The Mathematical Intelligencer

Note Lovely Math Wedding Cakes PAVEL PYRIH math baker, Jimmy Weddingcake, is preparing a brochure to advertise his lov...

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Lovely Math Wedding Cakes PAVEL PYRIH

math baker, Jimmy Weddingcake, is preparing a brochure to advertise his lovely math wedding cakes. He would like to prepare quite a large brochure, with uncountably many pages. Each cake has infinitely many layers; he can put glazing as he likes. He uses only two traditional happy wedding patterns, each layer being one of the two. See Figs. 1 and 2 for the patterns. Jimmy needs to prevent the situation where a cake ordered according to the brochure slips during the baking process into the shape of some other cake in

A

the brochure. For example, the infinitelevel cake A labelled by the sequence 12111... in Fig. 3 can, during the baking process, collapse into the shape of the infinite-level cake B corresponding to the sequence 111.... You see, the whole Level II can ooze down over Level I, with M going to the edge, with the whole section shown in blue going to f(K), the section p shown in brown going to the left portion of I, and the rest, the section q, going to the right portion of I. Well, Jimmy finally prepares an uncountable collection of cakes to be photographed for the brochure. They’re

Figure 1. Pattern 1.

Figure 2. Pattern 2. Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

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A

Wfcn g ¼ ftg [

1 [ n¼1

Wn [

1 [ n¼1

ln [

1 [

rn ;

n¼1

where t is a singleton (the top of the cake), Wn is a copy of Xcn of diameter 1/2n (the nth layer of the cake) and ln and rn are spirals from Wn to the corners of Wn+1, respectively (the glazing streams).

OBSERVATION If the cakes Wfcn g and Wfdn g are comparable, then the sequences {cn} and {dn} of digits 1 and 2 are equal. PROOF. Suppose that f is a continuous mapping from Wfcn g onto Wfdn g: Notice that:

B

1. X1 and X2 are not comparable under a continuous mapping. (Hint: Use the number of maximal arcwise connected subsets in both continua.) 2. The image of the first level in Wfcn g must meet the first level in Wfdn g. (Hint: Otherwise the mapping is not onto, due to the glazing streams and the definition of cakes.) 3. The first levels of both cakes have to be of the same pattern; this means c1 = d1. (Hint: Use the number of maximal arcwise connected subsets in both continua and their shapes.) 4. Similar arguments work on the other levels as well. This implies cn = dn for n 2 N by induction on n.

Figure 3. Cake A collapsing into cake B.

not only not the same, they are even protected from this sort of collapsing. Did he have to cheat?

The Cakes Jimmy will use the glazing between each two successive layers to prevent the layers getting confused during baking. The glazing streams will go from the bottom corners of the second layer lying on the first layer, and similarly on the other levels (see Fig. 4). (The glazing streams are in 3D and are only sketched in the figure.) Observe that the cakes for any two different sequences have different pictures. Moreover any continuous baking keeps the sequence of patterns unchanged. The brochure can have an uncountable collection of distinct designs. No cheating was needed. 2

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Math of the Math Cakes A continuum means a nonempty compact connected metric space. An arc is any space which is homeomorphic to the closed interval [0,1]. A set A X is said to be arcwise connected provided that, for any two points in A, there is an arc in A containing both points. For two disjoint continua X and Y, a spiral from X to Y is a space Z homeomorphic to (0,1) such that the disjoint union X [ Y [ Z is the smallest continuum containing Z. We say that a continuum X is comparable with a continuum Y if there is a (continuous) mapping f from X onto Y or a mapping g from Y onto X. Let X1 and X2 denote the continua introduced in Figs. 1 and 2. For a given sequence {cn} of digits 1 and 2, the cake Wfcn g is a continuum formed by the disjoint union

Then sequences {cn} and {dn} are equal.

COROLLARY There are uncountably many cakes which are not comparable by a continuous mapping.

Remark The first uncountable collection of incomparable continua was created in 1932 by Z. Waraszkiewicz in [W 1932]. The continua were formed by a spiral from a singleton to the unit circle, and the idea was to travel several times clockwise and several times counterclockwise around the circle. Replacing the circle with an arc, M. M. Awartani, in 1993, created another family in [A 1993]. Using these families, we can conclude that there are uncountably many ways to form each glazing

continued by C. Islas in [I 2007]. For another class of incomparable continua, see, for instance, [B 1971]. ACKNOWLEDGMENT

This research was supported by the grant MSM 0021620839. BIBLIOGRAPHY

[A 1993] M. M. Awartani, An uncountable collection of mutually incomparable chainable continua, Proc. Amer. Math.

Soc.

118

(1993),

239–

245. [B 1971] D. P. Bellamy, An uncountable collection of chainable continua, Trans. Amer. Math. Soc. 160 (1971), 297– [I 2007]

304. C. Islas, An uncountable collection of mutually incomparable fans, Topology Proc. 31, no. 1 (2007), 151–161.

[M 2006] P. Minc, An uncountable collection of dendroids mutually incomparable by continuous functions, preprint (2006), available online at http:// web.mst.edu/*continua/cdendr. pdf. [W 1932] Z. Waraszkiewicz, Une famille inde´nombrable de continus plans dont aucun n’est l’image d’un autre, Fund. Math. 18 (1932), 118–137.

Figure 4. A cake with the glazing streams.

stream in Jimmy’s cakes. I do not use this here (so I don’t prove it). Jimmy’s arguments are simpler (maximal arcwise connected subsets, and counting).

By the way, P. Minc in 2006, replacing the circle in Waraszkiewicz’s family by a continuum of shape ‘‘T,’’ constructed another collection of incomparable continua in [M 2006]. Awartani’s idea was

Department of Mathematical Analysis Charles University Prague 8, CZ 18675 Czech Republic e-mail: [email protected]

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Viewpoint

The Role of the Untrue in Mathematics CHANDLER DAVIS*

The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editors-in-chief endorse or accept responsibility for them.

* This article is the text of a talk presented to the Joint Mathematics Meetings, Washington, D.C., USA in 2009. 4

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e obtain perspective on any human activity by standing outside it. If mathematics were really concerned mostly with truth, or entirely with truth, then we may imagine that in order to appreciate it fully we might be obliged to position ourselves squarely in a world of fallacy: get a little distance on it. I am not proposing anything quite that quirky. I will speak just as unparadoxically as my subjects permit, but even so, there will still be slippery borderlines. There are plenty of ways in which untrue assertions demand our respect. Mathematical reasoning can be applied to untrue assertions in the same way as to true ones. We may say to an 18th-century geometer, ‘‘Let us assume that through any point not on line l there is more than one line which fails to intersect l,’’ and our interlocutor, no matter how absurd this assumption seems, will be able to scrutinize our deductions in the same way as if we had made less preposterous assumptions. It is necessary for us to be able to reserve judgement in this way — for consider this example: we may say, ‘‘Suppose if possible that p/q is a fraction in lowest terms equal to the ratio of a square’s diagonal to its side,’’ and we may want to establish that that supposition is not admissible. Then it is important that our interlocutor agree as to what reasoning is valid. If the rules changed, if there were one way accepted for reasoning about true assertions and a different way for all others, then there would be no way to prove anything by contradiction. Let me nail this observation down a little more snugly. It is not an observation about tertium non datur. Maybe our interlocutor is skeptical about the notion that every meaningful proposition must be either true or false; that’s all right: even if we allow that truth status need not be a binary alternative, still when we want to argue that a proposition fails to have some truth status, we

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may have to use methods of argument that do not depend on that truth status. Lawyers are clearer about this, perhaps. They frequently say, ‘‘Supposing, arguendo, that…’’ and proceed, arguing temporarily as though they were conceding a premise that they are not at all willing to concede. I like that notion of arguendo. Mathematicians used to use it more than they do today. In particular, the whole magnificent edifice of classical continuum mechanics seems to me to be a case of supposing arguendo that continuous media obey laws of particle mechanics which, however, the Bernoullis and Euler did not really expect them to obey: a dollop of matter has mass as though it were localized at a point, and Newton’s laws are invoked even in problems where the idealization to point masses would be nonsensical. Maybe I’m on safer ground if I cite a different example: the development of topology of manifolds in the 20th century. It was plain that certain aspects of manifolds deserved study, but it was not clear what they applied to — whether to certain chain complexes, or to certain abstract topological spaces, or what. The study proceeded arguendo by deductions as reliable as they could be made under the circumstances, and discrepancies between different entries into the subject were tidied up as well as might be. (The forging ahead and the tidying up are seen together in a book such as Raymond Wilder’s.) A chain of reasoning belonging to such an intellectual domain may turn out in the future to relate two chapters of truth, or it may turn out to be part of a great reductio ad absurdum; we deal with the deductive chain arguendo, independent of its ultimate fate; the tests of its soundness are the same either way. Even more persuasive for my purposes today is another centuries-old habit of mathematicians: to find what value of the variable makes a function zero, one pulls a guess out of the air

and substitutes it into the function, finding of course that the function fails to be zero there; then one extracts information about the problem from the failed guess. Such calculations by regula falsi were used off and on over the centuries to become systematized and exceedingly fecund from the 16th century on. Their naı¨ve motivation must have been, back then, like reasoning arguendo, and this doesn’t seem far off to me even in retrospect. One would be justified in 1100 (or in 1600) in trying 1.5 to see whether its square was about 2, even if one did not have an algorithm of root-finding and therefore did not know how taking this stab at H2 would lead one to a better guess. One would anticipate that working with the blind guess would teach something. And I observe in this context too, of course, that in order to hope to be taught anything, one would surely commit to reasoning the same for a wrong guess as for a correct one. The freakish notion that mathematics deals always with statements that are perfectly true would disallow any validity for this example, or for most discussion of approximation, for that matter. It would insist that ‘‘p = 22/7’’ be banned from mathematics as utterly as ‘‘p = 59’’; half of our subject would be ruled out. There is no danger that mathematicians of this or any other age would really try to live by this freakish doctrine, but it persists in everyday discourse about mathematics. I began with discussion unrelated to approximate answers in order to emphasize that restriction to true statements would be crippling to even the most finite and discrete branches. As we begin to examine the useful roles of less-than-true statements in our field, we have at once these two: • Falsehood is something to avoid. We find it useful to reason by contradiction. • Statements teach us something by their behaviour in reasoning arguendo, regardless of their truth value. Then to continue the examination of the subject, we must recognize and defy the tradition that mathematics is truth, the whole truth, and nothing but the truth. Permit me to call it ‘‘truthfetishism’’, though I accord it more

respect than the playful label suggests. This tradition has taken many forms, and you may not agree with me in lumping them together. By the 19th century, it had become clear that some true statements are contingent whereas others are essential to the cogency of human reasoning. (Thus it is merely a matter of observation that the South Pole is not in an ocean, and we can talk about it being in an ocean, even ask whether it once was; but we can not talk in any cogent way about the South Pole being on the Equator.) Truth-fetishism applied mostly to truths which were not contingent. In the decades after George Boole’s ‘‘Investigation of the Laws of Thought’’, it became conventional to hold that, at least in philosophical and mathematical discourse, all true statements were equivalent, so that any true statement implied every true statement, and a false statement implied every statement whatever. If you have ever tried to get a freshman class to swallow this, you have probably appreciated the trouble Bertrand Russell had in his day. Yes, I find it irresistible to retell the Russell anecdote: replying to the lay listener who objected that surely ‘‘2 = 1’’ does not imply that you are the Pope, Russell’s put-down was, ‘‘You will agree that the Pope and I are two; then if 2 = 1 it follows that the Pope and I are one.’’ Now his verbal cleverness is charming, but it is off the point of the listener’s objection, as the listener surely saw and we may hope Russell did as well. He was insisting that the only way to deal with truth and validity was the truth-fetishistic way, and that the only way to understand implication was material implication: that saying ‘‘A implies B’’ must be understood as saying ‘‘either B or not-A’’. Boole’s ambitious project of finding the laws of thought deserves the admiration it got. What kind of law should we hope for? We don’t really want a prescriptive law (‘‘thou shalt think in thy father’s way’’) or a normative law (‘‘here is the better way to think’’), we want an empirical law, one that refers to thinking that is actually done. On the other hand, we can’t insist that the laws of thought encompass our occasional pathology and our frequent simple blundering (to do that would be a formidable, never-ending task); so there is

some normative selection; let us ask, however, for laws that apply to thinking as well as may be done. That doesn’t mean surrendering to the truth-fetishists. Both before Boole’s time and since, when given propositions A and B that have nothing to do with each other, a thinker does not set about inferring B from A. It was natural, then, that even while truth-fetishism was extending its dominion, various resistance movements sprang up. Strict implication was distinguished from material implication, in the following sense. To say that A implies B could be regarded as a contingent statement even within logic, and there was a stronger statement that sometimes might hold: one distinguished the statement that A implies B from the statement that A must imply B, then one tried to elaborate rules of symbolic manipulation appropriate to thinking where both kinds of implication came into consideration. This ‘‘modal logic’’ of Langford and Lewis seemed to be a realistic strengthening of the vocabulary. Indeed, because there may be various bases for regarding an implication as necessitated, I even thought it worthwhile to allow for various strict implications within the same system. But in retrospect this program does not look like much of a success. None of the various algebraizations of strict implication seem to deepen one’s understanding of thought. At the same time there was some attention paid to allowing truth values intermediate between true and false, as by Jan Łukasiewicz. This is at least a start on embodying the notions expressed in everyday language by ‘‘sort of true’’ or ‘‘yes and no’’. It is a limitation to insist that the intermediate truth values be totally ordered—a limitation that could be overcome, and by the way, the corresponding limitation is not suffered by modal logic with multiple modal operators. In the later invention of ‘‘fuzzy logic’’ by Lofti Zadeh, it is claimed that still greater flexibility is obtained. In short, the 20th century brought us to an acknowledgement that truth may be of various strengths. The Go¨del incompleteness theorems suggested that this was even unavoidable, that no matter how faithfully one hewed to the

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line that truth was the goal, there could never be a notion of truth that would sort all possible mathematical statements into an army of true ones and an army of others (their negations). If every axiom system leaves undecided propositions, then it seemed that every mathematician on the corridor might make a different choice of what arithmetic facts were facts. Yet my deep discomfort with truth-fetishism is not addressed by making truth relative to a choice of axioms for set theory and arithmetic. We can agree that asserting ‘‘A’’ is distinct from asserting ‘‘A is provable (in some specified axiom system)’’ and distinct again from asserting ‘‘A is provable (in such-and-such other system)’’; certainly all are distinct from asserting ‘‘A is nine-tenths true’’ or ‘‘A is sort of true’’; and the list of options can be extended, as I will presently maintain. None of the options takes care of the big issue: logic based only on truth values is an impoverished logic, in that it sets aside intrinsic relations between concepts. I stipulate, in case it is not already plain, that by ‘‘relations’’ here I do not mean subsets of some direct product, as in many elementary developments of mathematics. I mean substantive relations. Let me turn to some other quarrels I have with truth-fetishism. Many spokesmen may say, since Boole, that all true theorems are equivalent and every true theorem is a tautology. As hyperbole, I understand this and endorse it; but oh, what it leaves out! First, it renounces any distinction between hard and easy theorems; second, it renounces any distinction based on what the theorems are about. Similarly, many spokesmen may say that our aim in mathematics is to simplify every proof to self-evidence — that the ultimately desired proof of a statement’s truth not only is necessarily tautological but also is plainly so. (I recently saw this thesis attributed to Gian-Carlo Rota, but many before him subscribed to it, and he, to my reading, did not.) Again, this ignores too much. Granted, we try to clear away the extraneous, and the Book proof never goes off on an unnecessary detour; but sometimes a theorem is valued for bringing together pieces from different 6

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conceptual sources, and this payoff may be reduced if we simplify to allow quick attainment of a conclusion. A third, grotesque example: some metamathematicians would have it that a mathematical theory is the set of propositions that are true in it. I have ranted against this interpretation elsewhere, maybe I don’t need to belabour it here, but please indulge me while I do. A theory that is any good says something, opens some door. It has high points and central points; it has beginnings (I don’t mean only axioms) and endings. Most likely it has avenues to other theories. The collection of all propositions that are true in it, on the other hand (if we could ever apprehend such a monster, which I doubt), consists mostly of banalities, so there can be no order, no revelation or insight. Such is not our science. We must try to get realistic about deduction, the deduction we actually do at our best. What may one say in practice about a statement? On just the dimension of truth value — one may assert it; or going farther, one may claim to be able to prove it; or going less far, one may say one tends to believe it. If this is within mathematics, however, one probably says something with more structure. Let’s take a realistic possibility. One may say, typically, ‘‘I think I can prove it with an additional hypothesis.’’ But stop right there! The truth-fetishist calls this vacuous; for of course the additional hypothesis could be the conclusion, and then of course it can be proved. We know this is irrelevant; we want to protest, ‘‘This is beside the point,’’ just as Bertrand Russell’s listener did; we feel the need for a sense to ascribe to the property ‘‘provable with an additional hypothesis’’. But needs such as this have not been described by modal logic or many-valued logic or fuzzy logic, and I suggest that enlarging the lexicon of truth-values is not the way to go to describe them. Truth — even truth understood in some new sophisticated way — is not the point. The point is pertinence. The point is relevance. In the last half-century, serious efforts have gone into analyzing relevance, but they commonly rely on deliberately ignoring the content. This approach is admittedly a sidetrack

from the direction of my quest, but I can’t brush it off. Today one may marshal computer power to discover which of a large population (of people; of factors in a plan; of propositions) are most related, but one often is looking just for the existence of some strong relation rather than for its nature. Only connect, as E. M. Forster said! This may be done in a search engine by looking for the singular values of a very sparse matrix of very large order. It is the few non-zero entries that guide us. At the intersection of Bacon and Shakespeare in the matrix appears a rather large number; people who deny that Bacon wrote the Shakespeare plays are there, right alongside those who affirm it, indiscriminately; and the weighting is upped by any discussion of the issue, including the present one, whether or not any new insight is achieved. In short, the relations are in the form of a weighted graph with positive integer weights. I can even give an example much closer to home: suppose two words are connected if they are often used in the same utterance; then ‘‘knife’’ and ‘‘bandage’’ will both have connections to ‘‘wound’’ although one causes the wound and the other is a response to it. This is an approach that suppresses syntax and even the distinction between yes and no. As psychologists speak of the impact of mere exposure to a stimulus regardless of positive or negative reinforcement, this way of boiling down intricate data uses mere association regardless of the nature of the association. The approach seemed wrongheaded to me at first, I confess. Architects said they broke down their design task by drawing a graph of which considerations were related and then analyzing the graph computationally to find subgraphs — sub-tasks — which could be carried out by separate teams. This was said to lead to efficient sharing of design effort; I was skeptical. But mere association can be a precious bit of knowledge these days, and I will pause to acknowledge it. What leads people to apply such blunt tools is the extremely large number of variables one may be trying to handle in many an application today. Take again minimization of an objective function. If there are fifty

thousand independent variables, inevitably most of them will be without effect on the function’s value, and in such a fix, the finding of one that does have an effect is a big part of the solution, even if one doesn’t find out at once what value one should best assign to it. What’s more, it may be that not one of the variables affects the value of the objective function enough to rise above roundoff error: only by a better choice of coordinate system, perhaps, can directions having noticeable effect be chosen. There are many contexts that impose hugely many interacting variables, but I want to mention one where this so-called ‘‘combinatorial explosion’’ sneaks up on us. In behavioural evolutionary theory, the traits whose selection one seeks to reconstruct are not life histories but strategies, that is, complete repertories of responses to life’s predicaments; thus any serious attempt must run up against the game-theoretic feature that the number of strategies grows with the size of the game faster than polynomially. Perceiving pertinence may be undertaken, then, by means having a family resemblance to the extensional characterization of properties, by methods in which the nature of relations between two things is banished from consideration. I do not cease to feel that pertinence should be respected as having structure, that what we employ, whether in reasoning or in observational science, should be not mere association but the structure of the association. Even if we call on high technology to explore a graph of connections between items, it will be natural to refine it to be a directed graph, a coloured graph, and surely much more. Only connect! — but there is such a wealth of ways that two nodes may be connected. Between two propositions there may subsist (aside from their truth or falsehood, as those may be in doubt) the relation that one entails the other, or the converse, or both, or neither. Though this may be all the truthfetishist recognizes, we see every day that the relations possible between propositions are much more diverse. Simple illustrations will make my point. Let me begin with a mantra of

20th-century math education: ‘‘ ‘but’ means ‘and’.’’ We all know that this makes partial sense: namely, if one says ‘‘John is poor but happy’’ one is asserting both ‘‘John is poor’’ and ‘‘John is happy’’. Nevertheless ‘‘but’’ is a major component in the structure of thought (like ‘‘nevertheless’’), and the version having ‘‘but’’ as the connective is not the same as the conjunction of the two simple assertions. Many English speakers would find ‘‘John is poor but happy’’ cogent but not ‘‘John is rich but happy’’. You will easily find more and subtler everyday examples. Examples within mathematics are subtler, inexhaustible, but more elusive; I will content myself with the one I already gave. Of course I do not maintain that natural languages contain all the precision we seek for our logic. On the contrary, their ambiguities are sometimes just concessions to imprecision. If one says ‘‘John is rich so he is happy’’, it is not clear whether one means to assert that every rich person is sure to be happy; it is clear only that something is being said beyond the conjunction of ‘‘John is rich’’ with ‘‘John is happy’’. Similarly for the connective ‘‘aussi’’ in French. I do maintain that syntax of natural languages and our experience with reasoning can yield a great enrichment of our logical conceptual resources. The reason proofs are expressed in natural language is not only our deplorable lack of facility in reading formulas (however large a factor that may be), it is also the great power of nuance in natural language. The proper continuation of Boole’s program is to do as much for relations as he did for truthversus-falsehood. There is gold in those hills. My prospector’s hunch is that the most promising underexploited lode is prepositions. By now I have surely advanced enough dubious doctrines for one afternoon, but if I stopped now you would feel the lack of any mention of probability theory. It must fit into my talk somehow, right? Just so: it is a part of mathematics, it deals throughout with propositions which may turn out untrue, and I do have some dubious things to say about it. I was just saving it for last. I have been discussing mostly the 19th and 20th centuries, but we must

glance back now to the 16th. At its inception, was probability regarded as a competing notion of truth? There is no doubt that the idea of probability was close to the idea of truth at that early stage—etymologically, ‘‘probable’’ is ‘‘provable’’, and even today, ‘‘probity’’ means utter reliability—and the emerging notion of something having positive probability had to be disentangled from the different notion of appearing credible. This fascinating story has been closely studied in recent years, especially by Ian Hacking and Lorraine Daston, and I have nothing to add to their work. I pick up the story with the incorporation of probability into physics in the 19th century and its reconciliation with mathematics in the 20th. The first development, the creation of statistical mechanics by Ludwig Boltzmann and others, and its success as a part of physics, has a consequence for the idea of truth in mathematics. Some statements about physical systems are definitively shown untenable if they are shown to hold with probability 0 — or just with probability extremely close to 0. An applied mathematician has an obligation to accept the conclusion that the ice cube melting in your glass of water is not going to separate out again into ice, because the molecular theory that assigns to that outcome a prohibitively low probability is successful. The theory also says that the sequence followed by the molecules during melting is reversible; the reverse process is nevertheless ruled out, and the argument uses probability. If G. H. Hardy or some other deplorer of applying mathematics wants to retain a notion of possibility for the ice cube to be reborn, fine; all I am saying is that a different notion of possibility and impossibility has emerged in statistical physics. That it happens to involve probability extremely close to 0 instead of probability 0 is just an aggravation of the antithesis. We have a hierarchy: something may be known untrue; or more generally it may be known to have probability 0; or still more generally it may be known to have so low a probability as to be ruled out. Let me emphasize that the consequences for relating predicted behaviour to observed behaviour are the same for all three. A physical theory

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which disproves a phenomenon we observe is refuted; but a physical theory which assigns a prohibitively low probability to a phenomenon we observe is refuted just as thoroughly. Statistical physics has extended its sway in the last hundred years and we must live with it. Its criterion of truth deserves our respect. Finally, a look at probability as a mathematical theory. With A. N. Kolmogorov’s wonderful little book (1933), probability seemed to have earned a place at the table of mathematics. Its special notions had been put in correspondence with notions of analysis and measure theory which were as clear as the rest. Aside from its application to gambling, insurance, and statistical physics, probability was now welcomed as a tool within mathematics. To speak of an event having probability 0 was exactly to speak of a subset having measure 0. A striking string of theorems came forth over the years. One did not conclude that a phenomenon was certain to happen by proving its probability was 1; but if one could prove its probability was 1, or merely that it had probability greater than 0, one could conclude that it was capable

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of happening: a set of measure greater than 0 had to have some elements in it. The first striking achievement of this sort long predated Kolmogorov, actually (and it was expressed in the probabilistic terms): Emile Borel proved that a sequence of decimal digits chosen at random represents a normal number with probability 1, and this provided the first proof that it is even possible for a number to be normal. (Many of you know the definition: a decimal expansion is normal provided that all sequences of k digits occur as subsequences of it with the same asymptotic frequency, and that for every k. Correspondingly, other bases than ten can be brought in.) Now I comment first on how sharply the relation of probability to mathematical truth here contrasts with what we saw in physics. The number-theorist contentedly deals with sets of measure 0, and moreover values the positive measure of a set primarily for its guarantee that the set is not empty. While we were talking physics, there was perhaps a temptation to live on the intermediate level of the hierarchy I mentioned: to disbelieve in events of probability 0. That doesn’t

work, of course. Yet there is a serious catch in the number-theorist’s usage too: to say that measure greater than 0 ensures that a set has members is to defy intuition. The intuitionist responds, ‘‘The set has members? Really? Show me one.’’ Today, after a century of debate, this catch is clarified, but, far from going away, it appears insuperable. The constructible numbers (by any appropriate definition) are a set of measure 0; yet they are the only numbers that might be shown. In the conventional terminology of 20th-century analysis, almost all real numbers are not constructible; in our experience, every real number that can be specified is constructible. Now Kolmogorov knew all of this, he understood it better than the rest of us do, yet it seems not to have bothered him. Shall I assume that he had confidence that we would be able to straighten things out after his death? That’s kind of him. By all means let us try. New College University of Toronto Toronto, Ontario M5S 3J6 Canada e-mail: [email protected]

A Nightmare Seminar1 Kenneth Falconer

When you go to a talk, speaker, blackboard and chalk, at the end of a hard day of teaching, It’s no great surprise that you half close your eyes and you can’t concentrate on the preaching. You very soon find that your wandering mind with ideas for strange theorems is teeming, But you’re somewhat bemused and your thoughts get confused and before very long you’re just dreaming. For you meet a long line of numbers, all prime, that pass by with a nod now and then, With big gaps in the queue, the primes become few, a proportion one over log n. The gaps vary in size and you want to know why, so you write as an infinite sum Riemann’s function called zeta, and what could be neater? A product o’er primes it becomes. You set out on a train ’cross the vast complex plane as you search for the zeros of Riemann, There are plenty to find on the critical line and you note them all down as you see them. Then the train hits a pole and you tumble and roll (near the pole it’s as cold as an icicle), But you pick yourself up and you find you’re in luck for nearby is a rusty old bicycle. You pedal like mad, though the weather’s turned bad, keeping lookout for rogue Riemann zeros, But before you get far you arrive at a bar full of great mathematical heroes. Next to Hilbert and Gauss, you see Erd} os and Straus who’re absorbed in some very hard thinking, Then there’s Riemann and Rayleigh, and Cauchy and Cayley, and Euler with Fermat is drinking. You buy Riemann a beer, and he says: ‘‘To be clear why my zeros all lie in a row, You will need to consider with very great rigour the way eigenvalues can grow.’’ Although you implore, he won’t tell any more and so you depart on your travels, You keep scratching your head at what Riemann said but the problem you still can’t unravel. By a river that’s deep you encounter a heap of some very large matrices random,

And their eigenvalues and the Riemann zeros wander off to infinity in tandem. You then get the point that if one’s self-adjoint eigenvalues all lie on a line, And the same should be so for the Riemann zeros – an idea that is somewhat sublime. You allow a faint smile as you search through the pile for such matrices, hoping you’ll see them, You spot one midst the trash - and you see in a flash how to prove the Hypothesis Riemann. You start work right away and by night and by day you fill hundreds of pages with writing, The lemmas are tricky, the details are sticky, and getting it right’s quite exciting. You break out in a sweat for you mustn’t forget any parts of the proof you’re recording, But at last comes the end and you put down your pen … and wake up to your colleagues’ applauding … You’re a regular wreck with a crick in your neck, Your hair’s in a mess and your head’s on the desk, Your mouth’s open wide and your tie’s to one side, The board’s covered with chalk and you’ve missed all the talk, You’ve forgotten your proof which must be a spoof, Your face has turned red, you’ve an ache in your head, With a throb that’s intense and a general sense That you’ll take a long time to recover. But the seminar’s past, you can go home at last, And the day has been long, ditto ditto my song, And thank goodness they’re both of them over!

Mathematical Institute University of St. Andrews St. Andrews, KY16 9SS UK e-mail: [email protected]

1

There may be some resemblance, not least in poetic metre and form, to the famous Nightmare Song from W.S. Gilbert and A.S. Sullivan’s ‘Iolanthe’. This is by no means the first rhyme relating to the Riemann Hypothesis. Tom Apostol’s song ‘Where are the Zeros of Zeta of s?’ is well known.

2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

9

Mathematically Bent

Colin Adams, Editor

Happiness is a Warm Theorem The proof is in the pudding.

COLIN ADAMS

D

octor: It’s okay, Craig. You can talk freely here. If you want to cry, you can cry.

Craig: I’m sorry, Doctor. It’s just so overwhelming, sometimes. Opening a copy of The Mathematical

Doctor: Why don’t you talk about it?

Intelligencer you may ask yourself

Craig: It’s Crushing.

uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.

crushing

depression.

Doctor: Here’s a tissue. Do you feel it all the time? Craig: Lately, yes. Lately, it is a total. I can’t see any way out. Doctor: But before that? Craig: Before, there were also the highs. Extreme euphoria. But almost always, the highs were again followed by extreme lows. Doctor: Yes, well, this sounds like classic bipolar disorder. What brings these feelings of depression on? Are you having trouble relating to your spouse? Craig: Spouse? She left me two years ago. Doctor: Oh, I am sorry to hear that. Did that make you very sad? Craig: Actually, I didn’t notice for a week. Doctor: What do you mean?

Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected]

10

Craig: Well, I was trying to prove a difficult theorem. I was really close. The solution was just out of reach. Couldn’t put it down for a second. I went home to check a theorem in a book I had left by the bed and I noticed she wasn’t there. Found a

THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

note on the kitchen table saying she had left. Doctor: And how did you react? Craig: It didn’t really register. I needed to get back to work. Doctor: Hmmm. Sounds defense mechanism.

like

a

Craig: Not really. I was at a crucial juncture in trying to prove that theorem…. We weren’t getting along anyway. Doctor: Why not? Craig: She was frustrated with me. Called me antisocial. Doctor: Were you antisocial? Craig: Not at all. I can be very social. I just didn’t want to be. Doctor: Why not? Craig: I was busy…. She used to drag me to these parties. A bunch of people standing around trying to come up with topics for conversation. The weather, politics, American Idol. And they were drinking alcohol to dull their intellects enough to enjoy it. Doctor: I take it you don’t drink? Craig: It gets in the way of my ability to do mathematics. Doctor: So what would you do at these parties? Craig: I would sit in the corner. Doctor: By yourself? You didn’t talk to people? Craig: I was thinking about mathematics. It’s one of the great advantages of my profession. Once you are thinking about a problem, you always have your work with you, wherever you go. Doctor: Okay…. Craig: But then her friends would spot me in the corner and they would feel sorry for me. They would assume I was shy. I’m not shy. But inevitably,

they would come over and try to engage me in conversation, and I would have to be rude to get them to leave me alone. Eventually, we stopped getting invited to parties. Doctor: And your wife was upset about that. Craig: Not only that. It wasn’t like I was keeping up my end of the conversations at home either. I had other things on my mind. She said it was like living with a zombie. My body was present, but my mind was far, far away. So she left. Doctor: Okay…. Well, then, I guess you don’t have a significant other at this point. Pets? Craig: Had a fish. Forgot to feed it, so it died. Doctor: And how did that make you feel? Craig: It was okay. At least then I didn’t need to remember to feed it anymore. Doctor: So you don’t have an emotional need to connect with another sentient being? Craig: Not really. Doctor: You don’t get lonely? Craig: Doc, I don’t have time to get lonely. Doctor: I see. It sounds like you are tremendously overworked. Craig: I choose to work this hard. Doctor: Nobody is forcing you to do this? Craig: Are you kidding? I’m tenured. If I wanted to, I could stop research and watch soap operas all day. Doctor: So what’s driving you? Ambition? You want to be at the top of your field? Craig: I do want to be at the top of my field. But for me, it’s not about the recognition. It’s about beating it. Doctor: Beating what? Craig: Whatever problem I am working on. Doctor: You make it sound like a battle.

Craig: It is a battle, but the long drawn out kind. A siege. Doctor: How so? Craig: It’s like this thing you are trying to prove, this theorem, it’s in its fortress, protected. And you are trying to break in, to find a crack in its defenses. And you try one thing, but it doesn’t work. So you try something else, and that doesn’t work. But you keep at it, camped outside the walls, trying to last longer than it does, trying to find a way into or over the walls. You lay siege to the problem. But after a while, you become discouraged, because nothing is working. You’ve struggled for so long with it. Tried every approach. There is nothing left to try. You’re out of ideas. So finally, you try to move on to something else. Find another problem to distract you. But it doesn’t work. Your mind returns to it even as you consciously try to think of other things. You might find yourself standing in the produce aisle of the grocery store, holding a bag of oranges and having no idea how long you have been there. And then one night you fall asleep at your desk. And you dream about the problem. And the next morning you wake up, and you just know you have solved it. You know you are right. Doctor: What does that feel like? To solve a difficult unsolved problem like that? Craig: Oh, Doc, you have no idea. It’s incredible. A euphoria. It makes you want to sing. It makes you want to kiss strangers. It makes you want to tell everyone. Doctor: Like I feel when one of my patients gets better? Craig: No, nothing like that. That’s like when I teach a student calculus, and they finally get it. This, this would be more like you having seen hundreds of patients over your career, and suddenly realizing there was a pattern to their behavior that no one else in your profession ever recognized before. You then discover a treatment that not only cures all of your patients, but everyone else’s

patients as well. You have solved this difficult problem that has plagued humankind for millennia. More like that. Doctor: Well, that does sound like a nice feeling. So this is the highs? Craig: Yes. Doctor: And how long does the feeling last? Craig: A week, maybe a bit longer. But then it’s time to work on the next problem. And the minute you begin, you get depressed. Doctor: This is the lows? Craig: Yes, you are pushing another rock up the hill, and it just keeps rolling down again. Doctor: But at least you are making progress. Craig: That’s just it. Maybe you are and maybe you aren’t. Doctor: What do you mean? Craig: Well, for example, one time 10 years ago, I was working on the Schmuel Conjecture, open since 1982. I had a great idea. Approach it through Kleiner bundles. No one had thought there was a connection between the two. But I saw it! Looked like I was about to solve the problem. But each time I thought I was almost done, something would go wrong at the last stage, and there would be this additional result that I would need to prove. This went on for a year. Intensely frustrating, but it still felt like I was getting closer and closer. Doctor: And? Craig: And it didn’t work. The last piece I needed happened not to be true. Dead end. Doctor: And what happened to the Schmuel Conjecture? Craig: Oh, it’s still open. Every once in a while, I find myself daydreaming about it, thinking of some alternative approach…, maybe through Lie groups…, maybe using the KacMoody algebra…. Doctor: Craig?

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11

Craig: …I wonder if the cohomology would tell me something…. Doctor: This is classic addictive behavior. And you are not the first patient in which I have seen such a pattern. Craig: Really? Doctor: Actually, I am treating several members of the mathematics department right now. But the rest are not as able to articulate the problem. Most just sit and stare off into space without talking. Craig: Probably just working on math.

12

THE MATHEMATICAL INTELLIGENCER

Doctor: Well, it is becoming apparent that this is a psychosis. A psychosis that afflicts the mathematics community. It has a variety of symptoms including addictive disorder, antisocial attitudes and obsessivecompulsive behavior. This is incredible! I have identified a new disorder! It may transform psychology! Craig: Yeah? Doctor: Yes, yes, and I will name it, umm, mathematitis. It is an unhealthy addiction to mathematics. I can see the papers already. This will be truly groundbreaking work. Perhaps we

can drug the patients. An amnesiac, that makes them forget the problem they are working on. Or perhaps we can hypnotize them into believing they have solved their problem, bringing on a virtually constant euphoria. Craig: Yeah, well congrats, then. Must feel good. Look, Doc, can we cut the session off early? I’m actually thinking I might have a possible approach to the Schmuel Conjecture. This could be a major breakthrough! Doctor: Yes, Goodbye!

certainly,

certainly.

The Continuing Story of Zeta GRAHAM EVEREST, CHRISTIAN RO¨TTGER

W

e can only guess at the number of careers in mathematics that have been launched by the sheer wonder of Euler’s formula from 1734, 1þ

1 1 1 1 p2 : þ þ þ þ ¼ 22 32 42 52 6

ð1Þ

AND

TOM WARD

1 22 þ 32 42 þ 52 þ ¼ 0:

ð4Þ

Readers doubting the validity of formula (4) will be reassured to note that it follows from 1 þ 22 þ 32 þ 42 þ 52 þ ¼ 0;

ð5Þ

2

Euler further obtained the generalization that for integral k C 1 the inverse 2k -th powers of the natural numbers sum 2k to a rational multiple of p , and identified that rational multiple. This identification involves the sequence of Bernoulli numbers (Bn), which is defined via the generating function 1 X x xn Bn : ¼ x e 1 n¼0 n!

ð2Þ

The first few Bernoulli numbers are shown below:

Euler showed that for k C 1

1þ

1 1 1 1 ð1Þkþ1 22k1 B2k 2k þ þ þ þ ¼ p : 22k 32k 42k 52k ð2kÞ!

ð3Þ

In particular, f(2k) is irrational for k C 1. Little is known about f(2k + 1) for k C 1; indeed it is only relatively recently that f(3) was shown to be irrational by Ape´ry. In his lovely paper [10], van der Poorten refers to Ape´ry’s theorem as ‘‘A proof that Euler missed’’. What follows is an even more stunning formula1 than (1) which Euler certainly found (in 1740, see [2, Section 7]), 1

after multiplying (5) by -7 = 1-2.2 . The concept of analytic continuation was developed partly in order to make sense of formulae such as (4) and (5). Here the concept is applied to Riemann’s zeta function f, which is defined for complex s with <(s) [ 1 by the absolutely convergent series fðsÞ ¼

1 X 1 : s n n¼1

ð6Þ

In this article we will report on recent work that allows f to be evaluated to the left of the line <(s) = 1 in an extremely elementary and natural way. If Euler’s ghost is sensed, then it is with good reason. In [2, Section 7], Ayoub comments on Euler’s article of 1740 in which he boldly evaluates divergent series to obtain formulae such as (4). The methods we espouse are in the same tradition, only taking care to articulate the convergence issues. The most interesting values of the zeta function occur outside the domain of convergence of the series in (6). For one thing, the formula describing f(-k) is simpler than that for f(2k), let alone the mysteries surrounding f(2k + 1). For another, the location of zeros of f(s) other than those for s = -2k (see Corollary 3) has a just claim to be one of the most important unsolved problems in Mathematics. Indeed, the Clay Mathematics Institute offers a prize of one million dollars for a proof that all these ‘‘non-trivial’’ zeros lie on the line <ðsÞ ¼ 12 – the famous Riemann Hypothesis. Special values of the zeta function, of interest in themselves, also hint at a possible route into the functional

At least, when one of us showed it to a final-year class in Analytic Number Theory, they were (to their credit) stunned.

Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

13

equation. Knowledge of the values of f(k) for all integers k might enable one to predict the shape of the functional equation of the zeta function. A comparison of (3) and (18) suggests that the function s 7! fðsÞ=fð1 sÞ can be represented by a simple combination of factorials (or Gamma functions) and exponentials. The extent to which Riemann might have been aware of Euler’s work on this subject is not clear; the interested reader might begin by consulting Ayoub’s paper.

Taking the Low Road Analytic continuation is easily illustrated using a simple example. Consider the power series f ðsÞ ¼ 1 þ s þ s2 þ ;

ð7Þ

which converges absolutely for |s| \ 1. The series diverges for |s| C 1, thus one could never evaluate f in that region using the definition (7). Nonetheless, inside the domain jsj 1 we have f ðsÞ ¼

1 ; 1s

ð8Þ

and the right-hand side can be evaluated everywhere on C n f1g: In light of this, Euler would have no compunction in using the definition of the left-hand side of (7) to describe the behaviour of the right-hand side of (8). On these grounds, it would be natural for him to write 1 1 1 þ 1 1 þ 1 ¼ : 2 For the example in (7), the analyticity of f (s) on C n f1g comes as a by-product; we simply recognize that the function in (8) can be differentiated by the usual rules of calculus. Nonetheless, the differentiability is important because it guarantees that the continuation of f (s) is unique in C n f1g (the region where it is analytic). In a similar fashion, the analyticity of the continuation of the zeta function will be understated in the text that follows. Actually, it is no harder than proving the analyticity in the halfplane <(s) [ 1. What is needed is the concept of uniform

convergence, and we refer the reader to any of the standard texts for a full account of this topic. To obtain the analytic continuation of f to the left of its natural half-plane of convergence requires more guile than for f (s) above. However, the principle is the same: an expression needs to be found for f(s) valid in a half-plane strictly containing <(s) [ 1. The high road, Riemann’s own [13], uses contour integration at an early stage, and leads directly to the functional equation. Many authors ([1, 4, 5, 9, 11, 16], and [17]) use this method, or variants of it. Other methods are known ([16, Chap. 2] lists seven) but a toll seems inevitable on any route ending with the functional equation. There are lower roads that give both the continuation to the whole plane and the evaluation at nonpositive integers but stop short of proving the functional equation. Our purpose in this article is to draw wider attention to these, often very scenic, roads. For example, Sondow [14] notes one way in which Euler’s argument can be made rigorous. Mina´cˇ [7] showed how to evaluate f at negative integers in an extremely simple and elegant way, by integrating a polynomial on [0, 1]. Other authors [3, 8, 12, 15], have shown how the continuation and evaluation of the Hurwitz zeta function can be obtained in a down-to-earth way that is applicable to the zeta function and to many L-functions. Their method, which uses little more than the binomial theorem and seems to be new, is presented here for the archetypal case of the zeta function itself. The main point of the article is to highlight how easily the continuation and evaluation of f can be obtained. The workhorse is (19), which can be viewed as the truncation of a formula of Landau [6, p. 274].

A Journey of a Thousand Miles.. Throughout, we use the standard notation s = r + i t with r; t 2 R: Notice that for r [ 1, Z 1 1 1 ¼ : ð9Þ x s dx ¼ 1 s s 1 1 The formula (9) yields a second example of analytic continuation. Clearly the integral in (9) can only be evaluated

AUTHORS

......................................................................................................................................................... GRAHAM EVEREST is a long-time faculty

¨ TTGER, after studying at Paris CHRISTIAN RO

member at the University of East Anglia, where he teaches number theory at all levels, and pursues research on it. He has three lovely children who do not share his interests in number theory.

VI and Augsburg, received a Ph.D. from the University of East Anglia in 2000. After postdoctorate positions at Go¨ttingen and Iowa State University, and a spell at the HypoVereinsbank, Munich, he returned to Iowa State as a lecturer. He works in number theory, in particular on asymptotic counting problems.

School of Mathematics University of East Anglia Norwich NR4 7TJ, UK e-mail: [email protected]

14

THE MATHEMATICAL INTELLIGENCER

Department of Mathematics Iowa State University Ames IA 50011, USA e-mail: [email protected]

for r [ 1. However, the right-hand side is analytic everywhere apart from a simple pole at s = 1. Thus we obtain the continuation to C n f1g of the function represented by the integral for r [ 1. In the half-plane r [ 1, Z 1 1 Z nþ1 X 1 ¼ x s dx ¼ x s dx s1 1 n n¼1 Z 1 Z 1 1 X X 1 1 x s ðn þ xÞs dx ¼ 1þ dx: ¼ s n 0 n n¼1 0 n¼1 ð10Þ All the sums converge absolutely for r [ 1. In the text that follows, we assume that r [ 1 and that |s| is bounded by K, a fixed arbitrary constant. The binomial expansion of the integrand in (10) yields x s sx ¼ 1 þ sE1 ðs; x; nÞ: ð11Þ 1þ n n

Letting s ! 0þ in (13), and noting the second part of (14), we obtain 1 1 ¼ fð0Þ ; 2 which yields the value fð0Þ ¼ 12 : The preceding argument begins with the binomial estimate (11), finds the analytic continuation of the zeta function to the half-plane r [ 0, and evaluates f(0) by a limiting process. What happens if more terms of the binomial expansion are included? An additional term in the binomial expansion gives x s sx sðs þ 1Þx 2 ¼1 þ þ ðs þ 1ÞE2 ðs; x; nÞ; 1þ n n 2n2 the higher binomial coefficients all include a factor (s + 1). Here, E2 is a function that satisfies jE2 ðs; x; nÞj

In (11) the function E1 satisfies jE1 ðs; x; nÞj

C1 x 2 C1 2; n2 n

ð12Þ

for all x 2 ½0; 1 and all n C 1, with C1 = C1(K) (since E1 is the error term of the Taylor series in x/n). Substituting (11) into the sum (10) and integrating with respect to x gives 1 s ¼ fðsÞ fðs þ 1Þ þ sA1 ðsÞ: s1 2

ð13Þ

The function A1(s) is analytic for r [ -1, and the proof of this, which we do not detail here, uses no more than uniform convergence alongside (12). It is precisely now that the crunch comes. The functions at both ends of (13) are defined for r [ 0, provided s = 1. Also, since f(s) is defined by a sum for r [ 1, it follows that f(s + 1) is defined by a sum for r [ 0. Therefore (13) may be taken as the definition of f(s) in this larger half-plane. Moreover (13) shows that the extended function is analytic in the half-plane r [ 0, apart from a simple pole at s = 1 with residue 1. In other words, (13) implies that limðs 1ÞfðsÞ ¼ 1 and therefore lim sfðs þ 1Þ ¼ 1: ð14Þ s!1

s!0

.........................................................................

for all x 2 ½0; 1 and all n, where C2 = C2(K). Substituting this into (10) and integrating as before yields 1 s sðs þ 1Þ ¼ fðsÞ fðs þ 1Þ þ fðs þ 2Þ þ ðs þ 1ÞA2 ðsÞ; s1 2 6 ð15Þ where A2 is analytic for r [ -2. Thus, (15) may be used to continue f to the half-plane r [ -1. As before, letting s ? -1+ and using (14) with s ? s + 2, we obtain 1 1 1 1 1 ¼ fð1Þ þ fð0Þ ¼ fð1Þ ; 2 2 6 4 6 1 yielding fð1Þ ¼ 12 :

General Method This method can be repeated in order to continue the zeta function further to the left in the complex plane. The method also yields the explicit evaluation at the nonpositive integers in terms of the Bernoulli numbers. To describe this, we record two well-known properties of these fascinating numbers in the following lemma.

L EMMA 1 With Bn defined by (2), N 1 X

TOM WARD received his Ph.D. from the

University of Warwick and has been at the University of East Anglia since 1992. He is currently Pro-Vice-Chancellor (Academic) and consequently does his mathematics (ergodic theory and interactions with number theory) only on alternate Sunday afternoons. School of Mathematics University of East Anglia Norwich NR4 7TJ, UK e-mail: [email protected]

C2 x 3 C2 3 n3 n

n¼0

N Bn ¼ 0 n

for all N 1;

ð16Þ

and Bn ¼ 0

for all odd n 3:

ð17Þ

P ROOF . The defining relation (2) can be written ðe x 1Þ

1 X n¼0

Bn

xn ¼ x: n!

For N [ 1 the coefficient of xN in the left-hand side is N 1 X

1 Bm ¼ 0; ðN mÞ!m! m¼0 Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

15

which gives (16) after multiplying by N!. The second statement follows from the fact that ex

x x xð1 þ e x Þ þ ¼ 1 2 2ðe x 1Þ

is an even function. Either (2) or (16) determines the Bernoulli numbers, but the latter allows them to be readily computed inductively.

T HEOREM 2 There is an analytic continuation of the zeta function to the entire complex plane, where it is analytic apart from a simple pole at s = 1 with residue 1. For all k C 1, fðkÞ ¼

Bkþ1 : kþ1

k ¼ 1; 2; . . .:

The proof of the Corollary follows from (17) and (18). The relation (18) is not true for k = 0, but our method has already given us the special value fð0Þ ¼ 12 : The case when k = 1 is an elegant interpretation of formula (5). OF

T H E O R E M 2. The analytic continuation of

the zeta function to the half-plane r [ -k arises in exactly the same way as before, by extracting an appropriate number of terms of the binomial expansion and using induction. For integral k C 0 and r [ 1, this gives the relation k X 1 ð1Þrþ1 sðs þ 1Þ. . .ðs þ rÞ ¼ fðsÞ þ fðs þ r þ 1Þ s1 ðr þ 2Þ! r¼0

þ ðs þ kÞAkþ1 ðsÞ

which transforms (20) to 0 ¼ fðkÞ þ

negative even integers:

PROOF

A simple manipulation of factorials gives ðk þ 1Þðk þ 2Þ k kþ2 kþ2 ¼ ¼ ; r þ1 kr þ1 ðr þ 1Þðk r þ 1Þ r

ð18Þ

C OROLLARY 3 The Riemann zeta function vanishes at fð2kÞ ¼ 0;

The term with r = k is known. Using the inductive hypothesis on the other terms gives k1 X 1 Bkrþ1 k 0 ¼ fðkÞ þ r k þ 2 r¼1 ðr þ 1Þðk r þ 1Þ 1 : ð20Þ 2ðk þ 1Þ

ð19Þ

k 2ðk þ 1Þðk þ 2Þ k1 X 1 kþ2

ðk þ 1Þðk þ 2Þ r¼1

kr þ1

Bkrþ1 :

ð21Þ

Now multiply by (k + 1)(k + 2) and apply (16) with N = k + 2. Only the terms for r = 0, k, k + 1 – missing in (21) – survive, yielding k 0 ¼ ðk þ 1Þðk þ 2ÞfðkÞ þ þ ðk þ 2ÞBkþ1 þ ðk þ 2ÞB1 þ B0 2 ¼ ðk þ 1Þðk þ 2ÞfðkÞ þ ðk þ 2ÞBkþ1 ; and this completes the induction argument. ACKNOWLEDGMENTS

Our thanks go to Robin Kronenberg, Ja´n Mina´cˇ and Sha Duan for helpful comments.

REFERENCES

1. T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, New York, 1976. 2. R. Ayoub, Euler and the Zeta Function, Amer. Math. Monthly 81

where Ak+1(s) is analytic in r [ -(k + 1), again because all higher binomial coefficients include a factor (s + k). Notice that k = 0 gives (13) and k = 1 gives (15). By induction, we may assume that the zeta function has already been extended to the half-plane r [ 1 - k so (19) is valid there, because the singularities at s ¼ 0; 1; . . . are removable. All the functions in (19) except f(s) itself are defined at least for r [ -k, which gives the analytic continuation of the zeta function to that half-plane. Let s ? -k+ in (19) and use (14), suitably translated, for the term with r = k to obtain k 1 X 1 fðk þ r þ 1Þ k ¼ fðkÞ þ r þ 1 kþ1 r þ2 r¼0 1 : ðk þ 1Þðk þ 2Þ Writing r for r + 1 simplifies this to k X 1 k fðk þ rÞ 0 ¼ fðkÞ þ þ : k þ 2 r¼1 r r þ1 16

THE MATHEMATICAL INTELLIGENCER

(1974), 1067–1086. 3. R. Dwilewicz and J. Mina´cˇ, The Hurwitz zeta function as a convergent series, Rocky Mountain J. Math. 36 (2006), 1191–1219. 4. G. R. Everest and T. Ward, An Introduction to Number Theory, Springer-Verlag, Graduate Texts in Mathematics Vol. 232, New York, 2005. 5. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, The Clarendon Press, Oxford University Press, New York, 1979. 6. E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 3rd edition, Chelsea Publishing Company, 1974. 7. J. Mina´cˇ, A remark on the values of the Riemann zeta function, Exposition. Math. 12 (1994), 459–462. 8. M. Ram Murty and M. Reece, A simple derivation of f(1 - k) = -Bk/k, Funct. Approx. Comment. Math. 28 (2000), 141–154. 9. S. J. Patterson, An Introduction to the Riemann Zeta-Function, Cambridge Studies in Advanced Mathematics 14, Cambridge University Press, Cambridge, 1988. 10. A. van der Poorten, A proof that Euler missed ... Ape´ry’s proof of the irrationality of f(3). An informal report. Math. Intelligencer 1, no. 4 (1978/79), 195–203.

11. K. Prachar, Primzahlverteilung, Grundlehren 91, Springer, Berlin,

15. B. Sury, Bernoulli numbers and the Riemann zeta function, Res-

1957. 12. V. Ramaswami, Notes on Riemann’s zeta function, J. London

onance 8 (2003), 54–62. 16. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function,

Math. Soc. 9 (1934), 165–169. 13. G. F. B. Riemann, U¨ber die Anzahl der Primzahlen unter einer gegebenen Gro¨sse, Monatsberichte der Berliner Akademie,

edited with a preface by D. R. Heath-Brown, second edition,

November (1859). 14. J. Sondow, Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series,

The Clarendon Press, Oxford University Press, New York, 1986. 17. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.

Proc. Amer. Math. Soc. 120 (1994), 421–424.

Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

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The Elementary Proof of the Prime Number Theorem JOEL SPENCER

AND

RONALD GRAHAM

rime numbers are the atoms of our mathematical universe. Euclid showed that there are infinitely many primes, but the subtleties of their distribution continue to fascinate mathematicians. Letting p(n) denote the number of primes p B n, Gauss conjectured in the early nineteenth century that pðnÞ n=lnðnÞ. In 1896, this conjecture was proven independently by Jacques Hadamard and Charles de la Valle´e-Poussin. Their proofs both used complex analysis. The search was then on for an ‘‘elementary proof’’ of this result. G. H. Hardy was doubtful that such a proof could be found, saying if one was found ‘‘that it is time for the books to be cast aside and for the theory to be rewritten.’’ But in the Spring of 1948 such a proof was found. Almost immediately there was controversy. Was the proof attributable to Atle Selberg or was the proof attributable to Atle Selberg and Paul Erd} os? For decades there seemed to be two mathematical camps with wildly different viewpoints. In the twenty-first century the controversy has finally subsided. Among previous discussions of the controversy, we mention particularly Goldfeld [1] (from which the previously mentioned quotation by Hardy is taken) and the book [3] of Paul Hoffman. Ernst Straus was in a unique position to observe the beginnings of the controversy. He then held a position at the Institute for Advanced Study as a special assistant to Albert Einstein. (We believe Straus is the only person to have joint papers with Einstein and Erd} os.) Straus had already worked a great deal with Erd} os, and this work would continue throughout his life. Sometime in the early 1970s (we aren’t sure of the exact dates), Straus wrote the account we present here. He did not want the notes to be published while the participants were still alive. Ernst Straus was, for us and for many of his friends, a man of great wisdom. He certainly attempted in these

P

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notes to give as faithful an account of the events as he could. Whether he succeeded is a judgment for the reader to make. Certainly, he was far closer to Erd} os than to Selberg. Let us be clear that we two authors both have an Erd} os number of one, and our own associations with Erd} os were long and profound. We feel that the Straus recollections are an important historic contribution. We also believe that the controversy itself sheds considerable light on the changing nature of mathematical research. In November 2005, Atle Selberg was interviewed by Nils A. Baas and Christian F. Skau [4]. He recalled the events of 1948 with remarkable precision. We quote extensively from that account. Selberg had first shown that X X ln2 p þ ðln pÞðln qÞ ¼ 2x ln x þ OðxÞ p\x

pq\x

Erd} os had heard about this through Paul Tura´n, and he wanted to see if he could use it to show that there exist prime numbers between x and x(1 + ), fixed and x sufficiently large. The case = 1 is known as Chebyshev’s Theorem. In 1933, at the age of 20, Erd} os had found an elegant elementary proof of Chebyshev’s Theorem, and this result catapulted him onto the world mathematical stage. It was immortalized with the doggerel Chebyshev said it, and I say it again; There is always a prime between n and 2n Clearly, Erd} os would be very keen to find an elementary proof that worked for an arbitrary positive . Selberg recalls: I had the Prime Number Theorem in my thoughts, that was my goal based on [the previous] formula that I had

THE MATHEMATICAL INTELLIGENCER Ó 2009 The Author(s). This article is published with open access at Springerlink.com

obtained. I told [Erd} os] that I did not mind that he try to do what he said he wanted to do, but I made some remarks that would discourage him. Erd} os succeeded in giving an elementary proof of the generalization of Cheybshev’s Theorem to arbitray positive . He showed some details of his proof to Selberg. Selberg continues: So I told Erd} os the next day that I could use his result to complete the proof, an elementary proof, of the Prime Number Theorem. [...] I really did not have in mind starting a collaboration with him. At this point Selberg travelled to Syracuse where he was to take a position. This was the critical time. As Straus puts it, ‘‘Alas, something had gone wrong.’’ The relationship between Selberg and Erd} os had soured, never fully to be repaired. This story of great mathematical discovery becomes all too human. Selberg writes: I started to hear from different sources that they only mentioned Erd} os’s name in connection with the elementary proof of the Prime Number Theorem, so I wrote a letter to Erd} os. Selberg proposes that they publish separately. [Erd} os] answered that he reckoned we should do as Hardy and Littlewood. But we had never made any agreement. In fact, we had really not had any collaboration. It was entirely by chance that he became involved in this - it was not my intention that he should have access to these things. Paul Erd} os and Atle Selberg were both giants of twentieth century mathematics. They (Erd} os 1913–1996, Selberg 1917–2007) were of the same mathematical generation.

AUTHORS

......................................................................... JOEL SPENCER is Professor of Mathe-

matics and Computer Science at the Courant Institute (New York). His research centers on probabilistic methods (aka Erd} os Magic) and probabilistic algorithms. He cofounded the journal Random Structures and Algorithms. He enjoys counting by nines with his grandson. Courant Institute, New York, USA e-mail: [email protected]

Figure 1. Paul Erd} os. 1962. Konrad Jacobs, photographer. Courtesy of the Archives of the Mathematisches Forschungsinstitut Oberwolfach.

......................................................................... ‘‘one of the principal architects of the rapid development worldwide of discrete mathematics in recent years,’’ has held the presidencies of both The American Mathematical Society and the International Jugglers’ Association, a feat that earned mention in Ripley’s Believe It or Not. He is Irwin and Joan Jacobs Professor at the Department of Computer Science and Engineering of the University of California, San Diego (UCSD) and Chief Scientist at the California Institute for Telecommunication and Information Technology. His Erd} os number is 130.

RONALD LEWIS GRAHAM

Department of Computer Science and Engineering University of California San Diego La Jolla, CA 92093-0404, USA e-mail: [email protected] Ó 2009 The Author(s) This article is published with open access at Springerlink.com, Volume 31, Number 3, 2009

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understanding theories. [. . .] If you are unsure to which class you belong then consider the following two statements. 1. The point of solving problems is to understand mathematics better. 2. The point of understanding mathematics is to become better able to solve problems For Gowers, Erd} os is the ideal member of the problemsolving group, who would select the second statement without hesitation. Selberg would naturally be placed in the theory-builder category. Gowers continues, It is that the subjects that appeal to theory-builders are, at the moment, much more fashionable than the ones that appeal to problem-solvers. Moreover, mathematicians in the theory-building areas often regard what they are doing as the central core (Atiyah uses this exact phrase) of mathematics, with subjects such as combinatorics thought of as peripheral and not particularly relevant to the main aims of mathematics. We agree about the dichotomy, but not the animosity. Indeed, Gower’s own well-deserved acclaim (including his receiving of the Fields Medal) from the entire mathematical community points to a less adversarial relationship. But it does seem to be a fair assessment of the mathematical landscape circa 1948, when the controversy began. We close with the words of Ernst Straus [6] himself, in a commemoration of Erd} os’s 70th birthday. Figure 2. Atle Selberg, 1981. Hermann Landhoff, photographer. Courtesy of the Archives of the Institute for Advanced Study.

Both were prodigies. Erd} os (so the stories go), at age five, realizing that 250 less than 100 is 150 below zero. Selberg, at age seven, showing that the difference between consecutive squares is an odd number. They were both winners of the prestigious Wolf Prize. But in (at least) two ways, however, their mathematical styles were antipodal. For Paul Erd} os, mathematics was a communal activity. Erd} os had more coauthors, roughly five hundred by current counts, than any other mathematician in history. He constantly worked with a crowd of mathematicians surrounding him. For Atle Selberg, mathematical results were to be perfected in solitude and then to be brought forth. Nils Baas [5] writes ‘‘Selberg wanted to work on his own, penetrating the problems by his own and at his own pace.’’ Selberg himself said, ‘‘I must say that I have never had any thought of collaborating with anybody. I have one joint paper, and that was with Chowla, but I must say that it was Chowla that first came to me with a question.’’ A deeper distinction is given in Tim Gower’s beautiful essay ‘‘The Two Cultures of Mathematics.’’ He explains [2]: I mean the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and 20

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In our century, in which mathematics is so strongly dominated by ‘‘theory constructors’’ Erd} os has remained the prince of problem solvers and the absolute monarch of problem posers. One of my friends - a great mathematician in his own right - complained to me that ‘‘Erd} os only gives us corollaries of the great metatheorems which remain unformulated in the back of his mind.’’ I think there is much truth to that observation, but I don’t agree that it would have been either feasible or desirable for Erd} os to stop producing corollaries and to concentrate on the formulation of his metatheorems. In many ways Paul Erd} os is the Euler of our times. Just as the ‘‘special’’ problems that Euler solved pointed the way to analytic and algebraic number theory, topology, combinatorics, function spaces, etc.; so the methods and results of Erd} os’s work already let us see the outline of great new disciplines, such as combinatorial and probabilistic number theory, combinatorial geometry, probabilistic and transfinite combinatorics and graph theory, as well as many more yet to arise from his ideas. Straus noted that Einstein chose physics over mathematics because he feared that one would waste one’s powers in pursuing the many beautiful and attractive questions of mathematics without finding the central questions. He goes on, Erd} os has consistently and successfully violated every one of Einstein’s prescriptions. He has succumbed to the seduction of every beautiful problem he has

encountered - and a great many have succumbed to him. This just proves to me that in the search for truth there is room for Don Juans like Erd} os and Sir Galahads like Einstein. To modify slightly, the twentieth century had room for Don Juans like Paul Erd} os and Sir Galahads like Atle Selberg.

A Note on the Controversy E. G. Straus Winter and Spring of 1948 were an exciting time in number theory at Princeton. Siegel was back from Germany, such relatively recent arrivals as A. Selberg, S. Chowla, and Paul Tura´n were at the Institute. Most stimulating for me was the presence of my - by then already long-term friend - Paul Erd} os, not only in Princeton, but for several months as a guest in my house. This led to a number of interesting and stimulating discussions, including the now celebrated and still unsettled Erd} os-Straus conjecture: 4 1 1 1 ¼ þ þ n x y z has a solution in positive integers x, y, z, for all integers n [ 1. Tura´n, who was eager to catch up with the mathematical developments that had occurred during the war, talked with Selberg about his sieve method and his now famous inequality. He tried to talk Selberg into providing a seminar, showing the power of his inequality by giving an elementary proof of Dirichlet’s Theorem on primes in arithmetic progressions; but Selberg, who was busy with other research and was also looking for a permanent academic position, declined. He suggested that Tura´n present the seminar, using the notes he had made for himself from his conversations with Selberg (perhaps even including some of Selberg’s own notes). This Tura´n did for a small group of us, including Chowla, Erd} os and myself and, I think one or two others that I cannot recall with certainty. After the lecture in which Tura´n completed the elementary proof of Dirichlet’s Theorem, there followed a brief discussion of the unexpected power of Selberg’s inequality. Erd} os said, ‘‘I think that you can also derive pnþ1 =pn ! 1 from this inequality. Some skepticism was expressed, also some question whether that result was more powerful than Dirichlet’s theorem. In any case within an hour or two Erd} os has discovered an ingenious derivation from Selberg’s inequality. After presenting an outline of his proof to the Tura´n seminar group, Erd} os met Selberg in the hall and told him that he could derive pnþ1 =pn ! 1 from Selberg’s inequality. In retrospect, I am sorry that I did not commit Selberg’s response to memory, but I remember its import exactly. He said, ‘‘You must have made a mistake, because with this result I can get an elementary proof of the Prime Number Theorem and I have convinced myself that my inequality is not powerful enough for that.’’ With that, excitement immediately reached a fevered pitch. Erd} os and Selberg checked and rechecked every step

of their respective proofs, and by about 10:00 p.m. they had convinced one another that the proofs were correct. An impromptu lecture was arranged in Fuld Hall. Since my wife was coming in from New York and was to arrive at Princeton Junction after the last shuttle train, the whole group (of nearly 50 people) was kind enough to wait until midnight. Then Selberg and Erd} os in succession produced their results. There appeared to me to be an atmosphere of great joy, even elation, in the room. Many of us, including myself, had the feeling of attending an important historic event. When we got home, too excited to go to sleep, Erd} os and I discussed for some time the best way to spread the word. We both realized that at that time Erd} os was far better known than Selberg and – at least in Erd} os’ mind – the elementary proof was a direct outgrowth of Selberg’s fundamental inequality, and Erd} os’ own contribution, although important, would not have been possible without that inequality. After lengthy discussion, we arrived at a formulation that Erd} os used in the scores of postcards that he sent all over the world. I believe I remember the formulation verbatim ‘‘Using a fundamental inequality of Atle Selberg, Selberg and I have succeeded in giving an elementary proof of the Prime Number Theorem.’’ A second problem in Erd} os’ mind was how the results were to be published. We discussed this repeatedly with one another and with other mathematicians, and I can no longer say how much of it arose from Erd} os, from myself, or from, say, Tura´n. However there was never any argument, just agreement that someone had made a good suggestion. The result struck me as fair to all concerned. Erd} os was to suggest that there would be back-to-back articles in the Annals. The first by Selberg alone, presenting his inequality and the derivation of Dirichlet’s theorem from that inequality; the second jointly by Erd} os and Selberg, presenting Erd} os derivation of pnþ1 pn from Selberg’s inequality and Selberg’s completion of the proof of the Prime Number Theorem. Thus anyone interested would obtain a clear picture of the respective contributions. Alas, something had gone wrong, and when Selberg returned from Syracuse – which institution was interested in him and where he gave a lecture – he was no longer willing to work with Erd} os at all. It has always been a source of great surprise and regret to me that two such superb minds and admirable human beings, whom I both consider my friends – although Erd} os is clearly a much closer friend than Selberg – have come to a permanent parting over a joint achievement that had been born with so much joy and hope. I have described Erd} os’ part in the matter and I am convinced that he did nothing intentional to hurt Selberg. I have already hinted at what may have been the source of unintentional pain, Erd} os was much better known, certainly among contemporary American mathematicians, and even the careful wording of the postcards may not have prevented recipients from passing on the news as: ‘‘Have you heard what Erd} os and some other guy have done?’’ In fact I was told this story (I forget by whom), which may well not be true, but which illustrates my point: When

Ó 2009 The Author(s) This article is published with open access at Springerlink.com, Volume 31, Number 3, 2009

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Ernst Straus Ernst Straus was born in 1922 in Munich. In 1933, after the Nazi rise to power, the Straus family emigrated to Palestine. He studied at the Hebrew University in Jerusalem. In 1941 he entered graduate school at Columbia University. He was the assistant to Albert Einstein from October 1944 through August 1948. He received his Ph.D. in 1948 from Columbia under the direction of F. J. Murray. He then accepted a position at UCLA, which he kept until his death of a heart attack in 1983. During his Princeton years, Straus was strongly influenced by E. Artin and C. L. Siegel, and his major mathematical interests shifted from relativity theory to number theory. Throughout his life, Straus’s mathematical interests continued to expand enormously and came to include geometry, convexity, combinatorics, group theory, and linear algebra, among other things. In a memorial issue,* from which the above information is taken, Straus’s colleagues, David Cantor, Basil Gordon, Al Hales, and Murray Schacher, add the following tribute: Ernst Straus was not only a great mathematician, but also a great human being. [. . .] His brilliance and enormous erudition in both humanities and science made a deep impression on all who were fortunate enough to know him. [. . .] This intellectual power was combined with a deep and radiant humanity which made Ernst truly beloved by his colleagues,

Selberg arrived in Syracuse he was met by a faculty member with the greeting: ‘‘Have you heard the exciting news of what Erd} os and some Scandanavian mathematician have just done?’’ In any case, Selberg was still in his twenties and perhaps not yet sure of the great position he would fill in the mathematics of the twentieth century. In my mind more blame attaches to the actions and inactions of the older mathematicians who should have worked to conciliate, but instead served to exacerbate the situation. In my opinion the chief culprit was Hermann Weyl, who held strong esthetic views about the work of both Erd} os and Selberg. The mathematics of Erd} os went very much against Weyl’s grain. This jumping from one challenging problem to the next, without first ascertaining its importance in the scheme of things, repelled Weyl, so that he never recognized the deep and important insights accumulated by Erd} os as if by accident. It was Weyl who had vetoed the renewal of Erd} os’ grant at the Institute in the 1930s, throwing Erd} os on the support of friends who were themselves hard pressed by the depression. Selberg on the other hand was a proud discovery of Weyl on his first tour of Europe after the war. I remember with some awe that after Selberg presented his beautiful result on the positive density of zeroes of the Zeta-function on the critical line in Weyl’s seminar, Weyl stood up to * Ernst G. Straus 1922–1983. Pacific J. Mathematics, vol. 118 (1985) v–vi.

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THE MATHEMATICAL INTELLIGENCER

Ernst Straus, c. 1949, in housing at the Institute for Advanced Study. Photo courtesy of the Straus family.

students, and friends. [. . .] Men of such talent, uncompromising integrity, generosity, and gentleness are among the world’s rarest and most precious treasures.

applaud – an event I have never seen before nor since in a mathematical seminar. If one divides mathematicians – as I sometimes do for myself – into Euler types and Gauss types, then Erd} os belongs to the former whereas Weyl and Selberg belong to the latter category. It was Weyl who caused the Annals to reject Erd} os’ article and published only a version of Selberg that circumvented Erd} os’ contribution, without mentioning the vital part played by Erd} os in the first elementary proof, or even in the discovery of the fact that such a proof was possible. I think everyone of us who could have improved things and failed to do so has a share in the blame. Chowla was senior enough, but was probably too polite and deferential. Tura´n and I felt too closely associated with Erd} os by longstanding friendships to be credible honest mediators. It may well be that nothing would have helped. The elementary proof has so far not produced the exciting innovations in number theory that many of us expected to follow. So, what we witnessed in 1948, may in the course of time prove to have been a brilliant but somewhat incidental achievement without the historic significance it then appeared to have. My own inclination is to believe that it was the beginning of important new ideas not yet fully understood and that its importance will grow over the years.

In any case, the two mathematicians involved have justified our high expectations for their mathematical contributions.

Postscript: A Note on the Effects of the Elementary Proof Carl Pomerance, Dartmouth College The Riemann zeta function seems perfectly suited to study the natural numbers since intrinsic in this one function are both their additive and multiplicative structure. But maybe it is too perfect! Although knowing the Riemann Hypothesis and its generalizations would have remarkable consequences for the primes, this approach seems stymied. Instead, many recent great results about the distribution of primes and related problems have come about through the use of elementary and combinatorial ideas. Thus, far from being an isolated intellectual challenge, the elementary proof of the prime number theorem was a signal that good ideas and strong tools are close at hand. We already had an inkling of this in Riemann’s era when Chebyshev used combinatorial methods to show that there is a prime in [n, 2n] for every natural number n. And a century ago, the elementary proof of Brun, stating that most primes are not part of twin-prime pairs, opened the door for combinatorial sieve methods and their many glorious consequences. Since the elementary proof, some of the most profound and exciting results in the field have had strong elementary and combinatorial leanings. After Roth used the (analytic) circle method to show that dense sets of integers must have 3-term arithmetic progressions, Szeme´redi used an elementary (and very complicated) proof to generalize this to k-term arithmetic progressions. This result became an intrinsic tool in the recent Green–Tao proof that the set of primes contains arbitrarily long arithmetic progressions. After Miller showed how a generalization of the Riemann Hypothesis allows a deterministic, polynomial-time

procedure for recognizing primes, a few decades later, Agrawal, Kayal, and Saxena showed the same with completely elementary (and rigorous) methods. I recall that both Erd} os and Selberg were astonished when Maier used elementary tools to show the existence of unexpected irregularities in the distribution of the primes. This and subsequent results raise the tantalizing possibility of turning the tables and using elementary and combinatorial methods to say something about the distribution of zeta’s zeros. For if these zeros tell us about the primes, why shouldn’t the primes say something about them? OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

REFERENCES

[1] Dorian Goldfeld, The Elementary Proof of the Prime Number Theorem: an Historical Perspective, in Number Theory (New York, 2003), 179–192, Springer, New York, 2004. [2] Timothy Gowers, The Two Cultures of Mathematics, in Mathematics: Frontiers and Perspectives, V. Arnold, M. Atiyah, P. Lax, B. Mazur, eds, American Mathematical Society, 2000. [3] Paul Hoffman, The Man who Loved Only Numbers: The story of } s and the search for Mathematical Truth (New York, Paul Erdo 1998), Hyperion. [4] Nils A. Baas and Christian F. Skau, The Lord of the Numbers, Alte Selberg. On his Life and Mathematics. Bulletin of the American Mathematical Society, vol. 45 (2008), 617–649. [5] Nils A. Baas, Speech at Atle Selberg Memorial at the Institute for Advanced Study, January 12, 2008. }s at 70, Combinatorica, vol 3 (1983), 243– [6] Ernst Straus, Paul Erdo 246.

Ó 2009 The Author(s) This article is published with open access at Springerlink.com, Volume 31, Number 3, 2009

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Years Ago

David E. Rowe, Editor

Mathematical Quiz: A Friendship of Lasting Value DAVID E. ROWE

M

athematicians have often competed with one another throughout history, though sometimes with disastrous consequences (think of Newton and Leibniz). Collaborations, on the other hand, were fairly uncommon prior to 1900, though the nineteenth century certainly witnessed a few striking examples of joint research (for example, Sturm–Liouville theory and the early work of Lie and Klein on transformation groups). But the proliferation of fruitful collaborations and friendships was a hallmark of mathematical research throughout the twentieth century. Hardy and Littlewood spring immediately to mind, as does Paul Erdo¨s, who famously collaborated with mathematicians all over the world. Do you recognize the two men pictured below, discussing (presumably) a mathematical manuscript?

Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

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THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

Here’s a hint: Their friendship, which began in 1931, would later have significant consequences for mathematical relations in Europe after the Second World War. Readers who have never seen this photo before (no insiders please!) are invited to write me and guess their names. The contest winner(s) will not only know who these two mathematicians were but also why their friendship is well worth remembering. Please address your reply in the form of a short essay. I will provide space for the best essay in a not-too-distant future issue.

Fachbereich 08, Institut fu¨r Mathematik Johannes Gutenberg University 55099 Mainz Germany e-mail: [email protected]

Mathematical Communities

Geometry of Numbers in Vienna CHUANMING ZONG

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of ‘‘mathematical community’’ is the broadest. We include ‘‘schools’’ of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater

eometry of Numbers, as a classic branch in mathematics, was formally born in 1891 when Minkowski gave a talk with this title in Halle, Germany. Roughly speaking, it is a subject dealing with relations between lattices and convex bodies. Before Minkowski, many great mathematicians had made contributions to particular problems of this subject. For example, Kepler studied the ball-packing density and made the Kepler conjecture; Newton studied the kissing number of the ball and raised the thirteen-sphere problem; Lagrange studied the reduction of positive-definite binary forms and therefore deduced the maximal lattice-packing density of circular discs; Gauss determined the density of the densest lattice-ball packing; Hermite studied the reduction theory of positive-definite quadratic forms and obtained the first general bounds for the constants named after him, etc. However, it was Minkowski who first studied its key problems in full generality. His fundamental theorem published in 1891 can be regarded as the cornerstone of the subject. It also shows the clear characteristic of this field.

G

than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.

Please send all submissions to Marjorie Senechal, Department of Mathematics, Smith College, Northampton, MA 01063, USA e-mail: [email protected]

MINKOWSKI’S FUNDAMENTAL THEOREM. Let C be an n-dimensional centrally symmetric convex body centered at the origin. If the ndimensional volume vðCÞ 2n ; then C contains an integral point which is different from the origin. Geometry of numbers is a branch of number theory, since its very original key problem deals with the arithmetic of positive-definite quadratic forms and it has applications almost everywhere in number theory, for example in Diophantine approximation, in algebraic number theory, in transcendental number theory, and even in the proof of the four-square theorem. It plays

important roles in the works of Alan Baker and Enrico Bombieri. In addition, many leading figures in geometry of numbers such as L. J. Mordell, Harold Davenport, Kurt Mahler, Edmund Hlawka and Wolfgang Schmidt have done great work in other areas of number theory. During recent decades, the geometry of numbers has developed great interaction with discrete geometry, convex geometry, computational geometry, coding theory and crystallography. In the history of its development, several schools –for example, the Manchester-London school and the Vienna school – have played crucial roles. In this article, I will discuss its development in Vienna.

The Start: Two Pioneers Our main story starts with Philipp Furtwa¨ngler. Furtwa¨ngler was born on 21 April 1869 in Elze, a small town close to Hannover, Germany. When the geometry of numbers was born, he was studying mathematics in Go¨ttingen. At that time, Minkowski was not yet in Go¨ttingen, but he was very well-known there since his work was highly appreciated by Hilbert and Klein, both of whom were professors there. Therefore, Furtwa¨ngler might have followed the development of this new area even as a student. In 1896, he received a doctorate for his dissertation ‘‘Zur Theorie der in Linearfaktoren zerlegbaren ganzzahligen terna¨ren kubischen Formen’’ under Klein’s supervision. In 1912, after Furtwa¨ngler had been a professor at Bonn and Aachen for eight years, he accepted a chair at the University of Vienna. At that time he was known as an algebraic numbertheorist, having made his name as a world-famous mathematician through his works in reciprocity principles in algebraic number fields and in the existence of class fields. However, he also published several important results in geometry of numbers. In 1896, Minkowski observed that every lattice-

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square tiling of the plane has two squares joined along an entire edge and every lattice-cube tiling of E 3 has two cubes joined along an entire facet. Based on this observation, he claimed that the higher-dimensional analogue should also be true and promised a proof. However, he was not able to give the proof before his sudden death in 1909. Let k be a positive integer, I n be an ndimensional unit cube centered at the origin of E n, K be an n-dimensional lattice, and let B denote the union of the boundaries of I n + p, p [ K. If every point x 2 E n n B belongs to exactly k of the cubes I n + p, p 2 K, we will call I n + K a multiple lattice-cube tiling of E n. In 1936, before Minkowski’s claim was proved, Furtwa¨ngler (Figure 1) considered the following generalization.

Figure 1. Philipp Furtwa¨ngler (1869– 1940).

FURTWA¨NGLER’S CONJECTURE. In every multiple lattice-cube tiling there is a twin. In other words, in every multiple lattice-cube tiling there are two cubes meeting each other along an entire facet. Like Minkowski, Furtwa¨ngler proved the two- and three-dimensional cases. However, the higher-dimensional cases were disproved by Gyo¨rgy Hajo´s in 1941. In the same paper, Hajo´s proved that Minkowski’s claim is correct. Furtwa¨ngler also did some work in Diophantine approximation using ideas from the geometry of numbers. For example, he proved that Let D denote the discriminant of any real, but not necessarily totally real, algebraic field of degree n + 1. Then for each number c [ jDj1=2 , there exist

n real numbers h1, h2, …, hn such that jhi xnþ1 xi jðcjxnþ1 jÞ1=n ;i ¼ 1;2;n have only solutions.

finitely

many

integer

As we will see, many years later, another mathematician educated in Vienna made his name in this field. Furtwa¨ngler was well-known not only as a leading mathematician, but also as a great teacher. He supervised several very well-known students including Nikolaus Hofreiter, Olga Taussky-Todd, Wolfgang Gro¨bner, and Henry Mann. Taussky-Todd is one of the best known female mathematicians in history. Gro¨bner is known for the Gro¨bner basis discovered by one of his

AUTHOR

.................................................................................................... CHUANMING ZONG received his Ph.D from TU Wien in1993. In 1997, he was promoted to a professor at the Mathematics Institute of the Chinese Academy of Sciences. In 2000, he moved to Peking University. He has been a visitor at ETH-Zurich, UCL, IHES, TU Berlin, MSRI, etc. Among several other prizes, he has received the S.S. Chern prize of the Chinese Mathematical Society (2007) and the von Prechtl medal from TU Wien (2008).

School of Mathematical Sciences Peking University Beijing 100871 China e-mail: [email protected]

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Ph.D students, Bruno Buchberger. Mann is known for his proof of the Schnirelmann-Landau conjecture, for which he was awarded a Cole prize by the American Mathematical Society. Hofreiter is the other pioneer discussed in this section. In addition, he had an important mathematical influence on Go¨del. For his great contributions to mathematics, Furtwa¨ngler was elected a member of the Austrian Academy of Sciences in 1927 and to corresponding member of the Prussian Academy of Sciences in Berlin in 1931. He died on 19 May, 1940, in Vienna. For geometry of numbers, of course, one of his most important contributions was inspiring Hofreiter’s interest in this subject. Nikolaus Hofreiter (see Figure 2). was born on 8 May 1904 in Linz, Austria. In 1923, he became a mathematics student at the University of Vienna. At that time, Hans Hahn, Furtwa¨ngler and Wilhelm Wirtinger were among the professors there. In 1927, Hofreiter earned a doctorate with a dissertation ‘‘Eine neue Reduktionstheorie fu¨r definite quadratische Formen’’ under Furtwa¨ngler’s supervision. This is one of the earliest dissertations in the geometry of numbers after Minkowski. Unlike Furtwa¨ngler, Hofreiter mainly worked in the geometry of numbers, especially in reduction theory and in the product of linear forms. Clearly, a unimodular transformation changes one positive-definite quadratic form to another such form. Therefore, unimodular transformations define an equivalence relation in the family of all positive-definite quadratic forms of n variables and thus the family can be divided into classes. By

choosing a suitable representative from each class, some arithmetic problems about quadratic forms can be much simplified. This is the philosophical principle of reduction theory. In the course of studying quadratic forms, several reductions were introduced and developed, for example, the Hermite reduction, the Minkowski reduction, the Voronoi reduction, the Korkin-Zolotarev reduction, the Venkov reduction, etc. In this context, Hofreiter proved, For n B 4, a positive-definite quadratic form in n variables is Minkowski reduced if and only if it is Hermite reduced. This result does not hold for all n [ 4: in 1971 Rysˇkov discovered a positive-definite quadratic form in seven variables which is Minkowski reduced but not Hermite reduced. To estimate the critical determinants of the products of linear forms is a classic topic in the geometry of numbers. It has attracted the interests of many prominent mathematicians like Mordell, Davenport, Mahler, and Rogers. Hofreiter published several papers on this topic. For example, he proved,

Let Dr,2s denote the critical determinant of the star body ) ( r s Y Y rþ2s 2 2 : xi ðxrþj þ xrþsþj Þ1 x 2E i¼1

j¼1

and let Dr,2s be the absolutely smallest discriminant of any algebraic number field of degree r + 2s, with r real and s pairs of complex conjugate fields. Then

d ðBÞ

fðnÞ ¼

Hofreiter attracted many students to the geometry of numbers. He supervised more than sixty Ph.D dissertations, most of them in that field. Among his students, Edmund Hlawka and Peter M. Gruber have become leading figures in this area, as we will see in later sections. As university professors, we often face the dilemma of whether to concentrate solely on our research, or to take time to supervise students. In some parts of the world, the latter is not counted in the annual work. However, for the development of our discipline, supervising a dissertation is certainly more important than publishing an article because a dissertation can lead to many papers. For this reason, Hofreiter is indeed a great force in the geometry of numbers. Hofreiter was also a distinguished administrator. He was the chief editor of the Monatshefte fu¨r Mathematik for many years and was once the President of the University of Vienna. He was elected a corresponding member of the Austrian Academy of Sciences in 1970.

To introduce the key result, first let us fix some notation. Let S denote an n-dimensional centrally symmetric bounded star body, let B denote an ndimensional unit ball, and let d ðSÞ denote the density of the densest lattice packing of S in E n. To determine or estimate the packing densities of certain objects is a fundamental problem in mathematics. First of all, it is one of the basic problems in understanding n-dimensional space. Second, many problems, pure or applied alike, for example, error-

fðnÞ ; 2n1

where

1

Dr;2s jDr;2s j2 :

The Highlight: The MinkowskiHlawka Theorem

Figure 2. Nikolaus Hofreiter, 1904– 1990.

correcting codes and crystal formations, can be reformulated as packing problems. In 1905, by studying positive-definite quadratic forms, Minkowski was able to prove

1 X 1 n k k¼1

is the Riemann zeta function. He then made a general conjecture for bounded star bodies. In 1944, by an ingenious mean-value method, Hlawka proved Minkowski’s conjecture, after which this fundamental result became known as the Minkowski-Hlawka theorem.

THE MINKOWSKI-HLAWKA

THEOREM.

For each n-dimensional bounded centrally symmetric star body S one has d ðSÞ

fðnÞ : 2n1

During the past 64 years, many prominent mathematicians, including Siegel, Davenport, Weil, Schneider, Mahler, Malysˇev, Rogers, Schmidt, Cassels, etc, gave new proofs of or improvements to this theorem. However, so far the best known improvement is Schmidt’s: d ðSÞ

n log 2 b ; 2nþ1

ð1Þ

where b is an absolute constant. On the other hand, Kabatjanski and Levensˇtein proved that d ðBÞ 20:599nð1þoð1ÞÞ :

ð2Þ

Comparing (1) with (2), we note that the truth is still far off, even for the ndimensional ball. Edmund Hlawka (Figure 3) was born on 5 November 1916 in Bruck, Austria. In 1934, he became a student at the University of Vienna. At first, he studied both mathematics and physics, but soon devoted his time solely to mathematics. In 1938 he obtained a ¨ ber die doctorate with a dissertation ‘‘U Approximationen von zwei komplexen inhomogenen Linearformen’’ under the supervision of Hofreiter.

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Impressed by his dissertation, Siegel invited Hlawka to be an assistant in Go¨ttingen, but he preferred to stay in Vienna. In 1948 he was promoted to professor at the University of Vienna, where he worked for 33 years. In 1981, he moved to the Technical University of Vienna, from which he retired in 1987. Besides the Minkowski-Hlawka theorem, Prof. Hlawka has obtained several other remarkable results in the geometry of numbers. For example, through the works of Mordell, Siegel, Hlawka, and Rogers, we have the following theorem. Let C denote an n-dimensional 0-symmetric convex body and let xn denote the volume of the n-dimensional unit ball. Suppose that 1 3 n vðCÞ 4500:5nðn1Þ n0:5n xn : 2 2 Then for each lattice K of determinant 1 there is a diagonal matrix D with positive diagonal elements and determinant 1 such that DK \ C has no other point except the origin. Prof. Hlawka is not only a great figure in the geometry of numbers, but also one of the pioneers in the theory of uniform distributions. For example, the following inequality is basic in discrepancy theory.

THE KOKSMA-HLAWKA

INEQUALITY.

If f(x) has bounded variation on the unit hypercube I d in d dimensions, then for any points x1 ; x2 ; . . .; xN in the unit cube we have

Figure 3. Prof. Edmund Hlawka, left, receiving a gold medal at the age of ninety-one (2007) for his great achievements. 1

Z 1X N f ðxi Þ f ðxÞdx V ðf ÞDN ; N i¼1 Id where V( f ) is the Hardy-Krause variation. Even the following elementary inequality bears Hlawka’s name: jaj þ jbj þ jcj þ ja þ b þ cj ja þ bj þ jb þ cj þ ja þ cj: He has published more than one hundred and fifty papers. In 1990, Springer-Verlag published his Selecta, which contains his most important mathematical works. As a great thinker, Prof. Hlawka has a sharp mind, an extraordinary memory, and a remarkable sense of humor. From 1991 to 1993, I was a Ph.D student in Vienna. At that time he was already seventy-six years old. I was often surprised by his ability to demonstrate complicated proofs on several blackboards without prepared notes. He is a great teacher as well. From the long list of his Ph.D students, one can see several famous names, Gruber, Niederreiter, Schmidt, etc. In 2001, at his 85th birthday conference, someone showed a picture of Prof. Hlawka taken more than thirty years earlier at a talk. In the picture he seems very old, even older than himself. Prof. Hlawka looked at the picture for a moment and then told the audience: ‘‘It is not me, it is only topologically equivalent to me.’’ I cannot be sure of the truth of the next story, which I heard several times when I was in Vienna. Nevertheless, it reflects Prof. Hlawka’s character. In the 1960s, once he took the train from Vienna to Berlin for a DMV conference. On the way, a mathematician from Munich got in. When both ascertained that they were going to the same conference, they started to chat. The Municher asked: ‘‘Ah, you come from Vienna. Do you know Hlawka? Is he still alive?’’ Prof. Hlawka answered: ‘‘Oh, yes, I know him. He is very well, and perhaps will also come to the conference.’’ Needless to say, the Municher could learn the truth easily when Prof. Hlawka gave a talk at the conference.

Prof. Hlawka died on February 19, 2009, at the age of 92.

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For his great contribution to mathematics, Professor Hlawka has received many honors. He is a member of both the Austrian Academy of Sciences and the (German) Leopoldina Academy, and is a corresponding member of the Bavarian Academy of Sciences, the Bologna Academy of Sciences, and the Rheinische - Westfa¨lische Academy of Sciences. He has had honorary doctorates conferred on him by the University of Vienna, the University of Erlangen, the University of Salzburg, the University of Graz, and the Technical University of Graz. He has also been the recipient of many prizes. For example, the Erwin-Schro¨dinger prize of the Austrian Academy of Scienses, the Gauß -medal of the Berlin Academy of Sciences, and the Dannie-Heinemann prize of the Go¨ttingen Academy of Sciences.1

From GN to Other Subjects In the past half century, mathematics experienced a period of interaction: its branches are no longer really independent. The development of the geometry of numbers especially reflects such interactions. On the one hand, computational techniques, ideas of coding theory and linear programming have shown great power for attacking some classic problems in the geometry of numbers. For example, Kabatjanski, Levensˇtein, Odlyzko, Sloane, and Musin’s works on sphere packings by linear programming methods, and Hales, Cohn, and Kumar’s works on packing densities, relied on computational methods. On the other hand, methods and ideas of the geometry of numbers have shown their power in other subjects like Diophantine approximation, convex geometry, discrete geometry, Crystallography, etc. The Vienna school has greatly contributed to the interactions between the geometry of numbers and Diophantine approximation, and between the geometry of numbers and convex and discrete geometry. From GN to Diophantine Approximation Let us take the following theorem as an example which will demonstrate what

Diophantine approximation is and why it is closely related to the geometry of numbers.

HURWITZ’S

THEOREM.

For each irrational number a, there are infinitely many rational numbers p/q satisfying a p\ pﬃﬃ1ﬃ : ð3Þ q 5q 2 Usually, this theorem is proved by continued fractions. However, the following argument shows its geometric nature. Clearly, (3) is equivalent to pﬃﬃﬃ 5qjaq pj\1; and therefore Hurwitz’s theorem is equivalent to the statement that for each irrational number a the star domain pﬃﬃﬃ Ca ¼ ðx; yÞ 2 E 2 : 5xjax yj\1 contains infinitely many points with integer coordinates. Comparing this with Minkowski’s fundamental theorem, one can easily see the similarity. In fact, Hurwitz’s theorem indeed can be proved by ideas from the geometry of numbers. Several contemporary Vienna mathematicians have made important contributions to Diophantine approximation and to the theory of uniform distribution, for example, Prachar, Schmidt, Niederreiter, and Tichy. Here we only focus on the work of Schmidt (Figure 4) since it is closely related to the geometry of numbers. In 1955, at the age of 22, Wolfgang M. Schmidt obtained a doctorate from the University of Vienna under Hlawka’s supervision. He started his career in the geometry of numbers and made some fundamental contributions to this area. At that time the Minkowski-Hlawka theorem was a very active topic. As I mentioned before, many prominent mathematicians tried to improve or generalize it. In that compaign, Schmidt was one of the driving forces. He published more than ten papers on this topic. Let l denote a normalized measure on the space L of all lattices of determinant 1 in E n. In 1945, C.L. Siegel proved the following result.

SIEGEL’S MEAN-VALUE THEOREM. Let f(x) be a Riemann integrable function defined on E n. Then

Z

X

f ðaÞdlðKÞ ¼

L a2Knfog

Z

a p \ 1 : q q nþ

f ðxÞdx:

En

ð4Þ

By taking f(x) to be the characteristic function of S one can deduce the Minkowski-Hlawka theorem. Clearly, this formula is important and elegant. Afterwards, several authors including Macbeath, Rogers, Schmidt, and Weil made contributions to sharpen this result and tried thereby to improve the Minkowski-Hlawka theorem. To this end, Schmidt generalized Siegel’s meanvalue formula to the following one.

This theorem has nothing to do with the geometry of numbers. Its proof is not difficult. However, it is very powerful – with it, one can construct uncountably many transcendental numbers. In more than one hundred years, through the works of Thue, Siegel, Dyson, and finally Roth, we now have the following theorem, for which Roth was awarded a Fields medal.

Let f(x1, x2, …, xk) be a Riemann integrable function of k vector variables defined on E n. Then Z X f ða1 ; a2 ; ; ak ÞdlðKÞ L Z Z Z ¼ f ðx1 ; x2 ; ; xk Þ

ROTH’S THEOREM. Let a be a real algebraic number. For any positive number there are only finitely many rational numbers p/q satisfying a p \ 1 : q q 2þ

dx1 dx2 dxk ; P where * denotes the sum over all independent vectors {a1, a2, …, ak} of K. In this way, he was able to obtain the improvement (1) which still holds the record. However, Schmidt’s best-known work is in Diophantine approximation rooted in the geometry of numbers. In 1844, Liouville proved the following theorem.

LIOUVILLE’S THEOREM. Let a be a real algebraic number of degree n. For any positive number e there are only finitely many rational numbers p/q satisfying

Comparing this with Hurwitz’s theorem, one can see that Roth’s theorem is the best possible. For many people Roth’s theorem is already the happy end. However, in 1970 Schmidt generalized Roth’s theorem to the following form.

SCHMIDT’S THEOREM. Let a1, a2, …, an be n real algebraic numbers and be any fixed positive number. If 1, a1, a2, …, an are linearly independent over the rational number field, then both ! !1þ n n Y X ai xi x0 ð1 þ jxi jÞ \1 i¼1

i¼1

and

Figure 4. Wolfgang M. Schmidt, 1933–. 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

29

1þ

ð1 þ jx0 jÞ

n Y ðai x0 xi Þ\1 i¼1

have finitely many integer solutions. Schmidt’s theorem is geometric not only in form, but also in proof ideas. As one can see from its proof, this theorem is deeply rooted in the geometry of numbers. Schmidt moved to the United States in the 1960s and today is a professor at the University of Colorado at Boulder. For his profound contributions to Diophantine approximation he was awarded the Cole prize in Number Theory by the American Mathematical Society in 1972. He is a member of the Austrian Academy of Sciences, the American Academy of Arts and Sciences, and the Polish Academy of Sciences. He has been honored by three doctorates honoris causa (that for both words) and has been invited to address the ICM three times. From GN to Convex and Discrete Geometry If one looks at Minkowski’s fundamental theorem carefully, one can easily discover that it deals with only two basic concepts — convex bodies and lattices. As we have seen in the preceding section, lattices naturally lead from the geometry of numbers to Diophantine approximation. In this subsection, we will see some connections between the geometry of numbers and convex geometry through the works of Peter M. Gruber. Convex bodies are very natural geometric objects. The subject is often mistakenly believed to be ‘‘easy’’. In fact, convex bodies have very rich structure and are fundamental to several other mathematical subjects such as functional analysis and optimization. Throughout the history of convex geometry, many great mathematicians made contributions to it, for example, Euclid, Archimedes, Kepler, Cauchy, Euler, etc. One gets an impression of convex geometry from the following result, by which one can deduce a lower bound for the lattice-packing densities of arbitrary convex bodies.

ROGERS-SHEPHARD

THEOREM.

Let K be an n-dimensional convex body and let D(K) be the difference body of K defined by {x - y : x, y [ K }. Then 30

THE MATHEMATICAL INTELLIGENCER

2n mðDðK ÞÞ ; 2 n mðK Þ

improve Swinnerton-Dyer’s theorem. However, Gruber’s next theorem tells us the opposite.

n

where the left-hand equality holds if and only if K is centrally symmetric and the right-hand equality holds if and only if K is a simplex. In 1941, Peter M. Gruber (Figure 5) was born in Klagenfurt, Austria. From 1959 to 1966, he studied mathematics and physics at the University of Vienna where, in 1966, he obtained a doctorate under Hofreiter’s supervision. In 1971, he became a professor at the University of Linz, and in 1976 he moved to the Technical University of Vienna. As one of the key members of the Vienna school, he has done important work in both the geometry of numbers and convex geometry. The kissing number b(K, K) of a convex body K in a lattice packing K + K is the number of the translates of K touching K. This is a very natural geometric concept. In fact, it is also an important arithmetic concept. Let D(K) be the difference body of K defined in the Rogers-Shephard theorem, and let uðxÞ denote the gauge function defined by D(K). Then the kissing number b(K,K) is exactly the number of lattice points on the boundary of D(K) and therefore is the number of lattice solutions to

Let b(K) be the maximal lattice kissing number of K. Then Minkowski and Swinnerton-Dyer’s results imply nðn þ 1Þ bðK Þ 3n 1: It seems that n(n + 1) is not the optimal lower bound when n is large. However, a proof or disproof is still missing. Like Schmidt, Gruber gradually shifted his research interest but, this time, to convexity. Approximating a given convex body by polytopes is a basic problem in convex geometry. Let d( ,) denote a metric defined on the space of all d-dimensional convex bodies, let P1n be the set of polytopes of at most n vertices inscribed in K, and let P2n be the set of polytopes of at most n facets circumscribed about K. In 1953, L. Fejes To´th posed the problem: show that If K is sufficiently smooth and has curvature function j(x) [ 0, then

uðxÞ ¼ 1: In particular, when K is an ellipsoid, the value of b(K, K) is nothing else but the number of integer solutions of a corresponding positive-definite quadratic form. Minkowski proved that bðK ; KÞ 3n 1

In the sense of Baire category, most n-dimensional centrally symmetric convex bodies C touch at most 2n2 others in every lattice packing of maximal density.

ð5Þ

and the equality holds only if K is a parallelopiped. In 1953, as a counterpart to (5), H.P.F. Swinnerton-Dyer proved the following result. Let C be an n-dimensional centrally symmetric convex body. In every lattice packing of maximal density the convex body touches at least n(n + 1) others. From the intuitive point of view, this theorem can be proved by inductively building up the lattice packings by layers. In fact, the proof is quite complicated. Comparing (5) with Swinnerton-Dyer’s result, one may think that there is a good chance to

inf dðK ; PÞ

P2Pni

si ; n2=ðd1Þ

i ¼ 1; 2

ð6Þ

hold for suitable constants si as n ! 1: There are several particular metrics defined on the space of all convex bodies, such as the Hausdorff metric dH( , ), the Banach-Mazur metric dBM ( , ) and the symmetric-difference metric dS( , ). For the Hausdorff case, Schneider and Gruber proved that If K is twice differentiable and has curvature function j(x) [ 0, then (6) holds with s1 ¼ s2 1 hd1 ¼ 2 xd1

!2=ðd1Þ

Z jðxÞ

1=2

drðxÞ

oðCÞ

where hk is the minimal ball-covering density of E k.

;

Figure 5. Peter M. Gruber, 1941–.

For the symmetric-difference metric, Gruber proved the following theorem. If K is twice differentiable and has curvature function j(x) [ 0, then (6) holds with !ðdþ1Þ=ðd1Þ Z jðxÞ1=ðdþ1Þ drðxÞ

s1 ¼

oðCÞ

; gd1

where gk is a constant related to the Delone triangulation in E k, and !ðdþ1Þ=ðd1Þ Z 1=ðdþ1Þ

s2 ¼

jðxÞ oðCÞ

drðxÞ

; ld1

where lk is a constant related to the Dirichlet-Voronoi tiling of E k. Besides his research work, Prof. Gruber has published several influential

books on both the geometry of numbers and convex geometry. Good books play a crucial role in the further development of the subject. First of all, they can inspire the interests of beginners. Second, they can provide graduates a broad view and a solid knowledge of the subject. Even for experts, a good book means a lot. Prof. Gruber’s books will affect the development of these two subjects for generations. Prof. Gruber’s contributions have been recognized by many honors, including two honorary doctorates and a grand decoration of honor in silver for services to the Republic of Austria. He is a member of the Austrian Academy of Sciences, a foreign member of the Russian Academy of Sciences, a corresponding member of the Bavarian Academy of Sciences and Humanities, and a member of several other distinguished societies.

for a coffee break. Sitting in a circle, drinking cups of coffee, we chatted about different things - from politics to mathematics. During that time I made several lasting friends and met many famous geometers, Boltjanski, Danzer, Fejes To´th, Larman, McMullen, Rogers, Rysˇkov, Schneider, Wills, …. I learned a lot besides mathematics. Above all, I understand much better what being a mathematician really means. ACKNOWLEDGMENT

For some useful suggestions and comments, I am grateful to Prof. Martin Henk, Prof. Marjorie Senechal and Dr. Iskander Aliev. For the kind permission to use the photos, I am grateful to the Mathematisches Forschungsinstitut Oberwolfach.

REFERENCES

[1] P.M. Gruber, Convex and Discrete Geom-

Vienna, Vienna In 1990, when I planed to go abroad for a Ph.D, I asked Prof. Hu (an expert in the history of mathematics) where is the best known place to study the geometry of numbers. He replied: Vienna — because both Gruber and Hlawka are there. Vienna is a great city. On the streets one can feel the presence of Bach, Beethoven, Mozart, Schubert, and Strauss; in the museums one can meet Raphael, Rembrandt, and Rubens; in the universities one can sense Boltzmann, Go¨del, and Schro¨dinger. On the other hand, one can enjoy the unique Vienna concerts, cafe´s, and Heuriger. When I was a Ph.D student in Vienna, colleagues would gather in Prof. Gruber’s large office after lunch

etry, Springer-Verlag, Berlin, 2007. [2] P.M. Gruber and C.G. Lekkerkerker, Geometry of Numbers, North-Holland, Amsterdam, 1987. [3] E. Hlawka, Nachruf auf Nikolaus Hofreiter, Monatsh. Math. 116 (1993), 263–273. [4] N. Hofreiter, Nachruf auf Philipp Furtwa¨ngler, Monatsh. Math. Phys. 49 (1940), 219–225. [5] A. Huber, Philipp Furtwa¨ngler, Nachruf, Jber. DMV. 50 (1941), 167–178. [6] C. Zong, Strange Phenomena in Convex and Discrete Geometry, Springer-Verlag, New York, 1996. [7] C. Zong, Sphere Packings, Springer-Verlag, New York, 1999. [8] C. Zong, The Cube: A Window to Convex and Discrete Geometry, Cambridge University Press, Cambridge, 2006.

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A Mathematical Analysis of the Sleeping Beauty Problem JEFFREY S. ROSENTHAL he Sleeping Beauty problem (Elga, 2000; see also Piccione and Rubinstein, 1997) is a philosophical dilemma related to conditional probability. It may be succinctly described as follows. Sleeping Beauty is put to sleep, and a fair coin (say, a nickel) is tossed. If the nickel shows heads, then Beauty is interviewed on Monday only, whereas if the nickel shows tails, Beauty is interviewed on both Monday and Tuesday (and given an amnesia-inducing drug between the two interviews so she does not remember the first interview during the second). In each interview, without access to any additional information (such as the result of the coin toss, or the existence of any previous interviews, or the day of the week), Beauty is briefly awakened and is asked to assess the probability that the nickel showed heads. The question is, what probability should she assign to this? One possible answer (e.g., Lewis, 2001; Arntzenius, 2002; Bostrom, 2007; Pust, 2008) is 1/2. After all, the coin is fair, so Beauty surely would assess the probability of heads as 1/2 before being put to sleep. Now, when Beauty is awakened and interviewed, she apparently does not gain any new information, because she knew in advance that she would be interviewed at least once no matter what. So, if the probability of heads was 1/2 before, it seems plausible that this probability should be unchanged during the interview, giving a final answer of 1/2. This answer seems intuitive (and was, in fact, the author’s first reaction upon hearing of this problem). But is it correct? Another possible answer (e.g., Elga, 2000; Dorr, 2002; Monton, 2002; Weintraub 2004; Horgan 2004; Neal, 2006; Titelbaum, 2008) is 1/3. One argument for this is that

T

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THE MATHEMATICAL INTELLIGENCER 2009 Springer Science+Business Media, LLC

Beauty will be interviewed twice as often when the nickel shows tails as when it shows heads. Thus, if during each interview she makes a bet in which she will win $1 on tails but lose $2 on heads, then she will break even on average (and also, by the Law of Large Numbers, break even over the long run if the experiment is repeated many times). But for this to be a fair bet, the probability of heads must be 1/3. This argument is somewhat convincing. However, one concern is that it does not appear to make explicit use of the amnesia aspect, that is, it appears to still apply if we instead assume that Beauty is permitted to remember any previous interviews. But under that assumption, the conclusion seems incorrect, for if Beauty remembers having a previous interview, she then would immediately know that the nickel was tails with probability 1. Some authors have accepted certain parts of both of these solutions. For example, although Arntzenius (2002) argues that Beauty should assign probability 1/2 to heads, he nevertheless agrees that ‘‘if she bets according to her degree of belief of 1/2, she can be expected to lose money against a bookie, and she and the bookie know this in advance’’. In other words, he feels that the question of Beauty’s assigned probabilities is distinct from the question of fair betting odds, the former being 1/2 and the latter being 1/3. A different approach is to appeal to the Principle of Indifference, which asserts that equal probabilities should be assigned to any collection of indistinguishable, mutually exclusive, and exhaustive events. But in this case, it is not clear to what ‘‘collection’’ this principle should be applied. If we apply it to ‘‘nickel heads’’ and ‘‘nickel tails’’, we obtain

an answer of 1/2. If we apply it to ‘‘nickel heads and interview Monday’’, ‘‘nickel tails and interview Monday’’, and ‘‘nickel tails and interview Tuesday’’, we obtain an answer of 1/3. If we apply it to ‘‘interview Monday’’ and ‘‘interview Tuesday’’ we conclude that with probability 1/2 the interview will be on Monday, and then a second application implies that the probability of heads is (1/2)(1/ 2) = 1/4. In short, the Principle of Indifference does not appear to resolve the problem satisfactorily. To a mathematical probability-theorist such as I, such controversy is frustrating. We are being asked to compute the conditional probability that the nickel showed heads, conditional on the fact that Beauty is currently being interviewed. Conditional probabilities are well understood and should be unambiguously analysable by straightforward mathematics, using the classic formula PðAjBÞ ¼ PðA and BÞ : So, how could this simple conditional probability PðBÞ problem create such controversy? The difficulty seems to be that a precise mathematical interpretation of ‘‘conditional on currently being interviewed’’ is unclear, thus creating an obstacle to direct mathematical calculation. This article attempts to reconsider the problem in such a way that precise mathematical reasoning can be applied. After such reconsideration, we then obtain the answer 1/3 through direct calculation. It is hoped that this mathematical approach avoids most of the philosophical ambiguities in some previous arguments.

A Subproblem: the Sleeping Peon Consider the following simple subproblem. We find a Peon and put him to sleep, and then flip a fair coin, say a nickel. If the nickel shows tails, we wake Peon and interview him (just once), asking him to assess the probability that the nickel showed heads. If the nickel shows heads, then we flip a second fair coin, say a dime. If the dime shows tails, we similarly wake the Peon and interview him once. If not (i.e., if the nickel and dime both show heads), then we do not bother to wake or interview Peon at all. In summary,

AUTHOR

......................................................................... JEFFREY S. ROSENTHAL , after writing his

doctoral thesis with Persi Diaconis at Harvard, found it fortunate that universities do not have strict nepotism rules; he became a professor at the University of Toronto, where his father Peter Rosenthal was entrenched in the Department of Mathematics. The younger Rosenthal is well known both as a theoretician and as a trouble-shooter in everyday applications of probability. His hobbies include improvisational comedy and watching Star Trek. Department of Statistics University of Toronto Toronto M5S 3G3 Canada e-mail: [email protected]

we interview Peon once if either the nickel or the dime shows tails, otherwise we interview him zero times. Hence, the overall probability that Peon is interviewed is equal to 3/4. Under these circumstances, what probability should Peon assign, upon being interviewed, to the event that the nickel showed heads? For this subproblem, the solution seems clear. Let Interviewed be the event that ‘‘Peon was interviewed (at all)’’, and let NickelHeads be the event ‘‘the nickel showed heads’’, and similarly DimeHeads, etc. Then Peon is being asked to assess the probability of NickelHeads, conditional on knowing only that the event Interviewed occurred. Now this subproblem involves no amnesia or multiple interviews, so all that Peon learns is whether or not he is interviewed at all, that is, whether or not the event Interviewed occurs; so it is mathematically clear that Interviewed is the event Peon should condition on. That is, Peon is being asked to compute the conditional probability PðNickelHeadsjInterviewedÞ: He would do so as follows: PðNickelHeadsjInterviewedÞ PðNickelHeads and InterviewedÞ ¼ PðInterviewedÞ PðNickelHeads and DimeTailsÞ 1=4 ¼ ¼ 1=3: ¼ PðNickelTails or DimeTailsÞ 3=4 Thus, for this simple subproblem, the correct probability that Peon should assign during the interview to the event that the nickel showed heads is equal to 1/3. I consider this answer to be correct and clear and unambiguous, following directly from straightforward mathematical laws of conditional probability. I shall now argue that the original Sleeping Beauty problem can essentially be reduced to this simple Peon subproblem.

The Original Problem Revisited To make use of the above Peon subproblem in analysing the original Sleeping Beauty problem, we add one additional element. We assume that in addition to the previous elements (the nickel, Sleeping Beauty herself, the amnesiainducing drug, etc.), we also have at our disposal another fair coin, say a dime. We make use of the dime as follows. If the nickel showed tails, then the dime is simply placed so that it shows heads during Beauty’s Monday interview, and then repositioned so that it shows tails during Beauty’s Tuesday interview. If instead the nickel showed heads (so Beauty will only be interviewed once), then the dime is instead simply flipped once in the usual fashion at the beginning of the experiment, and is allowed to show its actual flipped result (either heads or tails, with probability 1/2 each) during the one interview that will take place on Monday. Furthermore, we assume that Beauty is not allowed to see the dime at all, and might not even know of its existence. Thus the dime does not in any way affect or control or interfere with any aspect of the original problem. However, we shall see that the dime does permit a precise mathematical analysis of the problem. We reason as follows. Call an interview a ‘‘heads-interview’’ if it takes place while the dime shows heads. If the 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

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nickel showed tails, then there will certainly be precisely one heads-interview. If the nickel showed heads, then there will be either one or zero heads-interviews, with probability 1/2 each. So, the number of heads-interviews behaves just like the total number of interviews in the Peon subproblem. Now, if Beauty were told just before her interview that the dime showed heads (while still undergoing complete amnesia regarding any previous interviews), then she would learn that a heads-interview did indeed occur. This would then put her in precisely the same situation as that of the Peon in the subproblem previously described. Hence, just like the Peon, Beauty would then assign probability 1/3 that the nickel showed heads. In summary, if Beauty were told that the dime showed heads, then the correct answer to the problem would be 1/3. Similarly, if Beauty is informed (just before her interview) that the dime shows tails, then by identical reasoning, the answer would again be 1/3. In summary, the answer would be 1/3 if Beauty could see the dime, regardless of whether the dime was currently showing heads or tails. We can write this in mathematical terms as PðNickelHeadsjDimeHeadsÞ ¼ PðNickelHeadsjDimeTailsÞ ¼ 1=3: where now DimeHeads is the event that the dime shows heads during the particular interview under consideration, that is, that the interview is a heads-interview (and similarly DimeTailsÞ: In the actual problem, we assume that Beauty cannot see the dime. However, I maintain that, as far as probabilities for the nickel are concerned, that fact is irrelevant, and Beauty should still assign the probability 1/3 even if she does not know what the dime shows. To see this, write PðNickelHeadsÞ for the overall probability that Beauty should assign to the event of the nickel showing heads (but now without knowing about the dime). Then it follows by the Law of Total Probability that PðNickelHeadsÞ ¼ PðDimeHeadsÞPðNickelHeadsjDimeHeadsÞ þ PðDimeTailsÞPðNickelHeadsjDimeTailsÞ ¼ PðDimeHeadsÞð1=3Þ þ PðDimeTailsÞð1=3Þ ¼ 1=3; since PðDimeHeadsÞ þ PðDimeTailsÞ ¼ 1: Thus, the answer for this version of the problem seems unambiguously to be 1/3. And since the mere existence of the dime (which Beauty cannot see and has no knowledge of) cannot change Beauty’s probabilities, I submit that this argument shows unambiguously that the answer to the original Sleeping Beauty problem is also 1/3.

Some Related Issues Although the previous description completes my main argument, I now consider a few other related issues. A slight variant: randomised Sleeping Beauty Consider a slight variant of the original Sleeping Beauty problem. As before, if the nickel is tails, we will interview 34

THE MATHEMATICAL INTELLIGENCER

Beauty twice, once on Monday and once on Tuesday (with amnesia). And, as before, if the nickel is heads, we will interview Beauty just once. The only modification is that if the nickel is heads, then rather than necessarily interviewing Beauty on Monday, we will first flip another fair coin (say, a dime), and then conduct our (one) interview on Monday if the dime is heads, or on Tuesday if the dime is tails. (Assume as before that Sleeping Beauty cannot tell what day it is.) For this variant, if Beauty were told that her interview was taking place on Monday, then this would reduce precisely to the Peon subproblem mentioned previously. That is, as far as Monday interviews go, if the nickel showed tails then she would certainly have precisely one, whereas if the nickel showed heads then she would have one only with probability 1/2 (namely, only if the dime showed tails), otherwise zero. Furthermore, the fact that the interview is actually taking place on Monday tells her that she did indeed have one Monday interview. Thus, Beauty is in precisely the same situation as the Peon in the earlier subproblem. So, just as in the subproblem, the correct answer for the probability that the nickel showed heads would be 1/3. Similarly, if Beauty were told that her interview was taking place on Tuesday, the answer would again be 1/3. (The reasoning is identical as in the previous example, except that the roles of ‘‘heads’’ and ‘‘tails’’ for the dime are interchanged.) In summary, the answer would be 1/3 if she knew which day it was, regardless of whether that day were Monday or Tuesday. In the actual problem, Beauty is not told which day it is. However, by the Law of Total Probability just as before, it follows that since she would have assigned probability 1/3 upon being told either that it is Monday or that it is Tuesday, she should still assign probability 1/3 even if she does not know which day it is. Thus the answer for this variant of the problem again seems unambiguously and mathematically to be 1/3. It seems clear that this variant is probabilistically equivalent to the original Sleeping Beauty problem, for in the original problem it is not relevant whether the one interview (if the nickel shows heads) takes place on Monday or Tuesday. So this provides another (similar) argument for why the answer to the original problem is 1/3.

Yet another variant: sleeping twins Consider the following variant of the Sleeping Beauty problem. Suppose there are two twins, named Beauty1 and Beauty2. We put them both to sleep (in separate, soundproof rooms), and flip a fair nickel. If the nickel shows tails, we wake and interview each of them (separately). If the nickel shows heads, we flip a dime. If the dime shows tails we interview Beauty1 only, whereas if the dime shows heads we interview Beauty2 only. What probability should each of them assign, upon being interviewed, to the event that the nickel showed heads? It is clear that in this variant, the situation for Beauty1 is precisely the same as that of the Peon in the previous subproblem. Hence, as in that subproblem, Beauty1 should assign probability 1/3 to the nickel showing heads.

Similarly, Beauty2 should also assign probability 1/3 to the nickel showing heads. On the other hand, if we regard Beauty1 and Beauty2 as a ‘‘unit’’, then together they behave (probabilistically speaking) just as does Sleeping Beauty in the original problem. Indeed, the total number of times that Beauty1 and Beauty2 will be interviewed is two if the nickel is tails, and one if the nickel is heads. So, since each of Beauty1 and Beauty2 should assign the probability 1/3, this suggests that Sleeping Beauty in the original problem should also assign probability 1/3. This argument is very similar to, and perhaps more intuitive than, the argument I provided earlier. However, it is not completely definitive, owing to the possible confusion over conditioning on the same person being interviewed twice (in the original problem), versus two different people each being interviewed once (in this variant). A simple argument explaining why 1/2 must be wrong Another mathematical insight into the original Sleeping Beauty problem can be gained by conditioning on the day of the interview, that is, by considering how the probabilities would change if Beauty knew which day it was. Recall that, in the original problem, Beauty is interviewed on both Monday and Tuesday if the nickel showed tails, but is interviewed on Monday alone if the nickel showed heads. Suppose first that Beauty is informed that her interview is taking place on Monday. Then, since precisely one interview would be conducted on Monday regardless of whether the nickel showed heads or tails, she should at that point assign equal probabilities to the nickel showing heads or tails. In other words, we must have PðNickelHeadsjMondayÞ ¼ 1=2; where Monday is the event that ‘‘the interview is taking place on Monday’’. On the other hand, suppose Beauty is informed that her interview is taking place on Tuesday. Then, since it is impossible to have an interview on Tuesday if the nickel shows heads, she should at that point assign probability zero to the nickel showing heads. That is, we must have PðNickelHeadsjTuesdayÞ ¼ 0: It then follows, again by the Law of Total Probability, that PðNickelHeadsÞ ¼ PðMondayÞPðNickelHeadsjMondayÞ þ PðTuesdayÞPðNickelHeadsjTuesdayÞ ¼ PðMondayÞð1=2Þ þ PðTuesdayÞð0Þ ¼ PðMondayÞ=2: Now, it is not clear what value Beauty should assign to PðMondayÞ; the probability (without any additional knowledge) that her interview is in fact taking place on Monday. Is it 1/2, since she could be interviewed on either day? Or 2/3, since two of the three possible interview situations (heads-Monday, tails-Monday, tails-Tuesday) involve Monday? Or 3/4, reasoning that the probabilities of those three possible interview situations are respectively 1/2, 1/4, and 1/4, and 1/2 + 1/4 = 3/4? In any case, since sometimes interviews will take place on Tuesday, we must have PðTuesdayÞ [ 0; whence PðMondayÞ ¼ 1 PðTuesdayÞ\1; whence PðNickelHeadsÞ ¼

PðMondayÞ=2\1=2: We have thus concluded directly that the answer 1/2 cannot be correct. (Of course, after we agree that PðNickelHeadsÞ ¼ 1=3 is the correct answer to the original problem, then working backwards we can conclude that PðMondayÞ ¼ 2=3:Þ Generalisation to other numbers and probabilities After we accept the above reasoning, then it can also be applied to various generalisations of the original problem. For example, if Beauty will instead be interviewed n times (with amnesia each time) if the nickel showed tails, but just once if the nickel showed heads, then it follows (by replacing the dime by an n-sided die) that the answer becomes 1/(n + 1). The original problem corresponds to n = 2. Or, if the nickel actually was not a fair coin but instead had a priori probability q of coming up heads (and 1 - q of coming up tails), then the answer would become (q/2)/ (q/2 + (1 - q)) = q/(2 - q). The original problem corresponds to q = 1/2. If we combine both of the previous modifications simultaneously, then the answer would become q/(n (n - 1)q). The original problem corresponds to the values n = 2 and q = 1/2. Many other similar variations can be solved in a similar fashion.

Relation to Other Probability Puzzles The Sleeping Beauty problem is reminiscent of certain other well-known probability puzzles in which a conditional probability at first appears to be 1/2, but upon reflection is actually 1/3. I recall two such puzzles, and then consider herein their relation to the Sleeping Beauty problem. Bertrand’s Box The Bertrand’s Box probability puzzle was proposed by Joseph Bertrand in 1889. It is sometimes called (in an equivalent ‘‘drawers’’ version) the Three Desk Problem or Three Drawers Problem, or (in an equivalent ‘‘cards’’ version, see, e.g., Rosenthal 2006) the Three-Card Thriller or Three-Card Swindle. It can be stated as follows: There are three boxes. Box #1 contains two gold coins, Box #2 contains two silver coins, and Box #3 contains one gold and one silver coin. One box is chosen uniformly at random, and one coin is chosen uniformly at random from that box. Suppose the chosen coin is gold. What is the probability that the chosen box was Box #3? In this problem, many people will reason (correctly) that observing the gold coin immediately eliminates Box #2. They will then reason that Boxes #1 and #3 must (still) be equally likely, so the probability of Box #3 must be 1/2. However, what the question is really asking is for the conditional probability PðBox3jCoinGoldÞ: This can easily be computed (e.g., Rosenthal, 2006) as PðBox3; CoinGoldÞ PðCoinGoldÞ PðBox3ÞPðCoinGoldjBox3Þ ð1=3Þð1=2Þ ¼ ¼ 1=3: ¼ PðCoinGoldÞ 1=2

PðBox3jCoinGoldÞ ¼

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Hence the answer is 1/3. (If that seems counterintuitive, note that conditional on CoinGold; the chosen coin was equally likely to be any of the three gold coins available, only one of which is in Box #3.) Monty Hall Problem Another probability puzzle is the Monty Hall problem (vos Savant, 1990), which may be stated as follows: A car is equally likely to be behind any one of three doors. You select one of the three doors (say, Door #1). The Host then reveals one nonselected door (say, Door #3), which does not contain the car. At this point, you choose whether to stick with your original choice (Door #1), or switch to the remaining door (Door #2). What is the probability that you will win the car if you stick with your original choice? Most people, upon first hearing this problem, believe (vociferously!) that the car is equally likely to be behind either of the two unopened doors, so the probability of winning is 1/2 regardless of whether you stick or switch. However, in fact the probabilities of winning are 1/3 if you stick, and 2/3 if you switch. From a conditional probability point of view, this error arises because most people intuitively compute the wrong quantity. Specifically, most people compute PðCar1j CarNot3Þ; which is the conditional probability that the car is behind Door #1 given that it is not behind Door #3. This probability is easily computed (and intuitively seen) to be 1/2. This would indeed be the correct answer if we assumed that the Host just happened to reveal Door #3, by accident, and it just happened not to contain a car. (In Rosenthal 2008, this variant is called the Monty Fall problem.) However, in the original Monty Hall problem, what we actually want to compute is PðCar1jHost3Þ; that is, the probability that the car is behind Door #1 given that the Host elected to open Door #3. This is related to the Host’s motivations, so to compute this properly requires certain assumptions (which are implicit, though not explicit, in the original problem). Namely, we assume that the Host knows where the car is, and will always elect to open some door which is not the door you originally selected and which does not contain a car. Furthermore, we assume the Host will choose randomly (with probability 1/2 each) if there are two different such doors available. With these assumptions, we can compute (e.g., Rosenthal, 2006, 2008): PðCar1; Host3Þ PðCar1ÞPðHost3jCar1Þ ¼ PðHost3Þ PðHost3Þ ð1=3Þð1=2Þ ¼ 1=3: ¼ 1=2

PðCar1jHost3Þ ¼

Hence, the answer is 1/3. (If that seems counterintuitive, note that the strategy of sticking will only succeed if your original guess happened to be correct, which had probability 1/3. And since we knew the Host was going to open some door not containing the car, observing this doesn’t change the probability 1/3 that we were right in the first place.) 36

THE MATHEMATICAL INTELLIGENCER

Comparison to Sleeping Beauty Each of these two puzzles, like the Sleeping Beauty problem, involves computing a conditional probability which may at first seem to equal 1/2, but is in fact equal to 1/3. However, there are some subtle differences among the three scenarios. In Bertrand’s Box, the erroneous answer 1/2 arises purely from a misunderstanding of conditional probability. Once the rules of conditional probability are carefully brought to bear on the problem, the answer is clear and unambiguous. In the Monty Hall problem, there is some confusion about which conditional probability should actually be computed, and also about the implicit assumptions concerning the Host’s behaviour. However, after these points are clarified, then again the rules of conditional probability can be carefully brought to bear on the problem, again providing a clear and unambiguous answer. By contrast, with the Sleeping Beauty problem, even upon careful reflection, it remains unclear how to mathematically formulate the notion of ‘‘conditional on currently being interviewed’’. So, without some sort of reformulation, it seems that conditional probability cannot be brought to bear directly on Sleeping Beauty. That is just the aim of the present article: to make a slight reformulation of the Sleeping Beauty problem (by introducing a dime, while arguing that the dime does not affect the final answer), and to show that this reformulation can be analysed unambiguously by the mathematics of conditional probability. In summary, Bertrand’s Box and the Monty Hall problem, when formulated clearly, are unambiguous exercises in probability theory. By contrast, the Sleeping Beauty problem necessarily involves some sort of philosophical analysis, although in this article I attempt to keep such matters to an absolute minimum.

Final Discussion As mentioned in the Introduction, lots of articles have previously been published about the Sleeping Beauty problem, including many which argue (as I do) that the answer is 1/3. Thus, I see the main contribution of this article not as presenting a new result, but rather as providing a simple, mathematically-based justification for why 1/3 is correct. Specifically, my main argument requires only (i) the and BÞ ; mathematics of conditional probability, PðAjBÞ ¼ PðAPðBÞ and (ii) the ‘‘axiom’’ that people with identical relevant information will assign probabilities identically. The argument I gave is brief, and can be summarised as: • By (i), the Peon will assign probability 1/3 to the nickel being heads. • Hence, by (ii), after Beauty is informed that the dime shows heads (or, similarly, tails), she will assign probability 1/3 to the nickel being heads. • Hence, by (i), Beauty will assign probability 1/3 to the nickel being heads even before she is informed what the dime shows.

• Hence, by (ii), Beauty will assign probability 1/3 to the nickel being heads even if the dime does not exist. It seems that any reader who accepts mathematical probability theory together with the previous rather obvious ‘‘axiom’’, should be convinced that 1/3 is the correct answer. Of course, some readers may have already been convinced by various previously published arguments, and indeed, some of the previous arguments (e.g., the ‘‘Technicolor Beauty’’ variant described in Section 2.5 of the long article by Titelbaum, 2008), have some elements in common with mine. Conversely, some readers may continue to believe that 1/2 is the correct answer, for various philosophical reasons, even after reading my argument. So I do not expect the current aricle to completely resolve the controversy. Still, I hope to have provided a simple, short, direct argument that 1/3 is the correct answer, using solid mathematical foundations with few assumptions and little philosophical ambiguity.

Bostrom, N. (July 2007), Sleeping Beauty and self-location: a hybrid model. Synthese 157, 59–78. Dorr, C. (October 2002), Sleeping Beauty: in defence of Elga. Analysis 62, 292–296. Elga, A. (April 2000), Self-locating belief and the Sleeping Beauty problem. Analysis 60, 143–147. Horgan, T. (January 2004), Sleeping Beauty awakened: New odds at the dawn of the new day. Analysis 64, 10–21. Lewis, D. (July 2001), Sleeping Beauty: reply to Elga. Analysis 61, 171– 176. Monton, B. (January 2002), Sleeping Beauty and the forgetful Bayesian. Analysis 62, 47–53. Neal, R. M. (2006), Puzzles of anthropic reasoning resolved using full non-indexical conditioning. Technical Report No. 0607, Dept. of Statistics, University of Toronto. Piccione, M. and Rubinstein, A. (July 1997), On the interpretation of decision problems with imperfect recall. Games and Economic Behavior 20(1), 3–24. Pust, J. (January 2008), Horgan on Sleeping Beauty. Synthese 160, 97–101. Rosenthal, J. S. (2006), Struck by Lightning: The Curious World of

ACKNOWLEDGMENTS

I am grateful to Gary Malinas and Calvin Normore for discussing these issues with me, and to an anonymous referee for a helpful report.

Probabilities. Joseph Henry Press, Washington, D.C. Rosenthal, J. S. (September 2008), Monty Hall, Monty Fall, Monty Crawl. Math Horizons, 5–7. Titelbaum, M. G. (2008), The Relevance of Self-Locating Belief. Philosophical Review 117(4), 555–605. vos Savant, M. (Sept. 9, 1990), Ask Marilyn column. Parade Magazine,

REFERENCES

Arntzenius, F. (January 2002), Reflections on Sleeping Beauty. Analysis 62, 53–62.

p. 22. Weintraub, R. (January 2004), Sleeping Beauty: a simple solution. Analysis 64, 8–10.

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Mathematical Entertainments

Gaussian Integral Puzzle HIROKAZU IWASAWA This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA

Michael Kleber and Ravi Vakil, Editors

o evaluate a definite integral is sometimes a nice puzzle. Whether or not you agree with this proposition at the moment, when you want to succeed in evaluating definite integrals, one very important thing to recognize in advance is which integrals cannot be evaluated by the obvious methods. Otherwise, trying a change of variable, an integration by parts and so on you could take hours, perhaps in vain, trying to evaluate even a sweet-faced definite integral. Is such a puzzling integral just a troublemaker? Of course, sometimes yes especially when you must hurry. But while you are at leisure, playing with such a gremlin may be fun. In fact, at least for me, trying to find a nice solution for a challenging definite integral is a very amusing puzzle. For, in such integrals, some very elegant solutions may be concealed, and thus the solver sometimes can enjoy some nice ‘‘aha’’ experiences. Such a solver cannot even believe Sudoku is more interesting than the Definite Integral Puzzle, which was introduced as a puzzle genre in my presentation ‘‘Tricky Solutions for Integrals with Crazy Lazy Eight’’ at the Eighth Gathering for Gardner (G4G8). Its task is: Given a challenging definite integral whose antiderivative is not an elementary function, find a (presumably ingenious) elementary solution without Cauchy’s method. I don’t mean here that definite integrals whose antiderivatives are elementary functions cannot be puzzling. In fact, even if you have a formula, say,

T

Z

which is in fact a direct result of a useful formula (see Eq. (3) below). And I admit that some definite integrations with Cauchy’s method are also nice items for puzzles (see the section ‘‘Cauchy’s Method’’ below). Anyway, there are many nice definite integrals you can enjoy as puzzles. Among them, the Gaussian integral, or probability integral, is a pretty good example. It is the improper integral of the Gaussian function expðx 2 Þ over the entire real line. While this integral is very important both in theory and practice, it’s common knowledge that there’s no obvious way to show that Z 1 pﬃﬃﬃ expðx 2 Þdx ¼ p: ð1Þ I¼ 1

Can you show this very simply? Oh, you already know a very elegant way to solve it? That’s better. Then, why don’t you try to find a better one than the one you already know? For there are, I believe, several very simple and elegant solutions, some of which most (if not all) of you have never seen before. For convenience, however, since essentially the same equation in the form Z 1 pﬃﬃﬃﬃﬃﬃ 1 0 expð x 2 Þdx ¼ 2p ð2Þ I ¼ 2 1 is sometimes more useful than (1), we’ll say in what follows that

50 2 1 1 X ð2r 1Þp ð2r 1Þp cos logx 2xcos þ 1 dx ¼ x 100 þ 1 100 r¼1 50 100 50 1 X ð2r 1Þp cosðð2r 1Þp=100Þ x arctan ; sin 50 r¼1 50 sinðð2r 1Þp=100Þ

e-mail: [email protected]

38

you won’t be pleased to use it simply to get Z 1 1 p=100 ; dx ¼ 100 þ 1 x sinðp=100Þ 0

THE MATHEMATICAL INTELLIGENCER Ó 2009 Springer Science+Business Media, LLC

evaluating the Gaussian integral is completed when either (1) or (2) is shown. Therefore the theme of this article is to demonstrate several elegant ways to show either of (1) or (2) and to discuss what is the simplest way to evaluate the Gaussian integral. In the meantime, I’ll show a bunch of specific tricks in integration. I hope you’ll enjoy them and get interested in integral puzzles in general.

The Orthodox Way: Clever Coordinate Transformation Today’s most orthodox way to evaluate the Gaussian integral is a very clever transformation to polar coordinates. Indeed, multiplying I in (1) by itself results in Z 1 expðx 2 Þdx I2 ¼ 1 Z 1 expðy2 Þdy 1 Z 1Z 1 expfðx 2 þ y 2 Þgdx dy ¼ 1 1 Z 2p Z 1 expðr 2 Þr dh dr ¼ h¼0 r¼0 Z 2p Z 1 2 dh expðr Þr dr ¼ 0 0 1 1 2 ¼ ½h2p Þ expðr 0 2 0

Quadrature in High-School

Magic Integrand

Straight-A high-school students can calculate volumes in elementary cases. Thus they understand, or at least they think they understand, a quadrature problem to calculate the volume of a body which satisfies

The clue to the third way is only to evaluate the double integral Z 1Z 1 x expfx 2 ðy2 þ 1Þgdx dy D¼

0 \ z \ e ðx

2

1 \ x \ 1; 1 \ y \ 1:

þy2 Þ

;

and

¼

Z

D¼

¼

2

e x dx

1

Z

1

1 Z 1

Z

¼

x expfx 2 ðy 2 þ 1Þgdy dx

0

Z

1

expðt 2 Þdt

0

ln zÞ10

Z

1

expðx 2 Þdx

ðt ¼ xy Þ

0

On the other hand, they can note that Z 1Z 1 2 2 e ðx þy Þ dx dy V ¼ 1

1 1 p dy ¼ : 2 y2 þ 1 4

0

¼ p:

1

0 1

0

0

1

x expfx 2 ðy 2 þ 1Þgdx dy

And on the other hand, we also find easily

and

Then they can calculate the volume as Z 1 pð ln zÞdz V ¼

Z

1 Z 1

Z 0

0 \ x 2 þ y 2 \ ln z; 0 \ z \ 1:

¼ ½pðz z

0

in two different orders. In fact, on the one hand, we easily see D¼

On the one hand, as they are so smart, they can recognize that the above set of conditions is the same as

¼ p:

2

e y dy

1

¼

2 I I2 ¼ ; 4 2

where I is the Gaussian integral pﬃﬃﬃ of (1). Then we get I ¼ p: That’s it. Like magic, isn’t it? But how can I realize that special integrand, x expfx 2 ðy 2 þ 1Þg;

2

¼I ;

pﬃﬃﬃ From this we conclude that I ¼ p: That’s it. Great! Can there be any easier way? The answer looks like ‘no’ at first. And that may be why almost every textbook of calculus adopts this method. But, actually, to understand this technique you need know not only how to effect the coordinate transformation, but also how to calculate the Jacobian oðx; yÞ ¼ r; J ¼ oðr; hÞ

where I is the Gaussian integral of p (1). ﬃﬃﬃ Thus they can conclude that I ¼ p: Well done! I think this solution is rightly simpler than the orthodox one in that you don’t need to have knowledge about the Jacobian. But an acute observer may say it’s not simple enough because the students would get stuck on the calculation of

where x ¼ rcos h; y ¼ r sin h: Thus even a straight-A high-school senior can’t understand some of the above equations unless he is a true math enthusiast for his age. At least in this sense, this method is not so simple. Is there any way so simple that earnest high-school students can understand it?

in the first part, as they probably even don’t know l’Hoˆpital’s rule. Surely that makes sense. But, avoiding debating here on whether our second method is simple enough, let’s move on to the third one. For it’s so simple that every step seems to be understood by the students.1

1

0

lim z ln z ¼ 0

z!0þ

at first? Oh, please don’t ask the magician to give away the secret, ladies and gentlemen. Thank you. Anyway, every step above must be simple enough, right? But perhaps you need to suggest to the students, at the last step of the first part, changing the variable from x to h in which x = tan h. All in all, however, this method is supremely simple. Despite this, for long years before and after I found this solution, I hadn’t seen it in any other place. That’s possibly mainly because I was lazy about scouring the literature, but I can at least say that this way is surprisingly not popular in spite of its simplicity. This surprise increases when we get to know that this method can be tracked back as early as P.S. Laplace’s The´orie

2

In this section, I used e-x rather than exp(-x2) since high school students are more familiar with the former. But I won’t care about this kind of thing any more in the following sections.

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39

Analytiques des Probabilite´s, 1812.2 It’s true that his original method itself was not so simple. Partly because he evaluated the Gaussian integral in a broader context, he used a widely applicable, but apparently not at all elementary, formula, Z 1 1 p=a ða [ 0Þ; dx ¼ aþ1 x sinðp=aÞ 0 ð3Þ p 4:

to calculate D equals But, anyway, it seems to be a mystery that the above double integral has been much less popular than the one I introduced as the orthodox one. It is my guess that this method is unpopular partly because the technique of switching the order of integration without showing a justification is disliked by, among others, strict math teachers who write or choose textbooks. Yes, it’s dangerous to change the order of integration without care. But so is a coordinate transformation, although the reason is not quite the same. To see the danger of changing the order of integration, try to evaluate, for example, the following integral in two ways: Z 1Z 1 2 x y2 dx dy: ð4Þ 2 2 2 0 0 ðx þ y Þ You’ll get p4 and p4 : Here is a dangerous coordinate transformation: Z p Z 1 2 r sinðr 2 Þdh dr h¼0 r¼0 Z 1Z 1 sinðx 2 þ y2 Þdx dy ¼ 0 0 Z 1Z 1 ðsinðx 2 Þcosðy2 Þ ¼ 0

¼

p 4

0

þ cosðx 2 Þsinðy 2 ÞÞdx dy Z 1 * sinðz 2 Þdz 0 Z 1 cosðz 2 Þdz ¼ 0 pﬃﬃﬃﬃﬃﬃ 2p ; Fresnel integrals ¼ 4

The coordinate transformation here looks, in a sense, very natural. But, in fact, it’s not correct. For, to begin with,

R1

2 0 rsinðr Þdr found in the first line of the equations is obviously not convergent. For another example, how about the following seemingly nice transformation?: Z 1Z 1 2 x y2 dx dy 2 2 2 0 0 ðx þ y Þ ! Z pﬃ2 Z u 2 2uv ¼ dv du 2 2 2 0 u ðu þ v Þ ! Z pﬃﬃ2 Z pﬃﬃ2u 2uv þ pﬃ dv du pﬃﬃ 2 2 2 2 u 2 ðu þ v Þ 2 xþy xy u ¼ pﬃﬃﬃ ; v ¼ pﬃﬃﬃ 2 2 2uv ¼0þ0 Note that ðu2 þ v 2 Þ2 is an odd function of v:

¼ 0: Compare this result with the ones for (4). In any case, when a multidimensional integral is actually convergent, interchanging integration and transforming coordinates are both safe. And we are dealing with only this kind of integration for the Gaussian integral here.

Gamma and Beta Trick By the way, if we don’t require elementariness for a method to be simple, there are several other elegant ways, especially when we note that, for I in (1), Z 1 Z 1 1 expðx 2 Þdx ¼ t 2 e t dt I ¼2 0 0 1 ; ¼C 2 where C(s) is the Gamma function. Now, using the Beta function B(p,q), we can write 1 1 1 1 2 C ¼B ; I ¼C 2 2 2 2 ð5Þ Z 1 12 12 ¼ x ð1 xÞ dx: 0

This can be easily derived from the well-known facts, Bðp; qÞ ¼ CðpÞCðqÞ CðpþqÞ and

C(1) = 1. Anyway, if we realize (5), we can calculate Z 1 1 1 I2 ¼ x 2 ð1 xÞ2 dx: 0

Z

p 2

1 2sin h cosh dh sin h cos h ðx ¼ sin2 h: ) dx ¼ 2sin h cos h dhÞ Z p 2 ¼ 2 dh ¼ p:

¼

0

0

Then I ¼

pﬃﬃﬃ p:

The Formula But if we may use a well-known but apparently nonelementary formula, the best is, p : ð6Þ CðsÞCð1 sÞ ¼ sin ps Then we at once see, for I in (1), 1 1 p I2 ¼ C ¼ p: C ¼ 2 2 sinðp=2Þ pﬃﬃﬃ ) I ¼ p: Period. That’s all! This must be the simplest one, at least in terms of brevity. By the way, for us to evaluate the Gaussian integral, (6) for 0 \ s \ 1 is enough. And, if we have the formula (3), we can obtain (6) for 0 \ s \ 1 elementarily as CðsÞCð1 sÞ Z 1 Z ¼ x s1 e x dx 0

¼

Z

ys e y dy

0

1

Z

1

Z

0

Z

1

1

0 1

1 x s ðxþyÞ e dx dy x y

1=s du dv v1=s þ 1 s x u ¼ x þ y; v ¼ y Z 1 Z 1 1=s ¼ e u du dv v1=s þ 1 0 0 Z 1 1=s dv ¼ 1=s þ 1 v 0 p ; byð3Þ: ¼ sin ps

¼

0

e u

0

As a matter of fact, (3) is a pretty useful formula when we evaluate some

2 For this kind of information, see Peter M. Lee’s webpage, ‘‘Information on the History of the Normal Law (pdf file)’’ (http://www.york.ac.uk/depts/ maths/histstat/normal_history.pdf). It gives helpful data on the first appearance of each of the seven ways he gathered to evaluate the Gaussian integral, including most of the methods in the present article.

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kinds of challenging definite integrals. For instance, although the formula is not essential, if we use it, a Fresnel integral, Z 1 cosðx 2 Þdx; IF ¼ 0

can be easily evaluated as follows.3 On the one hand, Z 1Z 1 e y cosðx 2 yÞdx dy D¼ 0 0 Z 1 Z 1 ¼ e y cosðx 2 yÞdy dx 0 0 Z 1 1 p=4 ðbyð3ÞÞ dx ¼ ¼ 4 þ1 x sinðp=4Þ 0 pﬃﬃﬃ 2 ¼ p: 4 On the other hand, Z 1 Z 1 D¼ e y cosðx 2 yÞdx dy 0 0 Z 1 Z 1 1 e y cosðt 2 Þy 2 dt dy ¼ 0

0

ðt 2 ¼ x 2 yÞ Z 1 Z 1 1 ¼ y2 e y dy cosðt 2 Þdt 0 0 pﬃﬃﬃ pﬃﬃﬃ 1 *C ¼ p : ¼ p IF 2 pﬃﬃﬃﬃ Then IF ¼ 42p : And (3) will be indispensable when we evaluate a more general integral, Z 1 cosðx a Þdx ða [ 1Þ; IFa ¼ 0

in a very similar way. So I am, as a puzzlist, eager to find a short and elementary proof of (3) and indeed have tried for years. But I couldn’t get one so far. Please tell me if you know a solution which uses no nonelementary techniques or knowledge, including Cauchy’s method, facts about infinite products, etc. and which

is not as long as the one using the ‘‘Herglotz trick.’’4 Thank you.

Cauchy’s Method If we don’t need to be concerned with elementariness, we may try Cauchy’s method. It was long supposed, however, that the Gaussian integral could not be evaluated by Cauchy’s method, it’s having been said, ‘‘Cauchy’s theorem cannot be employed toR evaluate 2 1 all definite integrals; thus 0 e x dx has not been evaluated except by other methods’’5 or ‘‘quite simple definite integrals exist which cannot be evaluated by Cauchy’s method, R 1 x2 dx being a case in point.’’6 0 e This was wrong, however. There are several ingenious contour integrals which lead us to the value of the Gaussian integral. Indeed, to find one by yourself would be a nice puzzle.7 But here we’ll go in another direction than contour integrals. We need only Fourier’s integral formula and the identity theorem. We define two regular functions as Z 1 v2 exp ðzvÞexp dv; f ðzÞ ¼ 2 1 and gðzÞ ¼ I 0 expðz 2 =2Þ; where I 0 is the Gaussian integral of (2). Then, for u 2 R; Z 1 v2 exp ðuvÞexp dv f ðuÞ ¼ 2 1 Z 1 n ðv uÞ2 u2 o þ dv exp ¼ 2 2 1 2 Z 1 w2 u ¼ exp exp dw 2 2 1 ðw ¼ u vÞ u2 ¼ I 0 exp 2 ¼ gðuÞ:

Therefore, f(z) = g(z) for z 2 C; by the identity theorem. Now, applying Fourier’s integral formula to 2 expð x2 Þ; x2 Þ 2 Z 1

expð

1 expðixyÞ 2p 1 Z 1 t2 expðiytÞexpð Þdt dy 2 1 Z 1 1 expðixyÞf ðiyÞdy ¼ 2p 1 Z 1 1 y2 expðixyÞexpð ÞI 0 dy ¼ 2p 1 2 ¼

ð* f ðzÞ ¼ gðzÞÞ ¼

I0 I 02 x2 f ðixÞ ¼ expð Þ 2p 2p 2 ðsame as aboveÞ:

pﬃﬃﬃﬃﬃﬃ 02 Therefore, 1 ¼ I2p and hence I 0 ¼ 2p: We have looked at different simple ways to evaluate the Gaussian integral. Which way do you think is the simplest? To my eyes, the simplest one is the one which can be answered most briefly. So imagine that there is an international collect call from a very intelligent guy who is without Internet connection or any textbooks, asking how to evaluate the Gaussian integral. Then what will we answer? Well, what I might say is, ‘‘Evaluate Z 1Z 1 x expfx 2 ðy2 þ 1Þgdx dy 0

0

in two different orders. Period.’’

17-45 Mominokidai Aoba-ku Yokohama, Japan e-mail: [email protected]

3 Both iterated integrations in the following are actually valid and have the same value although that’s not obvious at all. For this kind of justification, a useful theorem is found in Paul Loya, ‘‘Dirichlet and Fresnel Integrals via Iterated Integration’’, Mathematics Magazine, Vol. 78, No. 1, 2005, pp. 63–67. 4 See, e.g., Martin Aigner and Gu¨nter M. Ziegler, Proofs from THE BOOK, 3rd ed., Springer, Berlin 2004, ch. 20. 5 G.N. Watson, Complex Integration and Cauchy’s Theorem, Hafner Publishing, New York 1914, p. 79. 6 E.T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Clarendon Press, Oxford 1935, p. 125. 7 When you give up (I’m sorry), you can find a bunch of solutions in Dragoslav S. Mitrinovic´ and Jovan D. Kecˇkic´, The Cauchy Method of Residues, D. Reidel Publishing Company, Dordrecht 1984.

Ó 2009 Springer Science+Business Media, LLC, Volume 31, Number 3, 2009

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The Mathematical Tourist

On Picturing the Past: Arithmetic and Geometry as Wings of the Mind

Dirk Huylebrouck, Editor

ne of the main delights awaiting the traveler to Strasbourg is the city’s magnificent Notre Dame Cathedral. This monument of late Gothic architecture, situated in the midst of the old part of town, enabled Strasbourg to gain its place on UNESCO’s World Heritage list. One cannot but

O

marvel at the architectural grandeur and the artistic splendors of the Cathedral, yet inside a unique source of wonderment appears in the form of a treasure from the realm of the mathematical sciences – the famous, and enormous, astronomical clock (Fig. 1).

VOLKER R. REMMERT Does your hometown have any mathematical tourists attractions such as statues, plaques, graves, the cafe´ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

42

Figure 1. The Strasbourg astronomical clock today. (taken from Roger Lehni: Die astronomische Uhr des Strassburger Mu¨nsters, Paris 1992, p. 10)

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The Strasbourg Astronomical Clock The original clock was built between 1352 and 1354, but when it stopped working a local mathematician, Christian Herlin ( 1562), was commissioned to design a new one. This was during the 1540s, a time when the Reformation in Strasbourg was consolidating its strength and reaching fruition.1 However, the imposition of Emperor Charles V’s Interim Settlement against the Reformation, alongside Herlin’s death, delayed the project until 1571 when it was taken up by Conrad Dasypodius (ca. 1530–1600). Dasypodius enlisted the support of the Swiss clockmakers, Isaac and Josiah Habrecht, and the well-known artist, Tobias Stimmer (1539–1584). Their joint efforts resulted in the completion of the new clock in 1574, and ever since this instrument has been considered a wonder of its time, not only as a stunning technological achievement but also for its remarkable art work, embellished with paintings, sculptures and automata.2 Although it was modified in the 19th century, a fine early 17th century engraving by Isaac Brunn gives a fair idea of the original decorations (Fig. 2).3 Let us step back in time to imagine what a contemporary observer would have seen. The slender tripartite tower on the left is particularly striking, its most appealing feature being the rooster on the top, which crows at noon. Below it were three portraits: a depiction of Nicolaus Copernicus (1473–1543), revered by Dasypodius, was in the lower compartment; above which one saw the colossus of King Nebuchadnezzar’s dream, as a reference to the coming triumph of Christ (cf. Daniel 2); and, at the top, some eight meters above the floor and therefore difficult to see, emerged an image of Urania, the muse of astronomy (Fig. 3). A closer examination of this picture of Urania reveals some interesting details. As was to be expected, she

Figure 2. The Strasbourg astronomical clock in the 1620s. (Isaac Bruun: The Strasbourg astronomical clock, Strassbourg, ca. 1621, by permission of the Herzog August Bibliothek, Wolfenbu¨ttel)

carries the insignia of astronomy – a celestial globe and a compass – and her

wings, too, are in line with 16th century representations. These wings, however,

1

On the context see Kenneth F. Thibodeau: Science and The Reformation: The Case of Strasbourg, in: Sixteenth Century Journal 7(1976), 35–50. On the history of the clock see Roger Lehni: Die astronomische Uhr des Strassburger Mu¨nsters, Strasbourg 1992; Gu¨nther Oestmann: Die astronomische Uhr des Straßburger Mu¨nsters. Funktion und Bedeutung eines Kosmos-Modells des 16. Jahrhunderts, Stuttgart 1993. 3 There are several earlier engravings, among them one of rather mediocre quality in Dasypodius’s description of the clock: Conrad Dasypodius: Wahrhafftige Auslegung und Beschreybung des Astronomischen Uhrwercks zu Straßburg, Strasbourg 1580. Nothing seems to be known about the origins of the original iconographic programme, cf. Paul Tanner: Die astronomische Uhr im Mu¨nster von Strassburg, in: Tobias Stimmer 1539–1584. Exhibition catalogue, Basel 1984, 97–106, on 98. 2

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Figure 3. Urania (detail from Figure 1).

incorporate a special feature: one symbolizes arithmetic, the other geometry, as can clearly be seen on the engraving (Fig. 4).

Philip Melanchthon, Supporter of the Mathematical Sciences We do not know who was responsible for this particular conception of Urania, 4

but there may well have been a link to Luther’s brother in arms, Philip Melanchthon (1497–1560), the ‘‘teacher of Germany’’ (praeceptor Germaniae). Melanchthon thought highly of the mathematical sciences and from the 1530s onward he wrote several prefaces to contemporary mathematical textbooks, among which were two editions

of Euclid’s Elements, Sacrobosco’s De Sphaera, and books by the mathematicians and astronomers Georg Peurbach (1423–1461) and Johannes Regiomontanus (1536–1476).4 Melanchthon was convinced that the mathematical sciences, and arithmetic and geometry in particular, had an immense propaedeutic value for the study of philosophy, and thus for theology. In the preface to the treatise on arithmetic published in 1536 by his Wittenberg colleague Joachim Rheticus (1514–1574), Melanchthon explained this propaedeutic role with reference to Plato: In the Phaedrus Plato invents two kinds of soul, one of which he says is winged, the other has lost its wings. Then he says that those that have wings fly up to heaven and delight in meeting and conversing with God […]. And, flying throughout all of heaven, these souls are charmed by the beauty of divine things and by the sweetness of the knowledge and of the virtue of that admirable order, and they wish to enjoy this one pleasure forever. […] Even if Plato thinks of the wings as heroic impulses of the minds, it is nevertheless not these impulses that lift up the minds, but the arts are also needed by which these impulses are raised up. Consequently, the wings of the human mind are arithmetic and geometry.5 Melanchthon’s conclusion was clear: those who seek higher knowledge ‘‘should attach those wings, that is, arithmetic and geometry, to themselves.’’ Once we have mastered them, we will be ‘‘carried up to heaven by their help.’’6 Melanchthon made a similar point in the preface to an edition of Euclid’s Elements, published in Basel in 1537 and reprinted in 1546. In emphasizing the necessity of geometrical instruction, he proclaimed that ‘‘the crowning glory of geometry is this, that it is not limited to the narrowness of earthly structures, but it soars to the sky, and it raises the human mind, cast down to earth, once more to its former heavenly haunts, showing to us the marvelous structure

On this see Charlotte Meuthen: The Role of the Heavens in the Thought of Philip Melanchthon, in: Journal of the History of Ideas 57(1996), 385–403, esp. on 386f. Quoted from the English translation by Christine F. Salazar in: Philip Melanchthon: Orations on Philosophy and Education, ed. Sachiko Kusukawa, Cambridge 1999, 90–97, on 93; Melanchthon refers to Phaedrus 246a-e. 6 Ibid. 5

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Philip Melanchthon (1497– 1560), Patron of the Mathematical Sciences

Portrait of Melanchthon by Albrecht Du¨rer, 1526 (by permission of the Heidelberger Graphische Sammlung)

and control of the universe.’’7 While Melanchthon argued along well-established lines for a thorough education in arithmetic and geometry, his exposition of their role as ‘‘wings of the mind’’ went beyond Plato’s account in the Phaedrus. Indeed, his preface to Euclid’s Elements may well be the early modern locus classicus of this topos. Even though Melanchthon did not make the connection between arithmetical and geometrical ‘‘wings’’ and astronomy, such a connection was surely in the air, given that arithmetic and geometry were fundamental for the study of astronomy. It cannot be proved that Christian Herlin, Conrad Dasypodius, or any of their collaborators were

Philip Melanchthon, born as Philip Schwarzerd in Bretten near Karlsruhe, taught ancient languages in Tu¨bingen until 1518 when he became professor of Greek at the recently founded University of Wittenberg (1502). There he soon emerged as the closest ally of Martin Luther and one of the central figures of the Lutheran Reformation in Germany. Melanchthon played an important role in the theological disputes of the time, codifying Luther’s ideas in the Augsburg Confession of 1530 and writing the first Lutheran theological textbook, the Loci communes of 1521. Soon he was known as the ‘‘teacher of Germany’’ (praeceptor Germaniae) by virtue of his engagement throughout Lutheran Germany in the reform of universities and the establishment of school systems that combined humanist education with Lutheran theological ideas. At Wittenberg he strongly supported the cultivation of the mathematical sciences, which were strongly represented on the

inspired by Melanchthon’s ideas in their design of Urania for the astronomical clock. Nevertheless, Dasypodius, as an editor of Euclid, probably knew Melanchthon’s preface, and in his description of the clock claimed that the ‘‘image represents the mathematical sciences’’.8 But no matter how it came about, the notion of arithmetic and geometry as wings of astronomy became common in the late 16th and 17th centuries.

Arithmetic and Geometry as ‘‘Wings’’ Among those who used this image was Nicolaus Reimers Ursus (1551–1600),

faculty. The distinguished astronomer Erasmus Reinhold (1511–1553) taught there from 1536 until his death in 1553. That same year 1536 Melanchthon hired Georg Joachim Rheticus (1514–1574), who later helped Copernicus see his De revolutionibus orbium coelstium into print. Rheticus was also the first to write a defence of heliocentrism against biblical arguments, On Holy Scripture and the Motion of the Earth.

References for Further Reading: Philip Melanchthon, Orations on Philosophy and Education, edited by Sachiko Kusukawa, translated by Christine F. Salazar, Cambridge, 1999. Hooykaas, Reijer: G. J. Rheticus’ Treatise on Holy Scripture and the Motion of the Earth, Amsterdam, et al., 1984.

who became Imperial Mathematician in Prague in 1591. In 1583 he published a short treatise on geodesy dedicated to his patron Heinrich Rantzau (1526– 1598) and entitled Geodaesia Ranzoviana. In his letter of dedication, Ursus presented a somewhat distorted version of Plato’s original argument about the wings of the mind, the source of which may have been Melanchthon’s preface to Rheticus’s booklet on arithmetic. According to Ursus, the great Plato had maintained ‘‘that arithmetic and geometry had been given to the human mind by divine wisdom as wings in order that the astronomer may fly to heaven.’’9 Ursus was succeeded by Tycho Brahe, who scorned him as an

7

Quoted from the English translation by Marian A. Moore: A Letter of Philip Melanchthon to the Reader, in: Isis 50(1959), 145–150, on 146. Conrad Dasypodius: Wahrhafftige Auslegung und Beschreybung des Astronomischen Uhrwercks zu Straßburg, Strasbourg 1580, 51: ‘‘[…] welches Bild die Disciplinas mathematicas anzeyget’’. An English translation of the Latin version of this text has just been published in: Conrad Dasypodius: Heron mechanicus. Ed. Gu¨nther Oestmann, Augsburg 2008. 9 Quoted from the modernized German version of Dieter Launert: Nicolaus Reimers Ursus Stellenwertsystem und Algebra in der Geodaesia und Arithmetica, Munich 2007 [= Nova Kepleriana, Neue Folge, Heft 9], 13: ‘‘es sagt der hocherleuchtete und weitberu¨hmte Philosoph Plato, dass die zwei freien Ku¨nste, die Arithmetik und die Geometrie, die Rechen- und die Messku¨nste, dem menschlichen Gemu¨t zwei von go¨ttlicher Weisheit angeborene Flu¨gel seien, damit der Astronom oder Sternkundige gen Himmel fliege und die oberen himmlischen Bewegungen gleichsam gegenwa¨rtig betrachte und beschaue.’’ 8

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Figure 4. Urania (detail from Figure 2).

impostor and thief. However, Brahe’s gifted assistant, Johannes Kepler (1571–

1630), employed the image of arithmetic and geometry as the two wings of

astronomy in his manuscript Defense of Tycho against Ursus.10 It is interesting that he used the image again during his controversy with Robert Fludd over Kepler’s Harmonice mundi of 1619.11 Yet another instance can be found in an unpublished little treatise on Sacrobosco’s Sphere by one Georg Horst of Torgau, composed in Wittenberg in 1604, in which we read that ‘‘by means of [… arithmetic and geometry], as if by wings, we raise ourselves up to the sky and traverse it in flight in the company of the sun and the other stars.’’12 The Dutch mathematician Martinus Hortensius (1600–1639), in his Oration on the Dignity and the Usefulness of the Mathematical Sciences (Amsterdam 1634), repeated Ursus’s blunder by claiming Plato said ‘‘that eyes were given to men to watch the stars, but also arithmetic and geometry were given as added wings, by which he might fly into the highest spaces of the world.’’13 Obviously Hortensius, like Ursus and others before him, had no idea where that conception originated. Both took it as commonplace, as did Robert Boyle (1627–1691) when he wrote in his essay on the Usefulness of Mathematicks to Natural Philosophy: ‘‘Arithmetic and geometry, those wings on which the astronomer soars as high as heaven.’’14 Occasionaly, though, these wings attached themselves to other allegorical Figures. Ursus, in his Geodaesia Ranzoviana, sought to make them into wings of geodesy, whereas the Jesuit Gaspar Schott (1608–1666) in his Cursus mathematicus of 1661 connected them, in assumed accordance with Plato, with the mathematical sciences as a whole.15 By the 17th century, Melanchthon’s wings were also flying in Catholic heavens. The Italian astronomer and astrologer Andrea Argoli (1570–1657) played with both notions— arithmetic and geometry as wings of the

10 Cf. Nicholas Jardine: The Birth of History and Philosophy of Science: Kepler’s ‘A Defence of Tycho against Ursus’ with Essays on Its Provenance and Significance, Cambridge, et al. 1984, 186, footnote 168. On Brahe, Ursus and Kepler see Edward Rosen: Three Imperial Mathematicians. Kepler Trapped between Tycho Brahe and Ursus, New York 1986. 11 Cf. Johannes Kepler: Gesammelte Werke, vol. 4: Harmonice mundi, Munich: C. H. Beck 1940, 393f. 12 Quoted from: Duhem, Pierre, To save the phenomena. An essay on the idea of physical theory from Plato to Galileo, Chicago/London: The University of Chicago Press, 1969, 97f. 13 Quoted from Annette Imhausen/Volker R. Remmert: The Oration on the Dignity and the Usefulness of the Mathematical Sciences of Martinus Hortensius (Amsterdam, 1634): Text, Translation and Commentary, in: History of Universities 21(2006), 71–150, on 103. 14 Robert Boyle: Of the Usefulness of Mathematicks to Natural Philosophy, in: Robert Boyle: Works III, London 1772, 425–434, on 429. 15 Kaspar Schott: Cursus mathematicus, Wu¨rzburg 1661, 1: ‘‘Alae mathematicae sunt Arithmetica & Geometria’’.

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Figure 6. Cesare Ripa: Iconologia, Rome 1603: Mathematica. (by permission of the Herzog August Bibliothek, Wolfenbu¨ttel)

Figure 5. Andrea Argoli: Primi mobilis tabulae, Padua 1644, frontispiece. (by permission of the Herzog August Bibliothek, Wolfenbu¨ttel)

mind as well as wings of astronomy—in the frontispiece to his Primi mobilis tabulae of 1644 (Fig. 5). Here Athena, as goddess of wisdom, rides up to the heavens in the service of astronomy on Pegasus, the winged horse. Pegasus, because of his close connection to Apollo, symbolized a source of eternal wisdom.16 But both were also connected to astronomy: Pegasus, as the northern constellation, and Apollo (since late Roman times) as god of the sun. Ascending to the heavens was a safe enough journey when a worthy person rode Pegasus and, as the image shows, even more so when the wings of arithmetic and geometry offered their support.

Concluding Remark: The Wings of Mathematics In the fine arts, mathematics has often been represented bearing wings on her head. Such a portrait appears in the Iconologia of Cesare Ripa ( 1622), an influential iconological treatise that served as vademecum for many artists throughout the Baroque period. In the Iconologia’s first illustrated edition of 1603, mathematica is supported by a compass, a table with numbers and figures, a sphere and a pyramid (Fig. 6).17 But Ripa also suggested the wings enabled her to fly to the ‘‘contemplation of abstract things.’’18 It is intriguing that when Ripa described mathematica in

this way, the imagery of wings had already been long established within the mathematical sciences, and it seems entirely possible that Philip Melanchthon played a noteworthy if not decisive role in this manner of picturing mathematics and astronomy in early modern Europe. If so, his influence has hitherto been overlooked.

Dept. of Mathematics FB 08 Physik, Mathematik und Informatik University of Mainz Staudinger Weg, Mainz D-55099 Germany e-mail: [email protected]

16

On the Pegasus myths in post-Antiquity see Elisabeth Decker: Pegasus in nachantiker Zeit, Frankfurt a.M., et al.: 1997, 13–36. Cesare Ripa: Iconologia, Rome 1603, 307–309. 18 Ibid., 307: ‘‘L’ali alla testa insegnano, che ella col’ingegno s’inalza al volo della contemplatione delle cose astratte.’’ 17

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Reviews

Osmo Pekonen, Editor

Complexities. Women in Mathematics Bettye Ann Case and Anne M. Leggett, editors PRINCETON, PRINCETON UNIVERSITY PRESS, 2005, 413 PP., US $37.95, ISBN 0-691-11462-5 REVIEWED BY MARIANNE KORTEN

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected]

48

T

his is an edited volume with over a hundred articles by different authors. The writers are mathematicians, most of them female, nearly all of them working in the United States of America. The articles are grouped into ‘‘past, present and future.’’ The historical articles discuss the life and work of pioneering women in mathematics, about half of them of the twentieth century. This is followed by a collection of contributions addressing institutional and political issues about women (and minorities) in mathematics. Among them one finds articles addressing the much publicized discussion about women’s mathematical abilities, the history of the Association of Women in Mathematics (AWM), affirmative action, women at the International Congress of Mathematicians (ICM), mathematics education and women migrating following their academic appointments. This is followed by a collection of articles addressing the experiences of African-American women, of female mathematicians in academic and nonacademic employment, plus a collection of articles about the conflicts and interweavings that occur when the female mathematician also has a personal life beyond mathematics. Then comes a short collection of articles in which women mathematicians write about what they

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enjoy in the mathematics they do. The book ends with another collection of articles in which the authors talk about how they came to be mathematicians. Reviewing Complexities was a more complex task than I anticipated. From past experience I know how difficult and oftentimes emotional it is for contributors to lay out for their reader the ways they personally found to negotiate the difficult equilibrium between our profession and our lives (this happens to many men, too). For this reason, I have a deep respect for the writers of the articles and for the giant task of the editors. To further clarify where I come from, I should say that where I grew up (Argentina), feminism sprang from the workers’ rights movement of the 1940s and 1950s. Accordingly, in priority it addressed rights such as health care, affordable day care, equal pay for equal work, equality of children regardless of the parent’s marital status, access to divorce, and maternity leave. In contrast, after quite a few years in the US – I am currently Associate Professor of Mathematics at Kansas State University – I seem to understand that here feminism had an intellectual origin. At least for some time, motherhood was considered instrumental to women’s submission and therefore to be avoided. As a result, feminism in the US has not had an impact on workers’ rights, and, for example, day care and family leave are far from obviously available to this very day, unlike in European countries. It seems to me that in spite of some non-US contributors, most of the articles in this book are framed on the backdrop of the status of American (female) workers’ rights, cultural issues, and gender expectations. In my reading I bypassed the historical articles, as there is plenty of material of this kind available. I was greedily seeking excitement and inspiration in stories showing how colleagues successfully navigated the intricacies of research and life, perhaps in order to learn something useful from

them. Working as I do at a public research university, I am evaluated in research, teaching and service; research is evaluated by research publications and grants. My experience is likely different from that of people working in teaching-centered colleges, and it would be reasonable to expect them to have a different view of the book. As a mid-career mathematician still wondering how far I will or won’t be able to go in my research, I found that my generation was not represented very extensively among the writers. So I looked for the senior researchers among the contributors, some of whom I know personally well enough to be aware of at least snippets of things they had to juggle, and what contortions and years it took for them to pursue their work in mathematics and have a life. Like our male colleagues, we went into mathematics because we madly liked it – enough to make huge personal sacrifices in order to do both our math and have a life. The senior researchers among the contributors have made an impact on their research area and, sure enough, had not only their second research grant but also many more thereafter. Except for one contributor (and I’ll come back to this), they write about their field in mathematics and what is hot and may become hot in it (presumably because they have been asked for articles of this kind). Of course, if they do not do it, who else would be qualified to write this type of article? But alas, I tend to read such articles only if they talk about my own field. So I did not get to learn how these senior researchers kept in good spirits along the way and what kept them pursuing their mathematical work. The one exception is Cora Sadosky’s article, which seems to be the only political article in the collection. With raving joy I read her lay out the reasons for the continued need of affirmative action policies (which are quite a bit more endangered now than when the article was written). Hers is concise, to the

point, and sparkling with fire: Enough to keep people like me working to make life better for the next generation, or maybe, if we are quick and thorough enough at it, even make life in our profession better for ourselves. This article left me with ideas to take home to put to use in my work and in any institution I do service for. Next, I went for the articles that dealt with ‘‘having a life,’’ for example, life’s expression in parenting and two-body problems. We all know that the culture of mathematics seems to view mathematics as a form of religion (this may hold for any science) to the point that, e.g., job candidates’ ‘‘devotion and commitment to mathematics’’ is seriously discussed by department members. I have seen job candidates being evaluated on this perceived loyalty as much as on their achievements and perceived potential. For a woman, having young children is perceived as an actual or potential disloyalty to mathematics, whereas for a man it is perceived as a sign of stability and hence an asset. Susan Landau’s contribution mentions some of this. I think this should be more frequently and clearly addressed. For example, one contributor, in discussing her two-cities relationship, states that the distance has kept it fresh. Having lost my daughter’s father after a hopeless illness when I was in my thirties, I cherished every second I got to have with him, and will do the same in any relationship, even more so after the experience of loss. I just hope departments won’t start to expect that job candidates enjoy living in two-cities relationships. I also read of the choice to work at a private teachingcentered school, as this environment was more accepting of the writer’s ‘‘cheating’’ on mathematics by raising children. In reading, it did not seem that her choice followed a calling to devote one’s life to teaching, but rather constituted a preemptive move to protect the author’s right to have a life along side mathematics. Moving on to the section about personal experiences (‘‘Into a New

Century’’), I found Karen Smith article’s last section, ‘‘The Worst Advice I Ever Got’’ to be a real gem. This section discusses the bad wisdom of postponing children (let me stretch this to say, living) until tenure. One issue that deserves to be addressed, given its huge incidence, but isn’t, is the need for fertility treatments to be covered by insurance and be accessible. This is minimal fairness after risking, or sacrificing, our best reproductive years on the altar of science. Helen Moore, writing in this section, describes her very unfortunate early career experience working at a liberal arts school after graduating from top-ranked schools with shining credentials. She has since mostly worked in the administration of mathematical institutions. A gem I found in her article speaks about the responsibility of department leadership concerning the fate of female graduate students and faculty within a department. I would like to see this responsibility issue addressed more frequently and emphatically. Awareness of this responsibility on the side of leadership is another notion I am taking home from my reading to put to use in my service work. An overarching cross-generational comment: I see the contrasting approach to the difficulties of women in mathematics between the generation that learned to fight for its rights growing up around the times of the Holocaust, endangered and aware that ‘‘they only had their chains to lose,’’ and we children of political correctness firmly believing that if we just try hard enough to fit in and make sacrifices (to the religion and the institution of ‘‘math,’’ in our particular case) all sorts of goodies will rain on us. It appears that today, as before, we never get more than we fight for. I’d dare to say, in math as in life, ‘‘I will not let you go unless you bless me.’’ Department of Mathematics Kansas State University Manhattan, KS 66506, USA e-mail: [email protected]

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Yesterday and Long Ago by V.I. Arnold TRANSLATED BY LEONORA P. KOTOVA AND OWEN L. DELANGE, SPRINGER-VERLAG/PHASIS, BERLIN/MOSCOW: 2007, XIV+229 PP., US $44.95 ISBN 978-3-540-28734-6 REVIEWED BY ROGER COOKE

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n their foreword to Felix Klein’s lectures on the development of mathematics in the nineteenth century, his students Neugebauer and Courant wrote, ‘‘A work by an historian will hardly ever exert such a strong attraction and open such deep insight into the essence of history as the thoughts and reminiscences of a great statesman who himself has been involved throughout a long life in the fate of the world in a leading position and combines a reflective intellectual personality with the power to shape an artistic narrative.’’ This assessment of Klein 80 years ago seems to apply to the author of the reminiscences under review. Prof. Arnold has indeed played a leading role in a number of areas of modern mathematics and has been at the forefront of his profession by virtue of his profound mathematical results, his excellent expository work, and the general cause of mathematics that he has championed in the Soviet Union where he was born and raised and in the Russia that has succeeded it. His writing is always thought provoking and often gives deeper insight into topics one may have thought familiar. The present collection of essays on miscellaneous topics differs from Klein’s reflections on the development of nineteenth century mathematics in that Arnold includes a number of essays on topics in which he is not a specialist. Courant and Neugebauer were probably not thinking of that possibility when they wrote the sentence quoted above. It is a distinction worth keeping in mind, even if one finds Arnold’s speculations intriguing. Being intriguing and being authoritative are different properties; both add to the attractiveness of a book, each in its own way. 50

Arnold has a great love for particular specimens of beauty, a love that he has generously shared in a number of books. I have read some of this writing and always with great pleasure. For many years, I treated my class to Arnold’s elegant proof of the focal property of an ellipse, that is, a ray of light from one focus will be reflected at a point P of the ellipse and travel to the other focus. The underlying assumption is that the ray will be reflected exactly as it would be reflected from the tangent to the ellipse at the point P. To see Arnold’s argument briefly, consider an ellipse with foci at F and G. By the definition of an ellipse, the distance d from F to a point P and thence to G is the same for all points P on the ellipse. It is visually obvious that the convex region inside the ellipse consists of points P for which this distance is less than d, while the outside of the ellipse consists of points for which it is greater. Now, a broken line from F to a point P of the ellipse and thence to G is obviously the shortest line from F to the tangent to the ellipse at P and thence to G, since any other such line must go outside the ellipse. Since light travels along minimal paths, a ray of light from F to P must be reflected in exactly this way. Arguments like the one just given have a beauty that no computational proof can match. Their only disadvantage is that they tend to be ad hoc arguments that apply only in limited circumstances, unlike, say, the principles of point-set topology, where the concepts of compactness and connectedness find application in a vast range of areas having seemingly little in common. To say that a person’s writings provide new insight and exert a fascination is not at all to say that one agrees with him in every respect. But even when one disagrees with details, one may agree with the overall picture presented. I often find myself in that rather ambiguous situation when reading Arnold’s writing. Arnold resembles Einstein in that he likes the world (including the mathematical world) to impose its own order on things. He is no friend of axiomatization, in which human beings create systems out of their own minds for the purpose of studying them, and has

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even debated J.-P. Serre over the value of the work of Bourbaki. As a particular example, when he asked his father why the product of two negative numbers is positive, he was told that this property was necessary to make the distributive law hold. That is the reason I would myself have given, and I would have illustrated it by showing how a rectangle with sides of lengths (a - b) and (c - d) can be built starting with a rectangle of dimensions a 9 c, subtracting rectangles that are a 9 d and b 9 c, and then putting back the c 9 d rectangle that was subtracted twice. Arnold did not like that argument. He prefers instead to derive this principle from oriented areas, noting that the positive x-axis and the positive y-axis, in that order, give a positive orientation to the plane, as do the negative x-axis and the negative y-axis, in that order. Arnold takes Descartes to task for the mechanical, prosaic quality of analytic geometry. It must certainly be dull for bright students who can get no satisfaction from the mechanical application of algebraic rules which command assent but hide the beautiful geometric structure that they analyze. Still, I think that the work of a mathematician is precisely to remove the mystery from the universe, even though a universe without mystery would be dull. The poet Paul Vale´ry described analytic geometry as ‘‘the most brilliant victory ever achieved by a man whose genius was applied to reducing the need for genius.’’ Reducing the need for genius is (in my view) the central aim of mathematics. While analytic geometry is dull, it led ultimately to differential geometry, which is exciting. The ultimate end of a river is to become a swamp, it is true; but swamps have a beauty of their own, quieter than the beauty of a raging river. And in any case, analytic geometry led to differential geometry and a series of higher geometries that are still exciting. There is no danger that the world of mathematics will ever be without intriguing mysteries. It has infinite reserves of mystery. The kind of synthetic Euclidean geometry practiced by Euclid, Archimedes, and Apollonius was bound to become a mathematical swamp. Its methods were too restrictive to support the study of complicated curves

and surfaces, and the later Greek mathematicians themselves realized the need for new concepts. The late commentators were reduced to going over old proofs and studying minutiae. In its defense, I used to say that without the Euclidean axiomatization, no one would have realized that nonEuclidean geometry was a possibility. I no longer believe that: The study of projective and differential geometry put non-Euclidean spaces right in front of the noses of nineteenthcentury mathematicians. The work of Schweikart, Taurinius, Lobachevsky, and Bo´lyai, all using methods that Euclid would have understood immediately, was really not in the main line of this development. This kind of synthetic non-Euclidean geometry served rather as a guide to thought. The models that brought the subject to life came from analysis. Just like the Greeks, modern mathematicians realized that synthetic non-Euclidean geometry was mined out except for minutiae, and they turned their attention to more fruitful approaches. Once again, what was difficult to do in the language of Bo´lyai and Lobachevsky became easy to do using Poincare´’s models. And once again, mathematics is not poorer because it became possible to do some geometry without thinking. The new methods simply allow more geometry to be done. That being said, I sympathize with Arnold in this matter. During the years that I taught, I was always somewhat wary when students told me they liked algebra in high-school but hated geometry. High-school algebra consists of rules, requiring no imagination to use, only memory. High-school geometry requires thought, visualization, and mathematical creativity. Although Euclid-style geometry is a mathematical dinosaur, it is pedagogically much more useful than algebra in teaching students to think mathematically. Arnold himself puts the issues in sharp focus in a discussion of the advantages and disadvantages of a written language: It conveys information and removes the need to overload one’s memory; but it also causes the memory to atrophy through lack of use, and thereby decreases intelligence. Having a sharp eye for the paradoxical and ironic, Arnold notices (and

writes about) things that catch his fancy. This book is a treasure trove of vignettes from history, such as the ‘‘Amazon Crusade’’ that Eleanor of Aquitaine led, the winding path taken by the Empress Catherine I to the Russian throne, and some aspects of the death of Julius Caesar that I had never heard before. And that brings me to another aspect of the book. Arnold is always interesting to read, even when one disagrees with him, as I do on a number of points. Let me list a few of them. Unfortunately, in my view, Arnold accepts as fact a great deal of unfounded conjecture about the high level of ancient Egyptian mathematical science, not only attributing Eratosthenes’s method of computing the size of the Earth to camel caravans centuries earlier, but even crediting these ancients with knowledge of Kepler’s laws. I was puzzled when I read this section, since after reading Arnold’s article ‘‘The antiscience revolution and mathematics’’ (Vestnik Rossiiskoi Akademii Nauk, (1999), 69:6, 553–558), I had believed Arnold shared my skepticism about such enterprises as the wild pseudohistorical theories of A.T. Fomenko. Arnold does not cite by name the historians who he says have known these alleged facts for a over a century. In general, I urge caution in reading Arnold’s general historical anecdotes, even those involving mathematical research, in which he does not know the creators personally. I found his account of Weierstrass and Kovalevskaya to be rather bumpy reading. He says that Weierstrass had asked Kovalevskaya to prove that the equations of motion of a rigid body are nonintegrable by showing that some of the solutions are multivalued. At the very least, that description gives the wrong emphasis. Kovalevskaya herself had long been intrigued by this problem, as she wrote to Mittag-Leffler in 1881. Weierstrass does not appear to have taken any position on the integrability of the general system, but he did tell her that if she didn’t succeed in solving the general case she could ‘‘invert the problem’’ and try to find all cases (body parameters and initial data) in which the solution can be expressed by abelian functions of time. Arnold says that Weierstrass was thinking

about Abel’s proof of the insolvability of the quintic equation. It is much more likely that he had in mind the doctoral dissertation of Carl Neumann, in which theta functions of two variables were used to solve a problem in mechanics. To describe Kovalevskaya’s work as ‘‘converting a failure to realize Weierstrass’s program into a positive result’’ is bizarre. If there was a Weierstrass program in this research, it was more likely his hope that theta functions could prove their worth by solving an old problem of mathematical physics, as Jacobi had used them to solve one special case of the problem in 1849. The ‘‘program,’’ in other words, would have been not to get an impossibility proof, but to do exactly what Kovalevskaya eventually did. Weierstrass may have worked on the problem himself during the early 1850s, when the Berlin Academy, inspired by Jacobi’s work, posed it for a prize. He suggested to Kovalevskaya 20 years later that she work on this problem, but she could not make progress on it at that time. I wonder if Arnold is conflating this paper of Kovalevskaya with two of her other papers, each of which matches some of his words. Kovalevskaya did convert a failure of one of Weierstrass’s conjectures into a positive result when she was his student. That is the content of the Cauchy–Kovalevskaya theorem. The springboard for her positive result was her discovery that Weierstrass’s conjecture—that a formal power series generated by applying the method of undetermined coefficients to a Cauchy problem with analytic initial data must have a positive radius of convergence— was incorrect. On another occasion, Kovalevskaya used a Weierstrass (divergence-theorem) technique to solve equations in the theory of elasticity and produced a paper having an error (discovered by Vito Volterra a few weeks after her death) resulting from the multivaluedness of the functions she was offering as a solution. Perhaps these two incidents insinuated their way into Arnold’s recollection of the paper on the rotation problem. It is also not quite accurate to say that Kovalevskaya ‘‘moved from Stockholm to Paris’’ after winning the Bordin Prize. She tried to move to Paris, but the obstacles in her way

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were formidable, and she went back to Stockholm a few months later. Arnold gives several examples of the apparent failure of educational systems: Americans unable to pronounce French, Vinogradov’s constantly saying ‘‘Diophantine equations’’ when he should have said, ‘‘differential equations,’’ students of English literature at Cambridge who did not know who Shelley was, and nineteenth-century Frenchmen blaming the Catholic Church for ‘‘burning Galileo,’’ corrected by British who said it was Tycho Brahe who was burned. However, I can assure him that the person who was actually burned (Giordano Bruno) is mentioned in all history books of the period in the United States and Europe. The story is cited in many books, including Bertrand Russell’s History of Western Philosophy and Morris Kline’s Mathematics in Western Culture. Arnold’s conclusion that Bruno is well known only in Russia is incorrect. In citing a string of mathematically absurd statements by supposedly welleducated people, Arnold reproaches Vladimir Nabokov for his explanation of Pushkin’s line in Evgenii Onegin describing the duel between Onegin and

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Lensky in Chapter 6, Stanza XXIX, line 3: (‘‘The bullets vanish into the faceted barrel’’). Nabokov said that the cross section of a gun barrel is a polygon. Arnold comments that Nabokov finished the gymnasium in Russia and hence presumably should have known better. Actually, it is Arnold who is wrong here. The bore of a gun barrel is circular, of course. Everybody knows that, and Nabokov certainly did. But the outside of the barrel of a dueling pistol was often hexagonal. Pushkin, who fought duels and eventually died in one, knew how to describe them. Like the hypothetical statesman referred to by Courant and Neugebauer, Arnold is at his best when describing his own experiences. He served on the program committee for the International Congress of Mathematicians in Kyoto and has met Pope John Paul II. For the anecdotes he tells about these experiences, the curious historical vignettes that have caught his fancy, and the interesting history of his own family that he has been willing to share, I highly recommend the book.

A note on the translation: I was asked to consider translating this work. I looked at it and decided it was too Russian for my Midwestern American background. I could see that there were features of Russian life and language in the book that would escape my notice. I recommended that the book be translated by someone with the relevant experience of Russia. I believe my decision was a sound one. I could not have provided the explanation of the indelicate word play on p.32, for example, and there are several other places where the translators have shown insight that I do not have. The price of getting a translator who fully understands the original book is that the English translation is not entirely idiomatic. ‘‘Honorable Legion,’’ for example, should be ‘‘Legion of Honor.’’ But the book is competently translated, and I humbly acknowledge the difficult job the two translators have done.

University of Vermont Burlington, VT 05405, USA e-mail: [email protected]

Episodes in the History of Modern Algebra (1800– 1950) Jeremy J. Gray and Karen Hunger Parshall, editors PROVIDENCE: AMERICAN MATHEMATICAL SOCIETY, AND LONDON: LONDON MATHEMATICAL SOCIETY, 2007, VIII + 336, US$69.00, ISBN:-13: 978-0-8218-4343-7 REVIEWED BY TONY CRILLY

he publishers boldly claim this book will be ‘‘essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics.’’ So a broadly based catchment of readers is envisaged—and hoped for. What is not in doubt is that this collection of episodes represents a valuable historical introduction into a mathematical subject which is notoriously technical and abstract. But will the ‘‘interested anyone’’ be able to digest it? Comprising 12 chapters written by individual authors, the episodes are arranged in chronological order. The book was a by-product of a semester course on commutative algebra hosted by the Mathematical Sciences Research Institute in Berkeley, California held in 2003—a few years ago now but still topical. (It contains a great deal concerning noncommutative algebra too.) Organised alongside this course was a week-long workshop on the history of nineteenthand twentieth-century algebra, and it was from this that the book was derived. From a nationalistic point of view the coverage is skewed: Two chapters deal with ‘‘British Mathematics,’’ one with ‘‘American’’ (the Chicago school), with the remainder primarily focussed on the German and French contributions to modern algebra. The essays focus on issues internal to pure mathematics. As lecturers at the workshop, the authors obviously configured their oral

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presentations to the audiences they had in front of them, primarily professional research mathematicians who were familiar with the technical side of algebra but would invariably be interested in the history of their subject. As a consequence of an assumed high level of ‘‘mathematical maturity’’ readers will have to be up to speed on such things as field extensions, algebraic number fields, group representations, crossed products, and the like. Some essays will be more accessible to the general reader than others, and it is not surprising that as they make the journey through the book towards the very modern theory the gradient becomes somewhat steeper. The editors set the scene. The first problem is to circumscribe the subject Algebra, and, once this has been done, to show how algebra so defined became ‘‘modern algebra’’ with its very definite set of meanings as understood in the present day. In the introduction, the familiar story of progress is propounded, from very early times (Mesopotamia, the works of Diophantus), through the ‘‘al-jabr’’ of alKhwa¯rizmı¯’s ninth-century text, to the story of the cubic and quartic equations in the Italy of the Renaissance, the progress continuing with such mathematicians as Franc¸ois Vie`te, Thomas Harriot, Pierre de Fermat, E´variste Galois and ultimately to leaders of the modern subject such as Bartel van der Waerden, who formalised modern algebra around 1930. A key idea to keep in mind when reading this book is that algebra meant different things in different time epochs. So algebra to al-Khwa¯rizmı¯ is not the same as algebra to Euler, which in turn is different from van der Waerden’s algebra. For instance, the algebraic groups of Galois are not the same as the groups written about by Cayley and still less identified with the group theory of the modernists. The aim of history is to understand each development within its own historical terms. The opening chapter by Eduardo Ortiz shows off Charles Babbage as a mathematician. We are perhaps more used to seeing Babbage as the flawed computer pioneer—a sort of grand failure who had the capacity to embark on ambitious ventures without the

ability to finish them. Accordingly, he promised much but delivered little, to end his days a cranky old man who turned his frustration on the organ grinders of Victorian England. What is often omitted from his biography is his youthful role in mathematics, part of the nucleus of those at the heart of the Cambridge Analytical Society. He was briefly the Lucasian professor of mathematics at Cambridge University (who famously never gave a lecture in the place), but well into the nineteenth century the rising generation of research mathematicians in England held his Calculus of Functions in high regard. In this first episode the French philosophical underpinnings of Babbage’s efforts to introduce a language to deal with this Calculus of Functions are explored. Of interest is the interface between mathematics and philosophy: Setting out on the discussion, a banner quotation from one Antoine-Louis-Claude Destutt de Tracy is used as a motif. De Tracy saw the meaning of algebra as language, and languages themselves as types of algebra. He was the leader of an intellectual group called the Ide´ologues who derived their ideas from the influential philosopher E´tienne Bonnot de Condillac and, by criticising his work, extended their own ideas. The work of Condillac which had stirred up the academically-minded circles in revolutionary France included La Langue des calculs (1798). Condillac’s bold statement that ‘‘All language is analytic method, and all analytic method is language’’ (p. 39) is synchronised with Babbage’s efforts, but in the first two decades of the nineteenth century Condillac’s conceptions were under attack by the Ide´ologues. Babbage’s conception of language as applied to the Calculus of Functions situated him within Condillac’s frame of reference, but by the time his own ‘‘Essay Towards the Calculus of Functions’’ was published (1815–1816), he had not kept up with the changing philosophical environment in Paris and seemed unaware of the criticisms of Condillac then being voiced. He made contact with the Ide´ologues, but this occurred only after he had written his Essay. Following a visit to Paris in

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1819, he learned and appropriated the latest thinking, and as Ortiz notes, he shortly afterwards gave up mathematical research founded on the development of analytical scientific languages. Based on Babbage’s experience this chapter questions the effectiveness of communication in philosophical and mathematical ideas between France and Britain in the period 1800–1820. The innovative Duncan Gregory and the founding of the Cambridge Mathematical Journal is explored by Sloan Despeaux in the following chapter. Gregory’s influence on modern algebra is as fleeting as Babbages’s treatment of the Calculus of Functions. Babbage turned away from mathematics to astronomy and mathematical instruments voluntarily, but Gregory was in the full flow of his algebraic researches when he was cut short, dead at only 30 years of age. Gregory was not only a mathematician (chiefly of local repute), but also a scientific allrounder. As so many of his generation, he proceeded to Cambridge having received a strong scientific training in his native Scotland. At Cambridge he graduated fifth student of his year but still managed to gain a fellowship at Trinity College. A cofounder of the Cambridge Mathematical Journal, he exerted an influence on the next generation of mathematicians, notably Robert Leslie Ellis, Arthur Cayley and George Boole. For his mathematical legacy, Gregory is known for his contributions to the Calculus of Operations, specifically the Calculus of Differential Operations closely connected with the Calculus of Functions. As with earlier work in France, he was concerned with setting up the theory d operator based on the of the D ¼ dx iconic symbolic form of Taylor’s Theorem, D = ehD-1, where D is the finite difference operator Df(x) = f(x + h)f(x). This presentation of the calculus was entirely algebraic, and, as several writers have noted, eschewed the notion of limits. Gregory had a view that the true algebra was ‘‘the science of symbols, defined not by their nature, but by the laws of combination to which they are subject’’ (p. 58). Chapters 4 and 5 set the scene for the future abstract algebra introduced by Emmy Noether and her collaborators. 54

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In the eighteenth and early nineteenth century, algebra principally dealt with undefined ‘‘quantities’’ and ‘‘magnitude.’’ Leonhard Euler said that in Algebra, ‘‘we consider only numbers, which represent quantities, without regarding the different kinds of quantity. These are the subjects of other branches of mathematics’’ (p.73), so that algebra in this view extended arithmetic which itself dealt only with numbers. Olaf Neumann sets out to describe divisibility theories in the history of algebra and of algebraic geometry. Of great importance in the transition to an abstract science is the theory of quadratic forms of the type ax2 + bxy + cy2 with a, b, c e Z and gcd (a, b, c) = 1. This theory was a cornerstone of algebra and contributed to by such luminaries as Euler, Joseph-Louis Lagrange, AdrienMarie Legendre, Gustav Peter Lejeune Dirichlet and Richard Dedekind. By far the longest chapter in the book, this chapter is both encyclopaedic and technically demanding. It leads to Emmy Noether’s theory of rings with ascending chain condition and her fundamental result that if D is a domain which satisfies the ascending chain condition for ideals (in which case D is ‘‘noetherian’’), then every ideal is the intersection of ‘‘primary’’ ideals (p.97). Noether regarded this theorem as a generalisation of a principal theorem of number theory which tells us that every number is the product of prime powers. In a short chapter by Harold M. Edwards (Chapter 5), Kronecker’s ‘‘Fundamental Theorem of General Arithmetic’’ is examined. The reader might well ask what fundamental theorem this is, given there are so many theorems so designated. A leading contender for the fundamental theorem of algebra is the one attributed to Gauss which states that a polynomial of degree n has n complex roots. Gauss pointed out that early proofs of this theorem (by Euler and Lagrange) were circular—they employed calculations with roots of the polynomial, tacitly assuming them to be complex, the very thing which was to be proved. What is required is computation with roots without making this assumption. This chapter beautifully explains how this can be done in terms of the cubic x3-2, which turns out to be meaty enough that the polynomial x6 +

108 = 0 has a root a - b, where a, b are two roots of the original cubic. This enables the three roots of x3 - 2 to be expressed as rational functions of this root of x6 + 108 and x3 - 2 to be factored into three linear factors. Kronecker’s Fundamentalsatz is the general statement: A polynomial F(x) with rational coefficients of degree n has n roots, each of which can be expressed rationally in terms of a single root of an irreducible monic polynomial G(x) with integer coefficients (p.114). Kronecker’s result can be used to repair Gauss’s proof of his fundamental theorem, though he did not appear to have done this. But he thought the theorem so fundamental that he celebrated it by publishing his result in the 100th volume of Crelle’s Journal of which he was an editor. The next four chapters (Chapters 6 through 9) deal with the kernel of what is now regarded as the subject of modern algebra. Physically and figuratively, these chapters are the heart of the book. A focus common to all of them (with varying degrees of emphasis) is the Brauer-Hasse-Noether Theorem, which states that the central division algebras over algebraic number fields are cyclic algebras. This was proved and published by the three authors in Crelle’s Journal in a joint paper in 1932 with the general title ‘‘Beweis eines Hauptsatzes in der Theorie der Algebren.’’ The first chapter in this sequence is dedicated to Johann Jakob Burckhardt on his hundredth birthday, the mathematician who translated Leonard Dickson’s path-breaking Algebras and Their Arithmetics (1923) into German. In a revised and enlarged edition, this influential book was republished in Zu¨rich in 1927. It adopted an abstract approach to noncommutative algebra and on its appearance was championed by Emmy Noether and Emil Artin. When Dickson’s text first appeared in English it had been largely ignored but was picked up by Noether. It was hoped that noncommutative algebras could be used to extend class field theory. The opening of this section by Gu¨nther Frei (Chapter 6) recalls the momentous day in October 1843 when Hamilton discovered the quaternions and then made them his life’s work. At

the same time but independently, Hermann Grassmann set out a theory of hypercomplex systems which were brought to international notice by W.K. Clifford in the 1870s. A different approach was made by the Peirces (father and son), who sought to classify algebra by introducing the notions of nilpotent and idempotent elements. Joseph Wedderburn proved a ‘‘main theorem,’’ a structure theorem of abstract algebra in 1907, and this threw the spotlight on the notion of skewfields (where the commutativity of multiplication in a field is not assumed)—but before 1906, the only known skew field was the quaternions. Dickson was able to construct a range of examples of skew fields distinct from the quaternions. Then came the focal Brauer-Hasse-Noether Theorem alluded to. In a letter to Hasse, Emil Artin wrote in 1931: ‘‘You cannot imagine how delighted I was about the proof finally happily achieved about the cyclic systems. This is the biggest advance made in number theory in recent years’’ (p.134). Of the essays in the book under review, this one is probably the most technically challenging (along with a following one on Noncommutative Methods in Algebraic Number Theory which covers much of the same material but from a different viewpoint) and its main appeal will be for the specialist. There are two principal paths to modern algebra traced out and compared in this book—the gradual development of hypercomplex systems of algebras stemming from Hamilton’s discovery and continued through the efforts of the American schools (of the two Peirce’s and the Chicago school of Wedderburn and Dickson) contrasted with the powerful abstract algebra being developed in Europe (with such as Emmy Noether and Emil Artin). In the chapter by Joachim Schwermer (Chapter 7), the Brauer-Hasse-Noether Theorem is seen as the culmination of a local and global principle applied to hypercomplex number systems. This avenue was laid down by Hermann Minkowski, Kurt Hensel, and Helmut Hasse. In this chapter, the arithmetic theory of quadratic forms ax2 + bxy + cy2 in two variables with integer coefficients is resumed. The local and global principle played a part in the

establishment of the Brauer-HasseNoether Theorem (p.169). The Chicago school of Leonard Dickson and his student A. Adrian Albert and the rivalry between it and the Brauer-Hasse-Noether collaboration in Germany is described by Della Fenster (Chapter 8). Dickson’s and Albert’s careers are contrasted, in particular, that of Albert, who tended to work alone but was highly skilled at finding initial approaches to difficult mathematical problems. He was seeking the same kind of Brauer-Hasse-Noether Theorem, but as Irving Kaplansky remarked he was ‘‘nosed out in a photo finish’’ (p.194). In the chapter by Charles W. Curtis (Chapter 9), the essay on Noncommutative Methods in Algebraic Number Theory concentrates on Emmy Noether’s 1932 Lecture to the International Congress of Mathematicians held in Zu¨rich. Once again, the BrauerHasse-Noether Theorem takes centre stage. Artin’s reaction that it represented ‘‘the biggest advance made in number theory in recent years’’ is reprised here (p. 201). The fruitful collaboration was split up in the fateful year of 1933 when Hitler came to power and the persecution of the Jews was extended. Richard Brauer and Emmy Noether emigrated to the United States while Helmut Hasse was appointed professor and director of the Mathematical Institute at the University of Go¨ttingen in succession to Hermann Weyl, who also emigrated to the USA. Hasse’s letter to Francesco Severi reveals his political affiliation and his encouragement of a mathematical research axis between the German and Italian mathematicians (p.277). From this preceding concentration on the establishment of the BrauerHasse-Noether Theorem, the next chapter by Leo Corry (Chapter 10) is of a different sort: A guided tour through the abstracting journal Jahrbuch u¨ber die Fortschritte der Mathematik set up in 1868, the first one of its kind. The changing face of algebra is seen through the classificatory schemes of the Jahrbuch between the years 1900 and 1941. This period bridges the critical era in the changing landscape from the publication of H. Weber’s expansive Lehrbuch der Algebra in 1895 to van der Waerden’s Moderne Algebra published in two volumes (1931–

1932)—the vital transition which one might now view as moving from ‘‘concrete algebra’’ to the modern axiomatic structural algebra. The final three chapters deal with algebraic geometry and its interplay with algebra. The chapter by Norbert Schappacher (Chapter 11) focuses on van der Waerden’s work in algebraic geometry for the 20 year period 1926– 1946 and his rejection of the Platonist philosophy. A key distinction is the difference between analytical and algebraic geometry. Analytical geometry is in the spirit of Descartes and is the application of algebra to geometrical problems; algebraic geometry begins with Riemann, who widened the field of study and studied varieties defined by algebraic equations. Alfred Clebsch and Max Noether in Germany and the famous Italian school which included Guido Castelnuovo and Federigo Enriques, and later Severi, defined early algebraic geometry. One of van der Waerden’s earliest contributions was to link a rigorous foundation of algebraic geometry to the theory of ideals. He made forays into the notorious Schubert calculus or enumerative geometry which has as its aim one of ‘‘counting.’’ Counting the number of twisted cubic curves tangent to quadratic surfaces and the classical problem of counting the number of circles tangent to three given circles are instances. The Schubert calculus had been popular study during the 1880s and attracted major contributors who came up with huge numbers often at odds with those found by others. The enumerative calculus rested on shaky foundations. In the modern era, Andre´ Weil wrote the ‘‘momentous treatise’’ entitled Foundations of Algebraic Geometry published in 1946 and it is a masterful introduction to the work of van der Waerden in algebraic geometry. When van der Waerden gained mathematical fame as the author of Moderne Algebra, he believed it was given for the wrong reason whereby his work on algebraic geometry, which he esteemed highly, had been neglected and largely forgotten. Later in life (in 1950), he was invited to give a lecture in Rome on abstract algebra, but he declined and, in doing so, showed as his first love the primacy of algebraic geometry: ‘‘I do not think I can give a

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really interesting talk on abstract algebra. The enthusiasm would be lacking. One knows me as an algebraist, but I much prefer geometry. In algebra, not much is marvellous. One reasons with signs that one has created oneself, one deduces consequences from arbitrary axioms: there is nothing to wonder about. But how marvellous geometry is!’’ (p. 271). We next pass to a chapter on the arithmetization of algebraic geometry—or geometry turned into algebra. This is the contribution by Silke Slembek (Chapter 12). Until the nineteenth century, algebraic geometry was shaped by members of the Italian school based in Rome. But as a decline in that school set in, Oscar Zariski, a student of Castelnuovo in Rome, raised the flag in the United States, adopting as his aim to make Harvard into the new Rome. His quest was to rewrite the subject in terms of the arithmetization, a quest which amounted to a rewrite of the whole subject of algebraic geometry in an algebraic language. The process of transforming a curve in order to remove its singularities marked the beginning of this arithmetization. Zariski turned to modern algebra and ideal theory as his method (p. 292). He remarked ‘‘after spending a couple of years just studying modern algebra, I had to begin somewhere.’’ The last chapter of the book, by Colin McLarty on the history of modern algebra, is in many ways the high point which brings us to the present. It provides the motivation for the design of the book’s eye-catching cover. The progress through the undercurrents of algebra in history is seen as leading to one vast simplification and ultimate generality: The rising sea. This amounts to the replacement of calculatory algebra with its excruciating detail to an abstract field composed of generalities. At the centre of this movement is Alexander Grothendieck and the broad canvas he adopted for the study of algebraic geometry. For this contribution, many prominent mathematicians

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place Grothendieck as the leading mathematician of the twentieth century. His ways of approaching mathematics are indeed novel, and this chapter exposes his thinking. It deals with matters close to the heart of a mathematician: Strategies for approaching mathematical problems. First, Grothendieck describes the hammer and chisel principle whereby the sought-for mathematical morsel is protected by the hard shell of the nut. The chisel is placed against the nut, and after some hammer blows the shell cracks and ‘‘you are satisfied.’’ In the second approach, the shell is gradually softened by immersing it in water, and after a time hand pressure is enough to open the shell. Grothendieck writes of the rising sea in which the theorem is immersed in some vast theory. His student Pierre Deligne said that a proof by Grothendieck is a long series of trivial steps where nothing seems to happen, but after a time a beautiful nontrivial theorem is uncovered. Grothendieck produces vast multivolume books while Jean-Pierre Serre with his hammer and chisel ‘‘cuts elegantly to an answer’’ (p. 303). Mathematics via the Serre method is what mathematicians have traditionally valued while Grothendieck could be (wrongly) regarded as producing pure waffle—and lots of it. As if to put Grothendieck to the test, the author examines the Weil conjectures on the nature of a zeta function (p.304) in topological terms and an amazing link between finite arithmetic and the topology of manifolds. Serre treated the conjectures in terms of cohomology. This initially hooked Grothendieck, who viewed them through sheaf theory. ‘‘The crucial thing here,’’ said Grothendieck, ‘‘from the viewpoint of the Weil conjecture, is that the new notion [of space] is vast enough, that we can associate to each scheme a ‘generalised space’ or topos’’ (p.312). So, according to Grothendieck, the whole approach depended on ‘‘schemes.’’ Grothendieck said ‘‘the very idea of scheme is of childish simplicity—so

simple, so humble, that no one before me thought of stooping so low.’’ It was so simple that many of the world’s leading mathematicians could not discover from Grothendieck what was actually meant by a ‘‘scheme.’’ No amount of sitting in Parisian cafes led them to a short and precise definition (now recognised as a mathematical construct affiliated to a ring in algebra). Around 1960–1961, Grothendieck summarized an 80 page paper into one of a thousand pages, whereby the study of algebraic geometry became the theory of schemes. In collaboration with Jean Dieudonne´ he traced this method back to Dedekind and Heinrich Weber (p.318). Grothendieck’s topology is now accepted as a powerful research tool and is slowly filtering down to graduate texts and reference books. The outline of this book under review brings us back to its presentation as an episodic history. A book I am reading, completely unrelated to the history of mathematics, delivers an appropriate quote on which to set out the high ground for a review of a book presenting history as a series of episodes: ‘‘To understand history we have to get inside episodes, which means setting ourselves to understand our subjects’ changing motivations and moods in their changing contexts, and to tracing the devious routes by which knowledge was acquired, understood, and acted upon’’ (Clendinnen, p. 287). To a large extent, the book under review is a welcome step in this direction. All students embarking on research in algebra will need to read it in order to appreciate the relation of their own work to mathematical developments in the big picture.

REFERENCE

Clendinnen, Inga, 2003, Dancing with Strangers, Canongate, Edinburgh.

Middlesex University London NW4 4BT, UK e-mail: [email protected]

The Double Twist: From Ethnography to Morphodynamics edited by Pierre Maranda

The Artist and the Mathematician: The Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed by Amir D. Aczel TORONTO, BUFFALO, LONDON: UNIVERSITY OF TORONTO PRESS, 2001. HARDCOVER 318 PP., ISBN: 0-8020-3524-8, US $71.00, AND NEW YORK: THUNDER’S MOUTH PRESS, 2006. HARDCOVER 240 PP., ISBN-10: 1-56025-931-0. ISBN13: 978-1-56025-931-2, US $23.95. REVIEWED BY OSMO PEKONEN

he one hundredth birthday of Claude Le´vi-Strauss, the founding father of structural anthropology and a member of the Acade´mie Franc¸aise, was celebrated with pomp in Paris on November 28, 2008. For reasons of health, Le´vi-Strauss did not participate in the main ceremonies held at the ethnographical museum of Quai Branly, but President Sarkozy paid a courtesy visit to his home. Math and myth intertwine in a subtle manner in the thought of Le´vi-Strauss. Mathematical tropes carry a heavy burden in his texts. They are more than metaphors. Structures of mathematics serve to express some of his most fundamental ideas about human societies. Mathematics may have played a bigger part in the genesis of structuralism than has generally been realized. I review two books that shed light on this interaction. The volume edited by Pierre Maranda is a technical and philosophical contribution to the more

T

sophisticated aspects of Le´vi-Strauss’s intellectual legacy, whereas Amir D. Aczel’s popular book on Bourbaki has been included because of the large place given to Le´vi-Strauss in it. Le´vi-Strauss is a bricoleur who during his extraordinarily long career has made use of the most diverse intellectual tools that happened to be available to him. After conducting ethnographic fieldwork in Brazil in 1935 through 1939, he became a foremost theoretician systematizing the mythologies of the world. In 1939, he returned to France to take part in the war effort. After the French capitulation, he was forced to flee – being of Jewish ancestry – to the United States. New York’s Greenwich Village became an intellectual hub where many European e´migre´s came together. Along with Jacques Maritain, Henri Focillon, and Roman Jakobson, he was a founding member of the E´cole Libre des Hautes E´tudes, a sort of university-in-exile for French academics. He was much influenced by the linguistic theories of Ferdinand de Saussure and Roman Jakobson. His basic notion of mythe`me, an irreducible kernel of a myth, can be traced back to the Saussurean concept of phone`me. Le´vi-Strauss was inspired by structures of music as well. His major work Mythologiques compares in its four-fold structure with the tetralogy of Richard Wagner. It is less known – and not always well-understood – that mathematics played a certain role in the unfolding of structuralism as an intellectual movement, to the point – as is vigorously claimed by Aczel – that we should count the mathematician Andre´ Weil as one of its founding fathers. Le´vi-Strauss rubbed shoulders with Weil in New York and helped to organize a temporary position for him at the University of Sa˜o Paulo. Weil stayed in the United States whereas Le´vi-Strauss returned to France and defended a thesis on elementary kinship structures at the Sorbonne in 1949.

Kinship Rules Modeled by Group Theory Le´vi-Straussian models are often rough versions of mathematical structures, assembled by bricolage out of pieces of ethnography. It seems that presenting

rough theories was enough for Claude. The act of creating structural anthropology was the precise mathematical resolution to the problem of kinship rules of certain aboriginal tribes of Australia. At the request of Le´vi-Strauss, Andre´ Weil wrote a famous appendix to his book Les structures e´le´mentaires de la parente´, which appeared in 1949. Weil applied abstract modern algebra to reveal the underlying mathematical structures that define the apparently complicated kinship rules of certain indigenous tribes. Aczel fails in clearly explaining why and how group theory should play a prominent role in the study of kinship. The group-theoretical analysis of the kinship rules of the Murngin tribe of Northern Australia, which Aczel tries to survey, is somewhat contorted and not easily summarized; moreover, its validity has been questioned (Cargal 1996). So let us consider another textbook example: A Western Australian tribe of aboriginals, the Kariera, is composed of four clans: the Banaka, the Karimera, the Burung, and the Palyeri. To simplify, let us denote A = Banaka, B = Karimera, C = Burung, D = Palyeri. The Kariera society strives to maintain and regenerate its system of clans. The possible marriages of everyone and the clans of the newborn are determined by time-honoured rules, which ethnographic fieldwork has revealed: Rules of marriage: • A and C may marry • B and D may marry Rules of descent: • • • •

father father father father

A & mother C ? child D C & mother A ? child B B & mother D ? child C D & mother B ? child A

Let S = {A,B,C,D} denote the set of clans. We shall consider various functions on S. Let e = IdS: S ? S denote the identity function. Let f : S ? S denote ‘‘the conjugal function’’. The marriage rules are expressed as: f(A) = C, f(B) = D, f(C) = A, f(D) = B. We readily notice that f f = e. To describe the rules of descent, let us introduce two more functions on S :

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The maternal (resp. paternal) function m : S ? S (resp. p : S ? S) associates to the clan of the mother (resp. father) the clan of the child. By inspection, we notice that m m = e and p p = e. Furthermore, it is not difficult to show that m f = p and f m = p. A mathematician now recognizes that the set of functions K = {e, f, m, p}, which contains all the information about the rules of kinship of the Kariera, has the structure of the classical Klein group – a remarkable discovery in its day. Many other tribes with analogous kinship rules have been identified. For instance, on the island of Malekula, in the Republic of Vanuatu, there is a tribe with six clans; their kinship rules are summarized by the dihedral group of order 6. In Northern Australia, a kinship structure determined by a group of order 8 occurs among the Warlpiri. Topics such as these are standard in ethnomathematics (Ascher 1994). Structuralism teaches us that the elements of a system under study are unimportant – only the relationships and structures among them are significant. This idea is at the heart of structuralism. To return to the example of the Kariera, we can analyze the kinship rules abstractly and read off various statements in ordinary language. For instance, the statement m(m(X)) = X = p(p(X)) means that ‘‘every child belongs to the clan of his or her maternal grandmother as well as to the clan of his or her paternal grandfather’’. As for marriage rules, f m = p implies that ‘‘a man may marry his daughter to a son of his sister’’.

The Le´vi-straussian Canonical Formula The next instance of group theory in Le´vi-Strauss’s thought was the introduction of the ‘‘Canonical Formula’’, (Le´vi-Strauss 1955). Sometimes also called the ‘‘Double Twist Formula’’, it is the topic of the volume edited by Pierre Maranda, a renowned Canadian anthropologist. The Canonical Formula (hereafter CF) is the boldest among Le´vi-Strauss’s applications of mathematics to anthropology. It was revealed to him as a potential unifying principle for a classification of South American myths.

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The CF is usually expressed as Fx ðaÞ : Fy ðbÞ :: Fx ðbÞ : Fa1 ð yÞ The statement is left deliberately vague. Countless interpretations have been offered over the years. Maranda’s volume – which contains one paper by Le´vi-Strauss himself – aims to bring order into the bewildering literature. In the literature of structural anthropology, the variables in parentheses (e.g., a and b on the left-hand side of the CF) are called characters whereas the variables appearing as lower indices (e.g., x and y on the left-hand side of the CF) are called functions. Typically, Fx and Fy represent antithetic oppositions, for example, virtuous action as opposed to evil action. The formula requires that one of the variables a has an ‘‘inverse element’’ a-1, which we might call, loosely speaking, its ‘‘twist’’. Moreover, the formula also requires that one of the functions y of the left-hand side can become a character on the right-hand side or, loosely speaking, its role can be ‘‘twisted’’. This explains the nickname ‘‘Double Twist’’ sometimes given to the formula. The function x remains invariant: its role does not change. The CF deploys a mediation mechanism through the ‘‘polysemic operator’’ b, which can subsume the antithetic modes Fx and Fy. In mythology, for instance, a ‘‘trickster’’ could play such a role. Le´vi-Strauss himself remained conspicuously silent about his formula of 1955 for a long while. In the 1960s, the topic was developed by Pierre Maranda together with his Finnish-born anthropologist wife Elli-Kaija Ko¨nga¨s, who passed away prematurely at the age of 50; the American Folklore Society has devoted a prize to her memory. They jointly conducted fieldwork among the Melanesian Lau people in the Solomon Islands. Their joint book on the CF was an outcome of Thomas A. Sebeok’s seminar at the University of Indiana, Bloomington (Ko¨nga¨s-Maranda, et al. 1971); a preliminary version appeared already in 1962. The CF has sometimes been dismissed as a pipe-dream. For Sir Edmund Leach, a British detractor of Le´vi-Strauss, his formula was but ‘‘a meaningless abracadabra’’. One may wonder, indeed, whether it belongs to

the realm of impostures intellectuelles, which is not entirely absent in some excesses of ‘‘French theory’’ (Bricmont, et al. 1997). Reassuringly, the CF can be interpreted from the point of view of formal mathematics. We shall follow here the exposition of Morava (2003). A first observation is that the CF is intrinsically unsymmetric. The Double Twist, whatever its interpretation, seems to work only in one direction: it is not required that the character b have a ‘‘twist’’, nor that the function x have a sensible interpretation as a character. Therefore, the double colon symbol : : can hardly be understood as an equivalence relation. Its proper interpretation should be as a transformation relation; in more standard mathematical notation, this might be written as Fx ðaÞ : Fy ðbÞ ! Fx ðbÞ : Fa1 ð yÞ The existence of such a transformation turning the left side into the right does not preclude the transformation from being an equivalence; all it does is to allow us to regard the axiom of symmetry as optional. Le´vi-Strauss says nothing about the quantifiers of his formula. It is left unspecified whether the formula is intended to hold for every object in an appropriate class, or perhaps only that some object exists, for which the relation holds. A mathematician would paraphrase the formula as follows: ‘‘In a sufficiently large and coherent body of myths we can identify characters a and b, and functions x and y, such that the mythical system defines a transformation which sends a to b, y to a-1, and b to y, while leaving x invariant.’’ We now turn to an interpretation of the CF in terms of a specific mathematical structure, namely as a certain anti-automorphism of the quaternion group of order eight Q ¼ f1; i; j; k g: An anti-automorphism of Q is an invertible transformation, which reverses the product of any two elements. For instance, the transformation k : Q ? Q, which sends i to k, j to i-1 = -i, and k to j, is an example of such a transformation. To check this, let us compute, for instance,

kði j Þ ¼ kðk Þ ¼ j ¼ ðiÞ k ¼ kð j Þ kðiÞ while kðj k Þ ¼ kðiÞ ¼ k ¼ j ðiÞ ¼ kðk Þ kð j Þ; etc. Once this is established, it is easy to check that the assignment x $ 1;

a $ i;

y $ j;

b$k

sends the anti-automorphism k precisely to the Double Twist transformation. We have thus established that the CF corresponds to a consistent mathematical system. A mathematician would point out, furthermore, that the group Q possesses many anti-automorphisms; indeed, together with the automorphisms they form a group of order 24. This means that we could consider many alternative formulations of the Double Twist phenomenon. Le´vi-Strauss himself in La potie`re jalouse invokes the following alternative version of the CF: Fx ðaÞ : Fy ðbÞ :: Fy ðx Þ : Fa1 ðbÞ At first sight, this looks very different. The transformation now sends x to y, a to x, and y to a-1, whereas b remains invariant. However, group theory reveals the underlying structure. The assignment x $ i;

a $ k;

y $ j;

b$1

expresses this transformation as another anti-automorphism of Q, say r, now defined by rðiÞ ¼ j; rð j Þ ¼ k 1 ¼ k; rðk Þ ¼ i: The two transformations differ by the cyclic transformation s : i ? j ? k ? i, which is an example of an outer automorphism of the quaternion group Q; in these terms, k = r s.

A Jivaro Myth Interpreted by the Canonical Formula Whatever the mathematical beauty of the Canonical Formula, the reader may wonder why myths should obey it or any other mathematical formula, for that matter. Le´vi-Strauss provides no comment; he merely rules it to be so. Hypotheses non fingo, could be his answer. Newton never explained why planets should obey his Law of Gravity.

To assess the meaningfulness of the CF, we shall discuss a concrete myth taken from La potie`re jalouse. For Le´viStrauss, variations of a myth are important because they correspond to the dynamics of his formula. The CF is at its most interesting when it displays predictive potential enabling the reconstruction of a forgotten, corrupted, or yet unborn myth. According to a Jivaro myth collected in Ecuador by the Finnish anthropologist Rafael Karsten in 1918, Sun and Moon once lived on the earth, sharing the same house and wife. The wife was called Goatsucker, and she preferred Sun over Moon. Sun was boastful of this, and Moon, upset, climbed up to the sky on a vine after having blown on Sun and eclipsed him. The Goatsucker set after Moon, taking with her a basket of potting clay. Moon got away from her by cutting the vine connecting the earth to the sky. Goatsucker fell and died, her clay was scattered over the earth, and she was transformed into the bird of the same name (Caprimulgus in Latin). Despite all his efforts, Sun, once in the sky, never succeeded in following the same orbit as Moon or in achieving reconciliation. This myth treats the origin of conjugal jealousy: if men cannot share the same wife, it is because Moon and Sun long ago argued over the favors of Goatsucker. Basing himself as much on mythology as on the ethnography of the Jivaro and their knowledge of bird mores, Le´vi-Strauss describes the various relationships existing between women, birds, pottery, and jealousy. Let us define the characters: a = goatsucker (bird), b = woman, and the functions: x = jealousy, y = pottery. According to the CF: The jealous function of the goatsucker relates to the potter function of the woman as the jealous function of the woman relates to the ‘‘inverse of goatsucker’’ function of pottery. The inverse element of the goatsucker a-1 is an unknown quantity, a bird species to be identified. Now the CF predicts that there should exist: 1) a bird species related to pottery

2) a congruence between the woman and the unknown bird with respect to jealousy Le´vi-Strauss observes that there is such a bird, namely the red ovenbird (Furnarus rufus). Well known to the Jivaro but not appearing in their myths, the bird is remarkable for its ability to use clay in nest building, whereas its conjugal habits are a model of harmony, the male and female building the nest together and constantly calling back and forth to each other. The inverse element of the goatsucker a-1 must be the red ovenbird. Consequently, there could exist a related myth explaining the origin of pottery, and we might even be able to predict its structure. Le´vi-Strauss’s own writings and the Maranda volume abound in other examples, some of them less convincing than the others. Uses of the CF fall into two categories: Le´vi-Strauss originally applied it to large families of myths to set out transformation relationships between variants of a given myth. On the other hand, Maranda and others have applied the CF to single myths in order to identify their ‘‘generative engines’’. The potential applications vary according to a vast range of disciplines in the humanities: philosophy, psychology, literature, the analysis of rituals, and so forth.

Contents of the Maranda Volume The volume edited by Pierre Maranda is a compilation of ten philosophical essays related to the CF, encompassing such diverse fields as architecture, ethnography, anthropology, mythology, religious studies, linguistics, economics, computer science, logic, and mathematics. Claude Le´vi-Strauss himself applies the CF to transformation relationships between patterns of indigenous architecture in connection with mythology. Fascinated by various ‘‘hourglassshaped’’ structures, the Master suggests that the CF would generate not only myths and mental structures but also sacred material buildings. Luc Racine provides some pedagogical case studies of the CF. He also comments on an alternative – in fact, impoverished – formulation of the CF suggested by

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Mosko (1991) and shows that such a reduction disserves certain Melanesian data. Eric Schwimmer tackles a basic controversial issue, namely, the ability of structuralism to cope with history. He also details his own use of the CF to analyze his field data, Orokaiva materials from Papua, New Guinea. Pierre Maranda applies the CF to explore the transformation of the ontological status of gender among the Lau people of Malaita, Solomon Islands, at the advent of Christianity. Lucien Scubla starts by reviewing diverse evaluations of the CF and comes up with a balanced conclusion. He then applies the CF to the Hesiodic myth of races to reveal its inner dynamics. Sa´ndor Dara´nyi applies factor analysis to demonstrate the correspondence of a geometric reformulation of the CF with a spatial distribution of myth variants related to Attis, a Phrygian deity from Asia Minor. Christopher A. Gregory sees the CF as a partial rehabilitation of the Ramist tradition of thought that flourished in the sixteenth and early seventeenth centuries. He argues that Le´vi-Strauss’s logic of binary oppositions is Ramistic, not Boolean. Alain Coˆte´ studies the transformations of the CF under the previously mentioned group of order 24 of symmetries including the anti-automorphisms of the quaternionic group of order eight. Andrew William Quinn’s connectionist philosophy focuses on the space of cognitive grammars. In his view, the CF offers a genuine mathematical formalism to investigate the isomorphism between the mind’s topological structures and representations of the world. Jean Petitot recasts the CF, viewed as a ‘‘structural equation’’, in the framework of Rene´ Thom’s morphogenetic theory as Petitot himself has developed it. The Double Twist is realized as a Double Cusp. This is perhaps the study with the most farreaching mathematical consequences, not devoid of interest for developers of a mathematical theory of music. Building on the work of Karsten, Ko¨nga¨s-Maranda, and others, Finland has developed a remarkable school of semiotics. The Finnish musicologist and semiotician, Eero Tarasti, a student of Le´vi-Strauss, has initiated the application of the CF in the music of Wagner and other composers (Tarasti 2003). Several Wagnerian characters, 60

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typically Siegfried or Bru¨nnhilde, can assume the role of the polysemic operator b. Yet a morphogenetic interpretation of Wagner0 s music in terms of the Double Cusp has never been performed to my knowledge. As a whole, the Maranda volume is a brilliant bouquet of pense´es sauvages, some of which might prove fertile across the disciplines. The book first appeared in 2001, but it has been revitalized by the celebration in many countries of Le´vi-Strauss’s centennial year, 2008, which has boosted the sales of his own books and created a plethora of new studies devoted to his thought.

Aczel’s not-so-successful Synthesis Andre´ Weil contributed, as we have seen, to the genesis of structural anthropology with his famous appendix to the published version of Le´viStrauss’s doctoral thesis. Weil having been the Zeus from whose head the Bourbaki group of young mathematicians of E´cole Normale Supe´rieure sprang around 1935, the question arises whether the Bourbakist striving for a radical reformulation of mathematics can be viewed as a prelude to the birth of structuralism as an intellectual movement sweeping across the humanities. Le´vi-Strauss must have been aware of Nicolas Bourbaki although he never cites him explicitly (he does cite numerous other mathematicians such as Claude Shannon and Norbert Wiener). The concept of structure was central in both undertakings, a fact that prompts Amir D. Aczel to declare that bourbakism was instrumental in the unfolding of structuralism. Alternatively, the two phenomena could be viewed as parallel, or totally independent. Aubin (1997) more cautiously calls Bourbaki a ‘‘cultural connector’’ between mathematics and structuralism. Aczel tends to overemphasize the impact of Bourbaki both inside and outside the mathematical community. For instance, he claims that Bourbaki ‘‘was the originator for the modern concept of a mathematical proof.’’ For the professional mathematician, Aczel’s book is irritatingly superficial. He most often fails when he needs to explain a mathematical concept. The

book is infested with errors and misconceptions. Many of the author’s savory anecdotes are untrustworthy. As I have proved, based on the archives of the Finnish counterintelligence agency (Pekonen 1991), Weil most probably was not going to be executed as a Soviet spy in wartime Finland even if he was detained for a fortnight (from 30 November until 12 December 1939, to be precise); consequently, the Finnish mathematician Nevanlinna (misspelled as Nervanlinna by Aczel) did not need to rescue him. Aczel’s account of Alexander Grothendieck’s dramatic life is illdocumented as well. Worse, he portrays Grothendieck as a hero and Weil as a villain. What do we make of sweeping statements such as: ‘‘Weil wanted to do only mathematics that was easy for him, and would not set his sights on overly difficult problems.’’ ‘‘Weil was a somewhat jealous person who clearly saw that Grothendieck was a far better mathematician than he was.’’ Nay, nay, this book is ill-researched. A more reliable popular account of the life and times of Bourbaki is Mashaal (2002). Even so, the diffusion of Aczel’s book among readers in humanities may do some good if it inspires them to study classics of mathematics instead of ‘‘fashionable nonsense’’.

REFERENCES

Ascher, Marcia (1994). Ethnomathematics: a multicultural view of mathematical ideas. New York/Boca Raton: Chapman & Hall/ CRC. Aubin, David (1997). The withering immortality of Nicolas Bourbaki: a cultural connector at the confluence of mathematics, structuralism and the Oulipo in France. Science in Context 10, pp. 297–342. Bricmont, Jean & Alan D. Sokal (1997). Impostures intellectuelles. Paris: Odile Jacob. English translation (1998): Fashionable nonsense. New York: Picador. Cargal, James M. (1996). An analysis of the marriage structure of the Murngin tribe of Australia. Behavioral Science 23, pp. 157–168. Ko¨nga¨s-Maranda, Elli-Kaija & Pierre Maranda (1971). Structural models in folklore and transformational essays. The Hague: Mouton.

Le´vi-Strauss, Claude (1949). Les structures e´le´mentaires de la parente´. Paris: PUF. Le´vi-Strauss, Claude (1955). The structural study of myth. Journal of American Folklore 78, pp. 428–444. Le´vi-Strauss, Claude (1958). Anthropologie structurale. Paris: Plon. Le´vi-Strauss, Claude (1964–1971). Les myth-

Le´vi-Strauss, Claude (1985). La potie`re jalouse. Paris: Plon. Mashaal, Maurice (2002). Bourbaki. Une socie´te´ secre`te des mathe´maticiens. Paris: Belin. English translation (2006): Bour-

maticiens 52 (avril 1992), pp. 13–20. Tarasti, Eero (2003). Mythe & musique: Wagner, Sibelius, Stravinsky. Paris: Maule.

baki: a secret society of mathematics. Providence, RI: AMS.

ologiques: Le cru et le cuit (1964). Du

Morava, Jack (2003). On the canonical formula of C. Le´vi-Strauss. arXiv:math/0306174v2.

miel aux cendres (1967). L’origine des manie`res de table (1968). L’homme nu

Mosko, Mark (1991). The canonical formula of myth and non-myth. American Ethnolo-

(1971). Paris: Plon.

Pekonen, Osmo (1991). L’affaire Weil a` Helsinki en 1939. Gazette des mathe´-

gist 18, pp. 126–151.

Agora Centre University of Jyva¨skyla¨ P.O. Box 35, Jyva¨skyla¨ FI-40014, Finland e-mail: [email protected]

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Mathematics and Democracy: Designing Better Voting and Fair Division Procedures by Steven J. Brams NEW JERSEY AND OXFORDSHIRE: PRINCETON UNIVERSITY PRESS, 2008, 390 PP, US $27.95, £19.95, ISBN 13: 978-0-691-13321-8 (US), ISBN 13: 978-0-691-13321-8 (UK) REVIEWED BY JONATHAN K. HODGE

n the last decade or so, the application of mathematics to the design and analysis of voting procedures has become a popular topic, both to mathematicians and more general audiences. Numerous books on the mathematics of voting and social choice have been published, and the Mathematical Association of America chose ‘‘Math and Voting’’ as its theme for the 2008 Mathematics Awareness Month. This being the case, Steven J. Brams’s new book, Mathematics and Democracy: Designing Better Voting and FairDivision Procedures enters the market at a time when interest is high. Brams, a professor of politics at New York University, is well known and highly regarded in this field, having made significant contributions throughout social-choice theory, and in particular to the study of approval voting and fair division. This book focuses on these two areas and is essentially a collection of papers and book chapters previously published by Brams and his co-authors. The upshot of this format is that it provides a level of depth and detail not found in other recent books. However, readers who are expecting a broader introduction to the mathematics of voting (and such an expectation would not be entirely unreasonable given the book’s fairly generic title) may be disappointed. That being said, Brams’s book is an excellent choice for readers who wish to deepen their knowledge of socialchoice theory and begin to delve into

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some of the recent research in the field. Although the main content of the book is focused on Brams’s own work, the nearly 400 references cited throughout the text provide ample fodder for further investigation. The book is divided into two parts, the former dealing with voting and the latter with fair division. Brams begins in Chapter 1 by introducing approval voting (AV) and presenting an empirical analysis of the use of AV by several major professional organizations. Brams argues that, in practice, AV almost always elects Condorcet winners when they exist, and that, contrary to popular belief, AV winners tend not to be simply humdrum, ‘‘least common denominators.’’ Chapter 2 focuses on theoretical aspects of approval voting, including a rebuttal to the argument advanced by mathematician Donald Saari (and others) that AV is indeterminate because the same ordinal preferences can lead to numerous AV outcomes (depending on where voters draw the line between acceptable and nonacceptable candidates). Brams counters that this ‘‘indeterminacy’’ actually speaks positively to AV’s ‘‘responsiveness’’ to voter preferences, and that the standard social choice framework should be expanded to incorporate a notion of acceptability. Brams goes as far as to suggest that acceptability should ‘‘replace the usual social-choice criteria for assessing the satisfactoriness of election outcomes,’’ an assertion that exemplifies Brams’s consistent public support of AV. (See [1] for example.) In Chapter 3, Brams analyzes preference approval voting (PAV) and fallback voting (FV), two procedures that combine aspects of approval voting with ordinal preferences. Chapters 4, 5 and 6 explore ways that approval voting and related systems can be applied to elections with multiple winners, such as boards, councils or legislatures. Chapter 7, which concludes the first part of the book, examines multiple elections, i.e., elections in which several pass/fail issues appear on the same ballot. Brams argues that in such elections, aggregating issues separately can lead to winning outcomes that receive little, if any, direct support from the electorate. Several examples of these ‘‘paradoxes of multiple elections’’ are explicated, and their relation to the familiar

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Condorcet paradox is explored in detail. Finally, Brams proposes a system called yes-no voting that may help to alleviate some of the problems associated with multiple elections. The fair-division portion of the book begins with two chapters pertaining to parliamentary systems of government. Chapter 8 proposes two similar mechanisms, termed fallback (FB) and buildup (BU), for forming governing coalitions. One of the more interesting results in this chapter states that the most likely BU coalitions are those that include all members (grand coalitions) and those that comprise a minimal majority. To lend credence to the viability of BU as a mechanism for modeling coalition formation, Brams relates this theoretical result to an empirical analysis of thousands of US Supreme Court cases, which reveals a similar bimodal distribution. Chapter 9 shows how divisor apportionment methods can be used to generate choice sequences for the purposes of allocating cabinet ministries to various parties in a coalition government. These methods are shown to be vulnerable to strategic voting, which can cause inefficient outcomes and violations of monotonicity. The last five chapters of the book focus on more traditional fair-division problems. Chapter 10 explores the trade-off between helping the worst-off player and avoiding envy when dividing indivisible goods. In Chapter 11, Brams tackles the problem of dividing a single homogeneous good (such as money) by proposing several modifications to the standard ‘‘divide the dollar’’ game, each of which aims to induce egalitarian outcomes endogenously by inducing egalitarian bids on the part of the players. Chapter 12 describes the adjustedwinner (AW) procedure for dividing multiple homogeneous divisible goods, specifically applying AW to the IsraeliEgyptian dispute that followed the 1973 Yom Kippur War. Brams argues that AW would have produced an outcome similar to that which resulted from the 1978 Camp David accords, but may have done so in a more expedient manner. In Chapter 13, Brams considers cake-cutting problems, extending the standard two-player ‘‘cut-and-choose’’ procedure to yield the surplus procedure (SP), which Brams argues is envy-

free, efficient, proportionally equitable, and strategy-proof. Brams then shows that for three or more players, the notions of envy-freeness and equitability may be incompatible. He proposes two new systems, each of which satisfies one, but not both, of these criteria. In Chapter 14, Brams introduces the Gap procedure to deal with problems that involve allocation of several indivisible goods and a single divisible good, such as money. As suggested by the title of the book, Brams’s focus is constructive in nature. That is, he seeks to use mathematics for the ‘‘prescription of new processes or institutions that are superior, in terms of specified democratic criteria, to those that arose more haphazardly.’’ Brams likens this process to engineering in the natural sciences in that it ‘‘translates theory into the design of political-economic-social institutions that better meet the criteria one deems important.’’ As noted by Brams, this approach is in contrast to that of other researchers who seek mainly to analyze and criticize extant procedures. Consequently, common voting procedures such as the Borda count and single transferable vote are presented mainly to highlight the ways in which they differ from the less familiar systems introduced by Brams. Throughout the text, Brams demonstrates a willingness to depart from the standard social-choice framework, defining new criteria when necessary to evaluate the systems he proposes. This flexibility tends to give Brams’s theoretical work a more practical flavor, a fact that is highlighted by the inclusion of numerous empirical examples that support his theoretical predictions. With that said, it should be noted that many of the procedures Brams proposes have failed to gain traction in terms of their adoption for widespread use within the public sphere. Some, such as the proportional representation systems in Chapter 6 that use integer programming to attempt to minimize misrepresentation, seem destined to remain squarely within the theoretical realm. Others, such as the surplus procedure from Chapter 13, seem more practical at first glance but nevertheless require an inordinate amount of information from the players involved. For instance, SP requires each player to submit a value function, which is essentially a finitely additive probability

measure. Calculating the allocations produced by SP involves integrating over these value functions, which of course requires that they be specified quite precisely. Brams acknowledges that SP is information-demanding, suggesting that ‘‘practically, players might sketch such functions, or choose from a variety of different-shaped functions.’’ It remains to be seen whether these potential solutions would provide a sufficient reduction in the burden placed on players while still offering enough flexibility to allow for the accurate expression of a full range of preferences. Brams seems to recognize the difficulties inherent in bridging the gap between the theory presented in his book and its practical application. In Chapter 1, he states: ‘‘One of the lessons I draw from my experience is that the adoption of AV, and probably any election reform, requires key support from within an organization. I never received this kind of support from politicians or political parties in my attempts to get AV adopted in public elections.’’ He goes on to note that the adoption of AV by several professional societies ‘‘would not have occurred without influential members of each society favoring reform.’’ In spite of these challenges, Brams seems eager to find ways to translate his theory into practice, exhorting readers in the preface to go ‘‘beyond showing that there are better procedures in principle’’ and take ‘‘the additional step of trying to implement these procedures to determine whether their advantages in theory translate into advantages in practice.’’ For this goal, Brams’s book is a good starting point. There are, however, two groups of readers that I think the book is unlikely to satisfy. The first consists of those who are primarily interested in the mathematics, and for whom the application of mathematics to the social sciences is simply a venue for discovering new and perhaps even elegant mathematical results. Indeed, the mathematics Brams employs is more practical than profound, a fact that in my mind doesn’t impugn its value, but rather underscores Brams’s more utilitarian use of mathematical analysis. Brams’s language is not always as precise as it could be, and some proofs seem more cryptic than necessary. There are also some expository inconsistencies

with regard to the amount of detail provided. For instance, Chapters 8 and 10 include numerous technical proofs, but Chapter 12 simply gives references to two papers in lieu of any kind of formal, theoretical analysis of the adjusted-winner procedure. While the exposition is generally adequate, those who seek deeper or more sophisticated approaches may be better served by other books. (As one example, see Saari’s work on the geometry of voting [2].) This book is also less than ideal for those who have a limited background in mathematics and are looking for a light, easy read. Although Brams states in the preface that ‘‘several chapters are accessible to those with little mathematical background’’ and in fact encourages ‘‘selective reading of the chapters,’’ much of the book’s content will present a challenge to readers who are not accustomed to working with precise mathematical definitions and logical arguments. Even those trained in mathematics may feel a bit lost if they are not at least somewhat familiar with the basic ideas and terminology from the field of game theory, which Brams draws upon heavily. To his credit, Brams does include a detailed glossary, which should be of some help to such readers. To summarize, Brams’s book is a good read for those who have some level of formal mathematical training and wish to see mathematics applied to the design and analysis of social-choice and fairdivision procedures. Brams is one of the world’s foremost political scientists, and this book provides a nice survey of some of his most significant contributions to the field over the past three decades. As such, it is a particularly worthwhile choice for anyone pursuing or planning to pursue a research program in the mathematical social sciences.

REFERENCES

[1] Steven J. Brams and Dudley R. Herschbach. The science of elections. Science 292 (no. 5521), p. 1449, 25 May 2001. [2] Donald G. Saari. Basic Geometry of Voting. Springer, Berlin, 1995.

Department of Mathematics Grand Valley State University Allendale, MI 49401, USA e-mail: [email protected]

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature by Tom Siegfried WASHINGTON, D.C.: JOSEPH HENRY PRESS, 2006, 272 PP., US$ 27.95, ISBN: 0-309-10192-1, ISBN: 978-0-309-10192-9 REVIEWED BY SANFORD SEGAL

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om Siegfried’s A Beautiful Math is informally and popularly written. The title, of course, is taken from Sylvia Nasar’s A Beautiful Mind, and John Nash (featured on the dust jacket) has a role to play in the book, though not as great as first indicated. In fact, Siegfried’s breezy pop-cultural style, while making the book easy to read today, may, in fact, detract from its influence in the nottoo-distant future, when off-the-cuff references to ‘‘Gilligan’s Island,’’ or other television remarks, are less familiar. In fact, the book is very dependent on American television for many of its side remarks. The book is an attempt to show how mathematical game theory influences the search for a ‘‘Code of Nature,’’ describing the laws of human behavior. There is no mathematics in this book (except in an appendix

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involving calculation of mixed strategies to yield a Nash equilibrium in two-person games; actually two different games). Nash equilibria are much talked about, but not computed; in fact in many situations where they are mentioned they cannot be computed. Siegfried’s book is about mathematics and his belief that game theory provides the glue cementing together the various attempts to understand (aspects of) human nature. At the very end of his book he quotes the neuroscientist Joshua Greene: The idea is really to have, in the end, a seamless understanding of the universe, from the most basic physical elements, the chemistry, the biochemistry, the neurobiology, to individual human behavior, to macroeconomic behavior—the whole gamut seamlessly integrated … . Not in my lifetime, though. Throughout the book, Siegfried shows how game theory might be linked to social networks, anthropology, quantum theory, statistics, biology, economics and neuroeconomics, and many other things. He discusses game theory ‘‘standards’’ like repeated Prisoner’s Dilemma, tit-for-tat strategies, mixed strategies, and others. For example, he quotes Schelling’s work, which began in the 1950s and produced the 1960 book, The Strategy of Conflict, but which won (one-half of) the Nobel Prize in Economics in 2005. Siegfried’s book is, in fact, very influenced by the behavioral game theory of Colin Camerer. Elucidations of some of the chapter titles, which are examples of his

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playfulness, are perhaps in order. Chapter 1, ‘‘Smith’s Hand,’’ refers to Adam Smith’s ‘‘invisible hand’’ of Wealth of Nations fame, but Chapter 4, ‘‘Smith’s Strategies,’’ refers to the biologist John Maynard Smith who was a pioneer in evolutionary game theory. Again, Chapter 8, ‘‘Bacon’s Links,’’ refers not to Francis Bacon or Roger Bacon but the actor Kevin Bacon and the now familiar notion of six links of connection. Siegfried’s book was inspired by Isaac Asimov’s Foundation trilogy, as he makes clear. Chapter 6, ‘‘Seldon’s Solution,’’ refers to the character Hari Seldon in Asimov’s creation, and Chapter 9, ‘‘Asimov’s Vision,’’ takes up the issue of whether Hari Seldon’s ‘‘psychohistory’’ or sociophysics is the appropriate discipline for discovering the ‘‘Code of Nature.’’ Chapter 10, ‘‘Meyer’s Penny,’’ refers to David Meyer, who is very interested in quantum computing and the possible merger of quantum theory with game theory. This is a charming and speculative book that is an ‘‘easy read’’ and can introduce people to ideas about game theory and its possible centrality in investigations of all sorts, leading perhaps to a ‘‘Code of Nature.’’ But no one should expect to learn any game theory from it.

Department of Mathematics University of Rochester Rochester, NY 14627, USA e-mail: [email protected]

Does Measurement Measure Up? How Numbers Reveal and Conceal the Truth by John M. Henshaw BALTIMORE, THE JOHNS HOPKINS UNIVERSITY PRESS, 2006, 248 PP., US $26.95, ISBN 0-80188375-X REVIEWED BY ERIC GRUNWALD

ops! On page 5, the author dismisses the entire readership of The Mathematical Intelligencer with a casual sneer: To a mathematician (in this case Fred S. Roberts), a measurement is a ‘‘mapping of empirical objects to numerical objects by a homomorphism.’’ A homomorphism is ‘‘a transformation of one set into another that preserves in the second set the operations between the members of the first set.’’ If this makes you glad you are not a mathematician, I suspect you are not alone. However, let’s not bear grudges. Let’s get on with the rest of the book. The issue of measurement and its uses and abuses is certainly interesting and important. Many readers would immediately think of some examples from their own experience. In my case, the following four instantly sprang to mind:

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* Does the use of shadow prices and marginal values in linear programming models of refineries lead to product prices being pared down to marginal costs to the detriment of the business? * When calculating the economics of a proposed project, to what extent do the use of a somewhat arbitrary discount rate and an estimate of the value of the project at the end of the calculation period invalidate the analysis? * The National Health Service in Britain was under pressure to reduce ‘‘waiting lists’’, that is, the length of

time patients have to wait before being treated by the appropriate specialist, so detailed measurement processes and targets were introduced: this led, among other things, to patients experiencing difficulties getting onto the waiting lists in the first place. * The UK government wishes to ‘‘lift people out of poverty’’. This involves first making a (rather arbitrary and dubious) definition of poverty, and then making sure that as many people as possible are helped to cross over the poverty line: a danger is that the easiest way of doing this is to find the large number of people who are just on the wrong side of the line and give them the small amount of money necessary to move them to the right side (thereby providing them with minimal help), while ignoring the much poorer people whom it would be expensive to assist. Not surprisingly, Henshaw does not deal with these specific issues. But to my mind the issues that he does deal with are on the whole less interesting and insightful. As an engineer, he discusses many examples of the use of measurement in engineering. But these hardly seem remarkable or particularly revealing: measurement and calculation are what engineers do, aren’t they? Worse, Henshaw is defeated by one of the problems of writing popular books on scientific subjects: how to find the appropriate level of technical detail that intrigues readers and gives them confidence in the author’s expertise without leaving them bored or bewildered. His solution is to omit all technicalities. So we are told that engineers have, by making detailed measurements, succeeded in designing concert halls with good acoustics and in building ships that don’t snap in half. The trouble is that most readers, once they are presented with these issues, would presumably assume that engineers take measurements in order to figure out what to do, so that merely stating that this is what happens without giving some more details doesn’t really help much. The author is concerned with the use of weighted averages, the problem being that by manipulating the weights one can often manipulate the answer

too. He writes with passion about the over-use of measurement in the U.S. university system (of which, thank goodness, I have had no direct experience), and I enjoyed this part of the book. Of particular concern is the use of the weighted average of various criteria to produce a national league table of U.S. universities. His own University of Tulsa hired a consultant. Henshaw asked him what one thing a university could do to improve its rankings. That was easy, he said: hire three or four very well-known people as professors – perhaps a Nobel laureate in literature, another in chemistry or physics or economics, and a law professor who was a former U.S. senator. They wouldn’t come cheap, but simply hiring them would create such a splash, in both the academic and the popular media, that the university’s ranking was sure to shoot up, because of the high weighting placed on reputation … in university rankings … . It really didn’t matter if these professors spent only a few weeks a year on campus, or even if they never taught a class at all. I think Henshaw is rather harsh on this consultant, whose answer he implies to be cynical. On the contrary, the answer seems to me to be exactly what one should want from a consultant: it is easily understood, it was not obvious before the question was asked, and is fairly clearly correct once stated. The answer seems cynical merely because the question was cynical. The strength of the book lies in the author’s wide knowledge. I appreciated his history of the measurement of global warming and other aspects of the environment, and was also interested in his discussion of the use of intelligence measurement, although this ground is perhaps too well trodden. On the other hand, there is a rather amateurish chapter on the use of measurement in sports. Much of the point of this seems to be to champion the author’s boyhood tennis heroes. After a lot of not particularly insightful statistics, we suddenly read that, ‘‘When we compare Drobny to Rosewall to Connors to Sampras, then, we have every reason to believe that a player like Drobny would have been able to succeed in today’s vastly different game were he given the same

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advantages that today’s players have, and a little time to assimilate them.’’ Quite apart from the question of what this can possibly mean – someone with Drobny’s DNA? With the same family circumstances as Drobny? With the same life experiences as Drobny? – it was disappointing that the conclusion seemed to come out of thin air. Compare this with Michael Lewis’ splendid MoneyBall, about the use of mathematical techniques in baseball, superficially more popular but actually more rigorous than this book. What I really wanted to see from Henshaw’s book, and what I think the reader has a right to expect, is some progress towards a criterion for deciding under what circumstances measurements should be taken seriously. On page 4, the author discusses the use of primitive ‘‘love meters’’, and whether these or other more sophisticated methods could be used to determine whether we are in love. He concludes, ‘‘Personally, with respect to love, I hope we never get to that point. It’s much more fun the old-fashioned way!’’ But why? Isn’t the whole point of the book to answer this kind of question?

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The nearest I could find to the kind of criterion I was looking for comes on page 101: It seems to me that at least some of the difficulty with the measurement of intelligence lies in the personal nature of the measurement. When our intelligence is measured, many of us assume that, ‘‘This is all I’ve got, and I can’t get any more, no matter what I do.’’ We can deal with the knowledge that, for example, we’ll never be any taller than our adult height. … Lots of happy, successful men are not very tall. But to learn as a parent that your young son’s innate, unchangeable ‘‘intelligence’’ places him in the bottom 20% of his peer group is pretty scary news indeed. But this seems not quite right. Lots of happy, successful men are surely not very intelligent either. If we’ll never be any taller than our adult height, then height sounds just as innate as intelligence. The important differences between height and intelligence seem to me to have more to do with the difficulty in making a meaningful and unbiased definition of intelligence and

the uses to which measures of intelligence (as opposed to measures of height) are likely to be put. Here is the final paragraph: …I spoke of a ‘‘vague inner desire for things that can’t be measured.’’ That desire is, I believe, somewhat misplaced. Instead, we should embrace measurement in all (well, okay, most) of its myriad forms. At the same time we must be constantly aware of the dual quantitative/qualitative nature of things. We have to be Mr Spock and Captain Kirk at the same time. To measure is to know? You bet. But the good cook both measures his ingredients and tastes the results. So there you are. We should embrace measurement in well, okay, most of its myriad forms. I am afraid that, after 216 pages on the subject of measurement, this just won’t do.

Mathematical Capital Ltd 187 Sheen Lane London SW14-8LE UK e-mail: [email protected]

Number and Numbers by Alain Badiou, translated by Robin Mackay POLITY: NEW YORK, 2008, 240 PP., US $24.95, ISBN 978-0-7456-3879-9. FIRST PUBLISHED IN FRENCH AS LE NOMBRE ET LES NOMBRES, EDITIONS DU SEUIL, 1990 REVIEWED BY REUBEN HERSH

he name ‘‘Alain Badiou’’ may be unfamiliar to some readers of The Mathematical Intelligencer, but Slavoj Zizek calls Badiou ‘‘much more than the most influential French philosopher at this moment,’’ and his work ‘‘announces a new epoch in philosophy’’ (back cover). Zizek, of course, is the ‘‘most formidably brilliant recent theorist to have emerged from Continental Europe’’ (The International Encyclopedia of Philosophy). To readers of the New Left Review, Badiou is well known as a post-Maoist revolutionary thinker. After retiring from the E´cole Normale Supe´rieure and the Colle`ge International de Philosophie, Badiou became affiliated with the European Graduate School in Saas-Fee, Switzerland. Although this isn’t a new book, it’s newly translated into English. As far as I have been able to learn, it has so far received little notice in the Anglophone world, either by mathematicians or philosophers. An excellent review by John Kadvany did appear in the Notre Dame Philosophical Review. Badiou isn’t what Anglophone academia calls a ‘‘philosopher of mathematics.’’ He pays no attention to old bickerings between Brouwer and Hilbert or Quine and Carnap. He’s after bigger fish, as they say. His question isn’t, ‘‘What is mathematics?’’ but rather, ‘‘What is Being?’’ And his answer is, ‘‘Being is Mathematics.’’ The big news, in brief, is that Alain Badiou is in love with John Conway’s surreal numbers! Badiou is not a deconstructivist or postdeconstructivist. He’s a metaphysician, a creator of speculative systems in the tradition of Leibniz, Hegel and Heidegger. His concerns are Being and

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Event. Being, I think, is roughly the same as ‘‘All that Is’’ or perhaps ‘‘Pure Existence’’ or simply ‘‘Absolute Reality.’’ ‘‘Event,’’ on the other hand, seems to mean, I think, the unpredictable inexplicable radical break from Being, which is exemplified by the sacred and ineffable highest moments of Art, Science, Love, or Revolution. Badiou’s earlier masterpiece, Being and Event, helpfully includes a dictionary. I found no entry for ‘‘Being,’’ but here is the definition of ‘‘Event:’’ An event—of a given evental site— is the multiple composed of: on the one hand, elements of the site; and on the other hand, (the event).— Self-belonging is thus constitutive of the event. It is an element of the multiple which it is.—The event interposes itself between the void and itself. It will be said to be an ultra-one (relative to the situation) (pp. 506–507). If this sounds quite unfamiliar, it may in part be because Anglophone philosophy has for a century or so been controlled by the descendents of Bertrand Russell, who practice something called Analytic Philosophy, which aspires to be Scientific, is obsessed with Logic and Language, and has long ago kicked Metaphysics, including Ontology, into the garbage can. But Badiou is practicing Ontology and Metaphysics! Not, however, in the traditional vein of Hegel or Heidegger—he does it with Mathematics. He is after a version of the surreal that doesn’t show any trace of human hands, a version that one can believe is eternal, extra-human—pure Being. This book starts out with interesting philosophical summaries and critiques of Frege, Dedekind, Peano and Cantor. There follows a careful and, so far as I can tell, correct presentation of the system of ordinal numbers, using the construction often attributed to von Neumann. Starting with the first ordinal, as represented by the singleton {U}, one then gets the second ordinal, with its representative {U, {U}}, and then continues to build the next ordinal by adjoining to any given ordinal a new, final element, namely, itself. After one gets up to the familiar ordered set ‘‘x’’ of natural numbers, comes the decisive step: One constructs the next ordinal by introducing a new element as the last

element—namely, x itself! Then begin again, and continue, defining, after any given ordinal X, the next ordinal, namely: {X, {X}}. And again, after doing this a countably infinite number of times, create a new ‘‘limit ordinal’’ by defining a new last element following this new countable infinity. This construction is explained in five chapters, with admirable detail and patience. I would be tempted to recommend it for beginning students of set theory, except that they would be deterred, not to say repelled, by Badiou’s extravagant Heideggerian metaphysical language. But, you say, what does this standard set-theoretic material have to do with Conway’s surreal numbers? As presented by Conway, the surreal numbers don’t seem at first to be about the ordinals. They’re about the ‘‘cut’’—Dedekind’s famous trick, by which he created the real numbers out of the rationals. Conway starts with NOTHING, and uses a kind of cut to create 0. Then, cutting away, he gets 1, and -1, and the integers, and the dyadic fractions, and finally, of course, like Dedekind, the real numbers. But why stop there? Make one more cut—0 on the left, and the positive reals on the right—and what do you have? An infinitesimal, of course. Contrariwise, make a cut with all the reals on the left, and what do you have? A positive infinite surreal number! Go all the way, cut as many times as there are ordinal numbers. You get a new incredibly rich and complex number system—the surreals. These surreals are what Badiou wants, but he doesn’t want them in this step-by-step, bottom-up ingenious and elegant constructive fashion of Conway. No, Badiou has a metaphysical ax to grind, an ontology to establish (as well as a political-social agenda). ‘‘Our philosophical project designates where Number is given as the resource of being within the limits of a situation, the ontological or mathematical situation. We must abandon the path of the thinking of Number followed by Frege or Peano, to say nothing of Russell or Wittgenstein. We must even radicalise, overflow, think up to the point of dissolution, Dedekind’s or Cantor’s enterprise.’’ (p.212) If we truly wish to establish the being of number as the form of the pure multiple, to remove it from the

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schoolroom (which means also to subtract the concept from its ambient numericality), we must distance ourselves from operational and serial manipulations. These manipulations, so tangible in Peano, project onto the screen of modern infinity the quasisensible image of our domestic numbers, the 1, followed by 2, which precedes 3, and then the rest. The establishing of the correct distance between thought and countable manipulations is precisely what I call the ontologisation of the concept of number. From the point at which we presently find ourselves, it takes on the form of a most precise task: the ontologisation of the ‘universal’ series of the ordinals. To proceed, we must abandon the idea of well-orderedness and think of ordination, ordinality, in an intrinsic fashion. It is not as a measure of order, nor of disorder, that the concept of number presents itself to thought. We demand an immanent determination of its being. And so for us the question now formulates itself as follows: which predicate of the pure multiple, that can be grasped outside of all serial engenderment, founds numericality? We do not want to count, we want to think the count. (p. 58) Since Badiou rejects the bottom-up constructive point of view, and since he dislikes, not to say despises, the view of mathematics as a calculus, he is lucky that Harry Gonshor provided an exposition of the surreals that takes the ordinals as given and then defines a surreal number as a mapping of an initial segment of the ordinals into the pair {+, -}. (The empty sequence is included as a possibility.) To understand this, first imagine the familiar binary expansions of the reals to be ‘‘continued’’ or ‘‘extended,’’ past omega, all the way through the ordinal numbers. Then, for example, we would get the first infinitesimal, as the binary sequence which is 0 in all the finite positions, and 1 in the final position that comes after all the finite positions. Now to get the surreals as Gonshor does them, replace 1 and 0 by + and -. Having presented the surreals in the Gonshor way, as {+, -} valued sequences (up to arbitrarily far out in the ordinals), Badiou is ready for his big coup. He defines the surreal numbers in a way intrinsic to Being, free 68

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from any construction or representation! How does he do it? I’m afraid his answer is quite a let-down. A simple trick. He says a surreal number is just a pair {A, B} where A is any ordinal number, and B is any subset of A. How does this work? Well, to get back from Badiou to Gonshor, take the subset B as the places in A which receive a +, and take the rest of A (the complement of B with respect to A) as the places which receive a -. In the opposite direction, starting with a surreal number given, a la Gonshor, as a mapping of an initial segment of some ordinal A into the pair {+, -}, you can elevate it to the high metaphysical level of a Badiou surreal {A, B} simply by choosing as B the subset of A which receives the value +. This simple relabeling is presented by Badiou as a deeply significant achievement in the understanding of not just Number, but of Being itself! Why? Because, for Badiou, instances of Being are Multiplicities, and Multiplicities are completely catalogued and described by ordinal numbers and surreal numbers. I have two distinct objections to this claim of Professor Badiou. My first objection is based on the fact, admitted by him, that the construction of the ordinals, as also the construction of the surreals, is not compelled by either experience or logic; it is a decision that ‘‘we’’ make. ‘‘We’’ could decide otherwise. My second objection is based on the fact that any well-ordered set, or number system based on a wellordered set, is grossly inadequate and insufficient to represent, describe or model Being, Reality or All That Is. As to the first objection, I quote Badiou: ‘‘There exists no deduction of Number, it is solely a question of a fidelity to that which, in its inconsistent excess, is traced as historical consistency in the interminable movements of mathematical refoundations.… These concepts arise from a decision whose written form is the axiom; a decision that reveals the opening of a new epoch for the thought of being qua being…’’ (p. 212–213). Prof. Badiou is right. Going beyond the countably infinite is just a decision, Cantor’s decision and ‘‘our’’ decision. ‘‘Our’’ choice, to extend Number by the repeated use of limit ordinals, thereby creating the whole

system of ordinal numbers, is not compelled, by either experience or logic. It is merely ‘‘our’’ decision, and ‘‘we’’ could decide otherwise. But this admission destroys the whole claim that Being is manifested in the ordinals and the surreals! If one had the privilege of presence at Prof. Badiou’s lecture, one would try to ask a question: ‘‘Since this decision is ‘our’ free, arbitrary choice, how can it claim to be a mirror or picture of Being, that which IS, regardless of ‘us’ and independent of ‘us’?’’ ’Now to the second, and even more fundamental objection. What is the basis for Prof. Badiou’s claim that the surreal numbers suffice to represent or depict Being itself? Here his dictionary is helpful. One of his important terms is situation. I take ‘‘situation’’ to stand for any concrete specific manifestation of ‘‘Being.’’ ‘‘Being’’ perhaps is just the sum or union of all possible ‘‘situations,’’ and ‘‘situation’’ is the ‘‘Being’’ that is present perhaps at any particular time and place. On p. 522 of Being and Event, we find a definition: ‘‘Situation. Any consistent presented multiplicity, thus a multiple, and a regime of the count-as-one, or structure.’’ We also have a definition of structure. ‘‘What prescribes, for a presentation, the regime of the count-as-one. A structured presentation is a situation.’’ And ‘‘Multiplicity, multiple: General form of presentation, once one assumes that the One is not’’ (p. 514). This seems to me to be the key fallacy of Badiou: This bare statement that the general form of every presentation is multiplicity. Badiou seems to actually say that the Multiplicity of a Situation is a complete description or specification of it! On the contrary, even a mathematical situation beyond abstract set theory is described mainly by the relations, the operations, which are defined on some set. So much more so is any ‘‘real-world’’ situation described by many more attributes than its mere multiplicity! Take these three ‘‘situations:’’ a bowl of 9 apples, or the first 9 prime numbers, or a session of the 9 members of the U.S. Supreme Court— all equivalent, with respect to multiplicity. But by saying that a situation is simply a multiplicity, Badiou can say that the surreal numbers are sufficient for a complete description of Being, and indeed are already an aspect of

Being—independent of ‘‘our’’ knowledge or understanding, now or ever at any time. He has characterized Being, above and beyond all human knowledge! To my mind, this one-dimensional reduction of reality to a well-ordered set is embarrassingly simplistic. Indeed, it is in essence already too familiar, as a way of caricaturing reality. Anyone acquainted with Marxist analysis will recognize its similarity to the onedimensional universal ranking of everything by Price, which is the essence of the ‘‘Free Market,’’ the reign of Capital. Yet it is the reign of Capital that Badiou imagines he repudiates! Badiou repeatedly explains the opposition between ‘‘number’’ (small n) and ‘‘Number’’ (big N). By Number (big N), he means the surreals, in his metaphysical-ontological representation. They are admirable and excellent. By the numbers (little n), he means the numbers that Capital uses to oppress us, in its commercial-militaristic degradation of humanity. ‘‘In our situation, that of Capital, the reign of number is thus the reign of the unthought slavery of numericality itself.’’ (p. 213) Well and good. But according to a well-established Marxist insight, Capital brings the commodification of all aspects of human life. It puts a price on everything, thus making any two entities comparable in value. If we reject this commodification, this putting a price on everything, then we escape the one-dimensional thinking of

our time, as manifested, for instance, in ranking everything (even mathematicians), or in the tyranny of grades and tests in school, of IQ in psychology, etc. Real situations are not one-dimensional, they are multi-dimensional, even infinite-dimensional. Mathematics used in a serious way to study nonmathematical reality cannot limit itself to any one-dimensional scale, no matter how extended or how refined! If it weren’t presumptuous to advise the most influential philosopher in France, I’d be tempted to suggest that Professor Badiou go beyond the set theory which he has mastered so well. How much benefit he would gain by learning some geometry, or even by just leafing through the beautiful new Princeton Companion to Mathematics! If he checks around among the mathematicians there in Paris or Switzerland, he will find that our best attempts to model Reality (or Being) require all the resources of mathematics as it has advanced so far. Set theory, and even the surreal numbers, impressive and beautiful as they are, constitute but one small sector of the vast field of mathematical tools and concepts that Being demands of us. I see Badiou as a modern Pythagorean using the latest incarnation of Number to provide objects of adoration. He calls himself a Platonist, but not a religious Platonist—a Materialist Platonist. (Multiplicity is a material phenomenon, you see.) Nevertheless,

to me his rhapsodic ‘‘Meditations’’ on Set and Number are a bit reminiscent of Georg Cantor, who knew that his mathematical infinite was the theological infinite of the Lord God.

REFERENCES

A. Badiou, Being and Event, Continuum, London, 2006. Originally published in French as L’Etre et l’e´ve´nement, Editions du Seuil, 1988. J. H. Conway, On Numbers and Games, Academic Press, London, 1976. J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1996. J. W. Dauben, Georg Cantor. His Mathematics and Philosophy of the Infinite, Harvard University Press, Harvard, 1979. H. Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, New York, 1986. T. Gowers (ed.), Princeton Companion to Mathematics, Princeton University Press, Princeton, 2008. J. Kadvany, Review of ‘‘Number and numbers,’’ Notre Dame Philosophical Review, (10:2), 2008.

Department of Mathematics and Statistics The University of New Mexico Albuquerque, NM 87131 USA e-mail: [email protected]

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Stamp Corner

Robin Wilson

Swiss Mathematics GERHARD MU¨LLER Jakob Bernoulli (1655–1705) was born in Basel (Switzerland) and became professor of mathematics there in 1687. He investigated infinite series, the cycloid, transcendental curves, the logarithmic spiral and the catenary, and introduced the term integral. His Ars conjectandi (1713), containing the law of large numbers, was an important contribution to probability theory.

Leonhard Euler (1707–1783) was also born in Basel. He studied mathematics there under Johann Bernoulli and then went to the St Petersburg Academy of Sciences as professor of physics (1731) and of mathematics (1733). In 1741 he moved to Berlin as director of mathematics

and physics in the Berlin Academy, but returned to St. Petersburg in 1766. He was a giant figure in eighteenth century mathematics, publishing over 800 books and papers on every aspect of pure and applied mathematics, physics and astronomy. His notations such as e and i have been used ever since. He had a prodigious memory that enabled him to continue mathematical work and to complex calculations in his head, even after losing his sight. He never returned to Basel.

Albert Einstein (1879–1955) was born into a Jewish family at Ulm, Germany. He was educated at Munich and Aarau, graduating in physics and mathematics from the Federal Polytechnic University (ETHZ) in Zu¨rich in 1900. He became a Swiss citizen in 1905, the year of his paper on special relativity, and was appointed examiner at the Swiss Patent office (1902–1909) in Bern. Professor in Prague in 1911 and in Zu¨rich in 1912, he then worked in Berlin as director of the Kaiser Wilhelm Physical Institute (1914–1933). In 1921 he was awarded the Nobel Prize

for Physics. He left Germany, and from 1934 lectured at Princeton, took US citizenship, and became a professor at the Institute for Advanced Study in 1940.

Max Bill (1908–1994) Swiss politician, artist and teacher, was born in Winterthur, Switzerland. He trained at the Zu¨rich School of Arts and Crafts (1924–1927) and the Bauhaus in Dessau (1927–1929). He developed the essential Bauhaus principles of cooperative design along purely functional lines. He built the Mo¨bius strip monument Continuity.

Wolf Barth (b. 1926) is an artist who uses mathematical designs in his work. This design, from a series on abstract art, shows a square standing on one of its corners slightly askew in a square frame. Gerhard Mu¨ller dpl. Ing. ETHZ Sennweg 5, CH-3012 Bern Switzerland

Jakob Bernoulli Leonhard Euler Albert Einstein

Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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Max Bill’s Continuity

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Wolf Barth