Writing Mathematics Well: A Manual for Authors

Writing Mathematics Well

Writing Mathematics Well A Manual for Authors

by Leonard Gillman

©1987by The Mathematical Association of America Library of Congress Catalog Card Number 87 061962

ISBN4-88385-443 0

CONTENTS PREFACE vii 1.

INTRODUCTION I

2.

GENERAL ORGANIZATION 3 21 The title 3 2.2 The introduction 3 2.3 Prerequisites 4 2.4 Notation and tenninology 4 2.5 Organization and p~ 4 2.6 The bibliography 5 27 The index (to a book) 5

3.

PRESENTING YOUR RESULTS 6 3.1 State first, prove second 6 3.2 Generality 6 3.3 Style 6 3.4 The character c:i a proof 7 3.5 Keeping the reader informed 8 3.6 Citing reasons 8 3.7 Citing references 9 3.8 Proofs by cootradiction 9

4.

MATHEMATICAL ENGLISH 11 4.1 Weandl 11 4.2 The grammar of symbols 12

4.3 4.4

If . . . , then . . . 13 Definitions 14

5.

SYMBOLS 5.1 5.2 5.3 5.4

5.5 5.6

5.7

15 You can do with fewer than you think 15 Choice of notation 16 Look 17 Mixing symbols with text 17 Your numbering system 17 Multiple indices 18 Suppress useless information 19

6.

ENGUSH USAGE 20 6.1 Style 20 6.2 Some distinctions in meaning 21 6.3 A few matters of grammar 23 6.4 In terms of 24 6.5 Some distinctions in spelling 25 6.6 Some ups and downs of punctuation 25 6.7 Latin words 26 6.8 Pre-plan ahead for the future in advance 27

7.

REVISE,REVISE 28

8.

MECHANICS 28 8.1 Submit a clean manu!l:ript 28 8.2 Spacing 29 8.3 Displays 29 8.4 Figures and tables 30 8.5 The conventions of the journal 30 8.6 Marking the manuscript 32 8.7 Proofreading your manu!l:ript 33 8.8 The odyssey of a manuscript 33 8.9 Marking the galleys 35

APPENDIX The Use of Symbols: A case study 37 ANNOTATED BIBUOGRAPHY 45

vi

PREFACE A few years ago, the MAA Committee on Publications recognized the need for a comprehensive writer's manual addressed to authors of articles for MAA journals. This booklet is the result. Most of the suggestions apply to books as well; but if you are planning a book, be sure to consult Halmos [2], particularly Sections 4, 6, and 7. I tis a pleasure to acknowledge the helpful counsel of a large number of individuals. In the embryonic stages, Alan C. Tucker, who chaired the Committee, and P. R. Halmos and Doris W. Schattschneider, journal editors, supplied ample doses of advice, encouragement, and prodding. As the manuscript progressed, Paul T. Bateman, Richard K. Guy, Meyer Jerison, Robert H. McDowell, and Ivan Niven, as well as Tucker, Halmos, and Schattschneider, read various drafts and provided conunents ranging from simple recommendations to stern admonitions, incisive observations, and detailed and substantial suggestions; several of these colleagues will recognize excerpts from their own prose. I enjoyed working with Peter L. Renz, who as editor contributed his own comprehensive list of insightful suggestions. I have accepted ideas from several published sources, particularly Halmos [2] and the AMS manual [4]. Here, too, I have incorporated several passages verbatim; I am grateful to the American Mathematical Society for permission to do this. Throughout, my wife, Reba, provided good-humored, perceptive help in the fine tuning. (Which word do I want here, may or might?) To all, my appreciative thanks. Leonard Gillman

February,I987, Austin, Texas

vii

This manuscript was prepared by the author on an Apple Macintosh with the help of MacEqn, a program for symbols and equations.

ix

1. INTRODUCTION This booklet offers suggestions on preparing manuscripts for the journals of The Mathematical Association of America. Most of the advice applies to books as well, and one or two remarks refer specifically to books; but if you are planning a book, be sure to consult Halmos [2], particularly Sections 4, 6, and 7. The publications of the Mathematical Association are expository publications, dedicated to the advancement of college mathematics. Unlike some research publications, ours are responsible first to our readers, then to the authors; but much of the advice is appropriate to research articles as well. Most of the suggestions are simple matters of common sense; some involve judgments on which opinion is divided; and others are just plain my own prejudices. The style is largely Do this and Don't do that, so the suggestions come across rather as rules. Each one has its counterexamples. But if you do break a rule, do it deliberately, not from carelessness. Articles for MAAjournals are expected to be well motivated and of wide interest The exposition should be clear and lively, and in fact the quality of expositioo counts as much as mathematical content in determining whether a paper will be accepted for publication. Articles showing a new slant on a familiar result are welcome; but don't merely reinvent the wheel in a well-known mode. Before you write up your results, be sure to search the literature and check with your colleagues. Many manuscripts are rejected out of hand because the author had not taken the trouble to consult even standard texts to see what was already available in print. TIIE AMERICAN MATHEMATICALMONTHLYis directed to readers in the range of levels from advanced undergraduate to professorial; think of the "average" reader as having had a year or more of graduate work. MATHEMATICS MAGAZINE and THE COLLEGE MATHEMATICS JOURNAL are devoted to undergraduate mathematics-with the JOURNAL concentrating on the first two

1. IN1RODUCITON

years-and many of tiE readers are students; think of an article as supplementary material for a standard course. To whom, then, is your article addressed? Keep a specific group in mindor, better, a specific person: how would you explain your results to your colleague down the hall, or to the student you were chatting with just now? Books. The Association publishes books at several levels. The MAA STUDIES IN MATHEMATICS series describes recent research. Each book is devoted to a single subject and contains expository articles from several contributors, written at the collegiate or graduate level. The CAR US MATHEMATICAL MONOGRAPHS are intended to make topics in pure and applied mathematics accessible to teachers and students of mathematics, and also to nonspecialists and scientific workers in other fields. The DOLCIANI MATHEMATICAL EXPOSITIONS are designed to appeal to a broad audience, challenging the talented high school student and intriguing the more advanced mathematician; topics thus far have emphasized elementary combinatorics, number theory, and geometry. The NEW MATHEMATICALLIBRARYis a series of paperbacks addressed to college and high school students who are interested in understanding and appreciating important mathematical concepts beyond those featured in traditional mathematics courses. The Association is always looking for good books, whether they fit into one of these series or not.

2

2. GENERAL ORGANIZATION 2.1. The title Keep your title short and include key words to make it informative; a long title sounds pompous and is a nuisance to refer to. On a theorem ofDavid Copperfield

is short but uninformative. Algebraic solutions of linear partial differential equations encroaches on the limit for length but includes key words and is good Steer clear of symbols: they will cause typesetting problems in the original article and thereafter in bibliographies and wherever else the title is quoted

2.2.

The introduction

The firSt paragraph of the introduction should be comprehensible to any mathematician Describe in general terms what the paper is about; and do it in a way that entices the reader to continue reading. Settle for a rough statement in \Wrds; eschew a precise statement loaded with symbols and technical terms. Your very f1rst sentence, in particular, must command the reader's immediate interest. Would you guess that most continuous functions are nowhere differentiable?

is excellent This paper describes an unusual application of the Mean-Value Theorem

is also good; it could say more about the application, but can stand as is because the Mean-Value Theorem is so familiar. Consider a sheaf of germs of holomorphic functions

is too technical (even in the late 1980s). Let f(x) = (arccos t arcsin 2t dt 0

3

2. GENERAL ORGANIZATION

is totally unmotivated and is atrocious.

2.3.

Prerequisites

It helps to mention any unusual prerequisites and to review a few key defini-

tions that might not be well known-with references. But don't do a snow job. If you feel the need for an entire page, you should stop and rethink the whole article; remember that the reader does not have to be familiar with each theorem you are going to invoke but need only be able to understand what it means when you quote il In any case, you don't have to list everything all at once: your paper will be friendlier if you weave the information unobtrusively into the text as you go along.

2.4.

Notation and terminology

You will certainly require some terms and symbols. Decide which ones you really need. Most mathematical articles (and lectures) use too many. (Proof: Have you ever encountered one with too few?) I once heard a very famous mathematician begin a talk by introducing two dozen symbols, one of which was so obviously unnecessary that I kept watch to see whether he would ever use it. Sure enough, 47 minutes later he wrote the symbol for the ftrst and only time. He did have the good sense to recall its definition-by the same stroke confirming that the symbol should never have been introduced at all. An introductory section listing every term and every symbol that are going to come up may serve as a handy reference, but will overwhelm the reader. Again, it is better to introduce such things in small doses. If you still feel a strong need to include a glossary, put it at the end. Chapter 5 and the Appendix offer further suggestions about symbols.

2.5. Organization and pace If your paper runs to several pages, divide it into sections with informative titles. This will help clarify the logical structure of the paper and make it easier to follow. Introduce one idea at a time. Make liberal use of examples, perhaps suppressing the most abstract formulation of the idea. Link ideas to what you ar sume is familiar to the reader.

4

2. GENERAL ORGANIZATION

2.6.

The bibliography

The purpose of the bibliography is to alert readers to the existence of books and articles that give history, priority, and attribution. List only published works, therefore, and make the list complete---don'tjustjot down a few references as an afterthought Never mind whether a work: is unavailable at most college libraries: that's irrelevant. Private communications should be woven into the text, not dignified with a number leading to useless flipping of pages. Likewise, "to appear" should not appear, unless the paper is already accepted by a journal that can be named.

2. 7.

The index (to a book)

The index is supposed to help the reader. It should therefore be something more than a permutation of the book into alphabetical order. The big job is selecting and classifying. This requires a feeling for the relative importance of the topics and hence can be done properly only by you. For example, in a book that quotes the axiom of choice, the index typically refers to the one place it is quoted. But if a central purpose of your work is to study the role of the axiom, then you might well decide to refer to every proof that invokes it. You can start this organizational work as soon as the manuscript is finished: it does not depend on knowing the page numbers. Don't ask the reader to guess what name you have listed a topic under; instead, you try to guess which name the typical reader will look for. If there are several, list them all. I have before me an index that references the CauchySchwarz-Buniakowsky inequality six times: each of the names appears twice, once in the main list and once under "inequalities" [20]. Excellent! Always err in the direction of helpfulness.

5

3. PRESENTING YOUR RESULTS 3.1.

State first, prove second

Keep in mind that you have to maintain the reader's interest at all times. An easy way to lose it is by stringing along a succession of statements with no discernible goal-the classic example being a sequence of arguments culminating in the triumphant cry: "We have proved the following theorem." RULE 42 (the oldest rule in the book) [26]: Always state the theorem before proving it.

(This has some pleasant exceptions.)

3.2.

Generality

It is good research practice to analyze an argument by breaking it into a suc-

cession of lemmas, each stated in maximum generality; it is poor expository practice to publish the results that way. Develop the argument in the special case that applies to your paper. If you want to mention the generalization for the record, don't state it as a formal theorem but as a passing remark; and postpone it until after you have stated and proved the special case. Do tell the reader, though, whether the general proof is a straightforward extension of the one you gave or requires new ideas; and mention where to find it.

3.3.

Style

State theorems concisely. Try for one short sentence--or at most two, one for the hypothesis and one for the conclusion. EXCEPTION. It is occasionally convenient to build modestly on related ideas, as in the form: Assume P. Then Q. If also R, then S. The statement of a theorem is not the place for definitions, proofs, or other discussion. EXCEPTION: If R is an immediate consequence of Q, then "discussion" at the level of If P, then Q; hence R

6

3. PRESENTING YOUR RESULTS

is acceptable, and in foct helps keep tre organization crisp. "Concise" does not mean short, but only that you not waste words. The common style that starts with ''The following four conditions are equivalent", and follows with a list, is undeniably concise; it is acceptable no matter how the sentence count comes out. But I would balk at eleven equivalent conditions, because at least six of them will surely be merely minor variants of the others. Confine the theorem to the basic list and relegate the others to an accompanying remark. The statement of a theorem is generally the wrong place to introduce notation for the proof: A differentiable function f is continuous. X A differentiable function is continuous. V

The first way is underhanded. (If not followed by a proof, it is gross.) Both versions illustrate a convention about quantifiers that we tend to withhold from our beginning students, whom we then scold for not getting it right: saying "a" (or "any") differentiable function to mean "every" differentiable function. Why not be safe and say "every"? Another common and acceptable form is, of course: Differentiable fWictions are continuous

V

-though I suppose an expert nitpicker could ask whether that means some or all. This way is at least safe from symbolitis: no one (I hope) would write: Differentiable functions f are continuous.

X

3.4. The character of a proof It is often good to illustrate the theorem before proving it-or even instead of proving it, in case the illustration contains the essence of the proof, or the proof is too hard. Strive for proofs that are conceptual rather than computational-the way you would describe the result to a colleague (or student) during a walk across the campus. If a proof is at all involved, begin by explaining the underlying idea. Leave all purely routine computations (i.e., not involving an unexpected trick) to the reader. Use a picture, perhaps as an important step in the proof it self MATHEMATICS MAGAZINE has for several years been encouraging this spirit with its de-

7

3. PRESENTING YOUR RESULTS

lightful series entidedProofwithout words. J. E. Littlewood, in his classic treatise [3], presents these ideas in telling style, first concocting a barbaric proof (of the Weierstrass approximation theorem), then presenting a civilized one with the help of a picture [pp. 30-36].

3.5. Keeping the reader informed Keep the reader informed of what you are doing and of how things stand. I have always enjoyed reading Sierpinski: first he tells you what he is going to do, then he does it, then he tells you he did it If you use a key term or symbol after a long spell without it, recall its definition or refer to where it was introduced Your readers will be grateful. Make sure the reader knows the status of every assertion you makewhether it is a ronjecture, the theorem just proved, something from the reader's background, a well-known result, a proposition you are about to establish. The last is the most important A void "roadblocks". If you write: But G is abelian. To see this, consider .. . , the reader is tempted to stop at the period and start thumbing back in search of the proof. It is better to say: But G is abelian, as we now show . . . -since the reader is likely to keep going past the comma For a more substantial improvement, put the hint of the forthcoming proof at the beginning:

Next, G is abelian. And why be parsimonious? Put in three additional words and say: Next we show that G is abelian. Now the flow is so smooth the reader won't even realize there was a problem. Tell the reader when a proof has ended. The best way is to say so outright "This completes the proof', or "D" or "+".(CAUTION: Some editors may disallow a symbol as too formal-looking.) An alternative is to say so indirectly-e.g., start the next paragraph with "It follows from this theorem that

"

3.6.

Citing reasons

Cite supporting reasons informatively. When you invoke a key hypothesis, identify it as such and tell which one it

8

3. PRESENTING YOUR RESULTS

is. Suppose for example your hypotheses are: a= 2, f is continuous, and G is abelian, and at some step in the proof you need just the second of these. Don'tjust say:

By hypothesis, we have . . . ; say:

By hypothesis, f is continuous; hence we have This helps both you and the reader see what is making things tick. Every development has mountains and valleys. Name the mountains andrefer to the names. Say "By the Fundamental Theorem of Calculus", not "By Theorem 5.7". Say "By the triangle inequality", not "By Axiom 3".

3. 7.

Citing references

When you quote a proposition from another work, tell whether it is hard or easy; this helps readers decide whether to read on without wasting time or try working it out for themselves. If the result is relatively unfamiliar, include a citation:

According to a deep theorem of Gulliver [31, p. 56], every such Lilliputian space is uncountable -and the reader can decide peacefully whether to continue reading or go look up Gulliver's proof. In case Gulliver's phrasing is much more general than yours, insert a hint on how to perform the transition:

(His theorem is stated in terms of arbitrary lattices and semi-Lilliputian spaces; to appy it here, note that our family of open sets forms a lattice, and that Lilliputian spaces are semi-Lilliputian.)

3.8.

Proofs by contradiction

I classify contradictions in proofs as essential, questionable, or spurious. In the familiar proof that ..J2 is irrational, the contradiction is essential: you assume the contrary, and you use that assumption in an essential way in the course of the proof. As an example of a questionable contradiction, consider the following proof

9

3. PRESENTING YOUR RESULTS

that if a2 is even, then a is even. Assume on the contrary that a is odd. • •

Then a is of the form 2b + 1, and a computation shows that a2 is of the same form and hence is also odd. This contradicts the hypothesis; therefore we must reject the assumption that a is odd. Hence a is even.

My concern is that the equivalence of a proposition with its contra positive is something we all take for granted-the two forms are often interchanged as a matter of course-and I see no reason to include yet another proof. Thus, I prefer to say that I will use the equivalent form-if a is odd then ~ is odd-and then prove that assertion directly (as is done in the marked lines). A proof by spurious contradiction sets up and knocks down a straw man. As an example, consider the following proof (which I have seen in print) that the functions ~ and tJx are linearly independent over the reals. Assume on the contrary that ex and e2x are linearly dependent. Then there exist constants cl and c2, not both zero, such that

• •

Then cl + c2eX = 0. Differentiating, we get c2eX = 0,

• whence c2 = 0. Then cl = 0. Thus, cl = c2 = 0. This contradicts the fact that cl and c2 are not both zero; therefore we must reject the assumption that the functions are linearly dependent. Consequently, they are linearly independent.

The point is that while the proof assumes that q

and c2 are not both ze-

ro, no use is ever made of that assumption (except to knock down the straw man~ To see this, delete everything but the marked lines (and begin the first of

10

4. MATHEMATICAL ENGUSH

them with "Assume that"). Y au now have a direct proof of linear independence. No one would start a proof of the formula writing:

Xl -

1 = (x- 1)(x + 1) by

Assume the contrary. Then there is a number XO such that

But when the setting is slightly more involved, that construction is harder to notice. When you first search for a proof, by all means begin if you wish by assuming the conclusion false. But at some stage before sending in your article, check your proofs for spurious contradictions.

4. MATHEMATICALENGLISH This chapter contains suggestions about the use of the English language in mathematical writing. Some discussion of English usage in general is given in Chapter 6.

4.1. We and I The use of "we" to mean you and the reader, or you with the reader looking on, is universal; to avoid gruffness, it is virtually indispensable: We see from (1) that . . . . Let us now prove that . . . . Joint authors have to think carefully, then, about how to refer to themselves:

We proved in [31] that . . . . ( ?) Safer:

The authors proved in [31] that . . .. People disagree about whether an individual author may use "I" without sounding egotistica~ or "the author" without sounding pompous. What counts

11

4.MATHEMATICALENG~H

is the spirit in which you write. If you need four references to yourself in the same paragraph, combine them into one: "These results are based on my papers [31,32,33,34]"; then proceed. You are still permitted crucial references as they come up, such as "[32, Theorem 6]". In the hands of the right person, "I" can be downright refreshing. My alltime favorite is a footnote in a 1950 paper by Paul Erdos [21, p. 137]: This remark is probably due to Sier pinski, but I do not remember for sure. What could be more warm and human? Try substituting "we" or ''the author'', or some circumlocution. Don't use "myself' to mean simply me; reserve it for emphasis: These results were obtained jointly by Hilbert and myself X These results were obtained jointly by Hilbert and me. ...; I made the same error myself ...;

4.2.

The grammar of symbols

A symbol represents a word or phrase. Writers do not always agree about whether it can represent more than one. My attitude here is permissive. I think it is all right to use"=" to mean either is equal to, just plain equal to (with no verb), be equal to, or which is equal to: We see that x = 0. Take x=O. Let x = 0. Then xZ + y2 = ,Z = 1. These expressions read smoothly, because the introductory words, "We see that", "Take", and "Let" in the first three, and the sequence in the fourth, prepare you for what is coming. I use "e ", "<", and other symbols in the same freeway. On the other hand, I would balk at were equal to as in:

If X= 0, then we would have . . . , (?) because (in the absence of a prefatory hint) the cue for how to read the equals sign to fit the grammar does not appear until after you have read it wrong. Solution: say it in words; or stick to simple tenses.

12

4.MATHEMATICALENGUSH

Finally, never suppress which after a conditional--e.g., don't use If a = b = c . . . to mean

If a = b, which is equal to c, . . .. Thus: Let 8 =

3 -e 4

> 0.

Then . . . .

X

Let 8 = 2-e. Then 8 > 0, and . . . . 4

v

Punctuate displays. A sentence that ends with a displayed formula still requires a period. (Some people, notably commercial book publishers, disagree.)

4.3.

If • • • , then . . .

I once got a paper to referee that stated a theorem in the following form: If a = b, c = d, e = f, g = h. (?) Include the "then"-and the comma preceding it-as a matter of habit. Do this even in the simplest case: If x > 0, then log x is defined; V your reader is seeking information, not mental calisthenics. (For an exception to the rule about the comma, see the last two lines of the proof in the Appendix.) Don't ask a historical fact to depend on a mathematical hypothesis: If x > 0, then Euler proved in 1756 that . X Euler proved in 1756 that if x > 0, then . . . . V Use "then", not "therefore", after an assumption: Suppose I lend you $10. Therefore you owe me $10. Assume x = 3. Therefore 2x = 6. X Assume x = 3. Then 2x = 6. V

13

X

4. MATHEMATICALENGUSH

The construction

Since . . ., then . . .

X

is discordant Instead of "then", say "it follows that", or "we have", or nothing: Since this limit exists, then the series converges. X Since this limit exists, the series converges. ...;

4.4.

Definitions

Don't use "if and only if' in a definition (except in formallogic)-it's too pompous. By tradition, "if' is sufficient: An integer > 1 is said to be prime

if its only positive

divisors are itself and 1. V ''Let" after the title DEFINmON is redundant and sounds gauche: Let lSI denote the cardinal number of S. V DEFINITION. Let lSI denote the cardinal number of S. X DEFINITION. lSI denotes the cardinal number of S. V

14

5. SYMBOLS 5.1. You can do with fewer than you think We mathematicians are trained in the use of symbols and tend to put them in liberally. That's not necessarily bad in itself-there they are in case we need them. What we have to remember later on is to take out those we don't need -and simplify any involved ones that remain. The Appendix presents a case study on the use of symbols, with examples of both types of editing. In an extreme case, a symbol can be removed the instant it is put in:

Obviously, every group G of prime order is simple. X Obviously, every group of prime order is simple. ..; Suppose you really "need" a whole menagerie of symbols-what then? Answer: your masterpiece does not qualify as an expository article. Rethink it and rewrite it. You may well end up simplifying the mathematical argument as well. The Appendix demonstrates a good example of that too. Sometimes one introduces a symbol for the sole purpose of reducing clutter; the question then is whether the exchange was advantageous. If :x2 + x + 1 shows up casually in the course of a computation, you may be tempted to call it u, or p(x), in order to cut down on clutter. I would say don't, even if it appears eight times: you would be asking the reader to keep track of something ephemeral in exchange for a saving of little consequence. On the other hand, if you are going to refer three or more times to a particular 4 x 4 matrix, or to an expression like Al-l A2-l .. ·An-1,

then I would say yes, display it and assign it a letter, or number it in the margin. Extra generality often requires extra symbols. Consider replacing a general formulation by an illustrative example. Extra generality often requires fewer symbols. Consider replacing a specialized formulation by a simpler, more general one-with a probable gain in insight For example, if you believe that two points determine a line even when the equation of the line is not written down, then you can derive the Mean-Val-

15

5. SYMBOLS

ue Theorem (from Rolle's Theorem) without bringing in the equation y=f(a) + m(x -a),

and certainly without calling up the detailed formula

F(x) = f(x)-f(a)-f{b)-f(a)(x-a). b-a Here f is continuous on [a, b] and differentiable on its interior. Let g be any function that has the same properties and agrees with f at a and at b. Then f- g satisfies the hypotheses of Rolle's Theorem, and there is a point z in (a, b) such that f'(z) = g'(z). The Mean-Value Theorem is the particular case in which the graph of g is a straight-line [22].

5.2.

Choice of notation

Use standard or familiar or suggestive notation that your readers can assimilate easily, allowing them to devote their energies to the mathematics. Use consistent notation. This may take some planning. No one would call the angles of a triangle B, L, and t 1. So don't get stuck with

a1x + a7J, or ax1 + bxz: (?) prepare the notation so that you end up with

Mathematicians use a variety of alphabets, and we have developed a number of helpful typographical conventions, such as the standard symbols e (italic), R (bold), x (Greek), L (stylized Greek), 0 (Danish), N (Hebrew). All these are at your disposal. But use them judiciously. Don't go out of your way to use lots of different fonts and sizes-that's an unnecessary invitation to typesetting errors. Finally, don't forget to review the symbols you have assigned-it is not uncommon to discover that the same one has been used in more than one way.

16

5. SYMBOLS

5.3.

Look

Pay attention to the visual impact of the printed page. A long stretch of symbols is hard to read Break it up with connectives. Phrases such as It follows that By hypothesis On the other hand,from (41), we know that are friendly and allow the reader to relax. If they seem too wordy, say "Hence", "Now'', "But". After a lengthy string of computations, start a new paragraph, with some introductory text Be sparing in the use of "3" (except in formal logic), and don't overdo "iff': write out the words. (Save the abbreviations for the blackboard when you are talking faster than you-or your students taking notes--can write.)

5.4.

Mixing symbols with text

Devote some thought to how you use symbols within a line of text Then every number on the left < every one on the right (?) and

We conclude that the two expressions are = (?) are grammatically impeccable, but most people would agree that they are inelegant at best. Editors will reject both. Editors strongly discourage starting a sentence with a symbol, as it can be counted on to confuse the copy editor, the compositor, and the reader. I myself recognize exceptions-for example, in a sentence displayed by itself. My irrevocable rule is: Never start a sentence with a symbol when the preceding sentence ended with a symbol: Then a > 4. b > 4 also, since . . .

5.5.

X

Your numbering system

If your manuscript is more than 8 typed pages (double spaced), divide it into numbered sections and number theorems serially within each section: 1.1, 1.2, ... , 2.1, 2.2, ... (unless there are very few theorems). Use a simple and self-explanatory system for marginal headings. My first prize for what not to do goes to von Neumann and Morgenstern [23], in which the following succession is typical [pp. 140-141]:

17

5. SYMBOLS

(16:15) (16:C) (16:16:a) (16:16:b)

16.4.2 (16:0)

By the way, their system is very logical.

Number only those expressions you refer to; the numbers then serve as clear signals. (This is likely to require checking and revising.) In addition, the copy editor will be free to relocate formulas, as convenient, between unnumbered displays and the text. (Some textbook authors number almost all displays, so that students can refer to them easily when asking questions.)

5.6

Multiple indices

Simple combinations of indices, such as

which are indispensable to the mathematician, are easy for the copy editor and the compositor to comprehend, and easy (in context) for the reader to assimilate. Involved combinations of higher order invite typesetting errors. I recall the defeat of a major research journal at the hands of the standard symbol

which it printed as 2Ka

2

;

the author protested, and the journal printed a "correction" [25]: 2Ka 2Ka For 2 read 2

One way out: exp exp N« (But see the remarks in Section 8.5.) A complex symbol, even when printed correctly, may require too much unraveling by the reader; moreover, second-order indices are necessarily small and often hard to read. Try to replace an elaborate symbol by a series of simpler ones; you may be able along the way to simplify the intricate mathematical ar-

18

5. SYMBOLS

gument that led to it My candidate for the all-time champion is the following concoction by Sierpinsk:i [24, p. 164]:

~

t ~ n~c

Actually, it was printed correctly-he must have climbed into the printing press for a last-minute check. The Appendix analyzes Sierpinski's paper and shows how this complexity could have been avoided

5. 7.

Suppress useless information

Do not routinely announce all the variables a temporary symbol depends upon. If you are deriving the formula 1 -rn+l 1-r

r-:1:

1,

you may wish to call the sum Sn. compute rSn• subtract, and solve. No. Just call it S. Quantifiers invite the same temptation. In the assertion

'V a, 'Vi, 3j, the dependence of j upon i and a is built into the symbol. When we shout at our students that 8 depends on e, it is for emphasis in a bewildering context, not from logical necessity. (If you report at the PTA that every teacher in the school drives a car, you don't add: "depending on the teacher''.) Don't go out of your way to make general remarks about all the

.a

1·. I

It may be that you are going to pick an a and an i only once; then on that occasion, you can call the corresponding j just plain j. The Appendix presents an outstanding example.

19

6. ENGLISH USAGE Use good English, writing carefully, clearly, and correctly. Your article does oot have to qualify as a literary masterpiece, but it should adhere to the accepted principles of correct diction, grammar, punctuation, and spelling. Among the several reasons for this are the very practical ones that incorrect English is distracting to the serious reader and is likely to be unclear. There are less benign possibilities. Richard Mitchell [8] is a forceful exponent of the thesis that ungrammatical writing and incorrect spelling betray a lack of precise thinking. Ben Jonson said the same thing several centuries ago [28]: Neither can his Mind be thought to be in Tune, whose words do jarre; nor his reason in frame, whose sentence is preposterous.

(A fuller quotation is given in [7, p. 3]). For convenience, I have assembled some do's and don't's, which I present here. Readers interested in pursuing these matters further can choose from among a variety of authoritative treatises. The bibliography lists several that are written with good humor and suitable for light and pleasant browsing (despite being informative).

6.1. Style Write simple, unaffected prose. Writing is harder then speaking because your tone of voice isn't available to help make your point clear. Keep sentences crisp-think of what you want to say and say it Mathematics is hard enough to read without convoluted writing that makes it harder. Use the active voice. All writing experts agree on this. Passive constructions leave the reader wondering who is doing what to whom; moreover, they encourage a verbose style, which makes things worse. It has been noticed that X We have noticed that ...; Occurrences were observed in which X I saw ...;

20

6. EN GUSH USAGE

Avoid ponderous nouns. Instead of long nouns ending in "tion", use verbs: for the preparation of (?) for preparing ..; to pre pare ..; for the acquisition of (?) to acquire ..; to get ..; Sentences that start off with ponderous nouns are doubly insidious, as they encourage a passive construction: Confirmation observation of civilization domination relationships are undergoing reexamination completion. (?)

A void abstract nouns that promise more profoundity than the context provides. The classic is "methodology" [the methodology of least squares (?), Newton's methodology (?)].Two current favorites are "objective" [the objective of my affections (?)] and ''motivation" [the profit motivation (?)]. Gilbert & Sullivan's Mikado did not announce: My objective all sublime (?) I shall achieve in timeTo let the punishment fit the cri~ The punishment fit the crime. That was his object (as he correctly asserted [27]); his objective was justice. (Fowler would even say the (grander) object was justice, reserving "objective" for contexts that do not strain the metaphor with military objective [14]). The newspaper quotes an entrepreneur: My motivation was simple: to make as much money as possible. X That was his motive; his motivation was greed. (Motivate means to provide with a motive.) Recommendation: Try the simple form first; if it works, keep it

6.2. Some distinctions in meaning Appraise vs. apprise: The editors will appraise your manuscript and apprise you of their decision

21

6. ENGUSH USAGE

Comprise vs. compose. A set comprises (embraces, consists of, is composed of) its elements; the elements compose (or constitute) the set. If you are not sure about "comprise", try "embrace" (or ''include''); if it's wrong, use "compose". R comprises the rationals and the irrationals.

v

R is comprised [embraced] of the rationals and the irrationals. X R is composed or the rationals and the irrationals. Comprise is all-inclusive: R does not comprise the rationals.

v

Ensure vs. insure. Reserve the second for taking out insurance; for guarantee, use the first Send me a calculator to ensure correct answers; and insure the package. Inclusion vs. containment. Reserve the second for military strategy; in mathematical writing, use the first If A c B and B c C, then A c C: inclusion is transitive. The Allies' containment of the enemy forces led to victory. Quickly vs. soon. I know more than one mathematician who writes, "I hope you will make up your mind quickly''. This allows you to put everything off for five months and then make a snap decision, but rules out deciding tomorrow after six hours of careful thought-in short, confuses t with d.xldt. Which vs. that. This distinction might seem a high-flown affair, but it rests on being able to recognize what a definition is, and mathematicians are good at that "That", present or understood, is the defming pronoun; "which", which invariably follows a comma, is the nondefining one:

v

Here's the calculus book that I bought yesterday. [Defmes which one.] Here's the calculus book I bought yesterday. [Defmes which one.] Here's the calculus book, which I bought yesterday. [Adds infonnation about the one already under discussion]

v

22

V

6. ENGLISH USAGE

False signals. You can't always guess the meaning of a word from its sound: bemuse: stupefy enormity: heinousness fortuitous: accidental

fulsome: gross hypothecate: mortgage mere!OCious: tawdry

noisome: disgusting notorious: infamous officious: meddlesome

To be safe, do not use fancy-sounding but slightly unfamiliar words without looking them up.

6.3. A few matters of grammar As far as. A popular solecism is

to drop the verb:

As far as algebra, I like it. X As for algebra, I like it. As far as algebra goes [is concerned], I like it.

v

Y

Before long, I suppose, we will be hearing: "As far as that, it's a good idea". (As far as I, it's a bad idea.)

Different than.

X

A is different than B. X A is different from B. Y Apparently, the "er" sound in "differ" suggests a comparative-like the slogan on the delivery truck: Faster than rail, regular than mail. (?)

She explained it to I. X They invited my husband and I to the colloquium. X They invited my husband and me to the colloquium. Y She explained the proof to Bob and I. X She explained the proof to Bob and me. Y It will be goodfor he and! to discuss it. X It will be good for him and me to discuss it.

Y

The test is easy-leave out the other object They invited me. Y She explained the proof to me. Y It will be good for him. It will be good for me.

23

Y

6. ENGLISH USAGE

One of those are. X A pernicious habit-a plural verb in response to the word it follows, although the actual subject is singular: The best one of those are in the book. The best one of those is in the book.

X

V

The character of the problems have changed a lot. The character of the problems has changed a lot.

X

V

The caliber of those people are not that of ours. X The caliber of those people is not that of ours. ..J

The test is easy-omit the phrase: The best one is in the book. V The character has changed a lot. V The caliber is not that of ours. V

One of those who. A standard trap: She is one of those who enjoys mathematics. She is one of those who enjoy mathematics.

X

V

The test is easy-invert the word order: Of those who enjoy mathematics, she is one. ..;

6.4. In terms of This phrase appears almost invariably as mere noise, at best displacing a simple in, to, for, of, or by. /low is he doing in terms of calculus? X /low is he doing in calculus? V /low shall we respond in terms of her letter? (Meaning ?) II ow shall we respond to [in view of(?)] her letter? V That's important in terms of getting information. X That's important for getting information. V That's important for information. V We should respond to their challenge in terms of inspiring the students. X (Meaning ?) We should respond to their challenge of [by (?)] inspiring the students. V

24

6. ENGUSH USAGE

In terms of schedule, let's meet Wednesday. X Let's meet Wednesday. ...; Express the roots in terms of the coefficients.

...;

6.5. Some distinctions in spelling 1/e complemented the lecture with slides and his audience complimented him for it. ...; She was discreet when criticizing the article on discrete mathematics. Y Their loose reasoning will lose the argument; that will be their loss. ...; Mathematical induction is the principal principle for proving theorems about the integers. ...; Do your students call it the "communitive" law too? Be sure to hold up your end: accidently, incidently X accidentally, incidentally ...; concensus X cons_ensus ...; correspond.e_nce, exist,e.nce correspondance, existance, inadvertant, occurrance X inadvert,e.nt, occurr,e.nce ...; indispensable ...; indispensible X

noone X no one ...; none ...; (The first two are easy: a "ly" adverb has to start from an adjective, not a noun. The third is easy, too: think of consent.).

6.6. Some ups and downs of punctuation Two "comman" pitfalls. Either set off a phrase with commas, or don't set it off; that means enclose it in an even number of commas. We will learn in the next chapter, how to solve it. X We will learn, in the next chapter, how to solve it. Y We will learn in the next chapter how to solve it. ...; Cultivate an affection for the semicolon; it was invented for combining two closely related sentences into one, for better flow. A comma is insufficient:

25

6. ENGLISH USAGE

This shows that x = 2, therefore y = 3. This shows that x = 2; therefore y = 3. EXCEPTIONS: I came, I saw, I conquered.

V

X

V

I think, therefore I am

V

The dllSh (-). I use this often in place of a colon or semicolon, because it stands out so well-but indiscriminate use devalues it. Do not type a hyphen when you intend a dash. Use two hyphens - - or type the underline a half -space up. Hyphens. I use hyphens to resolve otherwise ludicrous constructions: A non-PhD granting institution. X [Granting the non-PhD?] A non-PhD-granting institution. V An ex-college professor. X [A professor at an ex-college?] An ex-college-professor. V In the latter example, Bernstein [12] prefers to regard "ex" as an actual word and do without either hyphen. All authorities agree that Aformer college professor V is best of all.

The slash (!). The following actual quotes show the slash representing: and: at: in: or: a comma: a hyphen:

a list of hotels/motels; The University of Texas/Austin; our correspondent Leslie March/New York; study math and/or physics; algebra/topology/analysis and probability; a math! physics major.

And/or, I suppose, we could define the slash in "and/or" to mean and/or. The principal effect of a symbol with so many interpretations is to betray the writer's cloudiness of thought

6. 7 Latin words It is good to steer clear of foreign words unless you are sure of your ground.

26

6. ENGLISH USAGE

I.e. vs. e.g. Don't use "i.e." when you mean, e.g., "e.g.". The first stands for id est, =that is; the second is exempli gratia: "of example for the sake"-i.e.,for the sake of example. Curriculum vitae = "course of life". For short. use vita ("life"). But "my curriculum vita" X and "my vitae" X are illiteracies. The plural, by the way, is curricula vitae ("courses of life") or curricula vitarum ("courses of lives"); or, for short. vitae ("lives").

Noon is 12:00 m, for meridies, = "midday". Plurals. Data, like curricula, extrema, maxima, and minima, is plural, as are bacteria, media, and symposia, and, from the Greek, criteria (sing. criterion) and phenomena (sing. phenomenon). This data is interesting. X These data are interesting. ...; That criteria is the one. X That criterion is the one. ...;

6.8. Pre-plan ahead for the future in advance We used to reserve our seats, and packaged food came sealed. Today our seats are pre-reserved; and pre-packaged food is pre-sealed. Remember, then: Have your students pre-learn the material for the test Pre-prove your theorems before publishing them. Pre-prepare your lectures.

27

7. REVISE, REVISE When you have completed your manuscript, you may feel exhilarated, and impatient to rush it off to the editor. Contain yourself. Many manuscripts are rejected out of hand because the author has submitted a hastily written first draft The cautious procedure is k> set your manuscript aside for a day or two. Then give it a critical going over. Have one or more colleagues and maybe a student or two read the manuscript and comment on it When you get their comments, don't shout them down for being numbskulls: if they had trouble understanding you, others will too. Go over the manuscript one more time, editing carefully. Then, at last, you are ready to type up the final draft, to be submitted to the journal.

8. MECHANICS 8.1. Submit a clean manuscript Within a few years, authors will presumably be submitting manuscripts in finished form, ready for photographing and printing. Until then, every manuscript accepted for publication must pass through the hands of a copy editor, who marks it with detailed instructions to the compositor (typesetter) about styles of type (caps, small caps, boldface, italic, script); sizes of type; alphabets (Latin, Greek, German); spacing, particularly in formulas; leading (pronounced ledding): spacing between lines; displays; placement and layout of tables and diagrams; where to break an expression at the end of a line; style of fractions

28

8. MECHANICS

and differentiation symbols; indices (subscripts, superscripts, limits in summations and integrals); and so on. In order to do this, the copy editor needs a clean copy, with lots of room to make all those markings so that the compa;itor can read them. The manuscript has to withstand a great deal of handling, so use paper of good quality-at least that of standard xerox or ditto paper. (Steer clear of"erasable" bond: it's sticky and tends to smear.) Use white paper only. Use one side only. Submit clean copy with dark print Begin by replacing a worn ribbon. If you use a typewriter, clean the keys or typing element If you want to use a dot-matrix printer, be sure it produces text of good quality; the journal editors will return substandard stuff. Use pica type (10 characters per inch) or larger.

8.2. Spacing Double-space the texl Leave wide margins all around, at least 1~ ". Leave extra space between sentences. (That's standard typing practice.) Leave space between symbols; in elaborate expressions, consider a double space about equals signs: y=f(x)=Icosxdx X y = f{x) = I cos x dx V y = f(x) =I cosxdx

v

Leave a double space between a symbol and adjacent text; that helps the symbols stand out:

lienee u is even. This shows that x = y. X lienee u is even. This shows that x = y. V After a symbol at the end of a sentence, I always leave a triple space: This shows that x = y. lienee u is even. This shows that x = y. lienee u is even.

8.3.

(?)

V

Displays

The safest thing to do with a long or complicated expression is to display it -i.e., put it on a line by itself, with extra space above and below and with extra indentation. (Long means 15 typewriter characters or more.) The copy edi-

29

8. MECHANICS

tor is going to have to mark it with instructions for the printer, and can do so more clearly if it is surrounded by a lot of white space. State any special requests in the margin. Display any formula you don't want broken-i.e., continued onto the next line; the copy editor can then choose to run it into the line if there is room (and if it is not numbered).

8.4. Figures and tables Submit figures on separate sheets. For one thing, figures and text are handled by different people. For another, the exact location of a figure in the printed article cannot be determined until the editor knows how the pages will divide. But be sure to include sketches in your manuscript, for the benefit of the editor and the referees. Place them where you would like them to appear. Number every figure (if there is more than one}-Figure 1, Figure 2, and so on-and refer to each one by number at least once in the text; then the copy editor, at the time of making up pages, can find the best spot for it close to the first reference. Furnish finished drawings for camera copy-professional quality, done in india ink with an artists' pen-at 150% to 200% of the intended size. Send the original without lettering, and two copies, marked COPY, with lettering added. Computer-drawn graphics should be produced with nylon-tip (wet ink) pens, not ball-point If a finished drawing is not practical, send in a careful sketch with complete instructions; the journal editor will then arrange to have it redone prof essionally (at your expense). Treat any table that occupies more than four lines as a figure: don't try to fit it into an exact spot in the text, but submit it on a separate sheet, assign it a number as a figure, and refer to it as such in the text Source code for working computer programs should never be set in type, as no one can successfully proofread it; instead, submit good-quality camera copy. The same is true for lengthy, computer-generated numerical tables.

8.5. The conventions of the journal Consult the instructions at the front of the journal that specify how many copies to send in, am so on; they vary with the journal. Prepare your manuscript in a style consistent with the current format of the journal-e.g., are formula numbers at the left or the right? (Examine a recent

30

8. MECHANICS

issue to learn the conventions.) This will reduce the number of instructions the copy editor has to mark in and so will cut down on possible sources of error. Conventions change. We used to raise the "dot-dot-dot" both in operations and in lists, but the trend today is to indicate lists with ordinary periods: k = 1, 2, ... ; a 1 + a2 + · · ·+an; k = 1, 2, ... , n; check the journal you are sending your paper to. Remember in any case to precede the three dots by the appropriate symbol {plus sign, comma), and, in the finite case, to enclose them between a matching pair. The trend in definitions is to bold type, which stands out better than the timel10nored italic: A group is a nonempty . . .. A group is a nonempty . . .. Editorial tradition holds that pages look ragged if the spaces between lines are variable. On these grounds, authors have been asked to avoid layers of levels in the running text, by using such devices as xl(x + y), x(x + yrl, C n,r, exp(x2 + y2). Probably most editors still endorse these precepts. Others balk, maintaining that the architecture of the standard expressions X

X+Y

(~).

is instantly suggestive and that the convenience of the mathematician reader takes precedence over any set notions of page design (and moreover that an occasional irregularity of line spacing is refreshing). Check the current policy of the journal you are submitting your article to. In any event, don't ask to have fractions set in small type (except for simple ones like ~),as they are usually hard to read:

-. X

%+)'

There is no need for the expansive forms

31

8. MECHANICS

n

lim,

L,,

x--+0

k=l

(?)

within text. since the alternatives b

lim

X--+0

,

(cf.

Ja )

are simpler (and more logical).

8.6. Marking the manuscript Distinguish carefully between capitals and lower case. Err on the side of extra care. Include helpful notes in the margin, such as "cap P'' or "lc p". Distinguish likewise among other look-alikes, such as ' ' 1 (apostrophe, prime, superscript 1). The marking of italics in the text and of boldface in text and formulas is important If you cannot type them from your keyboard, use ordinary (roman) type and mark it as follows: umerli~ italics (words): italics (variables): do nothing boldface: use wavy uooerli~ Variables are set in italic-:-but do not underline them: let the copy editor handle that (An experienced mathematical compositor does not need nor even want them to be marked.) ExcEPTIONS: Underline every occurrence of the symbols o and 0, to distinguish them from 0 (zero). Also, underline the letter a when it appears as a symbol in the running text, to distinguish it from the indefinite article. By the way, constants-such as R, log, cos, 4, L. IT-are conventionally set in roman. (Log and cos are "constants" in the class of fuoctions-in contrast to f, for example.) EXCEPTIONS are the constant e, and the differential operators D, d, (), which are set in italic. Again, leave these questions to the copy editor. Indicate a Greek letter by stating its English name (e.g., "alpha") in the margin on the same line. For emphasis, underline the symbol itself in red If you have typed your Greek letters or written them legibly, it is sufficient to

32

8. MECHANICS mark the first occurrence. EXCEPTIONS: Mark every occurrence of the Greek letters e, II, and 1: to distinguish them from E, ll, and L (membership, product, summation). Whether Greek letters are set in italic depends on the convention of the journal. Leave the question in the hands of the editor. Identify every handwritten symbol the first time it occurs. Write its name (e.g., "implies" for"~") in the margin, on the same line as the symbol. Identify typed German and script letters the same way. Identify N ("aleph") and other troublesome symbols more often, perhaps at every occurrence. (Some writers fashion an aleph that is easily mistakable for a script N, thus: N.) If your notation is at all elaborate, provide the editor with a separate sheet containing a glossary and indicating any special conventions.

8.7.

Proofreading your manuscript

Before sending off your manuscript, give it one last careful, patient goingover. Neither the copy editor nor the compositor can assume responsibility for correcting mathematical misprints or resolving notational ambiguities.You wrote "euclidean" on page 3 and "Euclidean" on page 4; which do you intend? A little care on your part reduces the chances for error and avoids time-consuming correspondence that may delay the publication of your paper.

8.8. The odyssey of a manuscript Send your manuscript to one of the editors, as explained on the inside cover of the journal. The editor will acknowledge receipt and will forward the manuscript to one or more referees. The odyssey has begun. The complete path is outlined in TABLE 1. There are many variations. The editor may reject an article at any stage, original or revised, without sending it to a referee. The editor may consult a succession of referees. If you revise your paper (perhaps in response to a referee's suggestions), the editor may decide to have the referee look at the revision, thus starting a new round. If you disagree with a referee's suggestions, you may appeal to the editor, thereby also inaugurating a new (more controversial) round. The success of the journal depends upon good will among the author, editor, and referee. Reports from editors and referees should be scientific, factual, and helpful. Every editor has the right, and in fact the duty, to recommend changes

33

8. MECHANICS

TABLE 1: The odyssey of a manuscript Your manuscript goes from you:

(1) (2) (3) (4)

(5)

totheeditorofthejournal to one or more referees for opinions on its acceptability and suggestions for improvement back to the editor, who decides whether to consider it for publication (a) if not accepted, back to you (b) if accepted (i) conditionally (a) back to you for revisions (j3) back to the journal editor (ii) unconditionally to the copy editor, who marks it with instructions for the compositor to the compositor, who sets it in type and runs off the "galley proofs".

The galley proofs now go: (6)

(7) (8)

(9)

to the copy editor, who proofreads them and marks corrections to you for your proofreading and corrections back to the copy editor to the compositor, who makes the corrections and runs off the "page proofs".

The page proofs now go:

(1 0) (11) (12)

to the journal and copy editors, who check the page layout and proofread and correct the pages back to the compositor for final correction to the printer.

in a paper-in organization, in wording, in mathematical emphasis; but of course no one should actually incorporate any changes without the approval of the author. Referees work voluntarily and anonymously; the only public recognition for what is often a great deal of time and hard work is a list in the journal

34

8. MECHANICS

naming all the referees for the year. Still, they should be reasonably prompt; if you have not heard about your paper after two months, it is in order to make a polite inquiry. Finally, it is to your advantage as the author to be prompt, courteous, and cooperative. Accept advice, or, if you feel sure of your ground, stand up for your own opinion; but whichever it is, be gracious and keep the discussion on a scholarly level. Books. The procedure for a book is much the same. The main differences are the following. Usually, you begin by sending in some preliminary material, such as an outline and some sample chapters; these go to the editor or other representative of the book series, or, in case of doubt, to the Executive Director of the Association in Washington. You sign a formal contract You get to approve the marked manuscript before it goes to the compositor. You are given the opportunity to correct the page proofs.

8.9.

Marking the galleys

The copy editor is trained to catch technical errors, such as crooked lines, broken letters, or an integral sign that is too small; but the responsibility for mathematical accuracy rests with you. For journal articles, the galley stage is your only chance to catch typesetting errors. Keep in mind that even one misprint can render an entire proof incomprehensible. Muster your patience, and check the galleys word by word and symbol by symbol. Mark each correction twice, once at the spot in the text that needs changing, and once in the margin on the same line to catch the editor's eye and describe what is to be done. The main goal is that your instructions be clear. In case of any doubt, you can always state what you want in plain English. Table 2 illustrates the procedure for a mythical galley, furnished with an ample supply of errors. The abbreviations should be easy to figure out, except perhaps for "w.f." (=wrong font). For more details, see [6, Section 4.2.1 and Appendices D and E] or [4, Section 15 and pp. 17 -19]. Work carefully; but don't dawdle, especially with a journal article. The editor plans several issues of the journal simultaneously and operates on a tight schedule. Return the corrected galleys promptly: if you miss the deadline your article may be delayed for several issues. After you have returned them, the editors decide which issue the article will appear in; they make up the pages for that issue, send the corrected galleys and the page paste-up to the compositor, and correct and return the eventual page proofs. (As a matter of policy, you do

35

8. MECHANICS

TABLE 2: Correcting the galleys

Text to be corrected

wenow can cosnide r Al the given st In R2

The same text with your indicated corrections

fh/C4f/ d Jy nfw·~fe

t.

Is /•{.t jA\ '""

f.c.J h.f/ \31/ f)

The fmal copy, incorporating your corrections

We now consider AI' the given set in R2.

not get to see the page proofs.) The adventure ends when the compositor sends the necessary materials to the printer, and the printer prints, binds, and mails out the journal issue.

36

APPENDIX The Use of Symbols: A Case Study In this appendix, I take as ymbol-laden article and show how it can be drastically simplified Each expression to be replaced is enclosed in large parentheses and, for comparison, each suggested replacement follows immediately and is enclosed in square brackets. As a result, one can appreciate the simplification even without keeping track of the mathematical details. If you look at nothing else, at least take a moment to compare equations (11) and (12) with their counterparts (11') and (12'). The mathematics itself, while sophisticated, is elementary. There is no algebra, no geometry, no convergence; essentially, there is only bookkeeping. Recall the three important steps to take toward simplifying your notation: I. Use an uncomplicated symbol in place of an elaborate one. II. Discard any symbol that is just plain unnecessary. III. Simplify the mathematical argument itself. The article I picked is Sierpinski [24], as corrected by K. Kunugui. It is the source of the monster symbol displayed in Section 5.6, and contains examples of all three types of problems. In order to eliminate distractions, I have streamlined a great deal of the notation. In particular, I have recast the entire article in terms of R, the real numbers, so that the setting will be familiar to everyone and the argument elementary. (There are some comments about this at the end of the discussion.) The theorem now reads as follows:

THEOREM. Assume there exists a family (Aa>ae R of countably infinite subsets of R such that, for

a ::1- f3,

a sequence of functions fk: R

~

either

a e A p or f3 e A a- Then there exists

R such that every uncountable set is mapped

onto R by all but finitely many fk·

REMARK. Note that the assumed family is indexed by R itself.

37

APPENDIX

Proof. The only special fact needed in the proof is that R can be put into one-one correspondence with the set of all pairs of sequences n 1, n2, ... , t 1, t 2, ... , in which (nk) is an increasing sequence in N (the natural numbers)

and (tk) is a sequence in R. Consider any a E R. To indicate that sponds to the pair (nk), (t k), Sierpinsk:i writes

nk-- nka

and

tk

=

t a.

a corre-

(1)

k'

and he enumerates the countably infinite setA a as

(

Aa = {~?· ~q....

[

}.

)

(2)

]

(2')

(The alternative (2') is rid of the extra letter ~ .) (Type 1.) Then he defines fk(a) by the monster symbol. Sierpinsk:i had a pixyish quality and may have done all this partly because it was so much fun. But as it turns out, Sierpinski's argument is not complicated enough, and at the end of his paper, at the decisive step, he confuses a sequence with a subsequence. Kunugui corrects the proof as follows. First, inductively, he picks an integer ~a

~a

(

/.a E { n1 , n2 , . . . } I

(i

1, 2, ... )

)

(3)

[

q a; l.a E {n1 , n2, I

(i = 1, 2, ... )

]

(3')

... }

so that

38

APPFNDIX

f!'I

> l.a }

for allj < i. This is possible because the sequence in braces is increasing. (Note that (3') results from (3) by means of (2').) Then he points out that

(

a ~a l. = n , ·a

1

)

(4)

}j

]

[ where

(

(j.~ is a sequence of positive integers. I

)

]

[

The blank spaces signify that the author's expressions should be replaced by nothing at all-in other words, that what the author is writing is unnecessary. For, as it turns out. he is going to fix a, then determine i, and then, for the one and only time, pick .a 1·. I

In these circumstances, we can just call it j. (Type II.)

Next. Kunugui defines a new sequence of real numbers,

11?·

7]~.

39

. . .,

APPENDIX

as follows:

(

)

7Jf

[

]

and, fori > 1,

(

)

[

]

(5)

The reason none of this is necessary is that the only numbers

77~ that he uses in the proof are for tha;e k of the form

k = l.I

a

-but in that case we have, simply,

~.a I

[= a.], I

without further ado. (Type III.) Finally, the author defines

40

(6)

APPENDIX

(

(k

[

1, 2, ... ).

(k = l~;

'

1, 2, ... ).

)

]

(7)

(7')

In (7'), the remaining values offk(a) are arbitrary-for definiteness, say

This change from (7) to (7'), the most significant of all, proceeds in two steps. The first step is simply the substitution (6), which reduces (7) to

From here, a certain amount of experimenting leads to (7'). That's the interesting thing about this kind of editing: once you have simplified the mathematics and the notation so that the argument is easier to follow, you may see a way of simplifying the reasoning still further. Let us now follow through the argument that the functions fk, thus defined, satisfy the requirements of the theorem. Contrapositively, we consider any set S that infinitely many fk fail to map onto R, and show that it must be countable. (The original article argues by "questionable" contradiction; see Section 3.8.) By assumption, then, there exist an increasing sequence of indices n 1, n2, ... and a sequence t 1, tz, ... in R such that (j = 1, 2, ...).

(The values of li fori

-:F.

(8)

nj do not enter into the argument). By (1), there ex-

41

APPENDIX

ists

f3

e R such that

(

nk

=

nf'

[

nk

=

nk '

p

tk

=

t/3

tk

=

t{J

nk

k

)

(9)

]

(9')

fork= 1, 2, .... (The little twist in the second half of (9) requires some pondering. The more natural (9') is what leads to the simplification (7').) Now consider any a ~Ap ,with a :F- /3. (This is where we fix a.) By hy-

pothesis,

f3 e Aa. Hence (by (2))

[by (2')], there is an index i for which

(

f3

= ~.a. I

)

[

f3

a I..

]

(This is where we determine i.) Consider k = I.I

a .

In view of (5),

(

f3

= Tfka.

[

)

(10)

]

(Using (4) and (9), we get) [Using (3') and (9'), we get, for suitable j,]

42

APPFNDIX

)

(

[

k = l.

a

a;

= n.}

I

nP

=

}

]

= n .. }

(11)

(I I')

(This is the one time we pickj.)

Next. from ( (9), (1 0), and

(7)) [(9') and (7')], we have

)

(

[

ai

tk = tf

= fk(a).

tk

]

(12)

(12')

Substituting from ( (11) into (12)) [ (11 ') into (12')] gives us

(

tn

= ·a

fn

t nj

)

(13)

]

(13')

J;

J;

[

(a). ·a

fn.(a). J

43

APPENDIX

In view of (8), then, a ~ S. We have shown that fora

:ji!:,B,ifa~Apthena~S;that

then aeAp.Since Apis countable, Sis countable.

is,ifaeS

t

Comment on tbe theorem. The original article uses the language of transfinite ordinals; the hypothesis of the theorem is the continuum hypothesis (CH), which states that the uncountable sets in R, and the set of all countable ordinals, all have the same cardinal as R itself. The hypothesis as stated in this Appendix is an adaptation of (CH) to the particular problem. If we assume (CH), then the ordered set S consisting of the negative integers followed by the countable ordinals can be indexed by R, and the sets A a = {xES: x < a} are as stat ed. The converse can also be proved directly, but note that it is immediate from the conclusion of the theorem: every uncountable set in R is carried by some fk onto R, hence must have the cardinal of R.

44

Annotated Bibliography

MATHEMATICAL WRITING Essays [1] R. P. Boas, Can We Make Monthly 88 (1981), 727-731. Required reading.

Mathematic.~

Intelligible?, Amer. Math.

[2] Paul R. Halmos, llow to Write Mathematics, Enseign. Math. 16 (1970), 123-152; reprinted in Norman E. Steenrod, Paul R. Halmos, Menahem M Schiffer, and Jean A. Dieudonne, llow to Write Mathematics, American Mathematical Society, 1973, 19-48, and in P.R. Halmos, Selecta, Expository Writing, Springer, 1983, 157-186. Required reading. [3] J. E. Littlewood, A Mathematician's Miscellany, Methuen, 1953. A charming classic that every mathematician should read in its entirety in any case. The piece relevant to the present manual is described in Section 3.4.

Author's manuals [ 4] American Mathematical Society, A Manual for Authors of Mathematical Papers, Eighth Edition, pamphlet, 20 pp. American Mathematical Society, 1984. A basic manual, containing an excellent short section on organizing the paper (tactfully marked ''for beginners'') and a great deal of material on technical matters such as styles of notation and the form of abstracts. The manual has been updated at irregular intervals since the first edition in 1962; a precursor with the same title, style, format, and galley-proof example was issued by the Society in 1943 as a sup plement to the March issue of the Bulletin.

45

BffiUOGRAPHY

[5) Harley Flanders, Manual for MONTHLY Authors, Amer. Math. Monthly 78 (1971), 1-10. Good advice in no-nonsense style.

[ 6] Ellen Swanson, Mathematics into Type, Revised Edition, paperback, 90 pp. American Mathematical Society, 1979. An exhaustive work, as suggested by its subtitle: Copy Editing and Proofreading of Mathematics for Editorial Assistants and Authors.

THEENGLffiHLANGUAGE Essays and criticism [7] Richard Mitchell, The Leaning Tower of Babel, 279 pp. Little, Brown, 1984. Excerpts from The Underground Grammarian, Mitchell's hard-as-nails monthly attack on the evils of the education establishment The book is listed here for the passage by Ben Jonson referred to in the introduction to Chapter 6.

[8]

- - - - - . , Less Than Words Can Say, 224 pp. Little, Brown,

1979. The author at his brilliant and acerbic best versus the educationists. Chapters 3, 5, and 6 contain ideas that everyone, even mathematicians, might well ponder.

(9]

William Zinsser, On Writing Well, An informal guide to writing

nonfiction, Harper and Row, 1976. A lot of good sense served up in a friendly manner.

46

BIBLIOGRAPHY

Manuals of style [10] Claire Kehrwald Cook, Modem Language Association, Line by Line, //ow to improve your own writing, paperback, 219 pp. Houghton Mifflin, 1985. A low-key presentation of simple, sensible suggestions and clear ex planations. But the publishers get a D- for consistently crooked right hand text with nonexistent margins.

[11] William Strunk Jr. and E. B. White, The Elements of Style, Third Edition, paperback, 92 pp. Macmillan, 1979. A staple. The short paragraph under the heading Omit needless words is a gem.

Dictionaries of English usage [12] Theodore M. Bernstein, The Careful Writer, A Modern Guide to English Usage, paperback, 487 pp. Atheneum, Eleventh Printing, 1986. Far less ambitious than Follett or Fowler, and a warmer spirit: Large type, an open page, and an author who is talking to you rather than writing an essay.

[ 13] Wilson Follett, Modern American Usage, Edited and completed by Jacques Barzun, Thirteenth printing, paperback, 436 pp. Hill and Wang, 1986. Presumably an attempt to be the American Fowler. I do recommend the piece on possessives and the convincing argument for pronouncing "rationale" as ray shun ay' lee. But I am put off by two items in my field of greatest expertise: (i) he confuses playing by ear with playing by heart (well, that's only off by two letters), and (ii) he likens the proscription of the split infinitive in English to that of parallel fifths in music, whereas the first is mostly whim while the second has a sound basis (sorry about that). Finally, the pages do not lie flat and the inside margins are tiny, so that reading the text near the binding is a nuisance.

47

BIBUOGRAPHY

[14] R W. Fowler, A Dictionary of Modern English Usage, Second Edition, Revised and Edited by Sir Ernest Gowers, 725 pp. Oxford University Press, 1965. The peerless classic, revered for the wit and artistry of its writing as much as for its pungent advice. Must be sampled slowly; ideal for browsing: pick out any page and enjoy the language. Be sure to read the section on the "fused participle", especially the first column (and, for fairness, the last), as well as the discussion of the split infinitive.

Dictionaries of synonyms [15] J.l. Rodale, The Synonym Finder, 1359 pp. The Rodale Press. Valuable for suggesting possibilities or supplying the word that was at the tip of your tongue. Arranged alphabetically. [16] Roget' s International Thesaurus, Fourth edition, paperback, 1317 pp. Harper and Row, 1977. The classic reference for synonyms; immensely valuable. The material is arranged by categories of meaning, so that closely related ideas appear near one another. An alphabetical index of 500 pages, with entries subclassified by shades of meaning, gets you started.

[17] Webster's New Dictionary of Synonyms, 907 pp. Merriam Webster, 1984. Arranged alphabetically. Commentaries and quotations elucidate the dis tinctions among synonyms; these features are absent from Rodale and Roget. There are also lists of analogous words, contrasting words, and antonyms; Roget has an equivalent feature, but Rodale has nothing. The coverage seems skimpy. Examples. (1) "Laud" is here, but neither "laudable" nor "laudatory'', while the other two works give all three; (2) between "protect'' and "proud", Webster has 6 entries; Rodale, 12; Roget,35.

48

BffiUOORAPHY

English dictionaries [ 18] The American 1/eritage Dictionary of the English Language, First Edition, Unabridged, 1550 pp. Houghton Mifflin, 1969. In a class by itself; reputed to have been written as the answer to Web ster's permissive Third International. The usage notes are particularly valuable. But I understand that later editions have been watered down.

[19] The Random 1/ouse Dictionary of the English Language, 2059 pp. Random House, 1983. More comprehensive than Heritage. Includes helpful dictionaries to and from French, Spanish, Italian, and German. Exactly twenty years ago this month, I wrote the publisher that the interval [heimisch, Heimsuchung] should precede [heimttickisch, heissen] (taking the occasion to allude to the title of the work); the mislocation has withstood all in tervening printings.

MISCELLANEOUS REFERENCES [20] Edgar Asplund and Lutz Bungart, A First Course in Integration, Holt, Rinehart and Winston, 1966 [21] P. Erdos, Some Remarks on Set Theory, Proc. Amer. Math. Soc. 1 (1950),127-141. [22] Lester R. Ford, Sr. and Lester R. Ford, Jr., Calculus, McGraw-Hill, 1963. [23] John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1947. [24] W. Sierpinski, Sur une Certaine Suite lnfinie de Fonctionsd'une Variable Reelle, Fund Math. XX (1933), 163-165; XXIV (1935), 321-323. [25] Trans. Amer. Math. Soc. 64 (1948), 596. [26] Lewis Carrol~ Alice's Adventures in Wonderland, Chapter XII, Alice's Evidence. [27]

W. S. Gilbert and ArthurS. Sullivan, The Mikado.

[28] Ben Jonson, Explorata-Timber, or Discoveries Made upon Men and Matters.

49