of the interpersonal choices at each rank. Interpersonal meaning is not a delicate balance of a variety of unobtrusive strategies working in harmony as for example, in O'Toole's (1994) descriptions of the Primavera. The important Episodes, Figures and Parts are explicitly marked through choices for Labels, Position, Colour, Framing, Line Width, Line Solidarity and so forth. Second, the Modality or the truth-value of the visual image is high, but this is not because mathematical visual images faithfully depict material 'reality'. While mathematical visual images relate to our perceptual understanding of reality, there is not the same degree of correspondence as portrayed in photographs, for example. Rather the high level of certainty attached to mathematical and scientific representations has been culturally assigned: 'visual modality rests on culturally and historically determined standards of what is real and what is not, and not on the objective correspondence of the visual image to a reality defined independently of it' (Kress and van Leeuwen, 1996: 168). The perfection and exactness of the visual displays, the lack of contextual information, the Style of Production and the metaphorical and abstract nature of the Episodes, Figures and

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their Parts mean that the certainty of the mathematical and scientific visual images is difficult to counter. The interpersonal meaning of the mathematical visual images in fact replicates that found in mathematical symbolism, which similarly selects for a high value of Modality. The appearance of certainty in mathematical discourse arises from the nature of the selections for interpersonal and experiential meaning (the narrow range of participant, process and circumstance selections) and the truth-value which has been culturally accorded to these selections. These patterns of meaning are reinforced across the linguistic, symbolic and visual parts of mathematical discourse. Third, the nature of interpersonal engagement with mathematical visual images is sharp and direct, and the social relations which are consequently established between the writer/producer and the viewer are unequal. The writer directs the reader to certain parts of the graph so that the experiential and logical meanings become the dominant metafunctions once the short, sharp interpersonal impact subsides. The discourse creates a high level of certainty with respect to the representational meaning subsequently realized. Moving from the interpersonal orientation of mathematical visual images, the nature of the experiential meaning is explored with respect to Graphs (a) and (b) in Plate 5.2(1). 5.4 Visual Construction of Experiential Meaning

Visual images are typically conceived as dealing with the 'concrete real world' rather than the 'abstract world' of the symbolism (for example, Galison, 2002). The reason for this view is that graphs replicate our perceptual experience of the world. Mathematical visual images are, however, concerned with particular forms of experiential meaning. For example, the abstract graphs of functions and transformations such as those shown in Plate 5.2(1) display patterns of relations as lines, curves and threedimensional objects. Mathematical visual images are also concerned with perceptual reality in the form of geometrical displays, and mathematical symbolic reality in the form of statistical data graphs, Venn diagrams and other forms of diagrams which have no counterpart in the real world. While the form of the display, the curves, lines and three-dimensional shapes are intuitively accessible, it becomes evident in what follows that an understanding of the grammar of mathematical visual images is necessary for a reading of the experiential meaning realized by images. This is because the grammar of mathematical visual images is designed to function with the grammars of mathematical symbolism and language. Attention is directed to salient features of the mathematical visual images and the Titles, Labels and Captions. Once the immediate interpersonal impact subsides, representational meaning takes over. The viewer is engaged with the experiential and logical meanings of the graph or diagram which are pre-empted by the selections for interpersonal meaning. The conventionality of the visual display allows experienced viewers to

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access the representational meaning of graphs and diagrams at a glance. In what follows, the experiential meaning of Graphs (a) and (b) in Plate 5.2(1) is analysed. At the rank of Work, Graphs (a) and (b) each consist of an Episode involving two Figures, the Curve y = f(x) and the Line which depicts the Slope of the Curve at point P. The Episode is an Interplay of Relations between the two Figures. The Line appears in red (in the original version) which compensates for the lack of Labelling. The point of intersection arising from the interaction of the two Figures is marked with P. This Part of the Figures is prominent through the marking of the point with a dot and the italicized Label P. In Graph (a), the minor Episodes in the form of Circumstance are emphasized with the Labelling of points a and a + h on the x-axis and the distances h and/(a+ h) -/(«) which are marked by curly brackets. The hypotenuse of the right triangle is drawn to form a triangle where the lengths of the other two sides are marked as the distance between two points. The graph is partly accessible as it replicates our perceptual understanding of spatial relations; for example, the Line, Curve, and the lengths of the sides of the triangle as the distances between two points in the right triangle formed by the line segments. However, the experiential meaning of abstract Graphs (a) and (b) in Plate 5.2(1) requires a knowledge of the grammar of mathematical visual display and the grammar of mathematical symbolism and the grammar of language. The Captions, Titles and the surrounding co-text, for example, are linguistic and symbolic. To understand what the graph is depicting, the viewer needs to know the definition of the derivative which is given in the symbolic form: (a) and (b)

The symbolic

forms are re-instated linguistically in Plate 5.2(1) as the 'slope of tangent at P' and 'slope of curve at P\ Figure 1 Caption reads 'Geometric interpretation of the derivative'. The Labels on the Graphs (a) and (b) are symbolic. Mathematical visual images are multisemiotic texts. The viewer must also understand the grammar of mathematical abstract graphs. In this case, the Curve in Graph (a) represents the relation obtained by the mapping/: x—*f(x) where y =f(x). The Curve is the set of ordered pairs (x,f(x)) such that/(x) is the value of the function for x. Each value of x corresponds to a value of f(x) which is y. Strictly speaking, the x and y axes, the Curve, Line and the Axes in Plate 5.2(1) should also have Arrows to indicate ongoing continuity. The function f(x) in Plate 5.2(1) remains in generalized form, rather than a specific case (for example,/(x) = x3 + 1 or y = x" + 1). The general values /(a) and /(a + h) are indicated by the dotted vertical line segments which are drawn from a and a + h to intersect the curve y=f(x). In Graph (b), a + h has been replaced by the general value x, and this has repercussions for the lengths of the sides of the triangles which are conceptualized as the distance between two points. The relations captured symbolically appear in the form of a Curve and a

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Line in Graph (a). While there is an element of dynamism associated with the Curve and the Line, these visual displays may nonetheless appear as fixed entities, the Curve and the Line which intersect at point P. The symbolic description y = f(x) is dynamic; it represents the mapping/: x —> y, where f(x) is realized through configurations of the rankshifted Operative processes and participants,

The complex of the

relations captured by the symbolism is displayed in the form of an entity, the Curve. Despite appearances, the curve y = f(x) is dynamic as a set of relations which unfold temporally and spatially, and this dynamism is Framed by the axes x and y. Moreover, Graphs (a) and (b) in Plate 5.2(1) are frozen representations of the dynamic processes h —> 0 and x —> a. That is, the Graphs depict discrete instances of time for particular values of h and x. Further to this, the tangent line at point P represents the slope of the curve at the point when these limits h —> 0 and x —> a are reached. In essence, there are Multiple Time-Frames juxtaposed in Graphs (a) and (b) in Plate 5.2(1). These time frames are: 1 y—f(x) unfolds as a set of mappings/: x—*y. 2 The processes h —> 0 and x —> a unfold. 3 The limit is reached at point P. 4 The Line is the tangent line at point P. Variations in temporality, realized symbolically as h —> 0 and x —> a in , are given a spatial interpretation in Graphs (a) and (b). Temporal relations are visualized at instances of time in terms of decreasing distances. Continuity is made discrete in the form of spatial relations at different points of time. Once the limit at point P is reached, the tangent line may be drawn. The Interplay of Episodes and Circumstance result in Multiple Time-Frames with the Temporal Unfolding being realized through Spatiality. The repercussions in terms of logical meaning are discussed in Section 5.5. The interpretation of experiential meaning in abstract graphs relies on a knowledge of the grammar of mathematical visual display, symbolic notation and language, and the intersemiotic relations between the three resources. In the case of Graphs (a) and (b), this includes the notion of a graph as a set of points, algebraic functional notation, the notion of Cartesian co-ordinates whereby each point of the curve represents an ordered pair (x, f(x)), the notion of a function as a mapping from x —» f(x), the graph as the set of these mappings, the geometrical interpretation off(x) values, the algebraic definition of the derivative as a limit, and the notion of the tangent as a line which intersects the curve at one point only. In addition, the reader must recognize that the graphical display is a partial representation of the pattern of covariation described by f(x), and the graph shows discrete instances of time as the limits h —> 0 and x —> a are

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approached. The Inter-Visual Relations between Graphs (a) and (b) repeat the Interplay of Episodes with the exception that Circumstance changes from a and a + h to a and x. In this way, Circumstance is given prominence. The reason for Graph (b) becomes apparent in the linguistic text: Graph (b) establishes the equation of the tangent using the 'point-slope form of the equation of a line' (Stewart, 1999: 130). Although visual display in mathematics is more intuitive than the symbolic descriptions, the experiential meaning encoded within the visual display is complex. Plate 5.2(1) involves Multiple Time-Frames where the temporal unfolding of the processes is given a spatial interpretation at different instances of time. To access this experiential meaning, the grammar of the graph and the relations in the symbolic and linguistic text must be understood. The experiential meaning also depends on previously established mathematical results which have been derived symbolically and visually. The Inter-Visual Relations are significant in accessing the experiential meaning of Plate 5.2(1), which includes the Conceptual Development of the derivative of a function. The meaning of visual images replicates that found in mathematical symbolism. That is, reasoning depends on previously established results which are implicit. In what follows, the logical meaning of mathematical visual images is discussed in relation to Graphs (a) and (b) in Plate 5.2(1). 5.5 Reasoning through Mathematical Visual Images

Halliday (1994: 328-329) includes a linguistic category for spatial relations in terms of 'extent' and 'place'; for example, 'here', 'there', 'as far as' and 'wherever'. This category, corresponding to the semantics of spatiality, also functions as a rhetorical organizing device rather than a direct reference to actual space or place; for example, 'as far as I can see, that is not possible'. Martin's (1992: 179) classification for logical relations in language, however, does not include a category for spatial relations. The types of relations are: Additive (addition, alternation), Comparative (similarity, contrast), Temporal (simultaneous, successive) and Consequential (purpose, condition, consequence, concession, manner). It becomes apparent that logical meaning based on the semantics of spatiality is only minimally developed in language. While the grammar is richer in terms of temporal relations in the form of structural conjunctions and cohesive conjunctive adjuncts (for example, 'and', 'next', 'now', 'then' and 'simultaneously'), language did not develop the same potentiality for spatial logical relations, presumably because visual images are functional in this respect. Our perceptual apparatus permits 'logical deductions' based on spatiality to be performed through visual means rather than depending upon formalized linguistic and symbolic selections. The potential for logical meaning through visual images is discussed below. Graphs (a) and (b) in Plate 5.2(1) represent an instance of time in an unfolding dynamic process as the limits h —> 0 and x —> a are approached.

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This is only ascertainable from the linguistic Caption which explains that the derivative is given by (a)

and (b)

The symbolism captures the process of the derivative as the result of a limiting process. The visual image, however, gives a spatial interpretation of that limiting relation at an instance of time. Graphs (a) and (b) also depict the final result of that limiting process: the tangent at point P. Therefore, the temporal relations have a spatial dimension in Graphs (a) and (b). Consider the temporal frameworks represented in Graph (a): y =f(x) as a mapping/: x —> y, the limit as h —» 0, the reaching of that limit at point P, and the slope of the curve at the point P which is represented by the tangent line. The logical inference is that as h —> 0, a + h moves towards the value a over time with the result that the spatial distance h approaches zero. Thus the distance between the two points, displayed as lengths of sides of the triangle, approaches zero over time. The graph thus contains a visualspatial interpretation of relations over time. Furthermore, once the limit is reached at point P, the tangent line as the slope of the curve at that point appears. The visual display encodes a multidimensional Time-Frame. In other words, the graph is used to reason with. Graph (a) encodes logical meaning in terms of temporal relations which are expressed visually as spatial relations. The nature of this relationship permits logical meaning in Graph (a) in the form of elaborative type relations ('in other words' becomes 'in other pictures'); additive type relations ('this and this'); and causal relations (cause in terms of reason 'so'). The logical relation across Graphs (a) and (b) takes the form of causal relations 'therefore' to establish the equation of the tangent line in point-slope form. The dynamic unfolding of the process, as afforded by computer graphics, makes the ability to reason perceptually even more encompassing. The dynamic displays of the digital medium encode temporal relations through an unfolding spatial form, and also permit the manipulation of visual patterns in ways previously unimaginable. The potential for logical reasoning using computer graphics is explored in Section 5.7. In the next section, compositional meaning in terms of the organization of experiential, logical and interpersonal meaning in mathematical visual images is considered. 5.6 Compositional Meaning and Conventionalized Styles of Organization

The textual organization of the mathematical visual images is conventionalized in order to permit the viewer to engage immediately with the experiential and logical meaning which is encoded in precise and exact form. O'Toole (1994: 22) explains: 'decisions [in paintings] about the arrangement of forms within the pictorial space, about line and rhythm and colour relationships, have been made by the artist in order to convey more effect-

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ively and more memorably the represented subject and to make for a more dynamic modal relation with the viewer'. With mathematical visual images, the interpersonal meaning is direct, and this engagement is aided by the compositional arrangement of the Work, Episodes, Figures and Parts. As with mathematical symbolism, the narrow range of interpersonal meanings is accompanied by unambiguously encoded experiential meanings in condensed format in mathematical visual images. In what follows, the compositional meaning of Graphs (a) and (b) in Plate 5.2(1) is discussed. At the rank of Work, Gestalt is the term used for the complex relations between a whole visual image and its parts: ' "Gestalt theory" claims that we always have an overall perception of forms and objects and that when we focus on their parts we perceive them in relation to the whole' (O'Toole, 1994: 23). The Work is the Graphs (a) and (b), where the Episode is Framed by the x and y axes. These axes contribute to the stability of the visual display. The Curve and Lines appear as dynamic elements in the visual display because they are diagonally positioned, thus attracting the viewer's attention. Diagonal elements are similarly formed by perspective in geometrical representations, and diagonal alignments visually trace trends in the data and in bar and line graphs. Graphs (a) and (b) are organized through Labels which form Cohesive links to the main body of the text. The Curve and the Line are perfectly Positioned against the backdrop of the x and y axes, and Balance is achieved through these spatial positions and the curvature of the Curve. At the rank of Episode, the Interplay is explicitly marked through the Label for point P. At the rank of Figure and Parts, Labels are attached to the major participants and circumstance as discussed above. There is maximum Cohesion arising from the explicit ordering of each Part of Graphs (a) and (b). Inter-Visual Relations are established through the spatial position and organization of Graphs (a) and (b). As may be seen in Plate 5.2(1), the two graphs are placed next to each other, and they involve direct repetition of the Metaphorical Narrative. Inter-Visual Relations later establish the Line in Graphs (a) and (b) (which represents the derivative as the tangent of the curve at point P) as the instantaneous rate of change (Stewart, 1999: 132). The Conventionalization of visual images in mathematics is significant in establishing such Inter-Visual Relations. The Slope of the Line is conventionalized to correspond to the semiotics of physical/psychological perception. The steeper the Slope of the Line, the greater the rate of change, and thus the gradient of the tangent. A small change in x produces a large positive or negative change in y. That is, the derivative/' (x) as Slope of the tangent Line as the instantaneous rate of change at point P is compositionally organized to correspond to perceptual reality. At the ranks of Figure and Part, there is an exact relationship between compositional and experiential meaning of the Figures in abstract graphs. In Graphs (a) and (b), the experiential meaning of each point of the Curve corresponds exactly to its spatial location with respect to the x and y axes.

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Compositionally, Spatial Position has been connected to experiential meaning in terms of sets of co-ordinates; that is, the placement of a point is exactly determined by the x co-ordinate value and the y co-ordinate value. The resulting Curve or Line is the set of all such 'points' in the mappings /: x —> y. This means that Positioning, Size and Shape of the Figure are determined by the Scale selection on the axes. Choices of compositional meaning for the organization of the mathematical visual images function alongside selections for interpersonal meaning to direct the viewer to the significant parts of the visual text. In terms of the display stratum, compositional meaning is aided by Perspective, Colour and Spatial Arrangement. Experiential meaning is efficiently encoded, and the patterns are immediately ascertained. Often the visual image is consulted before the viewer attends to the symbolic and linguistic parts of the mathematics text. The analysis of interpersonal, experiential, logical and compositional meaning in mathematical visual images is incomplete as only abstract graphs are considered. Mathematical visual images also include numerous other genres in the forms of statistical graphs, diagrams and other forms of visual display in the form of computer-generated dynamic images. In what follows, the general nature of images realized through computer graphics is discussed, and the implications for a changing role of visualization in mathematics through the medium of computer technology are considered. 5.7 Computer Graphics and the New Image of Mathematics

The influence of computers is such that they have given rise to 'a new world view which regards the physical world not as a set of geometrical harmonies, nor as a machine, but as a computational process' (Davies, 1990: 23—24). One outstanding feature of the shift to computation is the ability of computers to generate, represent and manipulate the numerical results as dynamic visual patterns which unfold over time. Davis (1974: 115-116) predicted the resurgence of the visual image in mathematics through the development of computer graphics, and this is indeed proving to be the case. The visual image plays an increasingly important role in different branches of mathematics, as evidenced in the modelling of non-linear dynamical systems. The impact of increased computational ability is discussed in relation to the revolution which is taking place through computer graphics. Following Foley et al. (1990), computer graphics is denned as ' "the pictorial synthesis of real or imaginary objects from their computer based models" and covers working areas like rendering, scientific visualization, animation, graphics in documents, or interactive user interfaces' (GroB, 1994: 2). The visualization process consists of 'the transformation of numerical data from experiments or simulation [via mathematical models] into visual information' (Grave and Le Lous, 1994: 12). The other relevant process is 'image processing', which deals with 'the management, coding,

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and manipulation of images, the analysis of scenes, and the reconstruction of 3D objects from their 2D projectional presentations' (GroB, 1994: 2). Software programs encode the data displayed visually on the computer screen. The increasing sophistication of computer graphics may be appreciated through the images from the 1970s to the present which are displayed in Plates 5.7 (la-d). Computer graphics have developed from pictures consisting of (i) predominately lines and (ii) patterns arising from arrays of dots to (iii) coloured kaleidoscope images of complex patterns and (iv) displays that replicate the detailed intricacy of photographs. For example, the coloured fractal patterns in Plate 5.7 (Ic) (reproduced in black and white here) represent an entirely new 'image' of Newton's method for finding the solution to equations, and the coloured images and the textured surfaces of the three-dimensional graphs in Plate 5.7 (Id) (reproduced in black and white) illustrate the detail now possible in computer-generated images. The increased applications of computer graphics in applied mathematics and science for the interpretation of complex data sets relate to human capabilities of seeing visual patterns: 'Because visual analysis techniques are particularly well suited to the human cognitive capabilities, more emphasis has been placed on visual analysis tools for understanding computer simulations of complex phenomena' (Watson and Walatka, 1994: 7). Humans cannot process the information at the same rate if presented with the symbolic output generated by supercomputer simulations or high-powered scientific instruments. As Colonna (1994: 184) explains: Vision is the most highly developed of our human senses for reception, isolation and understanding of information about our environment. Vision provides a global reception of coloured shapes against a changing, moving, and noise-filled background. The idea of using the eye as the main tool in the analysis of numerical results is therefore quite natural.

Computer graphics are increasingly being used for a range of functions; for example, the visualization of large data sets (for example, in applied mathematics, geology, meteorology and engineering), the creation of threedimensional objects, the construction of view-dependent visualizations, multidimensional images of motion, and visualizations and reconstructions in medicine and biology (for example, see Moorhead et al., 2002). Mathematical ideas such as interpolation, approximation with polynomials, fractals and so forth have also rapidly moved into highly developed computer systems for military and industrial applications. Computer graphics are leading to new insights of complex mathematical problems (for example, topological problems). However, the symbolic still presides over the visual in higher dimensional mathematical theories. In addition, computer graphics have resulted in the growth of computer art and animated film, especially in the field of three-dimensional images (Danaher, 2001; Davis, 1974): 'Over the last 20 years, all phases of film production have been

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Plate 5.7(la) Stills from a computer-made movie: wrapping a rectangle to form a torus (Courtesy T. Banchoff and C. M. Strauss) (Davis, 1974: 126)

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Plate 5.7(lb) MATLAB graphics, circa 1985 (courtesy of Philip J. Davis)

changed by computer technology' especially postproduction which has 'been transformed by computer-generated imagery ("CGI")' (Bordell and Thompson, 2001:26). Davis (1974) suggested that 'visual theorems', where the traditional form of proof is not required, will become acceptable in mathematics: 'A figure, together with its rule of generation, is automatically and without further ado a definition, theorem and proof of "the perceived type" ' (ibid.: 122). Davis (2003) sees that this trend has yet to develop to its full potential. Whether what is suggested visually requires symbolic proof 'seems to depend on the particular mathematical culture within which the pictures have been derived' (ibid.). Colonna (1994) suggests that the notion of experiment rather than proof is more relevant in relation to computer graphics. Images generated through software programs are seen as tools for 'virtual experimentation' in a computerized environment. The traditional approach based on 'real experimentation' involves finding analytical or symbolic solutions to the model equations constructed from experimental data in order to explain the behaviour of physical systems. However, simple chaotic systems, for example, are difficult to capture analytically. The computerized numerical approach entails generating and visualizing numerical

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Plate 5.7(lc) The Complex Boundaries Of Newton's Method. (Gleick, 1987: insert)

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Plate 5.7(ld) Graphical and Diagrammatic Display of Patterns (Berge at al, 2003: 194)

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solutions, rather than finding an analytical solution to the model equations: 'Virtual experimentation, therefore, assumes that the underlying analytical models of the simulation are correct. . . and explores the range of behaviours produced by that model' (ibid.: 183-184). The researcher can change the parameters or request different views of the numerical results to explore the behaviour of the system: 'The researcher, the numerical calculations, and the displayed images form a feedback loop through which the complex behaviour of the simulated system is explored' (ibid.: 184). Colonna (1994) sees 'picture synthesis' as a scientific tool where coloured global patterns, rather than traditional point-by-point descriptions, may be generated: 'Complex forms will become distinct and the scientist may be able to ascertain a hidden order in the numerical results. As others have pointed out: Scientific visualization is the art of making the unseen visible1 (ibid.: 184). This statement echoes Newton's efforts at making the invisible 'visible' through mathematical descriptions. The semiotic in question is now the visual rather than the symbolic code. Through juxtaposition, connection, transformation and various other forms of manipulation, the various components 'come together to create a useful whole, the scientist, numerical calculation, and picture synthesis all work together to form a scientific instrument. . .' (ibid.: 184-185). The new functions of computer graphics arises from the expanded meaning potential afforded by these forms of visual image and the ease with which patterns can be generated, rearranged and combined. While static forms of graphs and diagrams in handwritten and print format have been time-consuming to produce, computer-generated visual images now encode dynamic representations with minimal effort. New systems of meaning such as Colour Saturation, Hue, Shading and Brightness are playing an important role in computer-generated images (Danaher, 2001; Levkowitz, 1997). The visual image has thus evolved into a dynamic display that can be easily manipulated in the same way that symbolic mathematics developed to be the semiotic that could be rearranged in print format. However, visualizations of the continuous patterns of relations have the advantage that they encode spatial and temporal dimensions. In addition, the patterns are generated with minimal effort. The development of mathematics is tied to the available technology, which has historically been limited to the pen, paper, the printing press and three-dimensional mathematical models. Computer technology extends the meaning potential of mathematics in the digitalized medium. New scientific methods arising from computer graphics may extend beyond those suggested by Colonna (1994: 191). Traditionally data is first filtered through the lens of numerical quantification (through, for example, the experiment). Following this, the data is distilled into generalized mathematical models which are solved and visualized. This led to a de-contextualization and reduction of the complexity of the phenomenon under study: 'The strength and novelty of seventeenth century science,

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both theoretical and experimental, was in its capacity to take things out of context and analyze their relations in ideal isolation' (Funkenstein, 1986: 75). However, the complexity of data can more easily be maintained in a computerized 'virtual' world, and reasoning can take place visually through the dimensions of space and time, rather being based on the ordering of space and time through language and symbolism. The potential for logical meaning in digitalized images is further considered below. The significant aspect of the digital computerization process is that visual images may be reconfigured in forms which unfold dynamically as a sequence of steps over time rather than a static display where instances of time are displayed discretely. Mathematical symbolism developed grammatical strategies to ensure that logical sequences could unfold progressively with ease. Now visual images in the form of computer graphics possess an added potential for sequential processing. Spatial and temporal dynamics are afforded through digital encoding of information in a computerized environment. A graph can be seen as an unfolding of relations through time and space on a computer screen. In addition, mathematical visual displays may be synthesized, rearranged and thus used to establish mathematical results in much the same way that symbolism evolved to do. The difference is that computer graphics combine spatiality and temporality. The mathematician is no longer working as someone deprived of the use of sight. The dependence on the grammar of mathematical symbolism (which evolved from the grammatical systems of language) is supplemented with visual forms of semiosis, which may develop further grammatical systems for encoding meaning. As visual images increasingly take their place alongside mathematical symbolism, this semiotic resource may be seen to offer more than an intuitive understanding of the phenomena and a means for experimentation and synthesis. Computer graphics may evolve formal systems for reasoning. The intuition arises naturally as the visual image relates to our perceptual understanding of the world. While this contribution is extremely productive for insights into the nature of the mathematics problem, language and mathematical symbolism have formalized 'intuition' through the development of systems for logical meaning. The linguistic and symbolic function to impose order, and one major contribution to that order, the part that is significant in mathematics, is logical reasoning. Language and mathematical symbolism possess systems for logical meaning in the form of elaboration-type relations (re-statement in the form of apposition or clarification), extension-type relations (additative- and variation-type relations) and enhancement-type relations (predominantly causal-conditional- and spatio-temporal-type relations). In order to make logical connections in what is seen to be a 'valid proof in mathematics, reasoning is established in a step-by-step fashion through choices from the available linguistic and symbolic systems (for example, 'if this', 'then this'). A sequence is seen to be necessary for logical reasoning. Indeed, a mathematical proof is considered to be 'a sequence of steps or statements

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that establish the truth of a proposition' (Wilkes, 1986: 1225). Mathematical proof is thus equated to what can be achieved using language and mathematical symbolism. Our perceptual apparatus lets us perceive patterns, but temporal- and cause-effect-type relations have been formalized using language and mathematical symbolism. With the advent of computer graphics, however, such step-by-step reasoning may be possible using visual images which have the potential to unfold sequentially over time. It remains to be seen if computer graphics will lead to the development of formalized systems for logical meaning in the grammar of the visual semiotic. The development of formal systems for logical meaning in visual images may not be so far-fetched. Galison (2002: 321-322), for instance, points out that the divide between the analytic and the visual is not so marked in a computerized environment: [i] t may be that the most significant development in the laboratory of the last fifty years has been the fusion of pictures and numbers into the production of the manipulable image. Controllable digitized images were built by computers from statistics and formed into pictorial renditions of non-visible worlds . . . After tracking the endless drive back and forth between images and data, it becomes clear that the powerful drive towards images and the equally forceful pressure towards analysis never completely stabilized scientific practice. Quite the contrary, neither the 'pictorialrepresentative' nor the 'analytical-logical' exist as fixed positions. Instead . . . we see that the image itself is constantly in the process of fragmenting and re-configuring . . . now, ever more intensively, the routinization of analog-to-digital and digital-to-analog conversions have made the flickering exchanges routine: image to non-image to image . . . every-day science propels this incessant oscillation: 'Images scatter into data, data gather into images.'

The continuous oscillation between visual and symbolic forms of display for digitalized data described by Galison may contribute to the development of grammatical systems for logical meaning in visual images. If so, the visual semiotic will possess an important advantage over the traditional semiotic resources of language and mathematical symbolism: computer graphics can display continuous spatial-temporal patterns of variation. The image is no longer the traditional frozen snapshot such as those displayed in the visualization of the derivative as the tangent to the slope of line in Plate 5.2(1). Traditional static visual images (such as those found in mathematics books) are limited compared to the potential of the dynamic and interactive complexity of computer graphics. The impact of this change remains to be seen, but one may predict with some degree of confidence that the functions of graphical visual images will increase, as evidenced by the rapid development of software programs and increased use of computer visualization across many disciplines and fields of study. Mathematical visual images are becoming increasingly functional alongside symbolic descriptions of patterns of relations. 'Just as the microscope showed us the "infinitely small" and the telescope showed us the "infinitely large", so the computer will enable us to regard

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our world in a new, richer way' (Colonna, 1994: 191). The 'newer richer' way suggested by Colonna arises in computerized environments through the capacity to model, experiment, manipulate and synthesize four dimensions of experience: three-dimensional spatial world and one-dimensional temporal world to create a dynamic rendition of real and imagined worlds. This recreation takes place on the two-dimensional computer screen. In addition, virtual reality has the potential to incorporate other dimensions for semiotic input, such as touch, smell and taste. A variety of software applications which can recreate perceived reality in space and time are already available as documented by Danaher (2001) in the recent book Digital 3d Design. The modelling process in a computerized environment permits noise, complexity and context (which are excluded in traditional approaches) to be incorporated for a more comprehensive understanding of 'reality'. Indeed 'reality' undergoes a transformation from 'real', that which can be perceived, to 'abstract', that which can be imagined. Both real and imagined realities can now be modelled and experienced. Computerized environments have the potential to move beyond that perceivable and conceivable by the human senses. The implications of this semiotic reordering of the world remain to be seen. This will depend on the functions which are assigned to computerized medium, and the purposes for which the reconstructions of reality are employed. Typically, advances made for the purposes of military concerns feed back into useful applications for other fields of human endeavour. Chapters 6 and 7 are concerned with intersemiosis, the process where new meanings arise integratively through transitions from one semiotic resource to another. The functionality of mathematical discourse does not only stem from accessing the three individual meaning potentials of language, mathematical symbolism and visual images. Rather, the functionality of mathematics also arises from intersemiosis between the three resources and the metaphorical construals which take place intersemiotically. In what follows, the nature of intersemiotic mechanisms and systems, and the resultant meaning expansions across language, visual images and mathematical symbolism are investigated.

6 Intersemiosis: Meaning Across Language, Visual Images and Symbolism

6.1 The Semantic Circuit in Mathematics

Language, mathematical symbolism and visual images in the form of graphs, diagrams and, more recently, computer graphics have together created a discourse which has transformed the face of the world, if technology is considered to be the direct result of mathematics and science. Mathematical discourse is effective because the systems of meaning for language, symbolism and visual images are integrated in such a way that the behaviour of physical systems may be described. Traditional mathematical descriptions break down, however, when the behaviour of the physical systems become non-linear and chaotic. In such cases, the entire system must be understood rather than the constituent parts. In practice, this means that the variety of constraints and conditions must be taken into account in order to effectively model and predict the behaviour of the system. These calculations and descriptions take place in digitalized form through the use of computers. The functions of language, the symbolism and the visual image may be summarized as follows. Patterns of relations are encoded and rearranged symbolically for the solution to the problem. The symbolism has limited functionality, however, so that language functions as the meta-discourse to contextualize the problem, to explain the activity sequence which is undertaken for the solution to the mathematics problem, and to discuss the implications of the results which are established. Visual images in the form of abstract and statistical graphs, geometrical diagrams, and other types of diagrams and forms of visual display, such as those generated through computer graphics, show the relations in a spatio-temporal format which involve multi-dimensional time-frames. As discussed in Chapter 5, the traditional role assigned to the mathematical visual image is changing with the increasing power of computers to generate and manipulate complex dynamic visual patterns. The metafunctionally based SFGs for language, mathematical symbolism and visual display in Chapters 3-5 provide the basis for the discussion of intrasemiosis, or meaning within the systems which constitute the grammar

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of each resource. However, the three separate grammars are not sufficient for the analysis of mathematical discourse because intersemiosis, the meaning arising across semiotic choices, must also be considered. The mathematics text, and elements within that text (mathematical graphs and the symbolic derivations to problems, for instance), are multisemiotic. The analysis of mathematical discourse therefore needs to take into account the meaning arising from intersemiosis at discourse, grammar and display strata both within and across elements in the text. Following Kok (2004), elements in mathematical texts which function as discernible units through systematic choices from the grammars of language, visual images and mathematical symbolism are called 'Items'. These include graphs, diagrams, tables, stretches of linguistic text and the symbolic equations, as well as photographs, maps and other forms of drawings. The Items are not called 'genres' because the use of this term is reserved for the communication plane which is concerned with the goals and cultural context of the entire mathematics text. The SF framework adopted for the analysis of mathematics texts is displayed in Table 6.1 (1). The expansions in meaning arising from intersemiosis in mathematics are investigated at the discourse, grammar and display strata. The register, genre and ideology of mathematical discourse are considered in Chapter 7. Royce (1998a, 1998b, 1999, 2002) refers to intersemiosis as 'intersemiotic complementarity' where Visual and verbal modes semantically complement each other to produce a single textual phenomenon' (Royce, 1998b: 26). As Royce and also Lemke (1998b) explain, the product is 'synergistic' or 'multiplicative' in that the result is greater than the sum of the parts. Language, symbolism and visual images function together in mathematical discourse to create a semantic circuit which permits semantic expansions beyond that conceivable through the individual contributions. The resultant meaning potential of mathematics therefore stretches beyond that possible through the sum of the three resources. Following this view, the success of mathematics as a discourse stems from the fact that it draws upon the meaning potentials of language, visual images and the symbolism in very specific ways. That is, the discourse, grammar and display systems for each resource have evolved to function as interlocking system networks rather than isolated phenomena. The ways in which the grammars of language, mathematical symbolism and visual display are organized to facilitate intersemiosis are explored in this chapter. The three semiotic resources fulfil different functions as the mathematics text unfolds. Therefore semiotic transitions, or movements between the semiotic resources, occur according to the required functions at the different stages of the text. ledema (2003: 30) refers to the transition process as 'resemioticization' or the translations from one semiotic resource into others as social processes unfold. Transitions are a feature of everyday dynamic discourse, which result in phases and sub-phases where there is a change in the semantic input from one or more of the semiotic resources (Baldry, 2004, in press; Baldry and Thibault, 2001, in press a, in press b;

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Table 6.1(1) SF Model for Mathematical Discourse IDEOLOGY GENRE REGISTER CONTENT (Items)

INTERSEMIOSIS

LANGUAGE MATHEMATICAL SYMBOLISM

MATHEMATICAL VISUAL IMAGES

OTHERS ITEMS

Discourse Semantics

Text

Inter-statemental relations

Inter-Visual Relations Work/Genre

Grammar

Clause complex Clause Word Group/ Phrase Word

DISPLAY

Statements Clause Expressions Components

Episode Figure Parts

For example: Photographs Maps Drawings ThreeDimensional Models Equipment and so forth

INTERSEMIOSIS

Graphology and Typography

Materiality

Thibault, 2000). For example, television advertisements combine the dynamic visual image, gesture, music, sound and spoken and written language into interwoven phases which favour metafunctional input from one semiotic resource over another. Baldry and Thibault (Baldry and Thibault, in press b; Thibault, 2000) investigate how different types of transitions give rise to phasal shift in video texts. However, transitions are not always clear cut as phases and sub-phases merge and blend in what amounts to an orchestral organization of semiosis.

INTERSEMIOSIS

161

Modern mathematics developed as a written discourse, and so the intersemiotic shifts or transitions are deliberate and 'calculated' to fulfil the required functions of the text at different stages. The increasing use of computers and computerized mathematics, however, is changing the static nature of semiosis which takes place in written mathematics. There is an increasing drive towards computation where the results of the numerical calculations are displayed visually through sophisticated software programs. Intersemiosis in mathematics may be viewed as a musical score, but one that is increasingly played in a computerized environment. The shift to the electronic medium is changing the nature of the theory and practice of mathematics, especially in the field of applied mathematics. However, the discussion presented here is concerned with intersemiosis in written mathematical texts. The analysis of the written format permits us to appreciate how language, visual images and the symbolism developed as integrated systems to create modern mathematical discourse. Despite advances in computerized mathematics, the three resources remain the primary tools for meaning in mathematics. Transitions are seen to take two forms in written mathematics. Macrotransitions occur at the rank of discourse, where Items which consist of predominantly one semiotic resource give way to Items consisting of another semiotic resource; for example, language to visual images (graph) to symbolism (symbolic solution to the problem) back to language. Macrotransitions are conceptualized as discourse moves across Items in the mathematics text. Theoretically speaking, the reading path of the text results from such discourse moves. In practice, readers scan multisemiotic mathematical texts to ascertain the important information according to their own requirements. On the other hand, micro-transitions in mathematics texts occur at the rank of grammar where functional elements of one resource are contained within Items which primarily consist of another resource; for example, symbolic elements appear in the graphs, and linguistic elements appear in the symbolic statements. More generally, macrotransitions involve discourse moves to access the meaning potential of a semiotic resource, while micro-transitions take place constantly because of the integrated grammars for language, mathematical symbolism and the visual image where it is possible to embed functional elements from one semiotic resource within a different semiotic resource. That is, microtransitions are the result of the interlocking nature of the system networks for the three resources. As seen in Sections 6.3 and 6.4, micro-transitions aid macro-transitions or discourse moves to another Item in the mathematics text. The nature of the macro- and micro-transitions for intersemiosis in mathematical discourse depends upon the metafunctional requirements at different stages in the text and the available choices in the discourse, grammatical and display systems. Regardless of the organization of the mathematics text and the explicit choices marking the transitions which should take place, often the reader will examine the diagram or graph and

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the mathematical equations independently; for example, before reading the linguistic text. The reading path is conceptualized as a recursive scanning process rather than the sequential path determined by the compositional organization of the mathematics text. The use of Caption, Titles and Labels aids this scanning process through the spatial separation and foregrounding of significant Items, such as the symbolic descriptions and the visualization of relations. The resulting recursive path depends on the needs and experience of the reader. As mathematical texts typically unfold in a non-sequential fashion, macro-transitions therefore are not always operational in practice. Intersemiotic transitions in mathematics offer the opportunity for metafunctional choices in system networks across the three semiotic resources to combine in new ways across different strata. For example, choices from system networks for experiential meaning in language and symbolism may combine with choices for interpersonal meanings in visual images. For instance, 'the derivative/' (a)' is drawn as two red lines in the original version of Figure 4 reproduced in Plate 6.3(2). The colour red in the visual image functions interpersonally to give salience to the experiential meaning of the derivative as a geometrical entity. One dimension of the 'multiplication of meaning' which takes place in mathematics arises from the combinations of system choices from different semiotic resources across different strata. The metafunctionally based systems for the three resources permit a range of different combinations. The number of possible intersemiotic relations or combinations increases with the number of semiotic resources which are involved. Mathematics thus succeeds through utilizing the meaning potential of language, visual images and the symbolism and the meanings which arise through intersemiosis. Significantly, intersemiotic shifts in mathematical discourse also permit metaphorical expansions of meaning beyond those which can occur within any one semiotic resource. The metaphorical nature of meaning expansion is considered in relation to the concept of semiotic metaphor in Section 6.5. These key ideas concerning intersemiosis in mathematics are developed through the notions of macro-transitions across Items and microtransitions within those Items. However, the strategies or mechanisms through which intersemiosis takes place need to be extended beyond the notion of transition. Thus, in addition to Semiotic Transition, intersemiotic mechanisms are conceptualized as Semiotic Cohesion, Semiotic Mixing, Semiotic Adoption and Juxtaposition in Section 6.2. In addition, semiotic metaphor accounts for the metaphorical meaning expansions which occur during transitions. This list of intersemiotic mechanisms is not exhaustive; on the contrary, further research is needed to understand how meaning expansion takes place intersemiotically. In what follows, previous approaches to the study of intersemiosis are discussed in order to contextualize what are conceived to be four main issues pertaining to intersemiosis in mathematics, namely: (i) the mechanisms for intersemiosis; (ii) the metafunctionally based systems at the discourse, grammar and

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163

display strata through which intersemiosis takes place; (iii) the semantics of intersemiosis; and (iv) the metaphorical expansions which take place in the form of semiotic metaphors. 6.2 Intersemiosis: Mechanisms, Systems and Semantics The underlying principle for the integration of semiotic resources in Thibault's (2000) theoretical framework for the transcription and analysis of television advertisements is metafunctional salience where meanings arising from choices from different resources function to contextualize each other. Thibault (2000: 362) explains that it is 'on the basis of cocontextualizing relations that meaning is created'. In order to analyse a dynamic multimodal text such as the television advertisement, Thibault segments the text in phases and sub-phases: 'A discoursal phase, following Gregory (1995, 2002), is a set of co-patterned semiotic selections that are co-deployed in a consistent way over a given stretch of text' (Thibault, 2000: 325-326). Thibault (2000: 325-326) deals with the complexity of multiple forms of dynamic semiosis through a transcription table where phases and metafunctional salience are marked in what is a seminal effort at handling the complexity of the integration of multiple semiotic resources. This approach is further developed into multimodal concordancing of patterns arising from different types of transitions (Baldry, 2004, in press; Baldry and Thibault, 2001, in press b). Lim (2004) conceptualizes the expanded meaning arising from intersemiosis as the 'Space of Integration' (Sol) in the Integrative Multisemiotic Model (IMM). The Sol is designed to capture the meanings which arise through the interaction between language and visual images. Royce (1998a, 1998b, 1999, 2002) formulates this space as the 'intersemiotic complementarity' between language and visual images: 'They [the visual images and language] work together to produce a coherent multimodal text for the viewers and readers, a text characterised by intersemiotic complementarity (Royce, 2002: 193). Royce (1998b) identifies a number of intersemiotic semantic mechanisms through which image and language orchestrate the meaning of a text in the analytical metafunctionally based framework reproduced in Table 6.2(1). Royce (1998b, 2002) adopts Halliday (1994) and Halliday and Hasan's (1976, 1985) categories of lexical cohesion to account for ideational meaning arising in a multimodal text. Royce's categories include intersemiotic repetition, synonymy, antonymy, meronymy, hyponymy and collocation across visual and verbal codes. For interpersonal meaning, Royce (1998b) is concerned with the relations established between the reader/viewer and the text through MOOD and MODALITY which function to reinforce address and attitudinal congruence or dissonance. In relation to textual or compositional meaning, the layout and composition through information value, salience, framing, inter-visual similarity and reading paths are considered. Royce attempts the difficult task of mapping intersemiotic

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Table 6.2(1) Analytical Framework for Visual-Verbal Intersemiotic Complementarity, Royce (1998b: 29) META-FUNCTION

Visual Meanings

Intersemiotic Complementarity

Verbal Meanings

IDEATIONAL

Variations occur according to the Coding Orientation. In the Naturalistic Coding we can look at: • Identification of represented participants • Activity portrayed • Circumstances of the mean, accompaniment and setting • Attributes of represented participants

Various lexico-semantic ways of relating the experiential and logical content or subject matter represented or projected in both visual and verbal modes through the intersemiotic sense relations of: • Repetition • Synonymy • Antonymy • Meronymy • Hyponymy • Collocation

Lexical items which relate to the visual meanings. These lexical items arise according to: • Identification (participants) • Activity (processes) • Circumstances • Attributes

INTERPERSONAL

Variations occur according to the Coding Orientation. In the Naturalistic Coding we can look at: • Address to the viewer • Level of Involvement of viewer • Power relations between the viewer and the represented participants • Social Distance between viewer and represented participants • Modality — believability or acceptability of the portrayal

Various ways of intersemiotically relating the reader/ viewer and the text through MOOD and MODALITY through intersemiotic semantic relations of: • Reinforcement of address • Attitudinal Congruence • Attitudinal Dissonance

Elements of the clause as exchange which relate to visual meanings. These arise according to: • The MOOD element in the clause realizing speech function • The MODALITY features of the clause • Attitude - use of attitudinal adjectives

TEXTUALCOMPOSITION

Variations in visual meanings occur according to choices made in: • Information Value intra-visual placement • Visual Salience • Framing of Visual elements

Various ways of mapping The body copy (verbal the modes to realize a element) as an coherent layout or orthographic whole composition by: realized by various • Information Valuation typographical on the page conventions: • General Typesetting • Salience on the page • Copyfitting • Degree of framing of • Other Typesetting elements on the page Techniques • Inter-visual synomymy • Also: Theme/Rheme, • Reading paths Given/New Structures

expansions across visual and verbal elements by proposing mechanisms for meaning expansion that largely depend upon linguistic conceptions such as those proposed for cohesion, and interpersonal dimensions of MOOD

INTERSEMIOSIS

165

and MODALITY. Royce also considers composition as the different mappings which result in the coherent layout of the whole text. Royce's analytical framework provides a point of departure for the description of intersemiotic mechanisms which extend beyond those proposed for language and visual images. Following Thibault (2000: 325-326) and Royce (1998b), Lim (2004) sees contextualizing relations as a significant aspect of the meaning arising from intersemiosis. Two types of contextualizing relations are proposed: 'In cases where the meaning of one modality seems to "reflect" the meaning of the other through some type of convergence, the two resources share cocontextualizing relations. On the other hand, in cases where the meaning of one modality seems to be at odds with or unrelated to the other, their semantic relationship is one that creates divergence or dissonance. In the latter case, the resources share re-contextualizing relations' (Lim, 2004: 239). From this perspective, the nature and the degree of the co-contextualizing or re-contextualizing relations are significant. Cheong (1999, 2004) provides a meta-language for the semantics of intersemiosis in terms of the degree of contextualization, and the implications of those contextualizing relations. Cheong conceptualizes the ideational meaning arising from the intersemiosis between visual and linguistic selections in print advertisements as the Bi-directional Investment of Meaning, which is measured through a scale known as Contextualization Propensity (CP). CP 'refers to the degree/extent to which the linguistic items contextualize the meaning of the visual images' (Cheong, 1999: 44). Cheong shows the CP has a direct influence on the Interpretative Space (IS), which results in the Semantic Effervescence (SE) of the text. For example, an advertisement with a high CP leads to a low IS resulting in a low SE. Further research is needed to theorize the range of intersemiotic mechanisms through which semantic expansions take place, and the implications of those semantic reconstruals. In what follows, the nature of intersemiotic mechanisms and the metafunctionally based systems through which intersemiosis takes place at the discourse, grammar and display strata are considered. The metafunctionally based systems for intersemiosis are developed through text analysis in Sections 6.3-6.4. The four major issues in relation to intersemiosis in mathematics are displayed in Figure 6.2(1). First, the means or mechanisms through which intersemiosis as a phenomenon takes place require investigation. A range of intersemiotic mechanisms is given below. The mechanisms reside within and across the systems for language, mathematical symbolism and visual images. The options within the system networks for the three resources function intrasemiotically as closed systems in theory only. In practice, the systems for language and other semiotic resources have the potential to function intersemiotically. Semiotic resources have evolved to be used in conjunction with other semiotic resources, and thus considering them in isolation gives only a partial picture of their functionality. The second issue

INTRASEMIOSIS (within one semiotic resource) Meaning through choices from systems forming intra-connected network options for each semiotic resource

INTERSEMIOTIC MECHANISMS

DISCOURSE, GRAMMAR AND DISPLAY SYSTEMS The functions of language, mathematical symbolism and visual images include the potential for intrasemiosis and intersemiosis

Meaning through choices from systems functioning as interlocking networks: 1 2 3 4 5

Semiotic Cohesion Semiotic Mixing Semiotic Adoption Juxtaposition Semiotic Transition

SYSTEMS AND SEMANTICS OF INTERSEMIOSIS Textual/Compositional meaning Interpersonal meaning Experiential meaning Logical Meaning Co-contextualization (parallelism) Re-contextualization (divergence or dissonance) Bi-directional Investment of Meaning. Contextualization Propensity (CP), Interpretative Space-(IS) Semantic Effervescence (SE) (Cheong, 1999, 2004) SEMIOTIC METAPHOR Metaphorical shifts across semiotic resources where functional status of elements is not preserved and new elements are introduced across discourse, grammar and display

Figure 6.2(1) Aspects of Intersemiosis: Mechanisms, Systems, Semantics and Semiotic Metaphor

167

INTERSEMIOSIS

therefore involves the description of the metafunctionally based intersemiotic systems according to the discourse and grammar strata and the display plane. A tentative description of these systems is proposed in Tables 6.2(2a-d).) The systems are discussed in relation to the analysis of two mathematics texts in Sections 6.3 and 6.4. The third issue is the semantics of intersemiosis which may be conceptualized in terms of co-contextualizing and re-contextualizing relations. The semantics of intersemiosis is considered in Sections 6.3 and 6.4. The fourth aspect is the metaphorical construals which result from shifts between semiotic codes. This is investigated in relation to semiotic metaphor in Section 6.5. Table 6.2(2a) Systems for Intersemiosis: TEXTUAL MEANING INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM AND VISUAL DISPLAY Metafunction Textual

Discourse

Grammar

Display

INTERSEMIOTIC IDENTIFICATION Cohesive devices for Intersemiotic Reference including elements which operate across resources through Direct Reference and Intersemiotic Repetition (for example, x) and

INTERSEMIOTIC SUBSTITUTION Substitution of one term for another (for example, x for AB + for addition) INTERSEMIOTIC ADOPTION Use of functional element across semiotic resources

JUXTAPOSITION (Textual and Compositional Arrangement) Use of spatial position and layout to juxtapose and separate selections and items from each semiotic resource

semantic reference

(for example, x)

(for example, Variable' and x) INTERSEMIOTIC MIXING Use of selections of different semiotic selections (for example, A ABC)

DEIXIS Use of deictics in language (for example, 'this' curve) compensated by generalized participants in symbolism and visual display

DISCURSIVE LINKS across text LABELS Use of Labels which CAPTIONS use multiple semiotic Use Captions which resources use multiple semiotic resources

FRAMING to organize text FONT Use font style, size and colour for cohesive purposes COLOUR Use of colour for cohesion across text

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Table 6.2(2b) Systems for Intersemiosis: EXPERIENTIAL MEANING INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM AND VISUAL DISPLAY Metafunction

Discourse

Experiential

INTERSEMIOTIC IDEATION Activity Sequences and relations which stretch across semiotic resources through direct repetition (for example, let AB terminate at C) and intersemiotic equivalence, synonymy, antonymy, hyponymy, meronymy and collocation (Royce, 1998b:29). Taxonomies which stretch across resources (for example, types of triangles)

Grammar

TRANSITIVITY JUXTAPOSITION RELATIONS (for Experiential The use of relation processes Relations) to set up identifying relations Use of space and across semiotic resources for position to create lexical, symbolic example, let AB = x and visual relations Transitivity selections which overlap, for example, A and B FONT have meaning in the grammar of visual images, Use font style, size and colour for and the grammar of language experiential meaning LEXICALIZATION, COLOUR SYMBOLIZATION and Use of colour for VISUALIZATION experiential Maintenance in process, participant and circumstance meaning and agency configurations through: (i) Lexicalization of symbolic and visual functional elements (for example,

'flirt^r^ fi=,' fr~*v ' 7»' ^»-«*-l ( '\ CAPTIONS Use Captions which use multiple semiotic resources

Display

(ii) Symbolization of lexical and visual functional elements (for example, 'h' for 'line' and ' — ') (iii) Visualization of lexical and symbolic functional elements (for example, ' — ' for distance and 'h') SEMIOTIC METAPHOR Shifts in functional status and introduction of new process, participants and circumstantial elements (for example, introduction of triangle visually which becomes symbolized and lexicalized), shifts in agency LABELS Use of Labels which use multiple semiotic resources

INTERSEMIOSIS

169

Table 6.2(2c) Systems for Intersemiosis: INTERPERSONAL MEANING INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM AND VISUAL DISPLAY Metafunction

Discourse

Interpersonal INTERSEMIOTIC NEGOTIATION The unfolding of discourse moves across semiotic resources using SPEECH FUNCTIONS, arrows, (to accommodate lack of gaze and so forth)

Grammar SPEECH FUNCTION, MOOD Speech functions (including commands to view parts of the text) MODALITY Consistency of Modality across visual, verbal and symbolism

LABELS and CAPTIONS POLARITY Use of Labels which Displays of Polarity use multiple semiotic across resources resources SEMIOTIC INTERSEMIOTIC METAPHOR APPRAISAL Shifts in functional Appraisal across status of expression semiotic resources of modality across semiotic resources

Display STYLE OF PRODUCTION Consistency in style of production SALIENCE as directing discourse moves across text PROMINENCE as directing attention to verbal, visual and symbolic romnonents FONT Use font style, size and colour for interpersonal meaning COLOUR Use of colour for interpersonal meaning

The mechanisms of intersemiosis are categorized as: 1 2 3 4 5

Semiotic Cohesion: System choices function to make the text cohere across different semiotic resources. Semiotic Mixing: Items consist of system choices from different semiotic resources. Semiotic Adoption: System choices from one semiotic resource are incorporated as a system choice in another semiotic system. Juxtaposition: Items and components within those Items are compositionally arranged to facilitate intersemiosis. Semiotic Transition: System choices result in discourse moves in the form of macro-transitions which shift the discourse to another Item consisting primarily of another semiotic resource, or alternatively macro-transitions within Items occur.

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Table 6.2(2d) Systems for Intersemiosis: LOGICAL MEANING INTERSEMIOSIS ACROSS LANGUAGE, MATHEMATICAL SYMBOLISM AND VISUAL DISPLAY

Metafunction Logical

Discourse IMPLICATION SEQUENCES Cohesive and structural devices across semiotic resources (for example, linguistic and symbolic strutural Conjunctions, Conjunctive Adjuncts, and cohesive ties, and arrows pointing to other semiotic resources)

Grammar

Display

LOGICO-SEMANTICS SPATIAL and POSITION INTERDEPENDENCY Alignment of Items Cohesive and in the text in conjunctive devices sequence across semiotic resources FONT Use font style, size INTERPLAY OF and colour for SPATIALlTYand logical meaning TEMPORALITY through visual, textual COLOUR and symbolic Use of colour to transformations direct the sequence for the construction of logical relations SEMIOTIC METAPHOR Shifts in functional status of logical relations across semiotic resources

Intersemiotic mechanisms generally involve a two-way directional investment of meaning as displayed in Figure 6.2(1). Semiotic transitions involve shifts in the discourse from one semiotic resource to another, and hence are indicated by one-directional arrows. However, the arrow pointing in the opposite direction shows that contextualization is a two-way process, despite the one-directional shift in the discourse. Intersemiotic transitions are particularly significant in mathematics where semiotic resources are seen to alternate between being primary and ancillary at different stages of the text. This alternation is explained by the functional requirements at different stages in the generic structure of the mathematics text. The intersemiotic mechanisms are realized through choices from the systems in Tables 6.2(2a-d), which are categorized according to the discourse and grammar strata and the display plane. The systems are organized metafunctionally, and thus form another semantic layer to the analysis of mathematics texts. Up to this point, the analysis of mathematical discourse has been based on the three SF frameworks proposed for language, mathematical symbolism and visual images in Chapters 3-5 respectively. The intersemiotic systems listed in Tables 6.2(2a-d), however, provide the

INTERSEMIOSIS

171

facility whereby choices from a metafunctionally based system in one semiotic integrate with choices from other systems in other semiotic resources. This includes the provision for integration of system choices across discourse and grammar strata and the display plane. The framework provides a semantic layer whereby the 'texture' of the text may be seen as the interaction between semiotic resource x metafunction x strata x system x choice through the mechanisms of interserniosis. The systemic framework for interserniosis in Tables 6.2(2a-d) is not exhaustive. Further research is needed to examine how semiotic resources are organized to function interserniotically, and how that functionality relates to system options in the grammars for each resource. However, the important points are that semiotic resources are functional with respect to the semantics of other semiotic resources, and that the systems through which interserniosis takes place require further theorization. The framework in Tables 6.2(2a-d) is discussed with reference to Newton (1736: 46) and Stewart (1999: 132) displayed in Plate 6.3(1) and Plate 6.3(2) in Sections 6.3 and 6.4. In the analysis of interserniosis, it becomes evident that the integration of semiotic resources is formalized in mathematics and science in a fashion which is not typically found in other forms of discourse. Following this discussion, the notion of semantic expansion through semiotic metaphor is illustrated through text analysis in Section 6.5. 6.3 Analysing Intersemiosis in Mathematical Texts Intersemiotic mechanisms through system choices at the ranks of discourse, grammar and display are examined in the extract from Newton's (1736) writings The Method of Fluxions and Infinite Series displayed in Plate 6.3(1). In Newton's text, '/' stands for the letter V in the linguistic text. 'Fluxion' is Newton's term (now obsolete) for the derivate as the rate of change of a function with respect to x, where the geometrical interpretation of the derivative is the tangent to a curve at a point. Newton's dy notation for the derivative x is used today along with other forms such as — dx f'(x), and Dxf(x). Newton is concerned with drawing the tangent to the curve in Plate 6.3(1). The Items in Newton's text are the linguistic text, the symbolic text in (Point 3) (which is embedded in the surrounding linguistic text), and the diagram. In Stewart's (1999: 132) description of the derivative as the instantaneous rate of change, the Items consist of Figure 4, the linguistic text which is concerned with the derivative, and Example 4. While the linguistic, symbolic and visual Items are spatially separated in the modern mathematics texts such as Plate 6.3(2), Newton's (1736) work is typeset in the style of running text in Plate 6.3(1). Newton's text may be compared to the contemporary rendition of the derivative as the slope of a tangent line displayed in Plate 5.2(1). Newton's work has been chosen for analysis, however, because the linguistic choices are largely congruent and thus intersemiotic shifts between linguistic, visual and symbolic parts are

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Plate 6.3(1) Newton's (1736: 46) Procedure for Drawing Tangents easier to track, unlike modern mathematical and scientific texts such as Stewart (1999: 132) which are highly metaphorical (see Section 6.5). Newton's writings in the eighteenth century illustrate how mathematics evolved to integrate three semiotic resources, and the examination of contemporary discourse reveals the results of that integration. In what follows, intersemiosis in Plates 6.3(1) and 6.3(2) arising from systems displayed in Tables 6.2(2a-d) is discussed according to metafunction.

INTERSEMIOSIS

173

From Equation 3 we recogni/e this limit as being the derivative o f / at .\i. that is. f'(x\). This gives a second interpretation of the derivative: The derivative /"(«) is the instantaneous rate of change of v -•/(.*) with respect to v when A = a.

FIGURE 4

The y-values arc changing rapidly al P and slowly at Q.

The connection with the first interpretation is that if we sketch the curve y = fix), then the instantaneous rate of change is the slope of the tangent to this curve at the point where .v = a. This means that when the derivative is large (and therefore the curve is steep, as at the point P in Figure 4). the y-values change rapidly. When the derivative is small, the curve is relatively flat and the y-values change slowly. In particular, it's — f ( t ) is the position function of a particle that moves along a straight line, then /'{«) is the rate of change of the displacement ,v with respect to the time t. In other words, fin) is I/if ir/«<7/y of the particle at rime t a (see Section 2,6). The speed of the particle is the absolute value of the velocity, that is. i / '(a) \, EXAMPLE « The position of a particle is given by the equation of motion i ~ /(?) — 1/i I + /), where t is measured in seconds and ,v in meters. Find the velocity and the speed after 2 seconds. SOLUTION The derivative o f / when / = 2 is

Thus, the velocity after 2 s is

m/s, and the speed is

m/s.

Plate 6.3(2) The Derivative as the Instantaneous Rate of Change (Stewart, 1999: 132)

Textual Meaning

At first glance, the primary resource for intersemiosis in mathematics appears to be textual; that is, the organization of the message for the enabling of interpersonal, logical and experiential meaning across semiotic resources. The importance of textual organization explains the traditional emphasis on compositional layout in graphical design where items are aligned, framed and juxtaposed in relation to other items to create certain effects in advertising, newspapers and magazines. From the systemic perspective, however, there are textual systems other than those on the display stratum which aid intersemiotic tracking of participants across linguistic, visual and symbolic items as displayed in Table 6.2(2a). These systems include INTERSEMIOTIC IDENTIFICATION where cohesion is achieved through Intersemiotic Reference. For example, in Plate 6.3(1) Intersemiotic Reference occurs through the selections 'tangents' and the actual line (TD) which is drawn to intersect the curve in the diagram. Similarly, 'BD' is 'a right line' which appears in the diagram. Often there is Direct Reference across the three semiotic resources; for example, x appears in the linguistic,

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visual and symbolic parts of the text in Plate 6.3(2). In addition, INTERSEMIOTIC MIXING aids the tracking of participants. For example, the graph in Plate 6.3(2) consists of visual, symbolic and linguistic selections which refer to participants in the linguistic and symbolic parts of the text. At the rank of grammar, textual resources include INTERSEMIOTIC SUBSTITUTION (for example, x for AB in Plate 6.3(1)) and INTERSEMIOTIC ADOPTION (x appears as an element in each resource in Plate 6.3(2)). DEIXIS in language aids textual organization across visual and symbolic modes; for example, 'the curve' in 'the curve y=f(x)' (which could have been labelled in Figure 4 in Plate 6.3(2)). As mathematical symbolism evolved from language, the grammars interlock in ways so that selections are almost interchangeable as seen in the example 'the curve y = f ( x ) \ INTERSEMIOTIC MIXING, LABELS and CAPTIONS ensure that the semantic realm of the linguistic and symbolic is included in diagrams and graphs. In terms of the display stratum, JUXTAPOSITION in terms of Textual and Compositional Arrangement aids intersemiosis. For example, the Items consisting of the graph, the linguistic text, and the symbolic solution to the problem are spatially organized in Plate 6.3(2) so that the reader may easily access the experiential and logical meaning of each Item. Further to this, COLOUR also aids textual organization; for example, in the original version of Plate 6.3(2), the definition of the tangent which is Framed (in red) coheres with the (red) line corresponding to the tangent lines in Figure 4. The choices which function intersemiotically for textual meaning result in a discourse where reference chains split and recombine in complex ways across semiotic resources (O'Halloran, 2000). The resulting texture is dense as linguistic, symbolic and visual participants are reconfigured and combined, especially within the symbolic parts of the text. Similarly, grammatical choices function to ensure that experiential and logical meanings arising from textual organization of the message are precisely encoded in ways that permit the reader to access these meanings across semiotic resources in the most immediate manner possible. While discoursal, grammatical and display choices for textual meaning function to ensure a coherent text, the problems of tracking participants across semiotic resources in mathematics are addressed in relation to pedagogy in Section 7.3. Experiential Meaning

Experiential intersemiosis is formulated in terms of INTERSEMIOTIC IDEATION at the discourse stratum. INTERSEMIOTIC IDEATION contributes to the construction of Activity Sequences which stretch across the text. For example, the reader is instructed in Plate 6.3(1) to 'first let BD be a right Line, or Ordinate, in a given Angle to another right Line AB, as a

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Base'. These participants correspondingly appear in the diagram. Furthermore, 'Calling AB = x and BD = / means that the symbolism is brought into play in the Activity Sequence. The sequences are realized through interpersonal commands and statements, and TRANSITIVITY RELATIONS which function intersemiotically to establish identifying relations across process, participant and circumstantial elements through LEXICALIZATION, SYMBOLIZATION, VISUALIZATION and the use of LABELS. In this process, Intersemiotic Repetition (where, for example, xis continually repeated in symbolic, visual and linguistic parts of the text) is important. In addition, semiotic metaphor is conceptualized as an important intersemiotic strategy for metaphorical meaning expansions in mathematics (see Section 6.5). Royce (1998b, 2002) uses the notion of cohesion across visual and verbal semiotic resource to conceptualize ideational intersemiosis (see Table 6.2(1)). Although associated with textual meaning in SFL, the concept of cohesion is nonetheless useful for tracing lexical relations in mathematics at the discourse stratum. For example, taxonomies are constructed in mathematics so that cohesive relations are realized through classification (x is a type of y relationship in the form of hyponymy; for example, the tangent is classified as the line which intersects the curve in one point only) and composition (whole/part relations in the form of meronymy; for example, a point is part of a line). The second main type of cohesive device for lexical relations is expectancy relations which are similarly direct given the limited semantic field with which mathematics is concerned. For example, Plates 6.3(1) and 6.3(2) are concerned with the geometrical interpretation of the derivative, and so particular linguistic, visual and symbolic choices are expected. At the display stratum, JUXTAPOSITION for Experiential Relations (for example, the relative position of the graphs and diagrams, and the accompanying Labels and Captions) and COLOUR (for example, the use of red lines for the derivative in Figure 4 in Plate 6.3(2)) aid the construction of intersemiotic experiential meaning. The use of space and position to create lexical, symbolic and visual relations is an important resource for experiential meaning in mathematics. In addition, FONT size, style and COLOUR function experientially to cohesively link, for example, symbolic variables, important parts of the linguistic text and significant Episodes, Figures and Parts in the visual display. Interpersonal Meaning

INTERSEMIOTIC NEGOTIATION for interpersonal meaning at the discourse stratum stretches across the three resources; for example, the reader is given a linguistic command 'first let BD be a right Line, or Ordinate, in a given Angle to another right Line AB, as a Base'. Compliance with this command takes place visually in the diagram in Plate 6.3(1). Also information is given through statements (for example, 'Calling AB — x and BD = y'

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in Plate 6.3(1)) which function intersemiotically to link the visual image to the symbolism. LABELS and CAPTIONS attract attention in order to aid INTERSEMIOTIC NEGOTIATION across resources. Further to this, FONT, COLOUR and the resulting visual SALIENCE and PROMINENCE function to assist discourse moves across and within the Items in the text. Interpersonally there is typically a consistency in terms of high MODALITY values across the discourse and grammatical selections (in terms of probability and obligation values) and the STYLE OF PRODUCTION at the display plane also suggests a high truth-value. Interpersonal relations of authority with a high truth-value are reinforced intra and intersemiotically in mathematics. Logical Meaning

IMPLICATION SEQUENCES function intersemiotically through linguistic and symbolic Conjunctive Adjuncts and structural Conjunctions in Plates 6.3(1) and 6.3(2) in combination with visual devices such as arrows (see Plate 5.2(1)). The interplays with spatiality and temporality in mathematical visual images are aided by linguistic selections such as those appearing under the CAPTION 'Figure 4' in Plate 6.3(2): 'The ^values are changing rapidly at Pand slowly at Q', so that intersemiosis permits a logical expansion beyond that possible with language and symbolic elements where typically space and time are separated into different elements. The INTERPLAY OF SPATIALITY AND TEMPORALITY in Figure 4 is glossed linguistically in terms of the manner of change, 'rapidly' and 'slowly'. Intersemiosis means that spatial and temporal dimensions are integrated in the visual image. In addition, the logic of spatial visual perception is translated into a text-based tool in the form of mathematical symbolism which permits reasoning to progress beyond that possible with language. The grammar of the visual image interlocks with the grammars of the symbolism and language so that these types of reasoning can take place. At the display stratum, SPATIAL POSITIONING and the alignment and sequence of Items also function to aid logical reasoning. For example, sequential steps realized through language and symbolism are placed alongside and underneath one another in Example 4 in Plate 6.3(2). Once again, FONT and COLOUR aid logical sequential processing in terms of organizing the symbolic and linguistic steps. Metafunctionally Based Co-Contextualization and Re-Contextualization

Textual, experiential, interpersonal and logical meanings are cocontextualized and re-contextualized through the intersemiotic mechanisms which are described above. However, the types of contextualization appear to be metafunctionally based. Interpersonal meaning in the form of dominating social relations with high modality values is co-contextualized across linguistic, visual and symbolic selections. The writer directs the

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Activity Sequences with commands and statements which possess high degrees of obligation and certainty which are reinforced through the nature of the de-contextualized visual representations and the modality found in the symbolic text. The Style of Production of the text and the abstract nature of the linguistic, visual and mathematical participants, processes and circumstances function to further co-contextualize the selections for interpersonal meaning. Experiential meaning, however, is re-contextualized as the Activity Sequences stretch across the different semiotic resources. This significant aspect of semantic expansion in mathematics is further developed in Sections 6.4 and 6.5. Similarly, logical meaning involves re-contextualization of spatial and temporal relations through intersemiosis across linguistic, visual and symbolic components. Aided by visual perception, the tool for reasoning to derive the mathematical results is ultimately the symbolism, although language functions to contextualize the results which are established. Textual and compositional meaning as the enabling function assumes a new realm of importance when viewed through the lens of intersemiosis. The organization of the message is critical in mathematics so that the meaning potential of each resource may be accessed and integrated at different stages in the unfolding of the text. New systems for textual meaning are called into play as participants, processes and circumstances are semantically realigned across the three semiotic resources. The identification and tracking of those functional elements and their subsequent semantic reconfiguration is important in mathematics. In what follows, the integration of language, mathematical symbolism and visual display is further examined in terms of the transitions which take place in order to investigate the semantics of re-contexualization in mathematical discourse. Newton's text in Plate 6.3(1) is examined for this purpose. 6.4 Intersemiotic Re-Contexualization in Newton's Writings

Newton's problem in Plate 6.3(1) is titled 'PROB. IV. To draw Tangents to Curves, First Manner', and the directions to achieve this objective take the form of a series of linguistic commands and statements. There are constant macro-transitions back and forth between the linguistic text and the diagram as the problem is visualized. Micro-transitions occur in that the functional elements within the linguistic text, for example, 'Line AB' and 'Curve ED', correspond to functional elements in the geometrical visual image, the points A, B, E and D. Once visualization is achieved, there are macro-shifts between the diagram and the symbolism to establish the relationship between entities; for example, Newton writes 'So that it is TB : BD :: DC (or B£) : cd'. Finally the functional elements within the geometrical image are given a symbolic algebraic form: 'Calling AB = x and BD = y\ Newton thus describes the relations using mathematical symbolism: 'let their Relation be Xs - ax2 + axy — yz = 0' and 'Therefore

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Newton concludes: Therefore the Point D being given, and thence DB and AB, or y and x, the length BT will be given, by which the Tangent TD is determined'. On the surface it appears that the intersemiotic transitions are largely a matter of giving functional elements in one semiotic resource an identity in another semiotic resource so that the meaning potential of that second resource can be accessed. For example, 'Calling AB = x and BD = y' means that functional elements in the visual image now have corresponding elements in the symbolism so that the problem can be solved algebraically. This certainly is the case as each semiotic resource has a unique meaning potential which permits particular functions to be fulfilled. Newton's problem in Plate 6.3(1) is introduced, contextualized and finalized using language. The visual image permits a perceptual understanding of the problem through re-contextualizing spatial and temporal relations at discrete instances of time. The relations of parts to the whole may also be ascertained through the visual image. The symbolism encodes the exact nature of the relations over time in a continuous and complete form. Newton's problem of drawing the tangent to a curve is resolved algebraically as a set of relations involving Operative processes and participants. The grammar of mathematical symbolism is such that these relations are easily reconfigured to solve the problem. The semantic circuit in the form of macro- and micro-transitions is seamlessly woven so that shifts back and forth are a constant feature of mathematical discourse. The result is a highly coherent text. The functional grammars of mathematical symbolism and visual display have evolved to ensure this is the case. However, the semantic circuit involves more than using relational clauses to give functional elements in one semiotic resource an identity in another. While this permits macro-transitions across resources and ensures representation of particular elements through repetition, (for example, x may appear in the linguistic, visual and symbolic parts of the text), the expansion of meaning also involves metaphorical exchange whereby functional elements do not always retain their original status when they are rerepresented in a second semiotic resource. The x, for example, functions as a participant (in the form of a generalized noun) in all three semiotic resources. However, close textual analysis reveals that functional roles are not always maintained across different semiotic resources, and that, further to this, new entities are created through intersemiotic transitions. For example, Newton (1736: 46) writes: Let this Ordinate [BD] move through an indefinitely small Space to the place bd, so that it may be increased by the Moment cd, while AB is increased by the Moment Bi, to which Deis equal and parallel.

The command for Actor/Agent ('you') to make 'this Ordinate [BD]' (Goal/Medium) 'move' (Process: material) 'through an indefinitely small

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Space' (Range) 'to the place bd (Location: Place) // so that 'it' [BD] (Goal/Medium) 'may be increased' (Process: Material) 'by the Moment cd (Range). First, the linguistic Material process 'move' BD (the goal) results in the introduction of new entities, the line segments foand B6, in the visual image. Second, the linguistic circumstance of location in 'to place bd becomes a major participant bd in the visual image. That is, the intersemiotic transition to the visual images from the linguistic involves the creation of new entities (foand B&), and a shift in the functional status of an existing entity (Circumstance 'to the place bd becomes a participant bd). Lastly, through the meaning potential of visual images, these entities (be, B£ and bd) enter into spatial relations with one another. In other words, Newton's command to 'Move X' introduces visually new participants Y and Z which are configured in the new semantic field of spatial relations with W. In a similar fashion, 'Let Dd be produced till it meets with AB in T' involves the introduction of the visual participant T from the linguistic circumstance 'in T'. Thus intersemiotic transitions in the semantic circuit do not only involve accessing the meaning potential of new semiotic resources, but also involve introducing new functional elements and changing the functional status of existing elements, both of which may be reconfigured in a new semantic realm. The semiotic interchange involves semantic recontextualization and metaphor. The significant phenomenon of intersemiotic metaphor is further explored through the notion of semiotic metaphor. 6.5 Semiotic Metaphor and Metaphorical Expansions of Meaning

Semiotic metaphor is the phenomenon where an intersemiotic transition gives rise to a metaphorical expansion of meaning (O'Halloran, 1996, 1999a, 1999b, 2000, 2003a, 2003b, in press) as demonstrated in the preceding discussion of Newton's text. In terms of experiential meaning, for example, the status of the functional element as a process, participant or circumstance undergoes a transformation through the shift or transition to another resource. Alternatively, the shift to another semiotic resource involves the introduction of a new process, participant or circumstance which did not previously exist. The phenomenon is best illustrated through an example from a secondary school mathematics lesson in trigonometry (O'Halloran 1996, 1999b). This lesson contains several instances of the types of semantic expansions which are made possible through semiotic metaphor. In this example, the metaphorical nature of the semantic expansion is traceable because the linguistic text includes the oral discourse in the classroom. Being spoken discourse, the classroom discussion is congruent rather than metaphorical (see Sections 3.6 and 6.6 for a discussion of grammatical metaphor in written mathematics and science). Cases of semiotic metaphor are therefore evident in the classroom discussion and board texts largely because meaning has not been pre-packaged metaphorically through grammatical metaphor.

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The teacher introduces and describes the trigonometric problem using language, and the diagrams in Figure 6.5(1) are produced in stages on the blackboard as a result of the classroom discussion led by the teacher. The problem concerns finding a generalized expression for the height of a cliff h and the width of a river r using two angle measurements. From the diagram, mathematical symbolism is used to solve the problem using the tangent ratio, which captures the relationship between the size of an angle and the length of the opposite and adjacent sides in the triangles. Extracts from the lesson which involve shifts between language, visual image and mathematical symbolism are given below. STEP ONE: Teacher's Linguistic Introduction to the Problem

// // // // // // // //

a man is actually at this point here he is climbing a cliff and /ahh doesn't know how high up he is and he looks down of course and looks at the river and doesn't know how wide the river is

// so with this information, he has a ten-metre rope and a device that measures angles, we are asking the question // how can the man determine, firstly, the height of the cliff at point A and, secondly, the width of the river

INTERSEMIOSIS

STEP TWO: Visualization of the Problem

(a)

HOW HIGH and HOW WIDE ?

HOW HIGH and HOW WIDE ?

Figure 6.5(1) Diagrams for Trigonometric Problem

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STEP THREE: Symbolic Solution to the Problem

In right A CBR (1)

In right &ABR. (2)

There are multiple cases of semiotic metaphor which occur in the process of solving the trigonometric problem through the macro-transitions between language, visual image and mathematical symbolism. These semiotic metaphors are described below. 1

The process realized by the verb 'look' in 'and he looks down of course' becomes an entity in the form of a line segment AR in the visual diagram on the blackboard (see Figure 6.5(la)). This new entity is later introduced in the verbal discourse as 'the line of sight'. 2 The Material process of 'measuring' becomes the participants a and 6 with arrows marking the direction through which the angles are measured and the dashed line representing the horizontal 'line of sight' (see Figure 6.5 (lb)). 3 The circumstance of 'how high' and 'how wide' realizing Extent (spatial distance) is transformed into participants h and later h — 10 which are marked with arrows indicating the relative distances in the diagram (see Figure 6.5 (lb)). 4 New entities in the form of the two triangles, AABR and ACBR, are introduced visually in the mathematical diagram. These entities did not exist prior to the visual semiotic representation of the trigonometric problem. The Figures of the triangles emerge as configurations of the participants h, rand 10, and a and 0. This means that these participants may be now viewed as a connected whole rather than as isolated entities. 5 The newly constructed Figures of the triangles permits the relations of sides and angles to be expressed symbolically as

and

Thus the new entities, tan a and tan 9 are introduced. From this point, the problem may be solved algebraically. The metaphorical shift from the linguistic processes and circumstance to the visual participants leads to the introduction of new participants in the form of two triangles in the diagram. From this point, the diagram becomes the point of departure for the expression of the mathematical relationship which exists between the sides and angles of a right triangle in the

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form of the tangent ratio. This relationship forms the basis for the symbolic solution of the problem. From this example, it becomes clear that mathematical discourse depends on the types of metaphorical shifts which take place with transitions between the three semiotic resources. From the semantic expansions made possible through semiotic metaphor, participants like 'the angle of elevation' and 'angles of depression' become choices in language. The important relationship between grammatical metaphor and semiotic metaphor is discussed in Section 6.6. Grammatical metaphor seems to have evolved as a response to the expansions of meaning which occurred through the use of other semiotic resources in mathematics and science, including those metaphorical expansions arising from semiotic metaphor. In the case of the trigonometric problem considered above, new visual entities are construed in the shift from language to visual image. The systems for language must necessarily expand to encompass these new visual entities, in this case in the form of the linguistic choices 'the angle of elevation' and 'the angle of depression'. However, such linguistic constructions have traditionally been theorized in SFL as grammatical metaphors rather than the product of meaning expansions through intersemiosis. There appear to be graduations in the types of semiotic metaphor according to the nature and the extent of the semantic expansion which takes place. The two opposite poles are conceived as parallel and divergent semiotic metaphors (O'Halloran, 1999a). A parallel semiotic metaphorhas 'an expanded semantic field but also one which is situated within the old' (ibid.: 348). Although there could be redundant meanings because of overlaps, 'new layers of meaning are [essentially] simultaneously added to the original representation' (ibid.: 348). Examples include the shift from process and circumstance to participant in Newton's diagram. The reconstrual of functional elements in a divergent semiotic metaphor, however, means that 'the functional element is reconstrued into a new semantic field' (ibid.: 348). The possibility for meaning expansions in divergent reconstruals is extensive as the functional element is relocated in a new semantic field which is not typically inter-textually related to the first. An example is the introduction of the tangent ratio in the mathematical symbolism which configures the relationship between an angle and the sides of the triangle. However, the meanings arising from divergent semiotic metaphors become naturalized over time, as has occurred with trigonometric ratios such as sine, cosine and tangent in mathematics. There are semantic redundancies involved in parallel semiotic metaphors, and thus these types of metaphorical shift serve a reinforcing and co-contextualizing function. On the other hand, divergent semiotic metaphors have the potential to involve conflicting meanings. These examples of 'ideological disjunction' are a possible result 'of the complex, often intricate, relations of inter-functional solidarity among the various semiotic resource systems that are co-deployed' (Thibault, 2000: 321). Divergent semiotic metaphors emerge to create re-contextualizing relations. In

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essence, semiotic metaphors are one further intersemiotic mechanism for the realization of co-contextualizing and re-contextualizing relations. The other mechanisms include Semiotic Cohesion, Semiotic Mixing, Semiotic Adoption and Juxtaposition developed in Section 6.2. The identification of the phenomenon of semiotic metaphor is important because it helps to explain how intersemiosis contributes to the expansion of meaning, including co-contextualization and re-contextualization. There remains much research to be completed in this field, which includes investigating semiotic metaphors for interpersonal, logical and textual meaning. From this perspective, using multiple semiotic resources is not just a matter of accessing different meaning potentials, rather the intersemiotic mechanisms prove to be an important tool for the semantic expansion. These types of semantic expansions feed back into the grammar of each resource so that not only do the transitions become seamless, but also new grammatical strategies evolve in each semiotic resource as a result of their co-functionality with other resources. As explained below, the evolution of grammatical metaphor in language may be seen as a response to the semantic expansions which occurred through semiotic metaphor and the use of mathematical symbolism and visual display in the construction of a scientific view of the world. The significance of semiotic metaphor is discussed in relation to mathematics pedagogy in Section 7.3. 6.6 Reconceptualizing Grammatical Metaphor

Halliday (1993a, 1993b, 1998), Halliday and Matthiessen (1999) and Martin (1993a, 1993b) describe the regrammaticization of experience which takes place through scientific language. There is a decided 'semantic drift' towards metaphorically reconstruing experience in terms of entities which enter into relations with other entities. The nominalization of process, quality, relator and circumstance takes place through the lexicogrammatical strategy of grammatical metaphor, which is described in Section 3.6. Halliday (1998: 209-210) and Halliday and Matthiessen (1999: 246-248) categorize the different types of grammatical metaphor and demonstrate that the general semantic shift is towards regrammaticizing of experience in the form of entities. Halliday (1998: 211) captures this shift in the following form: relator —» circumstance —> process —» quality —> entity Halliday and Matthiessen (1999: 263-264) distinguish two general motifs in the semantic shifts taking place through grammatical metaphor: '(i) The primary motif is clearly the drift towards "thing", (ii) The secondary motif . . . the move from "thing" into what might be interpreted as a manifestation of "quality" (qualifying, possessive or classifying expansions of the "thing")'. This secondary shift involves the expansion of the potential of the nominal group to encode experiential meaning in condensed format. Halliday (1993a, 1998) cites several examples from Newton's (1704) Opticks

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to demonstrate that the shift towards nominalization noted in modern scientific writing had begun to emerge in this work. Halliday (1998: 201-202) explains that although reasoning is carried forth congruently by conjunctions (for example, 'if, 'so', 'as' and 'and'), grammatical metaphor in the form of nominalization also carries forth Newton's argument. Grammatical metaphors may occur with relational processes to repackage experiential content, which Halliday (1998: 193) terms 'the favourite grammatical pattern ("syndrome" of grammatical features) in modern scientific English'. In this process, the semantic sequence of two clauses (forming a clause complex) is reconstrued as a single relational clause where the two original clauses are reconfigured into nominal group structures. The logical relations in the original clause complex are typically reconstrued as different forms of relational processes. Halliday and Matthiessen (1999: 239) summarize congruent and metaphorical realizations in Table 6.6(1). Halliday (1998: 202) uses an example from David Layzer (1990:61) to illustrate the metaphorical repackaging of experience: If electrons weren't absolutely indistinguishable, two hydrogen atoms would form a much more weakly bound molecule than they actually do. The absolute indistinguishability of the electrons in the two atoms gives rise to an 'extra' attractive force between them. (David Layzer, Cosmogenesis, 1990: 61)

In this example, the clause complex 'If electrons weren't absolutely indistinguishable, [then] two hydrogen atoms would form a much more weakly bound molecule than they actually do' is replaced by the clause 'The absolute indistinguishability of the electrons in the two atoms gives rise to an "extra" attractive force between them.' The logical relations 'if and 'then' are replaced by the causative relational process 'gives rise'. In addition, the clause Tf electrons weren't absolutely indistinguishable' is replaced by the nominalization 'the absolute indistinguishability of the electrons in the two atoms'. Close analysis of scientific writing reveals that this type of grammatical repackaging is typical of contemporary scientific writing (Guo, 2004a, 2004b; Halliday and Martin, 1993; Halliday and Matthiessen, 1999; Martin and Veel, 1998). Halliday (1998: 202) explains that the impact of grammatical repackaging is twofold. First, there is an increased potential for categorization and Table 6.6(1) Congruent and Metaphorical Realizations of Semantic Units: Halliday and Matthiessen (1999: 239) Semantic Unit

congruently

metaphorically

sequence figure logical relation

clause complex clause conjunction (relating clauses in a complex

clause nominal group verbal group

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taxonomic organization. Second, effective reasoning and logical progression are enhanced as the repackaging of a clause into a nominal group has a discursive textual function of carrying forth the momentum of the argument. Halliday (1998) sees the development of scientific language in terms of'technicalizing' and 'rationalizing'. Halliday and Matthiessen (1999: 239, 270-271) explain that the significance of grammatical metaphor is related to the new textual organization of the clause where participants may be foregrounded (and backgrounded) as packaged information. Halliday and Matthiessen (1999) also explain the gain is an increased potential to encode in condensed format experiential meaning in nominal group structures, which is essential for the construction of technical knowledge and taxonomies (Martin, 1993a, 1993b, 1993c). However, there is a subsequent loss of experiential meaning with the collapse of the clause into nominal group structures and the blurring of categories of experience where relators, circumstance and process become construed as entities. In effect, the loss of experiential meaning includes the loss of arguability which accompanies a congruent configuration of process/participant in a clause. For example, in addition to questioning the effect on Z, the reader/ listener can disagree that X transforms Y based on the following congruent clause complex sequence: X transforms Y, so Z will be affected However, the metaphorical construal does not admit such doubt concerning the relations between X and Y, and, further to this, the effect on Z seems much more certain with the absence of the logical conjunction: the transformation of X to Y affects Z

The packaging of experiential meaning through the use of grammatical metaphor and relational processes takes place in mathematical discourse. For example, the analysis of the linguistic text in Stewart (1999: 132) in Plate 6.3(2) reveals that the majority of clauses contain relational identifying processes with nominalized participants involving extended nominal group structures; for example, 'the derivative of/at x^, 'a second interpretation of the derivative', 'the instantaneous rate of change', 'the connection with the first interpretation', 'the instantaneous rate of change', 'the slope of the tangent to this curve' and so forth. The metaphorical nature of mathematical writing is evident in the following examples from Plate 6.3(2) where relational processes (in bold) configure entities in the form of nominalizations. In addition, logical meaning is re-packaged through process selections (for example, 'as being' and 'gives'). // this limit as being the derivative of /at xt// // this gives a second interpretation of the derivative// // The derivative/' (a) is the instantaneous rate of change ofy=f(x)// // the connection with the first interpretation is [[that if we sketch the curve y=f(x) \\ then the instantaneous rate of change is the slope of the tangent to this curve at the point 11 where x = a ] ] //

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Halliday (1998) and Halliday and Matthiessen (1999) thus explain how relational clauses and grammatical metaphors which expand the potential of the nominal group function in scientific language. However, if the symbolic text in Example 4 in Plate 6.3(2) is analysed, we may see that the opposite trend has occurred in the mathematical symbolism where process/participant configurations are preserved through the grammatical strategy of rankshift, as developed in Chapter 4.

However, if the symbolic statement

is verbalized,

it would read something like: the derivative of fat t = 2 is [[the limit as h approaches zero of the difference between the value of the function at 2+h and 2 [[divided by h]]]]. The linguistic version of the symbolic statement involves a relational process for '='. However, the rankshifted configuration of symbolic processes and participants on the right-hand side of the equation undergoes a reconstrual through the grammatical metaphor: 'the difference between the value of the function at 2+h and 2 divided by h'. In other words, in the transition to language, the meaning of the symbolic statement is not retained because process/participant configurations become entities in the form of nominalized participants. This is particularly significant in the context of the mathematics classroom where the symbolism is regularly verbalized. The metaphorical nature of such construals, however, is rarely if ever noted. The significance of semiotic metaphor for pedagogy is discussed in Chapter 7. It appears that scientific language evolved to construe a 'stable' view of the universe through relational processes with nominalized participants which carried forth the impetus of the argument because the function of the 'dynamic' description of the relations was allocated to the mathematical symbolism, which accordingly developed a grammar enabling it to fulfil this role. The functions of scientific language should be contextualized in relation to the context of its development. However, given the relative inaccessibility of mathematical symbolism to a general audience, scientific language has assumed functions which are not understood in relation to

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the limits of its applicability. For as Halliday and Matthiessen (1999: 271272) explain, the linguistic reconfiguration of reality through scientific discourse has repercussions: Metaphors are dangerous, however; they have too much power, and grammatical metaphor is no different in this respect. Because it leaves relations within a figure almost inexplicit, this demands that they should in some sense already be in place [beforehand] . . . It is this other potential that grammatical metaphor has, for making meaning that is obscure, arcane and exclusive, that makes it ideal as a mode of discourse for establishing and maintaining status, prestige and hierarchy, and to establish the paternalistic authority of a technocratic elite.

The mathematical and scientific view of the world is used in contexts which are not always appropriate. The metaphorical nature of the construction of mathematical and scientific discourse and the accompanying uses of the scientific view of the world are considered in Chapter 7. From the preceding discussion, it becomes evident that more research is needed in the study of the functions of language and other semiotic resources as an integrated phenomenon. Only from this perspective can semiosis be fully understood.

7 Mathematical Constructions of Reality

7.1 Multisemiotic Analysis of a Contemporary Mathematics Problem

Mathematical discourse consists of a range of genres such as the research paper, the mathematics lecture, the mathematics book and textbook which typically contain a range of sub-genres. For example, the mathematics textbook consists of chapters and subsections which contain definitions, theorems, explanations of theory, demonstration examples, practice examples and solutions. Thus the mathematics textbook is a multisemiotic genre consisting of sub-genres, which in turn consist of Items. In what follows, Example 2.24, a mathematics problem from Burgmeier et al. (1990: 76-77), an introductory university textbook on calculus, is analysed in order to investigate intersemiosis in contemporary mathematics discourse. Example 2.24 is reproduced in Plate 7.1(1). The theoretical frameworks for the analysis include SF models for language (Tables 3.1(1) and 3.1(2)), mathematical symbolism (Tables 4.3(1) and 4.3(2)) and visual images (Tables 5.2(1) and 5.2(2)), and the systemic framework for intersemiosis across the three resources (Tables 6.1(1) and 6.2 (2a-d). In Example 2.24, readers are asked questions concerning the 'receptivity of students', which is conceptualized as a function of the time elapsed in a lecture. Example 2.24 contains a linguistic description of the problem and the equation G(x) — - 0.1 x2 + 2.6x + 43 which describes receptivity, 'the ability of the students to grasp a difficult concept', as a function of time where x is the number of minutes which have elapsed in a lecture (Burgmeier et al, 1990: 77). The description of the mathematics problem includes a black and white photograph of a group of students in a university lecture. In Example 2.24, the questions concern (a) the times at which receptivity is increasing and decreasing, (b) the situation with regard to student interest after ten minutes, (c) the time in the lecture where the most difficult concept should be placed, and finally (d) whether it would be possible to teach the students a certain concept which 'requires a receptivity of 55' given 'the intelligence level of the students in this group' (ibid.: 77). The solutions to the questions (a)-(d) are given in the form of explanatory linguistic statements, mathematical symbolic statements and Figure

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2.5

Increasing and Decreasing Functions

Example 2.24 By studying the learning behavior of a group of students a psychologist determines that receptivity, the ability of the students to grasp a difficult concept, is dependent on the number of minutes of the teacher's presentation that have elapsed before the concept is introduced. At the beginning of a lecture a student's interest is stimulated, but as time passes, attention becomes diffused. Analysis of this group's results indicate that the ability of a student to grasp a difficult concept is given by the function

where the value G(x) is a measure of receptivity after x minutes of presentation. (a) Determine the values of x for which student receptivity is increasing and decreasing. (b) Is student interest being stimulated after 10 minutes or is attentiveness falling off? (c) Where in the presentation should the most difficult concept be placed? (d) For the intelligence level of the students in this group, a certain concept requires a receptivity of 55. Is it possible to teach the students this concept? Solution (a) Student interest is being stimulated when G(x) is increasing, and attentiveness is falling off when G(x) is decreasing. To determine where G(x) is increasing and where it is decreasing, we use the derivative of G(x) that is, G'(x) = -0.2* +2.6. We know that G(*) is increasing where G'(x)>0, so we solve the inequality which is equivalent to (inequality sign is reversed) Thus, student receptivity G(x) is increasing for x < 13 and decreasing for x > 13. (See Figure 2.24.) (b) Since G(x) is increasing at x = 10, student interest is still being stimulated 10 minutes into the presentation. (c) Receptivity is increasing during the first 13 minutes and decreasing after the first 13 minutes, maximum receptivity occurs 13 minutes into the presentation. Thus, the most difficult concept should be discussed 13 minutes after the presentation begins. (d) Since G(13) = 59.9 and the concept we want to present requires a receptivity of 55, it is possible to teach this concept to these students.

Figure 2.24 Graph of student receptivity function G(x) in Example 2.24

Plate 7.1(1) Mathematics Example 2.24 (Burgmeier et al, 1990: 76-77) 2.24, the graph of the function G(x) where different points of time have been marked. The solution is obtained by considering the derivative G (x) where receptivity increases for G (x) > 0 (i.e. an increase in x corresponds to an increase in G(x)) and decreases for G (x) < 0 (i.e. an increase in x corresponds to decrease in G(x)). In what follows, the linguistic, visual and symbolic choices in Example 2.24 are analysed with respect to the meanings which arise (i) intrasemiotically within the linguistic, visual and symbolic Items and (ii) intersemiotically within and across the Items. The register, genre and ideology arising from such choices are discussed.

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Compositional Organization of Example 2.24

Example 2.24 is a demonstration example, which is a common sub-genre in mathematical textbooks. These examples serve to illustrate standard solutions to typical mathematics problems. Such examples contain obligatory Items in the form of the problem and the solution. In mathematics problems involving the derivative (such as Example 2.24), a graph or sketch is typically included, although this is not always the case. The photograph is optional. The obligatory Items in Example 2.24, the question and the solution, are spatially separated from the optional Items, the photograph and graph. The compositional arrangement is marked by the justification of the linguistic/symbolic text in the column to the left of the two visual images. Within this text, the important symbolic statements are spatially separated through the use of line spacing and centre alignment. Each of the symbolic statements is marked through italicized font. Compositionally, the problem is organized so that the question and the solution (including the symbolic equations), the photograph and the graph are distinct Items. Example 2.24 is textually organized to accommodate a reading path for these four Items. Prominence is given to the question, the solution and the graph through Typography, where 'Example 2.24', the Questions '(a)-(d)', the 'Solution' for (a)-(d) and the Caption for Figure 2.24 appear in bold. At the rank of discourse semantics, intersemiotic patterns of meaning take place across the Items. The Photograph

At the rank of Item, the photograph is visually prominent because of its spatial location on the right-hand top corner of the page and the contrast which is provided by the density of black and white photograph against the white background of the printed text. The photograph is a picture of university students in a lecture theatre. Culturally, this is evident to the reader of the mathematics book (presumably a university student or lecturer) given the nature of the seating arrangements, the age of the students, the style of clothing and the activities which are depicted, such as listening and taking notes. The Gaze of the students in the photograph is directed downwards to the left, most likely towards the stage where the presentation is taking place. The students in the photograph are two male black students and four women. At least three of these women belong to ethnic minority groups. The two black male students physically lean sideward from each other as they look at and listen to the lecture presentation. The contrast provided by their light-coloured clothing and the dynamism provided by the angle formed by their arms and the arm of the woman sitting in the row above converge to an empty space that lacks a centrally framed Figure. However, given their Position and Size, the Figures in the photograph which emerge as prominent are the two male students whose body posture forms a

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dynamic 'V shape. At the rank of Part of the Figure, the facial expressions of these two students suggest a somewhat sceptical reaction to the performance which is taking place. The female students listen with varying degrees of concern, three looking to the left, and one looking down as though writing. The majority of the students in the photograph rest their chins upon their hands in what appears to be a studied reaction to the lecture. The Gaze and action of the students in the photograph do not beckon the reader of the mathematics textbook. Rather the reader's attention is drawn to the picture due to the position and the density of the black and white photograph. However, the students' Gaze and accompanying body posture in the photograph guide the reader's attention downwards to the left towards the block of text which consists of the question and solution in Example 2.24. There is a directional link which stretches from the photograph towards linguistic/symbolic parts of the text through the vector provided by the Gaze and posture of the student in the photograph. INTERSEMIOTIC NEGOTIATION thus includes a discourse move, however fleeting or recursive, from the photograph to the mathematics problem through Gaze and body posture of students in the photograph. The implications of this discourse move are considered below. Intersemiosis: The Photograph and the Linguistic/Symbolic Text

The linguistic text introduces the mathematics problem in the following manner: Example 2.24 By studying the learning behaviour of a group of students a psychologist determines that receptivity, the ability of the students to grasp a difficult concept, is dependent on the number of minutes of the teacher's presentation that have elapsed before the concept is introduced. At the beginning of a lecture a student's interest is stimulated, but as time passes attention becomes diffused. Analysis of this group's results indicate that the ability of a student to grasp a difficult concept is given by the function where the value G(x) is a measure of receptivity after x minutes of presentation.

INTERSEMIOTIC IDENTIFICATION at the rank of discourse involves identifying and tracking participants across the Items in Example 2.24. In the case of the photograph and the linguistic text, INTERSEMIOTIC IDEATION and TRANSITIVITY RELATIONS function to establish what appears to be a VISUALIZATION of the mathematics problem: 'By studying the learning behaviour of a group of students' (photograph of a group of students is provided) 'a psychologist' (a professional whose identity remains anonymous) 'determines that receptivity, the ability of the students to grasp a difficult concept' (photograph of students concentrating on the lecture), 'is dependent on the number of minutes of the teacher's presentation' (a second professional whose identity remains anonymous) 'that have elapsed before the concept is introduced.' 'At the beginning of a

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lecture a student's interest is stimulated' (the students in the photograph appear to be attentive), 'but as time passes attention becomes diffused' (the photograph is taken at some stage during the lecture, given the body posture and facial expression of the students). 'Analysis of this group's results' (the performance of 'this group' as measured by the psychologist) 'indicates that the ability of a student to grasp a difficult concept is given by the function G(x) = -O.lx2 + 2.6x + 43 where the value G(x) is a measure of receptivity after x minutes of presentation' (the notion of 'this group' is linked to the ability of an individual student). From the description of the problem, it becomes apparent that groups of students perform differently, and so the symbolic description of receptivity as a function of time elapsed in a lecture in Example 2.24 is a description of the results of one particular group. And the photograph does contain one group of students; that is, students from ethnic minority backgrounds. The results of the group are re-contextualized in relation to individual performance, 'the ability of a student to grasp a difficult concept', in Example 2.24. The added dimensionality to Example 2.24 which arises intersemiotically from the choices in the photograph and the linguistic text is further discussed below. In what amounts to a VISUALIZATION of the mathematics question, only some of the major participants introduced in the linguistic text are represented in the photograph. The professionals, the 'psychologist' who pronounces the relationship between time and student receptivity and the 'teacher', appear in the linguistic text only. On the other hand, the students are pictured as ethnic minority students. In this way, the linguistic construction 'receptivity', which is reconstrued as 'the ability of the students to grasp a difficult concept', becomes associated with one particular group of students. The view is reinforced that the ability of these minority students is related to how long they can pay attention because 'as time passes, attention becomes diffused'. Thus INTERSEMIOTIC IDEATION relations are established as to which group of students have problems grasping difficult questions given their attention span. Failure 'to grasp concepts' is linked to personal ability, attentiveness and, through the photograph, ethnicity and race. The problem of educational achievement becomes tied to internal psychological, mental and biological criteria such as the attention span, ability and race rather than the complex practices of educational institutions and society as a whole. A whole range of issues such as social, cultural and economic factors are thus distilled into what appears to be an extraordinarily simplistic and misleading construction of the factors underlying educational achievement. Further to this, the professional psychologist is seen to be able to determine the exact nature of the learning process, and the result is a concern with presenting lecture material at the right time. The organization of the lecture is a textual concern where ideational meaning is reduced to the notion of 'a difficult concept'. Features of the lecture and its presentation (for example, the actual content of the lecture, the language and visual images used, the interpersonal stance of the lecturer, the educational

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institution, and other social and cultural contextual factors) are excluded in the formulation of 'student receptivity'. Finally, the notion of 'receptivity' is converted into 'the intelligence level of the students in this group' in question (d). The relations between educational achievement and genetic makeup are reinforced along racial grounds. The implications of these constructions are further considered through an analysis of the linguistic and symbolic text in Example 2.24. Intrasemiosis in Language: The Use of Grammatical Metaphor

The linguistic text, which functions to introduce and contextualize the problem in relation to the photograph and the symbolic equations, is metaphorical as indicated by the numerous instances of grammatical metaphor, which appear in bold below: By studying the learning behaviour of a group of students a psychologist determines that receptivity, the ability of the students to grasp a difficult concept, is dependent on the number of minutes of the teacher's presentation that have elapsed before the concept is introduced. At the beginning of a lecture a student's interest is stimulated, but as time passes attention becomes diffused. Analysis of this group's results indicate that the ability of a student to grasp a difficult concept is given by the function G(x) =-0.1x2 + 2.6x+43 where the value G(x) is a measure of receptivity after x minutes of presentation.

After the explanation of the problem, questions (a)-(d) also contain grammatical metaphors which include: (a) the time when student receptivity is increasing or decreasing, (b) whether at different times interest is being stimulated or attentiveness is falling off, (c) where the most difficult concept should be placed in the presentation, and (d) whether or not a concept requiring a receptivity of 55 could be taught given the intelligence level of the students in this group. In terms of experiential meaning, grammatical metaphor functions to construct a situation where processes and attributes are reconstrued as entities. In this shift, the real participants (the lecturer and students as Medium and/or Agent) and circumstance completely disappear or they are buried within the nominal group structures of the metaphorical entities. For example, 'learning behaviour' is an entity (rather than the process of the students 'behaving' in some fashion) which can be measured by the trained expert, the psychologist. 'Receptivity' is an entity (rather than the process of the students 'receiving' material from the lecturer in some fashion) which is denned to be another entity 'the ability of the students' (rather than an attribute that the students demonstrate in some manner) to grasp the entity 'a difficult concept' (from the process of the students 'conceiving' something from the lecturer in some way). The mathematics problem in effect reduces the complex and dynamic practices of university education into metaphorical entities in the form of nominalized participants which are defined and aligned using relational processes. Following

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this formulation, these metaphorical entities are exactly described using mathematical symbolism in Example 2.24. The use of grammatical metaphor extends to interpersonal meaning in the form of modality metaphors. The series of statements are either unmodalized in terms of probability or usuality, or they are metaphorical in that the modality orientation is explicitly objective; for example, Ts it possible to teach the students this concept?' The interpersonal orientation of the text exudes a confidence and objectivity which is difficult to counteract, especially as the majority of clauses contain relational processes where identifying relations are established between the metaphorical entities. In addition, the textual organization of the clauses reveals marked choices such as foregrounding particular elements such as 'By studying the learning behaviour of a group of students', 'At the beginning of the lecture', and 'For the intelligence level of the students in this group'. Furthermore, logical meaning is metaphorically re-packaged as a process rather than being instantiated as a conjunctive relation; for example, 'Analysis of this group's results indicate that the ability of a student to grasp a difficult concept is given by the function'. In summary, the linguistic text is highly metaphorical, and, further to this, the shift to mathematical symbolic description involves a reconstrual which only includes one dimension of the context of the problem - the time elapsed in the lecture. Intersemiosis: Linguistic Construction of the Problem and the Symbolic Equations

The metaphorical entity 'receptivity' is symbolized as G(x). Through the relational process '=', G(x) is given a definitive description in the form of the mathematical equation G(x) = -O.I;*2 + 2.6x+ 43 where the independent variable is x, the number of minutes that have elapsed after the presentation has started. The mathematical symbolism thus provides a dynamic description of the metaphorical entity 'receptivity' in exact terms where the only variable is the time elapsed in the presentation. In the questions (a)-(d) which follow, 'receptivity' is related to another metaphorical entity, 'the intelligence level of these students' in question (d). As seen above, there are particular students in question as portrayed by the photograph. The nature of the experiential meaning in the linguistic construction of the problem and the intersemiotic shift to the symbolic description are rather incredible. First, the use of grammatical metaphor reduces the complex dynamic process of a university lecture into a series of metaphorical entities which are related to other metaphorical entities through relational processes. This is achieved with a high degree of certainty. The photograph functions to contextualize these relations along racial lines. Second, the symbolism permits the dynamic reconfiguration of that situation in terms of one variable only, the time which has elapsed in a lecture. Other variables concerned with teaching and learning are excluded in the symbolic formulation. In this process, the issue of the viability of measuring and

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describing such an entity as 'receptivity' in precise symbolic form is sidestepped. Grammatical metaphor and other linguistic choices construe metaphorical entities which are re-inscribed symbolically as a dynamic process which unfolds over time, but only within a semantic field which involves x, the time elapsed in the lecture. In the shift from language to the symbolism, the phenomenon of semiotic metaphor is invisible because of the layers of grammatical metaphor. For example, congruent linguistic constructions such as 'the lecturer uses language and other forms of semiosis to teach the students', or 'the student receives the lecture material from the lecturer in multiple semiotic forms' where the lecturer/student is the Medium or Agent are reduced to 'receptivity', 'the ability to grasp a difficult concept'. The mental/material/ behavioural dynamic process of teaching/learning shifts seamlessly from the metaphorical construct 'receptivity' to the symbolic participant G(x). In other words, semiotic metaphor occurs in that the process 'receive' (linguistic) is construed as an entity G(x) (symbolic). Once constituted as a symbolic entity, G(x) becomes a linguistic entity; that is, the 'measure of receptivity after x minutes of presentation'. The development of grammatical metaphor in language is related to the types of metaphorical expansions which take place through intersemiosis. Such grammatical and intersemiotic metaphorical constructions now appear as commonsense knowledge in contemporary discourse. In Example 2.24, the symbolic G(x) is reformulated in terms of a dynamic construal of mathematical participants and Operative processes in relation to the variable of time. In the shift to Figure 2.24, another form of semiotic metaphor occurs in that a new entity in the form of a curve is introduced visually. Thus through the use of language, mathematical symbolism and a graph, the complex dynamic process of teaching/learning is reduced to metaphorical entities and relations. This construal is hedged in rhetoric which implicitly relates educational performance to the innate characteristics of the students such as ability and intelligence which are literally 'viewed' along the lines of ethnicity and race. Intrasemiosis: The Mathematical Symbolism

To answer the questions (a)-(d) posed in Example 2.24, mathematical symbolism is called into play. The derivative G' (x) is calculated and equated to zero to find the times for which receptivity is increasing and decreasing, and the optimal time for the presentation of the most difficult concept. To determine when receptivity is increasing, the value for G' (x) > 0 is found as follows: -0.2x+2.6>0 -0.2x > -2.6 0.2% < 2.6 x<13

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The solution to the problem unfolds seamlessly in the form of an IMPLICATION SEQUENCE. The grammar of mathematical symbolism has been specifically designed exactly for this purpose (see Chapter 4). That is, the symbolism preserves the rankshifted process/participant configurations so that the equation can be reconfigured and simplified in order to find the solution to the problem. The algebraic inequality is rearranged so that the values for which the derivative is greater than zero are found. The solution to Example 2.24 depends upon previously established results which include the definition of the derivative, various algebraic laws, the definition of a negative number, and the Multiplicative Properly of Negative One (which is signalled in Example 2.24 by the statement 'inequality sign is reversed'). Thus the following answers are provided in Example 2.24: (a) student receptivity is increasing for x < 13 and decreasing for x > 13, (b) student interest is found to be still increasing when x = 10, (c) the optimal time for the discussion of the most difficult concept is x= 13 (although the fact that student receptivity must immediately start to decrease after this point of time is not considered) and (d) substituting x= 13 in the original equation G(x) = -0.1x2 + 2.6x+43gives 'the maximum value of receptivity' to be 59.9, which is higher than the 55 required for the concept to be taught to 'this particular group of students'. So apparently it is possible to teach the concept to these students, given 'their intelligence level'. Intersemiosis: The Mathematical Symbolic Solution and the Graph

The mathematical symbolic solutions to questions (a)-(d) are directly linked to the graph of the function, which serves to provide a VISUALIZATION of the results which are established symbolically. For example, INTERSEMIOTIC IDENTIFICATION is realized through direct reference in the caption for Figure 2.24 which reads 'Graph of receptivity function G(x) in Example 2.24'. Intersemiotic Reference takes place through the intersemiotic mechanism of Intersemiotic Mixing where the graph is Labelled G(x) = —O.lx2 + 2.6x + 43. A curved line pointing to the graph of the function accompanies the Labelling in Figure 2.24. The axes are Labelled linguistically 'receptivity' and 'time (in minutes)' and symbolically x and G. In addition, the scales are calibrated numerically. The use of Juxtaposition and compositional arrangement of the labels in Figure 2.24 means that there is no ambiguity as to the identification of the visual participants and circumstance in the graph. The mathematical symbolic relationship consists of rankshifted configurations of mathematical participants and processes indicated by [[ ]]: G(x) = [[[[-0.1[[*x x]]]] + [[2.6*]] +43]] This rankshifted configuration becomes a metaphorical visual entity in the form of a curve in Figure 2.24. The dynamic aspect of time is therefore related to spatiality in the graph. Significant values for (x, G(x)) for

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questions (b) and (d) are Labelled and marked by points on the graph as (10, 59) and (13, 59.9). Interpersonal attention is drawn to the critical turning point of the graph at (13, 59.9) through the dotted vertical line which extends from the x axis to the curve. There is no visual representation of the derivative as the slope of the tangent line at the points which are marked on the curve. The definition of the derivative is assumed given the preceding theory presented in Sections 2.1-2.5 in Burgmeier et al. (1990:45-76). The levels of assumption in this mathematics problem, in relation to its conception (for example, the method through which the psychologist determines that 'receptivity' for a group of students is given by G(x) — —0.1^ + 2.6x+ 43) and the solution, are not addressed in Burgmeier et al. (1990). In Example 2.24, the variable 'receptivity' is a measurable entity which is dependent on time. The terms 'receptivity' and 'intelligence' are reduced to a function of the time elapsed in the teacher's presentation. The problem is solved using mathematical symbolism. The main function of the problem is, however, to demonstrate how the derivative of a function may be used to solve mathematics problems involving increasing and decreasing functions, and this is perhaps the key reason for the formulation of the problem. Nonetheless, it is worthwhile to examine the types of semiotic choices in Example 2.24 in order to investigate the underlying ideological assumptions of the problem. Example 2.24: Register, Genre and Ideology

There are consistent patterns of registerial configurations across the linguistic, symbolic components of the Example 2.24. In terms of experiential meaning, the Bi-directional Investment of Meaning through intersemiosis with respect to the Contextualization Propensity (CP) (Cheong, 1999, 2004) is high. The linguistic items, visual images and mathematical symbolism directly function to contextualize each other so there is a low Interpretative Space (IS) which results in a direct Semantic Effervescence (SE). The formulation and solution to mathematics problems become straightforward standard procedures. Example 2.24, however, involves the application of mathematical concepts and theory to an educational context. The assumptions behind this application of mathematical theory and concepts are not discussed in Burgmeier et al. (1990). In terms of interpersonal meaning, the tenor, or the relation between the author and the reader, is unequal. Following generic conventions, choices from the systems for language, symbolism and visual images are such that the mathematics text appears objective and factual. Statements are made, directions are issued and the problem is solved. The abstract nature of the processes and participants and the style of production of the text contribute to the dominating tenor. Textually, the mathematics problem is tightly organized at the ranks of Item, discourse, grammar and display. There is no ambiguity arising from the conventionalized modes of presentation.

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Logically, the solution to the problem unfolds smoothly according to the meaning potential made available through the grammar of mathematical symbolism. The reasoned logic realized through the symbolism is mirrored in the surrounding linguistic text and the graphical display of the results. As seen in Example 2.24, the use of language, mathematical symbolism and visual representation in mathematics discourse has developed and expanded into new realms of human experience since the days of Newton and Descartes. In Example 2.24, order is imposed on educational practices such that identifiable entities such as 'learning behaviour', 'receptivity', 'the ability of students', 'teacher's presentation', 'difficult concepts' and 'intelligence levels' become commonsense constructs. Firmly entrenched in mathematical and scientific discourse, the use of grammatical and semiotic metaphor functions to order experience such that metaphorical constructs appear as real entities formulated in exact symbolic terms by accredited experts in the field. Once described, exact values may be calculated as they change over time. The nature of these exact relations may be viewed graphically. Within this discourse, certain groups are constructed as subjects in such a way as to provide causes and reasons for central inequalities in the educational system. The flux of experience is reduced to a matter of variables, in the case of Example 2.24, one variable in the form of the time which has elapsed in a university lecture. Explanations can be provided to legitimize and rationalize educational practices which continue to support privilege in society. Following Foucault (1991), this is done in a society that categorizes, measures, evaluates and normalizes. The tools for these descriptions are semiotic. Foucault (1980b, 1984) speaks of scientific discourse as the discourse of truth in contemporary society. Thibault (1997: 108) elaborates: 'This paradigm has provided the basis for approaching the problems of causality not only in physics, but in all other domains of enquiry in both the natural and social sciences.' The mechanisms through which semiotic mediation in mathematics contributes to this exercise are a central focus of concern in this study. Further interpretation of the complexity and the simplicity of the social-semiotic reality of mathematical and scientific discourse is needed, especially in the realm of mathematics and science education. In what follows, the implications of an SF approach to mathematics as a multisemiotic discourse for mathematics education are summarized. 7.2 Educational Implications of a Multisemiotic Approach to Mathematics

An SF perspective of mathematics as a multisemiotic discourse has many implications for mathematics education. In what follows, these are discussed in relation to language, mathematical symbolism, visual display, intersemiosis across the three resources, and semiotic and grammatical metaphor. In addition to providing a theoretical framework through which mathematics may be viewed in mathematics education, the SF approach also permits the analysis of discourse in mathematics classrooms. The

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nature of pedagogical discourse in mathematics classes is discussed in Section 7.3. The analysis reveals the complexity of classroom discourse and the difficulties which inadvertently arise from using English as the metalanguage to teach mathematics. Finally, concluding comments concerning the implications of the semiotic makeup of mathematics and science are made in Section 7.4. The SF Approach to Language

The SF approach to language offers a comprehensive model through which students may come to understand that language is a tool used to create order, and that meaning is a matter of choice. The metafunctional approach shows that content is only one aspect of the order which is imposed on the world. Equally important are the social relations which are enacted, the logical reasoning which takes place and the ways in which the message is organized and delivered. In addition, although meaning may be a matter of choice, students can appreciate that there are culturally specific ways in which language is used in different contexts. The understanding of those ways, and the interests served by such language selections, opens the way for a critical engagement with texts. An understanding that reality is enacted contextually through particular configurations of linguistic choices is critical for interpreting discourse in diverse contexts which span digital media, printed media and material lived-in-day-to-day reality. In contemporary times where information is increasingly a commodity, the ability to critically read, interpret and write is becoming a necessary resource for survival. This is particularly important in the context of education where groups of students are marginalized (Bernstein, 1971, 1973, 1977, 1990, 2000). The nature of linguistic selections in mathematics and science may be contextualized with respect to other possible choices using the SFL approach. Mathematical and scientific language involve particular types of linguistic choices which organize reality in particular ways. Mathematics and science are registers where particular configurations of experiential, logical, interpersonal and textual meanings are found. The choices relate to the functions of language in mathematics; that is, to contextualize the mathematical problem and to draw implications from the results. However, the nature of scientific writing and reading (Halliday and Martin, 1993; Lemke, 1990; Martin and Veel, 1998) need to be situated in relation to the functions of mathematical symbolism and visual images. For example, the semantic drift towards the constructions of metaphorical entities and the expansion of nominal group structures in language needs to be contexualized in relation to the functions and lexicogrammatical strategies found in mathematical symbolism and visual images. Mathematical and scientific language developed in particular ways because the dynamic construal of reality was delegated to mathematical symbolism where the relations could be displayed visually.

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In terms of experiential meaning, mathematical language is largely concerned with relational processes where identifying relations configure metaphorical identities. Logical meaning is concerned with elaborating, extending and enhancing relations for the description of problems and the prediction of outcomes. The interpersonal stance of mathematical language includes a high modality or truth value where statements and commands have maximal probability and obligation. The abstract and nominalized forms of the participants contribute to the authorative stance of mathematics texts, which are organized in generically specific ways. The typography and compositional choices in mathematics texts include the spatial organization of items according to significance; that is, important linguistic, symbolic and visual components are spatially marked so that the reader may easily access information through a recursive scanning-type reading path. Finally, to reiterate an important point, mathematical and scientific language cannot be viewed in isolation. The nature of the selections and the lexicogrammatical strategies for encoding meaning needs to be seen in relation to the mathematical symbolism and visual display. SFL offers a theoretical approach through which this is possible. The SF Approach to Mathematical Symbolism

The SF approach to mathematical symbolism demonstrates how this semiotic resource developed new grammatical strategies for encoding meaning, and the reasons for this development. A historical perspective reveals how realms of meaning were set aside in order to develop a semiotic resource concerned with the descriptions of relations in a de-contextualized environment. The algebraicization of geometry meant that spatial relations (the visual image) could be connected to temporal and logical relations (symbolic descriptions of relations over time) in an exact manner. A semantic circuit was created in that linguistic and mathematical descriptions were tied to visual images. Mathematical symbolism developed as a tool that could be used for reasoning, and the grammar thus developed special techniques where process/participant configurations were preserved for reconfiguration for the solution to problems. These techniques include the development of new grammatical systems, the simultaneous contraction and expansion of process types to include Operative processes, and the use of multiple levels of rankshift so that the symbolic processes and participants can be rearranged in the solution to mathematics problems. The grammar of mathematical symbolism functions differently from the grammar of language, and this needs to be made explicit in an educational context. Language functions to construe metaphorical entities through the expansion of the potential of the nominal group. These entities are related to other entities so that clause complex logical relations are reconfigured as single clauses, and clauses are reconfigured as nominal groups. In this process, logical deduction using language is realized metaphorically through the selection of causative relational processes. However, these

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grammmatical strategies took place in language because the responsibilty for the dynamic description of relations and logical deductions was allocated to the symbolism. Consequently mathematical symbolism developed new grammatical systems (such as the use of spatial notation, special symbols and the Rule of Order for operations) so that meaning was economically and precisely encoded in rankshifted configurations which could be easily rearranged for the solution to mathematics problems. Halliday (1985: 87) compares the 'density of substance' or the 'lexical density' of written text to the 'intricacy of movement' or the 'grammatical intricacy' of spoken language. However, the multiple rankshift or embedding of nuclear configurations of Operative processes in mathematical symbolic text gives what can be described as grammatical density. Even though the participants and processes are typically generalized variables and Operative processes, the symbolic mathematics is experientially dense because of the precision and economy with which meaning is encoded. The preservation of process/participant configurations involves an unprecedented flexibility through a range of grammatical systems which permit those configurations to be rearranged as the symbolic text unfolds. Mathematical symbolism combines the flow of spoken discourse with the density of written language. The difference is that density in mathematical symbolism involves specialized systems for economy of expression and multiple levels of rankshift, while written language packs meaning into extended word group structures. The price for the semantic expansion where mathematical symbolism provides exact descriptions of relations which are reconfigured to solve problems is a limitation of the semantic realm with which mathematics is concerned. Mathematical symbolism is largely concerned with relational and Operative processes with limited forms of circumstance. Logical meaning is aided by textual organization so that solutions of mathematics problems are organized in very specific ways which utilize spatial positioning. Interpersonal meaning is largely restricted to maximal values of modality and modulation where expressions of probablity and uncertainty are encoded through relational processes such as approximations and probability statements. Mathematical symbolism is functional, but only within a certain semantic realm compared to language. This reduction of meaning in the evolution of mathematical symbolism permitted semantic expansions so that the exact description of the relations can be displayed visually. In an educational context, students are typically presented with modern mathematics in a pre-packaged form where the functions and grammar of mathematical symbolism are not discussed. The discourse is presented in such a way that many students do not understand what mathematics is, or how mathematical symbolism developed historically as a semiotic resource in order to fulfil particular functions. Given that the grammar of mathematical symbolism is not taught from a linguistic perspective, the culmulative effect is that many students fail mathematics because they simply do not understand how (or why) mathematical symbolism functions as a resource

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for meaning. Students cannot write in the formats required by educational institutions if they cannot make use of the options provided by the grammar of language and, in a similar manner, they cannot solve mathematical problems without recourse to the grammar of mathematical symbolism, especially as the resource employs new strategies for encoding meaning. The functions and grammar of mathematical symbolism need to be addressed in the teaching and learning of mathematics. The SF Approach to Visual Images

The SF approach to visual images in mathematics includes a description of the functions and grammar of visual representation and how these relate to the algebraic and linguistic descriptions. Mathematical visual images developed to link spatiality with temporality so the visual relations could be exactly encoded using the symbolism. A semantic circuit with language, mathematical symbolism and visual images thus exists. While visual images are accessible as they correspond to perceptual reality, the grammar of visual images, however, needs to be understood because there are specific systems which permit links to be made to the symbolism. A historical perspective demonstrates how mathematics evolved as a discourse, and one means for understanding this development is the changes which occurred in the visual image. A de-contextualization of the visual display took place as human actors and material circumstances were removed as concern focused on the display of relations in the form of lines, curves and three-dimensional objects. Experientially, the participants and processes in visual images are the relations encoded symbolically, which are variously shown as intersecting lines, planes and other forms of visual representation. Visual images are typically multisemiotic as accompanying labels function to identify the relations with respect to symbolic descriptions. These types of selections in mathematical images mean that contemporary mathematical graphs and diagrams appear abstract with a high modality value. However, the development of computer technology permits digital data to be transformed into new forms of visual images. The increasing sophistication of computer graphics are leading to more complex forms of mathematical visual images. This includes the use of systems such as colour and three-dimensional displays that replicate and extend our perception of the world in what has become virtual reality. Modern mathematics developed as a written and printed discourse. However, because of advances in visualization and computation, mathematics and science are entering into a new era where new ways of construing reality are becoming possible. The relations between semiosis and technology could be incorporated into mathematics education.

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Intersemiosis across Language, Mathematical Symbolism and Visual Images

The SFL approach to mathematics as a multisemiotic discourse is significant because it provides a theoretical framework to explain how language, mathematical symbolism and visual images function intersemiotically. Apart from concerning itself with a limited semantic domain, mathematical discourse is successful because: 1 2 3

The meaning potentials of language, symbolism and visual images are accessed. The discourse, grammatical and display systems of each resource function integratively. Meaning expansions occur when the discourse shifts from one semiotic resource to another.

The functions and resulting grammars for language, symbolism and the visual images may be conceptualized as three integrated systems which permit intersemiotic transitions to take place. Intersemiotic transitions consist of macro-shifts from one resource to another across Items, and micro-shifts where choices from one semiotic resource are integrated within another semiotic resource. These transitions give rise to semantic expansions in mathematics. In addition to transitions, mechansisms for intersemiosis include Semiotic Cohesion, Semiotic Mixing, Semiotic Adoption and Juxtaposition. The examination of intersemiosis and intersemiotic mechanisms provides a conceptual framework which adds a new semantic layer to analysis beyond that provided by the SF frameworks for each resource. The systems for intersemiotic transitions and other intersemiotic mechanisms are metafunctionally based, and a new meta-language is useful for examining the semantic overlays which consequently arise. These overlays are formulated as co-contextualizing relations where there is a convergence of meaning, and re-contextualizing relations where divergence occurs. The extent and degree of these contextualizing relations arise through the Bi-Directional Investment of Meaning where the Contextualizing Propensity (CP) and Semantic Effervescence (SE) result in an Interpretative Space (IS) (Cheong, 1999, 2004). Traditionally, mathematics functions to co-contextualize and re-contextualize meanings intersemiotically in such a way that the Interpretative Space (IS) is limited. The potential for re-contextualization is changing with the advent of computer technology. This is not confined to the display of mathematical visual images where computer graphics display numerical rather than analytical solutions in rapidly changing dynamic formats. Computer technology is also revolutionizing mathematics in terms of approach (non-linearity versus linearity), method (computation versus analytic solution) and medium and materiality (computerized environment rather than print).

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Semiotic and Grammatical Metaphor

Semiotic metaphor is an important means for semantic expansions in mathematical discourse. Semiotic metaphor occurs when a choice from one semiotic resource undergoes a change of functional status in the transition to another semiotic resource. For example, a linguistic process ('to measure') becomes a symbolic or visual entity or circumstance (an xin the symbolism, or a distance in a diagram). In turn, these metaphorical semantic expansions are absorbed into the systems for each resource and thus contribute to the dynamic evolution of semiosis. The types of cocontextualizing and re-contextualizing relations which take place with semiotic metaphor extend beyond those possible within one semiotic resource. Co-contextualization in the form of parallel semiotic metaphors, and re-contextualization in the case of divergent semiotic metaphors, occur. Alternatively or concurrently, the transition to another semiotic resource may also permit new functional elements to be introduced. These metaphorical shifts which take place across language, symbolism and visual images are not always noticed because the high incidence of grammatical metaphors in mathematical and scientific language create a semantic layer over the semiotic metaphors occurring through intersemiosis. Grammatical metaphor in language is most likely a response to the dynamic function fulfilled by mathematical symbolism and the types of semantic expansions which take place intersemiotically. For example, if there is a shift from a linguistic process to a visual entity then the linguistic reconstrual of that visual entity results in a linguistic entity. However, that linguistic entity has traditionally been viewed as a case of grammatical metaphor. In other words, if one is only examining language, then the shift is perceived to be language -> language rather than language -» visual image/mathematical symbolism —> language. Thus grammatical metaphor may be viewed as the product of semiotic metaphor. Scientific and mathematical constructions appear as commonsense knowledge despite their metaphorical nature because these types of constructions have become 'the way' of establishing order. The scientific view of the world is the means through which truth is established, regardless of the context or the field of human endeavour. Semiotic metaphor is further discussed in relation to the nature of pedagogic discourse in mathematics classrooms. In this context where the oral discourse tends to be congruent, semiotic metaphors are traceable. 7.3 Pedagogical Discourse in Mathematics Classrooms

The theorization of mathematical discourse in this study concerns printed mathematical texts and the evolution of new forms of mathematics made possible through computer technology. In the context of the classroom, however, mathematics is taught using a variety of modes which include the black/white board, printed material including mathematical textbooks,

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practice sets of examples and test papers, computer software, graphics calculators, and mathematical models and equipment. The analysis of mathematical pedagogical discourse must necessarily take into account the multisemiotic nature of mathematics, and the shifts between the written/ spoken modes and the shifts between language, symbolism and visual display. From this perspective a new multimodal approach to discourse analysis emerges (O'Halloran, 1996, 2000), one that integrates the multisemiotic written and spoken modes of mathematics with the use of other systems of meaning such as gesture, body movement, proxemics and so forth. In what follows, the implications of the SF perspective to pedagogical discourse in mathematics classrooms are discussed. The constant shifts between the written and spoken modes in mathematics classrooms mean that the preceding theory of the functions and grammars for language, mathematical symbolism and visual display is relevant in terms of both intrasemiosis and intersemioisis. While mathematics evolved as a written discourse, the interaction between the teacher and the student in the context of the mathematics classroom involves spoken language. The meta-language for teaching and learning mathematics in this study is English, which includes the verbalization of the symbolic and visual descriptions. While gestures such as pointing and hand movements, facial expression, and body movement are significant in classroom interactions, the teacher and students inevitably resort to language at each stage of the lesson. Although teacher-talk dominates mathematical classrooms (O'Halloran, 1996; Veel, 1999), SF discourse analysis reveals that the patterns of interaction in classrooms are different (O'Halloran, 1996, 2004c). The nature of the discourse in three mathematics classrooms differentiated on the basis of gender and socio-economic status are summarized after the more general features of mathematical pedagogic discourse are outlined below. The texture of mathematical pedagogical discourse is dense as the spoken mode provides the meta-language for the action which takes place in the temporal material unfolding of the lesson which includes the written mathematics. For example, the INTERSEMIOTIC IDENTIFICATION of participants extends across language, symbolism and visual images in the written and spoken modes. As a result, major reference chains continually split and conjoin as the solution to the problem is derived. The grammar of mathematical symbolism contributes to the dense texture of classroom discourse. The constant reconfiguration of the symbolic rankshifted Operative processes and participants and the nature of taxonomic relations increase the difficulty of tracking participants. In terms of logical meaning, there are long chains of logical reasoning which typically involve a high incidence of consequential-type relations. In many cases these relations are based on previously established mathematical results which are often left implicit. Interpersonally, the authorative interpersonal stance of written mathematics may be replicated in the oral discourse, although analysis demonstrates that the nature of relations established between the teacher

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and the students varies across classrooms and schools (O'Halloran, 1996, 2004c). The use of English as the meta-language for the symbolic and visual texts involves metaphorical-type constructions that typically remain unnoticed (O'Halloran, 1996, 2000). This may be one important factor accounting for learning difficulties in mathematics. The verbalization of symbolic mathematics, for instance, often results in semiotic metaphor. For example, the teacher may refer to G(x) = -O.I;*2 + 2.6x+ 43 in Example 2.24 in the following manner: 'the function G(x) is negative zero point one x to the power of two plus two point six xplus forty-three'. The symbolic statement means G(*) = [[[[-0.1 [[xx *]]]] +[[2.6*]]+43]] where [[ ]] indicate rankshifted configurations of Operative processes and participants. However, the Operative processes become participants in the verbalization of the symbolic statement, for example, 'the function G(x) is [[negative zero point one x to the power of two plus two point six x plus forty-three] ]'. In other words, the linguistic entity in the form of the nominal group 'negative zero point one x to the power of two' and 'two point six x' replaces the process/participant configurations -0.1 X [[xx x]] and 2.6 x x Similarly, 'the sum of the square root of x squared and five x to the power of 4' is realized linguistically as an entity because it takes the form of a nominal group. However, there are multiple configurations of process/participants in the symbolic form. Jx2 + 5x* as indicated by [[^/[[x 2 ]]]] + [[5x[[xx x x xx x]]]]. This raises an important issue. Although verbal descriptions of the symbolic statements and visual images constantly take place in the mathematics classroom, the metaphorical nature of those linguistic constructions is not usually discussed. Although the verbalizations permit exact translations back to the mathematical symbolic statement, the impact of the metaphorical form of those constructions needs to be considered. Otherwise, students are presented with linguistic entities which are, in fact, complex configurations of mathematical processes with their associated participants. The consequent shifts in meaning that take place in oral pedagogical discourse need to be addressed so that students understand the metaphorical nature of those linguistic constructions in relation to the symbolic and visual forms. The nature of pedagogical discourse in mathematics varies greatly across institutions and teachers (O'Halloran 1996, 2004c). The analysis of discourse in an elite private school for male students reveals that the metafunctionally based choices in the pedagogical discourse and blackboard texts mirror those typically found in mathematical discourse. These students subsequently performed exceptionally well in university entrance examinations in mathematics and other subjects. Interpersonal dimensions of the interaction between the male teacher and female students in a similarly placed private school for girls were found to be orientated towards deferentiality compared to the patterns of dominance enacted in the lesson with male students. The female students, nonetheless, performed well in

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university entrance exams, although not to the same level as the male students. The most marked difference in classroom discourse was found in the lesson in the low socio-economic government school, where the pedagogical discourse was different in all respects from that found in the other two classrooms. The analysis indicated that the major concern of this lesson was the maintenance of interpersonal relations rather than the experiential content of the lesson. As far as the mathematical content was concerned, the register selections in the classroom discourse were colloquial rather than technical, and, more generally, the oral discourse and non-generic ellipsed format of the blackboard texts tended to be context dependent. Coherence in the lesson depended to a large extent on the immediate physical context of the classroom and accompanying physical gestures of the teacher and the students. In this classroom, '[t]he orientation towards particularistic/local/context-dependent meanings . . . does not accord with the universal, remote and seemingly context-independent meanings of mathematics' (O'Halloran, 2004c). The students who attended this school did not perform well in university entrance examinations. These results indicate that while we may speak of classroom discourse in mathematics classrooms, the reality is that the discourse differs across institutions. The practices and processes of teaching/learning mathematics are as diverse as the outcomes. This discussion has indicated some areas in which the study of semiotics of mathematics is useful for mathematics education. In particular, the SF approach offers a comprehensive social-semiotic theory of language through which mathematical symbolism and visual display may be viewed. Using this approach, the complexity of classroom interactions, including the oral discourse, the physical context and the non-verbal behaviour of the teacher and the students, may be analysed. The SF approach includes the development of SFL theory to account for the context of culture (education), the context of situation (school type and class), the school curriculum in the form of macro-genres (Christie, 1997), the lesson genre (the type of lesson), the micro-genres and activity sequences which constitute the lesson, and the use of language, visual images, language and other semiotic resources in those activity sequences (O'Halloran, 1996). In addition, the use of computers and software applications and the internet for teaching and learning mathematics may also be investigated. There are many applications of the SFL approach to mathematics education which require further research and development. 7.4 The Nature and Use of Mathematical Constructions

Mathematical and scientific discourses are semiotic constructions which have moved beyond the printed page into new realms of meaning. These semiotic constructions have proved enormously powerful for the reshaping of the physical world and developing technology. The uses of mathematics and science have also extended into other realms such as the social sciences

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where the problematic nature of the metaphorical constructions which consequently arise may be demonstrated through close textual analysis as undertaken in Section 7.1. Further to this, a commonsense reading of reality today is often based on a mathematical and scientific view of the world. The nature of that semiotic construction has been the subject of this study. In conclusion, a few final comments are made below. From the SF perspective developed here, critical concepts in mathematics and science such as abstraction, contextual independence, reason, objectivity and truth are viewed as particular types of semiotic choices made from the available systems from the grammars of mathematical symbolism, visual display and language. Abstraction is the re-contextualization which took place historically where 'superfluous' information was put aside in the pursuit of knowledge which entailed the description of relations in a form which could be visualized and rearranged for the solution to problems. Similarly, objectivity is the organization of particular experiential and logical realms of meaning which are accompanied by a contracted interpersonal stance. From this view, objectivity becomes a 'valued' cultural product which is enacted semiotically. Reason becomes the rearrangement of relations which can be undertaken with the available semiotic tools. And the nature of truth is reduced to the nature of the semiotic constructions found in the mathematical and scientific views of the world. The scientific method initially involved describing the physical world where new entities were introduced to explain physical phenomena. Modern mathematical and scientific constructions are effective in that they successfully model physical systems up to the point where they become nonlinear. From here, non-linear dynamical systems theory introduces new computerized techniques to describe and predict the behaviour of chaotic systems. However, mathematical and scientific descriptions have extended into the realms of the economic, the political, the social, the educational and the private. The re-contextualization which took place through Descartes, Newton and countless other mathematicians to produce modern mathematics has undergone further re-contextualization in what could only be described as the mathematicization of the human condition. Meanwhile, science and technology advance rapidly, especially in the fields of computer technology, the life sciences and the military. We need to come to terms with the functions and limitations of mathematical and scientific descriptions which increasingly appear to be harnessed to the demands of capitalism, before the political rhetoric surrounding such advances betrays us all.

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References

Anderson, M., Saenz-Ludlow, A., Zellweger, S. and Cifarelli, V. (eds) (2003) Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing. Ottawa: Legas Publishing. Archimedes (1615) Opera Quce Extant/Novis Demonstrationibus Commentariisque Illustrata perD. Rivaltum. Paris. Azzouni, J. (1994) Metaphysical Myths, Mathematical Practice: The Ontology and Epistemology of the Exact Sciences. New York: Press Syndicate of the University of Cambridge. Baldry, A. P. (2000a) 'English in a visual society: comparative and historical dimensions in multimodality and multimediality'. In A. P. Baldry (ed.), Multimodality and Multimediality in the Distance Learning Age. Campobasso, Italy: Palladino Editore, pp. 41-89. Baldry, A. P. (ed.) (2000b) Multimodality and Multimediality in the Distance Learning Age. Campobasso, Italy: Palladino Editore. Baldry, A. P. (2004) 'Phase and transition type and instance: patterns in media texts as seen through a multimodal concordancer'. In EL L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 83-108. Baldry, A. P. (in press) 'The role of multimodal concordancers in multimodal corpus linguistics'. In T. Royce and W. Bowcher (eds), New Directions in the Analysis of Multimodal Discourse. New Jersey: Lawrence Erlbaum Associates. Baldry, A. P. and Thibault, P. J. (2001) 'Towards multimodal corpora'. In G. Aston and L. Burnard (eds), Corpora in the Description and Teaching of English. Bologna: CLUEB, pp. 87-102. Baldry, A. P. and Thibault, P. J. (in press a) Multimodal Transcription and Text. London: Equinox. Baldry, A. P. and Thibault, P. J. (in press b) 'Multimodal corpus linguistics'. In G. Thompson and S. Hunston (eds), System and Corpus: Exploring Connections. London: Equinox. Barry, J. (1996) Measures of Science: Theological and Technical Impulses in Early Modern Thought. Illinois: Northwestern University Press.

212

REFERENCES

Barthes, R. (1972) Mythologies. London: Jonathan Cape. Barthes, R. (1974) S/Z. New York: Hill and Wang. Bateson, G. (1972) Steps to an Ecology of Mind. New York: Ballatine Books. Beevor, A. (2002) Berlin: The Downfall, 1945. London: Viking. Berge, L., Gouedard, C., Schj0dt-Eriksen,J. and Ward, H. (2003) Tilamentation patterns in Kerr media vs. beam shape robustness, nonlinear saturation and polarization states'. PhysicaD, 176(3-4), 181-211. Berger, P. L. and Luckmann, T. (1991) The Social Construction of Reality. London: Penguin. Bernstein, B. (1971) Class, Codes and Control: Volume 1: Theoretical Studies towards a Sociology of Language. London: Routledge and Kegan Paul. Bernstein, B. (1973) Class, Codes and Control: Volume 2: Applied Studies towards a Sociology of Language. London: Routledge and Kegan Paul. Bernstein, B. (1977) Class, Codes and Control: Volume 3: Towards a Theory of Educational Transmission. London: Routledge and Kegan Paul. Bernstein, B. (1990) The Structuring of Pedagogic Discourse: Volume PV: Class, Codes and Control. London: Routledge. Bernstein, B. (2000) Pedagogy, Symbolic Control and Identity: Theory, Research and Critique. Lanham: Rowan and Littlefield Publishers, Inc. Berry, M. (1981) 'Systemic linguistics and discourse analysis: a multilayered approach to exchange structure'. In M. C. Coulthard and M. Montgomery (eds), Studies in Discourse Analysis. London: Routledge and Kegan Paul, pp. 120-145. Bloor, T. and Bloor, M. (1995) The Functional Analysis of English: A Hallidayan Approach. New York: Edward Arnold. Bordell, D. and Thompson, K. (2001) Film Art: An Introduction. New York: McGraw-Hill. Borowski, E. J. and Borwein, J. M. (eds) (1989) Collins Dictionary of Mathematics. London, Harper Collins. Bourdieu, P. (1989) Distinction: A Social Critique of the Judgement of Taste. London: Routledge. Bourdieu, P. (1991) Language and Symbolic Power. Cambridge: Polity Press. Burgmeier, J. W., Boisen, M. B. and Larsen, M. D. (1990) Brief Calculus with Applications. New York: McGraw-Hill. Cajori, F. (1993) A History of Mathematical Notations (Two Volumes). New York: Dover Publications [first published 1928 and 1929]. Callaghan, J. and McDonald, E. (2002) 'Expression, content and meaning in language and music: an integrated semiotic analysis'. In P. McKevitt, S. O'Nuallain and C. Mulvihill (eds), Language, Vision and Music. Selected papers from the 8th International Workshop on the Cognitive Science of Natural Language Processing, Galway, Ireland, 1999. Advances in Consciousness Research, Volume 35. Amsterdam: Benjamins, pp. 205-220. Cameron, D. and Kulick, D. (2003) Language and Sexuality. Cambridge: Cambridge University Press. Carter, R. (1998) Vocabulary: Applied Linguistic Perspectives. London: Routledge.

REFERENCES

213

Chandler, D. (2002) Semiotics: The Basics. London: Routledge. Chen, Y. (2001) 'The use of grammatical metaphor by EFL learners and their language profiency: exploring the language maturity of Chinese university students'. PhD thesis, National University of Singapore. Cheong, Y. Y. (1999) 'Construing meaning in multi-semiotic texts — a systemic functional approach'. MA thesis, National University of Singapore. Cheong, Y. Y. (2004) 'The construal of ideational meaning in print advertisements'. In K. L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 163-195. Chomsky, N. (1965) Aspects of the Theory of Syntax. Cambridge, MA: MIT Press. Chomsky, N. (2000) New Horizons in the Study of Language and Mind. Cambridge: Cambridge University Press. Christie, F. (1997) 'Curriculum macrogenres as forms of initiation into a culture'. In F. Christie and J. R. Martin (eds), Genre and Institutions: Social Processes in the Workplace and School. London: Continuum, pp. 134160. Christie, F. (ed.) (1999) Pedagogy and the Shaping of Consciousness: Linguistic and Social Processes. London: Continuum. Christie, F. and Martin, J. R. (eds) (1997) Genre and Institutions: Social Processes in the Workplace and School. London: Continuum. Clerc, J. L. (2003) 'Special prehomogeneous vector spaces associated to F4, E6, E7, E8 and simple Jordan algebras of rank 3'. Journal of Algebra, 264(1), 98-128. Cobley, P. (ed.) (2001) The Routledge Companion to Semiotics and Linguistics. London: Routledge. Colonna,J. F. (1994) 'Scientific display: a means of reconciling artists and scientist'. In C. Pickover and S. K. Tewksbury (eds), Frontiers of Scientific Visualization. New York: John Wiley and Sons, pp. 181-212. Coulthard, R. M. (1993) 'On beginning the study of forensic texts: corpus, concordance, collocation'. In M. Hoey (ed.), Data, Description, Discourse. London: Harper Collins, pp. 86-114. Danaher, S. (2001) Digital 3d Design. London: Cassell. Dantzig, T. (1954) Number: The Language of Science. New York: Free Press. Davies, P. C. W. (1990) 'Why is the world knowable'. In R. E. Mickens (ed.), Mathematics and Science. Singapore: World Scientific, pp. 14-54. Davis, P. J. (1974) 'Visual geometry, computer graphics and theorems of perceived type'. Proceedings of Symposia in Applied Mathematics, 20, 113— 127. Davis, P. J. (1994) 'Mathematics and art: cold calipers against warm flesh?' In P. Ernest (ed.), Mathematics, Education and Philosophy: An International Perspective. London: Falmer, pp. 165-183. Davis, P. J. (2000) The Education of a Mathematician. Natwick, MA: A. K. Peters Ltd. Davis, P. J. (2003) 'The decline, fall and current resurgence of visual geometry'. Lecture, Brown University, 20 November.

214

REFERENCES

Davis, P. J. and Hersh, R. (1986) Descartes' Dream: The World According to Mathematics. New York: Harcourt Brace Jovan ovich. Derewianka, B. (1995) 'Language development in the transition from childhood to adolescence: the role of grammatical metaphor'. PhD thesis, Macquarie University. Derrida, J. (1978) Writing and Difference. London: Routledge and Kegan Paul. Descartes, R. (1682) Epistolae (Part 1). Amsterdam. Descartes, R. (1683) Geometria, anno 1673 GalliceEdita/Postea Cum Notts Fl. de Beaune in Latinam Linguam Versa/et Commentariis Illustrata Opera Francisci a Schooten: Volume 1. Amsterdam. Descartes, R. (1952) Descartes' Philosophical Writings (ed. and trans. N. Kemp Smith). London: Macmillan. Descartes, R. (1954) The Geometry of Rene Descartes. New York: Dover. Descartes, R. (1985a) The Philosophical Writings of Descartes: Volume 1. Cambridge: Cambridge University Press. Descartes, R. (1985b) The Philosophical Writings of Descartes: Volume 2. Cambridge: Cambridge University Press. Descartes, R. (1985c) 'Principles of philosophy'. In The Philosophical Writings of Descartes: Volume 1. Cambridge: Cambridge University Press, pp. 177-291. Eggins, S. (1994) An Introduction to Systemic Functional Linguistics. London: Pinter. Eggins, S. and Slade, D. (1997) Analysing Casual Conversation. London: Continuum. Eisenstein, E. L. (1979) The Printing Press as an Agent of Change. Cambridge: Cambridge University Press. Emmer, M. (ed.) (1993) The Visual Mind: Art and Mathematics. Cambridge, MA: MIT Press. Euclid (1956) The Thirteen Books of Euclid's Elements: Translated from the Text of Heiberg: With Introduction and Commentary by Thomas L. Heath: Volume 1 (Introduction and Books I, II). New York: Dover Publications, Inc. Fairclough, N. (1989) Language and Power. London: Longman. Febvre, L. and Martin, H. J. (1976) The Coming of the Book. London: Verso. Field, J. V. (ed.) (1997) The Invention of Infinity: Mathematics and Art in the Renaissance. New York: Oxford University Press. Foley, J., van Dam, A., Feiner, S. and Hughes, J. (1990) Computer Graphics, Principles and Practice. London: Addison Wesley. Foucault, M. (1970) The Order of Things. London: Routledge. Foucault, M. (1972) The Archaeology of Knowledge. London: Tavistock Publications. Foucault, M. (1980a) The History of Sexuality: Volume 1. New York: Random House. Foucault, M. (1980b) Power/knowledge: Selected Interviews and other Writings 1972-1977. London: Harvester Press.

REFERENCES

215

Foucault, M. (1984) 'Truth and power'. In P. Rabinow (ed.), The Foucault Reader. London: Penguin, pp. 51-75. Foucault, M. (1991) Discipline and Punish. London: Penguin. Francis, G. (1985) Anaphoric Nouns. Birmingham: University of Birmingham. Freud, S. (1952) Introductory Lectures on Psycho-analysis. London: Allen and Unwin. Freud, S. (1954) The Interpretation of Dreams. London: Allen and Unwin. Funkenstein, A. (1986) Theology and the Scientific Imagination from the Middle Ages to the Seventeenth Century. New Jersey: Princeton University Press. Galileo, G. (1638) Discorsi e Dimostrazioni Matematiche, Intorno a Due Nuoue Scienze Altenenti. Leida: appresso gli Elsevirii. Galison, P. (2002) 'Images scatter into data, data gathers into images'. In B. Latour and P. Weibel (eds), Iconoclash: Beyond the Image Wars in Science, Religion, and Art. Karlsruhe: ZKM, pp. 300-323. Gleick,J. (1987) Chaos: Making a New Science. New York: Viking. Goldhagen, D. J. (1996) Hitler's Willing Executioners: Ordinary Germans and the Holocaust. New York: Knopf. Grave, M. and Le Lous, Y. (1994) 'Visualization services in large scientific computing centres'. In M. Grave, Y. L. Lous and W. T. Hewitt (eds), Visualization in Scientific Computing. Berlin: Springer-Verlag, pp. 10— 28. Gregory, M. (1995) 'Generic expectancies and discoursal surprises: John Donne's "The Good Morrow" '. In P. H. Fries and M. Gregory (eds), Discourse in Society: Systemic Functional Perspectives. Meaning and Choice in Language: Studies for Michael Halliday. Norwood, NJ: Ablex, pp. 67-84. Gregory, M. (2002) 'Phasal analysis within communication linguistics: two constrastive approaches'. In P. H. Fries, M. Cummings, D. Lockwood and W. Spruiell (eds), Relations and Functions Within and Around Language. London: Continuum, pp. 316-345. GroB, M. (1994) Visual Computing: The Integration of Computer Graphics, Visual Perception and Imaging. Berlin: Springer-Verlag. Gumperz,J.J. (1982) Discourse Strategies. Cambridge: Cambridge University Press. Guo, L. (2004a) 'Multimodality in a biology textbook'. In K. L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 196—219. Guo, L. (2004b) 'A systemic-functional analysis of multisemiotic biology texts'. PhD thesis, National University of Singapore. Halliday, M. A. K. (1961) 'Categories of the theory of grammar'. Word, 17(3), 241-292 [reprinted in Halliday (1976): System and Function in Language]. Halliday, M. A. K. (1973) Explorations in the Functions of Language. London: Edward Arnold. Halliday, M. A. K. (1978) Language as Social Semiotic: The Social Interpretation of Language and Meaning. London: Edward Arnold.

216

REFERENCES

Halliday, M. A. K. (1985) Spoken and Written Language. Geelong, Victoria: Deakin University Press [Republished by Oxford University Press 1989]. Halliday, M. A. K. (1993a) 'On the language of physical science'. In M. A. K. Halliday and J. R. Martin (eds), Writing Science: Literacy and Discursive Power. London: Falmer, pp. 54—68. Halliday, M. A. K. (1993b) 'Some grammatical problems in scientific English'. In M. A. K. Halliday and J. R. Martin (eds), Writing Science: Literacy and Discursive Power. London: Falmer Press, pp. 69-85. Halliday, M. A. IL (1994) An Introduction to Functional Grammar (2nd ed). London: Arnold. Halliday, M. A. K. (1998) 'Things and relations: regrammaticising experience as technical knowledge'. In J. R. Martin and R. Veel (eds), Reading Science: Critical and Functional Perspectives on Discourses of Science. London: Routledge, pp. 185-235. Halliday, M. A. K. (2004) An Introduction to Functional Grammar (3rd ed), revised by C. M. I. M. Matthiessen. London: Arnold. Halliday, M. A. K. and Hasan, R. (1976) Cohesion in English. London: Longman. Halliday, M. A. K. and Hasan, R. (1985) Language, Context, and Text: Aspects of Language in a Social-Semiotic Perspective. Geelong, Victoria: Deakin University Press [Republished by Oxford University Press 1989]. Halliday, M. A. K. and Martin, J. R. (1993) Writing Science: Literacy and Discursive Power. London: Falmer. Halliday, M. A. K. and Matthiessen, C. M. I. M. (1999) Construing Experience through Meaning: A Language Based Approach to Cognition. London: Cassell. Hamming, R. W. (1980) 'The unreasonable effectiveness of mathematics'. American Mathematical Monthly, 87(2), 81-90. Hasan, R. (1984) 'Coherence and cohesive harmony'. In J. Flood (ed.), Understanding Reading Comprehension: Cognition, Language and the Structure of Prose. Newwark, DE: International Reading Association, pp. 181-219. Hasan, R. (1985) 'The structure of a text'. In M. A. K. Halliday and R. Hasan (eds), Language, Context, and Text: Aspects of Language in a SocialSemiotic Perspective. Geelong, Victoria: Deakin University Press, pp. 70-96 [Republished by Oxford University Press 1989]. Hasan, R. (1996a) The grammarian's dream: lexis as most delicate grammar'. In C. Cloran, D. Butt and G. Williams (eds), Ways of Saying: Ways of Meaning, Selected Papers of Ruqaiya Hasan. London: Cassell, pp. 73-103. Hasan, R. (1996b) The ontogenesis of ideology: an interpretation of mother-child talk'. In C. Cloran, D. Butt and G. Williams (eds), Ways of Saying: Ways of Meaning, Selected Papers of Ruqaiya Hasan. London: Cassell, pp. 133-151. Herman, E. S. and Chomsky, N. (1988) Manufacturing Consent: The Political Economy of the Mass Media. New York: Pantheon.

REFERENCES

217

Hjelmslev, L. (1961) Prolegomena to a Theory of Language. Madison, WI: University of Wisconsin Press. Hooper, A. (1949) Makers of Mathematics. London: Faber and Faber. Horkheimer, M. and Adorno, T. W. (1972) Dialectic of Enlightenment. New York: The Seabury Press, ledema, R. (2001) 'Analysing film and television: a social semiotic account of hospital: an unhealthy business'. In T. van Leeuwen and C. Jewitt (eds), Handbook of Visual Analysis. London: Sage, pp. 183-204. ledema, R. (2003) 'Multimodality, resemioticization: extending the analysis of discourse as a multisemiotic practice'. Visual Communication, 2(1), 29-57. Jewitt, C. (2002) 'The move from page to screen: the multimodal reshaping of school English'. Visual Communication, 1(2), 171-195. Joseph, G. G. (1991) The Crest of the Peacock: Non-European Roots of Mathematics. London: I.B. Tauris Company. Klein, J. (1968) Greek Thought and the Origin of Algebra. Cambridge, MA: MIT Press. Kline, M. (1972) Mathematics in Western Culture. London: Pelican. Kline, M. (1980) Mathematics: The Loss of Certainty. New York: Oxford University Press. Kockelkoren, J., Storm, C. and van Saarloos, W. (2003) 'Evidence for slow velocity relaxation in front propagation in Rayleigh-Bernard convection'. PhysicaD, (174), 168-175. Koesder, A. (1959) The Sleepwalkers: A History of Man's Changing Vision of the Universe. Harmondsworth: Penguin. Kok, K. C. A. (2004) 'Multisemiotic mediation in hypertext'. In K. L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 131-159. Kress, G. (2000) 'Multimodality'. In B. Cope and M. Kalantzis (eds), Multiliteracies: Literacy Learning and the Design of Social Futures. London: Routledge, pp. 182-202. Kress, G. (2003) Literacy in the New Media Age. London: Routledge. Kress, G., Jewitt, C., Ogborn, J. and Tsatsarelis, C. (2001) Multimodal Teaching and Learning: The Rhetorics of the Science Classroom. London: Continuum. Kress, G. and van Leeuwen, T. (1996) Reading Images: The Grammar of Visual Design. London: Routledge. Kress, G. and van Leeuwen, T. (2001) Multimodal Discourse: The Modes and Media of Contemporary Communication Discourse. London: Arnold. Kress, G. and van Leeuwen, T. (2002) 'Colour as a semiotic mode: notes for a grammar of colour'. Visual Communication, 1 (3), 343-368. Kuhn, T. S. (1970) The Structure of Scientific Revolutions. Chicago: University of Chicago Press. Latour, B. and Weibel, P. (eds) (2002) Iconoclash: Beyond the Image Wars in Science, Religion, and Art. Karlsruhe: ZKM. Layzer, D. (1990) Cosmogenesis. Oxford: Oxford University Press.

218

REFERENCES

Lemke,J. L. (1990) Talking Science: Language, Learning and Values. Norwood, NJ: Ablex Publishing Company. Lemke, J. L. (1998a) 'Metamedia literacy: transforming meanings and media'. In D. Reinking, L. Labbo, M. McKenna and R. Kiefer (eds), Handbook of Literacy and Technology: Transformations in a Post-Typographic World. Hillsdale, NJ: Erlbaum, pp. 283-301. Lemke, J. L. (1998b) 'Multiplying meaning: visual and verbal semiotics in scientific text'. InJ. R. Martin and R. Veel (eds), Reading Science: Critical and Functional Perspectives on Discourses of Science. London: Routledge, pp. 87-113. Lemke, J. L. (2000) 'Multimedia demands of the scientific curriculum.' Linguistics and Education, 10(3), 247-271. Lemke, J. L. (2002) 'Travels in hypermodality'. Visual Communication, 1(3), 299-325. Lemke, J. L. (2003) 'Mathematics in the middle: measure, picture, gesture, sign, and word'. In M. Anderson, A. Saenz-Ludlow, S. Zellweger, S. and V. V. Cifarelli (eds), Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing. Ottawa: Legas, pp. 215-234. Levine, P. and Scollon, R. (eds) (2004) Discourse and Technology: Multimodal Discourse Analysis. Washington, DC: Georgetown University Press. Levkowitz, H. (1997) Colour Theory and Modeling for Computer Graphics, Visualization, and Multimedia Applications. Dodrecht: Kluwer Academic Publishers. Lim, F. V. (2002) 'The Analysis of Language and Visual Images-an Integrative Multisemiotic Approach'. MA thesis, National University of Singapore. Lim, F. V. (2004) 'Developing an integrative multisemiotic model'. In K. L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 220-246. Lynch, M. and Woolgar, S. (eds) (1990) Representation in Scientific Practice. Cambridge, MA: MIT Press. Martin, J. R. (1992) English Text: System and Structure. Amsterdam: John Benjamins. Martin, J. R. (1993a) 'Life as a noun: arresting the universe in science and humanities'. In M. A. K. Halliday andj. R. Martin (eds), Writing Science: Literacy and Discursive Power. London: Falmer, pp. 221-267. Martin, J. R. (1993b) 'Literacy in science: learning to handle text as technology'. In M. A. K. Halliday and J. R. Martin (eds), Writing Science: Literacy and Discursive Power. London: Falmer Press, pp. 166-202. Martin, J. R. (1993c) 'Technicality and abstraction: language for the creation of specialised texts'. In M. A. K. Halliday andj. R. Martin (eds), Writing Science: Literacy and Discursive Power. London: Falmer, pp. 203-220. Martin, J. R. (1997) 'Analysing genre: functional parameters'. In F. Christie and J. R. Martin (eds), Genre and Institutions: Social Processes in the Workplace and School. New York: Cassell, pp. 3-39. Martin, J. R., Matthiessen, M. I. M. and Painter, C. (1997) Working with Functional Grammar. London: Arnold.

REFERENCES

219

Martin, J. R. and Rose, D. (2003) Working with Discourse: Meaning Beyond the Clause. London: Continuum. Martin, J. R. and Veel, R. (eds) (1998) Reading Science: Critical and Functional Perspectives on Discourses of Science. London: Routledge. Matthiessen, C. (1995) Lexicogrammatical Cartography: English Systems. Tokyo: International Language Sciences Publishers. Matthiessen, C. M. I. M. (1991) 'Language on language: the grammar of semiosis'. Social Semiotics, 1(2), 69-111. Messaris, P. (1994) Visual 'Literacy'. Boulder: Westview Press. Mirzoeff, N. (ed.) (1998) The Visual Cultural Reader. London: Routledge. Moorhead, R., Gross, M. and Joy, K. I. (eds) (2002) VIS2002: IEEE Visualization 2002. Boston, Institute of Electrical and Electronics Engineers, Inc. Newton, I. (1729) The Mathematical Principles of Natural Philosophy. London: Printed for Benjamin Motte. Newton, I. (1736) The Method of Fluxions and Infinite Series; with its Application to the Geometry of Curve-Lines; With Comments by John Colson. London: Printed by Henry Woodfall and sold by John Nourse. Newton, I. (1952) Opticks. New York: Dover Publications. Newton, I. (1953) Newton's Philosophy of Nature. New York: Hafner Publishing Company. Noth, W. (1990) Handbook of Semiotics. Indianapolis, IN: Indiana University Press. O'Halloran, K. L. (1996) 'The discourses of secondary school mathematics'. PhD thesis, Murdoch University, Western Australia. O'Halloran, K. L. (1999a) 'Interdependence, interaction and metaphor in multisemiotic texts'. Social Semiotics, 9(3), 317-354. O'Halloran, K. L. (1999b) Towards a systemic functional analysis of multisemiotic mathematics texts'. Semiotica, 124-1/2, 1-29. O'Halloran, K. L. (2000) 'Classroom discourse in mathematics: a multisemiotic analysis'. Linguistics and Education, 10(3), 359-388. O'Halloran, K. L. (2003a) 'Educational implications of mathematics as a multisemiotic discourse'. In M. Anderson, A. Saenz-Ludlow, S. Zellweger and V. V. Cifarelli (eds), Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing. Ottawa: Legas Publishing, pp. 185-214. ^ O'Halloran, K. L. (2003b) Tntersemiosis in mathematics and science: grammatical metaphor and semiotic Metaphor'. In A. M. SimonVandenbergen, M. Taverniers and L. Ravelli (eds), Grammatical Metaphor: Views from Systemic Functional Linguistics. Amsterdam: John Benjamins, pp. 337-365. O'Halloran, K. L. (ed.) (2004a) Multimodal Discourse Analysis. London: Continuum. O'Halloran, K. L. (2004b) 'Visual semiosis in film'. In K. L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 109-130. O'Halloran, K. L. (2004c) 'Discourses in secondary school mathematics

220

REFERENCES

classrooms according to social class and gender'. In J. Foley (ed.), Language, Education and Discourse. London: Continuum, pp. 191-225. O'Halloran, K. L. (in press) 'On the effectiveness of mathematics'. In E. Ventola, C. Charles and M. Kaltenbacher (eds), Perspectives on Multimodality. Amsterdam: John Benjamins. O'Toole, M. (1994) The Language of Displayed Art. London: Leicester University Press. O'Toole, M. (1995) 'A systemic-functional semiotics of art'. In P. H. Fries and M. Gregory (eds), Discourse in Society: Systemic Functional Perspectives: Meaning and Choice in Language: Studies for Michael Halliday. Norwood, NJ: Ablex, pp. 159-179. O'Toole, M. (1999) Engaging with Art. CD ROM. Murdoch University, Perth. Pacioli, L. (1494) Summa de Arithmetica. Venice. Page, T. (2001) The Mindful Moment. London: Thames and Hudson. Pang, K. M. A. (2004) 'Making history in from colony to nation: a multimodal analysis of a museum exhibition in Singapore'. In K. L. O'Halloran (ed.), Multimodal Discourse Analysis. London: Continuum, pp. 28-54. Poynton, C. (1984) 'Names as vocatives: forms and functions'. Nottingham Linguistic Circular, 13 (Special Issue on Systemic Linguistics), 1-34. Poynton, C. (1985) Language and Gender: Making the Difference. Geelong, Victoria: Deakin University Press. Poynton, C. (1990) 'Address and the semiotics of social relations: a systemic-functional account of the address forms and practices in Australian English'. PhD thesis, University of Sydney. Ravelli, L. J. (2000) 'Beyond shopping: constructing the Sydney Olympics in three dimensional text'. Text, 20(4), 489-515. Reisch, G. (1535) Margarita Philosophica. Basileae. Rotman, B. (1987) Signifying Nothing: The Semiotics of Zero. London: Macmillan Press. Rotman, B. (1988) 'Towards a semiotics of mathematics'. Semiotica, (72-1/2), 1-35. Rotman, B. (1993) Adlnfinitum. Stanford, CA: Stanford University Press. Rotman, B. (2000) Mathematics as Sign: Writing, Imagining, Counting. Stanford, CA: Stanford University Press. Royce, T. (1998a) Tntersemiosis on the page: a metafunctional interpretation of composition in the Economist magazine'. In P. Joret and A. Remael (eds), Language and Beyond. Amsterdam: Rodopi, pp. 157176. Royce, T. (1998b) 'Synergy on the page: exploring intersemiotic complementarity in page-based multimodal text'.JASFL Occasional Papers, 1(1), 25-49. Royce, T. (1999) 'Visual-verbal Intersemiotic Complementarity'. PhD thesis, University of Reading. Royce, T. (2002) 'Multimodality in the TESOL classroom: exploring visualverbal synergy'. TESOL Quarterly, 36(2), 191-205.

REFERENCES

221

Saussure, F., de (1966) Course in General Linguistics. New York: McGraw-Hill. Shea, W. R. (1991) The Magic of Numbers and Motion: The Scientific Career of Rene Descartes. Canton, MA: Watson Publishing International. Shin, S. (1994) The Logical Status of Diagrams. New York: Cambridge University Press. Simon-Vandenbergen, A. M., Taverniers, M. and Ravelli, L. (eds) (2003) Grammatical Metaphor: Views from Systemic Functional Linguistics. Amsterdam: John Benjamins. Smith, D. E. (1951) History of Mathematics. Volume 1: General Survey of the History of Elementary Mathematics. Boston: Ginn and Company. Staley, D. J. (2003) Computers, Visualization, and History: How New Technology will Transform Our Understanding of the Past. New York: M. E. Sharpe Inc. Stewart, J. (1999) Calculus. Pacific Grove, CA: Brooks/Cole Publishing Company. Sweet Stayer, M. (ed.) (1988) Newton's Dream. London: McGill-Queen's University Press. Swetz, F. J. (1987) Capitalism and Arithmetic. La Salle, IL: Open Court. Tannen, D. (1995) Gender and Discourse. Oxford: Oxford University Press. Tartaglia, N. (1537) La Nova Scientia. Venice. Tartaglia, N. (1546) Quesiti, et inventioni diverse de Nicolo Tartalea Brisciano. Venice: V. Ruffinelli. Thibault, P. (1987) 'An interview with Michael Halliday'. In R. Steele and T. Threadgold (eds), Language Topics: Essays inHonour of Michael Halliday: Volume 2. Amsterdam: Benjamins, pp. 599-627. Thibault, P. (1997) Re-reading Saussure: The Dynamics of Signs in Social Life. London: Routledge. Thibault, P. (2000) 'The multimodal transcription of a television advertisement: theory and practice'. In A. P. Baldry (ed.), Multimodality and Multimediality in the Distance Learning Age. Campobasso, Italy: Palladino Editore, pp. 311-385. Thibault, P. (2001) 'Multimodality and the school science textbook'. In C. Torsello-Taylor and N. Penello (eds), Corpora Testuali per Ricerca, Traduzione e Apprendimento Linguistico. Padua: Unipress, pp. 293— 335. Thomas-Stanford, C. (1926) Early Editions of Euclid's Elements. London: Printed for the Bibliographal Society. Thompson, G. (1996) Introducing Functional Grammar. London: Arnold. Tiles, M. (1991) The Image of Reason. London: Routledge. Unsworth, L. (ed.) (2000) Researching Language in Schools and Communities: Functional Linguistic Perspectives. London: Cassell. Unsworth, L. (2001) Teaching Multiliteracies across the Curriculum: Changing Contexts of Text and Image in Classroom Practice. Buckingham: Open University Press. van Leeuwen, T. and Jewitt, C. (eds) (2001) Handbook of Visual Analysis. London: Sage.

222

REFERENCES

Veel, R. (1999) 'Language, knowledge and authority in school mathematics'. In F. Christie (ed.), Pedagogy and the Shaping of Consciousness: Linguistic and Social Processes. London: Cassell, pp. 185-216. Ventola, E. (1987) The Structure of Social Interaction: A Systemic Approach to the Semiotics of Service Encounters. London: Frances Pinter. Ventola, E. (1988) 'The logical relations in exchanges'. InJ. D. Benson and W. Greaves (eds), Systemic Functional Approaches to Discourse. Norwood, NJ: Ablex, pp. 51-72. Ventola, E., Charles, C. and Kaltenbacher, M. (eds) (forthcoming) Perspectives on Multimodality. Amsterdam: John Benjamins. Vygotsky, L. (1986) Thought and Language. Cambridge, MA: MIT Press. Watson, V. and Walatka, P. P. (1994) 'Visual Analysis of fluid dynamics'. In D. F. Rogers and R. A. Earnshaw (eds), State of the Art in Computer Graphics. New York: Springer-Verlag, pp. 7-18. Webster,], (ed.) (2002-) Collected Works of M. A. K. Halliday (Volumes 1-3, forthcoming 4-10). London: Continuum. Wei, J. and Winter, M. (2003) 'Existence and stability of multispot solutions for the Gray-Scott Model in R2'. PhysicaD, 176(3-4), 147-180. Weitz, E. D. (2003) A Century of Genocide: Utopias of Race and Nation. New Jersey: Princeton University Press. Whitrow, G. J. (1988) 'Why did mathematics begin to take off in the sixteenth century'. In C. Hay (ed.), Mathematics from Manuscript to Print 1300-1600. Oxford: Clarendon Press, pp. 264-269. Wilder, R. L. (1981) Mathematics as a Cultural System. Oxford: Pergamon Press. Wilder, R. L. (1986) 'The Cultural basis of mathematics'. In T. Tymoczko (ed.), New Directions in the Philosophy of Mathematics. Boston: Birkhauser, pp. 185-199. Wilkes, G. A. (ed.) (1986) Collins Dictionary of the English Language. Sydney, NSW: Collins.

Index

abacus, use of 25 abstract theory 36 activity sequences 111,115,119,126, 176-7 Adorno, T.W. 18 advertising 8, 160, 163 agency, concept of 105-8 algebra 23-4, 27-8, 33, 45-6, 57, 96, 104-7, 130-2, 201 algorithms 25 anaphoric nouns 87 applied mathematics 33, 36 Aquinas, Thomas 34 Archimedes 31 Aristotle 34 Azzouni,J. 119 Baldry,A.P. 19, 124 Beevor, A. 1-7 Berlin, fall of (1945) 3-7 Bernstein, B. 9 Berry, M. 68 Bordell, D. 149-51 Botticelli, Sandro 139 Bourdieu, P. 9 brackets, use of 118, 125-8 BurgmeierJ.W. 189-90, 198 Bush, George W. 2 Cajori,F. 23,27,111 Carter, R. 73 certainty in mathematics, loss 119 Chandler, D. 6 chaotic systems 17, 58, 119, 151, 209 Cheong.YY. 8, 165 Chomsky, N. 62

Christie, F. 60 Chuquet, Nicolas 33 classroom discourse 205-8 Cobley, P. 6 Coca Cola (company) 8 code-switching 16 cohesion, concept of 175 Colonna,J.-F. 149-57 passim communication planes 89 compositional meaning 146-8 computational devices 25, 104 computer graphics applications of 149, 154—6 definition of 148 computer technology 15, 23, 58-9, 95-6, 119-21, 128, 132, 138, 146-58, 161, 203-5, 208-9 condensatory strategies 126—8 conjunctive relations 119-20,127 conjunctive reticula 80 context of text 89 contextualization propensity (CP) 165, 198, 204 contextualizing relations 165—7 coreness of lexical items 73,115 cosmology 55 culture 7-8, 89 Danaher, S. 157 Davies, P.C.W. 58, 148 Davis, PJ. 19, 34, 46, 129-30,132, 148, 151 deductive reasoning 119 Derewianka, B. 87-8 Descartes, Rene 17,22-3,33,38-48, 53-9, 72, 118, 129-31, 199, 209

224

INDEX

discourse systems 60-1, 64-8, 98-101, 128 dynamical systems theory 17, 58, 119 education, mathematical 16-17, 72, 199-208 Eisenstein, E.L. 33 ellipsis 122-6 encoding of meaning 97, 108, 110, 114, 118, 124, 155,202-3 condensatory strategies for 126-8 Euclid 27-31, 34, 46, 49, 130 exchange structures 68-72,115-16,127 existential graphs 130 experiential meaning 71,126-8,174-5, 179, 186,195,198 contraction and expansion of 103-14 encoding of 148 visual construction of 142-5 experimentation 58 virtual 151, 154 expression 108 Febvre, L. 31 female students 207-8 'field' variable 89 FoleyJ. 148 Fontana, Niccolo see Tartaglia Foucault, M. 91, 199 Francis, G. 87 Funkenstein, A. 154—5 Galileo 35-7, 58 Galison, P. 129-31, 142, 156 genre 20,89-91 geometry 45-8,130, 132, 201 Gestalt theory 147 Goldhagen, D J. 1 grammatical density 202 grammatical metaphor 83-8, 96, 128, 183-8,194-6,199, 205 grammatical repackaging 185-6 Grave, M. 148 Gregory, M. 163 GroB, M. 149 Guo, L. 134,138 habitus 9 Halliday, M.A.K. 6-7,13, 20, 60-90, 97, 105, 119, 128, 145,163,184-8, 202

Hamming, R.W. 24 Hasan, R. 20, 63, 90, 163 Hersh,R. 46 historical development of mathematical discourse 22-4 Hjelmslev, L. 61 Hooper, A. 58 Horkheimer, M. 18 ideational meaning 62, 88 ideology 91, 93 ledema, R. 11,20, 159 image processing 148-9 induction, mathematical 94 integration of semiotic resources 171 integrative multisemiotic model 91,163 internet resources 208 interpersonal meaning 62, 65, 67, 71-4, 88-91, 97,113-14,127,139,175-6, 198, 202, 206-8 contraction of 114—18 interpretative space (IS) 165, 198, 204 intersemiosis 11-12, 16, 65-6, 93, 96, 98,157,160-3,189-98, 204-5 and expansion of meaning 184 in mathematical texts 171-7 mechanisms of 169-71 semantics of 163-9 systemic framework for 171 intrasemiosis 16, 65-6, 88, 91, 94, 96, 158-9 Iraq war (2003) 18 'items' in mathematical texts 159-61, 189-91 Koestler,A. 24,55 Kok,K.C.A. 159 Kress, G. 20 Kuhn, T.S. 24 langue 62 Layzer, David 185 LeLous,Y. 148 learning difficulties 207 Leibniz, G. 33,45,57 Lemke, J.L. 13, 95, 122-4, 159 lexical density of text 202 lexical metaphor 88 lexicogrammar 63-7, 81, 92, 184, 200-1 Lim, F.V. 91, 134, 163, 165

INDEX

logical meaning 78-81,127 logical reasoning 128, 155-6 McDonald's (company) 8 macro-transitions 161-2, 178 Martin, H.-J. 31 Martin, J.R. 60-8, 73-93 passim, 97-8, 119, 128, 145, 184 mathematical spirit 19 mathematical books 24-33, 189-91 mathematics, status of 57-8 matrices 112-13 Matthiessen, C.M.I.M. 75, 184-8 medium, definition of 20 Messaris, P. 132 metafunctionality of language 7,13, 62-7,88,91,101,158,176,200 metaphorical terms 14-17, 57, 73-7, 188; see also grammatical metaphor; semiotic metaphor metaphorical exchange 178 micro-transitions 161-2, 178 mode 20,89 models, mathematical 94—7 multimodality 19-21,206 multisemiosis 10, 13-21, 59-60, 88, 91, 97-8 educational implications of 199-205 see also integrative multi-semiotic model natural language 97 Nazi regime 2, 6, 10 Newton, Isaac 17,22-3,31-3,40-58, 149, 152-4, 171-2, 177-9, 183-5, 199, 209 Nike (company) 8 non-linear systems 58, 209 notation, mathematical 23, 112-13, 127 Noth, W. 6-7 nuclear relations 77 numeration systems 24-7, 103-4 objectivity 209 O'Halloran, K.L. (author) 208 operative processes 103-13, 121-8, 178, 201-2, 206-7; see also rule of order O'Toole, M. 5, 13-14, 132-41, 146-7 Pacioli, Luca 33

225

Page, T. 2 parole 62 Peirce, C.S. 130 photographs, use of 191-3 picture synthesis 154 Plato 34 Poincare, Henri 130 positional notation 112-13 Poynton, C. 62 printing technology 27-33, 131-2 probability 72 rankshifting 66-7, 82, 105-9, 118, 128,131,187,197,201-2, 206-7 rationalism 18 realization, concept of 62, 65 reciprocity of choice 90-1 re-contextualization 177-9, 184, 204-5, 209 'register' of mathematical language 89-91 Reisch, G. 34 Renaissance mathematics 33-8 resemioticization 11, 159 Rose, D. 65, 73-4, 81 Rotman, B. 13 Royce, T. 11,159,163-5,175 rule of order for operative processes 118,124-8 Saussure, F. de 62 scientific language 14, 60, 96, 108, 128, 186-7, 200-1, 205 scientific view of the world 6,10,17, 22, 24, 94, 184, 188, 205, 209 secondary clauses 80 semantic drift 200 semantic effervescence (SE) 165, 198, 204 semantic shift 184 semiotic capital 9 semiotic metaphor 16, 88, 93, 179-84, 196, 199, 205, 207 September llth 2001 attacks 2 Shea,W.R. 44,46,54 Shin, S. 130 Smith, David Eugene 24 social distance 90 social-semiotic theory 6-10, 17

226

INDEX

space of integration (Sol) 163 spatial positioning of mathematical symbols 122,201-2 spoken discourse, status of 90 statistical graphs 141-2, 148, 158 Stewart, J. 67-83 passim, 88,186 Sweet Stayer, M. 57 Swetz,FJ. 24,27,33 syllogisms 119 symbolism, mathematical 6-7,10-17, 38, 44-8, 53-7, 60, 75, 78, 80, 89, 93-104,112-25,155-6 as distinct from visual images 129-32 framework for 97-103 grammar of 15, 22-3, 57-9, 65-7, 96, 108, 111, 126,131-2,142,178,187, 199-206 and intersemiosis 196-7 language-based approach to 96-7 operative processes in 106 as a semiotic resource 125 SF approach to 201-3 textual organization of 122-5 as a tool for logical reasoning 128 syntagmatic organization 121-2, 126-8 systemic functional (SF) approach 938,132-9,159,170,199, 204-9 and language 11-15, 60-5, 200-1 and mathematical symbolism 201-3 and visual images 133-8, 203 systemic functional grammar (SFG) 13-15, 18-20,128,134, 138, 158 systemic functional linguistics (SFL) 8, 60-1, 66-7, 70, 88-91, 98,101,175, 183,200-1,208 Tartaglia, Niccolo 34-42 taste 14 taxonomic relations 77

tenor 89-92 textual meaning 121-8 textual organization of language 81-3 Thibault, P. 19, 62,163,165,183, 199 Thomas-Stanford, C. 27-30 Thompson, K. 149-51 Tiles, M. 119 topological-type meaning 95 transitions in mathematics 159-62 Treviso Arithmetic 22-7 trigonometry 179-83 'true knowledge' (Descartes) 54—5 truth-values 117-18,201,209 typological-type meaning 95 van Leeuwen, T. 20 Venn diagrams 130,133,142 Ventola, E. 68 virtual experimentation 151,154 virtual reality 203 visual forms of semiosis 155-6 visual images grammar of 15 reasoning through use of 145-6 in SF framework 133-8, 203 visual theorems 151 visualization 129-33, 148-9, 203 Walatka, P.P. 149 war attitudes to 1-3 scientific impact of 18 Watson, V. 149 Weitz,E.D. 1 Whitrow, GJ. 33 Wilder, R.L. 23-4 Wilkes,G.A. 155-6