The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.
--The Happy (Non-Formalist) Mathematicianm This letter addresses some fallacies in J. Henle's article "The H a p p y Formalist" in the January 1991 issue of The Intelligencer. Hilbert [1] said that two things are given: " the sign and concatenation." The first implies recognition by shape, the second by rank in a sequence. Hans Freudenthal [2], w h o attempted to construct a language aimed at cosmic communication, wrote: We have agreed to abstain as much as possible from showing (concrete things or images of concrete things), but we cannot entirely abstain from it. Our first message will show numerals as an introduction to mathematics. Such an ostensive numeral meaning the natural number n, consists of n beeps with regular intervals; from the context [my italics] the reader will conclude that it aims at showing just the natural number n. Henle defines a formal system as consisting of " . . . a formal language (a collection of symbols together with unambiguous rules for forming these into statements of the language . . .)" etc. The symbols a n d expressions of the formal system are generally said to denote some arbitrary objects, be they concrete or abstract, w i t h o u t a n y further strictures as to h o w their denotees are to be recognized. It appears that the denotees are k n o w n by their names only, so that it h a p p e n s that two objects defined in two "disconnected" m o d e s are given the same n a m e (see Quine [5] below) a n d hence taken to be identical; elsewhere (e.g., Kleene [8] below), two objects that are considered to be distinct are given the same n a m e (and, in Kleene, are correlated to the same zero entity) and so become indistinguishable as formally denoted. To set up a formal system we would w a n t an apparatus to recognize the objects with which we start as designated by their n a m e s in the formal language: symbol and term, sign a n d expression, shape and rank. Two " c o m p l e m e n t a r y " m o d e s of recognition are needed to recognize which of the above objects is denoted by its designator in the formal language. One cannot avoid referring to the context in identifying the things we start from, whatever they are. Contrary to the claims of the formalists, the formal system is no more formal and context-free than are the systems that are to be 4
imbedded in it. Gian-Carlo Rota [3], in an article entitled "The barrier of m e a n i n g " in the Notices of the American Mathematical Society of February 1989 (see also m y comments [4] in a letter to A.M.S. of May that year), reported on a discussion between himself a n d Stan Ulam. Professor Ulam felt that artificial intelligence did not adequately address the all-important concepts of context and meaning. Specifically he asked, "Suppose I took a c o m m o n object, say a key, a n d s h o w e d it to you. Even the m o s t c o m p l e t e l y explicit d i c t i o n a r y d e f i n i t i o n would not help in recognizing the object as such unless you already had some familiarity with the w a y the object was u s e d . " He thus maintained that recognition of objects as such and such is only possible if definitions are given in terms of their function in context. Imagine n o w a person unfamiliar with the w o r d " k e y . " I s h o w her an object a n d n a m e it " k e y . " Thereafter a n y similar object will be so recognized. Alternatively, I m a y describe a key as something that can be placed in a lock to o p e n a door. Henceforth, anything performing that function will be recognized as a key. The two " k e y s " can be identified as naming one a n d the same thing since that which was s h o w n is recognized to function as described. A n d vice-versa, that which functions as a key is recognized as similar to the object that had been presented. The object d e n o t e d by " k e y " can then be identified as the same concrete comm o n object w h e t h e r it is recognized in one of these two m o d e s - - b y similarity or by description. By contrast let us turn to formal syntax. Quine [5, p. 287] writes,
Now all these characterizations are formal in that they speak only of the typographical constitution of the expressions in question and do not refer to the meanings of those expressions... We m a y thus, most simply, recognize formal symbols as written patterns of a totally arbitrary nature. Quine t h e n writes, But this explanation of formal is vague; we now turn to a more precise version. Let us use '$1', '$2', . . . . '$9,' as names of the respective signs or typographical shapes 'w', 'x', 'y', . . . ; thus:
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1 9 1992 Springer Verlag New York
S1 = 'w' = double-yu, (I) $2 'x' = ex and later (II) S~ = (~x)(3y)(yAx). -(3y)(xAy), where 'xAy' means x is the sign which alphabetically just succeeds the sign y . . . . In accordance with Quine's definition of description, x in (II) is the one a n d only object (entity) such that . . . . Here then '$1' is the name of an object which is recognized by its rank on a list. Now let us alter the typographical shapes of our objects: this w o u l d affect the recognition and the name, S1, of the object denoted by the first '$1' but not the recognition of that named by the second. Alternatively, let us change the order on the alphabetical list: this would leave recognition and the name of the object named by the first '$1' intact, but affect recognition and the name, $1, of the object named by the second '$1'. We are here dealing with abstract objects (see Bob Hale [6]) which have no identity other than that bestowed on them by their mode of recognition. The two '$I' denote "disconnected" (in the terminology of S. K6rner [7]) objects whatever they be. Nor does it help to equate $1 to double-yu. The first '$1' denotes the shape of the letter named double-yu; the second '$1' its alphabetic rank. The matter is not solved by claiming that the formal signs denote concrete objects--for instance, say, wooden things. The objects named by the first 'S 1' would still be recognized by their similarity in shape; those recognized by the second '$1' because, perhaps, they came out of the bin numbered 1. The two modes of recognition manifest themselves for objects denoted by formal signs in, say, Kleene [8, pp. 69, 70]: It must be possible to proceed regarding the formal symbols as mere marks on paper, and not as symbols in the sense of symbols for something which they symbolize or signify. It is supposed only that we are able to recognize each formal symbol as the same in each of its occurrences, and as distinct from the other formal s y m b o l s . . . Later he writes: 0 is a symbol and 0 is a term (a sequence of one symbol), where 0 are the names of entities of some kind. The first 0 is recognized as being the shape of a mark on paper or of a typewriter key; the second is recognized as a ranked entity within a recursive scheme. When Kleene maps these into his generalized arithmetic, he needs to map both of these into the same zero entity. (He erroneously stated [8, p. 249] that distinct objects were mapped onto distinct entities and recognized this error when it was pointed out to him.) He then can give one G6del number to the one entity correspond-
ing to both of these, so as to embed his formal system into a purely recursive scheme. In all these cases, things (whatever they are) that are recognized in two distinct ways are given the same name (Quine $1; Kleene 0) without justification; for unlike the case of the common concrete object " k e y , " the denotees cannot prove to be identical. Formal systems, contrary to Henle's claim (p. 14) do not have the same kind of reality as the chairs we sit on, because the things we start from (symbol, or term; sign, or expression as a sequence of one sign) cannot be recognized as would a chair or a key. Both Quine and Kleene later suggest a problem which is not confronted. Thus, Quine states [5, p. 287]: This concluding chapter will be unintelligible to those readers in whom there is a lingering tendency to confuse use and mention of expressions. I have not seen how to make the chapter less liable to misunderstanding except at the expense of a disproportionate increase in length. And Kleene [8, p. 250]: This problem of designation which is troublesome to treat explicitly, is extraneous to the metamathematics as mathematics. The issue can be avoided by using only names of the formal objects, and not claiming to exhibit the objects themselves. . . . (While we can thus avoid the problem of designation in our metamathematics, it would have to be faced in discussing the application of the metamathematics to a particular linguistic system.) The problem of designation, although troublesome, cannot be ignored. Kleene (private correspondence) maintained that the ambiguity shown above is harmless because it is resolvable by context. Quine [9, p.42] at one point suggested that we drop the whole notion of sets of inscriptions making up the universe of the protosyntactician, except for the single signs, and let variables range over sequences of signs. This does not solve the problem of recognition of the single sign which functions as a one-element sequence (if recognized as a sequence, the variables now range over sequences of sequences, etc.). Alternatively, he suggests that we consider only G6del numbers. This view is also implicit in others who assign different G6del numbers to the things denoted by say 0, ab initio, and then claim that they can use the same name "0" because the things (whatever they are) have different G6del numbers. This is putting the cart before the horse; we start, supposedly, with an unambiguous system of formal objects to which we then assign numbers. The fact is we don't. The a n a l o g y Henle m a k e s w i t h t h e i n t r i n s i c "meaninglessness" of the word "tree" is fallacious. A broad " K n o w l e d g e of the W o r l d " - - i n Fodor a n d Katz's [10] terminology--is already implied in recognizing the above "thing" as a word. We could just as well recognize it as a picture; as a sound; as a juxtaposition of letters (as shapes? as members of the alphabet?). THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992 5
It is also interesting to note that once the G6del number of, for example, a numeral is given, one can work backwards recursively to the G6del number of the numeral 0 as the first term in the expression. However if we merely write 0 (wishing to construct a numeral), the G6del number attributed to it would be that for the symbol 0; unless we consider the metamathematics prior to the formal language and let it tell us what we have written. Such a procedure negates the claim of formality. A computer could not be programmed to recognize 0 as a symbol or a term ab initio; nor could it be programmed (as Henle would want it) to arbitrate the dispute as to whether it is the one or the other, for that would create an infinite regress. We must do that and then give it the appropriate G6del number. Two, not one as Hente would have it, numbers are needed for coding. We may have to admit the inherent existence of two modes of recognition (knowing) and accept that we cannot escape from this duality.
References 1. Resnik, M. D., Frege and the Philosophy of Mathematics, Ithaca: Cornell Univ. Press (1980). 2. Freudenthal, H., Lincos, Amsterdam: North-Holland (1960), p. 17. 3. Rota, Gian-Carlo, "The barrier of meaning," Notices of the A.M.S. 36 (1989), p. 141. 4. Lipschtitz-Yevick, M., "Mathematizing the notion of similarity," Notices of the A.M.S., 36 (1989), p. 531. 5. Quine, W. 0., Introduction to Mathematical Logic, London: Norton (1950). 6. Hale, Bob, Abstract Objects, Oxford: Blackwell (1987). 7. K6rner, S., Philosophy of Mathematics, London: Hutchinson University Library (1960). 8. Kleene, S. C., Introduction to Metamathematics, New York: Van Nostrand (1952). 9. Quine, W. O., Ontological Relativity and other Essays, New York: Columbia University Press (1969). 10. Fodor, J. A., and J. J. Katz, Structure of Language, Englewood Cliffs, N.J.: Prentice-Hall (1964), p. 490. Miriam L. Yevick Department of Mathematics Rutgers University Newark Newark, NJ 07102 USA - A Letter to J. M. H e n l e , the " ' H a p p y F o r m a l i s t " Dear Professor Henle: I am very glad that you are so happy. I, along with many of my colleagues who do not find the reigning philosophy of mathematics nearly so comfortable as you do, are not quite as happy these days. Articles such as yours in the Mathematical Intelligencer (vol. 13, no. 1) add, in a small way, to my unhappiness. Let me try to tell you why. 1. Your discussion of the arguments against Formalism chases straw men. 2. Your discussion of metamathematical issues is weak. 6
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
1. Arguments Against Formalism In the "Ambiguity" section, you state, "I can imagine some worry about the loose description I have given of a formal system." Well, Professor Henle, I do not worry about it nor did any serious founding mathematical philosopher. The description of a formal system can obviously be made " m u c h more precise" as you state, and this has never been a point at issue. In the "Existence" section, you state, "Another objection has been raised: We at least regard formal systems as r e a l . . . Aren't we then Platonists, too." I have never heard anyone, not even the most radical finitist, argue that Formalism is inconsistent because it believes in formal systems themselves. Against whom are you arguing? In "Mathematical Aesthetics" you state, "The accusations here are similar to those above. Formalism says nothing about what mathematics is good and what is bad." What you do not state is that no one ever asked it to. Philosophy and aesthetics are two quite different things, in any field. In "Formalism in the Large" you discuss the book Descartes' Dream, stating, "[The authors] are careful not to blame the Holocaust on formalism, but they come very d o s e . " You then devote an entire half page to refuting this "argument." Maybe Davis & Hersh did claim that Formalism was somehow responsible for the Holocaust. I do not recall. But even if they did, do you really think that you are addressing any issues of mathematical foundations by arguing with them?
2. Metamathematical Issues To justify the assertion that your discussion of metamathematical issues is weak, I will analyze three statements of yours. From "Scope": "Of course, the logicists are right that mathematics can be reduced to logic, but the choice of logic is arbitrary. It can also be reduced to set theory, arithmetic, geometry, or knot theory." Can it indeed? First of all, as anyone versed in the history of mathematics can tell you, the Logicists were not right, at least not in the original form of their thesis due to Frege and Russell. The most valiant attempt was, of course, due to Russell-Whitehead. This attempt failed on at least three counts: their need for the Axiom of Infinity, as pointed out by Hilbert; the s y s t e m ' s i n c o m p l e t e n e s s , as pointed out by G6del; and their need for the Axiom of Reducibility, as pointed out by many, including Russell himself. But the failed attempt added immeasurably to mathematics. The choice of logic as a foundation was anything but arbitrary. They were trying explicitly to enlarge Kant's concept of "analytic" to include all of mathematics, thus showing that Kant's example of
mathematics as synthetic a priori k n o w l e d g e was faulty. It was a brilliant idea. And logic, in this Kantian sense of the analytic, was the only possibility. Arithmetic is adequate as a foundation for mathematics only insofar as G6del's provability predicate captures all mathematical activity. But this is no more than a technical restatement of the Formalist Thesis! I know someone w h o is (in a sense) attempting to reduce mathematics to geometry. Do you have any idea of how difficult such an undertaking is, how uncertain a program it represents, and how significant a result it would be if true? I have yet to see anything which purports to be a reduction of mathematics to knot theory. I would be very interested in such a thing. Also from "Scope": "What about intuitionist mathematics, for example, or model theory, or metamathematics? In fact all of these can be imbedded [in formal systems]." By definition, if, say, Intuitionists believed the truth of the above "fact," then all Intuitionists would be Formalists. But Intuitionists are manifestly not Formalists. They (and others) do not believe your imbeddings work. There are many reasons why, but I will only mention one. The statement that a particular mathematical construction represents an imbedding of an area of mathematics into a formal system ("the theorems of that system are exactly the theorems of the given area," as you define it) requires proof in any particular case. And that proof must itself be carried out within its own "area." Very often, the well-known "imbeddings" (such as your example of Kripke models for Intuitionism) require very strong reasoning principles such as Zorn's Lemma to prove that they have the defining property of an imbedding. Obviously, if one doesn't believe these strong principles are valid, then one does not believe an imbedding exists. Thus, for you to state your "fact," you are implicitly stating that you believe in Zorn's Lemma, etc. This is begging the question. From "The Incompleteness Theorem": "The formalism I affirm coexists (as we all do) with undecidability and uncertainty . . . . Rather than dismay, I face the situation with delight." Allow me to attempt to dismay you somewhat. As I discussed above, "the formalism you affirm" carries with it a great deal of logical baggage in the form of strong reasoning principles--principles often stronger than the mathematical area they are being used to study. David Hilbert, the founder of Formalism, was quite aware of this problem (and thus, was undoubtedly not quite as happy as you), and he attempted to use only the weakest possible system of logic for his metamathematical investigations. He restricted himself to what
he called "finitary reasoning," which is a weak subsystem of arithmetic. This was called Hilbert's Program. Now, a proof using only finitary reasoning that, say, model theory could be imbedded in a formal system really would be something remarkable; the vast majority of mathematicians would be forced to accept this proof. Model-theorists, for example, believe in finitary reasoning (whether they know it or not). They would have no choice but to accept the Formalist Thesis, at least with regards to model theory. G6del's Incompleteness Theorem (whose significance for formalism seems to elude you) destroyed this hope completely by showing that finitary reasoning was inadequate to imbed even arithmetic in a formal system, let alone model theory. In no area of significant mathematics (like arithmetic) can you prove (prove, not assert as you like to do) that an imbedding in your sense exists without using reasoning principles at least as strong as the area y o u are trying to formalize. Of course, there is nothing wrong with believing in the Axiom of Choice. Most mathematicians do. But most mathematicians are not Formalists, in spite of what they were taught to say. A true belief in Formalism is incompatible with a true belief in ideas like the Axiom of Choice. In your field of Large Cardinals, you don't care. This is a paradigmatic formalist study. You investigate a single axiom system (ZF) and consider the effect of adding various new formal axioms to it. If you spend all your time doing this, you may accept the Formalist Thesis. It would be analogous for a classical Galoistheorist to believe that all field extensions are separable. 3. In Conclusion You state, "The hallmark of the formalist is tolerance . . . . Its thesis guarantees the right of anyone to practice d e d u c t i o n and i n d u c t i o n in any formal system." I am sorry, Professor Henle, but this remark is comically similar to Henry Ford's reply to customers w h o wanted a color choice for their Model T's: "They can have any color they want as long as it's black." What rights does the Formalist Thesis give me if deduction in a formal system is inadequate for the kind of mathematics I want to do? I can assure you that there are many parts of mathematics (such as Intuitionism) which have eluded a generally accepted formalization. Strong arguments can be made that some of these can never be formalized. And your telling Intuitionists that they should be " h a p p y in a Kripke model" is inadequate, for reasons which I have explained. You say, "Beyond a brief definition of mathematics, [the formalism I have described] says nothing." THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
7
I agree that the Formalism you describe says nothing new.
But your "brief definition of mathematics" is precisely w h y F o r m a l i s m "'has b e e n a t t a c k e d so strenuously," as you claim. I personally have yet to see a "brief definition of mathematics" that I find at all satisfactory. You find it "ironic that formalism should suffer such abuse." I find it far more ironic that such a self-serving statement should appear in this issue of the Intelligencer. The previous issue featured a detailed account of the Intuitionist Brouwer's forced removal from the editorship of the Mathematische Annalen by the Formalist Hilbert precisely because he was afraid of Brouwer's philosophical influence on the journal. So much for formalist tolerance! Alan Paris Department of Mathematics Cornell University Ithaca, NY 18503 USA
9Jim Henle Replies I am sorry to make anyone unhappy. Despite previous experience, I am always startled by h o w deeply mathematical philosophy can be felt. I think that much of the disagreement I read in these letters can be traced to overestimating the claims of formalism. If I produce an axiom system, for example, such that all the k n o w n truths of arithmetic are exactly the known theorems of this system, I would claim this is evidence for the formalist thesis. Alan Paris would protest that I haven't formalized arithmetic. If I produce an axiom system that similarly mirrors our knowledge of the finite and infinite in mathematics, I would again claim this as evidence, but Peter Nyikos would object (Intelligencer, vol. 13 no. 3) that I have not captured the meaning of "finite." If I produce a formal system in which may be embedded the work of mathematicians in analysis, and I again offer this as evidence, Miriam Yevick would argue that my system is no more formal than the analysis I embed. My answer is that these objections are not relevant to formalism. Formalists do not claim to formalize the truth of arithmetic. We do not claim to formalize the meaning of "finite." And we do not claim our systems are more formal than others. What we claim is that all mathematical results can be discovered as the results of various formal systems. That's all. Paris believes I need the Axiom of Choice to prove that all intuitionistic results will fall into Kripke models, but I am not claiming this. The mathematical claim is that all results today do fall into Kripke models. The 8
THE MATHEMATICAL 1NTELLIGENCER VOL. 14, NO. 1, 1992
philosophical claim is that this will continue to be the case. And incidentally, belief in Choice is indeed independent of formalism, just as a staunch supporter of the First Amendment may or may not believe in God. Paris sees other weaknesses in my paper, but I think these may stem from our differing conceptions of arithmetic. When I note that mathematics can be reduced to arithmetic, I am merely noting that after G6detization, any system can be described arithmetically. Inasmuch as arithmetic can be recovered from geometry and knot theory, these fields can serve as well. The observation is trivial, I know, but as I say, formalism is not extravagant, and we formalists are simple folk. Alan Paris is especially upset by my arguments directed at what he calls "straw men." When I wrote the paper, I took as my charge the defense of formalism against all enemies, foreign and domestic. I assure him that all the attacks I cite are quite real. Indeed, the charge about formal systems is precisely Professor Yevick's. The charge of inconsistency comes from a philosopher of my acquaintance. The charge over aesthetics and ethics comes from the N e w Directions group. Finally, the charge involving the holocaust is in truth not mathematical, but it was made, and I felt compelled to address it. Yevick's concerns, by the way, seem quite real, though beyond both my formalism and my philosophical powers. Finally, there is tolerance. I claim formalism is tolerant, but I claim nothing about individual formalists, Hilbert included, although a careful reading of the article cited (Intelligencer, vol. 12 no. 4) leads me to guess that personalities played at least as large a role as philosophy. I notice also that Brouwer's intolerance came first (p. 19). I myself can only strive for tolerance. In face of charges that I am "self-serving," that I am metamathematically "weak," and that significant events have "eluded" me, I hope I am turning the other cheek. Jim Henle Department of Mathematics Smith College Northampton, MA 01063 USA
-Founding the European Mathematical SocietyIl faut qu'on ne puisse dire ni "il est mathdmaticien', ni 'pr~dicateur" ni "~loquent', mais "il est honnfte homme'. Cette qualit~ me plaft seule. - - . B l a i s e Pascal: les Pensdes
The European Mathematical Society came into existence on 28th October 1990. The auspicious birth took place at Madralin, a country residence of the Polish Academy of Sciences, located 20 kilometres from War-
saw, on a bright, crisp Sunday morning. The happy event was duly celebrated in time-honoured fashion by the fifty or so delegates from the twenty-eight European societies represented. The delegates were, perhaps, conscious of a sense of history as the event had taken place during a momentous period of European affairs, and were possibly also relieved, for the gestation had been protracted. But what were the origins and what had arrived? The genesis of the Society lay in efforts (1976) by the European Science Foundation to consider ways of improving European cooperation in mathematics. These efforts resulted in the creation at the International Congress of Mathematicians in Helsinki (1978) of a European Mathematical Council. This Council began to function, but political difficulties at the International Congress of Mathematicians in Warsaw (1983) inhibited the development; notwithstanding, the Council, while initially drawn mainly from the West, did evolve into a biennial forum for delegates from both Eastern and Western Europe. At Prague (1986) the first steps were taken to draw up a constitution for a society along the lines of the European Physical Society. The draft constitution was agonised over in Oberwolfach (1988) and subsequently, no doubt, by the various participating societies, until the final and eventually unanimous agreement was reached in Poland (1990). Professor Sir M. Atiyah, who had indefatigably chaired the Council since its inception, duly stepped down from office and Professor F. Hirzebruch (Bonn) was unanimously elected as first President of the nascent Society. By acclaim Sir Michael became the first individual member. The Society itself has been, for legal purposes, constitutionally established under Finnish law with its seat in Helsinki. Reflecting the manner in which the Society has been set up, the membership rules as given under Article 3 of the Constitution are somewhat complicated, namely:
1. Members of the Society may be either (a) corporate bodies with legal status, or (b) individuals. 2. Corporate bodies with legal status may join the Society in one of the following categories: (a) full members, (b) associate members, (c) institutional members. Full membership is restricted to societies, or similar bodies, primarily devoted to promoting research in pure or applied mathematics within Europe. Associate membership is open to all societies in Europe having a significant interest in any aspect of mathematics. Institutional membership is open to commercial organisations, industrial laboratories or academic institutes. 3. Individuals may join the Society in one of the following categories:
(a) individuals belonging to a corporate member of the EMS, (b) individuals not belonging to a corporate member of the EMS. Individual membership is open to all individuals who make a contribution to European mathematics. The founding mathematical societies are deemed to have joined as full members, other societies have joined, and interested societies are respectfully invited to join. Private individuals may either join directly or through a society which is itself a full member. The significance of the difference in the mode of membership is that by joining directly he or she will pay 280 Finnish marks annually, whereas by joining through a society he or she will pay only 70 Finnish marks annually (one US dollar = 3.6 Finnish marks, approximately). The supreme authority of the new Society is its Council, which is to meet every two years, and to which delegates are elected to represent the various categories of members. The Council, in turn, elects an Executive Committee which, at present, consists of the following: President: Professor F. Hirzebruch, Bonn, Germany Vice-Presidents: Professor Cz. Olech, Warsaw, Poland; Professor A. Figa-Talamanca, Rome, Italy Secretary: Professor C. Lance, Leeds, United Kingdom Treasurer: Professor A. Lahtinen, Helsinki, Finland Members: Professor E. Bayer, Besan~on, France; Professor A. Kufner, Prague, Czechoslovakia; Professor P.-L. Lions, Paris IX, France; Professor L. M~rki, Budapest, H u n g a r y ; Professor A. St. Aubyn, Lisbon, Portugal The Executive Committee had its first meeting at Oberwolfach (19-20 January, 1991), the members being keenly aware of the task involved in turning genuine aspirations into effective realities. Among the major issues considered were those of summer schools and research institutes. The Committee is not yet in a financial position to sponsor conferences, etc., but will consider requests for EMS support within frameworks that would emphasise the promotion of European integration or would assist younger research workers. Arising from the decisions of the European Mathematical Council meeting at Madralin, four sub-committees have been set up, on Publications, Education, the Applications of Mathematics, and Women and Mathematics. Publication proposals are advanced and members will receive a quarterly newsletter whose first issue has appeared in the summer of 1991. A major event to be held under the auspices of the EMS is a Congress which is to take place in Paris (6-10 July 1992) at the Sorbonne and on the Jussieu Campus. The objectives of the Congress are to present new and important aspects of pure and applied mathematics to a wide public, to promote consideration of the relation THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. I, 1992 9
between mathematics and society in Europe, and to stimulate European cooperation. As well as a more conventional programme, there will be "round tables" on a variety of topics of interest to mathematicians in their interrelation with the world around them, topics such as "European Countries, Harmonisation of Degrees," "Women and Mathematics," "Collaboration with Developing Countries," "Mathematics and Industry," "Mathematics and Computer Science." Further information is available from the contact address:
of the EMS! Other satellite conferences near the time of the Congress are planned. In particular the Luxembourg Mathematical Society is organizing a conference (29-30 June 1992) on "The development of mathematics during the period 1900-1950." For further information the address is:
Congr~s Europ4en de Math6matiques Coll~ge de France 3 rue d'Ulm Paris (Se) France. E-mail:
[email protected]
These are early and heady days for a young infant of a society. Growth will depend on support by the community of European mathematicians, who, it is hoped, will now rise to the challenge.
The registration fee will be reduced for an individual member of the EMS--a tangible benefit of membership
10
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
Soci6t6 math6matique du Luxembourg Centre universitaire de Luxembourg 162A, Avenue de la Fa'iencerie L-1511 Luxembourg.
D. A. R. Wallace (Publicity Officer, EMS) Department of Mathematics University of Strathclyde Glasgow, G1 1XH Scotland, UK
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreements and controversy are welcome. An Opinion should be submitted to the editor-in-chief, Chandler Davis.
To Guard the Future of Soviet Mathematics A. M. Vershik, O. Ya. Viro, L. A. Bokut' The Mathematical Intelligencer approached some leading Soviet mathematicians, soliciting their comments on the question: "'What must be done to make the Soviet Union a
12
place from which mathematicians will not want to emigrate?" Two replies are given here; we hope to have others in a later issue.
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1 9 1992 Springer Verlag New York
The first is in the form of excerpts from a joint presentation made by A. M. Vershik and O. Ya. Viro to a conference of scientific leaders on "'Science in the period of transition to the market" held in Leningrad, October 1990. A.V.: Our mathematics has gathered high prestige throughout the world. Mathematicians from the Soviet Union are very highly regarded. The point is not to congratulate ourselves, but to note the fact that the whole system of administrative-command economy and other such interference did not have for mathematics the kind of destructive effects they had (say) in biology. The reason for this is that mathematical work is specific to itself, and the mathematician can pretty much get along with nothing but pen and paper. Also, to be sure, contact with colleagues is necessary. International contacts, if only by correspondence, did not cease even during the most "stagnant" periods. This sustained us and gave a basis for determining what level we were at. There is an enormous amount of talent in this country. Mathematical aptitude always stands out, and maybe makes itself more visible than other kinds. Although only a portion of those who displayed talent were able to get a mathematical education, there were enough to maintain continuity. O.V.: On mathematical education in Leningrad it is probably unique in the world. We have a tradition of clubs going back to the thirties where the teachers are students who have only just graduated from such clubs. Leningrad students are regularly the best at AllUnion and international olympiads. At LOMI in particular, about 90% of the members under forty went through this system: clubs, olympiads, and mathematical schools. All this is very precarious, depending on the few individuals running the clubs in a given year. Just today I was talking with the young people teaching in these clubs; they say everything is going as w h e n they were students there. These clubs are n o w in the Pioneers' Palace. They need a base downtown. A.V.: Finally, I need to depict a most serious situation. Soviet excellence in mathematics may come to a very quick end. In general, we face problems connected with the transition to market economy and changes in the general situation of the country; but these are not our only problems. The problems of the market are new, to be sure, and must be considered, but there is baggage being dragged along from the administrative methods of past decades and it is in no way related to the market. If we are to understand what is going on, we have to keep this in mind. Permit me to bring up difficulties affecting mathematics that we have inherited from earlier times, and whose inertia may grow in times to come.
First, in our country mathematics (like all the sciences) is unnecessarily centralized. There is the Academy of Sciences, there are a few institutes of the Academy, and there are the universities. The best we have in mathematics is concentrated in two, three, maybe five places. Elsewhere there are some specialists, but they are exceptions. This seems quite unnatural if y o u compare it with the United States, for example, where alongside the ten or so universities of the first rank there are another thirty which stand almost as high. In Leningrad, our branch of the Academy, LOMI, is extraordinarily powerful, but it has essentially no competitors. Now, for all sorts of reasons, its role is diminishing. That means the whole diminishes because there is nothing else. The University, especially after the move to Peterhof, is losing all its influence on the scientific life of the city. Another problem needs to be considered. In the Academy of Sciences, and the mathematical section in particular, there have long been forces hindering the natural development. Fortunately, they are being replaced by new, younger leaders w h o are making changes. But such things as anti-Semitism, failure to admit talented people into teaching and decisionmaking, and blocking of degrees, leave consequences that will still be felt for decades. This has led many people to leave, and there will be others; it has led to distortion and collapse of scientific teams. There is no excuse for silence about this. At present we are at a critical juncture. There is real danger in the departure of specialists to jobs elsew h e r e - - n o t because they want to emigrate, but just so as to survive. Can they be blamed? It is only natural. The authorities have stopped putting obstacles in the way of foreign travel. Unfortunately, that leads to an incredible weakening of mathematics. O.V.: The middle generation of mathematicians, from 35 to 45, are rapidly emigrating. Where, formerly, one might leave for a year, n o w those w h o go for a year stay for another. This constitutes a great loss to the mathematical community as a whole. Some steps must be taken to preserve this part of world culture. Part of the problem with those w h o go to work in the West is that the present laws keep them from returning. Namely, income above about 3000 rubles is taxed at 60%. After conversion of any Western currency, this comes out still higher. People cannot afford to pay such a tax, and so they don't return. A.V.: Our first proposal is that there should be alternative institutes and universities. They should be based on new principles, including specialists from abroad, and they should offer competition to those we now have. One example is the projected International Mathematical University (not to be confused with the recently created International Euler Institute, which has different functions). In Leningrad there are m a n y talented and fully THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
13
trained mathematicians w h o do not have a proper position, but use their free time to work on theoretical mathematics at the University and the Institute. Among this under-used pool of talent are some of my students. We must find a place for them in alternative systems. The next proposal requires a historical remark. There was a time w h e n Leningrad mathematicians were much better integrated with applied work, and the other sciences. This same LOMI, w h e n Academicians Yu. V. Linnik and L. V. Kantorovich were there, would consult successfully with engineers in many different ways. This tradition is virtually extinct. With it we have lost something very important. To preserve science we need to improve its contacts with those engaged in production with scientific content. Unfortunately the Soviet Union does not have something found in the USA: a large intermediate group between the engineers and the theoretical mathematicians, who are able to formulate a question for a mathematician as well as understand the needs of technology. At Stanford or MIT, the interaction is strong and impressive; the university is not only a teaching institution, but in effect a producer. One of the relevant problems is education in the technical schools. The overwhelming majority of department heads (and others as well) are not professional mathematicians. This impedes the training of even a small number of people with both mathematical learning and knowledge of applications. One more desirable move is specialized financing with peer review. Our trouble has not been that funds for the sciences were inadequate, but that they were poorly allocated. In our funding arrangements, the person doing the work has no latitude in allotting the money. Let me refer again to America (for I'd rather imitate something sensible than think up something offhand and then find it bad). One can choose a topic and apply for a grant to the National Science Foundation. Reports are made by at least five referees. These do not work for the NSF. If you get enough positive reports, a grant is made to y o u - - t o you. (A clever feature is that your university takes off a certain percentage of the grant. Thus, it is profitable to the university to have its professors getting grants, whereas here the universities gain nothing from having scholars.) The scientist getting the grant uses the money to invite collaborators, to travel, for miscellaneous expenses. I asked people about their experience of the system, in particular about the objectivity of peer review. Most opinions were favorable. It is natural to be wary; one would have to have e n o u g h i n d e p e n d e n t evaluations to guard against favoritism. Our second commentary, dated ]une 1991, is from L. A. Bokut'. Thank you for the invitation to discuss the question 14
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
of what must be done to make the Soviet Union a place from which mathematicians will not want to emigrate. I am honored by the invitation, which I take as a recognition of the mathematical activity in Siberia in recent years. For example, there was the International Algebra Conference in memory of A. I. Mal'tsev, Novosibirsk, 1989; and from 1987 on there have been annual Siberian Schools of Algebra and Analysis (AA). I am very happy to accept your invitation, at the same time I know how difficult it is to say anything constructive about the massive outflow of Russian mathematicians to the West. The factors contributing to emigration are not only economic and political but also national. Will Russians in the West retain a feeling for their homeland and a wish to help their compatriots? Many of us here (including myself) are counting on it. It is altogether typical that the recently established Russian Academy of Natural Sciences (RANS) announced that it is hoping for help and contacts with Russian scholars and creative artists living abroad. What the RANS means by "Russians" can be seen from their naming Solzhenitsyn, Brodskii, Rostropovich, and Menuhin honorary members. Russian President Boris Yeltsin has repeatedly appealed to Russians (in the same sense) living abroad. There are n o w so many emigrants from Russia--Russians, Russian Jews, and others---living in such countries as Israel, France, the USA, and Canada. I was in Israel last November and I know many people there who would like to have special relations with Russia. This May, in the US, I met a number of first- and second-generation emigr6s from Russia, and I could see the warm feelings they retain toward our country. We must say that Western universities and mathematicians are doing a great deal to preserve mathematics in Russia, and to establish suitable conditions of work for some of the best Russian mathematicians. For those w h o do not emigrate, more or less regular trips abroad enable us not to feel cut off from world mathematics and help our economic position. Such trips are growing more frequent as a result of Mikhail Gorbachev's policies. I could cite the example that after the Mal'tsev Conference I mentioned, dozens of algebraists arranged foreign trips from university cities all over Russia and the other republics. This is a terrific help. It stimulates us to get significant new results, and it rescues many from the brink of poverty. It is hard to see h o w Western mathematicians could do more for Russian mathematicians. Yet this is not the whole story. As the Mal'tsev Conference (with more than 200 foreigners) illustrates, the more personal contacts there are between Russians and foreigners the more visits Russians will make abroad. So we in Siberia must not slacken our activity in organizing international conferences (I would even say we have no need for purely internal conferences). This August (1991) in Barnaul the Second International
Algebra Conference in memory of A. I. Shirshov will take place. Immediately afterward on Baikal will be the Fifth Siberian AA School (with foreign participants), and next year at the same place the Sixth International AA School. We hope to hold a Third International Algebra Conference in 1993 (maybe in Krasnoyarsk). The activity is hardly limited to Siberia: the International Euler Institute has opened in Leningrad; Minsk in May of this year had a small but singularly representative conference on algebraic groups; in Moscow from May to June, there was an International Jubilee Session of the Petrovskii seminar, etc. Currently we are looking for a permanent place to hold the annual AA Schools. We received a proposal from one of the Siberian cooperatives to build on Baikal a special hotel for conferences---a sort of Siberian Oberwolfach. A tempting prospect! This would go beyond mathematics, it would serve all sciences in Russia. Another international project we are beginning in Siberia is the organization of Siberian Higher Mathematics Courses (in the English language). The idea is for graduate students from both industrial and developing countries to be able, for a modest fee, to take courses and seminars at Novosibirsk and other university cities of Siberia. Many mathematicians have supported the project, and a bureau, the Siberian Branch of the Academy of the Ministry of External Economic Contacts, is ready to help implement it. We are preparing an announcement of it to submit to the Notices of the AMS. I have already said that contacts with the West have special importance to us. At the same time, we consider Siberia a natural place for contacts between Russian mathematicians and those of China, Japan, India, South Korea, Vietnam, Mongolia, and other countries of Asia. Let me extend to the readers of the Mathematical Intelligencer at universities in those countries an invitation to participate in one of the programs we announce and work together on further joint plans. For example, would some university or other institution in Japan help carry o u t the plan of the "Siberian Oberwolfach" I mentioned? As compensation for its contribution such an organization could send participants to mathematical or other conferences on Baikal. We are hindered in developing relations with Eastern countries by having so few personal contacts. We hope this situation will improve in time. Again, after the Mal'tsev Conference several Russain mathematicians were invited to H o n g Kong to the Asian Mathematics Conference (1990). Such contacts are very valuable to us. Turning to other continents, we recall that there is a tradition of Russian mathematicians teaching in various countries of Africa. So far these have gone only through administrative channels, but I think the mathemaical societies could do something too. Currently we are thinking of improving our relations with math-
ematicians of South America, Australia, and N e w Zealand. Something very valuable for our mathematicians is the Russian Translation Program of the American Mathematical Society. This publishes translations of journals, books, dissertations, proceedings of conferences, schools, and seminars. Translations are also undertaken by many publishers of the USA, Germany, Britain, the Netherlands, Singapore, etc. It should be emphasized that this work profits not only Russians but all those involved. There is one more idea: to establish in Novosibirsk an independent international university. Relative to the preceding plans, this project might be called the Project of the Century. Maybe with assiduous participation by leaders in all interested countries it could indeed be realized in this century. I am afraid I have been talking specifics where this publication might have preferred analysis of more general issues. Colleagues have criticized me for not bringing up the excessive teaching loads at Russian universities and institutes (20-25 hours a week and more!), for not mentioning the reduction in payments for science via the Soviet Academy of Sciences (the Academy is financed on the federal budget, so we now have no funds due to the "budget war" between the central government and the republics), and so on. These would be matters for an independent union of scientific workers and post-secondary teachers, if we had such a thing. (By the way, there is in Russia a Union of Scholars, founded during A. D. Sakharov's life.) However, I cannot state all the problems in this short "battlefield dispatch." I have concentrated on what can be done by mathematicians themselves to preserve mathematics in Russia. I share the view of O. Ya. Viro, expressed at the International Congress in Kyoto (1990), that the collapse of Russian mathematics would be a blow to all civilization. I hope most readers of the Intelligencer agree. A. M. Vershik Mathematics Department Saint Petersburg State University Saint Petersburg-Petrodvorets, 198904, USSR
O. Ya. Viro LOMI Fontanka 27 Saint Petersburg, 191011 USSR
Department of Mathematics University of California Riverside, CA 92521 USA
L. A. Bokut"
Institute of Mathematics Siberian Branch of the Academy of Sciences of the USSR Novosibirsk, 630090 USSR
THE MATHEMATICAL INTELL1GENCER VOL. 14, NO. 1, 1992 1 5
The Shape of the Ideal Column Steven J. Cox
The column stands both as the essence of an architectural order and as the first flexible body to fall to mathematical analysis. The aesthetic ideal, formulated and realized by the ancient Greeks, was recorded by Vitruvius in De Architectura (circa 25 B.C.). The result, a subtle variation on the cylindrical profile, calls for a bulge at approximately one third of the column's height and a diminution near its top. With a denunciation of this aesthetic ideal, Lagrange in 1773 formulated the first scientific criterion, one based on strength rather than appearance. A number of missteps in applying the calculus led him to the mistaken conclusion that the cylinder was the strongest hinged column. Though T. Clausen, in 1851, appeared to succeed where Lagrange had failed, C. Truesdell, troubled by "elements of mystery" remaining a century later, invited a fresh approach. In response, J. Keller recovered, in greater generality, the result of Clausen. Keller published his findings in 1960 and with I. Tadjbakhsh in a paper of 1962 tackled the remaining boundary conditions of interest. M. Overton and I have recently closed the longstanding debate over Tadjbakhsh and Keller's claim that the strongest clamped and clampedhinged columns possess interior points where the cross section vanishes. Here I trace the influence exerted by the aesthetic ideal through the early stages of the theory of elasticity and the subsequent formulation of the scientific ideal under the influence of Euler and Laugier. I indicate where the extension of Keller's successful analysis breaks down, tracing the cause to the lack of differentiability, indeed the lack of continuity, of Lagrange's measure of strength. Finally, in a setting in which an optimal design exists, I discuss the role of double eigenvalues and the consequent need for nonsmooth analysis in the construction of necessary conditions. 16
The Aesthetic Ideal The swelling of columns was but one of the optical refinements employed by the Greeks to counter perceived imperfections. This practice, which varied to a degree dictated by the proposed structure's size and surroundings, reached its zenith in the Parthenon where The delicate curves and inclinations of the horizontal and vertical lines include the rising curves given to the stylobate and entablature in order to impart a feeling of life and to prevent the appearance of sagging, the convex curve to which the entasis of the columns was worked in order to correct the optical illusion of concavity which might have resulted if the sides had been straight, and the slight inward inclinations of the axes of the columns so as to give
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1 9 1992 Springer Verlag New York
the whole building an appearance of greater strength; all entailed a mathematical precision in the setting out of the work and in its execution which is probably unparalleled in the world [5, p. 178]. For Vitruvius these refinements were direct consequences of the principle: Ergo quod oculus faUit, ratiocinatione est exequendum. "For what the eye cheats us of must be made up by calculation" [19, v. 1 p. 179]. Its application to the design of columns induced Vitruvius to warn that " . . . the sight follows gracious contours; and unless we flatter its pleasure, by proportionate alterations of the modules (so that by adjustment there is added the amount to which it suffers illusion), an uncouth and ungracious aspect will be presented to the spectators. As to the swelling which is made in the middle of the columns (this among the Greeks is called entasis), an illustrated formula will be furnished at the end of the book to show how the entasis may be done in a graceful and appropriate manner" [19, v. I p. 179]. That close inspection has turned up "no Roman columns without an entasis" [15, p. 121] suggests such warnings were indeed heeded. Though Vitruvius's illustration was lost, his text on this point differs so little from Alberti's discussion of entasis in De re Aedictoria (1450) that one expects the illustration (Figure 1) in Bartoli's 1550 Italian translation of this work to faithfully represent the ideal of Vitruvius. This despite Alberti's claim that his prescription "is not a discovery of the ancients handed down in some writing, but what we have noted ourselves, by careful and studious observation of the work of the best architects. What follows principally concerns the rules of lineaments; it is of the greatest importance, and may give great delight to painters" [1, p. 188]. It must be noted that here Alberti abandons the rationale of optical refinement for his much more abstract notion of lineaments ("the correct, infallible way of joining and fitting together those lines and angles which define and enclose the surfaces of the building" [1, p. 7]) and so obscures the motivation behind entasis. In addition, as with Vitruvius, Alberti's wooden prescription fails to encompass the full range of Greek examples, from the lack of entasis in the Temple of Apollo at Corinth to its overabundance in the Basilica at Paestrum. Though the correction of optical illusions is surely at work in these structures, the existence of a single theory embracing all cases "is liable to serious objections" [13, p. 103]. Subsequent architects, though keenly aware of the optical refinements as practiced, appear ignorant of, or at least unconcerned with, the causes that induced them. In the writings of the 16th-century Italian architects Palladio and Vignola, for example, one finds detailed illustrated prescriptions of entasis without discussion of the condition for which this remedy is being prescribed. Divorced from its inspiration the practice of entasis suffered instances of both exaggeration,
J 9
--_
ilil~
Figure 1. Illustration of entasis in Bartoli's 1550 Italian translation of De re Aedictoria. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
17
"even to a cigar shape" [5, p. 186], and neglect, "too delicate an ornament to be appreciated by the common man; columns more often were tailored to follow the form of the perfect cylinder" [18, v. 3, p. 495]. If this suggests a waning of the influence of Vitruvius, the decline of the Baroque would signal a return to the Greek models and the man in whose writings they were preserved. M. Blondel, the director of Louis XIV's Royal Academy of Architecture and member of the Paris Royal Academy of Sciences, endowed his 1675 treatise on architecture with the subtitle L'origine & les Principes d'Architecture, & les practiques des cinq Ordres suivant la doctrine de Vitruve. Blondel's treatment of e n t a s i s , d i f f e r i n g f r o m t h a t of P a l l a d i o or Vignola in his attempt to express it analytically, coincided with the announcements of R. Hooke, of the Royal Society of London, regarding both "The true Mathematical and Mechanical form of all manner of Arches for Building" and "The true Theory of Elasticity."
Early W o r k on Elasticity In his treatise on elasticity of 1678 [7] Hooke writes, "The Power of any Spring is in the same proportion with the Tension thereof . . . . The same will be found, if trial be made, with a piece of dry wood that will bend and return, if one end thereof be fixt in a horizontal posture, and to the other end be hanged weights to make it bend downwards." The latter remark, in stating the column's restoring force in terms of the loadinduced strain, contains the seed of the first constitutive law for a flexible body. It would remain for Euler and the Bernoullis to quantify the relevant notions of stress and strain and so flesh out this bending law of Hooke's. Clearly aware of the three-dimensional nature of the column, James Bernoulli, beginning in 1691, nonetheless sought to describe its bending in terms of the planar deformation of a "neutral axis," ~/. In particular, associating the strain in the column with the curvature ,: of ~/and the column's stress with the bending moment M, he attempted to derive Hooke's law, M ~ K. This program proved too ambitious, indeed the position of the neutral axis eludes us to this day, and it was not until 1732 that James's nephew Daniel Bernoulli first postulated M ~ K in a theory of bending. Euler, in an unpublished work on a special case, identified this proportion with the product of E, the modulus of extension, and I, the second moment of area of the column's cross section about a line through its centroid normal to the plane of bending, with the result M = EIK.
Young, but three years old w h e n Euler produced its precise definition, is that typically attached to the modulus E. In accordance with these measures of stress and strain D. Bernoulli, in a letter of 1738, posed to Euler the problem of finding that curve for which the stored energy fe M K ds was a minimum. In Additamentum I de curvis elasticis (1744) [6, s. 1 v. 24], an appendix to his text on the calculus of variations, Euler subsequently solved the problem of the inextensible elastica, i.e., for constant E and I he found that curve of prescribed length with prescribed terminal displacements and slopes and minimum stored energy. Here it will suffice to consider curves that are graphs of functions over the interval [0, 4~]. In this context, Euler succeeded in minimizing EIlu"12 + [u'12)1/2 foe (1 + lu'12)9/4 dx - ~. f e~ (1 dx,
where u and u' are prescribed at 0 and at f and h is the Lagrange multiplier associated with the length constraint. Identifying K with the axial load necessary to sustain a prescribed deformation, Euler found the precise load under which an initially straight column would commence to bend. This value, now known as the Euler buckling (or critical) load, is he = EI~r2/(4f2), where f is the length of a quarter period of the deformed curve. Euler singled out the hinged case, where the displacement and moment vanish at each end, for which f = f/2 and so ,IT2
Kc = E1 - ~ .
That the quadratures required to obtain this result owed their existence to the constant nature of E and I perhaps led Euler to the alternative characterization of )~r in Sur la force des colonnes (1757) [6, s. 2 v. 17]. In this work he observed, again in the context of hinged ends, that as )% marks the load under which deformation begins one could indeed restrict attention to the linearization of the first variation of (2) about the straight state. It follows that )~ is the least eigenvalue of (EIu")" + Ku" = O, u(O) = EIu"(O) = u ( O = EIu"(f) = O.
The corresponding first eigenfunction, u c, as above measures displacement of the neutral axis and contributes to the bending moment Mc via M c = EIu c. This choice of boundary conditions proved especially convenient, for in this case uc and M~ are each first eigenfunctions (with first eigenvalue Kr of E I y " + Ky = 0,
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. I, 1992
(3)
(1)
This is known as the Bernoulli-Euler formula for the bending of a column, while the name of Thomas 18
(2)
y(0) = y(e) = 0,
(4)
and hence u c = Me. With this formulation Euler pro-
F i g u r e 2.
The primitive hut: frontispiece from the second edition of the Essai sur l'Architecture, engraved by Ch. Eisen. THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992 1 9
column. With the fanfare: "'among those rules at the foundation of architecture there is but one that is fixed and invariable, and consequently susceptible to calculation: that is solidity," Lagrange offered the relative strength of (6). Though Lagrange cites no source of inspiration for this invective, his contempt for modern architects, his (5) misreading of Vitruvius, and his quest for fixed and he = ~ + (log(1 + b/a)) ~ " unchangeable rules are surely drawn from the ideas of then corresponds to columns of circular cross section Marc-Antoine Laugier, the anonymous author of the for which A either increases or decreases linearly with controversial, though very popular, Essai sur l'Architeclength. He indeed calculated Xc for a number of other ture (1753). Laugier, upset with an architecture that exponents but stopped short of formulating a basis of had "been left to the capricious whim of the artists comparison with which to distinguish the various w h o have offered precepts indiscriminately . . . fixed rules at random, based only on the inspection of anchoices. cient buildings, copying the faults as scrupulously as the beauty; lacking principles which would make them see the difference . . . . " s u m m o n e d the one who "will The Scientific Ideal undertake to save architecture from eccentric opinions This task was taken up by Lagrange in Sur la figure des by disclosing its fixed and unchangeable laws." [10, p. colonnes (1773) [9, v. 2]. Lagrange sought to maximize 2]. For his model Laugier took the primitive hut, a )~c, suitably normalized, over solids of revolution with rendering of which served as frontispiece for his prescribed length. In particular, he sought that func- work's 2nd edition (1755), see Figure 2. He begins his tion for which the "relative strength" first chapter with a list, first pronouncing correct methods in the design of columns then remarking on sevXc(A) (6) eral faulty methods. We recall o n e of each: "The column must be tapered from bottom to top in imitation V2(A) of nature where this diminution is found in all plants" achieves its maximum. Here A : [0, (?] --+ [0, oc) mea- [10, p. 14], and "Fault: to give a swelling to the shaft at sures cross-sectional area, V(A) = feo A dx is the col- about the third of its height instead of tapering the umn's volume, and hc(A) is the least eigenvalue of (4) column in the normal way. I do not believe that nature with E = 1 and I = A 2. With this I in (4) it follows for has ever produced anything that could justify this every positive oL that hc(o.A) = c,2K~(A), and, as the swelling" [10, p. 18]. In addition to parroting these volume obeys V(cxA) = oLV(A), if A maximizes (6) then opinions Lagrange goes so far as to adopt the vague so too does c~A. Consequently, maximizing (6) is in fact notion of soliditd, identified, though undefined, by equivalent to maximizing ~ over solids of revolution of Laugier as "the first quality a building must have" [10, p. 68]. We shall see that Lagrange, in answering this prescribed volume and length. Rather than arguing the efficacy of his relative summons with a cylindrical column, outdoes even strength in the design of columns, Lagrange instead Laugier by removing not only the swelling but also the attacks the legitimacy of the aesthetic ideal of the diminution. As preparation for the general case Lagrange first Greeks. Seeking to upstage Vitruvius, "le 16gislateur des architectes modernes," Lagrange claimed in his attacks the finite-dimensional problem of maximizing search for a rationale underlying the prescription of (6) over those functions of the form entasis to find nothing more sound than a resemblance A(x) = a + bx + cxL (7) to the human body, a profile he found, with reference to the primitive hut, inferior to that of the trunk of a tree. Noting the loss of Vitruvius's original illustration, Following Euler's lead, Lagrange finds Lagrange then denounced the prescriptions of Palladio, Vignola, and Blondel as arbitrary variations on an h =- :O ! dx )~c(A) = b2/4 - ac + "rc2/h2, A" already shaky theme. If Palladio, Vignola, and Blondel were not sufficiently critical in their reading of Vitruvius, Lagrange is clearly mistaken in his. For recall that which indeed reduces to (5) w h e n c = 0. In the case b2 Vitruvius prescribed entasis, not as mere decoration, = 4ac, i.e., A(x) = (Vaa + V~cx)2, he finds but as the subtle solution to a difficult engineering problem. Ignorant of this problem, Lagrange abanXc(A) Tr2(a q- V ~ e ) 2 doned the aesthetic ideal and sought instead a rational V2(A) e4(a + V ~ e + ce2/3) 2" basis from which one could judge the value of a given ceeded to compute Xc for the class of nonuniform columns in which E ~ 1 and I(x) = (a + bx/Q q at selected values of q. For columns with circular cross section, I is proportional to the square of the cross sectional area, A. The case q = 2, for which Euler finds
20 THEMATHEMATICALINTELLIGENCERVOL.14, NO. 1, 1992
For each a this is a decreasing function of c and therefore a maximum w h e n c = 0, i.e., among those columns for which A is a perfect square, the cylinder is the strongest. In his subsequent attempt to reduce (7) to a perfect square lies Lagrange's first misstep. In particular, after reducing the relative strength to the workable form that begins his section 24, he errs in setting its logarithmic derivative to zero and therefore arrives at an erroneous necessary condition. This condition implies that perfect square A are indeed to be preferred and hence that the cylinder maximizes (6) over those A given by (7). Offering up this result without physical interpretation Lagrange rushes into the general case of maximizing the relative strength over all functions A : [0, 2] ~ [0, oo). Again, he finds what he is looking for, the cylinder. The technical errors he was forced to commit at this stage were caught by J. Serret in editing Lagrange's Oeuvres. Had Lagrange had the courage to criticize the physical merits of his scientific design criterion, he would have been led directly to perceive his mistakes of calculus. For maximizing the relative strength is equivalent to maximizing the buckling load subject to fixed volume, and to raise a column's buckling load without changing its volume one should obviously increase A where large bending moment M is expected and decrease it in regions of relatively little bending. In short, A and [MI should be similarly ordered. As the differential equation (4) determines the qualitative properties of the bending moment, this meta-theorem has an immediate consequence. For M, being a, say positive, first eigenfunction of (4), must be a concave function vanishing at each end. Consequently, the buckling load of a hinged cylinder is increased w h e n material is removed from its ends and added to its middle. Finally, nowhere does Lagrange argue the relevance or indicate the role of the chosen hinged boundary conditions in the practical problem he has set himself. He appears to have followed Euler's use of these conditions as blindly as he followed the pronouncements of Laugier. Though Euler makes no reference to this work of Lagrange, T. Young, arguing that Lagrange possessed "the habit of relying too confidently on calculation, and too little on common sense," believed it "possible to assign a stronger form than a cylinder, since the summit and base must certainly contain some useless matter'' [20, p. 568]. T. Clausen in l~rber die Form architektonischer Siiulen (1851) [3], was the first to offer a correct solution to this problem of Lagrange. Clausen in fact solved the equivalent problem of minimizing volume subject to a fixed buckling load. I have seen this work only in the summary offered by Pearson [17, v. 2, p. 325], an assessment much clouded by Pearson's ringing endorsement of Lagrange's cylindrical solution. Unaware of Lagrange's historical, physical, and mathematical errors, Pearson credited him with
having "shaken the then current architectural fallacies" [17, v. 1, p. 67]. This prattle provoked Truesdell to surmise that "Pearson took [Lagrange] as a torch carrier for Victorian architectural practice, according to which, it seems, the ugliest forms turn out to be the most useful" [6, s. 2, v. 11.2, p. 355]. Confused by a solution which differed from Lagrange's, Pearson endeavored to "simplify" Clausen's analysis. Unfortunately he makes things too simple, for though he arrives at the correct conclusion, the path he takes is nonsense from the start. Rather than dwell on Pearson's mistakes I instead display Clausen's solution, Figure 3 (the exaggerated entasis of a cigar), and move on to J. Keller's derivation in The shape of the strongest column (1960) [8].
The Work of J. Keller and I. Tadjbakhsh Assuming, with respect to the nondimensionalized problem
y" + hA-2y = 0,
y(0) = y(1) = 0
(8)
fd A dx = We,
(9)
that (i) A ~ he(A) attains its maximum at A over those nonnegative A satisfying (9), and (ii) t ~-~ Kr + tAo) is differentiable for each variation A 0 satisfying fo1 Ao dx = 0, Keller succeeded in characterizing A via a firstorder necessary condition. In particular, the perturbed equilibrium equation
y" + ),~(A + tAo)(A + tAo)-2y = O, y(0) = y(1) = 0, when differentiated with respect to t at t = 0, yields
9"+ xc(A)A-~ (0) =
= 2x~(A)A -3 Aog, j)(1)
= 0,
(lO)
where I have used the fact that he(A) = 0 and denoted the first eigenfuncfion of (8) w h e n A = A by Y. For (10) to possess a solution, the Fredholm alternative requires that its right-hand side be orthogonal to each solution of the corresponding homogeneous equation, i.e., flA-392 A o dx = 0. This being necessary for every zero-mean A 0, there must exist a positive constant c for which
92 = cA3.
(11)
The form of A is now immediate, for in agreement with my simple meta-theorem, like 9 it must be a concave function vanishing at each end. Its precise form is found on solving the nonlinear differential equation, subject to (9), that results on substituting the necessary condition, (11), into the equilibrium equation, (8). ExTHE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
21
Figure 3. Solution for hinged end conditions.
Figure 4. Solution with clampedhinged end conditions.
plicitly, )~c(A) = 4~2V2/3~ 2, while the graph of A permits the parametrization
3(2
x(t) = G
"~(t - sint)
2 y(t) = 5(1 - cos t)
)
subject to either clamped-hinged end conditions, u(0) = u'(0) = 0,
u(1) = A2u"(1) = 0,
(13)
u(1) = u'(1) = 0.
(14)
or clamped end conditions, 0 ~ t ~2~r.
u(0) = u'(0) = 0,
This stunted cycloid, pictured in Figure 3, is stronger, by a factor of 4/3, than the cylindrical column of the same length and volume. In Strongest columns and isoperimetric inequalities for eigenvalues (1962) [16], Keller, with I. Tadjbakhsh, ext e n d e d his earlier findings to columns either free, hinged, or clamped at their ends. I follow their treatment of the nondimensionalized equilibrium equation (A2u")" + Ku' ' = 0 22
Figure 5. Solution for clamped end conditions.
THE MATHEMATICAL INTELLIGENCER VOL. I4, NO. 1, 1992
(12)
Unlike the hinged problem, the displacement u and moment M A 2 u '' do not coincide and it is only the m o m e n t that satisfies =
M" + K A - 2 M = O.
(15)
A s s u m i n g existence and s m o o t h dependence, Tadjbakhsh and Keller characterize the optimal design, A, in terms of its corresponding moment, /~4, via ?vl2 = CAB for some positive c. Continuity of M then implies continuity of A, and as/~I = A2~/" it follows that
A4I~"I2 = cA 3,
(16)
in perfect agreement with (11) when 9 is interpreted as moment. Where (11) led to a design with vanishing cross sectional area at its ends, we shall see that (16) with either (13) or (14) forces A to vanish at interior point(s). First note that any nontrivial C2(0, 1) function obeying (13) admits at least one inflection point, while (14) requires at least two, so in particular, if ~ E C2(0,1) then z/" (x0) = 0 for some xo E (0, 1). Equation (16) then implies that A cannot remain bounded in a neighborhood of x 0. As this contradicts the continuity of A (not to mention our meta-theorem), one must abandon the assumption that ~/ E C2(0, 1). The continuity of A and (16) then together force A to vanish at points where if' fails to exist. Indeed, the designs proposed by Tadjbakhsh and Keller as optimal under clamped-hinged and clamped end conditions possess, respectively, 1 and 2 interior zeros. Fifteen years passed before Olhoff and Rasmussen [12] discovered the buckling load of Tadjbakhsh and Keller's clamped column to be considerably less than advertised. As hard evidence, however, they cited numerical results with no discussion of the algorithm used. Their findings failed to convince those that have argued up through 1988 in favor of Tadjbakhsh and Keller's solution (see the references in [4]). In [4] M. Overton and I established that Olhoff and Rasmussen did however correctly identify the points at which Tadjbakhsh and Keller erred in (i) the calculation of the buckling load of their clamped column, and in (ii) their derivation of the necessary condition (16). At issue in the former is the fact that ~' need not even exist at points where A = 0. With this, we found [4, app.] both the clamped-hinged and clamped columns of Tadjbakhsh and Keller to buckle at loads significantly less than the associated cylinders. Regarding (ii), Olhoff and Rasmussen argued, again with supporting numerical data, that unlike second-order problems where eigenvalues can be at most simple, kc(A) may in fact be double. Under the assumption that k~(A) was indeed double for clamped ends, Olhoff and Rasmussen, and later Masur [11] and Seiranian [14], formally derived new necessary conditions. Applictltion of their conditions led, in each case, to the column of Figure 5. In spite of this consensus, doubt remained, for in addition to the formal nature of these derivations, a proof of existence was still lacking. I sketch below the resolution of these two remaining issues.
tions. This suggests the imposition of a uniform lower bound on those admissable A. In the interest of bounding kc(A) from above it is convenient to impose, in addition, a uniform upper b o u n d on A. This leaves us with the following set of admissible designs: ad = {A ~. L ~ : 0 < e~ <~ A(x) <~ [3, S~ A dx = 1}. And indeed, for each of the end conditions of interest, there exists an A ad for which kc(A)/> K~(A) for each A if: ad (see [4, w Regarding the differentiability of A ~-> kc(A), recall Rayleigh's characterization 01 a 2 1 u ' l
kc(A) = inf
dx
~111X'I2 dx
u E H2(0,1) A ((13) or (14)), and denote by %(A) those (eigen)functions at which this infimum is attained. It is not hard to show that the dimension of %(A), i.e., the multiplicity of k~(A), may not exceed two. Ah, but if the multiplicity is two, that is already enough to invalidate any derivation relying on smooth dependence upon A! As an infimum of smooth functions of A, A ~-> k~(A) of course need not be smooth. It is however Lipschitz and therefore amenable to the calculus of Clarke [2]. The generalized gradient of kr at A is by definition the collection of continuous linear functionals on L~(0, 1) subordinate to the generalized directional derivative of k c at A, i.e., ok~(A) =- {~ E (L~)*; k~ A) >i (~, A) V A if: L~}, where k~
A) =- lim sup B--* ,~ t,~0
kc(B + tA) - kc(B)
Where ok,(A) contains but a single function, k c is Gateaux differentiable and the formal arguments that began with Keller are justified. We found Okc(A) to be the derivative of the Rayleigh quotient evaluated at its various minimizers. In particular [4, w o~r
= co {A(af~. + b~)2 : a 2 + b2 = 1},
where co denotes convex hull, and {z/1, /12} spans %(A) and obeys S~ ~[f~; dx = 8ij. Zero is not an element of akc(A), but rather, for sufficiently large ~, an element of the generalized gradient of the Lagrangian kc(A) - ~2dist(A,ad),
Getting It Right In refuting the clamped-hinged and clamped columns of Tadjbakhsh and Keller we found evidence of the not surprising fact that A ~-> k~(A) need not be continuous in the sup norm topology over nonnegative func-
at A. Consequently, a member of akc(A) differs from a positive constant by an amount that is negative w h e n A = e~, positive when A = 6, and zero otherwise. More precisely, there exist a c > 0 and 8 i ~ 0, 8182 832/4, such that at almost every x E (0, 1), THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992 23
A = ~ ~
References
A ( ~ l l / ~ I 2 q- 83/d~/d~ q- 821/,l~12) ~ c
e, < A < 13 ~ A(811~/~12 + ~3t/i~ + 821t/~12) = c (17)
A = 13 ~ A(811all 2 + 83/~/~ q- 821a~12) -> c The difference 8182 - 82/4 should be interpreted as a Lagrange multiplier that measures interaction between the two buckling modes, ~1 and a2. W h e n this difference is zero, for example, the e i g e n f u n c t i o n ~ -= V ~ l U 1 q- V~2/~ 2 satisfies Ala"l 2 = c,
(16)
i.e., we recover the necessary condition of Tadjbakhsh and Keller. This clearly occurs w h e n hc(A) is simple, and it is not hard to s h o w that this is in fact the case w h e n the end conditions are either h i n g e d or clampedhinged. Regarding the former, (16) predicts singular behavior of ~" only at the ends and so Keller's calculation of the buckling load of his h i n g e d c o l u m n stands. With respect to d a m p e d - h i n g e d conditions, however, recall that (16) forces interior singularities of ~" and consequent zeros of A that invalidate Tadjbakhsh and Keller's calculation of the associated buckling load. Hence (16) in the context of clamped-hinged cannot hold over the column's entire length, i.e., there must exist portions of the column along which A is identically a or 13. Figure 4 depicts the strongest d a m p e d - h i n g e d c o l u m n for a particular choice of a and 13, obtained numerically in [4]. Though (16) m a y not hold over the entire column for d a m p e d ends, the same cannot be said for (17). That is, should X~(A) be double, so long as 8132 - 82/4 > 0 equation (17) in itself does not necessarily require infinite area near zeros of ~i~ or, conversely, zero area at points where ti~ fails to exist. The mixture of the two modes m a y compensate for the anomalies inherent in any single-mode formulation. Indeed one can choose ot sufficiently small a n d 13 sufficiently large so that (17) holds over the column's entire length. Figure 5 depicts the strongest clamped column for such a choice, again obtained numerically in [4]. This result vindicates the formal procedures invoked by Olhoff a n d Rasmussen, Masur, and Seiranian, in deriving the same profile.
Acknowledgments I thank Chandler Davis for this article's instigation as well as his careful scrutiny of an earlier draft. That draft also came u n d e r the eye of Clifford Truesdell. With pleasure I acknowledge the criticism a n d encouragement received from these two men. M y survey of relevant early w o r k in elasticity is but a gloss on Truesdell's The Rational Mechanics of Flexible or Elastic Bodies 1638-1788, comprising volume 11.2 in the second series of [6]. Mark Hall, Doug Moore, a n d Joe Warren are responsible for the software that rendered figures 3, 4, and 5. I thank t h e m for their help. 24 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992
1. L. B. Alberti, On the Art of Building in Ten Books, J. Rykwert, N. Leach, and R. Tavernor, trans., Cambridge, Mass: MIT Press, 1988. 2. F. Clarke, Optimization and Nonsmooth Analysis, Centre de recherches math6matiques, Montreal, 1989. 3. T. Clausen, "Uber die Form architektonischer S/iulen," Bull. cl. physico-math. Acad. St. P~tersbourg 9, 1851, pp. 369-380. 4. S. J. Cox and M. L. Overton, "On the optimal design of columns against buckling," SIAM J. on Math. Anal. 23 (1992), to appear. 5. W. B. Dinsmoor, "The architecture of the Parthenon," in The Parthenon, V. J. Bruno, ed., New York: Norton, 1974, pp. 171-198. 6. L. Euler, Leonhardi Euleri Opera Omnia, Scientiarum Naturalium Helveticae edenda curvaverunt F. Rudio, A. Krazer, P. Stackel. Lipsiae et Berolini, Typis et in aedibus B. G. Teubneri, 1911-. 7. R. Hooke, "Lectures de Potentia Restitutiva, or of spring explaining the power of springing bodies," London, John Martyn, 1678; reprinted, pp. 331-388 of R. T. Gunther, Early Sciences in Oxford 8, Oxford, 1931. 8. J. Keller, "The shape of the strongest column," Arch. Rat. Mech. Anal. 5 (1960), pp. 275-285. 9. J. L. Lagrange, Oeuvres de Lagrange, J. A. Serret, ed., Paris: Gauthier-Villars, 1867. 10. M. Laugier, An Essay on Architecture, W. and A. Herrmann, trans., Los Angeles, Hennessey & Ingalls, 1977. 11. E. Masur, "Optimal structural design under multiple eigenvalue constraints," Int. J. Solids Struct. 20 (1984), pp. 211-231. 12. N. Olhoff and S. Rasmussen, "On single and bimodal optimum buckling loads of clamped columns," Int. J. Solids Struct. 13 (1977), pp. 605-614. 13. F. Penrose, An investigation of the principles of Athenian architecture; or, The results of a survey conducted chiefly with reference to the optical refinements exhibited in the construction of the ancient buildings at Athens, Macmillan, London, 1888. Reprinted by McGrath, Washington, 1973. 14. A. Seiranian, "On a problem of Lagrange," Inzhenernyi Zh., Mekhanika Tverdogo Tela, 19 (1984), pp. 101-111. Mechanics of Solids 19 (1984), pp. 100--111. 15. G. Stevens, "Entasis of Roman columns," Mem. Amer. Acad. Rome W, 24 (1924), pp. 121-139. 16. I. Tadjbakhsh and J. Keller, "Strongest columns and isoperimetric inequalities for eigenvalues," J. Appl. Mech. 29 (1962), pp. 159-164. 17. I. Todhunter and K. Pearson, A History of the Theory of Elasticity and of the Strength of Materials, Cambridge, 1886. 18. E. Viollet-Le-Duc, Dictionnaire Raisonn~ de l'Architecture Fran~aise du XIe au XVIe si~cle, Paris" A. Morel, 1875. 19. P. Vitruvius, On Architecture, F. Granger, trans., London, W. Heinemann, Ltd.; New York: G. P. Putnam's sons, 1931-34. 20. T. Young, Miscellaneous Works, vol. 2, G. Peacock, ed., John Murray, London, 1855. New York: Johnson Reprint, 1972.
Department of Mathematical Sciences Rice University PO Box 1892 Houston, TX 77251 USA
Karen V. H. Parshall*
A Survey of Modem Algebra: The Fiftieth Anniversary of its Publication Garrett Birkhoff and Saunders Mac Lane
The " M o d e m Algebra" of our title refers to the conceptual and axiomatic approach to this subject initiated by David Hilbert a century ago. This approach, which crystallized earlier insights of Cayley, Frobenius, Kronecker, and Dedekind, blossomed in Germany in the 1920s. By 1930, relatively new concepts inspired by it had begun to influence homology theory, operator theory, the theory of topological groups, and m a n y other domains of mathematics. Our book, first published 50 years ago, was intended to present this exciting new view of algebra to American undergraduate and beginning graduate students. We had tried out our somewhat differing ideas of how this should be done in a course at Harvard for three successive years, before reorganizing and presenting them in textbook form. After explaining the conceptual content of the classical theory of equations, our book tried to bring out the connections of newer algebraic concepts with geometry and analysis, connections that had indeed inspired many of these concepts in the first place. The axiomatic (or postulational) approach to mathematics, which had been initiated in the preceding decades by Peano, Dedekind, Pasch, and others, received a decisive presentation in the context of Euclidean g e o m e t r y in Hilbert's 1899 Grundlagen der Geometrie. Two years later, this book was made the subject of a year-long seminar at the University of Chicago by E. H. Moore, and soon after that the thesis of Moore's student, Oswald Veblen, treated projective geometry in the same style. Hilbert's axiomatic approach soon became popular in the United States. By 1905, Harvard's E. V. Huntington and others had begun to study the i n d e p e n d e n c e of postulates for groups, Boolean algebras, and other algebraic and relational systems. Veblen expanded his thesis in collaboration with Moore's brother-in-law, J. W. Young, into their treatise Projective Geometry (1910, 1918); another * Column Editor's address: Departments of M a t h e m a t i c s a n d History, U n i v e r s i t y of V i r g i n i a , C h a r l o t t e s v i l l e , V A 22903 U S A . 26
THE MATHEMATICALINTELLIGENCERVOL. 14, NO. I 9 1992 Springer Verlag New York
indirect fruit of E. H. Moore's seminar was R. L. Moore's axiomatization of Euclidean geometry and the topological plane. The axiomatic method was also used by the active American school of finite group theory (as in the 1916 treatise by Miller, Blichfeldt, and Dickson). Meanwhile, in Europe, the word "ring" gradually became adopted for that n o w familiar concept; E. Steinitz expounded a general theory of fields in his Algebraische Theorie der K6rper, while Weyl characterized vector spaces by n o w standard axioms in a section on "affine geometry" in his Raum, Zeit, Materie (1918). Hilbert had f o r e s h a d o w e d these conceptual advances in papers published in 1888-90, where he proved that every system of polynomial invariants has a finite basis. At the time Gordan, a master of the art of manipulating invariants, dismissed Hilbert's paper as not mathematics but theology! In the 1920s, however, Gordan's student, Emmy Noether, who had moved to G6ttingen as an associate of Hilbert's, developed a lively and expansive school of algebra there. An early product of this school was her 1921 paper on "Idealtheorie in Ringbereichen", which derived many properties common to ideal decompositions in polynomial rings and rings of algebraic integers. Her enthusiasm-"Es steht alles bei Dedekind" and "Use homology groups, not Betti n u m b e r s " - - w a s contagious. Soon Emil Artin, R. Brauer, H. Hasse, H. Hopf, W. Krull, and many others, working in informal association with her and encouraged b y Hilbert, demonstrated the power of such general concepts as homomorphism and quotient-system for solving specific problems. A series of lectures by her and Artin were brilliantly written up by van der Waerden in his two-volume Moderne Algebra (1930-31), providing professional mathematicians with a masterful overview of what had been achieved by the "Emmy Noether school." We each learned about these developments in different ways from somewhat different sources. We now describe our respective backgrounds in chronological order, trying to bring out how they influenced us. Mac Lane (SM below) had attended high school in Leominster, Mass., where he had excellent tutelage in English composition from a remarkable teacher, Olive Greensfelder. As an undergraduate at Yale College in 1926-30, he learned calculus from Lester Hill, point-set topology from W. A. Wilson, a smattering of the use of algebra in geometry from a text by Snyder and Sisam on the analytic geometry of space; he had, in physics courses, long practice in the use of vectors in the sense of Gibbs. Egbert J. Miles, a dynamic teacher, had guided SM through advanced calculus. For the summer of 1930, Professor Miles asked SM to join him in preparing a readable text on advanced calculus. This summer project, though far too ambitious and never completed, turned out to be a splendid training in the writing of mathematics. Miles understood clarity. SM first encountered abstract algebra in the fall of
1929. Oystein Ore, fresh from Oslo and GOttingen, had just been made a full professor of mathematics at Yale. Ore's parallel courses on group theory and Galois theory were exciting, so SM audited both. They were indeed exciting, and exhibited the thrust of n e w views of mathematics. On Ore's advice, SM studied Fricke's rather ponderous volumes on algebra and then the brand n e w two-volume text (in German) by Otto Haupt, a professor at Erlangen w h o had learned the new approach from visits by Emmy Noether. Haupt's presentation was a bit heavy, but it did give a clear view of the conceptual formulation of Galois theory in terms of automorphisms. As a graduate student at Chicago, 1930-31, SM learned number theory from that redoubtable master, L. E. Dickson, and absorbed the Chicago view of vectors as n-tuples (or denumerable tuples) of numbers. But courses on the numerical limits for Waring's problem (Dickson), or projective differential geometry (Lane), or the second variation in the calculus of variat-ions (Bliss) seemed boring, lacking the excitement of the new ideas then arising in logic and algebra. E. H. Moore was the grand old man of the department. SM attended his seminar on the Hellinger integral; given his interests in logic, SM was assigned to report on a famous paper of Zermelo--the one with the second proof that the Axiom of Choice implies that every set can be well-ordered. SM thought he gave a brilliant presentation of this result, but Professor Moore took an hour to explain w h y this just was not so brilliant, which was, for SM, a remarkable learning experience. Seeking the fount of mathematical logic, SM left for GOttingen (1931-33). There his ideas were really shaken up. In a seminar under Professor Hermann Weyl he finally learned h o w to really use elementary divisors, and made the shocking discovery that a vector is better considered as an element in an axiomatically defined vector space---as he could have learned earlier by reading Weyl's Raum, Zeit, Materie. Lectures by Emmy Noether on non-commutative algebra exposed SM to her rapid and enthusiastic style (that of discovery in the midst of a lecture) and to facts about cyclic algebras, which he might have learned from Dickson and which he later used for group cohomology. SM also heard Emil Artin on class field theory and Herglotz on Lie groups, and wrote a thesis on logic with Paul Bernays [2, pp. 1-66], on which he was examined by Weyl after the dismissal of Bernays by the Nazis. SM thus witnessed the last great days of the original institute at GOttingen. In 1933-34, SM, as a post-doctoral student at Yale with advice from Oystein Ore, worked on the constructive factorization of primes in an algebraic number field. Clearly, Ore had a predominant influence in starting SM in algebra, even though Ore sharply disapproved of SM's continued interest in logic. But writing a good text requires above all experience THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
27
in teaching the material. During the depression years of 1934-36, SM felt very fortunate in being a Benjamin Peirce Instructor at Harvard. As such, he was invited to give a one-semester graduate course. He proposed as topics either logic or algebra; the department chairman, W. C. Graustein, recommended algebra. So SM taught a course based largely on van der Waerden's already famous Moderne Algebra. The next year, he chose the topic of algebraic number theory for a similar course. At Harvard, SM had as a colleague Marshall Stone, who was then applying topology to provide new and deeper foundations for Boolean algebra, using ideas from the spectral theory of linear operators on Hilbert space. Another colleague was Hassler Whitney, who was initiating the axiomatic theory of matroids, while giving a graduate half-course on topological groups, and an intermediate half-course in algebra (Math. 6) which was then given only sporadically. He also became a friend of Garrett Birkhoff (GB below), who was still in Harvard's Society of Fellows. When GB entered Harvard in 1928, all mathematics concentrators were exposed to three years of calculus and analytic geometry, in courses for which W. F. Osgood (the senior professor) had carefully designed the text. 1 During the previous summer, under parental edict, GB had mastered first-year calculus by selfstudy, and so he took sophomore calculus as a freshman. His section was taught by Marston Morse and Hassler Whitney, w h o was still a graduate student. Morse explained the relation between the numbers of pits, passes, and peaks in the graph of a smooth function z = F(x, y), thus introducing the students to Morse theory. He also showed h o w to construct an exception to the formula 32F/OxOy = 32F/OyOx. As a sophomore, fifteen months after beginning to study calculus, GB then entered the graduate course on functions of a complex variable. In it, Walsh emphasized the high points dramatically, and weekly exercises gave invaluable practice in rigorous theoremproving. In each of his two remaining undergraduate years, GB took three graduate half-courses in mathematics and one full graduate course in mathematical physics. But most absorbing was GB's tutorial reading, directed by Marston Morse, and the preparation of a related Honors Thesis, part of which concerned fractional-dimensional measure (now popularized under the name of "fractals"); part of this thesis was later published in the Bulletin of the AMS. By graduation, GB had been awarded a Henry Fellowship for Cambridge University. British universities did not then have active Ph.D. programs, and GB was admitted as a Research Student. He had planned to work on quantum mechanics, for which a satisfactory
1 For a d e s c r i p t i o n of H a r v a r d at t h a t t i m e , s e e [1, vol. ii, p p . 3-58].
28 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992
mathematical formulation had been given only a halfdozen years previously. However, shortly before graduating, he had also begun a secret love affair with the theory of finite groups. After learning a bit, he decided to try to work things out for himself. Most fascinating seemed the problem of determining all groups of given order n, for arbitrary finite n. Sensibly, he decided to first determine all such Abelian groups. Guided by techniques he had learned from Morse about matrices of integers, he worked out their unique decomposition into cyclic factors in Munich that summer. Fortunately, he also called in Munich on Constantin Carath6odory, who had been a visiting professor at Harvard in 1927-28. Carath6odory advised GB to read Speiser's Gruppentheorie and van der Waerden's Moderne Algebra if he wished to become more broadly informed about algebra. GB followed this advice. Indeed, when he found Hardy's brilliant lectures on number theory and other subjects fascinating, and Dirac's lectures on quantum mechanics over his head, he began to read voraciously about finite groups in the journal literature and to seek the advice of Philip Hall. Reflection on a series of papers on the structure of finite groups by R. Remak inspired GB to rediscover the concept of a lattice, about which (under the name of "Dualgruppen") Dedekind had written some 30 years earlier in two little-noted papers (see [3, p. 33]). P r e s i d e n t Lowell of H a r v a r d h a d m e a n w h i l e founded (and endowed) the Society of Fellows, with the aim of freeing a few exceptionally promising students from the usual Ph.D. requirements. GB, with W. V. Quine, B. F. Skinner and others, was one of the first of these junior fellows, and thus had three more years of free time to develop his ideas about algebra, including lattice theory, the logic of quantum mechanics, and "universal" algebra; see [3, pp. 1-198] for a review of the resulting developments. GB began teaching at Harvard in 1936. For his graduate half-course, he chose the sweeping title "Foundations of abstract algebra and topology." As a faculty instructor, he also taught first-year calculus, and tutored five undergraduate mathematics concentrators in lieu of giving the extra elementary half-course assigned to Benjamin Peirce Instructors; actually, he was a resident tutor in Lowell House where he was also on the intramural squash team. He felt strongly that Harvard's intermediate full-year course on g e o m e t r y should be paralleled by one on algebra that would emphasize basic ideas, while explaining the techniques of "college algebra" and the "theory of equations" then taught as such in most American colleges. After getting permission to inaugurate a very different and expanded version of Math. 6 in 1937-38, he prepared mimeographed notes for it. These notes began with the "algebra of classes," emphasizing the axioms for Boolean algebra and the reduction of Boolean polynomials to canonical form.
From the properties of finite sets those of the semiring of nonnegative integers were deduced, including the fundamental theorem of arithmetic. Following these preliminaries, the real field was constructed from the ordered semiring of positive integers, and its uncountability proved, as were the laws of exponents for the real function a~ (a > 0). The unique factorization theorem for polynomials in n variables over general fields was proved next, followed by the construction of the complex field and the fundamental theorem of algebra. The solution of cubic and quartic equations by radicals was then contrasted with numerical methods for computing all real roots of a general real polynomial equation. The first term was concluded with a somewhat recreational introduction to combinatory analysis. The second semester began with an axiomatic treatment of vector spaces over general fields, which made it easy to define algebraic numbers and algebraic functions. Cantor's proof that "almost all" real numbers are transcendental and the differentiation of real vector functions followed. Then matrices were introduced as linear operators on finite-dimensional vector spaces,
various canonical forms of matrices derived, and real determinants interpreted as volumes. However, other geometric applications were postponed until after group theory had been introduced as the "algebra of symmetry." Abstract groups (mostly finite) were also treated through Lagrange's theorem, before various discrete and continuous groups of transformations of Euclidean space of arbitrary dimension were taken up, somewhat in the spirit of Klein's Erlanger Programm. The course concluded with an introduction to algebraic number theory. While GB was giving the courses just described, SM had the privilege of teaching the standard graduate courses in algebra at two major universities. At Cornell in 1936-37, the text was B6cher's Introduction to Higher Algebra, which emphasized the connections of matrices with geometry. Projective and algebraic geometry were then predominant subjects of graduate study there. Then, back at Chicago in 1937-38 as an instructor, he taught from the brand new text Modern Higher Algebra by Adrian Albert. This text again emphasized matrices; in the process, SM learned much about Albert's work on linear algebras. Emil Artin visited Chi-
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992 2 9
cago, and gave a long, brilliant lecture on Galois the- w a r . Our preface began: "The most striking characteristic ory, which would later be reflected in the last chapter of modern algebra is the deduction of the theoretical of our Survey. Thus, when SM returned to Harvard in 1938, he had properties of such formal systems as groups, rings, already taught graduate courses four times in three fields, and vector spaces." It stated that its conceptual different universities. Moreover in 1938-39, while GB content is best conveyed "by illustrating each n e w was trying out his version of Math. 6 for the second term by as many familiar examples as possible." Our time, SM taught his fifth graduate course: this time on axiomatic approach was emphasized on p. 1: "Instead algebraic functions. We exchanged roles in the follow- of trying to define what the integers are, we shall start ing year: while GB gave a one-semester graduate by assuming that these integers . . . must satisfy cercourse on continuous groups, SM gave a very different tain algebraic laws." Chapter I, entitled "The Inteversion of the year-long intermediate course on alge- gers," then gradually deduced the Fundamental Thebra, Math. 6, relying primarily on his own previous orem of Arithmetic from the axioms for an ordered integral domain with well-ordered positive members, teaching experience. For this purpose, he prepared a 200-page set of emphasizing the notion of congruence modulo n and planographed notes, carefully typed up by his wife, introducing the notion of isomorphism--a far cry from Dorothy Mac Lane. These notes were organized into 11 GB's 1937-39 presentation. After constructing the rachapters: number theory, group theory, fields of num- tionals, Chapter II described ordered fields by axioms, bers, polynomials, real, complex, and infinite num- and gave the Peano postulates for the positive intebers, vectors, matrices and geometry, equations and gers. Chapter III characterized the real field R as the fields, rings, subsystems, and subclasses. Thus vectors only complete ordered field; its construction from the raand linear transformations were introduced conceptu- tional field by Dedekind cuts and Cauchy sequences ally, and their use in geometry was emphasized. In was deferred to two starred sections, again a far cry particular, the connection between linear transforma- from GB's 1937-39 presentation! tions and matrices and between vectors and rows (or Chapter IV distinguished polynomial forms from columns) was carefully developed, as subsequently in polynomial functions, presented the division a l g o Survey. The Galois group of an algebraic extension of a rithm and the axioms for commutative rings, and field was defined as a group of automorphisms of the proved unique factorization. The complex plane was extension field, but the fundamental theorem of Galois then constructed in Chapter V, followed by a topologtheory was not proved. Lattices and Boolean algebras ical proof of the Fundamental Theorem of Algebra and appeared. SM is now not clear how he managed to the solution of cubic and quartic equations by radicals. write all this up in one summer spent with Dorothy's In summary, our first five chapters developed the classical theory of equations from an axiomatic standpoint. family in Arkansas. The desirability of agreeing on a standard outline for Our pivotal Chapter VI first introduced groups of the course, so that it would fit well into Harvard's transformations with an example: the group of symcurriculum and could be effectively taught by our col- metries of a square. It then took up systems of axioms leagues, seemed obvious. We therefore agreed to write for "abstract" groups (with examples), subgroups and a joint text that would express our view of modern their cosets (Lagrange's theorem), and concluded with algebra and its important connections with other parts homomorphisms, quotient-groups, and general "'conof mathematics and science, while incorporating the gruence relations." (Groups had come near the beginbest features of our respective notes. Marshall Stone ning in SM's notes, and near the end in the notes of and Hassler Whitney were especially supportive of this GB.) effort, while Walsh later taught the course using our The next four chapters treated vectors, matrices (lintext in exchange for SM teaching Math. 13, the gradu- ear transformations), and determinants. After recalling ate course that GB had taken as a sophomore. Warn- a few familiar examples of vectors, we wrote: "The ings about the dangers of a possibly uncongenial col- algebraic properties of vectors will now be summarized laboration from our senior colleague, the geometer in a definition: A vector space i s . . . " . The main theJ. L. Coolidge, proved unfounded, and we did succeed orems about subspaces, bases, and dimension were in combining our respective notes after many lively then derived over arbitrary fields, and Euclidean vecdiscussions concerning the most effective arrangement tor spaces treated as real vector spaces with inner products having the usual properties. Matrices were of topics. Our resulting Survey of Modern Algebra, in its first introduced in the next chapter as linear transformations (1941) and later editions (all expertly typed by Dorothy of vector spaces---thus motivating matrix multiplicaMac Lane), presents a conceptual view of algebra tion. Invertible ("non-singular") matrices were treated which incorporates its classical background. Our book without determinants. The "full" linear, affine, orthogonal, Euclidean, and came out at the right time, and was ready for the surge of interest in mathematics that followed the end of the unitary groups, change of basis ("alias-alibi"), diago30
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
nalization and other "'canonical" reductions of quadratic and bilinear forms (mostly over the real and complex fields), with geometric applications, followed in Chapter IX. Then came determinants (interpreted as volumes), the characteristic polynomial, and the Cayley-Hamilton theorem. (The rational and Jordan canonical forms were added in later editions). The last five chapters (actually touched on lightly in Math. 6) branched out into more advanced topics. The Boolean algebra and cardinal numbers that had been used to construct the semiring of nonnegative integers in 1937-39 were taken up first, now including the Schr6der-Bernstein theorem with a neat diagram of the proof. Rings and their ideals were then discussed, illustrated by examples from algebraic geometry, followed by a definition of the characteristic of a ring or field. Chapter XIV introduced algebraic numbers and algebraic integers, with a proof that Gaussian integers could be uniquely factored into primes, and an example showing that factorization into primes was not unique for the integers of some other quadratic fields. Finally, finite fields and Galois theory were taken up, and the unsolvability of quintic equations by radicals shown to follow from the very different "unsolvability" of the symmetric group of all permutations of five symbols. Historically, presentations of Galois theory and the generality of the notions of homomorphism and quotient system had been obscure until they were illuminated by the insights of Dedekind, Steinitz, Emmy Noether, and Artin. Our book helped to popularize these insights at a time w h e n they were still novel. Moreover, it provided a logically homogeneous exposition of many algebraic methods, with copious exercises that stimulated students to develop their power to reason deductively. Assuming only high-school algebra, it finally proved the unsolvability of quintic equations by radicals. Modern algebra prospered mightily in the decades 1930-1960, from functional analysis to algebraic geome t r y - n o t to mention our own respective researches on lattices [3] and on categories [2]. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new. It began to sell well as soon as the war was over, in 1948-53 at about 2000--3000 annually. In 195153 we prepared a carefully polished second edition, in which polynomials over general fields were treated before specializing to the real field. Other more minor changes and additions h e l p e d to increase its popularity, with annual sales in the range 14,000-15,000. Our third edition, in 1965, finally included tensor products of vector spaces, while the fourth (1977) edition clarified the treatment of Boolean algebras and lattices. Our Survey in 1941 presented an exciting mix of classical and conceptual ideas about algebra. These ideas are still most relevant and worthy of enthusiastic pre-
sentation. They embody the elegance, precision, and generality which are the hallmark of mathematics!
References 1. Duren, Peter, et al. (eds.), A Century of Mathematics in America, 3 vols., Providence, RI: American Math. Society, 1988. 2. Kaplansky, Irving, (ed.), Saunders Mac Lane: Selected Papers, New York: Springer-Verlag, 1979. 3. Rota, G.-C., and J. Oliveira (eds.), Selected Papers on Algebra and Topology by Garrett Birkhoff, Boston: Birkh/iuser, 1987. Department of Mathematics Harvard University One Oxford Street Cambridge, MA 02138 USA Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL 60637 USA
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
31
The Riemann Mapping Non-Theorem I Nancy K. Stanton
I want to describe some differences between one and several complex variables, in particular, the failure of the Riemann mapping theorem in several variables. Let me begin with the statement of the Riemann mapping theorem ([1], Chapter 6, Theorem 1).
THEOREM (Carath4odory, 1913). Suppose ~ is bounded by a closed Jordan curve. Then the map_ping function extends to a continuous function that maps ~ one-to-one onto the closed unit disk.
RIEMANN MAPPING THEOREM (1851). Given any
Goluzin's book [7] contains a proof of Carath6odory's theorem in Chapter II, Section 3. That book also
simply connected region 1~ that is not the whole plane, and a point z o ~ fL there exists a unique analytic function f in 1~, normalized by the conditions f(zo) = O, f'(Zo) > O, such that f defines a one-to-one mapping of ~ onto the disk Iwl < 1. The function f is called the mapping function. Geometrically, the m a p p i n g theorem says that every simply connected region other than the plane is conformally, or biholomorphically, equivalent to the unit disk. Under reasonable hypotheses on the boundary of t , the mapping function behaves nicely at the boundary. The simplest such result is the following. THEOREM. Suppose the boundary ~1of f~ is a real analytic simple curve with nowhere vanishing tangent. Then the map~_ing function extends analytically to a one-to-one map on 1~, and this extension maps ~ onto an arc of the unit circle. If fl is bounded, ~/is mapped onto the circle. This theorem is a simple consequence of the Schwarz Reflection Principle. The proof is local, so the theorem really is a local theorem. For the precise statement of the local theorem see [1], Chapter 6, Section 1.3. Probably the most famous result about boundary behavior is the following. 1This is an expanded version of an Invited Address at the MAA meeting in Boulder, August, 1989. The title was suggested by Steven Bell [2]. The author is supported in part by NSF grant DMS 89-01547. 32
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. I 9 1992 Springer Verlag New York
Figure 1. The bidisk.
Figure 3. The ball.
Figure 2. {Iz[ < ~, Iwl < 1}.
has historical notes on the mapping theorem and the b o u n d a r y behavior of mappings. In the special case that the b o u n d a r y is a c o n t i n u o u s l y differentiable curve (with n o w h e r e vanishing tangent), this was p r o v e d by Painlev4 in 1887 and 1891 [8]. He also s h o w e d that one can conclude more if the b o u n d a r y has more differentiability.
T H E O R E M (Painlev4, 1891). If ~ is bounded by a C~ closed Jordan curve, then the mapping function extends to a C ~ function on F~. Steven Bell proves this result in his survey article [2], Theorem 3.1. N o w I want to discuss two complex variables. (Everything I say will a p p l y to n variables, with suitable modification, for n > 2.) I will begin with a striking difference b e t w e e n one and two variables. Let D denote the unit disk in the complex plane, D = {z : Izl < 1}. The function f(z) = 1/z is holomorphic in the punctured disk D\{0}, but it does not extend to a holomorphic function on D. N o w let D 2 be the bidisk, D 2 = {(z,w) : Izl, Iwl < 1} (see Figure 1). Let f b e holomorphic in D2\{(0,0)} (i.e., f is C 1 and satisfies the Cauchy-Riem a n n equations in each variable or, equivalently, f can be expanded in a convergent p o w e r series about each point of D2\{(0,0)}). Note that f(z,w) = 1/z is not such a function. It is only defined on D2\{{0} x D}}.
T H E O R E M (Hartogs). Suppose f is a holomorphic function on the punctured bidisk D2\{(0,0)}. Then f extends to a holomorphic function on the whole bidisk D 2. This is a special case of a general extension theorem due to Hartogs. A n y paper should include a proof, so I will prove this. For the proof of the general result s e e [10], Theorem 4.2.1.
Proof: Let 1 Ir F(z,w)
=
- -
2~i
fit,w) J= v2
-
z
at,
Izl < 1/2, Iwl < 1.
Then F is holomorphic in Izl < 89 Iwl < 1 (see Figure 2). (Differentiate under the integral to see this.) If w # 0, f(t,w) is holomorphic in Itl < 1, so for fixed w 0 # 0 1 ir f(t'w~ f(z,wo) = 2~r---i i=1/2 ~ ~ Z dr,
Iz[ < 1/2
(see Figure 2). Thus F(z,w) = f(z,w) on the open set 0 < Iwl < 1, Izl < 89(hence also on Izl < 89 Iwl < 1, (z,w) (0,0)). The desired extension is given by =
{f(z,w),(z,w) # F(z,w),Iz4 <
(0,0) Iwl < 1.
This completes the proof. The bidisk can be thought of as a two-dimensional analogue of the disk. There is another analogue, the unit ball B = {(z,w): Izl2 + Iwl2 < 1} (see Figure 3). Both B and D 2 are simply connected, and in fact contractible. They are topologically equivalent. However, they are not biholomorphically equivalent, i.e., there is no one-to-one holomorphic map f : D 2 --~ B with holomorphic inverse. There is no Riemann mapping theorem in two complex variables! Henri Poincar6 proved that the ball and the bidisk are not equivalent, although he did not explicitly state that fact. In 1907 Poincar6 [9] observed that if two domains fll and f12 are equivalent, then Aut(fll) -~ Aut(fl2), w h e r e Aut(lli) is the group of biholomorphic self maps of 1~i. The automorphisms of the unit disk D are all of the form z ~
e i0 -
Z -- a -
1-~z THE MATHEMATICALINTELLIGENCERVOL. 14, NO. I, 1992
33
with lal < 1, 0 E R. This is proved using Schwarz's persurfaces M and M' through the origin, is there a Lemma ([1] Section 4.3.4, Exercise 5). Thus, Aut(D) is biholomorphic map F defined in a neighborhood U of a 3-dimensional Lie group. It is obvious that Aut(D 2) the origin with F(0) = 0 and F(M A U) C M'? In one variable the corresponding local problem contains all automorphisms obtained by acting on always has a solution: given two analytic arcs ~ and ~/' each factor with an automorphism of the disk, through 0, there is a biholomorphic map F, defined in a neighborhood U of 0, which maps ~ A U to ~'. In ( Z , W ) --'* e iO - - , e i* - fact, if f:( - 1,1) ~ C parametrizes a neighborhood of 0 1 - -& 1 - -bw ' in ~/and g:( - 1,1) ~ C parametrizes a neighborhood of and also the automorphism that interchanges compo- 0 in ~/' with f(0) = 0 = g(0), then extend f and g to be nents, complex analytic in a neighborhood W of ( - 1 , 1 ) and let F = g o f-1 (see Figure 4). (z,w) --, (w,z). Poincar6 noticed that in general it is impossible to construct even a formal p o w e r series solution F to the In fact, these generate Aut(D2), so Aut(D 2) is a 6-ditwo-dimensional problem and that hypersurfaces mensional Lie group with 2 components. Also in [9], have an infinite number of "'geometric invariants." Poincar6 calculated the automorphisms of the ball. What are these "geometric invariants"? Let M be an They are given by analytic real hypersurface through the origin. Use the implicit function theorem and a linear change of vari( z , w ) _ . ~ ( a n z + a12w + bl_. a21z + a22w + b 2 ) ables to write the equation for M as \ clz + c2w + d " ClZ + c2w + v = f(x,y,u),
where
t
all
a12
a21 a22 C1
b~) ~ SU(2,1);
where z = x + iy, w = u + iv, and f(0,0,0) = 0. In the power series expansion for f, there are
C2
here SU(2,1) is the group of matrices of determinant 1 that preserve the quadratic form 14112 + 14212 - 14312. The subgroup of matrices that act trivially is generated by e2~'~3I and so is isomorphic to Z 3. Thus, Aut(B) SU(2,1)/Z3 is an 8-dimensional connected Lie group, and Aut(B) 9 Aut(D2). In the same paper, Poincar6 raised the question: when is there a local analogue of the Riemann mapping theorem in C2? Given two real analytic real hy-
a(n) =
(n + 1)(n + 2)(n + 3) 6
- 1
(real) coefficients of terms of positive degree ~< n. Now, if F is a biholomorphic map defined on a neighborhood of 0, with F(0) = 0, then in the power series for F there are 2( (n + 1)(n ) +22 ) - 1 complex coefficients of terms of positive degree ~< n or b(n) = 2n 2 + 6n
Figure 4. 34
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
real coefficients. If F : M --~ M', we can try to equate coefficients in power series to find the coefficients of F in terms of the defining functions for M and M'. This gives a(n) real equations in b(n) unknowns. N o w a(n) > b(n) for n >t 9, so as soon as we are above degree 8, there are more equations than unknowns. In general, there are an infinite number of conditions for solving the equations, hence an infinite number of geometric invariants. Poincar~ does not give much additional information about what the invariants are or how to find them. Some additional work on the problem was done by Beniamino Segre in 1931 [11] [12]. In 1932, Elie Caftan [3] used his " m e t h o d of equivalence" to study the pseudoconformal geometry of hypersurfaces in C 2. Cartan defined p s e u d o c o n f o r m a l geometry as the study of those local properties of hypersurfaces that remain invariant under local biholomorphic transfor-
mations. He found a complete set of local, intrinsically defined, geometric invariants for "non-degenerate" analytic real hypersurfaces in C 2. His approach was generalized to higher dimensions by Noboru Tanaka [13], [14] and Shiing-Shen Chern [4] in the late 1960s and early 1970s. The intrinsic approach is analogous to studying Riemannian geometry by constructing a connection on the bundle of orthonormal coframes. The Cartan-Chern-Tanaka invariants are a bundle and a connection. The bundle and connection, hence also the curvature, are real analytic. The notion of pseudoconformal geometry extends to C" real hypersurfaces - - i t is the study of those local properties of hypersurfaces that remain invariant under diffeomorphism by the boundary values of holomorphic maps. The definitions of the Cartan-Chern-Tanaka bundle and connection do not require analyticity, so the bundle and connection are defined in the C~ case as well. The construction is quite complicated; I will not describe it. In the early 1970s, Ji~rgen Moser [4] studied the equation of a "non-degenerate" real analytic real hypersurface. He showed that there is a biholomorphic map taking the equation of the hypersurface into a "normal form," and this normal form is unique up to the action of a finite-dimensional Lie group. This is a precise version of Poincar6's geometric invariants. Moser's approach to the problem is extrinsic. It is analogous to studying a geometric problem by choosing the best coordinates for the problem, e.g., analogous to rotating axes in the plane to simplify the equation of a conic section. Before I describe Moser's normal form, I will explain the non-degeneracy condition required by Cartan, Tanaka, Chern, and Moser in C 2. It is strict pseudoconvexity. A real hypersurface M in C 2 is strictly pseudoconvex if it is locally equivalent, by a biholomorphic map, to a strictly convex hypersurface, i.e., to a hypersurface whose defining function has a positive-definite Hessian. For example, any strictly convex hypersurface, such as a sphere or the boundary of an ellipsoid, is strictly pseudoconvex. The hyperquadric Q = {(Z,W) : V : Izl 2} is also strictly pseudoconvex. In fact, it
is locally equivalent to the unit sphere S3 via the linear fractional transformation (z,w)
(w
+i
w i) w+
"
Strict p s e u d o c o n v e x i t y is a pseudoconformally invariant property of a hypersurface. The hypersurface M = {v = 0} is the simplest example of a hypersurface that is not strictly pseudoconvex. To see that it is not, suppose there were a biholomorphic map F = (~1,f2) defined in a neighborhood of the origin and taking M to a strictly convex hypersurface M'. If necessary, make a linear change of coordinates so that 0 ~ M', the tangent space to M' at the origin is the hyperplane {v' = 0}, and M ' \ { 0 } C {v' > 0}. Then the function Im f2(z,0) would be a harmonic function on the z-plane with an absolute minimum at z = 0. This violates the maximum principle, so M is not strictly pseudoconvex. The boundary of the bidisk is not strictly pseudoconvex. You might wonder if a contractible domain with strictly pseudoconvex boundary is equivalent to the ball. The answer is still no. An explicit example is the ellipsoid x2 + 2y2 + u 2 + v 2 < 1 . In 1977, Sidney Webster [15] showed that an ellipsoid is equivalent to the ball if and only if it is complex linearly equivalent to the ball. Moser's normal form is defined for strictly pseudoconvex real analytic hypersurfaces. His idea is to look for the "simplest" possible equation for M, allowing a biholomorphic change of coordinates. The model is the hyperquadric Q. Make a complex linear change of variables (if necessary) so that M is given by the equation r = 0 with r(0) = 0,
rx(0) = ry(0) = r,(0) = 0,
rv(0) # 0 .
Use the implicit function theorem to solve r = 0 for v, so M is given by v = F(z,-d,u),
where F is real analytic, and F and its first partial derivatives vanish at 0. Here I am using conjugate coordinates z = x + iy and -~ = x - iy instead of x and y. THEOREM (Moser). Let M analytic real hypersurface in transformation rb, with rb(O) face M ' given by an equation
be a strictly pseudoconvex real C 2. There is a biholomorphic = O, taking M to a hypersurof the form
Fj.k(u)zJ- k,
v = Izl 2 + j,k~ 2 j+k~6
THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992
35
where Fj,k(U) is an analytic function of u, Fj,k(u ) = Fk,j(u) and F3,3 ~ 0. The hypersurface M' is uniquely determined up to a transformation preserving the origin and the hyperquadric Q. Thus, the h y p e r q u a d r i c Q osculates a n y strictly p s e u d o c o n v e x h y p e r s u r f a c e in C 2 t h r o u g h order 5. The coefficients in the Moser normal f o r m are geometric invariants of the hypersurface. The transformations that preserve the origin and the hyperquadric form a 5-dimensional Lie group. This t h e o r e m s h o w s that a strictly p s e u d o c o n v e x real analytic hypersurface M is locally pseudoconforreally equivalent to the sphere if a n d only if its normal form is v = Izl2. There is also a differential-geometric characterization of the sphere. THEOREM (Cartan). A strictly pseudoconvex real hypersurface M in C 2 is locally pseudoconformally equivalent to the sphere if and only if it is pseudoconformally fiat, i.e., if and only if the curvature of its intrinsic connection vanishes. In one complex variable, any simple C" curve ~ with nowhere-vanishing tangent is locally equivalent, via the b o u n d a r y values of a holomorphic map, to an analytic c u r v e - - i n fact, to an arc of the circle. To see this, pick P0 ~ ~/and a closed subarc ~0 of % with ~/0 contained in the interior of % Complete ~/0 to a C" simple closed curve ~/1 that b o u n d s a region fL Fix z0 ( ft. Then the b o u n d a r y value of the R i e m a n n m a p p i n g that takes z 0 to 0 m a p s ~/0 onto an arc of the circle. Is there an analogue of this result in C2? Is every strictly pseudoconvex C" hypersurface equivalent via the b o u n d a r y values of a biholomorphic m a p to an ana l y t i c hypersurface? As an application of both the Moser n o r m a l f o r m a n d the C a r t a n - C h e r n - T a n a k a connection, I will give an example d u e to James Faran [5] showing that the answer is no. Let 4) : R ~ R be a nonnegative C | function such that cb(u) = 0 for u ~ 0 and 6(u) = 1 for u I 1. Let M be the hypersurface in C 2 given by v = Izl2 + , 6 ( u ) ( z 4 ~ 2 + z2~4),
where e > 0 is sufficiently small that M is strictly pseudoconvex for Izl2 < 1. For u < 0 and u > 1, M is analytic and it is in Moser normal form. For u < 0 it is part of the hyperquadric and has curvature II -= 0. For u > 1 it is not the hyperquadric, since its normal form is not the hyperquadric. Thus, the curvature does not vanish for u > 1. Hence, there is a point p ~ M such that in every n e i g h b o r h o o d U of p the curvature vanishes on an open subset of U but not on all of U. Suppose a neighborhood of p in M were equivalent, via the b o u n d a r y values f of a biholomorphic map F, to an analytic hypersurface M'. Then the curvature II' = f-1 *II of M' w o u l d v a n i s h on an o p e n subset of a 36
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
neighborhood V of tip) but w o u l d not vanish identically on V. Since M' is analytic, II' is also analytic, so if it vanishes on a connected o p e n set, it must vanish identically. Thus, no such M' can exist. We see that there is no R i e m a n n mapping t h e o r e m in several complex variables. However, we can still ask about the b o u n d a r y regularity of biholomorphic maps w h e n such maps exist. Let me conclude with a very striking positive result on b o u n d a r y behavior. This result, proved by Charles Fefferman in 1974 [6], has inspired a lot of work in several complex variables during the last 15 years. ( F e f f e r m a n ) . Let f~ and ~ ' be bounded strictly pseudoconvex domains in C 2 with C~ boundaries, and let 4) : f~ ~ ~' be a _biholomorphic map. Then 4) extends to a C ~ map ~ : f~---~ 1~'. THEOREM
References
1. Lars V. Ahlfors, Complex Analysis, third edition, New York: McGraw-Hill (1979). 2. Steven Bell, Mapping problems in complex analysis and the 3-problem, Bull. A.M.S. (2)22 (1990), 233-260. 3. Elie Cartan, Sur la g6om6trie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, I, Annali di Mat. (4)11 (1932), 17-90; OEuvres, II, 1231-1304. 4. Shiing-Shen Chern and J~irgen K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271. 5. James J. Faran, V, Non-analytic hypersurfaces in C n, Math. Annalen 226 (1977), 121-123. 6. Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Inventiones Math. 26 (1974), 1-65. 7. Gennadii M. Goluzin, Geometric Theory of Functions of a Complex Variable, Providence: American Mathematical Society (1969). 8. Paul PainlevG Sur la th6orie de la repr6sentation conforme, Comptes Rendus Acad. Sci. Paris 112 (1891), 653657. 9. Henri Poincar6, Les fonctions analytiques de deux variables et la repr6sentation conforme, Rend. Circ. Mat. Palermo 23 (1907), 185-220. 10. R. Michael Range, Holomorphic Functions and Integral Representations in Several Complex Variables, New York: Springer-Verlag (1986). 11. Beniamino Segre, Intorno al problema di Poincar6 della rappresentazione pseudoconforme, Rend. Acc. Lincei Roma (6)13 (1931), 676-683. 12. , Questioni geometriche legate colla teoria delle funzioni di due variabili complesse, Rend. Sere. Mat. Roma 7 (1931), 59-107. 13. Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Japan 14 (1962), 397-429. 14. - - , On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967), 215-254. 15. Sidney M. Webster, On the mapping problem for algebraic real hypersurfaces, Inventiones Math. 43 (1977), 53-68.
Department of Mathematics University of Notre Dame Notre Dame, IN 46556 USA
Geometric Invariants for 3-Manifolds Robert Meyerhoff
To my father on his 65 th birthday. What is the shape of the 3-dimensional universe in which we live? To have a hope of answering this question, we need to understand 3-dimensional manifolds in general. This is an enormously difficult problem, because abstract 3-manifolds are inherently complicated objects and there is a bewildering array of them. Invariants have proven to be the most effective tools available for studying 3-manifolds. They take the unwieldy collection of information that defines a manifold and distill it into a manageable packet (some information may be lost in the process). However, the standard invariants, while useful, do not give us as much information as we might hope. A new class of 3-manifold invariants--geometrically defined invariants--reveals many insights about 3-manifolds. Specifically, if we restrict our attention to hyperbolic 3-manifolds, then we can use the hyperbolic structure to define new invariants. This is a reasonable approach because the work of Thurston indicates that most 3-manifolds are hyperbolic. The study of hyperbolic invariants for 3-manifolds is a new subject, and the goal of this paper is to explain its main ideas and motivations. 2-dimensional manifolds are a natural collection to warm up with before embarking on a study of 3-manifolds, and will be covered in Sections 1 through 4. A priori, 2-manifolds are hard to understand, but once an appropriate invariant has been introduced they prove easy to understand. This is covered in Section 1. Sections 2 through 4 discuss geometry in the 2-dimensional case, not because it is needed to answer the questions posed in Section 1, but because the basic ideas are the same as in the 3-dimensional case, yet easier to visualize. The basic facts about 3-manifolds are covered in Section 5. The standard invariants for 3-manifolds are
discussed in Section 6, where they are shown to be useful, but not up to the task of completely analyzing 3-manifolds (in contrast to the 2-dimensional case). Section 7 briefly discusses how hyperbolic geometry proves to be important in three dimensions. Hyperbolic geometry can be used to define invariants for 3manifolds, and the most natural of these invariants, the volume, is studied in Section 8. The Chern-Simons invariant and the -q invariant are mentioned in Section 9. Section 10 provides a summary of the paper. This subject is in its infancy, and open questions abound. This article starts out slowly and non-technically, but finishes up with a fair amount of machinery. A willingness to skip blithely over jargon should enable you to progress rather far into the paper.
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1 9 1992 Springer-Verlag New York
37
Figure 3: A list of the closed orientable 2-manifolds.
C) a
Figure 1: A 2-dimensional manifold locally looks like the xy-plane.
Figure 2: Trapped in a 2-dimensional universe.
S e c t i o n 1: 2 - D i m e n s i o n a l
Manifolds
Topology
We begin in the realm of topology. Two objects are said to be topologically the same if there is a homeomorphism from one to the other. A h o m e o m o r p h i s m is a one-to-one onto m a p that is continuous a n d whose inverse is c o n t i n u o u s . Naively, two objects are homeomorphic if one can be stretched a n d pulled into the other. Cutting a n d tearing are not allowed. Actually, cutting is acceptable as long as y o u mark the cut and then glue it back exactly later on. In this section, the phrase "looks like" means "is h o m e o m o r p h i c to." We will be naive a n d think of a 2-dimensional manifold M as a space that locally looks like R 2, the xyplane. If we stand at a point in M and look around, it looks like a piece of R 2. If we move over to another point and look a r o u n d , it also looks like R 2. See Figure 1. Our goal is to u n d e r s t a n d the entire 2-manifold M. The readily available information about M is local, but we want to come to a global understanding. Note that Figure 1 is m i s l e a d i n g - - i t implies that in 38
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
Z b ~....,o,,j
c
Figure 4: The vertical edges of the hexagon are pulled in front and glued to produce the cylinder with the comically enlarged vertices. The top and bottom of the cylinder are then glued to produce the torus, which has been laid on its side.
attempting to u n d e r s t a n d our 2-manifold M, we can leave it a n d look d o w n u p o n it. Since we cannot leave the 3-dimensional universe we live in, we should try to restrict ourselves to considering M from the intrinsic viewpoint. See Figure 2. If we imagine ourselves as 2-dimensional creatures constrained to move in our 2-manifold M, we quickly see that the task of u n d e r s t a n d i n g M is formidable. For us, u n d e r s t a n d i n g 2-manifolds will a m o u n t to two things: (1) Classifying t h e m - - t h a t is, producing a list, with no repetitions, of all the 2 - m a n i f o l d s - - a n d (2) Recognizing t h e m - - d e v e l o p i n g a usable means of determining where a given 2-manifold fits on the list. The recognition problem is subtle. That is, a given 2-manifold m a y be described, or constructed, in m a n y ways, some easy to deal with a n d some hard. Since a 2-manifold that one is apt to encounter in one's work m a y not be described in a convenient way, we w o u l d like our recognition scheme for (2) to be able to deal with all sorts of descriptions of 2-manifolds. The classification of 2-manifolds is well k n o w n (see [11], Chapter I or [5], Chapter 2). For convenience, we will restrict our attention to closed, orientable 2-manifolds. Closed means compact a n d without boundary. O r i e n t a b l e m e a n s that the 2 - m a n i f o l d contains no mirror-reversing path. All such 2-manifolds are listed in Figure 3. That each of these is a 2-manifold is easy to believe. That these are all of the 2-manifolds is less obvious. G i v e n the classification in Figure 3, 2-manifolds seem easy to r e c o g n i z e - - t a k e a look and count the holes. This is misleading. First, looking and counting is n o t i n t r i n s i c - - w e have to leave the manifold to
Figure 5: Going for a walk around a vertex.
Figure 6: Two 10-gons with the same gluing schemes. The one on the right has been decomposed into triangles.
look. Second, we are usually not so lucky as to have a manifold presented to us in such a convenient way. Consider the following two ways of describing 2-manifolds: (1) The Algebraic Geometry Approach: A 2-manifold is described as the solution set of polynomial equations. For example, x3 + y3 = z3 describes a 2-manifold in homogeneous co-ordinates in complex projective 2space. Which 2-manifold on our list is it? Naively, to us, it is not at all clear how to answer this question (see [6] for the answer). (2) The Combinatorial Approach: A 2-manifold is described as an identification space of a polygon. Take a 2n-gon in the plane and identify--glue topologically - - p a i r s of edges. For example, consider the gluing pattern in Figure 4a. The gluing is carried out in Figures 4b and 4c. Thus, Figure 4a is a 2-manifold, and is the second 2-manifold on our list. Figure 4a is superior to Figure 4c in that it is an intrinsic description of the 2-manifold. Figure 4c has a greater visceral appeal. In general, does this pairwise identification procedure for a 2n-gon yield a 2-manifold? We need to check whether all points in the identification space locally look like R 2. There are three types of points in the 2n-gon to consider: i. Interior points: These work fine. Simply put a small e n o u g h disk around the point so that the disk misses the edges of the 2n-gon. ii. Edge points: These work fine too. An edge point shows up on exactly two e d g e s - - p u t half of a disk around each copy of the point and identify to get a standard disk. iii. Vertex points: If we go for a walk near the vertex we will go from one edge of a polygonal corner to the other edge sharing that vertex in the polygon, then pass to the edge of a glued corner (Figure 5). Because all edges of our polygon are glued up, we always pass to a n o t h e r c o r n e r . Because t h e n u m b e r of comers in our polygon is finite, our walk must end. Because the only closed, connected 1m a n i f o l d is a circle, our walk a r o u n d the
vertex must have enclosed a disk. Thus, vertices locally look like R 2, and our construction always yields a manifold. Given a combinatorial description of a 2-manifold, how can we recognize it on our list of 2-manifolds? For example, which 2-manifold does Figure 6a yield? As in Figure 4 we could attempt to find out what 2-manifold it is by carrying out the indicated gluing, but gluing a 10-gon is an unpleasant task. Further, physically gluing is an extrinsic process, and because we are ultimately concerned with understanding 3manifolds intrinsically, we need an intrinsic recognition scheme for identification spaces. The Euler characteristic satisfies our need beautifully. The Euler characteristic, which is our first example of an invariant, is computed as follows: Triangulate the surface in a nice way ([11] page 16). Count up the number of vertices V, edges E, and faces F in the triangulation. The Euler characteristic of the surface S is x(S) = V - E + F, and it is independent of the triangulation. Independence of the triangulation is intuitively easy to explain, but a precise proof is troublesome. Imre Lakatos in [10] exposes the history of the Euler characteristic by having a fantasy class discuss its proof. The tortuous route the class takes towards a proof is actually the tortuous route mathematicians took. The Euler characteristic is different for each 2-manifold in our list of closed, orientable 2-manifolds, and thereby shows that our list has no r e p e t i t i o n s - - a point we ignored previously. In particular, X(sphere) = 2, x(torus) = 0, • torus) = - 2 , x(threeholed torus) = - 4 . . . . . x(n-holed torus) = 2 2n . . . . . Thus, given a closed, orientable 2-manifold, we can recognize which one on our list it is by triangulating it, computing V - E + F, and comparing numbers. This is an intrinsic process, and could be carried out by the inhabitants of that 2-dimensional universe. For example, Figure 6a is triangulated in Figure 6b, and the Euler characteristic is 2 - 12 + 8 = - 2. Thus, the 10-gon identified as in Figure 6a is actually a 2-holed torus. THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
39
The Euler characteristic gives us about as good a solution to the recognition problem as we could want. As such, we could n o w move on to the 3-dimensional classification and recognition problems. However, the 3-dimensional situation is much more difficult, and we will ultimately need to introduce geometry into the study of 3-manifolds. As such, let's w a r m up by studying the geometry of 2-manifolds; the ideas are the same, and the pictures are more easily understood. Remark: There are, of course, other questions to ask about 2-manifolds. For example, the fact that 2-manifolds are topologically simple indicates that they possess a rich class of diffeomorphisms, and the study of these diffeomorphisms is important. Geometry has proven very useful in this study. Section 2: 2 - D i m e n s i o n a l M a n i f o l d s - - G e o m e t r y
We enter now the realm of geometry. The word geometry comes from the Greek words for earth and measurement. A geometric 2-manifold is simply a topological 2-manifold with an appropriate metric (means of measuring distances) added on. Two geometric 2manifolds are geometrically the same if there is an isometry between them. That is, they must be homeomorphic by a map that preserves their metric properties. In attempting to add a geometric structure to a topological 2-manifold we should subject ourselves to constraints motivated by our beliefs as to the geometric nature of our 3-dimensional universe. Perhaps esthetic considerations enter in as well. For example, we will demand that our geometries be homogeneous. Homogeneity means that the space looks the same geometrically at all points; that is, surveying teams working in small neighborhoods of different points would come up with all the same measurements. Given the homogeneity constraint, there are only three 2-dimensional geometries: spherical, Euclidean, and hyperbolic. The standard models of these geometries are the 2-sphere, the Euclidean plane, and the hyperbolic plane. A 2-manifold is spherical if it has a metric locally isometric to the sphere. That is, at each point, its metrical properties in a small neighborhood are exactly those of the sphere. Similarly for the other geometries. Which closed, orientable 2-manifolds admit geometric structures? That is, which 2-manifolds admit metrics locally isometric to one of the three 2-dimensional geometries? We start with the simplest case: the 2-sphere admits a spherical structure. The torus admits a Euclidean structure. Take the unit square in the Euclidean plane and identify opposite edges via a translation of length one. This identification is an intrinsic process--it is impossible to physically glue up the square in 3-space without distorting 40
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
a
} I + b
Figure 7: The identified unit square as Euclidean torus.
the metrical properties. The first gluing--to get a cyli n d e r - w o r k s , but the second does not. Extrinsic observers should think of the identified square as a video screen--if a stick figure leaves the screen to the left, then it re-appears on the right. See Figure 7a. We show that this torus is Euclidean by studying each point. Interior points of the square have Euclidean neighborhoods. Edge points have neighborhoods formed by rigidly gluing two Euclidean halfdiscs. See Figure 7b. It remains to check the four vertices of the square. These four vertices are identified to one point in the torus, and going for a walk in a n e i g h b o r h o o d of this point reveals that the local neighborhood is a Euclidean disc. See Figure 7c. Thus, this torus is locally isometric to the Euclidean plane. We n o w consider hyperbolic structures. So far, I have been cavalier in describing the 2-dimensional geometric models. I have simply assumed that everyone is familiar with the Euclidean plane and the unit sphere. Hyperbolic space, being less well-known, will require a more careful exposition. Section 3: T h e H y p e r b o l i c Plane
Our goal in this section is to come to some understanding of the hyperbolic plane. One approach is the synthetic, or axiomatic, approach. The axioms for the hyperbolic plane are the same as the axioms for the Euclidean plane except that the Euclidean parallel postulate is replaced by the hyperbolic parallel postulate: given a line and a point not on the line, there exists more than one line parallel to the original line through the point. The synthetic approach builds up a body of knowledge about the hyperbolic plane by proving theorems from the axioms. The problem with this approach is that it is completely non-intuitive. The theorems that are proved seem bizarre.
We will shun the synthetic approach. Instead we will learn about the hyperbolic plane by studying a metric model. There are three standard models for the hyperbolic plane: the Poincar6 Disk model, the upper half-space model, and the Klein or projective model. They have different descriptions, but to the inhabitants they would be indistinguishable. We will focus on the Poincar4 Disk model. The Poincar6 Disk is simply the open unit disk in the Euclidean plane, together with a modified version of the standard Euclidean metric. Specifically, the in. . . . . . (dx)2 + (dy) 2 fimteslmal hyperbohc m e t n c i s (dSH) 2 . . (1. -. (x 2 T ~ya))2/4" The hyperbolic metric ds H is just the Euclidean metric ds E divided by 89 - (x2 + y2)). Thus, we can use our familiarity with Euclidean space to become familiar with hyperbolic space. This is an extrinsic process, and, as such, a rich source of future confusion. When we are near the center (0,0) of the Poincar6 Disk, we note that the Euclidean length of a short Euclidean line segment is roughly the same as its hyperbolic length (pretend that factor of 89in ds n isn't there --it's just needed to make some of the later calculations come out nicely). As we near the boundary of the disk, the infinitesimal hyperbolic metric blows up so that a short Euclidean line segment can become very long in the hyperbolic sense. But we will see later that to the inhabitants of the Poincar6 Disk all points locally look the same. A hyperbolic straight line segment is the shortest path, measured hyperbolically, between two points. A hyperbolic straight line, or geodesic, can be defined similarly. An understanding of the geodesics in the hyperbolic plane is the first step t o w a r d s understanding the hyperbolic plane in general. The construction of all geodesics will be carried out in the next six paragraphs. We begin by finding the geodesic segment running from (0,0) to (0,p). Consider the curve o~(t) = (0,t) = (x(t),y(t)) for 0 ~< t ~< p < 1. This has Euclidean length p, but its hyperbolic length r is computed as
r = I(o0 =
P X/(dx/dt) + (dy/dt) 2 ~ ~-f---~-~ dt
~(1
= 2 ~ k / -. .0. .+. 1 -
(x + y ))
1 dt = 2 fo"
t2
dt 1 -
t2
- In (1 + P/ . \1 - p/
Using straightforward calculations, it can be shown that this is the shortest path, in terms of hyperbolic length, from (0,0) to (0,p). Thus, the hyperbolic distance from (0,0) to (0,p) is r = ln(tl_--~p). In particular as p--~ 1, r--* o0. Similarly, the shortest hyperbolic path from (0,0) to (p cos 0, p sin 0) is the Euclidean straight line path, ~/(t). This can be seen either by doing the same calcula-
Figure 8: Hyperbolic geodesics through (0,0) in the Poincar4 D i s k m o d e l o f the hyperbolic plane. Recall that the Poincar~ D i s k is the interior of the u n i t disk.
tions as above, using polar coordinates, or by rotating the calculations back to the y-axis calculations alluded to above. In the latter case, if p(x,y) is a rotation about (0,0) of the unit disk, we need to compare l(7(t)) with l(p(~/(t))). We see that ds H is unaffected by p, because ds E and (1 - (x2 + y2)) are both unaffected by p. Thus, we can calculate I(7) by rotating ~/(t) to a path from (0,0) to (0,p). We n o w k n o w all hyperbolic geodesics through (0,0)--they are simply the Euclidean straight lines. See Figure 8. To find the shortest hyperbolic path from ( x v y l ) to (x2,Y2) w e would like to mimic the rotation trick, because comparing the lengths of all paths between two points would be a horror. We need to find a transformation of the Poincar6 Disk to itself taking (xl,yl) to (0,0), which does not affect d i s t a n c e s - - t h a t is, the transformation leaves ds n invariant. The desired transformations are the orientation-preserving MObius transformations of the extended complex plane that take the unit disk to itself. The general M6bius transformation takes the complex number z = x + i y t o c Taz+b ; - a, where a, b, c, d E C andad - bc = 1. To ensure that the unit circle is mapped to itself we have to restrict to z ~ az+r cz+---Xwhere tal2 - Icl2 = 1. We will not prove that MObius transformations preserve d s n - - a good source to consult is [9], Sections 5.3 to 5.5, although this text uses the upper-half-space model of hyperbolic space. We n o w exploit the MObius transformations. There exists a MObius transformation of the Poincar6 Disk to itself that takes (xl,yt) to (0,0). Because such a transformation preserves ds H, w e see that the geodesics through (0,0) yield the geodesics through (xl,y~). MObius transformations take circles and lines to circles and lines. Further, MObius transformations are conf o r m a l - - t h e y preserve angles. Thus, since the geodesics through (0,0) are Euclidean straight lines perpendicular to the unit circle, we see that the geodesics through (xi,yi) are circles perpendicular to the unit circle. Conversely, all Euclidean circles (or lines) perpendicular to the unit circle represent geodesics. See Figure 9. Keep in mind that the Poincar6 Disk is the THE MATHEMATICAL 1NTELLIGENCER VOL. 14, NO. 1, 1992
41
interior of the unit disk, and therefore the unit circle is not actually a part of the Poincar6 Disk.
Figure 9: Euclidean circles perpendicular to the unit disk represent hyperbolic geodesics in the Poincar~ Disk. Note that we have drawn three hyperbolic geodesics parallel to the geodesic through (0,0). The fact that two of these parallels intersect s h o w s that the Euclidean Parallel Postulate does not hold.
C) I
a
b
Figure 10: The ingredients for showing that the 2-holed torus admits a hyperbolic structure; a: This topological identification of an octagon yields a 2-holed toms; b: A regular octagon in the hyperbolic plane; c: Another regular octagon, but this one has very small angles; d: A hyperbolic regular octagon whose angles are very close to those of a Euclidean regular octagon.
Figure 11: A neighborh o o d of an edge under hyperbolic gluing.
42
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
We now have a complete understanding of geodesics in the hyperbolic plane. The hyperbolic parallel postulate is satisfied (see Figure 9), and the other axioms for hyperbolic geometry can be verified. Because MObius transformations preserve the hyperbolic metric, all points look locally the same metrically. Hence, the hyperbolic plane is homogeneous. Also, hyperbolic angles in the Poincar~ Disk model are simply the underlying Euclidean angles. If you lived in the hyperbolic plane, how would you know it? There are several approaches, but checking parallelness, apparently an infinite condition, is not the best choice. Here is the most basic approach. Simply construct a circle of radius r and measure its circumference. It is not hard to compute that if you are in the hyperbolic plane, then the circumference will be less than the Euclidean circumference of 2~rr. Of course, you must construct a large enough circle, or make extremely accurate measurements, to discern hyperbolicity. S e c t i o n 4: H y p e r b o l i c
2-Manifolds
A hyperbolic 2-manifold is a 2-manifold with a metric that is locally isometric to the hyperbolic plane. That is, at each point, its metrical properties in a small neighborhood are exactly those of the hyperbolic plane. I claim that a 2-holed torus admits a hyperbolic structure. A 2-holed torus can be obtained topologically by identifying the edges of an octagon (see Figure 10a, and check the Euler characteristic). Because all the vertices are identified, the identification cannot be carded out Euclideanly--there is too much angle at the vertex. The question now is w h e t h e r we can take an octagon in hyperbolic space and identify the edges via hyperbolic isometries to yield a hyperbolic 2-holed toms. Consider a hyperbolic circle of hyperbolic radius r centered at (0,0) in the Poincar6 Disk model (it is a Euclidean circle as well). Take eight evenly spaced points on this circle and connect them in series by eight geodesics segments. See Figure 10b. There are disk-preserving MObius transformations taking any geodesic edge to another. As such there are hyperbolic isometries that realize the combinatorial gluing of the octagon. We need to check whether all points in the glued octagon are locally hyperbolic. Interior points work trivially. Edge points work easily too, though from the Euclidean view this is surprising. From the hyperbolic view it is not surprising. See Figure 11. The only place where a problem can occur is at the (one) vertex. We need to have angle sum 2~ at the vertex. Because we have 8 equal angles contributing we need each angle to be ~E = ~ radians. Note that if r is very large then the angles are very close to 0 (see
Figure 12: We can understand what a neighborhood of an edge point looks like by taking a slice transverse to the edge. 3-dimensional filler has been removed from the pictures to ensure better viewing. Figure 12c is the transverse slice.
Figure 10c); while if r is very small, the octagon is almost Euclidean and the angles are all very close to 8~-2~r = ~ radians (see Figure 10d). By continuity, there is a radius r somewhere in between that has octagon angles of ~/4 radians. This octagon together with the hyperbolic identifications yields a hyperbolic 2-holed torus. The above technique works similarly for all n-holed toil where n > 1. For example, when n = 3 we can construct the 3-holed torus from a suitably glued 12gon. Again, all vertices will be identified: V - E + F = 1 - ( 6 + 9 ) + 10 = 1 - 1 5 + 1 0 = - 4 . Symmetrically centering the 12-gon at (0,0), we vary r until 12 angles of 2,, = ~r/6 radians are obtained. In summary, the sphere admits a spherical structure, the torus admits a Euclidean structure, and all other closed, orientable 2-manifolds admit hyperbolic structures. Is it possible that these 2-manifolds admit other structures as well? For example, could a 3-holed torus admit a Euclidean structure? The answer is no to the first question, and hence the second question, and the p r o o f is a g e m i n v o l v i n g the G a u s s - B o n n e t Theorem (see, for example [22], page 176). S e c t i o n 5: 3 - D i m e n s i o n a l
Manifolds---Topology
large collection of 2-manifolds, and this collection actually satisfied (1). Our ability to visualize 3-dimensional manifolds is more limited; the ability we sometimes have in the 2-dimensional case to leave our manifold and look down upon it is severely limited. We begin by describing three approaches to constructing 3-manifolds: (1) The Combinatorial Approach. We can take a (solid) polyhedron and glue it up face to face. Of course, to have a chance of getting a closed 3-manifold we need to have an even number of faces, and if two faces are to be glued then they must have the same number of edges. Further, the gluings can't all take place in 3space, so we are better off thinking in terms of abstractly identifying faces rather than physically gluing them. For simplicity, we will not consider gluing a polyhedron, but instead we will take some number of (solid) tetrahedra and identify them face to face. Does such a 3-complex yield a 3-manifold? We need to check that all points in the identification space locally look like R 3. There are four types of points to consider:
i. Interior points: These are easy. ii. Face points: Also e a s y - - j u s t glue together two half balls.
Once again we are in the realm of topology and two objects are the same if they are homeomorphic. As in the 2-dimensional case we will make heavy use of identifications--these are simply homeomorphisms. Our goal is to understand 3-dimensional manifolds-spaces that locally look like R3. For convenience we will assume, unless otherwise stated, that our 3-manifolds are orientable. We want to classify 3-manifolds and be able to recognize them. That is, we want: (1) A list, with no repetitions, of all 3-manifolds, and (2) A usable means of determining where a given 3-manifold fits on the list. As in the 2-dimensional case, (2) is s u b t l e - - a 3-manifold may be given to us in a confusing way. It is not at all clear where to begin. In the 2-dimensional case we were immediately able to visualize a
iii. Edge points: Edge points in our tetrahedra have little wedge n e i g h b o r h o o d s before gluing. Gluing identifies each open face of a wedge to another face. If we look at a slice transverse to the edge, then we get the same picture as occurred for vertices in the 2-dimensional case. Since this 2-dimensional slice is a disk, the wedges form a solid ball in R3. See Figure 12, where the 2-dimensional slice is in Figure 12c. iv. Vertex points: Vertex points in our tetrahedra have little tetrahedral neighborhoods before gluing. Actually, it is better to think of these n e i g h b o r h o o d s as p y r a m i d s on triangles, where the apex of the pyramid is the vertex in question. Gluing occurs along the non-base faces of the pyramids, and after carrying out THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
43
Figure 13: We can understand what a neighborhood of a vertex looks like by studying base triangles. The left-hand picture only shows three base triangles and their pyramids. The dots in Figure 13b indicate that more base triangles may be glued on.
Figure 14: A complicated rendering of a simple idea--two disks glued along their boundaries produce a 2-sphere.
the gluing ( a b s t r a c t l y - - w e m a y not be able to carry out the gluing in 3-space), w e see that the e x p o s e d base triangles f o r m a closed 2manifold (see Figure 13). It is quite possible that this 2-manifold is not a sphere! In fact, it is a s p h e r e if a n d only if the associated vertex in the identified 3-complex locally looks like R 3. H e n c e , if there is a sphere at each of the vertices t h e n w e have a 3-manifold, and conversely. C o m p a r e this with the 2-dimensional case, w h e r e the fact that the o n l y closed 1manifold is a circle enables us to conclude that vertices in the identified complex locally look like R 2. H o w can we tell w h a t the 2-manifold obtained by gluing these base triangles is? By using the Euler characteristic. The n u m b e r s of vertices, edges, and faces (V2,E2,F2) in the base triangles s u m m e d over all v e r tices in the 3-complex are related to the n u m b e r s of edges, faces, a n d tetrahedra (E3,F3,T3) in the 3-complex: V2 = 2E 3, E 2 = 3 F 3, F 2 = 4 T 3. S o V 2 - E 2 + F 2 = 2E 3 - 3F3 + 4T3; b u t F 3 = 4T3/2 = 2T 3, a n d we have V2 - E2 + F2 = 2E 3 - 2F 3 + 2T3. Each n e i g h b o r h o o d of the V3 vertices locally looks like a ball if and only if V2 - E2 + F 2 = 2V3, which h a p p e n s if a n d only if 2E 3 44
THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992
- - 2 F 3 q- 2T3 = 2V3. Thus, o u r 3-complex is a 3-manifold if a n d only if its Euler characteristic V3 - E3 + F3 - T3 is equal to zero! A t h e o r e m of Moise states that all 3-manifolds can be c o n s t r u c t e d b y g l u i n g t e t r a h e d r a . If t w o s u c h gluings p r o d u c e 3-manifolds, t h e n h o w can w e tell w h e t h e r or not t h e y are the s a m e 3-manifold? T h e Euler characteristic will not help us because the above a r g u m e n t shows that it is always zero for closed 3manifolds. At this stage, we have little intuitive feel for 3-manifolds. T h e combinatorial c o n s t r u c t i o n t e c h n i q u e is easily u n d e r s t o o d , but n o t really visualizable. H o w can w e construct closed 3-manifolds in a more intuitive m a n n e r ? (2) The Heegaard Gluing Approach. Take two unit 3balls i n R 3,(x 2 + y 2 + z 2 ~ l ) a n d ( ( x _ 3)2 + y2 + z 2 1), a n d glue t h e m together along their b o u n d a r i e s by (x,y,z) ~-~ (3 - x,y,z). This p r o d u c e s the 3-sphere S3. See Figure 14 for the 2-dimensional version. The 2-dimensional version of this process can be visualized as taking place in R a to p r o d u c e the 2-sphere x2 + y2 + z 2 = 1. The visualization process breaks d o w n in the 3-dimensional version, as w e w o u l d need a n o t h e r d i m e n s i o n to p r o d u c e the 3-sphere x2 J r y2 + z2 + w 2 = l i n R 4. Of course, the process a l m o s t works. H o w close could w e c o m e to gluing the t w o 2-balls (disks) in Figure 14a if we restricted ourselves to working in R27 If w e r e m o v e d the center of the s e c o n d 2-ball a n d sliced it o p e n as in Figure 15, w e could do it. We n e e d to a d d the r e m o v e d center point. This can either be d o n e concretely by going to R 3, or abstractly by a d d i n g a point at ~ to R 2. The two cases can be related b y stereographically projecting S2 in R 3 to R 2 U {o0}. The s a m e ideas w o r k for the 3-sphere: it can be t h o u g h t of as R 3 U {o~}, a n d this can be t h o u g h t of as the stereographic image of S3 in R 4. Note: W h a t h a p p e n s if w e r e m o v e a small neighborh o o d of the equator in each of o u r two balls in R37 This a m o u n t s to r e m o v i n g a solid torus from S3 = R 3 U {oo}. What remains? Two dumbbells glued together along their h e a d s and bottoms, w h i c h p r o d u c e s a solid torus (see Figure 16). Thus, S3 can be gotten by gluing together t w o solid tori, and conversely, if w e r e m o v e the usual boring solid torus f r o m S3 = R 3 U {~} w e get a solid torus. G e t t i n g back to o u r g l u i n g of t w o 3-balls, w h a t h a p p e n s if w e glue the two 3-balls together in a different way? That is, w h a t effect does changing the hom e o m o r p h i s m of the two b o u n d i n g 2-spheres have? It is not too h a r d to show that the resultant 3-manifold is once again the 3-sphere (see [18], page 10). After the case of gluing t w o 3-balls together, the next m o s t complicated case is that of gluing two solid tori along their b o u n d a r y tori. In Figure 16 we saw a scheme that p r o d u c e d S3. W h a t h a p p e n s if we place
Figure 15: Carrying out the g l u i n g in Figure 14 w h i l e restricting to R" doesn't quite work, but it a l m o s t does. In b through d, pay attention to the broken curves---they correspond; a: After r e m o v i n g the center of the second disk, we can n o w c u t . . , b: stretch . . . ; c: stretch (especially in a neighborhood of the removed point) . . . ; d: stretch . . . and exactly re-glue. We get all of R 2, but w e o n l y draw a finite piece. Putting back the removed p o i n t produces a point at infinity.
Figure 16: R e m o v i n g a solid t o m s boringly situated in S3 = R3 U 0o; a: Here we remove neighborhoods of the equator in two 3-balls. G l u i n g - - i n the required f a s h i o n - - t h e removed portions produces a solid toms; b: H a v i n g removed the equatorial neighborhoods, w e can start stretching before w e carry out the g l u i n g ( w h i c h is determined by our u s u a l g l u i n g of two 3-balls). The arrows indicate the gluing; c: After g l u i n g we get a solid toms.
two solid tori side by side in R 3 and glue together analogously to the ball case alluded to in Figure 14a? Is this S3 again? Our inclination would be to use the Euler characteristic, but this would prove futile. In general, we can glue two solid n-holed tori together and we can do so in many ways. It is a standard fact that all 3-manifolds can be constructed by this process (see [18], Section 9C). When are two such 3-manifolds homeomorphic? The Heegaard gluing approach gives us more of a feel for 3-manifolds, but the recognition problem still seems hopeless. (3) The Dehn Surgery Approach. This is m y favorite way of constructing 3-manifolds. However, when I explain it to mathematicians they don't believe it the first few times around. So, let's start slowly with the simplest examples. Figure 17: The set-up for doing D e h n surgery on the stanRemove from R 3 U {oo} = S3 the solid torus depicted dard solid t o m s in S3 = R3 O oo; a: We've removed the stanin Figure 16a. According to Figure 16, what remains is dard solid t o m s from S 3 = R 3 U o% and marked two curves also a solid torus. Glue the removed torus back in ex- on the boundary; b: Here's the removed solid toms. We've actly the way it was taken out. The manifold resulting designated a meridian disk. from this gluing is S3. This is the simplest Dehn surgery. the removal of this solid torus and call the curve laRemove the same solid torus. We will soon glue it belled as tx in Figure 17a the meridian and call the back in a more complicated way. Call the shaded disk curve labelled as h the longitude. (A cartographer in Figure 17b the meridian disk of the removed solid would probably call h a latitude, but longitude is estorus. Now look at the torus boundary left in S3 after tablished in the mathematical literature.) THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
45
Figure 18: Gluing back the removed solid torus in a funny way; a: Gluing the boundary of the meridian disk to the designated curve, b: N o w , w e work our w a y out from the boundary of the meridian disk.
Figure 19: A more complicated curve with which to start our D e h n surgery gluing. The lighter shaded portion of the curve is on the back side of the torus.
Begin gluing by identifying the boundary ~/ of the meridian disk in the r e m o v e d solid torus with the curve labelled p~ + h in Figure 18a. Continue gluing by identifying the little rectangles based on ~/as in Figure 18b with the little rectangles based on p, + h. If continued properly, this will determine an identification of the two boundary tori, and hence a perfectly legitimate 3-dimensional manifold. This gluing of solid tori cannot be carried out in R 3. Is the resultant 3-manifold the 3-sphere? The answer to this question is not obv i o u s - i t ' s difficult to grapple with an abstract identification. If we replace the curve ~ + ~ by a curve, denoted p~, + qM which wraps around the torus p times in the p, direction and q times in the ~ direction (see Figure 19 for the case p = 3 and q = 2), then we again get a 3-manifold, as long as p and q are relatively prime integers. We could also do various perturbations of these gluings, but we would only get 3-manifolds that had already been constructed for some p and q (see [18] Sections 9F and 9G). Let me n o t e - - w i t h o u t p r o o f - that if we've decided that ~/will be sent to p~ + qM then no matter h o w the gluing is extended over the rest of the boundary torus, the resultant 3-manifold is the same. We can complicate matters b y removing a solid torus neighborhood of an arbitrary knotted circle, K, from S3 and then obtain n e w 3-manifolds by gluing back the solid torus along the boundary in various ways. This process is called Dehn surgery on a knot in the 3-sphere. In Figure 20, the figure-eight knot is depicted. If we choose a meridian p, and a longitude k for our knot (see [18], pages 31 to 32), then we can parametrize the possible Dehn surgeries by relatively prime pairs of integers (p,q). The manifold obtained by performing (p,q) surgery on the knot K in S3 will be denoted (S3 - K ) ( p , q ) . A link is a collection of disjoint knots. Dehn surgery can be performed on a link one component at a time. The following surprising theorem (see [18], Section 9I for a proof) gives us a method for obtaining all orientable 3-manifolds. THEOREM (Lickorish, Wallace): All closed, orientable 3manifolds can be obtained by performing Dehn surgery on links in the 3-sphere.
Figure 20: The figure-eight knot is depicted w i t h the meridian and part of the longitude drawn. 46
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
We are faced now with the usual difficulties. When are two such manifolds the same? When are they hom e o m o r p h i c to the 3-sphere? These questions are hard. Dehn surgery and the Heegaard gluing approach have a similar visual appeal, but the Dehn surgery approach is superior because the complicatedness of the construction is spread more evenly between 3-dimen-
sional problems (what the knot or link is) and 2-dimensional problems (the gluing). In the Heegaard gluing approach, all of the complicatedness resides in the homeomorphism identifying the boundaries of the solid n-holed tori. Section 6: G o o d Invariants for 3 - M a n i f o l d s
We have several m e t h o d s for constructing 3-manifolds. What we lack are m e t h o d s for determining when two 3-manifolds are the same. In the 2-dimensional case we were faced with a similar problem and we were able to overcome it by using a single invariant--the Euler characteristic. This invariant assigned a number to each closed, orientable 2-manifold, and the assigned number enabled us to answer our basic questions about 2-manifolds. Specifically, two closed orientable 2-manifolds are homeomorphic if and only if they have the same Euler characteristic. In addition, the Euler characteristic is straightforward to compute (triangulate and compute V - E + F), and easy to use (compare integers). What we need in our study of 3-manifolds are good invariants, like the Euler characteristic for 2-manifolds. A good invariant is computable (mathematicians are able to compute it in reasonable cases, perhaps with the aid of a computer) and informative (it provides useful information; for example, it enables us to distinguish many 3-manifolds from each other). There are three standard invariants used in the topological study of manifolds: the Euler characteristic, the fundamental group, and homology (and cohomology) groups. We will not spend much time on these invariants, so little knowledge of them is needed. A brief discussion of their goodness follows. The Euler characteristic for 3-manifolds is not good because it is not informative--all closed 3-manifolds have Euler characteristic 0. It is, however, extremely computable. The fundamental group of a 3-manifold contains lots of information and is essentially straightforward to compute, but it cannot quite be considered a good invariant. The problem is that the information about the
3-manifold is hard to drag out of the group. This is because groups are hard to work with. For example, given two group presentations (a description of the group in terms of generators and relations), are they presentations of the same group or of different groups (see [11], page 106)? Homology groups are often able to distinguish between 3-manifolds, and they are relatively easy to compute. As such, they are good. However, there are broad classes of examples in which they fail to distinguish b e t w e e n 3-manifolds. For example, all knot complements in the 3-sphere have the same homology groups. Further, the homology groups of ( S 3 - K)(p,q) and (S3 - K)0~,,q,) are the same if p = p'. In particular, homology does not distinguish (S3 - K)(1,q) from the 3-sphere (S3 - /<)(1,0). So, homology is good, but not good enough. We need more good invariants, but the topological approach is running out of steam. Section 7: H y p e r b o l i c 3 - M a n i f o l d s
The Euler characteristic told us more about 2-manifolds than we had a right to know. As such, the introduction of geometry seemed s u p e r f l u o u s - - a t least from the viewpoint of the 2-manifold goals. The 3-dimensional situation is completely different. We will use geometry to help us understand 3-manifolds topologically. All closed 2-manifolds can be given geometric structures. That is, closed 2-manifolds admit metrics that are locally isometric to either the sphere, the Euclidean plane, or the hyperbolic plane. In fact, all but two of the closed, orientable 2-manifolds possess hyperbolic structures. The non-hyperbolic 2 - m a n i f o l d s - - t h e sphere and the t o m s - - a r e topologically simpler (have fewer holes) than the hyperbolic 2-manifolds. The 3-dimensional situation is similar. A hyperbolic 3-manifold is a 3-manifold with a metric locally isometric to hyperbolic 3-space. The work of Bill Thurston indicates that a typical 3-manifold is either topologically simple or possesses a hyperbolic structure. (Precise statements of Thurston's theorems and conjectures can be found in [20], Section 2.) For example, a knot in the 3-sphere has a hyperbolic c o m p l e m e n t - in which case we call it a hyperbolic k n o t - - i f and only if it is neither a torus knot nor a satellite knot (these terms will not be used again). Further, all but a finite number of the 3-manifolds obtained by performing Dehn surgery on a given hyperbolic knot possess hyperbolic structures. The computer evidence indicates that this finite number of non-hyperbolic surgeries is small (see [21], pages 9 to 15). A 3-manifold must be topologically complicated to admit a hyperbolic structure. As such, it is not surprising that the discovery of (closed) hyperbolic 3manifolds was a slow process. LObell constructed THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
47
the hyperbolic edge conditions are satisfied for the dodecahedron. The fact that our one vertex is geometrically nice follows for free, and we have constructed the SeifertWeber dodecahedral space. 9 The upshot of the work of Thurston is that hyperbolic 3-manifolds are the most abundant, the most complicated, and the most important class of 3-manifolds. As such, one approach to studying topological 3-manifolds is to restrict attention to 3-manifolds that admit hyperbolic structures, and to use the hyperbolic structure to s t u d y the 3-manifold. The following theorem of Dan Mostow is more than crucial. Figure 21: The Seifert-Weber dodecahedral space is gotten by gluing opposite faces via a ~-~(02~) clockwise twist. This is a non-ambiguous description--think about it.
some hyperbolic 3-manifolds in 1931. An easily described hyperbolic 3-manifold was discovered by Seifert and Weber in 1933 (see the example following this paragraph). In 1971, Alan Best found several other hyperbolic 3-manifolds. Shortly thereafter, Bob Riley put hyperbolic structures on several knot and link complements (not closed), and Troels J0rgensen constructed hyperbolic 3-manifolds that fiber over the circle. The Riley and J~rgensen examples were major motivations for Thurston's work. It should be noted that this brief account of the history is biased towards the geometric approach and away from the algebraic approach. Example: The Seifert-Weber dodecahedral space is a hyperbolic 3-manifold. We obtain this manifold topologically by taking a regular dodecahedron (12 pentagonal faces, 3 at each vertex) and identifying opposite faces via a i~(02"rr) clockwise twist. See Figure 21. For this to be a topological 3-manifold we must have the Euler characteristic equal to zero. By carrying out the gluing we see that our 30 edges are identified to 6 edges, and our 20 vertices are identified to 1 vertex; thusV- E + F- C = 1 - 6 + 6 - 1 = 0andwe have a topological 3-manifold. Can this manifold be constructed by isometrically identifying the faces of a regular dodecahedron in hyperbolic space? We need to check whether all points locally look like hyperbolic space. Interior points and face points of the dodecahedron will pose no problem, but edge points require some analysis. Each edge in the manifold comes from 5 identified edges in the dodecahedron. Thus each dihedral angle at an edge of the dodecahedron must be -~ radians. A regular dodecahedron in Euclidean space has dihedral angles of approximately 116.57 degrees. By the same technique as we used on the octagon in Section 4, we can construct regular dodecahedra in hyperbolic space having dihedral angles anywhere between zero degrees and approximately 116.57 degrees. In particular, 48
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
THEOREM: If a closed, orientable 3-manifold possesses a hyperbolic structure, then that structure is unique (up to isometry). Now, we can use the hyperbolic structure on a hyperbolic 3-manifold to define invariants for the 3-manifold. Mostow's theorem tells us that for closed, orientable 3-manifolds, these hyperbolic invariants are topological invariants for the underlying 3-manifolds. Three of the most natural invariants that come from the hyperbolic structure are the volume, the ChernSimons invariant, and the "q invariant. Of these, the volume is the most natural and will be discussed in detail in Section 8. The Chern-Simons and "q invariants will be mentioned briefly in Section 9. S e c t i o n 8: T h e V o l u m e
Is the volume of a hyperbolic 3-manifold computable? Let's consider the analogous 2-dimensional question first. The Poincar6 Disk model of the hyperbolic plane has infinitesimal metric ds = x/(ax)2+ (ay)2 . An infini(1 - (x2 + y2))/2 tesimal line segment parallel to the x-axis at the point (x,y) has hyperbohc length (1 (x~+ "))/2' similarly, such a segment parallel to the y-axis has hyperbolic length 92a.y_. Couvling these, an infinitesimal rectangle ll-(x~ +Y"~)/2" ,. ~ axay . . . . . . li nas nyperDo~lc a r e a ( l _ ( x 2 + y 2 ) ) 2 / 4 . l n u s , me nyperDo c area of a region in the Poincar6 Disk is computed by integrating the area form dA = (1-( x~X+d~2~, y/, 2/4over the region. The area of a hyperbolic triangle with angles ~x, B, and y is computed in this way and turns out to be "rr - (oL + B + y). Given this formula for the area of a hyperbolic triangle it is easy to compute the area of a hyperbolic surface S--simply decompose it into geodesic triangles, compute -rr - (o~ + [3 + y) for each, and sum over the triangles. The area turns out to be - 2,rrx(S). The game plan in three dimensions is much the same. The Poincar6 Ball model of hyperbolic 3-space -
dx
9
9
V ( d x ) 2 + (dy)2 + (dz) 2
has infinitesimal metric ds = ~ c ~-x~ ~ ~ ~;-z2)-~ , and dxdydz volume form d V = (l_(x2+y2+z2~)3/s. The volume of a
q = 11 q = 10 q=9 q=8 q=7 q=6 q=5 q=4 q=3 q=2 q=l
2.0065 2.0016 1.9950 1.9858 1.9725 1.9521 1.9186 1.8581 1.7320 1.3985 x.xxxx
2.0066 2.0017
x.xxxx
1.4407 x.xxxx
p=l
p=2
p=3
1.9727 1.9195
1.9862 1.9732 1.9120 1.8634
1.7371
~/'~
1.9738
1.9745 1.9557
1.9231
x.xxxx p--4
p=5
1.7571
1.9754 1.9287
1.8735 1.7714 1.5295 0.9814
I\
1.8243
1.2845
1.9321 1.8871 1.8058 1.6496 1.4638
1.5832
1.7521 1.6678
p=6
p=7
p=8
p=9
1.9027
Figure 22: Estimates of volumes of some hyperbolic 3-manifolds obtained by performing (p,q) Dehn surgery on the figureeight knot in the 3-sphere. For example, the volume of the hyperbolic 3-manifold obtained from (5,3) surgery is approximately 1.7714. The x.xxxx indicates that the surgery in question does not produce a hyperbolic 3-manifold. The first quadrant is all we need to fill in, because (p,q), ( - p , -q), (-p,q), and (p, - q ) surgeries all produce the same manifold (this is not hard to show).
hyperbolic tetrahedron can be computed by integration, and we get a formula for the volume of the tetrah e d r o n in terms of its dihedral angles (see [19], Chapter 7). As such, the problem of computing the volume of a hyperbolic 3-manifold reduces to the problem of decomposing it into hyperbolic tetrahedra. On the face of it, this seems very difficult, but Jeff Weeks has developed a computer program that will carry out this decomposition in an enormous range of cases (see [2]). The Weeks computer program takes a knot or link L in the 3-sphere; topologically decomposes S3 - L into tetrahedra; simplifies the tetrahedral decomposition via a series of moves; and solves by Newton's method the resulting edge-angle equations needed for hyperbolicity (compare with the Seifert-Weber example in Section 7). The solutions give the hyperbolic structure if S3 - L is hyperbolic. The program works very well in practice--the memory and time available are the only constraints--and we thereby get a decomposition of a hyperbolic knot or link complement into hyperbolic tetrahedra. As such, the volumes of hyperbolic knot and link complements in the 3-sphere are easily computable. The Weeks program is also good at calculating the volumes of 3-manifolds obtained by performing hyperbolic Dehn surgeries on hyperbolic knots and links in the 3-sphere. This involves solving a perturbation of the knot (or link) equations prior to the surgery. There are some minor problems that may arise, but in general the Weeks program makes the volume easy to compute when a description of the hyperbolic 3-manifold as Dehn surgery on some knot or link is given. The theorem of Lickorish and Wallace quoted at the end of Section 5 says that such a surgery description always exists. However, the surgery description may be hard to find: the Seifert-Weber dodecahedral space
is such an example. Fortunately, in the Seifert-Weber case, we can decompose it into hyperbolic tetrahedra by hand and the volume turns out to be approximately 11.199. Is the volume an informative invariant for hyperbolic 3-manifolds? I believe the answer is yes, for the simple reason that it is proving effective in distinguishing between 3-manifolds. For example, Figure 22 lists the volumes, to four decimal places, of some hyperbolic 3-manifolds obtained b y performing Dehn surgery on the figure-eight knot in the 3-sphere. Each relatively prime pair of integers corresponds to a 3manifold and if
(p,q) ~
{(+__4,___1), (---3,---1), (_+2,___1), (___1,+1), (0, _+1), ( _+1,0)},
then the associated manifold has a hyperbolic structure (see Sections 4.6 and 4.7 of [19]), and we denote it by M(p,q). By Mostow's theorem, different volumes imply non-homeomorphic 3-manifolds. Thus, for example, M(7,1), M(7,2 ), M(7,3 ), M(7,4 ), M(7,5) are not homeomorphic, even though they all have the same homology. The volume is not a complete invariant for hyperbolic 3-manifolds; that is, there are examples of nonh o m e o m o r p h i c hyperbolic 3-manifolds with equal volumes. One such example will be given later. The available computer evidence shows the volume to be very useful at distinguishing between 3-manifolds. Much information is being discovered about the volume function, and we present some of this information here. (i) Patterns abound in Figure 22. For example, it appears that the volumes increase as (p,q) gets large (p2 + q2 ~ ~). Do they approach a limit? The answer is yes, and, in fact, the behavior of the M(p,q) as (p,q) gets THE MATHEMATICALINTELLIGENCERVOL. 14, NO. I, 1992 49
and of finite volume, and Mostow's theorem still applies. It is shown in [19], Sections 5.11 and 6.5, that the volumes of the hyperbolic 3-manifolds obtained by performing surgery on a cusped hyperbolic 3-manifold approach (from below) the volume of the original J manifold as the surgery co-efficients get large. It is sometimes said that the limiting cusped manifold is obtained by performing "infinite" Dehn surgery on a the original manifold. For example, the cusped hyperb o bolic structure on the figure-eight knot, K, in the 3o o % sphere is often denoted (S3 - K)~. In studying closed hyperbolic 3-manifolds we are inexorably led to the study of cusped hyperbolic 3-manifolds. (i.i) The volume appears to be a good measure of the complicatedness of a knot or link. Thurston gives many examples of hyperbolic knots and links in S3 and observes that the volumes of their complements are closely related to a naive notion of how complicated the links are (see [19], pages 6.33 to 6.48). Further evidence is given by joining Bruce Ramsay's c d random-knot generator to the Weeks hyperbolic structures c o m p u t e r program. Ramsay's program uses Figure 23: A s o m e w h a t m i s l e a d i n g d e p i c t i o n of w h a t Fourier series to r a n d o m l y construct knots. If the happens to the core curve of the sutured-in solid torus as w e Ramsay program is set to low complexity, and the redo higher and higher order D e h n surgeries; a: A hyperbolic 3-manifold depicted as a blob. The sutured-in solid torus is sultant knots are plugged into the Weeks program, marked by an arrow, and the core geodesic of that solid then the associated volumes are small. But the knots torus is depicted as a broken curve; b: Fasten d o w n the do not necessarily have low crossing numbers (an apblobish 3-manifold firmly, grab the core geodesic (which is n o w a solid curve) and start pulling it out. (To do this geo- pealing, yet flawed, measure of complicatedness of metrically, the surgeries w o u l d have to be changed. That is, knots). For more on this material see [21], pages 6 to 8. the solid torus w o u l d end up being re-seated.); c: Continue Finally, S. V. Matveev and A. T. Fomenko, in [12], pulling on the core g e o d e s i c . . . ; d : . . . until it disappears. present evidence that volume is related to complicatedness. (iii) Figure 24 lists volumes, to four decimal places, large is well understood (see [17]). A qualitative depiction is given in Figure 23. The core of the sutured-in of hyperbolic 3-manifolds obtained by performing solid tube is a geodesic, and as (p,q) gets larger and Dehn surgery on one component of the left-handed larger this geodesic gets shorter and shorter until, in Whitehead link. The second component is cusped, the limit, it disappears. Further, the solid tube around that is, an "infinite" surgery is performed. There is this short geodesic gets bigger and bigger as the geo- some surprising symmetry in this set of volumes--2desic gets shorter and shorter. The region outside of fold symmetry would be expected, but the symmetry the solid tube stays pretty much the same for large group is actually the octic group. It can be proved that volumes that look the same (that is, they are the same (P,q). In the limit, the core geodesic is pulled infinitely far to four decimal places) are exactly the same, but it away from the "belly" of the manifold and pinched turns out that not all of these equal volume 3-manioff, so that what is left is a cusp (which is topologically folds are homeomorphic. For example, WL(1,I ) is not just (Torus) x [0,oo)) affixed to the belly. The limiting homeomorphic to WL(5 ,_ 1), but they both have volume manifold, made up of a belly and a cusp, is a perfectly 2.02988 . . . (the former manifold is the figure-eight legitimate hyperbolic 3-manifold. When we say that knot complement in the 3-sphere). Hence, volume is the figure-eight knot complement in the 3-sphere pos- not a complete invariant for hyperbolic 3-manifolds (as sesses a hyperbolic structure, we are referring to such was known long before these examples). How do we know WL(1,1) and WL(5 _1) are not hoa limiting cusped structure. This description of the limiting behavior of 3-mani- meomorphic? In this case it can be computed that their folds that have undergone Dehn surgery is completely homology groups are different. There are also proofs general. It applies to surgeries on all hyperbolic links using geometric invariants. The equal volume maniin the 3-sphere. The limiting hyperbolic structure is folds WL(4,I) and WL(4,_3) are considerably more diffinot closed, but it is complete (see Section 3.7 of [19]) cult to distinguish (see [8]). 50
THE M A T H E M A T I C A L INTELL1GENCER VOL. 14, N O . 1, 1992
pT ~ 39
3.4272
39
3.4935
3.5178
39
39
3.1640
3.2759
3.3567
39
39
3.4940
x.xxxx
29
29
29
39
3.2969
39
x.xxxx
x.xxxx
x.xxxx .___ 1.7778
2.5374
29
39
x.xxxx
x.xxxx
x.xxxx
x.xxxx
x.xxxx
2.0299
2.6667
3.0115
2.8281
2.6667
2.5690
2.5374
2.5690
2.6667
3.3831
3.3317
3.2759
3.2218
3.1773
3.1485
3.1386
39
I I
I
I
I
q=0
Figure 24: Estimates of volumes of hyperbolic 3-manifolds obtained by performing (p,q) Dehn surgery on one component of the left-handed Whitehead link in the 3-sphere. The other component is cusped. Volumes for surgeries with p < 0 can be filled in by noting that, for trivial reasons, (p,q) and ( - p , - q) surgeries yield the same 3-manifold. The pattern of volumes displays a surprising amount of symmetry. To show this symmetry, volumes of hyperbolic objects corresponding to non-relatively prime surgeries are displayed. As in the figure-eight knot table, x.xxxx denotes a non-hyperbolic surgery.
(iv) Jorgensen a n d Thurston proved the following theorem (see [19], Sections 5.10 to 6.6). THEOREM: Let ~ = {isometry classes of complete orientable hyperbolic 3-manifolds of finite volume}; then vol: ~ R+ is finite-to-one, and its image is well-ordered (that is, each subset has a smallest element), closed, and of order type ~o'~149 In particular, we can string out the set of volumes as follows, 0 < v 0 < v l < v 2 < . . .v,o
Idea of Proof: The l e f t - h a n d i n e q u a l i t y is g o t t e n as follows. In a hyperbolic 3-manifold, a short geodesic (length less t h a n 0.10686) possesses an e m b e d d e d solid torus neighborhood (solid tube), and the shorter the geodesic the bigger the volume of the solid tube (Figure 23 works again) 9 Thus a closed hyperbolic 3manifold either has a short geodesic and a good-sized tube, or it does not, in which case it has a good-sized e m b e d d e d ball. See Corollary 7.21 in [7] for details 9 The right-hand inequality is an explicit example found independently by Weeks and by Matveev-Fomenko.
Colin Adams, in [1], proved that the cusped hyperbolic 3-manifold of m i n i m u m volume is the Gieseking manifold. It has one-half the v o l u m e of the figureeight k n o t complement in the 3-sphere and is non-orientable. The orientable cusped hyperbolic 3-manifold of smallest volume is not k n o w n . The leading candidates are the figure-eight knot complement in S3 a n d its sibling, WL(5,_I), b o t h of w h i c h h a v e v o l u m e 2.02988 . . . . (v) It should be noted that it is not k n o w n w h e t h e r there is a hyperbolic 3-manifold with rational volume. In fact, it is not even k n o w n that a n y of the volumes are irrational.
S e c t i o n 9: B e y o n d
Volume
The v o l u m e is the best a n d most natural invariant for hyperbolic 3-manifolds, b u t it is n o t a complete inv a r i a n t - - t h e r e are distinct hyperbolic 3-manifolds that have equal volume. What geometric invariants should we turn to w h e n the volume fails? Two natural geometric invariants are the ChernS i m o n s i n v a r i a n t a n d the -q i n v a r i a n t . M o s t o w ' s theorem implies that they are topological invariants for closed hyperbolic 3-manifolds. The -q invariant is a real-valued invariant, while the C h e r n - S i m o n s invariant takes on values in the circle R/Z. This section will provide a brief description of their properties. More i n f o r m a t i o n - - f o r example, their d e f i n i t i o n s - can be f o u n d in the references m e n t i o n e d in this section. The Chern-Simons invariant grew out of attempts THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
51
by Shiing-shen Chern and James Simons to develop a combinatorial formula for the first Pontryagin number of a 4-manifold. It was an intractable term in their analysis, which seemed to warrant further study. It is defined for geometric 3-manifolds in general, and hyperbolic 3-manifolds in particular. The Chern-Simons invariant is difficult to compute. In fact, as of 1981, it was not known whether it was a non-trivial invariant for hyperbolic 3-manifolds. In [13], I developed some techniques for studying the Chern-Simons invariant, and proved the following. THEOREM: CS(Mr takes on a dense set of values in R/Z as higher and higher order surgeries are performed on the cusped hyperbolic 3-manifold M~. While I was doing this work on the Chern-Simons invariant, Walter N e u m a n n and Don Zagier were making a careful study of the volume function for hyperbolic 3-manifolds ([17]). Comparing their work and mine, they made a precise conjecture that there is a complex-analytic relation between volume and the Chern-Simons invariant for hyperbolic 3-manifolds! Yoshida, in 1985, proved the conjecture to be true (see [23]). Yoshida's theorem plus some work of Neumann and of Craig H o d g s o n has led to a ChernSimons calculator being affixed to the Weeks hyperbolic structures program. H o w informative is the Chern-Simons invariant? It is known that the Chern-Simons invariant enables us to distinguish large classes of hyperbolic 3-manifolds that have equal v o l u m e . D a n n y R u b e r m a n and ! proved the following theorem (see [16]). THEOREM: Given any rational number in the circle R/Z there exist hyperbolic 3-manifolds with equal volumes whose Chern-Simons invariants differ by that rational number. This is a nice result, but a systematic understanding of manifolds with equal volumes and different ChernSimons invariants seems to be a long way off. The ~q invariant was introduced by Michael Atiyah, V. K. Patodi, and Isadore Singer in [3]. It measures the extent to which the Hirzebruch signature formula fails for geometric 4-manifolds with boundary. This measurement turns out to depend only on the boundary of the 4-manifold, and therefore yields an invariant of geometric 3-manifolds. In [4], Proposition 4.19, it is shown that the ~ invariant and the Chern-Simons invariant are closely related. Specifically, 3~(M) = 2CS(M) mod Z. The "q invariant contains information that the Chern-Simons invariant does not. There are examples of hyperbolic 3-manifolds with equal Chern-Simons invariants but different ~q invariants (see [15]). Some
52
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
progress has been made towards calculating -q (see [23] and [14]).
Section 10: Summary The homology is a good invariant (computable and informative) for 3-manifolds, but it is not good enough for answering many of our questions about 3-manifolds. For example, it is easy to construct vast classes of distinct 3-manifolds with equal homologies. The fundamental group is almost a good invariant but is generally too hard to work with to provide us with answers to our questions. N e w invariants are needed. If we assume, as the evidence strongly suggests, that hyperbolic 3-manifolds are the most abundant, most important, and most complicated collection of 3-manifolds, then it makes sense to study invariants defined from the hyperbolic structure on a hyperbolic 3-manifold. Mostow's theorem tells us that such invariants are actually topological invariants (for complete hyperbolic 3-manifolds of finite volume). Three of the most natural hyperbolic 3-manifold invariants are the volume, the Chern-Simons invariant, and the "q invariant. Thanks to Weeks's hyperbolic s t r u c t u r e s p r o g r a m a n d s o m e theoretical w o r k , v o l u m e is e m i n e n t l y c o m p u t a b l e and the ChernSimons invariant is only slightly less so. There is hope for the -q invariant. Of course, the accuracy of these computer calculations will be restricted by the precision of the host computer's floating-point arithmetic. The volume has proven successful at distinguishing between manifolds with equal homology (see Figure 22), and the Chern-Simons invariant and the ~q invariant can distinguish between many hyperbolic 3manifolds with equal volumes. In addition, these invariants should be able to tell us information about the underlying manifolds--for example, the volume appears to be a good measure of complicatedness, while the Chern-Simons invariant appears to measure handedness. To what extent do the volume, Chern-Simons invariant, and ~ invariant taken together determine a given hyperbolic 3-manifold? I'm not sure. There are examples of cusped hyperbolic 3-manifolds that are not distinguishable by these invariants. For example, hyperbolic mutants have equal volumes, equal ChernSimons invariants, and the definition of the ~ invariant is problematic (see [15]). Similarly, the volume, the Chern-Simons invariant, and the ~q invariant do not provide a complete set of invariants for closed hyperbolic 3-manifolds. In [15] it is shown that certain mutations of closed hyperbolic 3-manifolds leave volume, Chern-Simons (mod 1), and "q unchanged. Paul Kirk has constructed examples of such mutants that are not homeomorphic. We need to construct new geometric invariants. To
do this, we n e e d to construct very similar m a n i f o l d s - ones that have all of the usual invariants e q u a l - - a n d s t u d y t h e m carefully (see, for example, [8]).
Acknowledgment I t h a n k Jeff Weeks for careful readings of several drafts of this p a p e r and for help with some of the figures. References 1. Colin Adams, The non-compact hyperbolic 3-manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987), 601-606. 2. Colin Adams, SNAPPEA: The Weeks hyperbolic 3-manifolds program, Notices of the Amer. Math. Soc. 37 (1990), 273-275. 3. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry: I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69. 4. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry: II, Math Proc. Camb. Phil. Soc. 78 (1975), 405-432. 5. Donald Blackett, Elementary Topology, New York: Academic Press (1967). 6. Raoul Bott, On the shape of a curve, Adv. in Math. 16 (1975), 144-159. 7. Fred Gehring and Gavin Martin, Inequalities for M6bius transformations and discrete groups, preprint. 8. Craig Hodgson, Robert Meyerhoff, and Jeff Weeks, Surgeries on the Whitehead link yield geometrically similar 3-manifolds, to appear in Topology '90, Proceedings of the Research Semester in Low-dimensional Topology at Ohio State University. 9. Gareth Jones and David Singerman, Complex functions, Cambridge: Cambridge Univ. Press (1987). 10. Imre Lakatos, Proofs and refutations, Cambridge: Cambridge Univ. Press (1976).
11. William Massey, Algebraic Topology: An Introduction, New York: Harcourt, Brace, and World, Inc. (1967). 12. S. V. Matveev and A. T. Fomenko, Constant energy surfaces of Hamiltonian systems, enumeration of 3-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds, Russian Math. Surveys 43 (1988), 3-24. 13. Robert Meyerhoff, Density of the Chern-Simons invariant for hyperbolic 3-manifolds, Low-Dimensional Topology and Kteinian Groups (D. B. A. Epstein, ed.) Cambridge: Cambridge University Press (1986), 217-239. 14. Robert Meyerhoff and Walter Neumann, An asymptotic formula for the -q invariants of hyperbolic 3-manifolds, preprint. 15. Robert Meyerhoff and Daniel Ruberman, Mutation and the ~ invariant, J. Diff. Geometry 31 (1990), 101-130. 16. Robert Meyerhoff and Daniel Ruberman, Cutting and pasting and the "q invariant, Duke Math. J. 61 (1990), 747- 761. 17. Walter Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), 307-332. 18. Dale Rolfsen, Knots and Links, Berkeley: Publish or Perish (1976). 19. William Thurston, The geometry and topology of 3-manifolds, Princeton Univ. preprint (1978). 20. William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (2)6 (1982), 357-381. 21. Jeffrey Weeks, Hyperbolic structures on 3-manifolds, Princeton Univ. Ph.D. thesis (1985). 22. Jeffrey Weeks, The Shape of Space, New York and Basel: Marcel Dekker, Inc. (1985). 23. Tomoyoshi Yoshida, The -q invariant of hyperbolic 3manifolds, Inventiones Math. 81 (1985), 473-514.
Mathematics Department Boston University Boston, MA 02215 USA
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
53
David Gale* For the general philosophy of this section see Vol. 13, no. 1 (1991). Contributors to this column who wish an acknowledgment of their contributions should enclose a self-addressed postcard.
We All Make Mistakes II In response to my request for examples of of mistakes by famous mathematicians several people mentioned a well-known error of Lebesgue. The following nice description was supplied by Doug Lind. "Lebesgue was trying to prove that the projection of a Borel set in the plane is a Borel subset of the line [Sur les fonctions repr6sentable analytiquement, Journal de Math~matiques, Series 6, Volume 1 (1905), page 195]. His argument uses the 'fact' that if you have a decreasing sequence of sets in the plane, then the projection of their intersection is the intersection of their projections (this is contained in the third paragraph, starting with "Supposons que e soit F de classe 1 ou 2"). Of course this is wrong (no first-year graduate student should make such a mistake!), and led to the Souslin-Lusin theory of analytic sets." [20-second time-out while the reader finds the obvious counterexample.] "Lusin even asked Lebesgue to write a preface to his book on analytic sets [Lefons sur les Ensembles Analytiques et leurs Applications]. There Lebesgue says that this was the most fruitful error that he had ever committed!"
order with card 1 on the top and placed face down on a (finite) table.
Definition A perfect n-shuffle consists in picking up the top n cards of the deck and interlacing them with the next n. Thus, if one executes a 5-shuffle on the deck in its initial ordering, the resulting ordering will be 6,1,7,2,8,3,9,4,10,5,11,12,13 . . . . Consider now executing a sequence of shuffles, first a 1-shuffle, then a 2-shuffle, then a 3-shuffle etc. CONJECTURE. In the course of this sequence of shuffles every card will come to the top of the deck infinitely often.
This conjecture is a mild modification of a conjecture of Richard Guy, as will be explained later, so I will refer to it from now on as Guy's conjecture. Here are the orderings given by the first eight shuffles:
Imagine (if you can) a countably infinite deck of cards. Each card is marked with a different natural number and initially the cards are arranged in their natural
0 1 2 3 4 5 6 7 8
* Column editor's address: D e p a r t m e n t of Mathematics, University of California, Berkeley, CA 94720 USA.
Note that at this point cards 1 through 6 have made their way to the top of the deck, 1 having been there already three times. However card 7 won't get there
Careful Card-Shuffling and Cutting Can Create Chaos
1,2,3 . . . . 2,1,3,4,5 . . . . 3,2,4,1,5,6 . . . . 1,3,5,2,6,4,7,8,9 . . . . 6,1,4,3,7,5,8,2,9,10 . . . . 5,6,8,1,2,4,9,3,10,7,11,12 . . . . 9,5,3,6,10,8,7,1,11,2,12,4,13,14 . . . . 1,9,11,5,2,3,12,6,4,10,13,8,14,7,15,16 . . . . 4,1,10,9,13,11,8,5,14,2,7,3,15,12,16,6,17,18 . . . .
54 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1 9 1992 Springer VerlagNew York
until shuffle 78, a first indication of the sort of erratic behavior to be described shortly. Getting back to Guy's conjecture, what basis is there for believing it may be correct? There are two arguments for it, the first "empirical," the second "probabilistic" (quotation marks here mean that what we are about to say is, rigorously speaking, nonsense). We consider first the probabilistic argument suggested by Raphael Robinson. Note first that card c stays in its place until the [c/2]th shuffle after which it may bounce around in a seemingly turbulent manner, but it can never reach a position further than 2n from the top of the deck on the nth shuffle, so the position of c on the nth shuffle is some number between 1 and 2n. N o w assuming (here's where the nonsense sets in) that the position of c is random, the probability that it is not at the top of the deck is 1 - 1/2n, and (more nonsense) assuming successive positions are independent, we see that the probability that c never reaches the top of the deck is
1-[(1 - 1/2n), n=l
which is zero because of the divergence of the harmonic series. As for "empirical evidence," Ilan Adler has checked all cards from I to 5000, listing for each card the number of the shuffle at which it reaches the top of the deck, and the results are interesting. Things go along comparatively quietly until we get to card 39, which takes 13,932 shuffles to reach the top, after which things settle down again, although card 43 requires 30,452 shuffles; but then there is a major explosion. After card 53 takes a mere 30 shuffles, card 54 goes on a wild rampage (card 54, where are you?), finally making it to the top on shuffle 252,992,198. Collecting all the data took about 80 hours of computer time on a NeXT Work Station, but most of that time was taken up with three "monsters," 4546, 3729, and the current world champion 3464, which took respectively 2,263,846,432 and 15,009,146,841 and 21,879,255,397 shuffles. Of course, this sort of behavior is what one would expect from the fallacious probability argument. The longer a card stays away from the top, the longer it is likely to continue to do so, i.e., the chance of "choosing" a I on the millionth shuffle is one in a million. Since perfect shuffles behave so wildly, perhaps one should look at something simpler. Instead of shuffling one might try simply cutting the cards. The traditional cut takes the top, say, n cards and places them on the bottom of the deck. In our model, however, the bottom is too far away, so instead let us define an n-cut to interchange the top n with the next n cards. Thus a 5-cut on the original order produces
6,7,8,9,10,1,2,3,4,5,11,12,13 . . . . If we now perform n-cuts in consecutive order starting with a 1-cut then a 2-cut, etc., it is trivial to show that Guy's conjecture is true, for card c remains in place until the [c/2]th cut, after which it moves up the deck by one on every other cut until it hits the top, whereupon it jumps down and then again proceeds to work its way back up to the top. So to make things interesting, instead of merely cutting, we will cut, but after each cut we discard the top card. The question is then whether every card is eventually discarded. The statistical behavior here is somewhat more restrained than for the shuffles, although there are occasional spurts. For example, card 752 survives over nineteen million cuts before being discarded. It seems though that this problem may be tractable. The orbit of a given card has a clear pattern which the reader will easily find by working a few examples, and the question of how long a card will survive boils down to a question in number theory--which, however, as of this writing has not been settled. A little bit about the origin of these questions: It all began with a problem proposed by Clark Kimberling which appeared in Crux Mathematicorum volume 7, number 2 (Feb. 1991). Kimberling considers the following array: 12345678910... 234567891011... 42567891011 12... 62748910111213... 879210611121314... 621191271381415... Here each row is obtained from the previous one by a sort of leap-frog procedure. Start with the number to the right of the diagonal term, which is underlined. Then go to the number to the left of the diagonal, then back to the 2nd number to the right, then the 2nd number to the left, etc., until you reach the first number in the row. Then jump back to the right and leave the remaining numbers in their natural order. Once a number appears on the diagonal it is expelled. Kimberling n o w asks, "(a) Is 2 eventually expelled? (b) Is every number eventually expelled?" The procedure is easily interpreted as a shuffle, which I will call the Kimberling Shuffle (sounds like the name of a nineteen-thirties dance craze), in which on the nth round one discards card n, then reverses the order of the first n - 1 cards and interlaces them with the next n - 1. Richard Guy noticed right away that the answer to (a) was yes, and in fact 2 is expelled on row (shuffle) 25, as a fairly easy hand calculation shows. Guy then conjectured that (b) is also true, and with the help of his grandson Andy Guy, who is studying computer science at Cambridge, verified the conjecture for all THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992 5 5
numbers up to 1200. Their table shows the same sort of wild behavior as the one for the perfect n-shuffles. In a private communication Guy has written, "I'd guess all numbers are expelled, but I also guess that no one's going to prove it." So he has actually made two conjectures, with the interesting property that if either one is confirmed the other one won't be.
A Spanish Self-Descriptor Readers may recall that last spring Lee Sallows gave a recipe for constructing self-descriptive sentences. Obviously the procedure is language-independent, so here is a Spanish version constructed by Miguel A. Lerma of the computer science department of the Universidad Politecnica of Madrid. ESTA FRASE CONTIENE EXACTAMENTE DOSCIENTAS TREINTA Y CINCO LETRAS: VEINTE A'S, UNA B, DIECISEIS C'S, TRECE D'S, TREINTA E'S, DOS F'S, UNA G, UNA H, DIECINUEVE I'S, UNA J, U N A K, DOS L'S, DOS M'S, VEINTIDOS N'S, CATORCE O'S, UNA P, U N A Q, DIEZ R'S, TREINTA Y TRES S'S, DIECINUEVE T'S, DOCE U'S, CINCO V'S, UNA W, DOS X'S, CUATRO Y'S, Y DOS Z'S. I understand Sallows also has a Dutch example.
Problems Products of two cycles (91-6) by column editor (These are three problems in one, a q u i c k i e , a not-so-quickie, and an unsolved) The first thing one learns about permutations is that any permutation is the product of disjoint cycles. (a) (quickie) Prove that if disjointness is not required then any permutation is the product of at most two cycles. (All group theorists seem to know this but I have been unable to find any reference to it in the literature.) (b) (unsolved, as far as I know) Is every bijection of a countable set the product of at most two cycles? A cycle here means a bijection which has exactly one orbit which is not a fixed point (so infinite cycles are included, as in infinite cyclic groups). (c) Prove that the bijection on 77 which maps n to n + 3 is the product of two cycles. 56
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. I, 1992
A Re-view of Some Reviews Among his many noteworthy accomplishments, Paul Erd6s may well hold the all-time world record for the number of papers which he has co-authored. It is interesting, therefore, to note that there is at least one paper of which he is the sole author which was, nevertheless, in some sense a collaboration. Irving Kaplansky had just finished writing a review of a (joint)
paper by Erd6s w h e n he encountered the author himself and m e n t i o n e d that he h a d admired the result but w o n d e r e d w h e t h e r the proof of the main theorem, which ran over a page a n d a half, couldn't be substantially shortened. Erd6s took another look a n d quickly found that indeed it could. The excerpt from Math. Reviews [Vol. 7, 1946, page 164] reproduced below provides the full story.
Computational Modelling of Free and Moving Boundary Problems Proceedings of the First International Conference, held 2 - 4 July 1991, Southampton, U.K. Edited by L.C. Wrobel and C.A. Brebbia
So here is a rare example of a paper published in its entirety in two different j o u r n a l s - - a n d n o w this makes it three (perhaps another world record).
MOWNG?
We need your new address so that you do not miss any issues of
MATHEMATICAL INTELLIGENCER. Please fill out the form below and send it to: Springer-Verlag New York, Inc., Journal Fulfillment Services 44 Hartz Way, Secaucus, NJ 07096-2491 Old Address
Name
(orlabel) Address City/State/Zip Name New Address Address City/State/Zip Please give us six weeks notice.
Vol. 1: Fluid Flow
Vol. 2: Heat Transfer
1991. 15,5 x 23 cm. 464 pp. Cloth DM 198,ISBN 3-11-013172-2 US $116.00 ISBN 0-89925-908-1
1991. 15,5 x 23 cm. 332 pp. Cloth DM 198,ISBN 3-11-013173-0 US $116.00 ISBN 0-89925-909-X
Set price: Cloth DM 336,- ISBN 3-11-013174-9 US $198.00 ISBN 0-89925-910-3 These two volumes contain submitted contributions and invited lectures presented at the First International Conference on Computational Modelling of Free and Moving Boundary Problems, held in Southampton (UK) in July 1991. The contributions are classified in the following sections: Vol. 1: Flow through porous media. Wave propagationCavitational flow 9 Free surface flow 9 Mathematical problems and computational techniques Vol. 2: Solidification and melting 9 Metal casting and welding- Electrical / Electromagnetic problems- Scientific applications. Walter de Gruyter & Co., Genthiner Str. 13, D-1000 Berlin 30, FRG, Phone (30) 2 60 05-0, Telex 184 027, Fax (30) 2 60 05-2 51 Walter de Gruyter, Inc., 200 Saw Mill River Road, Hawthorne, N.Y. 10532, USA, Phone (914) 747-0110, Telex 64 66 77, Fax (914) 747-1326
Ian Stewart* The catapult that Archimedes built, the gambling-houses that Descartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the
famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.
Salem's Bowditch Joe Albree Nathaniel Bowditch (1773-1838) is honored in the history of American mathematics and science for the applications he made of the works of others. The first fifty years of Bowditch's life revolved around Salem, Massachusetts, a compact seafaring town along the picturesque north shore, sixteen miles north of Boston. * C o l u m n Editor's address: M a t h e m a t i c s I n s t i t u t e , University of Warwick, C o v e n t r y CV4 7AL E n g l a n d .
Bowditch's life and renown may be explored in concert with Salem's 300-year heritage in pleasant walking tours around the town. The house in which Bowditch was born is a threestory frame dwelling common to the Salem merchants and sea captains of the mid-eighteenth century. Originally located just west of the Salem Common on Brown Street, this house was moved, at about the turn of the twentieth century, fifty yards back to the end of
58 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. I 9 1992 Springer VerlagNew York
T h e C h a r l e s O s g o o d p o r t r a i t of B o w d i t c h (courtesy P e a b o d y M u s e u m of Salem). THEMATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992 59
a cul-de-sac called Kimball Court. The home is now a delightful bed and breakfast. On March 28, 1773, less than a week after his birth, Nathaniel was baptized at St. Peter's Church, two blocks further west of the Common, at the corner of Brown and St. Peter Streets. From the first church building, erected in 1733, the present-day church has preserved the Bowditch family pew; and from several gravestones on the grounds, we can see that the IngersoUs (Nathaniel's mother and his second wife were members of this large family) were also communicants. Bowditch was apprenticed at about age eleven to the Ropes and Hodges ship-chandlers, and he began his campaign of self-education with the Ropes's family library. The Ropes home and gardens at 318 Essex Street, about seven blocks west of the Common, is open to visitors today. Bowditch soon graduated to the Philosophical Library Company. This was reportedly the finest scientific library north of Philadelphia because it featured the library of the Irish chemist Dr. Richard Kirwan (1733-1812). Kirwan's invaluable collection had been captured by a Yankee privateer dur- A Lissajous figure. These were in fact invented by Bowing the American Revolutionary War. We can follow ditch. Bowditch's scientific progress in the first few of his "Commonplace Books," housed at the Boston Public On November 1, 1803, on his fifth and final sea voyLibrary. Bowditch began his study of surveying on age, Bowditch began his study of Laplace's M~canique March 7, 1787, and his study of algebra on August I of cdleste. On his return he was elected president of the that same year. On January 4, 1790, he began to teach Essex Fire and Marine Insurance Company, and here himself Latin, and by 1793, he had read Newton's Prin- Bowditch became America's first actuary. However, cipia, and found an error in it. between 1814 and 1820 his major work was his transFrom January 11, 1795, to September 16, 1800, lation and annotation of Laplace, "an epoch in AmerBowditch made four overseas voyages on Elias "King" ican science," according to Simon Newcomb (p. 211 of Derby's ships, the Henry and the Astrea. Derby may [1]). One of the Bowditch family homes during this have been the n e w nation's first millionaire. Derby's time was a typical federalist structure at 9 North Street, wharf, minus its warehouses, and his home (1762) are about five blocks west of the Common and now a Regprominent landmarks of Derby Street, facing the har- istered National Historic Landmark. bor two blocks south of the Common. Bowditch's four When Bowditch moved from Salem to Boston in voyages formed the crucible wherein he found the er- 1823, he moved 2,500 books, more than 100 maps and rors in John Hamilton Moore's Practical Navigator and charts, and 29 volumes of his o w n manuscripts. As "concluded to take up the subject anew," composing president of the Massachusetts Hospital Life Insurance his New American Practical Navigator (first edition 1802, Company, he enjoyed enough material success so that and the latest edition, the 70th, 1984). he could afford the $12,000 it cost to have his translaBowditch was the 49th member of the East India tion of Laplace published (1829-1839). Marine Society, founded in 1799 by Salem mariners One of Bowditch's final acts was to write his dediwho had sailed b e y o n d Cape Horn, and he served the cation of the Laplace translation to the memory of his Society in many capacities including President (1820 to second wife, Mary (1781-1834). He died in Boston on 1836). The Peabody Museum, descended from the So- March 17, 1838, and he and his wife are buried side by ciety, contains a permanent Bowditch exhibit whose side in the family plot in Mt. Auburn Cemetery, Camfeatures include the h a n d s o m e portrait (1835) by bridge. There is an attractive statue of Bowditch in Charles Osgood (1809-1890); the bust of Laplace looks another part of the cemetery. on from the upper left corner. There are also several of Bowditch's navigational and surveying instruments, R e f e r e n c e s and the copper plate of his splendid chart (1806) of 1. R. E. Berry, Yankee Stargazer, the Life of Nathaniel Bowditch, Salem harbor. The P e a b o d y M u s e u m is on Essex New York: Wittlesey House (1941). Street, about two blocks south and west of the Com- Department of Mathematics mon and almost directly across the street from the Es- Auburn University at Montgomery sex Institute and Museum. Montgomery, AL 36117-3596 USA 60
THE MATHEMATICAL [NTELLIGENCER VOL. 14, NO. 1, 1992
The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. An Opinion should be submitted to the editorin-chief, Chandler Davis.
I. R. Shafarevich's Essay "Russophobia" There has been widespread discussion of I. R. Shafarevich's essay "'Russophobia," published in 1989 after circulating for years privately.1 There is no doubt it is extremely nationalistic, and many have attacked it as hate literature. An especially poignant view is that of Shafarevich' s colleagues, such as Boris Mo~shezon, his student and collaborator, who emigrated to Israel in 1972. To share some of Mo~shezon' s reactions with readers of the Mathematical Intelligencer, we turn to a long interview with him by Mark PopovskiL 2 The following excerpts are translated and reprinted with permission. . in the summer of 1958 he [Moishezon] met the one who would be his teacher and friend for many years, who would determine his scientific course, Igor' Rostislavovich Shafarevich. The 22-year-old student met the 35-year-old Shafarevich on the same day that the scientist was named a Corresponding Member of the Soviet Academy of Sciences. "He received me very warmly," Moishezon recalls. "He tested my knowledge with some questions. The answers seemed to satisfy him. When I left he gave me .
.
1 See Smilka Z d r a v k o v s k a ' s i n t e r v i e w with Shafarevich in t h e Intelligencer vol. 11 no. 2 (1989) a n d the letter from S h e p p a n d Veklerov in t h e Intelligencer vol. 12 n o . 3 (1990). The p u b l i s h e d v e r s i o n of " R u s s o p h o b i a " is in Nash Sovremennik 1989, nos. 6, 11; Professor Shafarevich considers t h a t text of t h e e s s a y definitive a n d a s k s that a n y future d i s c u s s i o n s of " R u s s o p h o b i a " be b a s e d o n it rather t h a n the s a m i z d a t version. 2 M a r k Popovskii, "Teacher, w h o m are y o u t e a c h i n g ? " , in Novoe Russkoe Slovo, 5 October 1990.
his manuscript on algebraic number theory . . . . " [Returning to Tadzhikistan later with "homework" from Shafarevich] he found a new simpler proof of a known theorem. Shafarevich wrote the student, "The fact was known before, but your proof is very simple and clear. If you come to Moscow I will be happy to direct your work for the degree." "This was a fantastic success for me," says Moishezon. "Along with Israel Gelfand and Andrei Kolmogorov, Igor' Shafarevich was considered one of the most brilliant mathematicians in the country. In m y wildest dreams I had never imagined that such a person would be my thesis director . . . . "I literally worshipped Igor' Rostislavovich. Though Gelfand and Pyatetskii-Shapiro were my advisers, m y real teacher throughout those years was Shafarevich. Honest, generous, honorable, he was loved by all his students, among whom, it may be pointed out, were a number of Jews. " . . . At a banquet (this was in 1964), I raised the q u e s t i o n of his religious beliefs. He a n s w e r e d brusquely, 'The notion of the second coming of Christ seems as nonsensical to me as saying that Stalin would be resurrected and lead us.' " "But in his current public statements Shafarevich takes a clear Orthodox Christian position, doesn't he?" "Yes; evidently now he has reconsidered his opinion." The date of the "reconsideration" [Popovskii's article continues] is not hard to determine . . . . Sakharov
THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1 9 1992Springer-VerlagNew York 61
in his Memoirs remarks, "'The general position of Shafarevich [by 1971, w h e n he asked to participate in Sakharov's Committee on H u m a n Rights] was very close to that of Solzhenitsyn. I can't tell which of the two was the leader." . . . In the following years the mathematician's chauvinism separated him completely from Sakharov's position . . . . Once teacher a n d s t u d e n t spent a w e e k rooming together at a mathematical seminar in the mountains of Armenia. In those d a y s they often r e t u r n e d to the "Jewish question." Moishezon remembers Shafarevich saying, "Jews make problems for themselves by complaining too much. For example, they say 6 million Jews died under Hitler, whereas it is k n o w n that only 600,000 died." [Moishezon sees his teacher as attracted to] "'the idea of a 'strong h a n d , ' which comes out so clearly in his recent statements. He is tending toward this idea already in his 1971 work 'Does Russia have a future? '3 . . . I don't k n o w w h a t influenced Shafarevich, but it seems plain to me that all his present positions are united by one c o m m o n concept: the thirst for a powerful regime, which always almost presupposes the need for an e n e m y . The presence of a permanent, strong, dangerous e n e m y (the 'little people') provides the principal justification for the 'iron h a n d . ' r . . " "'What are y o u r feelings toward y o u r teacher today?" "The samizdat version of 'Russophobia' was first brought to the West by Soviet mathematicians in aut u m n 1988. W h e n I got to the words The chosen people introduced the concept of the Messiah in order to gain sovereignty over the world I felt a real shock. Shafarevich is really delving in the political-philosophical dregs. A n d yet it is h a r d for me wholly to set aside m y friendship for Igor' Rostislavovich. It is the same w i t h him: giving an interview, he recalls in the most positive w a y my n a m e a n d those of his other Jewish students, including those w h o are in Israel. "After the appearance of 'Russophobia' in print I was asked to write Shafarevich a letter of protest. I refused. I would rather he quietly and w i t h o u t prejudice . . . r e t h o u g h t the grotesque interpretation of world history he has built for himself over the last two decades. There's little hope of this, to be sure . . . . "But after all, one's teacher is almost like a father. He is associated with y o u t h , with the time of hope. In that time Igor' Rostislavovich gave so m u c h to us, his students .... I w a n t to believe. After all, he is m y teacher."
3 In Solzhenitsyn's collection Iz-pod Glyb, YMCA Press, Paris, 1974. 62
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO, 1, 1992
Now in its Third Revised P r i n t i n g - -
To Infinity and Beyond A Cultural History of the Infinite
by Eli Maor From the reviews: "Fascinating and enjoyable... Maor has written a book that places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics." - - Science "The work of a gifted w r i t e r . . , this book is must reading for anyone who wants to find out what mathematicians do, how they think, and what turns them on." - - SIAM News
"A splendid and lavishly designed and illustrated book for the general reader about all aspects of infinity. It is destined to become a classic." - - Mathematics Magazine
"Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M.C. Escher, six of whose works are shown here in beautiful color plates." - - Los Angeles Times
Order Your Copy Today! To Infinity and Beyond: A Cultural History of the Infinite by Eli M a o r
Third Revised Printing, 1991
1987/304 pp., 162 illus., 6 color plates Hardcover/ISBN 0-8176-3325-1/$49.50 9 Call: Toll-Free 1-800-777-4643. In NJ please call (201) 348-4033. Your reference number is Y495. 9 Write: Send payment plus $2.50 for postage and handling to: Birkh~iuser, Attn. S. Gebauer - Dept. Y495, 175 Fifth Avenue, New York, NY 10010. 9 Visit: Your Local Technical Bookstore. Visa, MasterCard, American Express and Discover Charge Cards, as well as personal checks and money orders, are acceptable forms of payment. All orders will be processed upon receipt. If an order cannot be fulfilled within 90 days, payment will be refunded. Payable in U.S. currency or its equivalent.
Birkhiiuser Boston
Basel
Berlin
The Uses of Set Theory* Judith Roitman
The distinguished mathematician Saunders Mac Lane has titled his talk at these meetings Algebra as a means of understanding mathematics. Had I known about his title, I would have called this talk Set theory as a means of doing mathematics. These titles represent quite a historical change. Speaking simplistically, and exaggerating somewhat, seventy-five years ago algebra was something mathematicians did largely for its o w n sake, while set theory belonged to that quasi-philosophical branch of mathematics k n o w n as foundations, whose main purpose seemed to be to convince the philosophically inclined that m a t h e m a t i c i a n s should be allowed to continue doing what they had always done, and which most mathematicians felt they could safely ignore. As with all historical generalizations, this is, of course, not entirely accurate. I will let the algebraists speak for algebra (such matters as groups and crystall o g r a p h y come i m m e d i a t e l y to mind). As for set theory, its roots lie much more in mathematics than in philosophy. Cantor invented ordinals while trying to solve a question about Fourier series on which mathematicians are actively working t o d a y - - K e c h r i s and Louveau have an excellent book on the subject. Early on, Polish and Russian mathematicians invented descriptive set theory, the study of the structure of the reals in set-theoretic fashion. And point-set topology, since its inception, has been inextricably intertwined with set theory. Even Hilbert's problems did not avoid all reference to set theory. But it remained true that
most mathematicians could get by quite well with union, intersection, and (if necessary) the axiom of choice, learned as quickly as possible, preferably in the preface to a book about some other area of mathematics. The thesis of this talk is that this situation has changed drastically. Hard-core, mainstream mathematics (defined as: the mathematics most mathematicians think doesn't use set theory) is beginning to see results using modern set theoretic techniques. The purpose of this talk is to introduce you to a few of
* This p a p e r is freely a d a p t e d from the a u t h o r ' s talk, of the s a m e title, jointly s p o n s o r e d b y t h e Association for W o m e n in M a t h e matics a n d the M a t h e m a t i c a l Association of America, in h o n o r of t h e M A A ' s 75th b i r t h d a y , at t h e 1990 S u m m e r J o i n t M a t h e m a t i c s Meetings. THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1 9 1992Springer-VerlagNew York
63
these results. These are by no means the only results of this nature. There are many more out there, and many, many more to be obtained. Before getting into the specific mathematical results ! want to discuss, I should tell you the basis of set theory. Set theory is based on three truths and two beliefs. The first truth is due to Georg Cantor, and is the root of set theory:
Truth #1. Not all infinite sets look alike. The simplest way infinite sets can differ is their size - - t h i s is Cantor's great discovery. Another way sets can differ is the e-relation. Consider the e-relation restricted to a set X, that is, all pairs (x, y) with x, y e X and x e y. Different sets X will have e-relations that look quite different. Both ordinals (in which the e-relation is a linear order) and models of set theory are grounded in the e-relations of sets. Yet a third w a y to distinguish sets is u n d e r mild combinatorial relations--partial orders, say, or the relations of Ramsey theory. You could claim, with some justification, that everything set-theorists do boils down to seeing all the ways in which infinite sets can differ. Historically, certain infinite structures have come under the rubric of settheoretic combinatorics, whereas others are labelled algebra, topology, etc. It is the arbitrariness of this division that lends set theory much of its power. The second truth comes from modern logic:
Truth #2. Not all formulas look alike. Another way to put this is: the linguistic complexity of a concept matters. Computer scientists are familiar with this truth. If you are not familiar with it, it means
if you know the simplest way of defining a set, you already know quite a bit about the set. In the particular application of this truth we shall see, we will be concerned with what are k n o w n as analytic sets. The topological definition of an analytic set is: the continuous image of a Borel subset of a complete separable metric space. But much of the power of the notion of analytic sets comes from the logical equivalent, which basically says that the analytic sets are those with particularly simple definitions. The full theorem is: X is analytic iff there is a formula #(x,z,r) where r is a fixed real number, ~ quantifies only over natural numbers, and x e X iff 3 z such that ~(x,z,r) holds. Remember that parameter r. We will return to it later.
Truth #3. Not all approximations to the mathematical universe look alike. Mathematicians without a background in logic often have trouble with this, so let me try to sketch it carefully. 64
THE MATHEMATICAL INTELLIGENCER VOL. I4, NO. 1, 1992
"Approximation to the mathematical universe" is a somewhat flamboyant means of describing models of set theory. We begin with a fairly small, easily described set of a x i o m s - - t h e axioms of set t h e o r y - which we believe the mathematical universe satisfies. These axioms really are pretty simple: for example, if x, y are sets, so is {x,y}. Any structure satisfying these axioms is called a model of set theory, just as a structure satisfying the axioms of group theory (i.e., a group) is a model of group theory, or a field a model of field theory. The point is, these models don't agree on what's t r u e - - f o r example, they may disagree on how many real numbers there a r e - - a n d we really have no way of knowing which are better approximations to the mathematical universe (and at this point I have to add: if it exists) than others. The process of going from one model to a bigger one is analogous to adding, say, V'2 to the field of rationals to get a new field. You can also go from bigger to smaller models; in particular, you can move from a given model to a countable submodel that looks a lot like the original. In addition to these truths, there are two beliefs that enter into the applications of set theory.
Belief #1. Set theory is consistent. This is a belief, because, by GOdel's second incompleteness theorem, set theory can't prove its own consistency. G6del's second incompleteness theorem applies to any theory in which we can embed elementary number theory, so set theory is no worse than any other theory ambitious enough for
Belief #2. Mathematics can be embedded into set theory. This is actually a program, partially carried out in many undergraduate textbooks, in which we find representatives in the set-theoretic universe for most of our familiar mathematical objects. The program is too tedious to carry out fully, and there are some areas, mostly involving objects too big to be sets, where some fudging is necessary, but even if you withhold full belief in Belief #2, actual applications of set theory just use small, mathematically standard pieces of this belief.
Notation and Definitions A statement is consistent iff it cannot be proved false by set theory. Equivalently, by the completeness theorem of mathematical logic, a statement is consistent iff it holds in some model. A statement is independent iff both it and its negation are consistent. Thus, you can show that a statement is independent by showing that it holds in some model and that its negation holds in another.
I have tried to avoid superfluous notation, b u t some is inescapable. to is the set of natural numbers, a n d also the first infinite ordinal. 001 is the first u n c o u n t a b l e ordinal. 2 ~ is the cardinality of the set of reals. R is the set of reals, N the set of natural n u m b e r s , C the set of c o m p l e x numbers. AB is the collection of functions from A to B. (This notation distinguishes ~,: = the set of functions from ~, to K, from the cardinal n u m b e r KK) ZFC is the axioms of set theory. CH is the c o n t i n u u m hypothesis: 2" is the first uncountable cardinal. MA is Martin's a x i o m . f MA + --7 C H is: M A holds but C H fails.
Some Things I Will Not Talk About I will not talk about fields of mathematics p e r m e a t e d b y set theory. Such fields include general topology, Boolean algebra, a n d m e a s u r e theory, as well as certain subfields of algebra a n d n u m b e r theory. I will also n o t talk a b o u t t w o i m p o r t a n t a p p l i c a t i o n s of set theory, because t h e y have excellent and l e n g t h y expositions elsewhere. Those two applications are The Kaplansky conjecture. Let X be compact. Is every h o m o m o r p h i s m from C(X,C) into a Banach space B continuous? Dales a n d Esterle s h o w e d that u n d e r C H the a n s w e r is no. Solovay f o u n d a model in which the a n s w e r is yes. F o r m o r e details, see [Dales a n d Woodin]. The Whitehead conjecture. Is every W h i t e h e a d group free? Shelah s h o w e d that u n d e r MA + --1 C H the answer is yes, while u n d e r the combinatorial principle the a n s w e r is no. For m o r e details, see [Eklof]. I will discuss examples from analysis, algebra, and algebraic topology. I take all responsibility for the choice of e x a m p l e s - - t h e r e w e r e m a n y results f r o m which to choose, and time is short.
Example 1: The Ideal of Compact Operators Let us define Hilbert space to m e a n infinite-dimensional, separable, c o m p l e x Hilbert space, a n d an ideal to be a two-sided ideal in the ring of b o u n d e d linear
f If you are unfamiliar with M A that's okay. To u n d e r s t a n d this talk you n e e d only k n o w that b o t h M A and MA + -q C H are provably consistent with the axioms of set theory. The exact statement of MA is:
If X is compact Hausdorff w i t h n o uncountable family of disjoint open sets, and ~ is a family of d e n s e open sets w h e r e ~ has size less than 2% then N ~ is n o n e m p t y . Thus, MA is a variant of the Baire category theorem. In particular, CH implies MA.
operators o n Hilbert space. Brown, Pearcy, and Salinas asked
Question 1. Is the ideal of compact operators on Hilbert space the s u m of two p r o p e r l y smaller ideals? This p u r e l y analytic question turns out to be equivalent to p u r e l y set-theoretic combinatorics.
Definition 2. F is a non-principal ultrafilter on a set X iff 1. a e F i m p l i e s a C X 2. a e F implies a is infinite 3. F is closed u n d e r finite intersection and u n d e r superset 4. for e v e r y a C X either a ~ F or X \ a ~ F. The t h e o r y of non-principal ultrafilters o n ta is a rich subject in b o t h set t h e o r y a n d point-set topology. The principle NCF (near c o h e r e n c e of filters) states that if F, G are distinct non-principal ultrafilters on ta, then there is a finite-to-one function f: ta ~ to so that {fla]: a e F} = {fla]: a e G}. That is, some fairly simple function o n ta lifts to take the image of F onto the image of G. The a n s w e r to Question I is d e t e r m i n e d by the following t h e o r e m of Blass.
Theorem 3. The ideal of compact operators on Hilbert space is the sum of two properly smaller ideals iff NCF fails. W e l l - k n o w n t h e o r e m s a b o u t ultrafilters give, as easy corollaries, the failure of NCF u n d e r MA. Blass and Shelah f o u n d a model in w h i c h NCF holds. So w e have
Theorem 4. The statement "The ideal of compact operators on Hilbert space is the sum of two properly smaller ideals'" is independent.
Example 2: A Characterization of Free Groups A n o r m v o n an additive g r o u p G is like a measure. That is,
v: G--~ R v(g) = 0 iff g is the identity v(g + h ) ~ v(g) + v(h) v(mg) = Imlv(g), for m an integer. v is discrete iff, for some r > 0, v(g) > r for all g # identity. Theorem 5. (Steprans) An Abelian group G is free iff it has a discrete norm. Notice that this is a t h e o r e m of ZFC, not an indep e n d e n c e result. Also, its s t a t e m e n t makes no reference to set theory. Why, then, is it included here? Steprans's proof is based on infinitary combinatorics. We will sketch the combinatorics, a n d t h e n the proof. Consider a linearly o r d e r e d set X and consider the order t o p o l o g y on it. Because X is o r d e r e d we have the notion of an u n b o u n d e d subset of X. Because X is a THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992 65
space we have the notion of a closed subset of X. Thus we have the notion of a club, i.e., a closed unbounded subset of X. For example, N is a club in R. A stationary set is defined as a closed set that intersects every club. For example, S is stationary in R iff S contains an interval of the form (a, oo); S is stationary in N iff S if cofinite. Both R and N contains disjoint clubs, hence each of their stationary sets is a (fairly large) club. For uncountable ordinals, the situation is quite different. If our ordered set X is the ordinal oo1 then the clubs on X form a filterbase and there are not only stationary sets, but stationary sets that are not clubs. Furthermore, ool is not the only ordinal with this property: any ordinal with no countable unbounded set (we say it has uncountable cofinality) has the property that the collection of its clubs forms a filterbase, and it has stationary sets that are not clubs. Clubs and stationary sets of ordinals are extremely important set-theoretic objects. An important subclass of ordinals is the regular cardinals: K is regular iff every unbounded subset of n has size K. Now we are ready to sketch Steprans's proof. Suppose there is an Abelian group G that is not free but that has a discrete norm v. By a previously known theorem (due i n d e p e n d e n t l y to J. Lawrence and F. Zorzitto) G is not countable. G turns out to be the increasing union of subgroups {G~: o~ < K} where K is regular uncountable, and there is a stationary set S and a function f: S ~ K SO Gf(,~)/G,~is not free when c~ 9 S. The interplay between clubs and stationary sets is then used to derive a contradiction. The reader familiar with mathematical logic will norice the mark of model theory on this proof--G is not free, and has, roughly speaking, arbitrarily large approximafions that are not free as well. Steprans's proof is based on a technique of Shelah, who has brilliantly and prolifically exploited the use of model theory w i t h i n set t h e o r y . By u s i n g this example we are cheating a little--Shelah's technique comes from work related to Whitehead groups, which we promised not to talk about, However, since at least in some models the Whitehead groups are exactly the free groups, if you are to talk about free groups you might as well help yourself to techniques stemming from the Whitehead problem. Another example of this model-theoretic point of view (also due to Shelah) is E x a m p l e 3: T h e F u n d a m e n t a l G r o u p
What groups can be f u n d a m e n t a l g r o u p s of nice spaces? For the purpose of this talk, let's define nice to mean: compact, metric, path-connected, locally pathconnected. 66
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
Question 6. Can the additive group of the rationals be the fundamental group of a nice space? Shelah's answer is no. This answer is a corollary of the following theorem of Shelah that is as obviously set-theoretic as question 6 is not.
Theorem 7. The fundamental group of a nice space is either finitely generated or has cardinality 2 =. Again we have a theorem that is not a consistency result. However, the proof uses methods related to consistency results. Let us sketch the proof. The first two steps are completely within algebraic topology: members of a certain subclass of nice spaces always have a finitely generated fundamental group; every other nice space has a point x so that each neighborhood of x contains an essential loop at x.
/ Each neighborhood of x contains an essential loop at x.
Suppose our space has such a point x. Fix an infinite countable collection of essential loops attached to x, and index this collection by co. We now associate, to each A Coo, the path fA we get by taking the first loop named in A, then the second, then the third, and so forth. This gives rise to an equivalence relation: A ~ B iff fA is homotopic to fB. The relation = turns out to be analytic. Let r be the parameter. Because r is a real number (hence coded by a countable sequence of natural n u m b e r s - - 0 ' s and l's suffice) there are lots of countable elementary submodels M containing r. Here, "elementary submodel" means that M is a model nearly indistinguishable from the universe: any sentence true in the universe and not true in M must refer to some set not in M. In particular, anything true of ~ that doesn't mention anything outside of M is also true in M. Such an M knows essentially everything there is to know about ~. Furthermore, because the definition of = is so simple, any model extending M also knows essentially everything there is to know about ~. So let M be such a countable elementary submodel, r 9 M. Now consider the partial order of finite functions from 00 into {0,1} given by reverse inclusion. This is a pretty simple object, so simple that it sits in every model, hence in M. But it is a very powerful object: it can be used, via the method of forcing, to add a new
element to M, known as a Cohen real. It is well known that there are 2~ m a n y Cohen reals over a countable model M. When we add Cohen reals to M, we have an extension that still knows all about -=. In particular, in such an extension N, if A, B are distinct Cohen reals over M, then "A = B" is true in N iff A ~- B. But we can show that " A m B" is true in N for distinct Cohen reals A, B over M. So fA is not homotopic to fB for distinct Cohen reals A, B; there are 2~ many distinct Cohen reals over M; and the theorem follows.
the answer to Question 9 is no, so is the answer to Question 8. Marde~i4 and Prasolov proved that the first strong homology group of ~f2 is a particular group G, which is itself defined as a derived limit. So Question 9 has a positive answer iff G = 0. Now comes a step that should seem familiar by now: the reduction of infinite combinatorics. Suppose we have a function K with domain A and a function J with domain B. Define
Example 4: The Hawaiian Earring
In other words, K =* J iff they are almost equal on their common domain. If K =* J we say that K, ] almost cohere. The K's and J's we are interested in are defined on particular subsets of 0)2. The subsets of 0)2 we are interested in are defined as follows. Given f: co ~ 0), define A I = {(n,m): m ~ f(n)}; i.e., Af is the set of all points in 0)2 that lie on or below the graph of f.
The n-dimensional Hawaiian earring is a countably infinite collection of n-spheres of decreasing radius glued at a single point. Here's a picture of part of the 1-dimensional Hawaiian earring.
K = * I iff {p ~ A • B: K(p) ~ / ( p ) } is finite}.
f
f . The 1-dimensional Hawaiian earring. The Hawaiian earring in some sense lies behind the preceding example: its homotopy group is easily seen to have size 2~ since the paths fA are distinct. Shelah's result can be interpreted as: nice spaces whose homotopy groups are not finitely generated look a lot like the Hawaiian earring. In this example we look at homology. The Hawaiian e a r r i n g is c o m p a c t m e t r i c , so a n y u s u a l (e.g., Steenrod) homology can be applied. It is connected, so its first homology group is 0. Lisica and Marde~i4 invented strong h o m o l o g y (which we will not define here) to extend the Steenrod homology to all spaces in a way that satisfies the Eilenberg-Steenrod axioms on a large class of spaces. In particular, the disjoint topological sum of countably m a n y compact metric spaces belongs to that class. Strong h o m o l o g y is a conservative e x t e n s i o n of Steenrod homology: the strong homology group of a space equals its Steenrod homology group if the latter is defined.
Af is the set of points on or below the graph of f.
~
~[~[ .9 . .9. . 9
9
9
Let K(J) be the collection of all functions K: Af ~ 0). The combinatorial principle ACF says let ~K be an almost coherent family picking an element from each K(J). There is a single fixed function o n 0) 2 that almost coheres to each element of
J~. Now we are ready to return to homology. Marde~i~ and Prasolov proved
Theorem 10. G = 0 iff ACF. They also proved
Theorem 11. CH implies ACF fails. So CH implies that strong homology is not additive. It is not known if strong homology really is not additive, but a negative answer to Question 9 is not the way to decide, since Dow, Simon, and V a u g h a n proved
Theorem 12. It is consistent that ACF holds. Hence it is consistent that G = 0, and at least plausible that strong homology is (consistently) additive.
Question 8. Is strong homology additive? A particular instance of this question is
Question 9. Does the first strong homology group of ~2 equal 0? Here ~f2 is the disjoint topological sum of countably many copies of the 2-dimensional Hawaiian earring. If
Example 5: A Banach Space w i t h Few Operators If you have a type of object and a bunch of maps, it is reasonable (and traditional) to ask whether there is an example of the object in which the only maps are THE MATHEMATICAL
1 N T E L L I G E N C E R V O L . 14, N O . 1, 1992
67
those absolutely necessary. The following theorem, due to Shelah a n d Steprans, is in this tradition.
Theorem 13. There is a non-separable Banach space on which every operator has the form S + pI, where p is a scalar, S is an operator with separable range, and I is the identity operator.
tive law. In particular, we w a n t to consider A = the free left-distributive algebra on one generator, call it x. For exposition purposes we'll also look at C = the free left-distributive algebra on three generators 9 A w o r d in an algebra is just a legal, u n a m b i g u o u s string of symbols. For example, xxx is not a word in A, and x(x is not a word in A, but (x(xx))(x) is. Once y o u have any k i n d of algebra you w a n t to k n o w which words are equivalent, where w -= v iff there is a method, using only the laws satisfied by the a l g e b r a (in this case t h e l e f t - d i s t r i b u t i v e law) of starting at w and arriving at v. For example, in C, x(yz) - (xy)(xz) =- ((xy)(x))((xy)(z)) 9 . . ; a n d x(yz) v~ (xy)z. The collection of equivalent pairs from one of these algebras is of very low complexity 9 In fact it is recursively e n u m e r a b l e - - y o u can write a c o m p u t e r program that, given infinite time, will spew out all the equivalent pairs from A (or from C) and no others. So one might hope that A has a fairly simple structure.
In other words, y o u can't avoid operators with separable ranges; a n d y o u can't get rid of scalar multiplication; but maybe those are essentially the only operators possible9 (The original question is whether there is a Banach space on which every operator has the form S + pI where p is a scalar a n d S is a compact operator. As far as I k n o w this question is unsolved.) Like our second a n d third examples, this is a real result. Its connection to set theory again lies in the dependence of the proof on infinite combinatorics, this time on ~o1. The idea is to construct a particular " n o r m " H" [[ on '~ (the quotation marks are because this " n o r m " can take on the value oo) a n d consider Y = {y: [[y[[ < oo}. arising This turns out to be a Banach space, although it still D e e p a n d t e c h n i c a l c o m b i n a t o r i c s f r o m q u e s t i o n s f a r r e m o v e d f r o m a n a l y sis end has too m a n y operators 9 N o w restrict yourself to B = cly span({-& o~ < 001}), where ~ is the function that is 1 up b e i n g a p p l i e d to B a n a c h spaces. exactly at ot a n d 0 everywhere else. This space B turns out to be the desired space. Now, given two words in an algebra, let's define w The " n o r m " II " I[is quite complicated, a n d I will omit < v iff there is some u ~ v, u # w, with w a c o m p o n e n t its definition. I will only tell you that ][-[[ d e p e n d s on of u. Here a component is just w h a t you think it is: for the existence of a function F:[~01]2 ~ 001 interacting example, x, y, z, and yz are the components of x(yz) in richly with C. Thus, in C, xz < x(yz) (even t h o u g h xz is not a com(a) every uncountable disjoint family of pairs of p o n e n t of x(yz)), because x(yz) = (xy)(xz), while zy countable ordinals xO/z). The relation < on A (or o n C) is also recursively and e n u m e r a b l e - - t h e r e is a c o m p u t e r program that, if left (b) every pair of functions from some finite n into ~o1. to r u n forever, w o u l d s p e w o u t exactly those pairs In other words, [[ 9 [[ is defined by quite technical in(w,v) where w < v. Let's say that w ~ v iff w < v or w -= v, and let's form finite combinatorics. The origins of F are i n t e r e s t i n g - - i t comes out of the the quotients of our algebras by the equivalence relaconstruction of a graph (due i n d e p e n d e n t l y to Shelah tion -=. and to Todorcevic) that embeds all small graphs on Question 14. Is ~ a linear order on A/=-? any large set of vertices 9 F is also closely related to ToLaver has recently s h o w n that the answer is yes if dorcevic's work on partition calculus, a particularly important field of infinite combinatorics. Thus deep there is an extremely large cardinal 9 a n d technical combinatorics arising from questions far H o w did Laver come u p o n his result? Laver was not originally interested in A. He, a n d r e m o v e d f r o m a n a l y s i s e n d u p b e i n g a p p l i e d to other m a t h e m a t i c i a n s , w e r e i n t e r e s t e d in the folBanach spaces. A n y o n e w h o complains about the excessively technical nature of set theory a n d its failure lowing large cardinal axiom. to relate to the rest of mathematics should contemAxiom 15. There is a cardinal k and a non-trivial eleplate this particular application 9 m e n t a r y e m b e d d i n g from Vx into itself.
Example 6: The Free Left-Distributive Algebra on One Generator The left-distributive law is: a(bc) = (ab)(ac). Let's consider free left-distributive algebras, w h e r e " f r e e " means that no other laws hold, not even the associa68
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
Axiom 15 needs some explanation. W h y is it called a large cardinal axiom? Large cardinals are those whose existence implies the consistency of ZFC. Hence, by G6del's second incompleteness theorem, they cannot be proved to e x i s t - - t h e y are so large we can't be sure they're there 9
What is V~? Vx is an initial segment of the set-theoretic universe, constructed by transfinite induction: V0 = {~}; if cx is a limit ordinal then V= is the union of the previous V~'s; and V~+ 1 is the power set of V=. This construction was first carried out by Mirimanoff in 1917. The statement that the universe is the union of the V,'s is equivalent to the axiom of regularity. What is an elementary embedding? An elementary embedding can be thought of as a kind of homomorphisrn preserving the relations of set theory. Usually the only elementary embedding from some V x to itself is the identity. To say there is another one implies that is very large. Laver's theorem becomes Theorem 16. If axiom 15 holds, then <, is a linear order on A/=-.
H o w is theorem 16 proved? Suppose you have such a non-trivial elementary embedding, j. Then j gives rise to other embeddings, both by composition and by a more complicated operation. In studying the algebras generated by j under these operations, Laver saw that the algebra A* generated by using only the second operation is exactly isomorphic to A. The major problem in answering Question 14 is to show that ~< is irreflexive: symmetry and transitivity are clear, and it can be shown from ZFC alone that the trichotomy law holds. Thus ~ is a linear order on A/=iff ~ is irreflexive. By the definition of A* irreflexivity is easy to show. In fact, Laver uses irreflexivity to find a canonical form from which the t r i c h o t o m y law can be derived. Without the assumption of Axiom 15 it is not known if irreflexivity holds, or if there is such a canonical form. A word about the ontological status of Theorem 16. Adding a large cardinal hypothesis strengthens set theory considerably. There is always the possibility that a reasonable sounding large cardinal hypothesis will turn out to be false. (This has actually happened.) So whenever a large cardinal hypothesis is used, one would like to either remove it or prove that some large cardinal (possibly a w e a k e r one) is n e e d e d . Thus, Question 14 is not yet fully answered. The possibilities are: 1. Linearity of ~ is a theorem of ZFC. 2. Linearity of ~ implies the consistency of the existence of large cardinals. 3. Linearity of ~ is independent, without the use of large cardinals. 4. Linearity of ~ is refutable in ZFC, hence Axiom 15 fails. Remark: Just before I sent the final copy of this paper to the Intelligencer, Laver told me of an improvement on his r e s u l t - - t h e irreflexivity of ~ follows from an extremely strong large cardinal axiom that is not, however, as strong as Axiom 15. (The axiom is "there is an n-huge cardinal for every finite n," where n-huge cardinals are also defined in terms of the existence of certain elementary embeddings.)
Conclusion I have presented a few theorems of mainstream mathematics that have been proved by set-theoretic techniques. In some cases we know that set theory is necessary; in other cases it has certainly proved convenient. The t h e o r e m s p r e s e n t e d are just a small percentage of such applications. One suspects that the existing applications are just a small fraction of the applications to be found in the near future. My thesis has been that set theory is an important tool of mathematics, whose use extends far outside the obvious. My motivation has been to encourage such use by the simple expedient of showing you some examples. If you are interested in reading more about the specific examples I've talked about, an abbreviated reading list follows. If you are interested in learning set theory, books at all levels and meeting all tastes are available.
Acknowledgments I would like to thank Fred McClendon, Rich Laver, and Juris Steprans for helpful comments and stimulating discussions.
Selected Reading List Algebraic topology A. Dow, P. Simon, and J. Vaughan, Strong homology and the proper forcing axiom, PAMS 106 (1989), 821-828. S. Marde~id and A V. Prasolov, Strong homology is not additive, TAMS 307 (1988), 725-744. S. Shelah, Can the fundamental (homotopy) group of a space be the rationals?, PAMS 103 (1988), 627-632.
Banach and Hilbert spaces A. Blass, Applications of superperfect forcing and its relatives, Set Theory and its Applications, (J. Steprans and S. Watson, eds.) Lecture Notes in Math. 1401 (1989), 18-40. S. Shelah and J. Steprans, A Banach space with few operators, PAMS 104 (1988), 101-105.
Free groups J. Steprans, A characterization of free Abelian groups, preprint.
Universal algebra R. Laver, The left distributive law and the freeness of an algebra of elementary embeddings, preprint.
Well-known applications H. G. Dales and W. H. Woodin, An introduction to independence for analysts, Cambridge: Cambridge University Press (1987). P. C. Eklof, Whitehead's problem is undecidable, Amer. Math. Monthly 83 (1976), 775-788. A. Kechris and A. Louveau, Descriptive set theory and the structure of sets of uniqueness, Cambridge: Cambridge University Press (1987). Department of Mathematics University of Kansas Lawrence, KS 66045 USA THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
69
Jet Wimp*
The New Physics Edited by Paul Davies Cambridge: Cambridge University Press, 1989. ix + 516 pp. US $49.50
Reviewed by Philip W. Anderson I am not a buyer of books on physics, hardly even a reader--so it represents high enthusiasm for Davies's effort that I strongly recommend acquiring this book, at least for your local library if not for yourself. Serious amateurs of other peoples' sciences, historians, sociologists, and philosophers of science need this book. And, particularly, so do physicists: those concerned with what other physicists are doing or those interested in presenting physics to general audiences or in teaching a modern physics course. It is not, probably, going to be a blockbuster bestseller for the lay audience, because the essays are not superficial and, I am afraid, are in most cases at something above the Scientific American level of difficulty (which is considered the standard). Nonetheless almost any chapter could be recommended to a bright undergraduate as a source for a term paper. The thesis from which Professor Davies starts is one that historians of science should take to heart: that the breakthroughs in relativity, q u a n t u m theory, and atomic physics of 1900-1930 were only the background for an extraordinary explosion of revolutionary discoveries; that the " N e w Physics" of 1990 is at least as different from that of 1930 as 1930 was from 1870. Historians and philosophers of science are fixated on every morsel of information about Einstein and Bohr, but the qualitative leaps which are embodied in gauge theory, broken symmetry, asymptotic freedom, scale invariance, chaos, localization, cosmology as an experimental subject, neutron stars, and many other examples discussed in this book, amount to at least as much of a revolution as relativity and quantum mechanics, and I fear that the history of these things is slipping away from us. It is these multiple revolutions that are chronicled in a series of chapters of variable but high quality and differing natures. Malcolm Longair and Frank Close have provided superb, lengthy reviews of * C o l u m n Editor's A d d r e s s : D e p a r t m e n t of Mathematics, Drexel
University, Philadelphia, PA 19104 USA.
the entire subjects of astrophysics and particle physics, respectively; one would wish a comparable review chapter had been commissioned as an overview of condensed matter physics to supplement the specialty chapters of Thouless, Leggett, Bruce-Wallace, and Nicolis, in the way that the Longair and Close reviews provide solid background for chapters on inflation by Guth and Steinhardt and on relativity by Will, and for chapters on particle field theories by Georgi and Taylor, as well as an overview by Salam. The recognition that there are three frontiers, the large, the ultrasmall, and the complex, is a pleasant surprise. All too often, the oversupply of underemployed particle physicists and relativists as possible authors focusing on their specialties leads to popular books' ignoring complexity as a subject. But now for the bad news. Davies is nowhere near as at home in the subject of complex behavior of matter (what has come to be associated with condensed mat-
There are three frontiers, the large, the ultrasmall, and the complex. ter physics, because those w h o call themselves condensed matter theorists are the physicists with the skills to make the relevant breakthroughs) as he is in the more glamorous and simpler realms of astro- and particle physics. Only a tyro could produce such a condescending remark as this: "It is only comparatively recently that complex systems have received systematic study as a physical science. In large part this is due to the advent of electronic c o m p u t e r s . . . ". Does he not realize the extent to which astrophysics is a computerized science? Has not particle experimentation led us all in use of computer time? Four out of five of his condensed matter chapters cover subjects where computers have played little role in discovery. The naive trust of a Stephen Hawking or Ken Wilson in the power of the computer does not resonate in the breast of those of us who actually deal in complex systems. We have developed new concepts and principles, just as have our friends in the large and small. If space was a consideration, we did not need two chapters on quantum gravity, a subject which is not yet physics in the sense that physics is an experimental
70 THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1 9 1992 SpringerVerlagNew York
science; we did not need so many repeats of the basic facts of the standard model, in three essentially historical chapters on particle theory. We did need, in addition to the overview of complex physics at least a chapter on the n e w p h y s i c s of d i s o r d e r e d m a t t e r - localization, percolation, and especially spin glass ideas with their opening to neural networks, computational complexity, evolution theory, and so on. This field is full of superb expositors, such as Toulouse, Sherrington, and Ramakrishnan. We needed an overview on the grand idea of broken symmetry, with its origins in condensed matter physics and its key role both in particle physics and in early universe theories, bringing out the unity in such concepts as topological singularities of order parameters, Higgs fields, and general relativity; of phase transitions in the early universe and in condensed matter; and of Higgs fields' origin in superconductivity theory. It is a pity that the lavishly illustrated and wellwritten chapters on nonequilibrium phenomena were not more balanced and open to other parts of this fascinating world. Nicolis's modest chapter gives no justification for Davies's extravagant e n c o m i u m that "these processes have been brought to fame by the work of Ilya Prigogine . . . . " and gives some, but not enough, coverage of other schools and views, of history, and of recent relatively rigorous work such as that of Langer, Cross, Levine, and others. I must resist the temptation to rewrite a good book. Nonetheless, once having realized that complexity belonged in it, Davies ought to have made the effort to give it a satisfactory treatment. One has to look to the next attempt for the definitive job of chronicling "the N e w Physics."
Department of Physics Princeton University Princeton, NJ 08544 USA
Joseph Liouville 1809-1882, Master of Pure and Applied Mathematics (Studies in the History of Mathematics and Physical Sciences 15) by Jesper Liitzen New York: Springer-Verlag, 1990. xix + 884 pp. US $98.
Reviewed by J. Dieudonn~ Among the famous French mathematicians out of the past centuries, there are two Cinderellas: D'Alembert and Liouville. They are the only ones whose complete works have not been collected in a scholarly publication (although m a n y efforts have been m a d e for D'Alembert, with no success due to lack of money); a n d t h e r e is as y e t no t h o r o u g h b i o g r a p h y of D'Alembert. The publication of a full-scale biography of Liouville is therefore very welcome. In this bulky volume, the first 260 pages are devoted to the life and career of LiouviUe, the remaining 490
pages to a description of his mathematical work. The author has spent 8 years in a painstaking and exhaustive examination of all the material he has been able to find on Liouville. Two of his sources are particularly noteworthy: 340 notebooks, which have been preserved in the Archives of the Institut de France, and the handwritten proc~s-verbaux (records of meetings) of the Bureau des Longitudes, which have never been published. Liouville's father was an army captain w h o had retired to Toul in Lorraine, the part of France where both his family and his wife's family were reared. The young Liouville thus had his early schooling in Toul, until he went to Paris to prepare for the entrance examination for the Ecole Polytechnique, which he successfully passed at the age of sixteen. He was schooled in analysis and mechanics by Andr6 Marie Ampere, who inspired Liouvflle's early research. On leaving the Ecole Polytechnique in 1827, Liouville entered the Ecole des Ponts et Chauss6es, as Cauchy had done 20 years earlier. But after 2 years of practical training, he was, like Cauchy, dissatisfied with the travels and physical hardships which the job of an engineer involved. He resigned in 1830 in order to live in Paris, as the only city where he could devote himself to research in mathematics. He also married in 1830, and during the next 10 years he actively sought academic positions, as prestigious as possible, and lucrative enough to support his family. Salaries were pitiful, and one had to accumulate several jobs to earn a decent living. Liouville of course started at the bottom of the ladder, and had to take several simultaneous teaching jobs in secondary schools. At the same time he was doing much mathematical research, and sent many notes and papers to the Acad6mie des Sciences and to the few scientific journals then in existence (Crelle's Journal was one of them, which had only been created in 1826). His first success came in 1833, when he was appointed to a professorship at the Ecole Centrale (an engineering school). In that same year he was also, for the first time, a candidate to the Acad6mie, and in 1835 to a professorship at the Ecole Polytechnique. However, he was defeated in both attempts. During those years Liouville was teaching at least 34 hours per week. Nevertheless, it is between 1830 and 1847 that he was most active in research, and the value of his work soon began to be highly regarded. Furthermore, in order to replace Gergonne's Annales de Mathdmatiques pures et appliqu&s, which ceased publication in 1832, Liouville decided in 1835 to create a new journal, of which he remained the sole editor until 1875. The Journal de MatMmatiques pures et appliqu~es, soon called "Liouville's Journal," immediately attracted papers by the best contemporary mathematicians. In the next few years, Liouville finally reached the top of the academic establishment. He was appointed professor THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992 7 1
at the Ecole Polytechnique in 1838, and was able to resign from the Ecole Centrale. Then, on the basis of his work in astronomy, he was elected in 1839 to the section of Astronomy in the Acad6mie, and in 1840 to the Bureau des Longitudes (members of both drew salaries). During the next ten years Liouville was a very busy man, and he complained that he had little time for research. However, he added politics to his activities, welcomed the 1848 revolution, and served a term in the Assembl6e Constituante; but he was not reelected in 1849. In 1850 he was chosen as professor at the Coll6ge de France (where he had taught as substitute for Biot from 1837 to 1843); he then resigned from the Ecole Polytechnique, where the teaching was much less to his liking. Apparently all these duties were not enough for him, and in 1857 he applied for a professorhip in mechanics at the Facult6 des Sciences, and was elected to that post--but he soon regretted the added burden he had taken. After 1870, poor health compelled him to leave the teaching position at the Facult6 to substitutes (Bouquet, Darboux, and Tisserand), although he kept the larger part of the salary. But he taught (with some interruptions) at the Coll6ge de France until his death in 1882. In his last years, from 1876 on, he suffered very painful attacks of gout and lack of sleep, and in 1880 he had to mourn the death of his wife and his son. Liouville u n d o u b t e d l y was the foremost French mathematician of the generation between Cauchy and Hermite. Although he was keenly sensitive (like so many mathematicians) to priority questions, on the whole his behavior towards colleagues and students w a s b e n e v o l e n t (especially in c o m p a r i s o n with Cauchy). He had an archenemy, the disreputable Count Guglielmo Libri (a mediocre man w h o finally had to be expelled from the Acad6mie for thefts of documents). He also repeatedly quarreled with the authoritarian and bad-tempered astronomer Urbain LeVerrier, w h o m he had supported as a young beginner. In competitions for seats in the Acad6mie or professorships, he refused to be a candidate while his best friend Charles Sturm was being elected to the Acad6mie; he acknowledged Cauchy's genius, but did not hesitate to compete with him for academic positions. He was a close friend of Dirichlet and greatly admired his work and that of Jacobi. As editor of his Journal, he was in contact with many foreign mathematicians, and he befriended beginners such as William Thomson (later Lord Kelvin) and Pafnuti Chebyshev. He also helped the younger French mathematicians such as Joseph Bertrand, Alfred Serret and Charles Hermite. And finally it must be recalled that he went out of his way to publish Galois's papers, which had remained unknown for 15 years. The second part of this biography consists of a description of Liouville's mathematical papers, arranged 72
THE MATHEMATICALINTELLIGENCERVOL. 14, NO. I, 1992
in chronological order, and including the notebooks and the records of the Bureau des Longitudes. Due to a lack of time, many published papers of Liouville are very sketchy, and the unpublished papers provide much light on his thoughts and methods. There are eleven chapters, each of which covers one of the parts of mathematics to which Liouville has contributed. For each one, the author carefully sketches its status prior to Liouville, then goes into much detail to explain Liouville's contributions. Often he had to reconstruct missing parts of arguments. Finally, the biographer sketches the evolution of the theory after Liouville, and the influence of his results. Liouville's range of interests is matched in the French school only by those of Cauchy and Poincar6. His position as editor of his Journal acquainted him with the work of all European mathematicians. In this review, there is only room to emphasize the main results for which he still is remembered. The most original ones, in my opinion, are the Sturm-Liouville theory of differential equations, the existence of transcendental numbers, and the so-called "integration in finite terms." The concept of an eigenvalue for a second-order linear differential equation with two boundary conditions can be traced back to D'Alembert. Fourier and Poisson had realized that there is an infinite sequence of eigenvalues, that they are real numbers tending to + 0% and that the trigonometric Fourier series are special cases of expansions in series of mutually orthogonal functions. However, their papers are in the style of the 18th century, without any attempt at rigorous proofs. The great progress brought by Sturm and Liouville was based on n e w ideas: first, Sturm's comparison theorems between two differential equations, which allowed him to prove the existence of eigenvalues; then Liouville's celebrated introduction of his transformation of both variable and function, which enabled him to transform the equation into the first example of a Volterra integral equation, yielding asymptotic expressions for the eigenvalues and eigenfunctions; and finally his proof of convergence of the "generalized Fourier series" by the method of successive a p p r o x i m a t i o n s ( c e r t a i n l y i n d e p e n d e n t of Cauchy's work, not to speak of Picard!). The only aspect of the theory which eluded him was the completeness of the system of eigenfunctions, only proved much later. Before Liouville, it had been proved that e, e2, ~r and ~ 2 a r e irrational numbers, and many mathematicians believed that e and ~r were transcendental. Liouville first tried to prove that e was transcendental, using Lagrange's method of approximation of roots of polynomials of Z[X] by continued fractions. He did not succeed, but that method led him to the fundamental idea that such roots are "badly" approximated by rational numbers: if a sequence of rational numbers PJqn
(in irreducible form) tends to an algebraic number x of degree M, then there is a positive constant c such that x
~
c Pn ~-~nn
for every n. It was then easy to create transcendental numbers as limits of more rapidly converging sequences of rational numbers. This discovery inaugurated the theory of transcendental numbers. It was marked by the proofs of the transcendence of e by Hermite (1872) and of the transcendence of ~ by Lindemann (1882), and has been very actively developed since then. The general problem of "integration in finite terms" is the following: g i v e n a d i f f e r e n t i a l e q u a t i o n F(x,y,y',y", . . . ) = 0 where F is an "elementary f u n c t i o n , " is t h e r e a s o l u t i o n w h i c h is also "elementary"? The problem had been raised by Condorcet and Laplace, w h o did not succeed in obtaining conclusive results. Abel, in connection with his famous theorem on abelian integrals, also studied it in a paper (which unfortunately was lost) in which he proved special cases of Liouville's later results. Liouville worked intermittently on the problem from 1832 to 1844. He first had to define by induction on an integer k, what is to be meant by "elementary functions of order k": functions of order 0 are the algebraic functions of x; a function of order k is one that is an algebraic function, or an exponential, or a logarithm, of a function of order k - 1. By ingenious arguments, he could then give conditions for a function to be "elementary." For example he could prove that f~ __r dt is not elementary, and that the Riccati equation + ay 2 = bx m is not "soluble by quadratures" unless dx 4, 1 for an integer n. However, with one has m 2n_+ the exception of Chebyshev, the subject did not attract 19th-century mathematicians, and it is only recently that it has been taken up in a much more algebraic context. Of Liouville's other important results, his global theory of doubly periodic functions, later developed by Hermite, won high praise from Weierstrass. In geometry, he pioneered the use of isothermal coordinates on surfaces, and he discovered the surprising fact that in R 3 the only conformal mappings are those composed of an inversion and a similitude. In mechanics, Liouville is remembered for his theorem on the invariance of volume in phase space for a Hamiltonian system; Jacobi had earlier found an equivalent result in a different formulation, but Liouville's proof was done without knowledge of that fact. There is an interesting discovery made by the author in Liouville's unpublished notes, written around 1845. He develops the idea of finding the eigenvalues of the Laplacian in a b o u n d e d domain by successive minimizing of the Dirichlet integral (what we n o w regard as the
generalization of the determination of the axes of an ellipsoid by extremal properties). This did not appear in print before a paper of 1869 by H. Weber, and is mistakenly called the Rayleigh-Ritz method. The only part of Liouville's publications that is not examined by the author is the endless series of short notes (sometimes 30 in a single year) which he published in his Journal after 1860, on the arithmetic of quadratic forms (mostly special cases of quaternaries). The author excuses himself for that omission, on the ground that he has not enough knowledge of number theory. Specialists in that field w h o have looked at these notes do not consider them as containing any worthwhile results, although I think it would have nicely concluded the book to have enlisted one of them to write a short appendix which would have explained the question in some detail. Except for this missing part, the book is likely to remain the definitive account of Liouville's life and work. One slight defect concerns the French language. The spelling of French words is completely erratic, and there are at least two rather funny mistakes in the translations: "aggregated" for agr~g~ (a certification for teachers) on p.29, and "regrettable" for regrettd (lamented!) on p.40.
120, avenue de Suffren 75015 Paris, France
A Budget of Trisections by Underwood Dudley New York: Springer-Verlag, 1987; xv + 169 pp.
Reviewed by Ian Stewart If I had before me a fly and an elephant, having never seen more than one such magnitude of either kind; and if the fly were to endeavour to persuade me that he was larger than the elephant, I might by possibility be placed in a difficulty. The apparently little creature might use such arguments about the effect of distance, and might appeal to such laws of sight and hearing as I, if unlearned in those things, might be unable wholly tO reject. But if there were a thousand flies, all buzzing, to appearance, about the great creature; and, to a fly, declaring, each one for himself, that he was bigger than the quadruped; and all giving different and frequently contradictory reasons; and each one despising and opposing the reasons of the others--I should feel quite at my ease. I should certainly say, My little friends, the case of each one of you is destroyed by the rest. With his characteristic dry h u m o u r and c o m m o n sense, so did Augustus De Morgan introduce his classic work A Budget of Paradoxes [2]. The word "paradox" is not used in the sense of, say, Russell's Paradox; but with its alternative meaning: "something contrary to general opinion." A paradoxer is a perpetrator of paradoxes; and the Budget collects, and ruminates about, paradoxers. Not every paradoxer is a crank, b u t - - t o quote U n d e r w o o d D u d l e y - - " u s u a l l y they are." In THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992 7 3
support of which thesis he offers his own Budget. Most of De Morgan's mathematical cranks were circlesquarers. "I like to think the Budget was the main cause of their decline." Dudley's are trisectors, and he would dearly like to do the same for them. Using a garotte if need be. Cranks are tiresome creatures. Most of mine are Fermat's Last Theoremers, I assume because I wrote a book about it. But I also wrote one about trisections, cube-duplications, and circle-squarings, and nobody ever sends me those. Most mathematicians throw crank literature away--Dudley collects it. "I want it all." It would be fitting if "Underwood Dudley" were an anagram of "Augustus De Morgan," but the nearest I can get is "Oryudled De Wodun." It's closer than most trisectors get to their goal. The facts of the matter are simple. The Greek geometers offered no ruler-and-compass construction to trisect an angle: others subsequently wondered if such a thing were possible. In 1837 Pierre Laurent Wantzel proved that you can't trisect an arbitrary angle with a ruler (unmarked, as is the rule----ha ha in such games) and compasses. The gist of the proof is that trisecting an angle of 60~ is equivalent to constructing a root of the irreducible cubic x 3 - 3x - 1, by starting with the rationals and forming a succession of square roots. In a modernised version, a field whose degree over Q is a power of 2 cannot contain a subfield of degree 3. The matter would seem to be conclusively resolved, inasmuch as trisection is equivalent to proving that a number that is not a multiple of 3 is a multiple of 3. But awkward facts carry no force for trisectors. The Greeks were probably clever enough to have guessed that no solutions are possible, but they lacked the techniques (coordinate geometry and algebra) that are needed to prove it. They did produce a whole series of solutions by transcendental methods, and A Budget of Trisections opens with a survey of these. The simplest and most elegant is that of Archimedes (Fig. 1). Where do trisectors come from? How can you recognise one? There are some typical characteristics. First, they are almost all male. "There are only two females in the Budget of Trisections, and, from what they wrote, it is possible that they are not genuine trisectors. They sounded a bit tentative and may not have known the trisection is impossible." The archetypal trisector is an elderly gentleman who heard about the problem as a schoolchild, but did not begin to work on it until many years later-----often in retirement. The trisection bug resembles AIDS---it is a deadly disease with a lengthy incubation period. There are some young trisectors, but they generally accept that they are wrong, if only out of deference to authority. The true crank never admits to being in error. "My teacher said that it was the opinion of mathematicians that a solution was impossible. For over 55 years that puzzle 74
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
J A Figure 1. To trisect B O A draw a circle centre B through O to cut one leg at A, and align a ruler O T such that B T is parallel to O A and T X = OB. Observe that this construction requires a m a r k e d ruler.
has b u g g e d m e . " On w h i c h D u d l e y c o m m e n t s : " 'Opinion,' did the teacher really say that? You would hope not, but what did he or she say?" Trisectors are persistent: "In more than 12,000 working hours I have in the course of 40 years found this solution . . . . " Six years of productive time wasted. Teachers should be careful what they say to impressionable pupils. It seems to have been the same in De Morgan's day, although there are only two trisectors in the original Budget. His entire review of one reads: "The consequence of years of intense thought," very likely, and very sad. Trisectors do not u n d e r s t a n d w h a t " i m p o s s i b l e " means in mathematics. " H o w could men of science be so stupid? Any scientist or mathematician who declares that a thing is impossible is showing his limitations before he even starts." Trisectors, says Dudley, are a menace. They waste their own time, and they waste the time of professionals to w h o m they write. The only useful thing you can do with a trisector is write a book about him and hope to sell lots of copies. Budget deserves huge sales: Underwood Dudley writes with the same dry humour as De Morgan, and this is the first mathematics book for a long time that I have been unable to put down. Buy it, get your department to buy it, get your institution to buy it, and tell all the local High Schools to buy it. Back to trisectors. Do not give one a testimonial, especially if it wriggles hypocritically on the verges of half-truth: "I was not able to detect any fallacy in your work." (Translation: I never made the effort to.) Do not give this person anything that can even be represented as a testimonial. Never wish him or her good luck in the quest. "[To get rid of a trisector] by encouraging folly is not right." Remember that there is no need to be polite, and remember that a crank cannot be taught. It is not easy for professional teachers to accept this, but if
0 tan ~ = 2 sin 0, J
Figure 2. Typical trisector's diagram. you don't, you'll learn the hard way. One mathematician corresponded with a trisector for seven years. Most trisectors are ignorant of mathematics, but not all. One used Desargues's Theorem in his proof. One had a Ph.D. in mathematics and made a living as a mathematics teacher. Even if ignorant of mathematics, many are well educated, often "self-made men." Having succeeded with Mammon, they imagine Mathernatics will succumb to a similar assault. Trisectors think that the construction has practical importance. They are convinced that the world will beat a path to the successful trisector's door. They draw complicated diagrams, nearly always too complicated: Figure 2 is typical. Equivalent constructions are usually simpler, and it's usually easier to locate the mistake by simplifying first. Maybe that's w h y trisectors' diagrams are complicated. There is some kind of natural hierarchy of trisections. Any plausible trisection has to look reasonable to the eye, so it must yield some approximate formula for 9 0 sm ~0 or tan ~. One of the commonest such formulas was found (implicitly) by the Very Reverend J.J.C., who in 1931 was "the President of a third-rank but respectable u n i v e r s i t y . " The m a t h e m a t i c s faculty m a y have thought twice before explaining the error! His approximation is sin
0 sin 0 3 2 + cos O"
Computation reveals the disagreement between the two expressions to be no worse than 1%. Elementary trigonometry shows that they are unequal. If you nod in sympathy, recognising that it is difficult to spot fallacies w h e n the formula is only implicit in a geometric construction, here are two formulas that are explicit in Father C's pamphlet:
0 cot ~ = 2 cosec 0.
(Corollary: 1 = 4.) Trisectors do not understand logic. "Those who are skeptical should offer something more than rhetoric or argument in order to disprove geometrical facts . . . . If it be denied that the trisectors pass through these p o i n t s . . , let them show by ruler and compasses where these lines and points are.'" (To prove my construction is wrong, you've got to find one that's right.) They invoke special pleading: "The solution, as it stands, will divide any given angle into three equal component angles, yet it might not be a true trisection." (I know trisections are impossible, but my construction just divides angles equally into three.) "My construction is not a trisection in the traditional sense. It is the expansion of angles in the ratio 3:4." (I construct 40/3, not 0/3, so that's all right.) "One of the implications of G6del's great work must then be that all negative proofs of impossibility are limited to the extent that a process is impossible only within the boundary of presently k n o w n mathematics." (Some statements can't be proved, therefore the impossibility of trisection can't be either.) They do not understand the interconnectedness of mathematics. "[I will prove] the constructions by Plane Geometry. They cannot be proven by trigonometry." (Someone disproved my construction using trigonometry: no fair!) They ask advice: "Prof. Harry Phelps, United States Naval Academy; Prof. Thomas S. Barrett, London, England, and other geometers show that trigonometric calculations fail to agree with above Trisection of Angles." But they don't take it: "Present Text Books of Mathematics are fallacious and new Tables and Text Books are required." Another, looking into a text that gave a modern version of Wantzel's proof, similarly concluded that "It becomes a necessity to revise ring theory." Some are plain daft. "The following construction will prove that the 60 degree angle can be trisected." Method: Start with an angle and triple it. Some choice of initial angle "may by chance trisect a 60 degree angle." Others are plain weird: "Select 3 = unit O's series o 'successive approxim8ns' by arriving @ I radius taken 3 times from pt o intersection o bisector-subtended arc--1 leg o angle recognized according 2 completeness property as it applies 2 plane geometry . . . . " Trisectors lack the normal mathematician's distrust (which to my mind is overdone, but that's another story) of the media. They positively court the press. The Associated Press carried stories about Father C's trisection in August 1931, and again on December 12, 15, 19, and 29. Some readers wrote to point out that the problem was already known to be impossible; several more sent their own trisections (one had five methTHE MATHEMATICAL 1NTELLIGENCER VOL. 14, NO. 1, 1992 7 5
ods). Trisectors can be masters of publicity: one advertisement claimed to have won "Top Awards at the 1959 Chicago City and Illinois State High School Science Fairs!!", to which Dudley responds: "Hard to believe, and dismaying if true." The Cincinnati Post and TimesStar of 1966 did say that " m a n y mathematicians say it just can't be done," but failed to make it clear that all of them do. The obligation to provide a balanced viewpoint presumably means that 2 + 2 = 5 is as deserving of editorial space as 2 + 2 = 4. The media may perhaps be forgiven for not knowing about Wantzel, but in 1935 Associated Press put out a story about "DETROIT MATHEMATICIANS STIRRED BY YOUTH'S METHOD." The construction uses multisple bisections to construct what is plainly and visibly/~ of the angle (hint: 5 = 1 + ~). Even a newspaper ought to be able to carry out the division. This is pathetic. To whet your appetite, here are three trisections, a sample from the final chapter that makes up the bulk of Budget. Try to disprove them (not a bad exercise for a mathematics class, but you'd better consult the book if you want a lot of constructions that are simple enough to tackle in high-school or freshman trigonometry).
Trisection 1. Trisection of an A n g l e w i t h Ruler and Compass (R.J., 1986) To trisect/_AOB trisect chord AB and join to O. Yes, really. Accompanied by the claim, "More than 50 professors of mathematics (many of them Ph.Ds) have evaluated my paper and approved it." As Dudley says, "What could he mean? Was he lying, or was it that anything short of physical violence counted as approval?"
B
O
D
E
A
Figure 3. The fruit of six years' hard labour. where D = 52200 + 27600 cos 0 - 20750 cos20 + 3650 cos30 - 200 cos40. You do get a superior kind of crank for 12,000 hours of work. Don't set this one to high-school students.
Trisection 3. T h e o r e m - - A n y Angle Can Be Trisected (K.B.S., 1972) For the diagram see Figure 4. Underwood Dudley says he has lost the detailed description of the construction; he'd like to hear from anyone who can reconstruct it from the diagram, even if they don't have proof that it's wrong. Write to him at the Department of Mathe-
A
H
Trisection 2. The Thrice D i v i s i o n of an Angle b y the S i m p l e Use of Compass and Ruler (J.E., 1973). V
T
This is the one that took 12,000 hours. In Figure 3 draw OC bisecting BOA. Divide OB and OA into 9 equal parts, to get D, E, F, I. Draw a line through D parallel to OB to get G and extend OG to get H. Draw IJ parallel to AB and find T, the trisection point, by drawing an arc centre I radius JC. The construction is accurate to within 8' for angles between 0~ and 145~. Trigonometrically, it amounts to asserting that
tan
0 3
D
I
-
(sin 0)(1608 - 676 cos 0 + 68 cos 2 0 (6 - cos 0)(168 + 100 cos 0 - 68 cos 2 0 + V ~ ) 76
J ~ /
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
"
S
Figure 4. The mystery trisection.
M
N
C
matics and Computer Science, DePauw University, Greencastle, IN 46135-0037 USA. Incidentally, he'd love to receive all of your mathematical crank mail, on trisections or not. Successful books usually demand sequels. Watch out for A Budget of Fermatters. "You may as well knock your head against a stone wall to improve your intellect as attempt to controvert my proofs," the circle-squarer James Smith wrote to De Morgan. "I thought so too, and tried neither," August-us wrote. Dudley echoes him: "Thus ends the Budget of Trisections. Reader, do not add to it." But even if trisectors cannot be corrected, newspapers can. As a profession we owe it to ourselves and the rest of the human race to correct abuses of mathematics wherever they occur. A common thread running through trisection tales is the belief that mathematical theorems are merely matters of opinion. G6del notwithstanding, we need to make it clear that the nature of mathematical truth runs deeper than that. Otherwise the trisectors will be following the creationists and demanding equal time in our textbooks. That mathematics is not a matter of opinion, that mathematical truth cannot be decided by votes, but that innumeracy is rampant in our society and many people think that it can be, was driven home recently b y a bizarre episode in the syndicated column Ask Marilyn [4,5]. It is written by Marilyn vos Savant (listed in the Guinness Book of World Records as the person with the world's highest IQ). "Suppose you're on a game show, and you're given a choice of three doors. Behind one door is a car, behind the others goats. You pick a door; then the host opens another door behind which he knows there is a goat. Should you now switch to the single remaining door, or stay with your first choice?" Marilyn's answer: the optimal strategy is always to switch. Switching gives you a 2/3 probability of winning; staying only 1/3. Many of her readers, at universities, colleges, or research institutes, objected strongly: 9 "You blew it! As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful." 9 "'There is enough mathematical illiteracy in the world, and we don't need the world's highest IQ propagating more. Shame!" 9 "'Your answer to the question is in error. But if it is any consolation, many of my academic colleagues also have been stumped by this problem." 9 "Your answer is clearly at odds with the truth." 9 "May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again?" 9 "'How many irate mathematicians are needed to get you to change your mind?" 9 "I am in shock that after being corrected by at least
three mathematicians, you still do not see your mistake." 9 "Maybe women look at math problems differently than men." 9 "You're wrong. But look on the serious side. If all those PhD's were wrong, the country would be in very serious trouble." Marilyn received "thousands of letters, nearly all insisting that I'm wrong, including one from the Deputy Director of the Center for Defense Information and another from a research statistician at the National Institutes of Health." In all, 92% of letters from the general public disagreed with her; the same was true of 65% of the letters from universities. Trouble is, Marilyn was right. (Exercise for the reader: w o r k out why.) All the comments quoted above were from men, incidentally, and most of them were mathematicians. Fellow mathematicians, do not despise the ignorance of trisectors! Many of you made even wilder accusations with far less justification. Trisectors at least find good approximations, whereas your favoured answer (equal probabilities whether or not the contestant switches) is wrong by more than 16%. Indeed you fell into a trap that dates back at least to J. Bertrand's Calcul des Probabilit~s of 1889. It is usually known as Bertrand's Box Paradox, and as Eugene Northrop [3] observes, it "has been used as an illustrative example in almost every subsequent textbook.'" The reader w h o suggested consulting a probability text would have been well advised to follow his own advice. (The chauvinist pig is beneath contempt.) A version of the problem involving three prisoners and a prison governor is described in detail by Martin Gardner [1]. So not only did many of Marilyn's critics let their sense of outrage and hyperbole run away with their good taste: they didn't do their homework as professionals. "If all those Ph.D.s' were wrong, the country would be in very serious trouble." Hmmmm. References
1. Gardner, Martin, More Mathematical Puzzles and Diversions from Scientific American, London: Bell & Sons (1963). 2. De Morgan, Augustus, A Budget of Paradoxes, 1st edition 1872; 2nd edition edited by David Eugene Smith, Freeport, New York: Books for Libraries Press (1915), reprinted 1969. 3. Northrop, Eugene P., Riddles in Mathematics, Harmondsworth, U.K., Penguin Books (1960). 4. Savant, Marilyn vos, "Ask Marilyn," Parade Magazine (Dec. 2, 1991), 25. 5. Savant, Marilyn vos, "Ask Marilyn," Parade Magazine (Feb. 17, 1991), 12. Mathematics Institute University of Warwick Coventry, CV4 7AL United Kingdom THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992 7 7
Polynomials (Problem Books in Mathematics) by E. J. Barbeau New York: Springer-Verlag, 1989. xxii + 441 pp. US $59 R e v i e w e d by George Szekeres
When I was y o u n g I was told that a polynomial was a function (whatever that meant, the word "mapping" hadn't entered our vocabulary as yet) described by some numbers called coefficients and a mysterious something called a variable, which seemed to lead a Jekyll-and-Hyde-like existence; occasionally it materialised into a respectable number and supplied a value for the function, but other times it remained an undisclosed something, ready to pounce when the opportunity presented itself. Sometimes it did not materalise into the right sort of number, for instance when one looked for the roots of a polynomial. Later on, when I learned about complex polynomials (from Knopp's Funktionentheorie), this particular mystery disappeared; the polynomial became a complex analytic mapping with beautifully simple properties, paving the way to the wondrous world of algebraic functions. It was much later (I was already twenty-eight) when I finally found out what a polynomial really was. It happened on board the refugee ship Victoria, sailing towards some distant destination (not Australia yet) whose only merit was to be as far away from European "civilisation" as geographically possible. As I huddled with my wife in the corner of a crowded boat, clutching our only worldly possession of value, a copy of Van der Waerden's Moderne Algebra (a parting present from a dear friend), suddenly the truth emerged: a polynomial is a member of an algebra R[x] over a ring R and basis a commutative infinite semigroup with identity, generated by a single element x. (Actually Van der Waerden used somewhat dated terminolo g y - w h o would ever speak of a hypercomplex system nowadays?) Here, then, was a beautifully crisp definition of great generality, ready to cope with any eventuality. Polynomials could be over another polynomial domain, or a rational function field, or integers, or a finite field, or even matrices. Ed Barbeau, on the first page, maps out a more mundane course, reminding me of my days of innocence: A function of t is a polynomial if it can be put in the form a,t n + % _ i t ~-1 + . . . + alt + a o. Barbeau's definition is refreshingly simple and perfectly suited for his purposes, but also sets the tone for all that follows. This book is not a systematic text but a problem book, to teach the reader not what polynomials are but how to find out things about them. The message of the book is clear: First you must become interested in and learn what to do with your subject--understanding will come with experience. Who are the prospective readers of this book? Bright and motivated high-school students, or so it seems. But just look at the first set of exercises in Chapter 1, 78
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1, 1992
Section 1: 1. State the degree and the constant, linear, and leading coefficient of the following polynomials: 7t5 - 6t 4 + 3t 2 + 1, etc. 2. Give examples of a monic polynomial of degree 7, etc. Not exactly stimulating exercises for a bright student. As we drift towards Exercise 20, the exercises acquire more bite. Then comes a surprise. Straight after Exercise 20, under the heading Explorations, you are hit on the head with problems such as Find a polynomial with more than one term whose square has exactly the same number of terms as the polynomial itself. Are there polynomials whose square actually has fewer terms? (What is the answer? Numbers at this stage are just ordinary real numbers.) The next problem in this section leads straight into the wilderness of the TarryEscott problem. Clearly this section speaks to an entirely different audience than the exercises. N o w the reader may start wondering again; you may be bright and motivated but not yet in a position to face research-type problems. But don't give up. If you persevere to the end of the chapter--and there are enough snippets to keep your interest alive---then you will be richly rewarded. At the end of each of the seven chapters there is a section headed Problems, and here Ed Barbeau builds on his vast experience with competition and olympiadtype problems. These sections are (to my mind) the real strength of the book. A treasure trove of challenging problems, they are taken from various sources, mainly from Canadian and American problem papers such as the Putnam Competitions, and journals such as Crux and the American Mathematical Monthly. They give the reader plenty of opportunity to test his skill and score victories, and in the process whet the appetite for further explorations. Hints are given at the end of each chapter, and if these don't suffice, solutions are given at the end of the book. On the way you acquire, mainly through the exercises and your own efforts, a store of basic information about polynomials that will stick to the mind more effectively than if it had been acquired ready-made from a standard textbook. Strangely, I found the Explorations sections less attractive. Sandwiched between sets of Exercises, they seemed to be out of place. They are neither Erd6s-type avalanches of unsolved problems, nor a Szeg6-P61ya-type Aufgabensammlung with its purposefully systematic collection of problems, progressively leading to monuments of classical analysis. I think I would have preferred to see all Explorations put in a single final chapter. This is exactly what h a p p e n e d w h e n I was reading through the book at first skipping the Explorations altogether, and then going back to them (together with the notes added at the end of the book) bit by bit. This made for somewhat jerky and not very convenient reading,
which is a pity because these sections are full of interesting material. You learn, in the form of problems often left up in the air with a question mark, about such things as rook and chromatic polynomials (not knot polynomials), the range of real polynomials in several variables, Fibonacci numbers and Hilbert's tenth problem, second-order recurrences, Sturm sequences, and so forth. The material is presented, not as accomplished theories, but as food for the investi-
gating mind. Ed Barbeau deserves our gratitude for bringing all this together in a particularly stimulating forIn.
School of Mathematics The University of New South Wales P.O. Box 1 Kensington, NSW 2033 Australia
THE MATHEMATICALINTELLIGENCERVOL. 14, NO. 1, 1992 7 9
Robin Wilson*
Probability and Statistics (1750-1850)
There has been a number of stamps relating to probability and statistics. Important early texts on probability theory were written by Marie-Jean Condorcet (17431794) and Pierre-Simon Laplace (1749-1827), both of w h o m have b e e n featured on French stamps. At around the same time, George Louis Leclerc, Comte de Buffon (1707-1788) described his "needle experiment" for approximating the value of "rr, while Carl Friedrich Gauss (1777-1855) w o r k e d on the m e t h o d of least squares and the normal (or Gaussian) distribution; this latter distribution has appeared on a mathematical postmark. Noteworthy, too, was the distinguished Belgian statistician Lambert Adolphe Jacques Qu~telet (1796-1874), w h o proposed the notion that naturally occurring distributions tend to follow a normal curve, and who established a central statistical bureau that was imitated around the world.
If you are interested in mathematical stamps, you are invited to subscribe to Philamath. Details may be obtained from the Secretary, Estelle A. Buccino, 135 Witherspoon Court, Athens, GA 30606 USA. *Column editor's address: 80
Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA England.
THE MATHEMATICAL INTELLIGENCER VOL. 14, NO. 1 9 1992 Springer Verlag New York
