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:
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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
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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
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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
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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
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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
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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
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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,
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(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~ 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
