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. 0
G
G
=
The Fundamental Theorem of Algebra, Improved These ideas make possible much better versions of the Fun damental Theorem of Algebra: not only do fields of char acteristic 0 no longer need degree axioms for composite degrees, but the theorem now applies to fields of all char acteristics.
COROLLARY 2 .if a field K has characteristic 0, if all odd prime-degree porynomials in K [x] have roots in K, and if all elements of K have square roots in K(i), then K(i) is alge braicalry closed. PRooF. We are able to replace "odd" with "odd prime" by applying Corollary 1: for any odd composite [d], the primes dividing dare odd and there is a sufficiently large odd prirae.
© 2007 Spnnger Science+ Business Media, Inc., Volume 29, Number 4, 2007
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For completeness, I give an argument which does not depend on the proof in [vdW] . Assume K (i) has square roots for all elements and K has roots for polynomials of odd prime degree. Applying Corollary 1 , all odd-degree polynomials have roots. If f in K [x:l has even degree, its Galois group has order 2rm for m odd. Corresponding to the 2-Sylow subgroup, which has index m, is an exten sion of degree m; but there are no irreducible polynomi als of odd degree, so m 1 and 2r. Since p-groups have subgroups of index p, we can build a chain of ex tensions of degree 2 to reach the splitting field of f; but since K (i) has square roots for all elements, each extension comes from a degree-2 polynomial with coefficients in K, so f splits into linear and quadratic factors. Any polynomial in K (i )[x] can be multiplied by its "conjugate" to get a poly nomial in K [x], and from the resulting factorization into lin ear and quadratic factors we can get a complete split into linear factors in K (i )[x] . 0
G
lei
=
=
THEOREM 2 Any field which satisfies [ p] for allprimesp sat isfies [n] for all natural numbers n. PROOF. If the field K has characteristic 0, this follows di rectly from Corollary 1 and the existence of infinitely many primes. The only place where the assumption of charac teristic 0 was needed in the proof of Theorem 1 was to ob tain primitive elements for algebraic extensions of K; but we have [p] for all primes p, so every element of K has a p-th root in K; this holds in particular for the characteristic of the field, so K is a perfect field, and all algebraic ex tensions are separable and they have primitive elements anyway. 0 Theorem 2 allows us to delete all axioms [ n] for com posite n from our axiomatization of algebraically closed fields. Can we go further? No!
THEOREM 3 Theorem 2 is not trne if we omit any single prime from the hypothesis. PRooF. Let K be the field generated by all algebraic num bers whose degree over Q is not divisible by a given prime p. This K contains no numbers of degree p over Q, because we can write K as an expanding union of fields of finite de gree over Q, where each field is obtained from the previous one by adjoining the "next" algebraic number whose degree is not divisible by �at each stage we have a finite exten sion whose degree over Q is not divisible by p, so no num ber of degree p can ever get in. Therefore there are polyno mials of degree p in Q [x] (and so also in K [x]) with no roots inK For any other prime q, every polynomial in K [x] of de gree q has an irreducible factor of degree not divisible by p, and so has a root r whose degree over K is not divisible by p. But r has the same degree over the subfield of K gener ated by the coefficients of its irreducible polynomial, which has a finite degree over Q that is not divisible by p; so r also has such a degree and is therefore in K by construction. 0 We have thus obtained an "optimal" axiomatization for algebraically closed fields: ACF {AF, [2], [3], [5], [7], [ 1 1 ] , =
12
THE MATHEMATICAL INTELLIGENCER
. . . I, where each axiom is independent of the others. Adding the axioms {COz, C03, COs, C07, . . . ) gives an op timal axiomatization for algebraically closed fields of char acteristic 0, while adding the single axiom �cop gives an optimal axiomatization for algebraically closed fields of characteristic p. However, omitting any set of primes is no worse than omitting one, as long as we still have infinitely many "good degrees" for which all polynomials have roots:
THEOREM 4 For any field K, if there are arbitrarily large "good degrees" d such that all polynomials of degree d have roots, then either K is algebraically closed, or there is exact�y one "bad prime" which is the degree of a rootless polynomial, and a degree is ''good" ifand only if it is not a multiple of that prime. PRooF. We know there can be at most one "bad prime," because if two primes were bad then all sufficiently large degrees could be expressed as a sum of those primes and so would have a rootless polynomial, contradicting the as sumption of arbitrarily large "good degrees." Corollary 1 im plies that if infinitely many primes are "good degrees" then any number only divisible by "good primes" is a "good de gree." If there are no bad primes, the proof goes through to show that K is algebraically closed. D
Sufficiency for Characteristic p
Theorem 1 gives us the best possible version of the Fun damental Theorem of Algebra, but it can itself be made stronger: the sufficient condition is also necessary, and the characteristic 0 assumption can be dropped. First, let's look at some examples. Suppose n is odd. We know the alternating group An is a possible Galois group, and it contains subgroups of index n, G), ('_3), . . . , (d), where d (n- 1)/2. These subgroups are intransitive and arise from partitioning { 1 , . . . , n} into two pieces. When n = 2k is even, there is also a transitive imprimitive sub group of index (�)/2 containing those even permutations which permute { 1 , . . . , kl and {k + 1 , . . . , n } indepen dently OR switch the two blocks. It is not difficult to prove (see [DM, section 5 .2]) that, with a few small exceptions where n < 10, any other subgroup of An is smaller than these or is contained in one of them. What degree axioms do we need to ensure [ 1 5]? The largest subgroups of A15 have indexes 1 5 , 105, 455, 1365, 3003, 5005, 6435. The semigroup is therefore gen erated by these numbers plus some others larger than 6435. However, it is not hard to see that < 1 5 , 455, 3003> in cludes 1 0 5 , 1 365, 5005, 6435, and all larger indexes of sub groups of A15, so = < 1 5 , 4 5 5 , 3003>. This means that to derive the degree axiom [ 1 5] , we will need either [ 1 5kl for some k, or at least [4551. And [455] by itself isn't enough, because it only eliminates the possibility of A15 as a Galois group, but we also need to get rid of the prime 3. It turns out (I omit the details of the derivation from The orem 1) that [ 1 5] follows from any set of degree axioms where the degrees include a multiple of 3, a multiple of 5, and an element of the semigroup (of which 3533 is the first prime) . =
Now let's see if the proof of Theorem 1 can fail in char acteristic p. If a "degree implication" (i1 ]& . . . &[iml ==> [ n] holds in characteristic 0, we know that it holds in charac teristic p also if p divides one of the ii, because the proof fails only in the case of "inseparable extensions," which can not occur in characteristic p when every element has a p-th root. But if p does not divide any of the �, it doesn't divide n either, for in the preceding section, "The Fundamental The orem of Algebra Improved," we constructed a characteristic0 field in which [ n] was true iff n was not a multiple of p. So we may assume p does not divide n. If n divides any of the �, the degree implication is trivially true, so we may rule out this possibility. Purely inseparable extensions have degrees that are powers of the characteristic, which means we may assume there is an irreducible polynomial of de gree pr for some r; furthermore, pr must be < n if we are going to have a degree-n polynomial give an inseparable extension. So if there is a counterexample, we have root less polynomials of degree p r and degree n. This means we can construct rootless polynomials of all degrees in <pr, n>, and since p doesn't divide n, this semigroup includes all sufficiently large degrees, in particular, all degrees r(n - 1) or greater. If n is even, then pr is odd, and <pr, n> includes n(n - 1)/2 as well, because n ( n - 1)/2 = (n/2)•(n - 1) = (n/2)•(n - 3) + n (n/2)•(n - 5) + 2n = = (n/2)•Pr + ((n + 1 - pr)/2) • n. But we saw above that, for n > 9, the smallest element of that is not a multiple of n is n(n - 1)/2, if n is even, and for odd n is at least CD= n( n - 1)(n - 2)/6, which is greater than n ( n- 1 ) since n > 9. Therefore, <pr, n> contains the entire semigroup , so at least one of the & must be in <pr, n> and there is a rootless poly nomial of that degree. Thus we can't get a counterexam ple to our degree implication, because one of the degree axioms on the left-hand side must fail. We can deal with the remaining cases n < 10 by direct calculation. When n is prime, the only valid degree impli cations have a multiple of n on the left-hand side, and they are trivially valid in all characteristics. For n 4, 6, 8, 9 , we calculate the following semigroups: =
·
·
·
Begin by constructing fields K and L such that L is the splitting field over K of a polynomial j(x) of degree n, with Galois group Gal(UK) G. (This can be done so K and L are both algebraic over Q.) Let z be a primitive element for this extension, so L = K(z) and z satisfies an irreducible poly nomial of degree G over K Let Krnax be a maximal alge braic extension of K with the property that Lmax = Kma:x:(z) has degree over Kmax- (We can construct this by succes sively adjoining algebraic numbers that don't kill any of G, because there is an enumeration of the algebraic numbers.) Since we haven't disturbed G, j(x) still has G as its Ga lois group, and no roots in Kmax, but any further algebraic extension of Kmax will fail to extend Lmax by the same de gree-that is, for any new algebraic number y, Kmax(y, z) = Lmax(y) has a degree over KmaxCY) that is smaller than We need to show that all the degree axioms (hl, . . . [iml are true for Kmax-then, since j(x) is still rootless, [n] is false and thus (* ) is also false, as required. So suppose that we have a polynomial g (x) of degree � over Kmax, where by assumption i1 is not in the semi group < G> . g is a product of irreducible polynomials, and at least one of these must not have a degree in < G> (for if they all did, their product would). So we now have an irreducible polynomial h(x) whose degree i is not in < G > . Let y be a root o f h . Then KmaxCy) has degree i over Kmax, since h is irreducible. Consider the intersection M of KmaxCy) and Lmax = KmaxCz). Let d1 be the degree of this field over Kmax· Since M is a subfield of Lmax, the subgroup of G fix ing it must have index d1 , so either d1 = 1 or d1 is in < G> . =
I l
lei
lei.
Lmax(y)=Kmax(Y ,z)
=
< � > = < 3, < � > = 1 5 , 28, 35> 84, 280>.
In each case, for any prime power pr less than n and not dividing n, the generators of the semigroup (and so the whole semigroup) are in <pr, n>, so we can't get a coun terexample to the degree implication. Therefore the char acteristic 0 assumption in Theorem 1 can be eliminated.
COROLLARY 3 ([3]&(10]) COROLLARY 4 ([2]&[15])
==>
[6] is true in all fields.
==>
(8]
is
1
L
K(z)
true in all fields.
PRoOF OF NECESSITY. Reversing the direction of Theorem 1 is trickier. Suppose ( ) is false, so we have G acting on < 1 , . . . , n> with none of the fs in < G> . We need to falsify (*) , so we must construct a field where [id, . . . [inJ **
are true but [ n] is false.
K
Fields defined in proof of Theorem 5
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But d1 also divides i because M is a subfield of Kmaly), which means we must have d1 = 1 , because we know i is not in < G > . Thus M= Kmax: the extension fields KmdY) and KmaxCz) have only Kmax in common.
( ** ) for every subgroup G of Sn which acts withoutfixed-points on {1 ,2, . . . , n), the semigroup < G > contains one of the ii . (Compared with Theorem 1 , Theorem 5 eliminates the characteristic 0 hypothesis and works in both directions.)
way, and led ultimately to the formulation of Theorem 1 (which is not hard to prove once it is formulated just right!). Although Theorem 5 may appear definitive, there are several directions for further investigation. The algorithm implicit in (**) is slow, but it can be sped up by making certain assumptions about permutation groups; however, verifying these assumptions will require careful analysis of the O 'Nan/Scott Theorem on maximal subgroups of An (see [DM]) and the Classification of Finite Simple Groups. There is also a rich theory for several kinds of weak ened degree axioms, such as [ n] ' : "all polynomials of degree n are reducible," or [ nk] : "all polynomials of degree n have a factor of de gree li' (when k = 1 this is the standard degree axiom [n]). These weakened axioms are still expressible in the lan guage of field theory, but they translate differently into the language of Galois groups. Finally, the "finite choice axioms'' deserve further inves tigation. The great progress in finite group theory over the last 35 years ought to make it easier to calculate the rela tionships between these axioms, including weakened ver sions which identify subsets or partitions of {1 , . , n) in stead of elements.
Conclusion
ACKNOWLEDGMENTS
Theorems 2 and 3 establish the minimum algebraic condi tions necessary for a field to be algebraically closed, and they can therefore be said to "optimize" the Fundamental Theorem of Algebra. But each specific "degree implication" is a first-order consequence of the axioms for fields, and could have been discovered two centuries ago; the exis tence of these finitary relationships appears to have been unsuspected by practically everyone, with one important exception. The inspiration for Theorem 1 was the work done by John H. Conway on "Finite Choice Axioms" in 1 970, de tailed in [Co] . Conway, building on earlier work of Mostowski and Tarski, identified a necessary and suffi cient condition for effective implications between axioms of the form "Every collection of n-element sets has a choice functio n . " Conway's group-theoretic condition is very similar to ( * * ), the difference being that one could use the semigroup < H > for any subgroup H of a group G acting fixed-point-freely on { 1 , . . . , n), rather than re quiring G = H. The present article also borrows some ter minology, notational conventions, and proof ideas from Conway's work. Theorem 2 was originally proved by a difficult combi natorial argument that generalized Gauss's original proof. Corollaries 3 and 4 emerged during discussions with Con-
I am grateful to Dan Shapiro, Alison Pacelli, Harvey Friedman,
But this means that every automorphism of Lmax fixing Kmax extends to an automorphism of LmaxCy) fixing KmaxCy), because it doesn't matter which of the conjugates of z we use when forming Kma.xCy,z) = KmaxCz,y) = Lma.xCy) . Therefore the Galois group of Lmax(y) over Kma.x(Y) is still G; but we constructed Kmax so that any algebraic extension would collapse some of G. Therefore KmaxCy) is not really an extension: y must already be in Kmax. which means that h(x) is of degree 1 , and g(x) has a root, as was to he shown. We have now established Theorem 5 .
I Gl
THEOREM 5. The statement ([il)&[izl& . . . &[inJ)
(*)
=>
[ n]
is true in all fields if!
14
THE MATHEMATICAL INTELLIGENCER
Frank Morgan, Simon Kochen, Noam Elkies, and Jonathan Co hen for verifications, suggestions, and encouragement. I would especially thank Professor John Conway for many instructive and enjoyable conversations over the last 20 years, as well as for his inexhaustibly inspiring writings and per sonality. REFERENCES
[Co] Conway, John H., "Effective Implications between the 'Finite ' Choice Axioms, " in Cambridge Summer School in Mathematical Logic (eds. A. R . D. Mathias, H . Rogers), Springer Lecture Notes in Math ematics 337, 439-458 (Springer-Verlag, Berlin 1 971 ). [DM] Dixon, John D . , and Brian Mortimer, Permutation Groups, Springer Graduate Texts in Mathematics 1 63, Springer-Verlag, 1 996.
[FR] Fine, Benjamin, and Gerhard Rosenberger, The Fundamental The orem of Algebra, Springer-Verlag, New York 1 997.
[G] Gauss, Carl Friedrich, Werke, Volume 3, 33-56 (In Latin; English translation available at http://www.cs.man.ac.uk!�pt/misc/gauss web.html). [T] Tarski, Alfred, A Decision Method for Elementary Algebra and Geom etry, University of California Press, Berkeley and Los Angeles, 1 951 .
[vdW] van der Waerden, B. L. , Algebra (7th edition, Vol. 1 ), Frederick Ungar Publishing, U .S.A. , 1 970.
MatheiTII atically Bent
The proof is in the pudding.
Colin Adam s , Editor
North North Wester n State U n iver sity M athe mat ics Departme nt Safety M anua l COLIN ADAMS
Opening a copy of The Mathematical Intelligencer you may ask yourself
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 disorienta tion 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.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267 USA e-mail: [email protected]
elcome to North North Western State University (NNWSU). We here at the Office of Office Safety (OOS) are happy to be one of the first offices on campus to welcome you to the Department of Mathematics. This manual is but one of a pile of documents that you have just received as a new member of the fac ulty. BUT IT IS THE MOST IMPORTAN71 Because safety is our number-one concern here at NNWSU. You might have thought it was education, or research, or bringing in grant dollars, or showing support for our surprisingly large football team. But, no, it is safety. Of course, new faculty members might not associate safety issues with a mathematics department. They might think that safety concerns should be rel egated to chemistry departments, where an exploding beaker can send shards of glass streaking toward an unprotected eyeball, or physics departments, where an errant laser can b urn holes in the seats of pants and what they contain. New teachers may believe that they need not fear for their physical safety when work ing in a mathematics environment. But some once new fac ulty members are no longer employed at NNWSU. In fact, some of them are n ot employed any where, because they no lnnger walk this earth! So please read this docu ment carefully. It could save your life!
1. What is the number-one safety con cern in a mathematics department? This is an exceUent question. We made it up, but it is an excellent ques tion nonetheless. The number-one safety concern is eye strain. Did you know that? We bet you didn't. Strained eyes cause more lost work days than any other single mathematics office in j ury. Often, we see faculty members dri ving erratically, on their way home af ter a debilitating eye strain injury. They are pulled over by police officers who believe they are intoxicated, and who ask them to walk a straight line. And often they fail, because of eye strain. Then it's off to the pokey for them. Don,t let this happen to you!
2.
How do eye strain injuries occur?
Another excellent question. And yes, again, we made it up. There are three main categories of eyestrain inj ury. A. Eye Fatigue Syndrome (EFS): Just as we can strain a leg muscle from overexertion, we can strain our eyes by staring in one direction too long, say at a c omputer screen or at a par ticularly enchanting fractal poster. What can be done to prevent or al leviate EFS? Here are some Eye Strain Prevention Exercises (ESPE): 1 . Look away from the computer screen, say, at the corridor outside your office door. Cup the palms of your hands over your eyes and stare for 60 seconds, counting out l oud. Then slowly twist your wrists to alternately cup your eyes and create blinders while staring at anyone looking in from the cor ridor. Continue for 60 seconds. Then cup with one hand while making a blinder with the other. Alternate hands back and forth for another minute. 2. Close one eye . Moving the op p osite hand in a clockwise circle of diameter one foot, follow the index finger with the open eye. Do this for three revolutions and then change eyes. If anyone is staring at you through your open door, scan from their feet up to their head and then back down to their feet. Repeat six times. Then return to work.
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B. Sudden Eye Movement Injury (SEMI): This injury may occur while you are in your office working at your computer and a student comes to the open door. "Excuse me pro fessor, but I don't think I should have lost this point on the home work . " Startled, you jerk your eyes from the screen to the doorway and feel an immediate explosion of pain caused by tearing of the muscles that control eyeball direction. What can be done to prevent SEMI? 1. Keep the office door closed. 2. When there is a knock at the door, do not swivel the eyes quickly to the door, but rather, move them slowly in that direction, taking time to peruse the items you see along the way. Avoid jerky eye movements at all times. C. Repetitive Eye Roll Injury (RERI): This injury typically occurs at de partment meetings. Excessive rolling of the eyes can cause severe fatigue of the muscles under the upper eye lid, and leave one incapable of look ing anywhere but down without pain. (The common misconception that mathematicians are shy is a di rect result of this phenomenon.) If, however, you are a representative of the faculty senate, you have had ample opportunity to condition your eyes to rolling. So roll on!
3. Is it safer to work with a pencil or with a pen? Here at Double-North Western, pen injuries exceed pencil injuries by six to one. Yes, six to one! We have had three pen injuries and about half a pen cil injury. How to explain the discrep ancy? We don't know, but there it is . You choose pen and you may be writ ing your own obituary. 4.
Must I wear loose-fitting clothes?
Yes, you must. Constricting clothing can be a constant distraction, causing you to lose focus on safety issues, and risk calamity. But not so loose that they fall off, causing a distraction for your fellow workers as they are handling sharp implements. 5.
What is the most dangerous object in a mathematics department?
Great, great question. This is an easy one, too. It is the cups of coffee. Many
16
THE MATHEMATICAL INTELLIGENCER
people have claimed that styrofoam cups were the greatest invention in history. We here at the Double N Double-D SU Dou ble 0 S beg to differ. The styrofoam cup is perhaps the most dangerous invention ever, as measured by the Steiner Hot Cof fee Burn Index. They are tippy and they retain the heat in the coffee. This is an obvious recipe for disaster. More mem bers of the Mathematics Department have been treated for coffee burns than for any other single kind of burn, with the ex ception of rug burns, which really don't deserve to be called burns at all. We actually keep spare pairs of pants at the OOS for members of the Mathe matics Department who spill coffee on themselves. We have men's cuffed tan pleated slacks, in a size 34 waist, and women's gabardine plaid slacks in a size 6, in case you want to plan ahead. Well, actually, we don't have the men's slacks right now. We are waiting to get them back from a certain someone who we suspect is purposely spilling coffee on himself just to get the use of the slacks. 6.
Why is mathematics so dangerous?
Mathematics is perhaps the most ab stract of subjects. To study and do mathematics, you must remove your mind from the real world around you. In the process, you lose touch with re ality. You don't see those stairs that you are approaching. You don't see the open file cabinet drawer. You don't see the students milling around after that exam with angry looks on their faces. 7. But isn 't mathematics goodfor your brain development?
Is all exercise good for you? Is it good for your back to lift heavy boxes filled with safety goggles for hours at a time, day in and day out, for no apparent pur pose? I think you know the answer is no. Well, it's the same with brain exer cise. You do difficult problems day in and day out, you could blow out your medulla oblongata, rip your brainstem , o r split your hemispheres permanently. Then where would you be? And even if that didn't happen, you might overdevelop your brain, and it could end up looking like Arnold Schwarzennegger used to look, when he was dressed in a speedo and slathered in oil, glistening in the bright lights. Forgetting about the fact he mar ried Maria Shriver and is now the Gov-
ernor of California, be honest: Is that how you want your brain to look? So, after doing your share of math, kick back at the end of the day. Forget about that lemma that's been driving you crazy. Go home, have a soda and watch some reality 1V. You'll be glad you did. 8. Why is safety the most important is sue at NNWSU?
Each college and university strives to be the best it can be. Those of us at Northie have realized that we don't have a chance in hell of being the best in any academic discipline. So, instead, we have decided to focus on safety. Our goal is nothing less than to be the safest educational institution in the country. Better than Harvard. Better than MIT. Better than Bluebonnet Hill Community College, that pretender to the safety crown right down the road. Remember when you were a candi date for a faculty position in mathemat ics here? You probably thought you were being evaluated on the basis of your abil ity to teach and do research. Nothing could be farther from the truth. In fact, you were being evaluated solely on the basis of your previous safety record and your future safety potential . During your job talk, we had a checklist. Low-heeled shoes? Laces tied in a double knot? Zip per up? Pens carefully capped? You must have satisfied all the criteria on the check list, as otherwise, you would not be read ing this document. But don't think you can rest on your laurels. Tenure and pro motion are also contingent on your at tention to safety procedures. Well, that concludes this initial dis cussion about safety in an arithmetic en vironment. We hope we haven't scared you with all of this talk of the dangers of mathematics. If you approach math ematics with an eye to safety, you may find it productive, and yes, perhaps even enjoyable. We look forward to meeting you per sonally at our mandatory weekly math safety seminars, which begin soon. Look for our multi-colored notice com ing in your mailbox shortly. But in the meantime, remember. Safety: It's not just a word anymore.
This document brought to you compli ments of the North North Western State University Office of Office Safety (NNWSUOOS).
M a t h e m atical C o m m u n ities
Re l ig io us H er esy and M at he mat ica l Cr eat iv ity i n Russ ia LOREN G RAHAM AND JEAN-MICHEL KANTOR
Ibis 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 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.
uring the last several decades we have frequently gone to the So viet Union and, after its collapse, to Russia, where we have often worked with scientists. Gradually, a remarkable story about the relationship of science and religion in Russia has emerged from conversations with Russian colleagues. The story helps explain the birth of the Moscow School of Mathematics, one of the most influential modern movements in mathematics. The conflict at the cen ter of the story persists today and raises fundamental questions about the nature of mathematics, not only in Russia, but throughout the world. Since the history of the issue dates to the early years of the last century, we must begin with a single event from that time. This event kicked off a movement that is still alive. Early in the m orning of July 3, 1 91 3 , two ships from the Imperial Russian Navy, acting on orders from Tsar Nicholas II, steamed into the azure wa ters surrounding the holy mountain of Mt. Athas in Greece, a center of Or thodox Christianity for a thousand years. The ships, the Donets and the Kherson, anchored near the Pantalei mon Monastery, a traditional center of Russian Orthodoxy and residence of h undreds of Russian monks. Small boats loaded with armed Russian marines made their way to the dock, where the men disembarked. The marines pro ceeded to the cathedral of the monastery, at that moment nearly empty. There, the officer in charge met with several of the religious ascetics and told them that they were to inform all their brethren to leave their cells and assemble in the cathedral. When the other monks learned of the order, they barricaded the doors of their cells with furniture and boards. Inside they fell to their knees and began crying "Lord, Have Mercy!" ( Gospodi pomiluz) and many of them launched into a unique prayer,
one causing controversy in the Church, called "The Jesus Prayer. " (We will say more later about the Jesus Prayer.) The Russian officer demanded that the monks come out. When he was ig nored, he ordered his marines to tear down the barricades and aim water from fire hoses at the men inside. The marines flushed the recluses from their cells and herded them into the cathe dral. There, the officer announced to the soaked and terrified monks that they must either renounce their heretical be liefs or be arrested. Only a few stepped forward and promised to obey. The oth ers remained obstinate, crying that the marines represented the "Anti-Christ. " The officer commanded the marines to force the recalcitrant crowd onto the waiting ships, which took them to the Ukrainian-Russian city of Odessa, on the Black Sea. In all, approximately 1000 monks were detained in this fashion. (The sources differ on how violent this operation was; according to some, the marines at one point used a machine gun and killed several monks; official ac counts deny this, but it was certainly a bloody affair, with many wounded.) In Odessa the religious believers were told that the Holy Synod in St. Pe tersburg-the highest a uthority of Rus sian Orthodoxy-had condemned them as heretics for engaging in the c ult known as "Name-Worshipping . " They were forbidden to return to Mt. Athas or to reside in the major cities of St. Pe tersburg and M oscow. They were also warned that they m ust not practice their deviant religious beliefs in Russian Or thodox churches on penalty of excom munication. Otherwise they were free to go. The unrepentant monks dis persed all over rural Russia, where they often lived in remote monasteries, far from central a uthorities, and continued there to practice their heresy and to propagate their religious faith.
Please send all submissions to Marjorie Senechal, Department
of Mathematics, Smith College, Northampto n , MA 01063 USA e-mail: senechal@min kowski.smith.edu
The authors are writing a book on this subject titled Naming God, Naming Infinities and scheduled for publi cation by the Harvard University Press. The article also draws heavily on Loren Graham and Jean-Michel Kan tor, "A Comparison of Two Cultural Approaches to Mathematics: France and Russia, 1 890-1930," /sis, Vol.
97 (No. 1 , 2006). pp. 56-74.
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Instead of dying out, as the tsarist a uthorities obviously hoped that it would, the heresy continued to spread surreptitiously. With the outbreak, a year later, of World War I the attention of the tsarist government shifted else where. The practice of Name-Worship ping quietly increased in strength, grad ually moving from the countryside to the cities, where it attracted the atten tion of the intelligentsia, especially mathematicians, some of whom be lieved it contained profound insights for their field. Among the leading mathe maticians who became interested in Name-Worshipping were Dmitri Egorov ( 1 869-1931) and Nikolai Luzin ( 1 8831 950), later the founders of the Moscow School of Mathematics. In seeing c on nections between mathematics and Name-Worshipping, they were aided by a heretical priest, Father Pavel Floren skii (1882- 1937), a former mathematics student at Moscow University, where both Luzin and Florenskii studied un der Egorov. While at the university, Flo renskii and Luzin served, one after the other, as secretary of the Student Circle of the Moscow Mathematical Society, of which their professor, Egorov, was later president. In subsequent years they car ried on an 1 8-year correspondence, of ten about mathematics and religion. Both the Russian Orthodox Church and the new Communist regime perse c uted Name-Worshipping after the Rev olution of 1 9 1 7, but the practice never died out. Following the collapse of the S oviet Union it has been enjoying a small resurgence in Russia. B ut even n ow it remains a "heresy, " equally op posed by intellectual camps so differ ent that their followers usually agree on very little: Marxists, the leaders of the Russian Orthodox Church, and secular rationalists. What was " Name-Worshipping" and h ow could this religious movement have anything to do with mathematics? B oth mathematicians and religious be lievers try to grasp concepts that seem inexpressible, ineffable, or even incon ceivable. The history of mathematics demonstrates a number of such mo ments. "Infinity" was first denoted by the Greeks as apeiron ("endless, un limited mass , " "primal chaos"), irrational numbers ("alogoi , " absence of logos) were unspeakable or unthinkable at the time of Pythagoras, and imaginary n um bers were only reluctantly accepted in 18
THE MATHEMATICAL INTELLIGENCER
Seated: N. N. Luzin (1883-1950); standing on right: D. F. Egorov (1869-1931); standing on left: W. Sierpinski, well-known Polish mathematician.
Plaque which is now located on the building on Anbat St. in Moscow where Luzin lived.
the Henaissance . In modern times "ideal theory'' began with numbers that were only supposed to exist "ideally. " In the period 1 H90-1930 a great de bate was occurring among mathemati cians over the new field of set theoty, a controversy that became connected in the minds of some leading Russian mathematicians to Name-Worshipping. A "set" is a collection of objects shar ing some property and given a "name. ·· For example, the set of all giraffes in South Carolina could be named "SCG" for "South Carolina Giraffes." This set obviously has a finite number of ele ments . The set 1 , 2,3, . . . of whole num bers has an infinite number of elements; Georg Cantor names it � 0 . The birth o f set theory at the end of the nineteenth century brought with it new debates about the nature of "in finity . .. Is the "infinity" of points in a line segment just another description for �0• or is it an infinity of another type? A new theory of infinities was born in De cember 1873 when Cantor proved that these are different: one cannot "count" the number of points on a line. He then defined an infinity of infinities, the alephs. and another infinity of other numbers corresponding to ordered sets, and he gave new names to all these in finities. for example � 0 and � 1 . A cru cial point here is the idea of naming. In retrospect, we can see that after Can tor assigned different names to differ ent infinities. these infinities seemed to take on a reality that they earlier had not possessed. A new world of transfi nite numbers was being created. More . over, the concept of ''naming, . as we will see, became the link between reli gion and mathematics. Even many leading mathematicians were reluctant to accept this new world. How do we define these new infinities? Is it possible to postulate the existence of a mathematical entity before it is de fined? According to most monotheistic reli gions "God'' is also beyond the com prehension of mere mortals, and can not be defined. Is it possible to postulate the existence of a deity he fore it is defined? If God is, in princi ple, beyond human comprehension (and in the Christian and Jewish scrip tures there are many such assertions). how. in complete ignorance of his na ture, can human beings worship him? What does one worship? Traditionally,
religious believers have side-stepped this question through the use of sym bols: prayers, names, rituals, music, relics. scents. tastes, etc. Symbolism is the term given to a perceptible object or activity that represents to the mind the semblance of something which is not shown but is realized by associa tion with it. And the importance of sym bols both to religion and mathematics is one of the many bonds that brought mathematicians and religious believers together in Russia in the early decades of the last century. Both mathematicians and religious believers use symbols they do not fully master. Names are symbols, and the signifi cance of assigning names to objects has been controversial throughout the his tory of philosophy and religion. One of the great theological disputes of the mid dle ages, that over nominalism, revolved around it. When one invents a name. does one at the same time create some thing new, or does one merely give a label to an existing thing? For example, we might ask. ''I s the term 'virtual real ity,' so commonly used in computer sci ence, a human construction or a tag at tached to something already existing?" The issue goes back to the begin ning of human thought. In Genesis we are told, "God said, ·Let there be light. ' and there was light.'' He gave the thing a name before he created it. The an cient Egyptian God Ptah is described in Memphite theology as creating with his tongue that which he first conceived in his head. Naming God is forbidden in the Jewish tradition, and in the mysti cal Kabbala (Book of Creation, Zohar) a large role is assigned to language in the act of creation. In the first verse of the gospel according to St. John we read, "In the beginning was the Word, and the Word was with God, and the Word was God . " Words are names, and one of the leaders of the Russian Name Worshippers. the monk Ilarion, said "the name of God is God!'' ("Imia Bozhie est ' sam Bog"). Intellectual and artistic Russia at the end of the nineteenth century and in the first decades of the twentieth was seized with the question of the signif icance of symbols. The Symbolist Movement affected ballet, music, liter ature, art, and poetry, as the names Di aghilev, Stravinsky, Belyi, Stanislavsky, Nemirovich-Danchenko, and Meyer hold remind us. Now we should add
the mathematicians Egorov and Luzin to such lists. Indeed, there was even a connection between the literary and mathematical movements. Andrei Belyi, the symbolist poet, was the son of a Moscow mathematician, and he ma jored in mathematics at Moscow Uni versity, where he studied under Egorov and together with Luzin. He was fa miliar with Name-Worshipping. Belyi once wrote an essay called "The Magic of Words, " in which he claimed, "When I name an object with a word, I thereby assert its existence. " We can ask, " Does this apply both to mathematics and to poetry? If the object is a new type of infinity, does that infinity exist just af ter you name it?'' At the heart of the Name-Worship ping cult was the "Jesus Prayer" Ciis usovaia molitua), a religious practice with ancient roots. In the Jesus Prayer the religious believer chants the names of Christ and God over and over again, hundreds of times, until his or her whole body reaches a state of religious ecstasy in which even the beating of the heart and the breathing cycle, are sup posedly in tune with the chanted words "Christ" and "God." According to Name Worshippers, the proper practice of the prayer brings the worshipper to a state of unity with God through the rhythmic pronouncing of his name. Franny ob served in J D. Salinger's novel Fran ny and Zooey that in this state of ecstasy "you get an absolutely new conception of what everything's about. " The Jesus Prayer has always been part of the Russian Orthodox tradition, but it took on an unusual prominence in the late nineteenth century after the publication in 1 884 of a hook titled The Way qf the Pilgrim, later translated into many languages, in which the potency of the prayer was acclaimed. The prayer became popular throughout Russia. Ac cording to some sources , the Empress Alexandra and her notorious advisor Rasputin sympathized with the heresy and unsuccessfully tried in 1 9 1 3 to stay the hand of Tsar Nicholas II in arrest ing the heretical monks in Athos. But the establishment Orthodox Church won out with its view that the Name Worshippers were pagan pantheists who confused the symbols of God with God Himself. Church officials advised Nicholas to squelch the heresy before it hopelessly split the faith and the na tion. Since that time the position of the
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Church on Name-Worshipping has re mained the same. On the question of whether more than one kind of infinity exists, each of which can be given a name, not all mathematicians agreed with Cantor. For some of them, set theory can not apply to the line, what they called "the con tinuum. " The debates became very complex and sometimes very heated. French and Russian mathematicians were leaders in this debate. The French who wrestled with set theory included Emile Borel (187 1-1 956), Rene Baire (1874--1 932), and Henri Lebesgue (18751 944); they were the inheritors of a great and powerful mathematical tradition, and at first they taught the Russians more than they learned from them. Both Egorov and Luzin traveled repeatedly to Paris to talk with their French col leagues. They usually lived in the aca demic heart of the city, in the Hotel Parisiana, near the Pantheon. The concierge of the building remembered many years later both the devotion of the Russian visitors to their studies and their religious piety. The French tended to be skeptical of set theory, or at least the furthest exten sions of it into discussions of new types of infinities. A few of them, such as Borel, were at first attracted to it but gradually became more hesitant. The old French establishment of mathematics, represented by Emile Picard, stoutly re sisted. Picard acidly remarked, "Some be lievers in set theory are scholastics who would have loved to discuss the proofs of the existence of God with Saint Anselm and his opponent Gaunilon, the monk of Noirmoutiers. " Picard thought that he could dismiss set theory by link ing it to discussions of religion, exactly the way the Russians thought they could strengthen it. The French worked within the tradition of Cartesian rationality; the Russians were speculating within the tra dition of Russian mysticism. A contrast between the cold logic of the French and the spirituality of the Rus sians is not new in the history of cul ture. Leo Tolstoy, in War and Peace, compared Napoleon's Cartesian logic in his assault on Russia with his opponent Kutuzov's emotional religiosity. After the critical battle of Borodino, the novelist described the Russian general Kutuzov kneeling in gratitude before a holy icon in a church procession while Napoleon rationalized his "miscalculation." Tolstoy 20
THE MATHEMATICAL INTELLIGENCER
Emile Borel
Rene Baire
saw Borodino as a victory of Russian spirit over French rationalism. Eventually the French mathematicians lost their nerve and yielded the field to their Russian colleagues. The French could not stomach the thought that new infinities could be created simply by naming them, and that these new in finities then became legitimate, and even necessary, objects of study by mathe maticians. Some of the French actually feared that one could lose one's mind pursuing the problems of set theory ap plied to these infinities. They noticed that the founder of the field, Georg Cantor, had a series of attacks of depression af ter 1884. Baire, who already had some digestive problems, fell seriously ill in 1898, as if being punished for his flirta tion with the new ideas; eventually, he killed himself. Borel, after referring to the illnesses of Cantor and Baire, told his friend Paul Valery that he had aban doned set theory "because of the fatigue it caused him, which made him fear and
set theory was to Florenskii a brilliant example of how naming and classifying can bring mathematical breakthroughs. To him a "set" was simply a naming of entities according to an arbitrary mental system, not a recognition of real objects existing in nature. When a mathemati cian created a "set" by naming it, he was giving birth to a new mathematical be ing. Mathematicians who created sets by naming them, according to Florenskii, were performing an intellectual and re ligious act similar to what Name-Wor shippers did when they named and wor shipped God.
foresee in himself serious illness if he persisted in that work." The Russians did not have these problems. They rejoiced in what they saw as the fusion of mathematics and religion. At the time of the Russian Rev olution in 1917 Father Florenskii was liv ing in a monastery town near Moscow and he translated the religious ideas of Name-Worshipping into mathematical parlance. He stated his goal as creating a "synthesis between religious and sec ular culture. " He expounded the view that "the point where divine and human energy meet is 'the symbol', which is greater than itself." The development of
Henri Lebesgue
When Egorov, Luzin, and their stu dents created a new set, they often called it a "named set, " in Russian imen noe mnozhestvo. Thus the root word imia (name) occurred in the Russian language in both the mathematical terms for the new types of sets and the reli gious trend of imiaslavie ("Name Praising," or "Name-Worshipping"). In Luzin's personal papers in the Moscow archives, the historian can see today how obsessed he was with "naming" as many subsets of the continuum as he could. Roger Cooke studied Luzin's pa pers and noted that he "frequently stud ied the concept of a 'nameable' object and its relationship to the attempted cat alog of the flora and fauna of analysis in the Baire classification . . . . Luzin was trying very hard to name all the count able ordinals. " At one point Luzin scrib bled in infelicitous but understandable French "nommer, c'est avoir individu" ("naming is having individuality"). 1 The circle of eager students at Moscow University which formed around Egorov and Luzin at about the time of the beginning of World War I and continued throughout the early twenties was known as "Lusitania. " This group caused an explosion of mathe matical research that still affects the world of mathematics. Lusitania was at first a small secret society, and the place of religion in that society is illustrated by the names the members gave one another; Egorov was "God-the-father," Luzin was "God-the son'' and each of the students in the so ciety was given the monastic title of "novice. " They all went to Egorov's home, an apartment not far from the university, three times a year: Easter, Christmas, and Egorov's Name-Day (again the emphasis on "names. ") But how long could such a religiously oriented group exist in the Soviet Union, where the campaign against religion was gathering force? In their effort to com bat religion, the Communists made no distinction between orthodox believers and heretics. The three men most in volved in the effort to link religion and mathematics followed different paths in responding to this threat. Florenskii was the most defiant, refusing to take off his priest's robe, causing the Soviet leader Trotsky to inquire at a meeting they both
attended, "Who is that?" Egorov also continued his religious practices and worked closely with Florenskii in in spiring the "True Church" movement aiming at a religious revival in Russia despite the Soviet efforts to suppress re ligion. Luzin was much more cautious, refusing to attend meetings of the Name Worshippers, and concealing his reli gious convictions. Meanwhile, the Moscow School of Mathematics flourished. It grew until it included dozens of young mathemati cians, many of them now prominent in the history of mathematics (e. g . , Andrei Kolmogorov, Pavel Aleksandrov, Alek sandr Khinchin, Mikhail Lavrent'ev, Lazar Lyusternik, Petr Novikov) . It was inevitable that as the group increased in size it would lose its earlier ethos. Some of the students of Egorov and Luzin were out of sympathy with their teachers' religious impulses. A few were even members of the Young Commu nist League. Divisions, rivalries, and ide ological disputes began to develop among Moscow mathematicians. In 1 930 Ernst Kol'man, a militant Marxist mathematician who was never a member of Lusitania himself, attacked Florenskii and Egorov in an address to mathematicians, castigating their use of "mathematics in the service of religion, " "mathematics i n the service of priest craft." He continued the attack in pub lished articles, saying "Diplomaed lack eys of priestcraft right under our noses are using mathematics for a highly masked form of religious propaganda. " Responding t o such denunciations, starting around 1 930 the Soviet authori ties moved heavily against the Name Worshippers. Fortunately, the most im portant mathematical work had already been done. They arrested Father Floren skii, the main ideologist of mathematical Name-Worshipping, and eventually sent him to a labor camp in the Solovetsky Islands, far north in the Arctic Ocean, where he continued to do scientific work. On December 8, 1937, he was executed by firing squad. In one of his last letters to his grandson, who lives in Moscow to day, Florenskii wrote, "Above all I think about you, but with worry. Life is dead." All Florenskii's voluminous writings were removed from Soviet libraries, and even mentioning his name was forbidden.
Tombstone of Egorov in Kazan.
Dmitrii Egorov, president of the Moscow Mathematical Society, was ar rested in 1 930 and exiled to a camp near Kazan, on the Volga River. There he went on hunger strike because the prison guards would not permit him to practice his religious faith. Near death, Egorov was sent to a local hospital, where he was recognized by a physi cian, the wife of a mathematician named Nikolai Chebotaryov. The two Chebo taryovs did everything they could to save Egorov's life, but it was too late. We are told that he died in the arms of Dr. Chebotaryova . Egorov's name, like Florenskii's, was not to be mentioned in Soviet society. The Name-Worshippers became the object of name censorship. The most talented of the mathemati cians connected with the religious movement, Nikolai Luzin, was subjected to a show trial, known even today as the "Luzin Affair. " One of the ideologi cal charges against him was that he "loved" capitalist France, where he of ten worked, and was a friend of the French mathematician Emile Borel. Borel was at that moment Minister of the Navy in the French government, and therefore was obviously a "militarist" ea ger for aggression against the Soviet Union. In a great act of heroism, one of the most famous physicists in the Soviet Union, Peter Kapitsa, wrote a confiden tial letter to the Soviet leaders Molotov
' Roger Cooke, "N. N. Luzin on the Problems of Set Theory," unpublished draft, January 1 990. pp. 1 -2, 7. Luzin's notes are held in the Archive of the Academy of Sci
ences of the USSR, Moscow, fond 606, op. 1 , ed. khr. 34.
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Emile Borel
and Stalin, pleading for mercy for "one of our greatest mathematicians, known throughout the world." Luzin was repri manded but miraculously saved, and he continued mathematical work until his death in 1 950, although no longer in set theory but instead in applied mathe matics, and no longer in communication with his French friends. The persecution of the Name-Worshippers continued throughout the Soviet period, with ar rests as late as the 1 980s up to the time of the Gorbachev years, starting in 1 985. In the summer of 2004 Loren Gra ham met with a prominent mathemati cian in Moscow known to be in sym pathy with Name-Worshipping. The mathematician implied he was a Name Worshipper without stating it outright. His a partment was decorated with the symbols of Name-Worshipping, includ ing photographs of its leaders. His li brary was filled with rare books and ar ticles on Name-Worshipping. Graham asked if it would be possible for him to witness a Name-Worshipper in the je sus Prayer trance. "No," replied the mathematician, "this practice is very in timate, and is best done alone. For you to witness it would be considered an intrusion. However, if you are looking for some evidence of Name-Worship ping today I would suggest that you visit the basement of the Church of St. Tatiana the Martyr. In that basement is a spot that has recently become sacred to Name-Worshippers. " Graham knew o f this church; forty five years earlier, as an exchange stu dent, he had attended a student dance in the building after the church itself had been eliminated by Soviet authorities 22
THE MATHEMATICAL INTELLIGENCER
and converted into a student club and theater. Now, in the post-Soviet period, it has been restored as the official church of Moscow University, as it had been be fore the Revolution. It is located on the old campus near the Kremlin, in a build ing next to the one that housed the De partment of Mathematics when Egorov and Luzin dominated that department. It is the church where they often prayed. Graham asked the mathemati cian, "When I go into the basement, how will I know when I have reached the sa cred spot?" The mathematician replied, "You will know when you get there." The next day Graham went to the Church of St. Tatiana the Martyr, and made his way to the basement. There he found a particular corner where the photographs of Father Florenskii and Dmitri Egorov, founders of mathemati cal Name-Worshipping, faced each other, and he knew that he was in the place where Name-Worshippers liked to come, alone, to practice the jesus Prayer. But six months later, in Decem ber 2004, he visited the basement again and found that the sacred spot had been eliminated by the Church, which had fi nally realized that Name-Worshippers were coming to the basement to cele brate their "heresy. " Now an official chapel of the Church occupies the base ment, with a priest guarding it and en suring the orthodoxy of all worshippers. The Jesus Prayer is not practiced there any more. Thus, the struggle over Name Worshipping continues today. This story is a tragic and dramatic one, like many stories about Stalinist Russia, but this one also contains a deep philosophic question about the nature of mathematics. Where do the concepts and objects used by mathematicians come from? Are they invented in the brains of mathematicians, or are they in some sense discovered, perhaps in a platonistic world? Florenskii, Egorov, and Luzin believed that the objects of mathematics are invented not through analysis but through mystical inspiration and naming. They thought that French mathematicians like Baire, Borel, and Lebesgue were mistaken in their com mitment to Cartesian rationalism. We were trained in the tradition of Western rationalism, and we do not share the mysticism of the Russian founders of the Moscow School of Mathematics. We would point out that naming is not identical with creating.
We can name "unicorns·· but that does not make unicorns real. We also note that the basic idea behind Name-Wor shipping is not new; there are many similarities between Name-Worshipping and other types of religious and medi tation practices, including variants of Hinduism, Buddhism, Judaism, and Is lam. The practice of "'talking in tongues·· of Protestant evangelicals is also related. The endpoint, as in Name-Worshipping, is a state of glottal ecstasy. We do not see this state as one usually conducive to scientific creativity. But the reason that this episode in Russian history is different is that in this case mysticism may actually ha\'e helped science. In the early twentieth century, mathematicians truly differed among themselves about the existence of vari ous infinite sets. The French, with their secular, rationalist worldview, had neither the courage nor the motivation to enter the frightening world of the hierarchy of infinities. The French feared what the Russians exalted. And in the hands of the Russians what earlier seemed like fanci ful unicorns became useful mathematical objects. (A similar situation may have oc CI.med more recently in string theory, when Anglo-Saxon and Russian mathe maticians and physicists were ahead of French scholars.) If we had been mathematicians in the period 1 900-1 930 we surely would have hesitated along with the French mathe matical establishment, constrained by our rationalism. The Russians, however, be lieved that they had absolute freedom to invent mathematical objects and to give their inventions names. Following their approach, the Russians created a new field, descriptive set theory, at a time when mathematicians elsewhere faltered. And the Moscow School of Mathematics, founded by Egorov and Luzin, still exists today. And the significance of their achievement is still with us. Loren Graham Program in Science, Technology, and Society Room E5 1 - 1 28 Massachusetts Institute of Technology Cambridge, MA 02 1 39 Jean-Michel Kantor lnstitut de Mathematiques de Jussieu, Case 247, 4 Place Jussieu 75252 Paris Cedex, France [email protected]. fr
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Th e H eptago n to the Sq uar e, and Ot her W i l d Tw ist s G REG N . FREDERICKSON
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, suprising, or appealing that one has an urge to pass them on.
M i c hael Kleber and Ravi Vaki l ,
geometric dissection is a cutting of a geometric figure into pieces that can be rearranged to form another figure [ 1 2 , 22]. Such visual demonstrations of the equivalence of area span from the times of the ancient Greeks [3, 7] to the flowering of Arabic Islamic mathematics [ 1 , 4, 27, 29] to the heyday of mathematical puzzle columns in newspapers and magazines [8, 9, 24, 25] to the appearance of articles on the \\'orid-wide web [31]. It has long been known that two polygons of equal area can be dissected in a finite number of pieces [5, 18, 23, 32]. During the last 100 years. the emphasis has generally been on minimizing the number of pieces for any given dissection. As dissection methods have become more sophisticated, attention has also been paid to special properties. Most notable is the property that all pieces of a dissection be connected by hinges, so that when the pieces are swung one way on the hinges, they form one fig ure. and when swung the other way on the hinges, they form the other figure.
Editors
A hundred years ago, Henry Dudeney demonstrated a hinged dissection of an equilateral triangle to a square [ 10]. Since then, there have been an increas ing number of such dissections [2, 6, 13, 14, 20, 2 1 , 30, 33] culminating in a whole book on the subject [ 1 5] . An open problem is whether for any two polygons of equal area there is a swing hinged dissection in a finite number of pieces. Other types of hinges have also drawn attention. A twist hinge has a point of rotation on the interior of the line segment along vvhich two pieces touch edge-to-edge. It allows one piece to be flipped over relative to the other, using 1 80° rotation through the third di mension. Pieces A and B (with exag gerated thickness) are twist-hinged to gether in Figure 1 . The twist-hinged dissection o f a n el lipse to a heart (Figure 2) is a direct ap plication. We mark any piece that ends up turned over with an ·· * ·· on one side and a " * " on the other. A few isolated dissections [ 1 1 , 26, 28] were the only
Contributions are most welcome.
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Figure I . A twist hinge for pieces A and B .
Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford Univers ity,
Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-m a i l : [email protected] nford.edu
Figure 2. Twist-hinged dissection o f a n ellipse to a heart.
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examples of twist-hinged dissections prior to a more concerted search [13, 14, 1 5] . It is also open whether for any two polygons of equal area there is a twist-hinged dissection in a finite num ber of pieces. In this article we shall concentrate on extending the range of twist-dissec tions of one polygon to another. Com pletely unanticipated are our 10-piece twist-hinged dissection of a regular hep tagon to a square and our 6-piece (!) twist-hinged dissection of a regular hexagon to a square. Our other new twist-hinged dissections are surprising as well, using a variety of techniques that allow us to produce the first twist hinged dissections for certain pairs of figures or to reduce the number of pieces compared with previous twist hinged dissections. We shall go well be yond the approach of converting a swing-hinged dissection that is "hinge snug" to be twist-hinged [15]. Taken to gether, these wild dissections should make you flip! In the next section we shall review the technique of crossposing strips to produce dissections with some swing hinges and also a technique to replace swing hinges with twist hinges. In the remaining sections, we shall identify twist-hinged dissections of certain reg ular polygons, first to squares, then to equilateral triangles, and finally to reg ular hexagons. These dissections were first presented in March 2006 at the Sev enth Gathering for Gardner (G4G7), for which the twist-hinged dissection of the heptagon to the square was particularly apropos. Animations from that pres entation are posted on the webpage: http://www . cs.purdue.edu/homes/gnf/ book2/rni_anims.htrnl
Crossposing Strips and Converting Swing Hinges Let's first review two fundamental tech niques for creating swing-hinged and twist-hinged dissections. An effective method for dissecting one polygon to another is the strip technique [12, 22]. We first cut each polygon into pieces that we can rearrange to form a strip element. We then fit copies of this element to gether in a regular fashion, forming a strip that stretches infinitely in two op posite directions. We then crosspose the strip for one polygon on top of the strip for the other polygon, so that the com-
24
THE MATHEMATICAL INTELLIGENCER
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Figure 3. Crossposition of triangles and squares.
Figure 4. Henry Dudeney's swing-hinged dissection of a triangle to a square.
Figure 5. Stealing isosceles triangles in the dissection of a triangle to a square.
Figure 6. Twist-hinged dissection of a triangle to a square.
Figure 7. Intermediate configurations for a twist-hinged triangle to a
square.
Figure 8. Heptagon to square.
mon area is precisely the area of one of the two polygons, or is double that area. When the common area is double, then crossposition should have rotational symmetry about certain points. We see the strip technique applied to an equi lateral triangle and a square in Figure 3. The crossposition leads to the swing hinged dissection in Figure 4, which Henry Dudeney first described [10] . The pieces are swing-hingeable because every line segment in one strip that crosses a line segment in the other strip is either on the boundary of the strip or has the crossing at a point of rotational symmetry in its strip. The small dots in Figure 3 identify points of rotational symmetry where such crossing occurs. Two pieces connected by a swing hinge are hinge-snug if they are adjacent along different line segments in each of the figures formed, and each such line segment has one endpoint at the hinge [1 5). This property enables us to replace a swing hinge by two twist hinges, by stealing an isosceles triangle from each piece, unioning the two pieces together, and attaching the twist hinges to the new piece. We call this technique hinge con version . Using it on the swing-hinged dissection in Figure 4, we steal isosceles triangles in Figure 5, where the dashed edges indicate the bases of the triangles. We then glue the isosceles triangles to gether to produce, in Figure 6, a 7-piece twist-hingeable dissection of an equilat eral triangle to a square [15]. Intermediate configurations are in Figure 7. On the left, we see the lower left corner of the triangle flipped up, us ing a pair of twists. Then on the right, we see the lower right corner of the tri angle similarly flipped up. Flipping the right corner of what results will then give us the square. With the fundamental techniques of crossposition and hinge conversion as our base, we are now set to introduce additional techniques that will help to handle a variety of challenging problems.
Regular Polygons to Squares
Figure 9. Twist-hinged heptagon to square.
We shall first consider dissections of various regular polygons to squares, and immediately tackle a dissection that should be rather challenging: a heptagon to a square. There is an unhingeable dis section (solid lines in Figure 8) due to Gavin Theobald which uses 7 pieces [12, Figure 1 1 .30]. At first it doesn't look too
© 2007 Springer Science + Business Media,
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25
promising, since there are just two po sitions where we can place swing hinges: at the end of a shared edge between pieces E and F, and at the end of a shared edge between pieces C and E. Amazingly, we can convert this dis section to a 1 0-piece twist-hinged dissec tion. First, enlarge piece G by annexing an isosceles triangle from piece F. Next, we carve an isosceles trapezoid out of piece B. We twist-hinge this trapezoid with pieces A and C and with what re mains of piece B. Using the trapezoid al lows pieces A and C to be interchanged. Dotted lines indicate the isosceles trian gle and the isosceles trapezoid. Next, convert the swing hinge be tween pieces E and F. Then convert the swing hinge between pieces C and E . A remarkable byproduct of this latter conversion is that we can twist-hinge piece D with the little right triangle cre ated by the conversion. The resulting dissection is shown in Figure 9. We next attempt to find a twist hingeable dissection of a regular hexa gon to a square. There are any number of 5-piece dissections of a hexagon to a square, and even more 6-piece swing hingeable dissections. For inspiration we turn to the previous dissection of a heptagon, in which piece B is adjacent to pieces A and C, allowing those pieces to switch positions. Most promising is Gavin Theobald's 5-piece dissection (solid lines in Figure 10) [31]. Although no piece is swing-hinge able with any other, we can add twist hinges rather easily. Because piece B is an isosceles triangle, we can twist-hinge it with pieces A and C. Furthermore, we can cut an isosceles trapezoid out of piece D , and twist-hinge this piece with both pieces C and E. Finally, we can then twist-hinge the isosceles trapezoid with what remains of piece D , produc ing the 6-piece twist-hinged dissection in Figure 1 1 . With the heptagon and the hexagon being so cooperative, can we use sim ilar techniques to find a twist-hingeable dissection of a pentagon to a square? We have found a 7-piece unhingeable dissection (Figure 1 2) of a pentagon to a square that is similar in some ways to Theobald's previous dissections. Actu ally, it's difficult to see the seventh piece, piece B, which is a long and very
26
THE MATHEMATICAL INTELLIGENCER
Figure I 0. Partially twist-hingeable hexagon to a square.
Figure I I . Twist-hinged hexagon to a square.
Figure 1 2. Pentagon to square.
Figure 1 3 . Pentagon to a rectangle.
Figure 1 4. Twist-hinged pentagon to rectangle.
\
\ \ \
\ \
\
\ \" \
\ \ \
\ \ \
Figure I 5. Crossposition of a
\ \
\
{ 1 2/2} to a square.
Figure 1 6. Swing- and twist-hingeable
Figure 1 7. Twist-hinged
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\
{ 1 2/2} to a square.
{ 1 2/2} to a square.
thin right triangle that is to the right of pieces A and C in both the pentagon and the square. The long leg of the tri angle has length equal to the sidelength of the square, and the length of the short leg of the triangle is .002789 of the length of the side of the square. We can convert this dissection into an 1 1piece twist-hinged dissection, much the way that we did in Figure 9 . T o b e able t o visualize the process that produces the twist-hinged dissec tion, we will dissect a pentagon to a rectangle whose length is 1 . 1 559 times its width, as shown in Figure 1 3 . Pieces A and D can readily be swing hinged together, as can pieces C and E , and pieces E and F. Furthermore, there is an isosceles triangle that we can re move from piece F and combine with piece G that would allow us to twist the new piece G and what remains of piece F. Next, we can carve an isosce les trapezoid out of piece B, which we can twist-hinge with pieces A, C, and what remains of piece B. The isosceles triangle and the isosceles trapezoid are indicated by dotted lines in Figure 1 3 . Finally, w e convert the three swing hinges, producing the 1 1 -piece twist hinged dissection in Figure 1 4 . The cor responding twist-hinged dissection of the pentagon to the square is com pletely analogous. We will tackle one final dissection to a square, namely that of a { 1 2/2}, which is a 1 2-pointed star with every second vertex connected. This will involve somewhat different techniques from the previous three dissections. The swing hingeable dissection in [ 1 5 , Figures 1 1 .53 and 1 1 .54] is not hinge-snug. However, we can create a strip element as a result of doing certain twists, and from its crossposition (Figure 1 5) with a square strip get a 1 2-piece swing- and twist-hinged dissection that is hinge snug (Figure 16). The six pieces A through F are con nected by swing hinges, and we can convert each of the five swing hinges to a twist hinge. This would give a 1 7piece twist-hinged dissection. We can do better by cutting a zigzag piece out of pieces C and D by taking a rectangle from C and a rectangle from D and combining them. We use this one piece to twist-hinge pieces B, C , D, and E together. As before, we finish by con-
© 2007 Springer Science+ Business
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27
verting the swing hinge between pieces A and B, and the swing hinge between pieces E and F. We thus get a 1 5-piece twist-hinged dissection (Figure 17).
Regular Polygons to Equilateral Triangles Let's next attempt twist-hinged dissec tions of regular polygons to equilateral triangles. Gavin Theobald [31] gave an unhingeable dissection of a regular hep tagon to an equilateral triangle which is based on a crossposition very similar to that shown in Figure 18. The resulting 8-piece dissection is given in solid lines in Figure 19. Note that piece F is an equilateral triangle of one quarter the area of the given equilateral triangle. We can convert this dissection to a 1 5-piece twist-hinged dissection. First, enlarge the small isosceles triangle (piece B) by annexing an isosceles triangle from piece A. Next, convert the swing hinge between pieces A and C and then con vert the swing hinge between pieces C and E. As we have seen previously, a byproduct of the latter conversion is that we can twist-hinge piece D with the lit tle right triangle created by that conver sion. Then convert the two swing hinges between piece F and pieces E and G. To handle piece H in the heptagon, we swing-hinge it to piece E and then convert the swing hinge. To make room for this piece in the full-size equilateral triangle, we cut an obtuse triangle out of piece A. Finally, we use a rectangu lar piece to twist-hinge the new obtuse triangle onto piece F. The new obtuse triangle takes the original place of piece H in the equilateral triangle. We see the resulting 1 5-piece twist-hinged dissec tion in Figure 20. We find the next dissection, of a reg ular pentagon to a triangle, a bit easier. Lindgren [22] observed how to use a crossposition to derive Goldberg's dis section of a pentagon to a triangle [19]. We see a slight variation of that cross position in Figure 2 1 . The triangle strip is what Lindgren called a T-strip, and small dots indicate the points of 2-fold rotational symmetry. Edges of pieces in the strips cross only at strip boundaries or at points of symmetry. Thus pieces A, B , C, and D in Figure 22 can be swing-hinged. Pieces E and F can be swing-hinged as well. To connect all of the pieces with
28
THE MATHEMATICAL INTELLIGENCER
'
'
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'
'
'
'
'
'
'
'
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Figure 1 8. Crossposition of a heptagon to triangle.
Figure 1 9. Heptagon to triangle.
Figure 20. Twist-hinged heptagon to triangle.
'
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,
---
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Figure 2 1 . Crossposition for a pentagon to a triangle.
Figure 2 2 . Partially-hingeable pentagon to a triangle.
Figure 23. Twist-hinged pentagon to a triangle.
hinges, we would like to have piece E connect with the top of piece D, rather than touch the bottom of piece D. We can accomplish this by cutting an isosceles trapezoid from D , which in terchanges top and bottom of the left side of piece D, using a twist hinge. Besides the twist hinge, there are five swing hinges, each of which converts
to an additional piece plus two more twist hinges. This produces the 1 2-piece twist-hinged dissection in Figure 23.
Regular Polygons to Regular Hexagons Finally, let's attempt twist-hingeable dis sections of regular polygons to regular hexagons. Gavin Theobald [31] gave an
8-piece unhingeable dissection of a reg ular heptagon to a hexagon. To be able to produce a twist-hinged dissection, we shall use his heptagon strip with a hexagon strip in the crossposition in Figure 24. This gives the 9-piece par tially-hinged dissection (solid lines) in Figure 25. We can convert this dissection to a 1 5-piece twist-hinged dissection. First, enlarge the small isosceles triangle (piece G) by annexing an isosceles tri angle from piece F. Next, convert the swing hinge between pieces E and F and then convert the swing hinge be tween pieces C and E. Again, a byprod uct of the latter conversion is that we can twist-hinge piece D with the little right triangle created by that conversion. Finally, convert the four remaining swing hinges, between pieces A and B , B and C, F and H , and H a n d I . W e see the resulting 1 5-piece twist-hinged dis section in Figure 26. We next dissect an {8/3} to a hexa gon. (An {8/3} is an 8-pointed star with every third point connected.) There is a 10-piece strip dissection of an {8/3} to a hexagon. We use the same strip for the hexagon as we did in the preced ing dissection. We form the strip for the {8/3} by cutting off four of its points and nestling them between the other four. The crossposition (Figure 27) treats the hexagon strip as a T-strip and produces a partially-hingeable dissection (solid lines in Figure 28). Note that pieces A and B swing-hinge together, as do pieces B and C, and C and D . Also, pieces D and I twist-hinge together, as do pieces B and ] . Assuming we have positioned the hexagon strip appropriately, we can then steal small isosceles right triangles from pieces E and F and attach them to piece B, allowing us to twist-hinge what remains of E and F onto B. We can steal the same-size isosceles right triangles from pieces G and H. We need an in termediate piece that results from merg ing the two isosceles triangles with a rectangle from piece D . This new piece effects the appropriate shift of pieces G and H relative to piece D . We get a n 1 1-piece swing- and twist hinged dissection that is still hinge-snug with three swing hinges. Converting the the three swing hinges, we get a 14-piece twist-hinged dissection (Figure 29).
© 2007 Springer Science+ Business Media, Inc . . Volume 29, Number 4, 2007
29
I I I
I I I
Figure 24. Crossposition for a heptagon to a hexagon.
Figure 2 5 . Partially-hingeable heptagon to a hexagon.
Figure 26. Twist-hinged heptagon to a hexagon.
30
THE MATHEMATICAL INTELUGENCER
....
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Figure 29. Twist-hinged
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to a hexagon.
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to a hexagon.
to a hexagon.
© 2007 Springer Science+Business Media, Inc., Volume 29, Number 4, 2007
31
Our last dissection is of a regular pentagon to a hexagon. There is a swing-hinged dissection of a hexagon to a pentagon in [ 1 5 , Solution 1 1 .2] which uses 10 pieces and is hinge-snug. A direct conversion would give a 19piece twist-hinged dissection. However, we can do better if we use the cross position from [22, Appendix D], as in Figure 30. We thus get the 7-piece dissection (solid lines) in Figure 3 1 . We then need to hinge piece G (from the pentagon) onto the rest of the pieces. We do this as follows: We split an isosceles trape zoid from piece F and place a twist hinge between the trapezoid and what remains of piece F. If piece G is swing hinged to the trapezoid, then we are able to bring piece G from its position in the pentagon to the appropriate po sition in the hexagon. Noting that pieces A and B form an isosceles trapezoid in the hexagon, we flip these pieces over in the hexagon and connect piece B to piece C with a twist hinge. There are five swing hinges, between pieces A and B, C and D, C and E, E and F, and G and the trapezoid. Each of the five swing hinges con verts to an additional piece plus two additional twist hinges, producing a 13piece twist-hinged dissection (Figure 32). Note that the new triangle between pieces C and E is so small that there is no room to mark one side with an "*" and the other side with a " * " .
'
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Figure 30. Crossposition for a hexagon to a pentagon.
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Figure 3 I . Partially-hingeable hexagon to a pentagon.
Conclusion We have identified some surprising new twist-hinged dissections, using special purpose techniques. Further examples and adaptations of these techniques appear in a companion paper [17]. All of these dissections, and a few others, are summarized in Table 4 . 1 of [16, chap. 4], in which it is discussed how to convert them into yet another type of hinged dissection, namely, piano hinged dissections. REFERENCES
Figure 32. Twist-hinged hexagon to a pentagon.
[1 ] Abu'I-Wata.' ai-BOzjanT. Kitab fTma yahtaju al-sani' min a' mal al-handasa (On the
deney
Geometric Constructions Necessary for the
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Artisan). Mashdad: Imam Riza 37, copied
Discrete
in the late 1 Oth or the early 1 1 th century.
Japanese Conference, JCDCG'98, LNCS,
Dublin, 1 889. [4] Anonymous. FT tadakhul al-ashkal al-mu
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volume 1 763, pages 1 4-29. Springer-Ver
tashabiha aw al-mutawafiqa (Interlocks of
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THE MATHEMATICAL INTELLIGENCER
dissection and
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Computational
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[5] Farkas Bolyai. Tentamen juventutem. Typis Collegii Reformatorum per Josephum et Simeonem Kali, Maros Vasarhelyini, 1 832. [6] Donald L. Bruyr. Geometrical Models and Demonstrations.
J. Weston Walch, Port
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Inquirer, October 23, 1 898-1 901 .
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Vorlesungen
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in nickel-plated aluminum, limited edition
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of 80 produced by Bayer, in Germany,
[1 7] Greg N. Frederickson. Unexpected twists in geometric dissections. Graphs and Combi
Geschichte der Mathematik, volume 1 . B .
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in the medieval Islamic world. Historia
[1 8] P. Gerwien. Zerschneidung jeder beliebi
G . Teubner, Stuttgart, third edition, 1 907.
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gen Anzahl von gleichen geradlinigen
[8] Henry E. Dudeney. Perplexities. Monthly
Figuren in dieselben StUcke. Journal fOr
puzzle column in The Strand Magazine from
die reine und angewandte Mathematik
forming geometric shapes. U . S . Patent
May, 1 9 1 0 through June, 1 930.
(Grelle 's Journal), 1 0:228-234 and Taf. I l l ,
4,392,323, 1 983.
[9] Henry E. Dudeney. Puzzles and prizes. Col
[29] Aydin Sayili. Thabit ibn Ourra's general
1 833.
umn in the Weekly Dispatch, April 1 9 ,
ization of the Pythagorean theorem. Isis,
[1 9] Michael Goldberg . Problem E972: Six
1 896-March 27, 1 904.
51 :35-37, 1 960.
piece dissection of a pentagon into a tri
[1 0] Henry Ernest Dudeney. The Canterbury Puzzles and Other Curious Problems. W.
Heinemann , London, 1 907.
angle. American Mathematical Monthly,
adapted to define a plurality of objects or shapes. U.S. Patent 4,542,63 1 , 1 985. [ 1 2] Greg N . Frederickson. Dissections Plane & Fancy. Cambridge University Press,
New York, 1 997 .
[30] H . M. Taylor. On some geometrical dis sections.
59: 1 06-- 1 07, 1 952. [20] Anton Hanegraaf. The Delian altar dissec
[ 1 1 ] William L. Esser, Ill. Jewelry and the like
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[22] Harry Lindgren. Geometric Dissections. D .
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[23] (Mr.) Lowry. Solution to question 269,
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[2 1 ] H . Lindgren. A quadrilateral dissection . 1 960.
Messenger
35:81 -1 01 ' 1 905. [3 1 ] Gavin Theobald. Geometric dissections.
Van Nostrand Company, Princeton, New
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[28] Erno Rubik. Toy with turnable elements for
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Glendinning, London, 1 8 1 4.
Mathematical Sketch and Model Book.
Educational Publishers, St. Louis, 1 949.
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[24] Sam Loyd . Weekly puzzle column in Tit
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uing into 1 897.
e-mail: [email protected]
West Lafayette, Indiana 47907
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33
Ricc i Flow and the Poincare Conjecture SIDDHARTHA GADGIL AND HARISH SESHADRI
he field of Topology was born out of the realisation that in some fundamental sense, a sphere and an el lipsoid resemble each other but differ from a torus. A striking instance of this can be seen by imagining water flowing smoothly on these. On the surface of a sphere or an ellipsoid (or an egg), the water must (at any given in stant of time) be stationary somewhere. This is not so in the case of the torus. More formally, in topology we study properties of (classes of) spaces up to certain equivalence relations. For instance, one studies topological spaces up to homeomor phism, or smooth manifolds up to diffeomorphism. A fun damental problem in topology is thus to classify a class of topological spaces, say smooth manifolds of a given di mension, up to the appropriate equivalence relation. The first interesting case is that of dimension 2, i.e., sur faces. In the case of surfaces (more precisely closed sur faces), there are two infinite sequences of topological types. The first sequence, the so-called orientable surfaces, con sists of the sphere, the torus, the 2-holed torus, the 3-holed torus, and so on (see figure 1). The non-orientable surfaces are obtained from the sphere by removing interiors of dis joint discs and gluing Mobius bands to the resulting bound ary components-with the surfaces differing according to how many discs have been replaced by Mobius bands. One would like to have a similar classification in all dimensions. However, due to fundamental algorithmic issues, it is im possible to have such a list in dimensions 4 and above. Manifolds of dimension 3 are also too complex to be re duced to such a list. Nevertheless, one may hope that some features of the classification of surfaces continue to hold in higher di mensions. I n particular, there is a simple way to charac terise the sphere among surfaces. If we take any curve on the sphere, we can shrink it to a point while remaining on
T
34
THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Mecia. Inc.
the sphere. A space with this property is called simply con nected. A torus is not simply connected, as a curve that goes around the torus cannot be shrunk to a point while remaining on the torus. In fact, the sphere is the only sim ply connected surface. In 1 904, Poincare raised the question whether a similar characterisation of the (3-dimensional) sphere holds in di mension 3. This has come to be known as the Poincare conjecture. It can be formulated as follows.
CONJECTURE (POIN CARE) Any closed, simply connected, smooth 3 -manifold is diffeomorphic to the 3 -dimensional sphere s3. Note that by a theorem of Moise, every 3-manifold has a unique PL (piecewise-linear) structure, that is, it can be homeomorphic to only one complex of 3-dimensional poly topes, up to equivalence. Further, work of Kervaire-Milnor, Munkres, Hirsch, Smale, and others gives a very good un derstanding of the relation between PL and smooth struc tures, which in particular implies, using the theorem of Moise, that every 3-manifold has a unique smooth struc ture. Hence the above conjecture is equivalent to the state ment that every closed, simply connected topological 3manifold is homeomorphic to the 3-dimensional sphere 53. As topology exploded in the twentieth century, several attempts were made to prove this (and some to disprove it). However, at the turn of the millennium this remained
Figure I . The first three orientable surfaces.
unsolved. Surprisingly, the higher-dimensional analogue turned out to be easier and was solved by Smale and Freed man. For a brief history of the Poincare conjecture, see [8] . In 2002-2003, three preprints ([10] , [ 1 1] , and [ 1 2]), rich in ideas but frugal with details, were posted by the Rus sian mathematician Grisha Perelman, who had been work ing on this in solitude for seven years at the Steklov Insti tute. These were based on the Ricciflow, which had been introduced by Richard Hamilton in 1 982. Hamilton had de veloped the theory of Ricci flow through the 1 980s and 1 990s, proving many important results and developing a programme [6] which, if completed, would lead to the Poin care conjecture and much more. Perelman introduced a se ries of highly original ideas and powerful techniques to complete enough of Hamilton's programme to prove the Poincare conjecture. It has taken three years for the mathematical community to assimilate Perelman's ideas and expand his preprints into complete proofs. Recently, a book [9] containing a com plete and mostly self-contained proof of the Poincare con jecture has been posted. An earlier set of notes which filled in many details in Perelman's papers is [7]. Another article [2] regarding the proof of the geometrisation conjecture (see below) has also appeared (see also its erratum [3] and a note from the editors in the same issue). In this article we attempt to give an exposition of Perel man's work and the mathematics that went into it. ACKNOWLEDGMENTS
It is our pleasure to thank Kalyan Mukherjea for several help ful comments that have considerably improved the exposition, and Gerard Besson for inspiring lectures on Perelman's work. We also thank C. S. Aravinda, Basudeb Datta, Gautham Bha rali, and Joseph Samuel for helpful comments.
Why the Poincare Conjecture Is Difficult Both the plane and 3-dimensional space are simply con nected, but with an important difference. If we take a closed, embedded curve in the plane (i.e., a curve which does not cross itself), it is the boundary of an embedded disc. However, an embedded curve in 3-dimensional space may be knotted (see figure 2). As we deform a knotted curve to a circle, along the way it must cross itself.
Figure 2. A knotted curve.
Thus, an embedded curve in a simply connected 3-man ifold M may not bound an embedded disc. Furthermore, such a curve may not be contained in a ball B in M. While embedded discs are useful in topology, immersed discs (i.e. , discs that cross themselves) are not. I t is this which makes it hard to use the hypothesis of simple connectivity, and thus to prove the Poincare conjecture (in dimension 3). The analogue of the Poincare conjecture in dimensions 5 and above is easier than in dimension 3 for a related rea son. Namely, any (2-dimensional) disc in a manifold of di mension at least 5 can be perturbed to an embedded disc, just as a curve in 3-dimensional space can be perturbed so that it does not cross itself. What made Perelman's proof, and Hamilton's pro gramme, possible was the work of Thurston in the 1 970s, where he proposed a kind of classification of 3-manifolds, the so called geometrisation conjecture [ 1 3] . Thurston's geometrisation conjecture had as a special case the Poin care conjecture, but being a statement about all 3-mani folds, it could be approached without using the hypothe sis of simple connectivity. However, most of the work on geometrisation in the 1 980s and 1 990s was done by splitting into cases, so to prove the Poincare conjecture, one was still compelled to use the simple connectivity hypothesis. An exception to this was Hamilton's programme. Interestingly, Perelman found a nice way to use simple connectivity within Hamilton's programme, which simplified his proof of the Poincare con jecture (but not of the full geometrisation conjecture). To introduce Hamilton's approach, we need to refor mulate the Poincare conjecture as a statement relating topol ogy to Riemannian geometry: namely, that a compact, sim-
SIDDHARTHA GADGIL received his PhD from
HARISH
the California Institute of Technology and then
work at the Indian Institute of Technology Kan
taught at SUNY Stony Brook before returning
pur, and received his PhD from SUNY Stony
to India. His main area of research is low
Brook, USA He works in differential geometry.
dimensional topology. Department of Mathematics Ind ian I nstitute of Science
SESHADRI did
undergraduate
He is now an assistant professor at the Indian In stitute of Science. Department of Mathematics
Bangalore 5600 1 2
I ndian Institute of Science
India
Bangalore 5 600 1 2
e-mail: [email protected]
his
India e-mail: [email protected] isc.ernet.in
© 2007 Springer Science+ Business Media, Inc., Volume 2 9 , Number 4, 2007
35
ply connected 3-manifold admits an Einstein metric. To make sense of this we need a summary of some Riemannian geometry.
Some Riemannian Geometry Intrinsic differential geometry and curvature
I n intrinsic differential geometry, we study the geometry of a space M in terms of measurements made within the space M This began with the work of Gauss, who was involved in surveying large areas of land where one had to take into account the curvature of the earth. Even though the earth is embedded in 3-dimensional space, the measurements we make do not take advantage of this. Concretely, one has the question whether one can make a map of a region of the earth on a flat surface (a piece of paper) without distorting distances (allowing all distances to be scaled by the same amount). This is impossible, as can be seen by considering the area of the region consist ing of points with distance at most r from a fixed point P on the surface M. The area in case M is a sphere can be seen to be less than 7T'r2, which would be the area if we did have a map that did not distort distances. In fact, for r small the area of the corresponding region on any surface is of the form 7T'r2(1
-
.!!_ r2 12
+
. . . ) , and K is
called the Gaussian curvature a t P. Intrinsic differential geometry gained new importance because of the general theory of relativity, where one stud ies curved space-time. Thus, we have manifolds with dis tances on them that do not arise from an embedding in some !R n . This depended on the higher-dimensional, and more sophisticated, version of intrinsic differential geome try developed by Riemann. Today, intrinsic differential geometry is generally referred to as Riemannian geometry. To study Riemannian geometry, we need to understand the analogues of the usual geometric concepts from Eu clidean geometry as well as the new subtleties encountered in the more general setting. Most of the new subtleties are captured by the curvature. Tangent spaces
Let M be a k-dimensional manifold in !R n and let p E M be a point. Consider all smooth curves y : ( - 1 , 1) � M with y(O) p. The set of vectors v y'(O) for such curves y gives the tangent space TpM. This is a vector space of dimension k contained in fR n . For example, the tangent space of a sphere with centre the origin at a point p on the sphere con sists of all vectors perpendicular to the radius ending at p. If a particle moves smoothly in M along the curve a(t), its velocity V(t) = a'(t) is a vector tangent to M at the point a(t) , i . e . , V (t) E Ta(t)M =
=
Riemannian metrics
I f a : (a, b) � [Rn is a smooth curve, then its length is given by l(a) J!; lla ( t) ll dt. In Riemannian geometry we consider manifolds with distances that are given in a similar fashion in terms of inner products on tangent spaces. A Riemannian metric g on M is an inner product speci fied on TpM for each p E M. Thus, g refers to a collection =
36
'
THE MATHEMATICAL INTELLIGENCER
of inner products, one for each TpM. We further require that g varies smoothly in M. For a point p E M and vectors V, W E TpM, the inner product of V and W corresponding to the Riemannian metric g is denoted g( V, W). A Riemannian manifold (M,g) is a manifold M with a Riemannian metric g on it. Recall that near any point in M, a small region U C M can be given a system of local coor dinates x1 , . . . , Xk· At every point p in U we denote the corresponding coordinate vectors by el ' ek; then the inner product on TpU is determined by the matrix giJ g(e1, e1). This is a symmetric matrix. The first examples of Riemannian manifolds are mani folds M C !R n, with the inner product on TpM taken to be the restriction of the usual inner product on !R n. This met ric is called the metric induced from fR n. A second important class of examples are product met rics. If (M,g) and (N,h) are Riemannian manifolds, we can define their product (M X N, g E9 h) . The points of M X N consist of pairs (x,y), with x E M and y E N The tangent space 1ix,y)(M X N) of the product consists of pairs of vec tors ( U, V) with U E TxM and V E TyN The inner product (g E9 h) is given by 0
0
0
'
=
(g E9 h)(( U, V), ( U', V')) = g( U, U') + h ( V, V ) '
We can identify the space of vectors of the form ( U,O) (respectively (0, V)) with TxM (respectively, TyN). Distances and isometries
Given a pair of points p, q E M in a Riemannian manifold (M,g), the distance d(p,q) between the points p and q is the minimum (more precisely the infimum) of the lengths of curves in M joining p to q. For p E M and b > 0, the ball of radius r in M with cen tre p is the set Bp( r) of points q E M such that d(p,q) < r. Note that this is not in general diffeomorphic to a ball in Euclidean space. Two Riemannian manifolds (M,g) and (N, h) are said to be isometric if there is a diffeomorphism from M to N so that the distance between any pair of points in M is the same as the distance between their images in N In Rie mannian geometry, we regard two isometric manifolds as the same. Geodesics and the exponential map
Geodesics are the analogues of straight lines. A straight line segment is the shortest path between its endpoints. A curve with constant speed that minimises the distance between its endpoints is called a minimal geodesic. More generally, a geodesic is a smooth curve with con stant speed that locally minimises distances; i.e., it is a smooth function y : (a, b) � M such that II'Y'(t)ll is constant and having the following property: for any p y(to), there is an E > 0 so that the segment of the curve y from time to E to to + E has minimal length among all curves join ing y(to E) to y(to + E). Let p E M be a fixed point. Then we can find r > 0 such that if d(p,q) < r, then there is a unique minimal geodesic y joining p to q. This follows from the fact that a curve is ge odesic if and only if it satisfies a certain second-order (non linear) ordinary differential equation (ODE), as explained in =
-
-
Appendix A. The existence and uniqueness of solutions for ODEs, together with some geometric arguments, then give us the corresponding statements for geodesics. We can parametrise y (i. e . , choose the speed along y) so that y(O) = p and '}'(1) = q. Then the initial velocity y'(O) gives a vector in TpM with norm less than r. This gives a one-to-one correspondence between points q in M with d(p,q) < r and vectors V E TpM with norm less than r. The point that corresponds to the vector V is denoted by expp( V), and this correspondence is called the exponential map. As an example, consider the exponential map at the north pole of the 2-sphere p. This map is one-to-one on BoC7r) and it maps I3o(7r) to the sphere minus the south pole. The supremum of the values of r for which there are unique geodesics as above is called the injectivity radius at p E M. This term comes from its equivalent description as the largest r such that expp is injective on Bp(r). Sectional, Ricci, and scalar curvatures
Let p E M be a point and let g C TpM be a 2-dimensional subspace. Choose an orthonormal basis { U, V) of g and con sider the following family of closed curves in M:
Cr(8)
=
expp (r cos(8)
U+
r sin(8) V), 8 E [0,27Tl.
It can be proved that the length of expansion:
Cr has the following
REMARK: It is important to note that in local coordinates these curvature quantities can be expressed in terms of giJ and its first and second derivatives. This follows from the more standard way of defining curvature in terms of the Levi-Civita connection associated to g. See Appendix A for details. The fact that curvature can be so expressed provides a link between Riemannian geometry and partial-differential equations. We consider some examples. This is just !Rn with the usual inner product. In this case, all the sectional curvatures are zero. Hence so are the Ricci tensor and the scalar curvature.
(1) Euclidean space. (2)
Sphere sn(r) of radius r with the metric induced from
jR n+ l . In this case, all sectional curvatures are equal
( n - 1)r - 2g(U, V), and R(p) point p. Here g( · ; , ) is (the restric tion of) the standard inner product in !Rn (3) There is an analogue of Example 2 , called hyperbolic space, for which the sectional curvature is - r- 2. The underlying manifold can be taken to be !Rn. We will not describe the metric since we won't need it. r-2, Ric(U, V) ( n - 1) r-2 for any to
=
=
We have the following important converse of the above examples: Let (M,iJ be a simply-connected complete Rie mannian manifold of constant sectional curvature k. Then M is isometric to Euclidean space, the sphere of radius Vi/k, or hyperbolic space according as k 0, k > 0, or k < O. =
We define the sectional curvature of (M,lf) along g to be the number KCp,g) above. Other notations for sectional curvature include Kg(p,g) to clarify what metric we consider, and K(p, U, V> to indicate that g is the linear span of U and V In the latter notation, we put K(p, U, V> 0 if U and V are linearly dependent. We often omit the point p in the notation if it is clear from the context. Averaging all the sectional curvatures at a point gives the scalar curvature ap). More precisely, let {E1 , . . . , En) be an orthonormal basis of TpM. Then we define =
ap) =
L i,j
K(E;,Ej).
There is an intermediate quantity, called the Ricci ten which is fundamental in our situa tion. The Ricci tensor Ric(U, V) at a point p E M depends on a pair of vectors U and V in TpM. Further, it is linear in U and V and is symmetric (i.e., Ric(U, V) Ric(V, U)). It is
sor or Ricci curvature
=
defined as follows: If U is any unit vector in TpM, then we extend U to an orthonormal basis { U, E , . . . , E ) and define
2
n
By linearity, for a general vector aU, with U a unit vector, a2Ric(U, U). Further, by linearity and symme try, if U and V are any two arbitrary vectors in TpM, then we put Ric(U, V) = (Ric(U+ V, U+ V) - Ric(U- V, U- V)) in analogy with the formula
Ric(aU, aU)
=
±
(a + b)2 - (a - b)2 = 4ab.
(4) A product Riemannian manifold (M X N, g g1 EB gz): If g is a plane in Tp(M X N) that is tangent to M (re spectively, N), then K(p,g) K1(g) (respectively, K2 (g)). Here K1 and K denote the sectional curvatures with 2 respect to g1 and g . On the other hand, if g is the span 2 =
=
of a vector tangent to M and one tangent to N, then 0. (5) As a special case of the above, consider a surface M which is the product of two circles, possibly of differ ent radii, with the product metric. Then the tangent plane at any point is spanned by a vector tangent to the first circle and one tangent to the second circle. Hence the sectional curvature of M at any point is zero. (6) Another example of a product metric that we need is that on M 52 X R In this case, the sectional curva ture K(x,g) is 1 if g is the tangent plane of 52 and 0 if g contains the tangent space of R
K(g)
=
=
Manifolds with non-negative sectional curvature
We have defined sectional curvature in terms of the growth of lengths of circles under the exponential map. In other words, sectional curvature measures the divergence of ra dial geodesics. In particular, if a Riemannian manifold has non-negative curvature, geodesics do not diverge faster than in Euclid ean space. This has strong consequences for the geometry and topology of these manifolds. In fact, if a simply con nected 3-manifold (M,lf) has non-negative sectional curva ture, it has to be diffeomorphic to one of IR3, S3, and
52 X IR.
© 2007 Springer Science+ Business Media, Inc., Volume 2 9 , Number 4, 2007
37
Scaling and curvature
Suppose (M,g) is a Riemannian manifold and c > 0 is a con stant. Then the sectional curvature K' of the Riemannian manifold (M, elf) is related to the sectional curvature K of (M,g) by
for every point p E M and every tangent plane g C TpM at that point. Note that if c is large, then K' is small. Hence, given a compact Riemannian manifold (M,g), we can always choose c large enough so that (M, cg) has sectional curvatures ly ing between - 1 and 1 .
Einstein Metrics and the Poincare Conjecture An Einstein metric is a metric of constant Ricci curvature. More precisely, g is said to be an Einstein metric if, for all p E M and U, VE TpM, we have
To get a feeling for the analytical properties of this equa tion, we first consider the simpler case of the heat equa tion, which governs the diffusion of heat in an insulated body. The heat equation is
au = l:, u. at The temperature i n an insulated body becomes uniform a s time progresses. Further, the minimum temperature o f the insulated body increases (and the maximum temperature decreases) with time. This latter property is called a max
imum principle. To see the relation of the Ricci flow with the heat equa tion, we use special local coordinates called harmonic co ordinates (i. e . , coordinates {xi} such that the functions xi are harmonic: /:,xi = 0) . We can find such coordinates around any point in a Riemannian manifold M In these co ordinates we have
Ric( U, V) = a g( U, V), for some a E IR. In general relativity, one studies an action functional on the space of Riemannian metrics called the Einstein-Hilbert action, which is the integral of the scalar curvature of a metric. Einstein metrics are the critical points of this functional among Riemannian metrics on a manifold with fixed volume. To relate Einstein metrics to the Poincare conjecture, one notes that an Einstein metric g on a 3-manifold necessarily has constant sectional curvature (in all dimensions metrics of constant sectional curvature are Einstein metrics). Hence, by the remark after Example (3) above, one concludes that if (M,g) is closed, simply connected, and Einstein, then (M,g) is isometric to S3 with a multiple of the usual met ric. Note that we can rule out Euclidean and hyperbolic space because they are not closed. In particular, M is dif feomorphic to S3. Hence the Poincare conjecture can be formulated as say ing that any closed, simply connected 3-manifold has an Einstein metric. More generally, Thurston's geometrisation conjecture says that every closed 3-manifold can be de composed into pieces in some specified way so that each piece admits a so-called locally homogeneous metric. This means that any pair of points in the manifold have neigh bourhoods that are isometric. Metrics of constant sectional curvature are locally homogeneous.
Hamilton's Ricci Flow In the 1980s and 1 990s Hamilton built a programme to prove geometrisation, beginning with a paper [5] where he showed that if a 3-manifold has a metric with positive Ricci curvature then it has an Einstein metric. By positive Ricci curoature we mean that if p E M and if U E TpM is non zero, then Ric( U, U) > 0 . Hamilton's approach was t o start with a given metric g and consider the 1-parameter family of Riemannian metrics g(t) satisfying the Ricci flow equation (1)
"*
=
-2 Ric(t),
g(O) = g,
where Ric(t) is the Ricci curvature of the metric g( t).
38
THE MATHEMATICAL INTELLIGENCER
where Q is an expression involving g and the first partial derivatives of g, and where RiciJ = Ric(e i,e1). Hence the Ricci flow resembles the heat flow
agiJ = l:,gif, at
leading to the hope that the metric will become symmetric (more precisely, the Ricci curvature will become constant) as time progresses. However, there is an extra term (tg, a g) of lower order. Such a term is called the reaction term and equations of this form are known as reaction-diffusion equations. In order to understand such an equation, one needs to understand both the nature of the reaction term and conditions that govern whether the reaction or the dif fusion term dominates. Let us consider some examples: If g is the induced metric on the sphere s3 of radius 1 , then g(t) (1 - 4 t)g is the solution to ( 1 ) . Note that the radius of (S3, g(t)) is =
V1
-
4t and the sectional curvatures are
1
t � 4 , these curvatures blow up.
1
� 4t .
As
More generally, if g(t) is an Einstein metric, the Ricci flow simply rescales the metric. In fact, if Ric = ag, then g(t) = (1 2at)g satisfies (1). Note that (M,g(t)) shrinks, ex pands, or remains stationary depending on whether a > 0, a < 0, or a = 0. On the other hand, if the metric is fixed up to rescaling by the Ricci flow, then it is an Einstein metric. Let (M1 X M2, g1 EB g2) be a product Riemannian mani fold. Then the Ricci flow beginning at g1 EB g2 is of the form g(t) = g1(t) EB g2(t) , where g1(t) and gz(t) are the flows on M1 and M2 beginning with g1 and g2: Ricci flow pre serves product structures. In particular, the flow beginning with the standard product metric g0 EB g1 on S2 X IR is g(t) = (1 - 2 t)g0 EB g1 , i . e . , the S2 shrinks while the IR direction does not change. This example is crucial for understand ing regions of high curvature along Ricci flow. We now consider some analytical properties of Ricci flow. One of the first results proved by Hamilton was that, given any initial metric g(O) on a smooth manifold M, the -
Ricci flow equation has a solution on some time interval [O,E) with E > 0. Furthermore, this solution is unique. It fol lows that a solution to the equation with initial metric g(O) exists on some maximal interval [0, T), with T either finite or infinite. Further, if T is finite then the maximum of the absolute value of the sectional curvatures tends to oo as we approach T The main idea of Hamilton's programme is to evolve an arbitrary initial metric on a closed simply connected 3-man ifold along the Ricci flow and hope that the resulting met ric converges, up to rescaling, to an Einstein metric. Hamil ton showed that this does happen when g has positive Ricci curvature. It is convenient to analyse separately the cases where the maximal interval of existence [0, T) is finite and infinite. It turns out that if the manifold is simply connected, then this time interval is finite. In particular, the curvature blows up in finite time on certain parts of the manifold. The central issue in Hamilton's programme was to un derstand, topologically and geometrically, the parts of the manifold where curvature blows up along the Ricci flow.
Curvature Pinching The first major steps in understanding the geometry near points of large sectional curvature were due to Hamilton and Ivey, using maximum principles. In the simple case of the classical heat equation, the maximum principle implies that if the temperature is ini tially greater than a constant a at all points in the mani fold, then this continues to hold for all subsequent times. In the case of the Ricci flow, we have a similar maximum principle for the scalar curvature. This is because the scalar curvature also satisfies a reaction-diffusion equation with the reaction term positive. Indeed, the evolution equation for scalar c urvature is
where /':,. denotes the Laplacian with respect to the metric (see Appendix A) and IRiq denotes the norm of the Ricci tensor. An immediate consequence of this equation is that if R > 0 on M at t = 0, then R > 0 at any subsequent t E (0, T]. This is seen as follows: Let Xm(t) E M be such that RCxm(t),t) = minxEM R(x, t) , where R(x, t) denotes the scalar curvature of (M,g(t)) at x. Assume, for the sake of brevity, that Xm(t) varies smoothly in t. We then see, from the evolu-
-
aR . . cXmc t),t) � 0. This implies that the t10n equat10n, th at at
positivity of R is maintained. More generally, scalar curva ture R is bounded below, along the Ricci flow. Hamilton also developed a maximum principle for ten sors. Using this, Hamilton and Ivey independently obtained an inequality for sectional curvature, which we mention and use in the next section. A consequence of the Hamilton Ivey inequality is that if, at a point p E M, there is a 2-plane E TpM for which the absolute value is large, then Hp) is large. Furthermore, the Hamilton-Ivey inequality im plies the following crucial fact: At a point of high curva-
g
IKCp,g)l
ture, there are 2-planes of positive sectional curvature much bigger than any negative sectional curvature. All these maximum principles amount to showing and using positivity properties of the reaction term.
Blow-up and Convergence of Riemannian Manifolds To study points of high curvature, we use a version of a classical technique in PDEs called blow-up analysis. Roughly speaking, this involves rescaling manifolds near points of high curvature. We then study the limit points, if any, of the rescaled manifolds. For this approach to be fruit ful , we need a theorem guaranteeing the existence of the limit points, i . e . , a compactness theorem. Such a theorem, addressing convergence of Riemannian manifolds, was proved in the 1 970s by M . Gromov, R. Greene, H . Wu, and others, following the pioneering work of J. Cheeger. First we need to define the distance between Riemann ian manifolds with respect to which we consider conver gence. This is the so-called Lipschitz distance, on the space of all Riemannian manifolds of a given dimension with given basepoints. It is defined as follows. Let (M,g) and (N, h) be Riemannian n-manifolds and let x E M and y E N be chosen as basepoints. Then the dis tance between M and N is the infimum of real numbers E > 0 such that there is a diffeomorphism f from the ball Bk of radius 1/E in M to the ball B of radius 1/E in N with j(x) = y so that for p, q E B�;, -e
< log
(
d([C[iJJCq))
d(p,q)
)
< E.
Note that our notion of distance, hence limits, depends on the choice of basepoints. We call the manifold (M,g) with basepoint x the pointed manifold (M,g,x). Consider the set o f pointed Riemannian manifolds o f a fixed dimension n, equipped with the above notion of dis tance. One would like to understand when a given se quence (M;, g;, p;) in this set has a subsequence which con verges to a pointed Riemannian manifold of the same dimension. Let us note two necessary conditions: First, as the example in Figure 3 shows, if the curvature of (M;, g;, p;) is not bounded, then the limit may not be a smooth manifold. Second, the injectivity radius at p; should be bounded below by a constant not dependent on i: A sequence of manifolds with bounded curvature need not have limiting manifolds (of the same dimension), as the manifolds may collapse to lower dimensions. For example, let M; = 51 X 51 be the 2-torus, g; = r 1g0 E9 g0 and p; = Cp,q), where g0 is the usual metric on the circle. Observe that (M;, g;) is the torus, with the product metric obtained by viewing the
Figure 3. A sequence without bounded curvature with the
limit singular.
© 2007 Springer Science+ Business Media. Inc. . Volume 29. Number 4. 2007
39
@00
Figure 4. An example of collapsing.
torus as a product of a circle of radius 1/ i with a circle of radius 1 . In this case the sectional curvature of (M;, gf) is zero for any i. On the other hand, the limit of this sequence of metrics is the degenerate metric 0 EB g0. Hence the limit of the Riemannian manifolds (in the appropriate sense) is a circle (see Figure 4). It turns out that these two conditions are also sufficient to guarantee convergence: For K E IR and r > 0, the space .M(n,K,r) of c� pointed Riemannian manifolds (M,g,x) with sectional curvature bounded above by K and injectivity ra dius at x bounded below by r is pre-compact in the topol ogy given by the Lipschitz distance. More precisely, any se quence of pointed manifolds in .M( n,K, r) has a subsequence which converges to a manifold with a Riemannian metric g which is C1 . Note that the limit may not be in .M(n,K, r), since the metric may not be C"'. We sketch briefly a key idea in the proof of the com pactness result. Suppose that we have both a lower bound on the injectivity radius and an upper bound on the cur vature of a Riemannian manifold (M,fi). As mentioned ear lier in this article, we can choose harmonic coordinates x1, . . . , Xn , i.e., coordinates such that each Xk is a har monic function, near each point in M. The bounds on cur vature and injectivity radius guarantee that these coordi nates exist on balls of fixed radius. Furthermore, the bound on the curvature gives a bound on the C1·a norm, for any a < 1 , of giJ. Hence, by the Arzela-Ascoli theorem, a sub sequence of the giJ and their first derivatives converge. The limiting local metrics can be patched to give a global C1 metric. Now let us return to the case of a Ricci flow. Suppose that (M,g(t)) is a Ricci flow on a closed 3-manifold whose maximal interval of definition is a finite time interval [0, T). Since T < oo, we know that limt -. T lemax(t) = oo, where kma.lJ) = suPiKCx,g)l is the maximum of the absolute values of sectional curvatures of (M,g(t)). Choose a sequence t; � T We rescale g by km.a:f.. tf) to get manifolds (M, kma.£tf)fi) with bounded sectional curva ture (see "Scaling and curvature" above and figures). In or der to apply the compactness theorem to this sequence, one needs to know that the injectivity radius at p is bounded below (independent of i). One of the major results of Perel man was that, for these manifolds, there is indeed a lower bound on the injectivity radius (Perelman's non-collapsing theorem). Thus, by the compactness result, some subse quence of the manifolds has a limit. In order to extract special properties of the limit, recall that the Hamilton-Ivey pinching estimate implied that at any point of high curvature on (M, g(t)), there are 2-planes of positive sectional curvature much bigger than any negative sectional curvature. This implies that the limiting manifold is non-negatively curved.
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THE MATHEMATICAL INTELUGENCER
So far we have been considering limits of the Riemann ian manifolds (M, kmaol.tJ/i). In fact, as explained in Ap pendix B, one can rescale not only the metric but the en tire Ricci flow at these times and at suitable points, and consider the convergence of manifolds with Ricci flows. A compactness theorem for flows, similar to the compactness theorem above, was proved by Hamilton, and using this, one gets a limit manifold with a limit 1 -parameter family of metrics. In fact, as a result of the smoothing properties of heat-type equations, one has a stronger conclusion than the general compactness theorem. Namely, one can deduce that the limiting 1 -parameter family is C"' and again satisfies the Ricci flow equation. The nonnegativity of curvature along with Perelman's non-collapsing result shows that the flow for the limiting manifold is what Perelman calls a K-solution. Perelman proved that points in a K-solution have canonical neigh bourhoods (which we explain below). Furthermore, he proved a technical result giving a bound on the derivative of curvature for K-solutions, which was crucial in under standing behaviour near points of high (but not necessar ily maximum) curvature.
The Canonical Neighbourhood Theorem By considering limiting manifolds as above, it follows that small neighbourhoods of the points of maximum curvature are close to being 'standard' . However, this procedure does not work if we want to understand points with high cur vature which are not the maximal curvature points. The problem is that rescaling with respect to these points does not give metrics with curvature bounded independent of i. A surprising and remarkable result of Perelman's, which overcomes this difficulty and can be considered to be one of the central results in his proofs, is the canonical neigh bourhood theorem. This says that either M is diffeomorphic to S3!G, with G a finite group acting freely, or every point of high scalar curvature has a canonical neighbourhood which is an E-neck or an E-cap. An E-neck is a Riemann ian manifold which is, after rescaling, at distance less than E to the product of a sphere of radius 1 and an interval of length at least
.!. .
An E-cap is either an open ball or the E complement of a ball in the real projective 3-space, with a metric such that the scalar curvature is bounded and every point is contained in an E-neck on the complement of a compact set. In case M is simply connected, we must have M = S3 in the first case and M an open ball in the third. This result is surprising in many ways. Normally, by the kind of rescaling argument sketched above, we can study a neighbourhood of a point of maximal curvature. How ever, one expects that near points of high (but not maxi mal) curvature, there are nearby points where the curva ture is much higher. This means that the curvature can be fractal-like, and the resulting system has behaviour at many scales (as happens with complex systems). To study a neighbourhood of a point of high scalar cur vature, Perelman used the bounds on the derivative of the curvature of standard solutions in an ingenious inductive ar gument (which proceeds by contradiction) to show that the curvature of the appropriate rescaled metric is bounded near
C> 4 One correspondent objected that reviewers sometimes make valuable suggestions for exten sions of results and this important in put would be lost under Vandiver's suggestion. Bellman fully supported Vandiver's initiative as he had a very low opinion of the current state of the refereeing system: I think that the only intelligent and efficient technique is one based upon a board of associate editors
52Vandiver to Bellman: May 1 2 . 1 960 (HSV). 53Vandiver to Bellman: May 1 2 , 1 960 (HSV). Emphasis in the original. 54Stone to Vandiver: January 2 1 , 1 962 (HSV). ssvandiver to Montgomery: January 1 3, 1 962 (HSV). 56Vandiver to Grad: February 27. 1 962 (HSV). 570n this matter, see [Fenster 1 999].
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THE MATHEMATICAL INTELLIGENCER
empowered to present any paper which they think fit. The system of anonymous refereeing which we use now in most journals has so many defects and so many abuses that I think any unprejudiced observer would say that it had failed almost completely. Oddly enough, it serves the purpose of passing the mediocre paper along with no difficulty, and almost completely hindering the novel paper with original and un conventional results and ideas. Vandiver summarized the reactions and his own responses to them in a de tailed, formal letter to the President of the AMS, Deane Montgomery (19091992), and expected Montgomery to raise this matter in a forthcoming meet ing of the AMS Councii.55 It seems that his initiative did not reach any further, and his ideas on reviewing were never adopted in the Reviews, although Zen tralblatt often uses "Autor-referats". Vandiver was involved in a second undertaking that shows how he tried to turn his views on mathematical schol arship into a concrete plan of action. In 1961 William ]. LeVeque submitted a proposal to the National Science Foun dation calling for the publication of "A General Survey of the Theory of Num bers Leading to the Compilation of a Topical History and Critical Review of the Theory of Numbers, 1 9 1 5-1960." Not surprisingly, Vandiver was enthusi astic about this project and wrote a highly positive report.56 LeVeque mentioned three main top ics not originally covered by Dickson that should be included: Analytic The ory of Prime Numbers, Diophantine Ap proximations, and Algebraic Numbers. Vandiver suggested that a chapter on Bernoulli and Allied Numbers should also be included, as well as the very important topic of Higher Reciprocity that Dickson had left for a fourth vol ume but never published.57 He insisted that only abstracts of articles should be included, with somewhat longer ones when the original paper had appeared in an out-of-the-way journal. If Dickson
had included criticisms in his book "such material would now be worth less." Finally, he referred to the inten tion to rely on the Mathematical Re
views:
Since experienced reviewers are hard to obtain in order to write re views for the Math. Reviews, I regard most of the reviews appearing in that journal as quite inadequate. And
from what I have seen of the other review journals, I do not think they are much, if any, better. 58 The NSF decided not to fund the pro ject, and it was postponed and even tually abandoned.59 Grad explained to Vandiver that although most reviews were favorable ''the bibliography was considered to be of second importance as compared with research of the usual type. "60 Vandiver replied, "if the NSF continues to support 'research of the usual type' to the exclusion of support of bibliography projects, then as time goes on it will be supporting the pub lication of the results of research which already are described in the litera 1 ture. ''6
Parting Company in Silence Both Moore and Vandiver remained ac tive until a very advanced age. For decades, Austin ' s two leading mathe maticians hardly exchanged a word so cially, if at all , and their careers ended quite differently. Not everyone in Austin welcomed Moore's "volunteering spirit" when he continued to work at the de partment after 1952 under a "modified service" contract. On becoming Dean of Arts and Sciences in 1967, John R. Sil ber "made no attempt to conceal his view that Moore's very presence and reputation hindered the recruitment of new faculty."62 Silber, formerly chair of the Philosophy Department, thought mathematics should be taught by ex perienced teachers in fewer sections with more students in them. This, of course, ran counter to Moore's peda gogical philosophy.
Silber brought in visiting scholars to evaluate the performance of various de partments and attempted to introduce mandatory retirement at the age of sev enty-five. This only raised tensions be tween the administration and Moore's still large and influential group of sup porters. A lengthy and rather nasty process ensued that finally led to Moore ·s forced retirement in September 1 969 at age eighty-seven. Almost seventy-one years after he arrived as a freshman and less than three years before his death, R. L. Moore walked off the University of Texas campus for the last time, refusing to attend any events to "honor" him. When, in 1973, the ne\v mathematics building was named the Robert Lee Moore Hall, he was noticeably absent at the dedication ceremony 63 Moore's long-time rival, Henry S. Van diver, voluntarily took emeritus status in 1966. Despite poor health in his later years, he continued to do research and even received a research grant at the age of seventy-six. Yet the only public hon ors conferred on him at the end of his career were quiet affairs that largely es caped notice. In 1961 he was invited to deliver the keynote address at the Texas Section of the Mathematical Association of America.61 Five years later, a few friends and collaborators put together in his honor a special issue of the Journal
time-was finally proved in 1994, it un leashed a t1urry of publicity inside and outside the mathematical community, hut Vandiver's noteworthy achievements were completely overlooked. Some fifty years before their passing, Moore and Vandiver had begun their mathematical careers at the University of Texas together. Each went on to be come distinguished in his own particu lar way, but their paths parted quickly and never again crossed. Vandiver died on January 4, 1 973, aged 9 1 ; Moore was dose to 92 when he passed away on October 4, 1974 . But both are buried in Austin's Memorial Park Cemetery.
of Mathematical Analysis and Applica tions, a publication otherwise devoted to
material and more. and the writing of an
topics unrelated to his own research.65 No buildings were named after Vandiver, nor did he leave a mark as a teacher at the University of Texas. None of his five doctoral students went on to become a leader within the American mathemati cal community. Of his many interesting contributions to mathematical research, only the conjecture of 1 934 bears his name, and this remains barely known, except to specialists. But most ironic of all, when Fermat's Last Theorem-the
without the basic work done at the
problem to which he devoted so much of his energy and on which he became the world's leading expert during his life-
ACKNOWLEDGMENTS
Albert C. Lewis and David Rowe read earlier versions of this article. I thank them for their critical remarks, which led to
significant
improvement.
Editorial
comments by Marjorie Senechal were also highly useful in preparing the final version. Primary sources used here were taken from the Archives of American Mathematics, Center for American His tory, the University of Texas at Austin, and cited with permission. Also some of the important secondary sources cited re lied on documents taken from the same archive. Albert. C. Lewis played an im portant role in putting together all of this article like this one would be impossible archives. REFERENCES
Anderson,
Richard D. and Ben Fitzpatrick
(2000), "An interview with Edwin Moise," Topological Commentary 5 (http ://at.yorku.
ca/t/o/p/c/88.htm). Corry, Leo (2004), Modern Algebra and the Rise of Mathematical Structures, Boston and
Basel, Birkhauser. --
(2007), " Fermat comes to America: Harry
Schultz Vandiver and FLT (1 9 1 4-1 963) , " Mathematical lntelligencer 29 no. 3 30-40.
58Vandiver had considered not including this latter comment so as not to jeopardize the prospects of the project's being approved, but he obviously changed his mind. See Vandiver to Grad: February 7, 1 962 (HSV). 59LeVeque to Vandiver: March 9, 1 962 (HSV). 60Grad to Vandiver: March 5, 1 962 (HSV). 51 Vandiver
to Grad: March 1 3, 1 962 (HSV).
62[Parker 2005 , 322]. 63[Parker 2005, 332].
64[Greenwood, et a/. 1 973, 1 0939]. See [Vandiver 1 961]. 65[Corry 2007].
© 2007 Spnnger Science+Business Media, Inc., Volume 29, Number 4, 2007
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-- (2007a), "FLT Meets SWAC: Vandiver, the Lehmers, Computers and Number Theory," IEEE Annals of History of Computing (Forthcoming). Fenster, Della D. (1 999), "Why Dickson Left Quadratic Reciprocity out of His History of the Theory of Numbers," Am. Math. Monthly 1 06 (7), 61 8-629. Greenwood, Robert E. (1 983), "History of the Various Departments of Math ematics at the University of Texas at Austin: 1 883-1 983. " unpublished manuscript in the Greenwood archive, Archives of American Mathe matics, Center for American History, The University of Texas at Austin. -- (1 988), "The Benedict and Porter Years, 1 903-1 937 , " unpub
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lished oral interview (March 9, 1 988) (Oral History Project , The Legacy
cluding
of R . L. Moore, Archives of American Mathematics, Center for Amer
electronic form (offline, online) or other reproductions of similar nature.
ican History, The University of Texas at Austin). Greenwood, Robert E., et a/. (1 973), "In Memoriam. Harry Schultz Van diver, 1 882-1 973," Memorial Resolution, Documents and Minutes of the General Faculty, The University of Texas at Austin, 1 974, 1 0926-1 0940. Lewis, Albert C. (1 989), "The Building of the University of Texas Math ematics Faculty, 1 883-1 938 , " in Peter Duren (ed.) A Century of Math ematics in America - Part Ill, Providence, R l , AMS, pp. 205-239.
Moore, Robert Lee (1 932), Foundations of Point Set Theory, Provi dence, Rl, AMS (revised edition, 1 962). McShane, Edward J. ( 1 957), "Maintaining Communication, " Am. Math. Monthly 64 (5), 309-31 7 .
Parker, John (2005), R. L. Moore. Mathematician & Teacher, Wash ington, DC, Mathematical Association of America. Raimi, Ralph (2005) , "Annotated Chronology of the New Math , " unpub lished manuscript available at the site Work in Progress, Concerning the History of the so-called New Math, of the Period 1 952-1975 (http:// www.rnath .rochester.edu/people/faculty/rarrn/the_new_math.htrnl).
Reid, Constance (1 993), The Search for E. T. Bell, Also Known as John Taine, Washington, DC, Mathematical Association of America.
Traylor, D. R . , Creative Teaching: The Heritage of R. L. Moore, Univer sity of Houston, 1 972. Usiskin, Zalman (1 999), "The Stages of Change" (1 999 LSC PI Meet ing Keynote Address), downloaded from http://lsc-net.terc.edu/do.cfm/ conference_material/6857/show/use_set -oth_pres. Vandiver, Harry Schultz (1 924), " Review of Volumes II and I l l of Dick son ' s History of the Theory of Numbers ," Bull. AMS 39, 65-70. -- (1 960), "On the desirability of publishing classified bibliographies of the mathematics literature," Am. Math. Monthly 67(1 ), 47-50. -- (1 961 ) "On developments in an arithmetic theory of the Bernoulli and allied numbers , " Scripta Mathematica 25, 273-303. -- (1 963), "Some aspects of the Fermat problem (fourth paper) , " Proc. NAS 49, 601 -608. --
(unpublished 1 ) , "On the relation between academic teaching and
academic research and the most effective working conditions for eac h , " unpublished manuscript (HSV), 1 2 pp. --
(unpublished 2), "On the training in universities which is desirable
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R. L. M oore: Mathematician & Teacher by john Parker WASHINGTON, DC, MATHEMATICAL ASSOCIATION
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has been written about Robert Lee Moore, the Texas topologist who is best known for the "Moore Method" of teaching. An early biography, D . R. Traylor's Creative Teaching: Heritage qf R. L. Moore [1], was published in 1972, two years be fore Moore's death. Its author was not unbiased; Traylor is an academic grand son of Moore-R H Bing was his the sis advisor-who attempted to get Moore hired at the University of Hous ton when Moore was forced into re tirement from the University of Texas at age 85. This hagiography is also a bit outdated; over half of the book consists of a list of publications by Moore and his academic descendants as of the early 1970s. A more current resource for Mooreabilia is The Legacy of R. L. Moore Project at http://www.discov ery.utexas.edu/rlm/. Its mission, as stated there, is "To disseminate the In quiry-Based Learning methodology of R. L. Moore and to further its imple mentation throughout the educational systems in the United States and abroad. " For readers who are uncon vinced of Moore's importance to math ematics, one can click there in the pho tos section to see all six MAA Presidents who were Moore students. How many can you name? MAA Presidents R. D . Anderson, R H Bing, Edwin Moise, R. L. Wilder, and Gail Young were doc toral students of Moore's, and Lida Bar rett was a master's student who received her Ph. D. under John R. Kline, Moore's
very first doctoral student. And of these six, Bing and Wilder were also Ameri can Mathematical Society (AMS) Presi dents, as was Moore student G. T. Why burn; Anderson and Moise were also AMS Vice-presidents. John Parker, the author of the new Moore biography, is a British journalist and eclectic writer who has authored 1 4 military o r investigative books, among them, Commandos, Tbe Gurkhas, and Inside the Foreign Legion. and 16 criti cal biographies including Prince Philip, Tbe Trial qfRock Hudson, and King of
Fools: Tbe Duke of Windsor and his Fas cist Friends. Given the author's back
ground one wouldn' t expect to see much of Moore's mathematics in the book, and in fact there is little detail about his specific mathematical contri butions. For a survey of Moore's math ematical work, see R. L. Wilder's trib ute "Robert Lee Moore, 1 882- 1974'' [2] . Wilder classifies Moore's 68 research pa pers into three categories, namely Geometry, Foundations of Analysis, and Point Set Theory. This last category, Wilder points out, was so-named in def erence to Moore's own preferences, as many would call it set-theoretic topol ogy. In the Preface to the Moore biogra phy, Parker tells of an incident when Moore was being photographed for his Presidency of the AMS. When the pho tographer offered to airbrush a wart from his face, Moore replied, "Warts and all. " The author goes on to say, "And thus, in this account, I have followed the same guidance. This then is the ex traordinary story of R. L. Moore and how he developed the Moore Method, which was bigger than the man (with all his faults and idiosyncrasies), how it equipped its beneficiaries to excel in fields of excellence other than mathe matics, and how it has been modified to meet the educational requirements of today. " For the convenience o f the reader, in the remainder of this review I'll try to separate the Moore Method from Moore's life, although the book doesn't
© 2007 Spnnger Sc1ence +Business Media, Inc., Volume 29, Number 4, 2007
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do this and in real life it wasn't easy to do. In both parts I'll include some in formation and anecdotes that may be new even to the cognoscenti. I'll con clude with a few particulars about the book itself, and a bottom line on whether you might want to read it.
Moore-the Method In the 1 966 MAA film Challenge in the Classroom, which is still available as a Classic Video from the MAA , Moore himself singled out a defining moment in the genesis of his method, when he taught himself calculus at age 1 5 . He would read the statement of a theorem and try to prove it, keeping the text's proof covered. If he needed to uncover too many lines of the proof to finish it, he felt that he had failed. Later, as a graduate student at the University of Chicago, Moore was intrigued by the laboratory system of instruction in mathematics that was advocated there by E . H. Moore (no relation). In his 1902 presidential retirement address to the AMS, E. H. Moore described how, in education, "The teacher should lead up to an important theorem gradually in such a way that the precise meaning of the statement in question, and further, the practical truth of the theorem is fully appreciated and the importance of the theorem is understood and indeed the desire for the formal proof of the propo sition is awakened before the formal proof itself is developed. Indeed, in most cases, much of the proof should be secured by the research work of the students themselves. " What then i s the Moore method? Parker summarizes it nicely: "Its prin cipal edicts virtually prohibit students from using textbooks during the learn ing process, call for only the briefest of lectures in class and demand no col laboration or conferring between class mates. It is in essence a Socratic method that encourages students to solve prob lems using their own skills of critical analysis and creativity. Moore summed it up in just eleven words: 'That stu dent is taught the best who is told the least. ' " The biography is permeated with discussions, often from old interviews with or articles by famous Moore stu dents, of the details in applying the method. For example, in 1959 Raymond Wilder, an early Moore student (Ph.D.
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THE MATHEMATICAL INTELLIGENCER
1 923), gave six essentials of the Moore method as it was used in the mid-1930s [3]. (Parker lists just the first five.) 1 . Selection of students capable (as much as one can tell from personal contacts or history) of coping with the type of material to be studied. 2. Control of the size of the group par ticipating, from four to eight stu dents probably the best number. 3. Injection of the proper amount of in tuitive material, as an aid in the con struction of proofs. 4. Insistence on rigorous proof, by the students themselves, in accordance with the ideal type of axiomatic de velopment. 5. Encouragement of a good-natured competition. 6. Emphasis on method, not on subject matter. Wilder, later in his article, gave a sev enth feature, "Selection of material best suited to the method," but he down played it because of the vagueness of its terms and its ambiguity. Readers interested in a more recent appraisal of the method, including ob servations on its difficulties and draw backs, as well as who can use it and where, can consult the Monthly article [4] by another Moore alumnus, F. Bur ton ]ones (Ph.D. 1935). Jones notes that Moore wanted his topology classes to be as "homogeneously ignorant (topo logically) as possible," thus insuring a level playing-field where "competition was one of the driving forces." "Moore sternly prevented heckling," but it was rarely a problem as "the whole atmos phere was one of a serious community effort to understand the argument. " In contrast, Wilder, in talking about earlier Moore classes [3] , said that "good-na tured 'heckling' was encouraged. " Many Moore alumni single out his selection process, the filtering step, and his special ability to spot mathematical talent as key factors in his success. Mary Ellen Rudin met Moore at the registra tion table on her first day at the Uni versity of Texas. At this meeting, as de scribed in [5], Moore discovered in questioning her that she used "if'' and "then" , and "and" and "or", correctly, and decided to make her into a math ematician. In fact, according to [6], "It appears now that he [Moore] rearranged his schedule to teach her a mathemat ics course every semester she was in
college. " Parker's book describes how chemistry students F. Burton Jones and Gordon T. Whyburn, along with Ray mond Wilder, who intended to become an actuary, were all "hijacked into math ematics by Moore. " Moore was a master a t social engi neering and had, according to Rudin, a "terrific ability as a psychologist. " One of many techniques he employed was to add a student's name to a theorem, even when the student might only have slightly generalized it or had just found a better proof. Gail Young described Moore as an "incredible father substi tute," and pointed out that "We [Moore's students] were all fighting for approval." Young also singled out another impor tant Moore classroom technique, his us ing an "Inverse Order of Likelihood" cri terion for calling on students in class. An advantage in calling on students in reverse order of ability, F. Burton Jones remarked, was that this gave "the more unsuccessful students first chance when they did get a proof. " However, some times psychological bruising occurred. In [5] Mary Ellen Rudin described a class in 1945 with R. D. Anderson, R H Bing, Ed Moise, Ed Burgess, and "a sixth, whom we killed off right away. He was a very smart guy-1 think he went into computer science eventually-but he wasn't able to compete with the rest of us. " She added, "It builds your ego to be able to do a problem when some one else can't, but it destroys that per son's ego. I never liked that feature of Moore's classes. Yet I participated in it. " Another female Moore student, Mary Elizabeth Hamstrom, described her as tonishment at receiving a very long let ter from Moore even before her arrival at Texas for graduate school. In it Moore regrets her having already taken a course in real variable theory, and ad vises her not to read any point-set the ory in her summer reading, since for his course this "would make you too much of a spectator" and "an onlooker watch ing others work on theorems which are new and interesting to them but which you have already heard or read about." When students did make it into Moore's point-set topology course on the Foun dations of Mathematics they were vir tually forbidden from taking courses containing related material. Moore even once wrote a publisher, "I have even gone so far as to remove, or have re-
moved, from our university library a copy of my own book Foundations of Point Set Theory [7). On one occasion this university copy was lost or mis placed and I was inclined to be sorry afterwards that it was found and again shelved in our library." Many years after experiencing the Moore method, Hamstrom noted that it fostered the power and ability to do re search, as well as good research atti tudes. But there were limitations: "You do not get any training in learning the kind of mathematics other people do. It's something that you need; you need to learn to read the literature. " Mary Ellen Rudin observed in [5] that the mathematical language Moore used was his own, "completely different from the language of the mathematical litera ture. " For example, he used region for open set, and Rudin didn't learn the standard definitions of compact and limit point from his courses. When she received her Ph.D. she "didn't know any algebra; literally none. I didn't know any topology. I didn't know any analy sis-! didn't even know what an ana lytic function was . . . I never took an examination in mathematics in my life." Moreover, when she took her first fac ulty position she had never read a math ematical research paper, since in Moore's opinion reading other mathe maticians' work could destroy a stu dent's confidence. In his Monthly arti cle [4] F. Burton Jones discusses the problem of reading, and how difficult it is to get students to read mathemat ics after they have been working out their own proofs for a few months. He comments that even Moore himself complained that "when he wanted (finally) the students to read, they couldn't.'' Jones offered one effective technique he had found useful in his beginning graduate courses in general topology: to have students not read un til Christmas, then buy Kelley's General Topology for bedtime reading, and to prepare for the final examination. According to Gail Young, Moore be lieved algebraic topology was "the work of the devil. " Some Moore students who went on to direct their own doctoral students made certain that they learned some algebra. For example, Norman Steenrod wrote his first paper on point set topology after taking topology from Raymond Wilder at Michigan. But
Wilder then guided Steenrod towards algebraic topology and to Princeton, where Steenrod became Solomon Lef schetz's star Ph. D. student, with a the sis on "Universal Homology Groups," and later introduced what is now known as the Steenrod algebra. One Moore alumnus noted, "The Moore method is not for the faint hearted." Other students singled out Moore's ability to ask just the right ques tion at the right time, and also the pa tience required in doing this. In [3] Wilder compared the Moore method with a Socratic approach but with ca sualties, since only the fittest students survived. Wilder reported on his own use of a modified Moore method in classes of 30 or more students, although ''inevitably, a few, sometimes only two or three, students, would star in the pro duction. " Chapter 1 6 o f the biography de scribes Moore's calculus courses after World War II, when more and more en gineering and physics students took them. Moore made few concessions for
. I never took an examination in mathematics in my life . " these students, lecturing little and, for example, focusing on existence theo rems for extrema but avoiding word problems involving them and tech niques for finding the extrema. Mary Ellen Rudin observed that Moore's stu dents didn't end up learning much cal culus, which was disappointing for technical students who needed it as a tool. In addition, most poorer students dreaded and hated the classes and ended up with a feeling of defeat. They even became antagomsttc towards Moore himself, in spite of his being "a brilliant psychologist, " according to one student. John Green, a successful Moore student who went on to get two Ph.D.s, asserted, "He told us early on that he had no use for the university guidelines stating that we should expect three hours of outside class work for each hour in the classroom. He said he wanted us to think about his class all day, every day, to go to bed thinking about it, to wake up in the night think ing about it, to get up the next morn ing thinking about it, to think about it
walking to class, to think about it while we were eating. If we weren't prepared to do that, he didn't want us in his class. '' Moore used his calculus classes to discover mathematical talent. One such student was John Worrell, who went on to a very successful career in medicine and applied mathematics. Worrell recalled that in Moore's calcu lus courses, "he would give you no re ward for rote memorization. And in fact, he gave the veiled threat that if you try to memorize from the calculus I will de
tect it, and I will excommunicate you . You will not be allowed in my class. "
[italics Worrell's] According to former Moore student Edwin Moise [8], the Moore Method re quired the instructor to dominate the environment that his students lived in. At places like Harvard, Princeton, and Chicago, said Moise (then teaching at Harvard), no one, not even Moore him self, could prevent students from work ing together or getting help from other professors, as Moore had done at Texas. Many of Moore's colleagues and con temporaries tried the method, but with mixed success. H. S. Vandiver, a dis tinguished number-theorist also at Texas, tried to teach as Moore taught but without using his very carefully or ganized structured sequence of ques tions. Vandiver eventually gave up, and went back to using texts in his classes. At Princeton both Steenrod and Lef schetz, an old Moore rival, used the method in topology seminars; Lefschetz even tried using it in an algebraic geom etry seminar. In his 1 977 Monthly arti cle [4] F. Burton Jones discusses areas other than topology where the Moore method works very well. These in cluded areas that start with an axiomatic development, such as group theory, foundations of geometry, and even an introduction to Hilbert Space. Paul Halmos in his "Automathogra phy" [9] discussed experimenting with a modified Moore Method and being converted after solving tactical prob lems. The method is not just for pro ducing research mathematicians: "The Moore Method is, I am convinced, the right way to teach anything and every thing-it produces students who can understand and use what they have learned. " Halmos used it at the Univer sity of Michigan in a beginning calcu lus course where, he found, 20 students
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is the absolute maximum for which the method works, and in an honors linear algebra class in which he gave students on the first day a nineteen-page hand out with a bare-bones statement of fifty theorems for the course. For the latter, he noted that it took him several months of hard work to prepare for it, and com mented that teaching with the Moore Method "takes a lot out of you. " Hal mas also argued that the criticism that the method covers less material is a "red herring, " and that "with a little elastic ity you can even adapt the Moore Method to 'covering' a previously fixed amount of material. You'll be covering a lot of it a lot better than your lectur ing colleague (namely, the part that your students were led to discover and prove by themselves) . . . . " Elitism is a common criticism of both Moore and his method. Moore himself, being primarily interested in producing research mathematicians, spent more time and effort on the AMS than the MAA ; he once described the MAA as the Salvation Army of Mathematics. R. D . Anderson reported, "I was told that he [Moore] had joined the MAA three times, which meant that he must have resigned at least twice. " In spite of the elitism of the Moore Method itself, the biography points out its tremendous ef fect on both collegiate education and K-1 2 education in mathematics. The lat ter is due primarily to first-generation Moore students like Moise and second generation students like E. G. Begle, a Wilder student who was a founder of the School Mathematics Study Group (SMSG) in 1958. For collegiate educa tion, Anderson observed, while Moore did not believe in cooperative learning, in his classes "the overall learning ex perience was very similar to that now thought of as cooperative learning. " A footnote in the book credits the coop erative (or small-group) learning move ment in mathematics to Neil Davidson, a student of Bing's at Wisconsin, and cites the article "The Texas Method and the Small-Group Discovery Method" by Jerome Dancis and Neil Davidson at www. discovery.utexas.edu/rlm/refer ence/dancis_davidson.html. Davidson "started with the idea established by R. L. Moore that bright students can de velop mathematics, " but he changed the social environment. Instead of having students work individually within a
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THE MATHEMATICAL INTELLIGENCER
competitive system, students worked to gether in a small group within a coop erative system. The article presents a nice overview of both methods, along with offering practical suggestions for using them.
Moore-the Man and the Mathematician Robert Lee Moore was born in Texas in 1882 and, as his name suggests, was raised in the southern tradition, in Texas. After an unexceptional boyhood he en tered the University of Texas at age 16, where he studied mathematics and was mentored by George Bruce Halsted, who had been a research assistant for]. ]. Sylvester at Johns Hopkins. Moore blossomed under Halsted's guidance, as had L. E. Dickson, an earlier student at Texas, and both proceeded to the Uni versity of Chicago for doctorates in mathematics. Moore's Ph.D . , which was received when he was just 20, was on "Sets of Metrical Hypotheses for Geom etry" and was supervised by E. H. Moore and Oswald Veblen. An indicator of Moore's single-mindedness is that his personal diary, written in this hotbed of mathematics, had, according to Parker, "virtually no mention of any social ac tivity, sports, girlfriends or other inter ests that might otherwise amuse and en gage someone of his age." The decade after Moore left Chicago was a sterile research period for him, but he did experiment with and refine his teaching method at Princeton, North western, and the University of Pennsyl vania. According to a Texas historian and future colleague of Moore's, at Penn, Moore developed a reputation "as a cur mudgeon that would last until his death. " But Moore's complaints about the office space there were clearly justified: he shared a basement room with eight other instructors or assistant professors, right next door to an odoriferous stable! While at Penn, Moore shifted his in terest from pure geometry to point-set topology. In 1 920 he returned to the University of Texas, where he stayed until his forced retirement in 1969. In 1931 he was elected to the National Academy of Sciences, and the follow ing year the AMS published his opus, Foundations of Point Set Theory [7]. Af ter that Moore gradually turned his at tention away from his own research to training future research mathematicians,
with remarkable success. From 1 940 un til his retirement, in what the biography calls his "golden age" for discovering mathematical talent, Moore produced 35 of his 50 Ph.D.s, 24 of them between 1953 and 1 969, when he was-not by choice-in modified service (half-time work with half-time pay). Moore ig nored the conditions and the spirit of this modified service, teaching 15 hours a week and 5 or 6 classes a semester. His doing so spurred a bizarre Faculty Council proposal in 1954 to restrict modified service to no more than half the regular duty of a full-time employee. In his eloquent argument against this proposal and its "union shop attitude," Moore amusingly referred to himself as "Professor X, " asserting, "I have full au thority to speak for him. " Moore won the battle against this proposal to limit his time in the class room but eventually lost the fifteen year-long war to forcibly retire him. (One of several reasons that the uni versity wanted Moore retired was that his presence was considered to be a hindrance to hiring new mathematics faculty.) Along with the messy details of this sad ending to a remarkable ca reer, the book includes some lighter moments, such as the strong rumor, when Moore started his mandatory modified service at age 70, that he sub scribed to an annuity that would not be gin paying out until he was 1 20! Parker does not shy away from dis cussing all aspects of Moore's person ality. In Chapter 1 5 he makes it clear that Moore was not a misogynist, al though Mary Ellen Rudin did observe in [5] , "He always pointed out that his women students were inferior." And Moore was a bigot, both in and out of class. He would not allow blacks in his class, and he once walked out of a topology lecture by a student of Bing's when he discovered that the student was black. Scott Williams, a black math ematician at the State University of New York at Buffalo, maintains a webpage documenting Moore's racism at www . math.buffalo.edu/mad/special/RLMoore racist-math. html. Moore's anti-Semitism was less pronounced than his racism but, according to Rudin, he claimed that Jews were inferior and enjoyed baiting and arguing in class with Edwin Moise, who was Jewish. Moore was also out spoken, both in class and as AMS Pres-
ident in the late 1930s, about the " in
R. L. Moore-the Book
vasion" ofJewish emigre mathematicians
I particularly enjoyed the seventy pho
and his personal loathing of President
tos Parker included in his biography
Generalizing from this admittedly small
Roosevelt
pictures of Moore, his students, and his
sample, can we conclude that investiga
cies. Anderson and Rudin, members of
contemporaries
of
tive journalists are perhaps best suited
Moore's renowned Class of '45,
and
his
New
Deal
poli
at
the
Sylvia Nasar for John Nash (A Beaut[fu.l
University
/'vfind) and now Parker's book for Moore.
Chicago and the University of Texas.
for authoring critical biographies of out
served that students \vould try to come
Many of these photos are informal, al
standing mathematicians?
up with mathematical results for class
though Moore is never seen smiling, as
ob
to prevent him from digressing to so
he thought he looked silly when he
REFERENCES
cial or political issues.
smiled. Forty-four pages of Appendices
[ 1 ] Traylor, D.
I n these days of enforced collegial ity in academia,
in the book include descriptions of four
I am fascinated by
courses commonly taught by Moore at
Moore's feuds with several mathemati
Texas, as well as short biographies of
cal colleagues at Texas. In 1 944, when
each of his fifty doctoral students, along
Edwin F. Beckenbach and Moore were
with their thesis titles. Many of these
competing
a
theses were in subject areas that are no
heated confrontation ended in fisticuffs!
for
graduate
students,
longer, if they ever were, part of main
The next year Beckenbach left Texas
stream mathematics. ( R H Bing once
for UCLA.
remarked . regarding his thesis "Con
A longer and more significant feud with
number-theorist
Harry
cerning simple plane webs , " that if any
Schultz
one \\anted a journal article based on
Vandiver is discussed in the chapter
it he still had forty-eight of the fifty
"Clash of the Titans ( 1 944-1950). " 'If Van
copies he had been give n . ) The book
diver and Moore had been colleagues
has a useful nine-page index, and for a
in the same department for 2 4 years
hardcover book with a quality binding
when their personal differences came
it is very reasonably priced.
Reginald Creative Teaching:
Heritage of R. L. Moore, University of Hous
ton, 1 972. [2] Wilder, R . L. Robert Lee Moore 1 882-1 97 4 , Bulletin
o f the
American
Mathematical
Society 82 (1 976), 4 1 7-427 Available on
line at http://www.discovery.utexas.edu/rlm/ reference/wilder2 . html [3] Wilder, R. L. Axiomatics and the develop ment of creative talent. In The Axiomatic Method with Special Reference to Geome try and Physics, L. Henkin, P. Suppes, and
A Tarski, Eds. North-Holland, 1 959, pp.
474-488. Available on-line at http://www. discovery. utexas.edu/rlm/reference/wilder1 . html [4] Jones, F.
Burton. The Moore Method.
to a head in 1 946. Vandiver threatened
We have reached the bottom line
to resign but instead transferred from
should you read this book? Probably
the Department of Pure Mathematics to
not, especially if you've already read
that of Applied Mathematics, not the ob
some of its high points in this review
vious place for a renowned expert on
and aren't just peeking at this conclu
Mary Ellen Rudin in More Mathematical
Fermat's Last Theorem. Vandiver and
sion. The biography is readable and
People, D. Albers, G. Alexanderson , and C .
Moore never exchanged a single word
well-documented, but it has too much
Reid, Eds.
again.
minutiae for many if not most readers.
Boston, 1 990.
American Mathematical Monthly 84 (1 977),
273-278. [5] Albers, D., and Reid, C. An interview with
Harcourt Brace Jovanovich,
B y 1969 the two departments of Pure
In addition. as important as Moore was,
Mathematics and Applied Mathematics
he is not the most pleasant person to
of Women and Minorities in Mathematics,
had combined into one. Its members
read about. However. if you would like
voted 27 to 1 , with 9 abstaining, in fa
to read more about the Moore Method
American
vor of .\1oore·s retirement. That summer
I suggest chapters: 6, 8, 9 . 1 1 ( "Moore
Moore taught his last class. He refused
the Teacher'' ) . 1 3 ( "Class of '45" ), 1 5
any symposium or other celebrations to
( "His Female Students " ) , 1 6 ( "Moore's
"honor'' his retirement. He also-at age
Calculus"), and 1 8 . Alternatively, you
85 !-turned down a professorship at the
might ignore this book altogether and
University of Houston. But he accepted
just read the descriptions of the method,
the title Professor Emeritus and spent
including its modifications and limita
the remaining five years
tions, hy Halmos [9], Jones [4], Wilder
of his life
within walking distance of the Texas
[3], and Moise [8] .
campus.
I have a final comment. Many years
The book concludes with a nice sum
ago when I read Constance Reid's en
maiy of Moore's legacy to mathematics,
grossing
including not only the Moore Method in
Courant I knew that she had much math
biographies
of
Hilbert
and
[6] Kenschaft, P. Change is Possible: Stories Mathematical
ory, American Mathematical Society Collo
quium Publications 1 3, 1 932, revised ed. 1 962. [8] Moise, E. E. Activity and Motivation in Math ematics. American Mathematical Monthly 72 (1 965), 407-4 1 2 . [9] Halmos, P. I Want To Be a Mathemati cian: An Automathography in Three Parts ,
Springer-Verlag, 1 985 . pp. 255-265. [re printed as a Mathematical Association of America paperback, 1 988]
teaching but also the concept of Moore
ematical expertise
such as that of her sister Julia Robinson
numerous academic descendants. Ac
and other distinguished mathematicians
cording to The Mathematics Genealogy
at Berkeley and elsewhere. More re
Project at
cently, there have been several well-writ
Santa Clara, CA 95053
nodak.edu/, there were 2 , 1 1 0 such de
ten and mathematically accurate biogra
USA
scendants as of July 2007.
phies of mathematicians by outsiders like
e-mail: [email protected]
�Al so see the article by Leo Corry in this issue, pp. 62-7 4.
- The
Provi
[7] Moore, R . L. Foundations of Point Set The
spaces in topology. It also mentions his
http:/ Igenealogy. math.ndsu.
Society,
dence, R . I . , 2005, p. 62.
available for help,
Department of Mathematics and Computer Science Santa Clara University
Editors
© 2007 Spnnger Science-.- Bus1ness Med1a, Inc., Volume 29, Number 4. 2007
79
The Poincare Conjecture: In Search of the Shape of the U niverse by Donal O 'Shea NEW YORK, WALKER
ix +
&
COMPANY, 2007,
293 PP, U S $26.95, ISBN-10: 0-8027-1532-X,
ISBN-13: 978-0-8027-1532-6
Poincare1s Prize: The H undred-Year Quest to Solve One of Math1s G reatest Puzzles by George G. Szpiro NEW YORK, DUTION, 2007,
ix + 320
PP
US $24.95, ISBN-10: 0525950249, ISBN-13: 978-0525950240
REVIEWED BY JOHN J. WATKINS
t is not often when something im portant in mathematics makes head lines around the world. But that did happen just this past year with the proof-after almost exactly one hun dred years-of the famous Poincare Conjecture by a reclusive Russian math ematician Grigory Perelman. Newspa pers everywhere ran articles trying to explain to their readers the significance of this brilliant achievement. Science magazine officially designated it the "Breakthrough of the Year" for 2006. 1be New Yorker magazine published a highly controversial article by Syvia Nasar (author of A Beautiful Mind ) and David Gruber, in which they portrayed a large cast of mathematicians behav ing like characters in a melodrama; the magazine has been threatened with a lawsuit for its trouble. Perhaps one of the clearest indica tions that the Poincare Conjecture had indeed captured the attention of the general public came during an episode of an American television series Studio 60 on the Sunset Strip when the fictional head comedy writer, wonderfully played
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by Matthew Perry, had a running gag throughout the show as he struggled to write a comedy sketch about turning a bunny into a sphere. He knew this had something to do with the Poincare Con jecture, and that the conjecture itself is truly important, and he sensed that he should be able to make it funny. He just couldn't write the sketch. The script writers for Studio 60 are very smart people and they certainly de serve a lot of credit for inserting such a topical mathematical reference into one of their shows, so they can be excused if they don't have some of the details of the Poincare Conjecture itself exactly cor rect. They know it has something to do with topology and that topology in turn has something to do with deforming ob jects. For example, a topologist views a solid chocolate bunny as essentially the same as a solid sphere of chocolate be cause one of these objects could be grad ually deformed into the other. But they like almost everyone else in the world--don't have the vaguest idea what the Poincare Conjecture itself is really all about, much less why it might be im portant to anyone. For example, they clearly don't real ize that the Poincare Conjecture is about the three-sphere-that is, the three dimensional sphere-which, confus ingly, doesn't exist in three-dimensional space at all, but in four-dimensional space. So, if Matthew Perry or anyone else in the general public wants to know what it was that Grigory Perel man achieved when he proved the Poincare Conjecture and why anyone cares, they are clearly going to need some serious help. Fortunately, that help is at hand. In a truly remarkable book, The Poincare
Conjecture: In Search ofthe Shape ofthe Universe, Donal O'Shea tells the com plex story of the Poincare Conjecture in a way that provides just the help that any interested reader needs in order to gain an understanding of how Perelman did what he did, why it took so long, and why it is so important. O'Shea begins his story quite dra matically in a crowded lecture hall in April 2003 at MIT with Perelman at a blackboard explaining how he had used a very clever idea called Ricci flows that treat curvature in space as if it were heat. This allowed Perelman to recast the Poincare Conjecture in terms of a type
THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media. Inc.
of equation known as partial-differen tial equations, an extremely well-studied branch of the area of mathematics that makes up most of mathematical physics. All of Perelman's training had prepared him for this particular method of attack on the Poincare Conjecture, from his high-school days at Leningrad Secondary School in Saint Petersburg, a school that specialized in mathematics and physics, to his later joining a group at the Saint Petersburg branch of the Steklov Insti tute that did legendary work in the field of partial-differential equations. As O'Shea lets his leisurely story un fold, we learn that there are many math ematicians whose work ultimately went into the solution of the Poincare Con jecture. It was Richard Hamilton who developed the method of Ricci flows in the early 1980s-and even the term itself-and did much of the real work behind solving the Poincare Conjecture. Within a decade Hamilton and others had achieved sensational results for two dimensional surfaces, and for dimen sions higher than three, but Hamilton had also showed that in three dimen sions there were genuine difficulties still to be overcome when using Ricci flows. It was by overcoming these seemingly insurmountable difficulties that Perel man was eventually able to crack the century-old Poincare Conjecture. Actually, and this may seem strange, Perelman and Hamilton were not really working on the Poincare Conjecture at all, they were working on a much more ambitious and much harder problem known as the Geometrization Conjec ture. In two dimensions we all know that there are exactly three different geometries: spherical, flat, and hyper bolic. But in three dimensions it turns out that there are eight different geome tries. In the 1 970s Bill Thurston made the extraordinary conjecture that in a natural way each individual piece of any three-manifold has to have one of these eight geometries! This became known as the Geometrization Conjecture. In turn, then, the Poincare Conjecture has long been known to be a rather simple consequence of Thurston's truly bold new vision of the way geometry works in three dimensions. While Perelman appears to have knocked off something much bigger than the Poincare Conjecture-namely the Geometrization Conjecture-it is of
course the Poincare Conjecture that is getting all the attention. It is the Poin care Conjecture after all that was listed by the Clay Institute in 2000 as one of its seven millennium problems and for which they offered one million dollars for a solution. It is for solving the Poin care Conjecture that the International Mathematical Union, in the summer of 2006 in Madrid, awarded Perelman the Fields Medal, the equivalent in mathe matics of a Nobel Prize. (Perelman, however, declined to accept it.) The Clay Institute will soon announce its de cision on whether they will award the full million dollars to Perelman or split it in some manner. In any event, it seems likely that, remarkably, Perelman will also decline to accept this award. The Poincare Conjecture is the fol lowing deceptively simple assertion: the only compact three-manifold on which any closed loop can be shrunk to a point is the obvious one, namely, the three sphere. O'Shea does a beautiful job of bringing the reader to the point of fully understanding this conjecture by talking about how we came to learn the shape of our own earth long before we could ever view it in its entirety from space and how we use maps-think about how Google Earth overlays a series of rectangular pictures to represent the en tire two-manifold that is the surface of our planet-and then he effortlessly takes the reader up one dimension to explain how the Poincare Conjecture is in fact one of the simplest and most fun damental questions we could ask about the shape of our own universe. We live in a three-manifold, we just don't know what it looks like. If we could view it in its entirety from some vantage point in four-dimensional space, we would know instantly what it looks like, but we can't. But, as O'Shea patiently ex plains, we can still represent our uni verse-just as was done with maps in the day of Columbus for the earth-with a series of pictures, each picture a three dimensional rectangular box, and the entire universe can be represented by overlaying this collection of pictures to form a map of the universe. Now, because the Poincare Conjec ture is at last a theorem, we know that if our universe is finite, which seems likely, and if it is also true that any closed loop in our u niverse can be shrunk to a point, then our universe must look like
a three-sphere. That's why the Poincare Conjecture is so important! Of course, it might not be true that any closed loop in our universe can be shrunk to a point. Just like a loop on the surface of a torus (a doughnut) that passes through the hole of the torus can't be shrunk to a point, it might be the case that some loops on the three dimensional surface in which we live might similarly get stuck if we try to shrink them. We just don't know-yet. Donal O'Shea takes the reader on a fascinating journey from the ancient world more than 2500 years ago when Pythagorus taught that the earth was a sphere to the present day, and provides the reader a solid intuitive understand ing of the complex details of what Grig ory Perelman and others have accom plished. While O'Shea takes great pains to explain carefully and skillfully all mathematical ideas along the way-in deed, were the fictional Matthew Perry to read The Poincare Conjecture: In Search of the Shape of the Universe, he would have not the slightest bit of trou ble finally being able to write that com edy sketch about turning a bunny into a sphere-this book is equally good reading for even expert mathemati cians. Anyone's understanding of the mathematics involved here will be en hanced by O'Shea's thoughtful discus sions and many readers will also ap preciate the many additional themes he manages to weave into his basic plot (although some may find these "tan gents" distracting) . I, for one, gained a far better feel for three-manifolds than I had before read ing this book, even though strictly speaking I have seen it all before, sim ply because O'Shea writes so well and so vividly. And I also found myself with a refreshing new awareness of Poin care's true greatness as a mathematician who, in O'Shea's words, "shaped the form of twentieth-century mathematics." I enjoyed his discussion of Euclid's Elements; his biographies of Gauss, Bolyai, and Lobachevsky; his placement of Riemann at the very heart of this story; his retelling of one of the most famous moments in intellectual his tory-when Poincare had an instanta neous flash of mathematical inspiration completely out of the blue just as he was stepping on a bus to go on an ex cursion, and his description of the key
role in this drama played by the stun ning example of the three-manifold pro duced by Poincare that we now know as the Poincare dodecahedral space. I especially enjoyed a joke that French schoolchildren recited gleefully about the most famous mathematician in France: "Qu'est-ce un cercle? Ce n'est point carre . " What is a circle? It is not a square. Which makes little sense in English, but becomes very clever in French where "point carre" is pro nounced the same as "Poincare . "
I d o have a few minor complaints. In such a fine book it is unfortunate that some of the computer-generated figures are of a rather poor quality by today's standard. For the most part this causes no difficulty whatsoever, it is merely unattractive. I might add that, by con trast, some of the hand-drawn figures are quite lovely. Half of one figure, however, is utterly incomprehensible, which seems strange since a portion of a subsequent figure could serve ad mirably if suitably adapted. There are also quite a few typos and several ref erences to the wrong figures, but none of these is too serious and each is rel atively easy to spot. It is not surprising that an event as earthshaking as a proof of the Poincare Conjecture would prompt the appear ance of several books. One book which takes a decidedly more sensational tack on this event is Poincare 's Prize: 1be
Hundred- Year Quest to Solve One of Math 's Greatest Puzzles by George G . Szpiro. The back cover carries a blaz ing headline worthy of a tabloid an nouncing in large red type: "AMAZ INGLY, THE STORY IS TRUE . " Well, much of Szpiro's story in the book is true, but not quite all. More troubling is that the front cover for this book, which after all is about
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one of the great conjectures in topol ogy, contains a glaring topological blun der. The front cover features a sphere and a horse (it could have been a bunny, but a horse is much more graceful) and each of these objects has been given a very attractive checkerboard coloring. This coloring and a dashed line between the horse and the sphere form a visu ally compelling way to show that the horse and the sphere are equivalent ob jects from a topological point of view. So far, so good. The only problem is that it has been known since roughly the time of Euler that you can't color a sphere with a checkerboard pattern. The basic idea-see the book Across the Board for the details-is that in 1752 Euler found a formula for polyhedra that relates the number of vertices, edges, and faces, and this easily gener alizes to figures drawn on surfaces (where instead of counting faces you count regions). Remarkably then, it turns out that surfaces all have an "Euler characteristic, " and you can only draw a checkerboard figure on a surface whose Euler characteristic is 0. So, for example, you could draw a checker board figure on a torus (or a Klein bot tle if you know what that is) , but you can't ever draw such a figure on a sphere or a horse (or a bunny), simply because the Euler characteristic for a sphere is 1 . Ironically, Szpiro himself talks at some length in Poincare 's Prize about Euler's formula, so he could have no ticed this blunder on his own cover. He even makes matters worse by crediting Euler with proving his own formula in 1 7 5 2 . It is true that Euler did publish what he believed to be a proof in that year, but (see Graph Theory 1 7361936) Euler used a slicing procedure that does not work in all cases. Legendre did prove Euler's formula in 1 794. A somewhat similar historical inaccuracy occurs when, while not explicitly saying so, Szpiro leaves the reader with the im pression that the mathematical field of graph theory began with Euler and his solution to the famous problem of the seven bridges of Konigsberg, but graphs such as Szpiro has in mind did not ap pear until the second half of the nine teenth century. In another instance Szpiro misleads the reader mathematically as he tries to illustrate an early disagreement about
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convergence between Poincare and an astonomer by discussing infinite series and, in particular, the harmonic series , which 1 + 1/2 + 1/3 + 1/4 + 1/5 + of course diverges to infinity. Szpiro wants to emphasize how slowly this se ries diverges so he says "no less than 1 78 million entries must be added to ar rive at the sum of 20 . " But this is com pletely false. One of the most beautiful properties of the harmonic series is that while it is lazily strolling its way to in finity it somehow manages to miss ab solutely every integer except 1 , the in teger where it started! In order to be truthful Szpiro would need to say some thing like " 1 78 million entries must be added just to get beyond 20. " (Paul Hoffman claims in The Man Who Loved Only Numbers that you need about 300 million terms in order to exceed 20.) Elsewhere Szpiro talks about "the Eu clidean three-dimensional space that we live in" and while this phrase makes its appearance merely as part of a state ment whose proof we later learn is wrong, an unwary reader is again left with an incorrect impression, namely that the universe is Euclidean. In spite of the breezy style of the au thor, Poincare 's Prize is probably not the best choice for a reader who is seeking to understand the mathematics behind the Poincare Conjecture. Szpiro clearly made the decision not to include any di agrams in this book, but he pays a high price in terms of what he is able to com municate to a novice reader. Often he is effective, but not always. How many of his young readers will have seen an au tomobile tire that has an inner tube? Or know that a pretzel-when made in the traditional old way from a single long roll of dough with a quick twist and the two ends joined-has three holes? And even though his teaching instincts are well-meaning, the resulting metaphoric images can sometimes be amusingly dis concerting, as when he has us imagine ·
·
·
a fly taking a step back to lift off and look at a bagel at arm 's length. One very promising approach to con structing a counterexample to the Poin care Conjecture did involve pretzels, or at least it involved knots that can look a lot like pretzels. The idea is to start with a cube with a knot inside and then remove the knot so that you have a cube with a knotted hole. Then you attempt to put things back together again so that
loops shrink to a point and yet you man age to come up with something other than the three-sphere. Needless to say, this never quite worked.
Szpiro does have a flair for imagi native comparisons that can be in structive as well as entertaining, as when he uses LEGO bricks to help ex plain the Geometrization Conjecture or when he uses BOTOX injections to il lustrate the fundamentally important idea of how Ricci flows work. But he can also show a lamentable tendency to wander off into gratuitous and irrel evant comments, such as complaining about FDA approval of BOTOX, or when, in the middle of an already overly long biography of the man be hind the Clay Institute, he gets further sidetracked to lecture the reader about the right of certain museums to keep ancient artifacts of dubious provenance simply because they display them in a "more seemly manner." There are a good many things· I learned and liked in Poincare 's Prize. Szpiro traces in great detail a story he nicely frames from 9 October 1492, with Columbus standing on the bridge of the Santa Maria, to 22 August 2006, with Juan Carlos, the King of Spain, awarding the Fields Medal to Grigory Perelman who of course wasn't present in Madrid for this grand-opening cere-
mony for the Twenty-Fifth International Congress of Mathematicians. I'm sure each individual reader would find many things of interest and many new ideas in Poincare 's Prize; I will just mention a few that came my way, in order to suggest that there is indeed a genuine richness that lies be tween the covers of this book. Much to my amazement I learned that the Poincare dodecahedral space, which he discovered in 1904 as a coun terexample to the "theorem" he had claimed in 1 900, to this day remains the only known counterexample to this false theorem. I learned that RH were not the initials of that giant of topol ogy, R H Bing, but were in fact his ac tual first name. More importantly, I also learned that Bing, who himself made prolonged serious attempts on the Poincare Conjecture, in the end may have concluded that the conjecture was false. This , for me, was one of the very best moments in Szpiro's book because it showed so dramatically how hard it is in mathematics to know what is true and what is false if someone of R H Bing's stature can be so completely wrong about such a fundamental ques tion as the Poincare Conjecture. From a more personal point of view, I was delighted to learn that james Alexander, who is well known to most mathematicians as the discoverer of the famous and fabulous "Alexander horned sphere, " has a classic technical climbing route up 14,255 foot-high Longs Peak in Colorado, named Alexander's Chimney, after him. Simi larly, I found a rather lengthy history of the Ecole Polytechnique-which Poincare attended from 1 873 to 1875, graduating second in his class-ab solutely fascinating because my wife di rected a program at Colorado College for several summers during the mid1980s. That program was designed for students from the Ecole Polytechnique to guide them in the study of English and to acquaint them with American culture. Here is one final nugget in the same vein that takes on special irony now that Perelman has rejected the Fields Medal: at his death in 1 9 1 2 , Poincare had received the largest number of nominations for a Nobel Prize of any non-winner. Recall that there is not a mathematics category for Nobel Prizes
and that it is for this reason that the Fields Medal is considered to be the mathematical equivalent of a Nobel Prize. Poincare's nominations were all in physics. I must admit that, in the end, Szpiro does deliver on the tabloid promise from the back cover of his book. The most gripping, hard-to-put-down read ing are Szpiro's last two chapters, when he finally gets down to discussing the very messy controversy surrounding the solution of Poincare's famous con jecture. It is hard not to be intrigued by this controversy. There are some very serious issues here : What share of the credit for the solution does Hamil ton deserve' That the Clay Institute may well award him a large share of the mil lion-dollar prize is just one measure of the fact that many people believe that he and Perelman share equally in ar riving at the final solution. Another of the serious issues is the way in which Perelman bypassed traditionally ac cepted methods for publishing mathe matical proofs by placing his unrefer eed proofs on the Internet. It has taken three years and several teams of heav ily financed experts to conclude that Perelman's work is correct. Meanwhile, the really messy part of the controversy arose from a claim made by a team of Chinese mathematicians that they had published the first complete proof of the Poincare Conjecture. The article in Tbe New Yorker greatly inflamed this controversy by including a full-page drawing with Shing-Tung Yau looking as if he is about to rip the Fields Medal from the neck of Perelman (who looks a bit like Vincent Van Gogh in this drawing). Szpiro sorts through the details and complexities-including the ethical ones-of this controversy quite thor oughly and with what seems to be con siderable fairness and a great deal of wisdom, both concerning human na ture and with respect to maintaining al ways the highest regard for the well being of mathematics. Thus, he is able to bring us beyond the controversy to the point where we can celebrate the solution of the Poincare Conjecture, perhaps dream of solutions to one of the six remaining millennium prob lems, and find other ways-in the words of the mission statement of the Clay Institute-"to further the beauty,
power, and universality of mathemati cal thinking. " REFERENCES
1 . N. L. Biggs, E. K. Lloyd, and R. J. Wilson, Graph
Theory
1 736- 1936,
Clarendon
Press, 1 976. 2 . P. Hoffman, The Man Who Loved Only Num bers: The Story of Paul Erdos and the Search for Mathematical Truth , Hyperion, 1 998.
3. D . Mackenzie, The Poincare Conjecture Proved, Science 3 1 4 (22 December 2006), 1 848-1 849. 4 . S. Nasar and D . Gruber, Manifold Destiny, The New Yorker (August 28, 2006), 44-57 .
5. J . J . Watkins, Across the Board: The Math ematics of Chessboard Problems, Prince
ton University Press, 2004.
More math comics by Courtney Gibbons are available online at: brownsharpie. courtneygibbons.org Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail : jwatkins@coloradocollege .edu
Fixing Frege john Burgess PRINCETON AND OXFORD: PRINCETON UNIVERSITY PRESS, 2005. PP. xii + 257. ISBN 0-691·12231-8, US$ 39.95
Reason's Proper Study: Essays Towards a NeoF regean Phi losophy of M athematics Bob Hale and Crispin Wright OXFORD, CLARENDON PRESS, 2001. PP. xiv + 455. US$ 45.00 ISBN 0-19-823639-5
REVIEWED BY 0YSTEIN LINNEBO
e know that there are infi nitely many prime numbers and that every natural number
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has a unique prime factorization. What sort of knowledge is this? Unlike our knowledge that Mogadishu is the capi tal of Somalia or that electrons have negative charge, arithmetical knowl edge does not seem to be empirical; that is, it does not seem to be based on observation or experiment. The German mathematician, logician, and philoso pher Gottlob Frege ( 1 848-1925) devel oped a bold new account of the nature of arithmetical knowledge: He argued that pure logic provides a source of such knowledge, and that arithmetic there fore is a priori rather than empirical. This view is now known as logicism and is one of the main philosophical accounts of mathematics (alongside for malism, intuitionism, conventionalism, and structuralism). Frege's defense of his logicist view of arithmetic proceeds in two steps. The first step consists in an account of how numbers are applied and of their iden tity conditions. Frege argues that count ing involves the ascription of numbers to concepts. For instance, when we say that there are eight planets, we ascribe the number eight to the concept " . . . is a planet". Let '#' abbreviate the op erator 'the number of'. Frege's claim is then that '#' applies to any concept F to form the expression '#F ', meaning "the number of Fs". Next Frege argues that the number of Fs is identical to the number of Gs if and only if the Fs and the Gs can be put in a one-to-one cor respondence. This principle (which is typically associated with Georg Cantor) is known in the philosophical literature as Hume's Principle (since it may have been anticipated by the philosopher David Hume). In order to formalize this principle, Frege makes essential use of the fact that his logic is second-order; that is, in addition to the ordinary first order quantifiers Vx and 3x, which range over some domain D, Frege's logic also has second-order quantifiers VR and 3R, which range over relations on D (of some particular adicity). Let 'F= G ' abbreviate the pure second order statement that there is a relation R that one-to-one correlates the Fs and the Gs. Hume's Principle can then be expressed as: (HP)
#F = #G � F = G
This makes sense because equivalence relation.
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=
is an
The second step of Frege's defense of logicism provides an explicit defini tion of terms of the form '#F'. Frege does this in a theory that consists of second-order logic and his "Basic Law V," which states that the extension of a concept F is identical to that of a con cept G if and only if the Fs and the Gs are co-extensional; or, in contemporary notation (V)
lxiFx) lxi Gx) � Vx(Fx � Gx). =
In this theory, Frege defines #F as the extension of the concept "x is an ex tension of some concept equinumerous with F." That is, he defines
#F = lxi3 G(x = lyiGy) l\ F= G)}. This definition is easily seen to sat isfy (HP). More interestingly, Frege proves in meticulous technical detail how this definition and his theory of ex tensions entail all of ordinary arithmetic. However, just as the second volume of his magnum opus was going to press in 1902, Frege received a letter from the English logician and philosopher Bertrand Russell, who reported that he had "encountered a difficulty" with Frege's theory of extensions. The diffi culty Russell had encountered is the paradox now bearing his name. Frege's theory of extensions is in effect a naive theory of sets. We may thus consider the set of all sets that are not members of themselves. In Frege's theory we can then prove that this set both is and is not an element of itself. Frege's re sponse to Russell's letter is remarkable. Sixty years later Russell described it as follows. As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege's dedication to truth. His en tire life's work was on the verge of completion, much of his work had been ignored to the benefit of men infinitely less capable, his second volume was about to be published, and upon finding that his funda mental assumption was in error, he responded with intellectual plea sure, clearly submerging any feel ings of personal disappointment. Russell's paradox eventually led Frege to give up on logicism. Until the 1 980s both logicians and philosophers regarded Fregean logicism as a dead
end, and people attracted to the idea of logicism pursued other versions of it, such as Russell's very complicated "type theory." However, over the past two decades there has been a resurgence of interest in Fregean logicism. A variety of con sistent fragments of Frege's theory have been identified and explored, and their possible philosophical significance has been vigorously debated. The two books under review are without doubt among the most important products of this resurgence. Reason 's Proper Study is the most extensive philosophical ar ticulation and defense to date of a spe cific neo-Fregean programme, whereas Fixing Frege offers the deepest and most comprehensive technical investi gation of a variety of different neo Fregean approaches. Neo-Fregeanism began with Crispin Wright (Frege's Conception of Numbers as Objects, 1983) who suggested that the problem posed by Russell's paradox be evaded by making do with the first step of Frege's approach, abandoning alto gether the second step and its incon sistent theory of extensions. This ap proach is made possible by two relatively recent technical discoveries. The first discovery is that (HP), unlike (V), is consistent. More precisely, let Frege Arithmetic be the second-order theory, with (HP) as its sole non logical axiom. Frege Arithmetic can then be shown to be consistent if and only if second-order Peano Arithmetic is. The second discovery is that Frege Arith metic and some very natural definitions suffice to derive all the axioms of sec ond-order Peano Arithmetic. This result is known as Frege's Theorem. It is an amazing result. For more than a century now, informal arithmetic has almost without exception been given some Peano-Dedekind style axiomatization, where the natural numbers are regarded as finite ordinals, defined by their po sition in an omega-sequence. Frege's Theorem shows that an alternative and conceptually completely different ax iomatization of arithmetic is possible, based on the idea that the natural num bers are finite cardinals, defined by the cardinalities of the concepts whose numbers they are. Technically speaking, the neo Fregean foundation of arithmetic is thus a success: it is consistent and strong
enough to prove all of ordinary arith metic. But what about its philosophical significance? Reason 's Proper Study, which brings together 1 5 essays by the two foremost neo-Fregeans, is an extended argument that neo-Fregeanism is a philosophical success as well. It is argued that this ap proach enjoys most of the philosophical benefits promised by Frege's original but sadly inconsistent form of logicism. I can here mention only three of the questions that Hale and Wright grapple with in their defense of this claim. The first question is whether the first of the two steps of Frege's approach (which I described above) can stand on its own. Frege himself thought it could not because (HP) fails to settle all ques tions about the identity of numbers. For instance, (HP) fails to settle whether the number of planets is identical to the Ro man emperor julius Caesar! In order to settle such pesky questions, Frege thought it necessary to proceed to the second step and give an explicit defin ition of the natural numbers. Hale and Wright disagree, arguing in stead that (HP) gives the criterion of identity for numbers; that non-numbers have different such criteria; and that this implies that Caesar cannot be a number. The second question concerns the epistemological status of Hume's Princi ple. As Hale and Wright admit, (HP) does not particularly "look like" a logi cal principle. They defend instead the slightly weaker claim that (HP) can serve as an explanation of the meaning of the #-operator and thus be known a priori. If correct, this claim would be very sig nificant, as it would establish that arith metic-with its infinite ontology of num bers-can be known a priori. This would be almost as good as what was promised by Frege's original logicism. The third question concerns the de mand for a deeper and more general understanding of the kind of explana tion that (HP) is supposed to provide. The demand is made particularly acute by the structural similarity between (HP) and the inconsistent principle (V). What if it is just a happy accident that (HP) is consistent? If so, the neo-logi cists can hardly claim that merely lay ing (HP) down as an explanation of the meaning of the #-operator yields a pri ori knowledge that (HP) is true. For surely a belief cannot count as know!-
edge if it is just a happy accident that it is true! Hale and Wright respond by arguing that knowledge does not re quire any kind of antecedent guarantee against error. It seems to this reviewer that this can at most postpone, not elim inate, the need for a deeper explana tion. After all, it is part of the very na ture of both mathematics and philosophy to seek general explana tions whenever such are possible. Whereas the agenda of Reason s Proper Study is predominantly philo sophical, that of Fixing Frege is pre dominantly mathematical. For instance, Fixing Frege has little to say about the first two questions mentioned above but a lot to say about the third. The opening chapter provides a very readable introduction to the mathemat ical aspects of neo-Fregeanism. Burgess first provides a useful summary of Frege's own constructions, of Russell's paradox and Frege's response to it, and of Russell's competing form of logicism. He then develops a sophisticated frame work in which various neo-Fregean the ories can be analyzed and their strengths compared. Particularly useful is his explanation of a hierarchy of mathematical theories, ranging from very weak subsystems of arithmetic up to very strong systems of set theory. This hierarchy provides a unified sys tem of targets for neo-Fregean recon struction, which enables us to measure the strength of a neo-Fregean theory in terms of how much of this hierarchy the theory allows us to reconstruct. The remaining two chapters explore the two main ways of ensuring the con sistency of Frege-inspired theories. The standard way-already encountered above and the topic of Chapter 3-is to abandon Frege's Basic Law V in favor of related but weaker principles such as (HP). An alternative way-which forms the topic of Chapter 2-is by placing re strictions on which open formulas can define relations. In order to explain this alternative way we need some defini tions. A comprehension axiom is an ax iom which states that an open formula ¢, with free variables x1 , . . . , Xn, suc ceeds in defining an n-adic relation R, under which n objects fall if and only if they satisfy the formula ¢, or in symbols:
3RVxl . . . Vxn[Rxl, . . . , Xn � ¢Cx1 , . . . , x�]
A comprehension axiom is said to be predicative if ¢ contains no second order quantifiers and impredicative otherwise. If regarded as definitions, predicative comprehension axioms have a nice property of non-circularity, namely that the relation R is defined without quantifying over a totality that includes R itself. The philosopher Michael Dummett has proposed an exciting but contro versial analysis of "the cause" of Rus sell's paradox: He blames the contra diction not on Basic Law V but rather on the presence of impredicative com prehension axioms in the background theory. To substantiate this analysis, it must be shown that restricting oneself to predicative comprehension restores consistency. Chapter 2 gives a nice pre sentation of some earlier theorems which show that this is indeed the case. But for the analysis to be plausible, it must also be shown that this restriction leaves the character of the relevant the ories intact; otherwise consistency will be restored not by excising a precisely circumscribable "cause of paradox" but more bluntly by rendering the theories impotent. But some new results from Chapter 2 show that the resulting the ories are very weak. So this bodes ill for Dummett's claim that impredicative comprehension is "the serpent that en tered Frege's paradise. " The final chapter examines t h e stan dard way of restoring consistency. Call a principle of the logical form
() *
§F = § G � F - G
an abstraction principle. Some abstrac tion principles-such as (HP)-appear to be acceptable, whereas others-such as (V)-clearly are not. Can a sharp and well-motivated line be drawn between abstraction principles that are accept able and those that are not? A natural thought is that (V) is made unaccept able by the fact that it requires a one to-one map from the concepts on a do main into the domain itself, which we know by Cantor's theorem to be im possible. Say that an abstraction prin ciple ( ) is non - inflationary on a do main D if the equivalence relation - is such that there are no more --equiva lence classes of concepts than there are objects in D. One easily sees that every non-inflationary abstraction principle has a model. *
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The question of when a system of abstraction principles is acceptable is harder. For even if the principles that make up the system are individually non-inflationary, the system as a whole need not be. This question receives a penetrating analysis in Kit Fine's Tbe Limits ofAbstraction (2002), where it is shown that a system of abstraction prin ciples is acceptable provided that each principle is individually non-inflationary and based on an equivalence relation that satisfies a certain "logicality con straint'' (involving invariance under per mutations of the domain D) . In Fixing Frege, Burgess gives a nice exposition of this analysis and substantially ad vances the discussion by pinpointing the strength of the resulting system: that of "third-order arithmetic . " Although the strength o f this sys tem is thus substantial, it falls far short of the stronger target theories in Burgess's hierarchy. In order to reach higher, Burgess proposes a new way to motivate the axioms of ordinary set theory. He first uses "limitation of size" considerations to motivate a so called reflection principle, from which he derives (drawing on earlier work by Paul Bernays) most of the axioms of ZFC set theory, as well as some large cardinal hypotheses. Although this is an impressive feat, Burgess ad mits that the motivation and the re sulting theory are no longer particu larly Fregean. Summing up, it is now clear, in a way it was not two decades ago, that a wealth of philosophical and techni cal insights can be rescued from the ruins of Frege's logicism. Whether these insights add up to a coherent and attractive philosophy of mathe matics is still (in my opinion) an open question. But Reason 's Proper Study and Fixing Frege are warmly recom mended as the best places to start for an examination of, respectively, the philosophical and technical insights to be learnt from Frege and the prospects for a neo-Fregean philosophy of math ematics. Department of Philosophy University of Bristol 9 Woodland Rd Bristol BS8 1 TB UK e-mail: oystein . [email protected]
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Letters to a Young Mathematician by Ian Stewart NEW YORK, BASIC BOOKS, 2006. HARDCOVER US $22.95 ISBN: 9780465082315
REVIEWED BY REUBEN HERSH
Dear grandchildren David, Jessica, and Ze'ev, As is customary, I list you in chrono logical order, by your year of birth. As all of you probably know, your grand father in New Mexico is a retired math professor. If it should some day hap pen, by some good book or some good teacher, that one or two of you get turned on to math, I'll be more than ea ger to help you, with my information and my advice. In all fairness then, I am letting you know that the new book by my friend Ian Stewart already offers such information and advice. Therefore, for you three, and also for any other potential readers of this prestigious pe riodical, I am going to tell you about Stewart's book. First I'll tell you about his informa tion. Then I'll tell you about his advice. And finally, I'll tell you my own advice. The book consists of 21 letters to "Meg . " The quotation marks inform us, I suppose, that "Meg" is imaginary or fictitious. In the first letter "Meg" is "at school." Since Ian doesn't hesitate to speak seriously and deeply to "Meg", I guess she's already in what we here in the States call "high-school." Meg is wondering what mathematicians do, and how her Uncle (??) Ian became a mathematician. As time passes, Meg chooses to study math(s) at University. By the last letter, she's concerned with what a tenure-track Assistant Professor is concerned with: the mad struggle to attain tenure. ("Tenure" is the college professors' word for "job security.) I am sure you , or anyone interested in mathematical life today, will find the "Letters" interesting and enjoyable. Stewart freely confides to Meg some of his own personal story, of how he was drawn to mathematics, and of some of his pleasures and successes as a math ematician. There is a really wonderful account of how an investigation into the abstract theory of groups turned out to
be of great use in analyzing the gait of four-footed creatures, like dogs! Who knew that there was even such an aca demic specialty as "Gait Studies"? Several of the chapter titles tell enough to make clear their messages: "The Breadth of Mathematics," "Hasn't it All Been Done?" "How Mathemati cians Think," "How to Learn Math," "Fear of Proofs," "Can't Computers Solve Everything?" "Impossible Prob lems. " Every sentence is clear and com prehensible. The love of mathematics that impels Stewart is always there; if the reader is susceptible at all, she or he may well become infected. Starting with Chapter 14, and going on to Chapter 20, the next to last, there is a noticeable change of tone and focus. "The Career Ladder," "Pure or Applied," "Where Do You Get Those Crazy Ideas?" "How to Teach Math," "The Mathemati cal Community," "Perils and Pleasures of Collaboration. " Ian is no longer talking to a child, sharing his enthusiasm and en joyment. He is talking to someone who is committed to mathematics, and is wor ried about how to make a living at it. Of course, this is a very realistic kind of conversation to imagine. In fact, many senior mathematicians, responsi ble for guiding advanced undergradu ates, graduate students, postdocs, and faculty just starting to teach, do have such conversations many times over their teaching careers. I imagine that while the "Meg" of the first half of the book is partly based on real acquain tance with school children, and partly a creation of Stewart's imagination, the "Meg" of the second part may well be an amalgam of many young mathe maticians Stewart has counseled. So what kind of advice does he give? I would say, sound and sober advice. Realist advice, how to get on in the math ematical world as it really is. (Meaning, of course, not necessarily as we would most, in our heart of hearts, desire it to be.) Stewart knows what's what, and he most kindly and sincerely wants Meg to make it, to get that job and that tenure. That means, knowing what hiring com mittees look for, and what promo tion and tenure committees look for,
etc.etc.etc. "THE REAL WORLD. " So, very good, what could be wrong with that? Nothing at all. And yet I can't help remembering a
young English mathematician I met years ago who did some crazy things. For instance, he wrote and illustrated COMIC BOOKS about fractals and chaos! And since, I guess, he couldn't get them published in English, he pub lished them in France, IN FRENCH! He even gave me copies of those two won derful works of his. That wasn't all that he did which some would have considered ill-ad vised. A "hot" new specialty in mathe matics appeared with a wonderful name: "Catastrophe Theory." My English friend became an active worker in this new field, and an active public advo cate of it, even though everyone knew that it was controversial. Many influen tial senior mathematicians disliked it, considered it a shallow fad, vastly over publicized because of its exciting name. Many of his older friends must have had doubts whether this was really the wis est career move he could make. Of course, he also did fine work in other noncontroversial specialties. But he also did something much more un wise. He knew, of course, that many re search mathematicians don't have the highest admiration or respect for jour nalists. And that to many people high up in mathematics, popular books that can be understood by anybody are hardly above journalism. And yet, what did he do but write lots of popular books! Over 20 are listed in the front matter of "Let ters to a Young Mathematician." I would say that the career of Ian Stewart is the career of a supremely suc cessful mathematician whose first con cern does not seem to have been "play ing it safe . " S o , now I a m ready t o offer m y own advice. I have tried to tell you about Stewart's advice. You may know, from experience or from hearsay, about a parent who admits that he isn't in all ways an ideal role model for his child. He may admit to some weaknesses or even vices. But, "Child," says he, "do as I say, not as I do . " I am enough of an optimist to look at things the opposite way. Read Stewart's book, enjoy it, hut do as he Does, not as he Says. 1 000 Camino Rancheros Santa Fe New Mexico 87505 USA e-mail: [email protected]
Funf M inuten Mathematik by Ehrhard Behrends WIESBADEN. FRIEDR. VIEWEG 2006, 256 PP,
&
SOHN VERLAG,
22.90, ISBN-10: 3834800821,
ISBN-13: 978-3834800824
REVIEWED BY VAGN LUNDSGAARD HANSEN
inner is served: One hundred well assorted German mathe matical "tapas·· online. Each of them takes five minutes to consume, and what an enjoyable five minutes! The recipes for the mathematical tapas are collected in a book by Ehrhard Behrends based on the first one hun dred of his columns ''Fiinf Minuten Mathematik," written for the German daily newspaper Die Welt in the years 2003 and 2004. The tapas are easily di gestible and well prepared, indeed. Behrends is an experienced popu lariser of mathematics. His columns in Die Welt have attracted much attention, and by reading his book you can see why. There is something for everybody with the slightest interest in mathemat ics in the book. And Behrends is not compromising with his presentation of mathematical results and methods. One gets real and valuable information about mathematics in small doses. This is, I think, a fruitful way of reaching out to the general public. Having read the col umn one week, I am sure that many readers have looked forward to seeing what fascinating mathematics the next column would bring the following week. It is a pleasure to notice that Behrends covers almost all subjects of mathemat ics: number theory, algebra, geometry, analysis, probability theory, stochastics, and pure as well as applied topics, like coding theory. The hook demonstrates that, with the right care, almost anything from mathematics can be presented so that a lay person can get some feeling for it. Space prevents me from describing the 100 columns in detail; I can only present a few of my own personal favourites. In a nice column on mathe matics and music, Behrends explains the Pythagorean and the chromatic tone scales, revealing his insights and strong
interest in music. His mathematical re search field is probability theory, and there are several interesting columns on topics from combinatorics and the the ory of games-among others, a column on mathematics and chess. I also liked columns on the old classics from geom etry about constructions with ruler and compass. In a column titled 'How much mathematics do human beings need?' Behrends shows that he is also prepared to take up this kind of discussion, which I suppose all mathematicians engage in from time to time. The book is attractively prepared by Vieweg, and several illustrations are in colour. The book would make a good gift for a lay person with an interest in mathematics. The hook is also valuable for the mathematician who needs ideas for the occasional, unprepared mathe matical conversations at dinner parties. Given the opportunity, why not offer your guests some mathematical tapas from Behrends' book "Fiinf Minuten Mathematik"? They will probably be sur prised hut they will enjoy it. And those five minutes spent with mathematics will truly be remembered! Department of Mathematics Matematiktorvet, Building 303 Technical University of Denmark DK-2800 Kgs. Lyngby Denmark e-mail: [email protected]
Images of a Complex World The Art and Poetry of Chaos by Robin Chapman and julien Clinton Sprott SINGAPORE, WORLD SCI ENTIFIC PUBLISHING CO. PTE. LTD., 2005, 175 PP. PLUS CD, $34.00 ISBN-13: 978-9812564016
REVIEWED BY JOHN PASTOR
he goal of both mathematician and poet is to seek clarity and beauty of expression about the world around us through elegant use of their respective languages. While nature
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is a common source of inspiration for both mathematician and poet, the poet examines the human response to nature while the mathematician explores the logical order of nature. Despite these similarities, creative use of both mathematics and poetry together is uncommon. Several mathematicians have written poetry-]. ]. Sylvester and James Clerk Maxwell sometimes incor porated poems into their papers, al though these are now forgotten (perhaps with good reason) while their mathe matics continues to inform new research. Many of us have enjoyed the light verse of Ralph Boas, Jr. [1] and poetry has of ten graced the pages of 7be Mathemat ical lntelligencer. But few poets have dared to incorporate mathematical themes in their exploration of the hu man condition, although Anne Michaels has captured Kepler's life and thought superbly in her long first-person poem, A Lesson from the Earth [2), which be gins "I begged scraps from the Rudol phine Table-the rinds of orbits, stars scattering like pips spat from Tycho's chewing mouth . . . " and continues with " . . . We must learn this lesson from the earth, that the greater must make room for the small, just as the earth attracts the smallest stone . . . " and " . . . I used to think that we escape time by disap pearing into beauty. Now I see the op posite. Beauty reveals time." In Images of a Complex World, a poet and research psychologist (Chapman) and a physicist and dynamicist (Sprott), both at the University of Wisconsin, col laborate on exploring the beautiful world of dynamical systems and nature through poems, illustrations, and thumbnail es says. Although billed as an addition to your coffee table, this book really be longs in your classroom instead. By adding depth and dimension to many dynamical ideas and concepts, Chapman's poems enrich our and our students' understanding of them. The accompanying poem, Fixed Point, is from a set of poems titled Stillness (in the second column). The clarity and depth of her poem is not, of course, the pithy clarity and depth of :x! = f (:x!). But just as we teach our students how to unpack an equation to discover its hidden meaning, so does Chapman unpack the concept of a fixed point to uncover its hidden poetic beauty. The poems in this book also explore
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The Rxed Point The dot: how it stops everything. Finishes the thought. Ends t.h sentence. \\7h r everything vanishes in the end. P riod. l3u r it is not the all of it, though all come to it. It is only the idea of no di m ens i on over which \Ve exclaim, the van ishing point that lends the observer perspective.
a fiction of the eye too far away to see speck, mote. egg.
the human condition at the same time they illustrate mathematical ideas. How many of us have had difficulty ex plaining to students the meaning behind the property of nonlinear systems that f(x + y) =I= f(x) + j(y)? Here's Chap man's illustration of nonlinear systems, poignant not only for capturing the essence of this property but also for de scribing an all-too common condition of contemporary human existence:
Def 2: One in which does not equal f (x)
f
(x
+
+ f (y)
y)
This is easily enough understood By an} child of divorce .\l om's house And Dad's house are not the same
As the house with both Dad and
'\t om bdo rc . Or think off as h a ppine ss , And kn
