Letters to the Editor
The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
The Composition Identity I write to protest an appalling mathe matical scandal: the lack of a standard notation for the composition identity. The nearest thing to a standard no tation is the abbreviation "id," assum ing the domain is clear from the con text. Note that "id" is two letters which won't do in an introductory cal culus text where the need first becomes urgent. The notation x� x is even worse. One "solution" (in the context of single-variable calculus) is to make ex plicit the often tacit identification of the independent variable with the compo sition identity. Then the common gaffe f= f(x) is correct, since the variable x represents the identity function on the domain off Unfortunately, the inerad icable habit of always thinking x' = 1, which this convention engenders, naively but inexorably mutates to u' = 1, thereby subverting the chain rule to (f( u))' = .f'Cu). Students then view cor rections to this faux pas as exceptions to the rule, notably when u represents a constant, or when x represents a func tion of t in a related-rates context. Adoption of a universal symbol for the identity function would do away with much if not all of this type of con fusion. I have seen the pound sign, the dol lar sign, and other special characters used in various contexts. For example, if # represents the identity function, then you can write (e#)' = e# instead Of the CUmbersome (X� eX)' =(X� ex), or (e-")' = e'", which is not true un less x' = 1, i.e., only when x represents the identity function. Uppercase i or "I" for identity is another possibility for more general application. The identi cally equal symbol with three horizon tal dashes (not possible to write in this text editor) is another possibility. I propose that Tbe Mathematical In telligencer take nominations for two or three years and then hold a vote. Our foresight will be taken for granted in future generations as we now take for granted the use of "0" for zero, the ad-
ditive identity, though it, too, was courageously adopted only when long overdue. My choice would be�. the Greek let ter iota, if it were available on standard keyboards. Forest W. Simmons Portland Community College Portland, Oregon USA e-mail:
[email protected] Reply Does the identity function need a sym bol of its own? Fifty years ago, Karl Menger made the case for a variable free calculus ( Calculus: a modern ap proach, Ginn and Co. , 1 955), but there is still no consensus. In this issue, For rest Simmons reopens the discussion. We hope his letter will spark a debate. Please send us your thoughts-and your suggestions. (Two candidates Menger j, Simmons �-are already in the running). We'll invite you to vote in two years or so. -The Editors
The Road to Reality In the Summer 2006 issue you pub lished two reviews of Roger Penrose's book Tbe Road to Reality. They bring to mind the standard politics of two party Anglo-Saxon democratic systems as trivialized by journals such as Newsweek, or rather, the "good cop bad cop" approach to criminals. Did you do that by mistake, or on the con trary, as a matter of pride, to try to im plant that approach into science? In the less than fortunate latter case, one can wonder why only two opposing views were presented. Why not, indeed, three, or even more opposing views? After all, why not bring some sort of circus into rather arid realms like mathematics? And now back to the two reviews. The first, shorter and quite sparse in detail, finds the book highly meritori ous and readable. The second finds quite a number of outstanding features,
© 2007 Sprtnger Science+ Business Media, Inc., Volume 29, Number 3, 2007
5
but that is totally and hopelessly drowned in a manifestly vicious over all prejudiced attitude and judgement. One can only wonder how a third, or perhaps, fourth and so on, review might have looked, had The Mathe matical Jntelligencer gone one better than the trivial Newsweek approach. I myself have had some arguments with Penrose on certain strictly mathe matical issues; thus I cannot be counted as one of his unconditional admirers. But I would like to say here, first and above all, that the subject of the book is by far the most fundamental and con sequential of the last few centuries. Second, for more than half a century now, science has discouraged scholar ship, especially wide-ranging and deep scholarship, in favour of narrowly spe cialized research production. Penrose happens to be one of the very few scholars, if not in fact the only one nowadays, with truly impressive depth and breadth. Consequently, even if his latest book were rather poor, which clearly it is not, one should appreciate his scholarship and his willingness to make the considerable effort to bring it into the public domain. Penrose, in this book, has given us a grand and most fascinating view of a fundamental and all-important field of science. A view that, hopefully, will tempt many in future generations to try to complete. For others who care to look at it, or to browse it, or read parts of it, the book may help them connect to things beyond, and no less impor tant than, day-to-day concerns or events. Elemer E Rosinger Emeritus Professor Department of Mathematics and Applied Mathematics University of Pretoria
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THE MATHEMATICAL INTELLIGENCER
2007 Springer Science+Business Media, Inc.
UM'ii
A Mathematician Called Bourbaki
H
ieremia Drexel (1 '581-1638) was a jesuit and professor of "the clas sics" at Augsburg ( Germany). He wrote many books in Latin on history and theology. Among these books is Aurifodina Artium et Scientiarum om-
nium; Excerpendi Sollertia Omnibus lit teraru m amantibus monstrata [ Gold Mine of Arts and Sciences, judiciously Chosen Extracts to Be Shown to Cul tured Amateurs] (Figure 1). In this book the author describes the
FRANCOIS l.AUBIE
Figure I.
In the frontispiece author.
is probably the
the
writer working by lamplight, at the right,
© 2007 Springer Science+Business Media, Inc .. Volume 29. Number 3, 2007
7
Figure 2.
The passage in question.
state of the arts, literature, the sciences, religion . . . as if for a scholar's guide. On page 265 of the second edition (Antwerpen, 1641), he lists the eighteen mathematicians he considers to be the best in the world. In this list we find ARCHIMEDES, COPERNICUS, KEPLER, . . . and "Georgius BURBACHIUS"! We note that BURBACHIUS is the natural latinized version of BOURBAKI. Let us translate the quotation de picted in Figure 2: " . . . in my opin-
On Two Fellows Who Wanted to Mal<e Money on Fluctuations JACEK MI�KISZ
8
THE MATHEMATICAL INTELLIGENCER
ion, the most remarquable doctors in mathematics, amongst the most recent, are Johannes REGIMONTANUS and his professor Georgius BURBACHIUS; the Scot Alexander ANDERSONIUS, the Prussian Nicolaus COPERNICUS, . . . " The puzzle is easily solved: Johannes Regimontanus (1 436-1476)-his real name was Johan Muller-is quite well known; he was the student of Georg Von Purbach ( 1423-1461), Viennese mathematician and astronomer. Thus it is simply an author's error: "BUR BACHIUS" instead of "PURBACHUS". One of my colleagues (D. Roux) no ticed that the coincidence is furthered by the first name of Nicolaus Coperni cus, which, in the printed text, lies di rectly under the surname Burbachius. But is it really a coincidence? A founding member of the Bourbaki group, and lover of old books, could have read this one and remembered this quotation (perhaps subconsciously).
I
write this to warn readers about two swindlers who hang around this neighborhood. One of them intro duces himself as a Mathematician and the other as a Physicist. They want to involve people in enterprises that look extremely promising. I will not soon forget the day when I was approached by the Mathemati cian. Listen-he said-I'll throw a coin. If heads comes up you will pay me a dollar. If tails comes up I'll pay you a dollar. Fair enough, I said. The proba bility of winning is 1/2 for each of us. After a while both of us agreed that the game was boring. Here is another one of my games said the Mathematician-! wonder what you will think of it. In addition to the true coin, I have two weighted coins; the probability of heads for the second coin is 3/4, and for the third coin it is 1/10. Which of these two coins I throw depends on my "financial status" (which may be given by a negative number). If the figure describing my funds is di visible by 3, then I throw the third coin, otherwise the second coin. As before, heads means a dollar for me and tails a dollar for you. Well? said the Mathe matician, and looked at me expectantly. At first glance the proposal didn't look too good for me. I reasoned that,
Xlim, UMR 61 72 CNRS Universite de Limoges 87060 Limoges Cedex France e-mail:
[email protected] Comment Back in 1994, I reported the existence of the name Bourbaki in a book printed in Berlin in 19 18, and I bet 10,000 Pold evian crowns that this was the earliest occurrence of the name in any book of mathematics (see The Mathematical In telligencer 16 ( 1 994), no. 1 , 3-4). My challenge has been superbly met by Fran2 = 3/4- E,
where E is a fixed, small positive num ber. For then, as is easy to verify (please, kind Reader, do this!), the probability of your winning, in either game, is greater than 1/2. Of course r said-I'll be glad to play. Then came the suggestion I won't for get for a long time. The Mathematician suggested that, for the sake of variety, we should switch from one game to the other randomly. To avoid boredom he said-we'll play the first game with probability 1/2 and the second game with the same probability. After a few hundred runs, I realized with horror that my balance was deep in the red. The Mathematician left in a burry, and I em barked on a post mortem examination. The second game, like the first one, favors me. But if the figure of the Math ematician's funds is divisible by 3, which happens more or less in half the number of cases, he would not throw the third coin-which favored me-but the first coin. But then I could win or lose a dollar with equal probabilities. This may have been the source of my problem. I began to compute. I ana lyzed the Markov chain corresponding to the random combination of the two games with E 0. I concluded that the limiting frequencies of being in states 0, 1, or 2 were 245/709, 180/709, and 284/709, respectively. In the end, the probability of my winning in a move is
This means that if we play long enough, my funds will decrease in pro portion to the elapsed time with pro portionality coefficient - 18/709. Now the frequencies of visits of states of our Markov chain, and hence the expected values, depend continuously onE it fol lows that, for an appropriately small E, though each of the two games favors me, their random combination spells fi nancial disaster for me. Thus we are dealing here with an example of two random dynamics for each of which the expected value of a certain random variable goes up in time, whereas for the random combination of the two dy namics the expected value of this ran dom variable goes down in time. I urge you to check these results (ei ther analytically, on a piece of paper, or by simulating coin throws on a com puter). This may build up your resistance to other tricks of probabilistic swindlers. () ()()
I barely managed to come to after the encounter with the Mathematician when the Physicist knocked on my door. He wasted no time on preliminaries and showed me a sketch of his new device (see Figure 1 ) . It consisted of an axle with a fan at one end, a ratchet wheel with a pawl at the other end, and a spool with a thread in the middle. There was a little weight at the end of the thread. All components of the device were tiny. Due to random fluctuations-said the Physicist-there are moments when more particles hit the fan blades on one
side rather than on the other side. The situation is similar to Brownian motions of a particle in a suspension accidentally hit by particles of the surrounding fluid. But for the pawl, the fan blades would move now clockwise and now coun terclockwise. My device-the Physicist summed up--replaces variable fluctua tions with a single selected direction of rotation. The fluctuations supply work, and so we have a free source of energy. The device can be yours, hut, of course, not free. This time I was cautious. The en counter with the Mathematician taught me that the composition of two random dynamics, for which the state of the sys tem remains on the average unchanged, can result in unidirectional motion. But the pawl, like the little fan, is subject to impacts of surrounding air particles, per forms analogous Brownian motions and, as a result, every now and again goes accidentally up and lets the weight drop. In effect, the average shift of the weight is zero: the asymmetry of the pawl won't work miracles. Had the temperature around the pawl been lower than the temperature around the little fan, then the number of fluctuations of the pawl would have been smaller than that of the fan, and the Physicist's device would have really done work-at the expense of the energy drawn from the warmer environment and transmitted in part to the colder environment. 2 This is an ex ample of the Brown engine. Alas, there is one more difficulty: we would have had to maintain steady air temperatures
=
245/709( 1/2 . 1/2 + 1/2 . 9/10) + 180/709(1/2 . 1/2 + 1/2 . 1/4) + 284/709(1/2 . 1/2 + 1/2 . 1/4) = 691/1418, which is less than 1/2. Then the expected value of my profit for one move is 691/1418 - 727/1418
=
- 18/709.
Figure I
© 2007 Springer Science+Bus1ness Media, Inc., Volume 29, Number 3, 2007
9
around the fan and the pawl, and this, of course, would have involved an ad ditional cost. Yet another example of constructing perpetuum mobile (of the second kind) went down the drain. In the evening I talked with a Biol ogist friend. I told him about my math ematical and physical "adventures" ear lier in the day. The Biologist made an interesting comment. Maybe-he said Nature had found a way of exploiting microscopic fluctuations of particles in cells, and of using the energy liberated in biochemical reactions to transport useful cell components. Maybe the mol ecular cell motors function like Brown engines. I was glad the Biologist made no attempt to sell me molecular cell mo tors. We focussed instead on consum ing tasty food items, that is, transform ing the offerings of the head chef into simple organic compounds-an activity I recommend to all of my readers. 3
With my very best wishes, ]acek Mi�kisz, a physical-biological mathemati cian. NOTES
1 . A minicourse on Markov chains In our game we are dealing with a system that can be in states 0, 1, or 2. These states
�Springer
the language of science
correspond to the remainders in dividing the Mathematician's funds by 3. In the second game, the transition probabilities between these states, pij; i, j 0, 1, 2, are, respectively, 0, Po1 0.1, P o2 0.9, Poo P11 P22 . 75, P21 0.25. P12 0. 75, P10 0.25, P2o =
=
=
=
=
=
=
=
=
=
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=
5/13,
=
6/13,
11"0 + 71"J +
11"2
=
1'
and this gives the asymptotic frequencies of vis iting the states of our system. In particular, the
memory loss, which can be briefly character
figure of the Mathematician's funds is divisible
ized as follows: if we know the present, then the future does not depend on the past. Let 1r1j
by 3, on the average, in 5/13 of all throws. 2. The idea of a device utilizing Brownian mo tions to do useful work was first discussed in
denote the probability of the system being in state i at time t. Then the formula for total prob ability implies that
The above evolution of our system is an ex ample of a Markov chain. Note that we can go from one state to another in a finite number of steps. For such chains, with the passage of
1912 by Marian Smoluchowski, and was sub sequently developed by Richard Feynman (see Feynman, Lectures on Physics, vol. 1 , part 2, ch. 46). In 1996 Juan Parrondo wrote an (unpub lished) article titled "How to cheat a bad math ematician," in which he proposed certain para doxical gambling games. 3. The Polish original of this note appeared in
time, the frequencies of visiting particular states
Delta, a publication of the University of War
of the system tend to values that are indepen dent of the initial state. In our case, the limiting
saw, and is used by permission. Translation by A. Shenitzer.
probabilities satisfy the following system of lin ear equations:
Institute of Mathematics, Polish Academy
1ro
=
1r2
=
11"1
=
0.2571"1 + 0. 7571"2, . 1 7ro + 0.25 71"2 , 0.9 7ro + 0. 7571"1.
of Sciences 00956 Warsaw 10 Poland e-mail:
[email protected] springerlink.com
.,. Journals, eBooks and eReference Works integrated on a single user interface .,. New powerful search engine .,. Extensive Online Archives Collection
THE MATHEMATICAL INTELUGENCER
7T2
2/13,
The probability of being in state i at time t + 1 depends only on the state we were in at time t. This is the so-called Markov property of
The world's most comprehensive online collection of scientific, technological and medical journals, books and reference orks
10
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satisfies the condition
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Viewpoint
Poetic Metaphor and Mathematical Demonstration: A Shallow Analogy M IRIAM LIPSCHUTZ-YEVICK
The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. Viewpoint
J
an Zwicky in "Mathematical Anal ogy and Metaphoric Insight" [Zw] says that understanding a poetic metaphor feels like understanding cer tain mathematical demonstrations. She investigates the correspondences he tween the notion of metaphor primar ily as it is used in poetry and that of analogy in the development of mathe matical demonstrations. Although she clearly states that metaphors and math ematical analogies are not the same thing, she maintains that there are such fundamental similarities that they should both be considered as species of "analogical reasoning. " She posits that the sense of understanding, the "flash of insight" (the "I get itl" , the "Eu reka moment" ) on grasping a metaphor or a demonstration, is closely related in the two domains. Analogy, the drawing on associa tions, is all-pervasive in our thinking, in our language, and in our creative en deavors, be they artistic, scholarly, or everyday: concocting a new Italian recipe; racial stereotyping; formulating a legal opinion; making a medical di agnosis; the "coup de foudre" estab lishing a romantic link; and so on. As soCiation appears as an essential concept in the Fourier logic represen tation of brain function proposed by Karl Prihram [Prl, [LYll. Similes are imbedded in our language, carrying much of our meaning ( as proverbs, be fore they were so displaced by techni-
cal jargon, used to do). Zwicky's article abounds in them: "the field of reso nance", "lift off the page", "has no pur chase on", "cede pride of place" . . . . Hannah Arendt [A] wrote that "all con ceptual or metaphysical language is ac tually and strictly metaphorical. " Zwicky argues that metaphor and mathematical demonstration have spe cial kinship, in that in both, the new in sights derive from discovery of unsus pected analogies between facts long known but wrongly believed to be strangers to each other. But this kinship extends to all creative thinking! I maintain that analogical reasoning, being a generally present feature of thought, can not prove mathematical reasoning any closer to poetry than say ing that both are thought. Going beyond this commonality, we follow the divergent aspects in the further use of analogy in the two domains-"Points of Non-Correspon dence", as Zwicky calls them-and find two different "Languages of the Brain". To me, they look complemen tary (as the word is used in physics). Mathematical thinking analyzes; it is modelled, perhaps, by digital logic of networks [vN] . Poetical thinking embellishes; it more resembles holo graphic pattern recognition. Let us look at the dichotomy. Though the two use analogy differently, their symbiosis may suggest a more insightful mode of thought.
should be submitted to the editor-in chief, Chandler Davis.
Metaphorically Valid? While my husband and I were graduate students at M.I.T. during WWII, the young Walter Pitts. a brilliant protege of the great mathematician Nor bert Wiener, offered to deliver a lecture on "Sinkiewicz's Theorem" to an eager audience of graduate students. Pitts gave, as usual, a dazzling per formance. He proved the theorem moving seamlessly through a maze of lemmas and analogies, with frequent hand-waving to bypass the "obvious. " His ( almost poetic) presentation was received with applause and admiring comments. The lemmas were profound, the theorem still more so. Even though the lecture had the form of-and felt like-a proof, unbeknownst to us it was fiction. ( Sinkiewicz was in fact a Polish novelist.)
© 2007 Spnnger Science+ Business Media, I n c . , Volume 2 9 , Number 3, 2007
11
Proof vs. Gestalt: Two Modes of Creative Endeavor Poincare (P] defined Discovery as "ap pearances of sudden illuminations, ob vious indications of a long course of pre vious unconscious work. All that one can hope for from these inspirations which are the fruit of unconscious work, is the point of departure for such calculations. They must be done in the second pe riod of conscious work: results must be verified and consequences deduced." This is the dichotomy: on the one hand Zwicky's visual intuition, the "see ing as"; on the other, a rigorous proof, which requires analytic validation, a de rivation from axioms, or an algebraic computation. Creating a mathematical proof
Professor Norbert Wiener's lectures dur ing my graduate studies at M.I.T. ( 1943-1947) were revelations of the re searcher at work. The classroom had blackboards on three sides. Wiener, starting on the left wall, would write the theorem he intended to prove. He pro ceeded to accomplish this by assem bling a chain of valid deductions from various lemmas he had previously given in class. Talking to himself as much as to us-though with an occasional "just a minute, just a minute"-he would pro ceed from blackboard to blackboard, never losing his thread, though perhaps leaving us well behind. After much trial and error he might exclaim, "Let us do a Cesaro job on this" or the like. At the end (often close to the last space on the right wall blackboard, near the door) the proof would be complete. To the students he appeared to be pulling the proper tools from his toolkit,
as an experimental scientist would reach for instruments. If he thought of his tools in an analogic way, like the "Cesaro job", this was not the defining feature of his work. It is hardly necessary to insist on the magnitude of Wiener's discoveries [M]. No doubt he had to perceive un suspected analogies between facts long known but wrongly believed to be strangers to each other, as Zwicky says; but then came the checking; his lectures both displayed the checks and displayed his conviction of their importance. Creating a poem
Allow me to give some impressions of the experience of writing a poem. This poem was inspired by what I had learned in 1948 about the fate of my Jew ish classmates at a genteel girls' school in Antwerp. The school had a hateful dis ciplinarian Principal with a strict rule for the wearing of gloves. I had not thought of her for years. Only on recently en countering an acquaintance with similar background, and seeing her gloves and how she removed them to shake hands, as we were taught, did a chain of asso ciations start up which led to the poem [LY2]. Some lines from the end: "This one and that one," They picked out my former class mates. One by one they gathered them. Even the blonde, blue-eyed Berthe Perelman Whose name betrayed her. "A Jewess!" The Principal stood by her post at the head of the passageway, As the girls walked by to the trucks waiting outside.
MIRIAM LIPSCHUTZ-YEVICK was bom in Scheveningen, Holland,
and arrived in the U.S. in 1940 after a three-months-long fiight from the Nazis. She eamed her doctorate at M.I.T. in 1947 (one of the few
women in mathematics up to then). She was at Rutgers (University Col lege) from 1964 until retirement She has published
on
probability, on
her invention holographic logic, and on other areas, including a text
Mathematics
(or the Billions
for her remedial students. She is a deeply
devoted grandmother.
Miriam Lipschutz-Yevick
22 Pelham Street Princeton, NY 08540 USA
e-mail:
[email protected] 12
THE MATHEMATICAL INTELLIGENCER
Did she check to see if the girls wore their gloves? The writing of the poem seemed to arise from a consciousness distinct in character from mathematics. After read ing Jan Zwicky's article I subjected my poem to critical scrutiny and noted nu merous metaphors. I might have started with a problem: my feelings of guilt for having escaped the fate of my friends; my buried wish for revenge on my Principal; the contrast between the en forcement of good manners and the brutality of the Nazis . . . but then-no poem. Yet the poem I did write holds all of this "compact in one." A poem, even a long one, may some times be grasped as an emotional whole; a lengthy proof can be recon structed only step by step, even if one has first grasped the general thrust. The thought processes in the writing of a poem do not take the conscious form "this reminds me of this reminds me of this . . . ", though they may do so in an attempt to implement an intu itive mathematical perception. Rather the search for "the right word" for use as poetic metaphor is often a tip-of-the tongue phenomenon, and in all cases feels almost the reverse of mathemati cal puzzling.
Conclusion Associations, "atoms" of analogies, guide our discoveries, be they poetic, mathe matical, judicial, culinary, amorous, or whatever. Sometimes the "right" analogy is discovered by a chain of steps. On other occasions it pops up sponta neously as though by "ghosting" the ob ject with another stored jointly in an as sociative hologram, or by the emergence
of a cluster of unconnected associations which together recreate the object. Comparing the domains on which analogy acts, I find another contrast: po etic analogy casts a wide net (the search for the telling metaphor runs through a wide web of relevant associations); mathematical analogy deals with con cepts appropriate to a particular theory, and the validation is deductive and se quential in character. When mathemat ical research does widen its scope, it is by generalization, which "by condens ing compresses into one concept of wide scope several ideas which seemed widely scattered before" (P6lya [Pol, p. 30) . Alas, this sometimes relies on for malism which obscures the ideas re lated, thereby impoverishing the mean ing and insight. "Mathematics does notfit all. " Cloth ing humanistic and social sciences in
mathematical garb is a technique fre quently used by scholars in the non quantitative fields ( The Phillips Curve, The Bell Curve, and so on) to over-awe a quantitatively uneducated population. Perhaps we agree that the technique can be pernicious. I have tried to argue that, likewise, poetry has nothing to gain from overstating its resemblance to mathematics-and vice versa. Rather our aim should be to teach the general public to appreciate the insights of the two domains. And beyond that, to un derstand and act upon the problems of our world with rational thought and em pathic feeling.
Jan Zwicl k. This results in the trivial solid torus braid. D
Constructing the Generators
There is a well-known method of constructing new juggling patterns from old ones. Let (h(O) , h(l), h( n - 1)) be a ·
the " 1 " throw.
In other words,
the map J is defined only
·
· ,
up to reflections of
links.
0
wn at
•
I =
1 =
I, .
==
==
·
·
·,
•
paration of 34 uni
juggling sequence. Let a and h be integers such that 0 :S a < b :S n - 1 and b - a :S h(a) . We construct a new se quence which coincides with h(i) on all beats except at a and b. Namely, define g(i) h(i) for all i of. a, h, and let g(a) b( h) + ( h - a) and g(h) = h( a) - ( h - a). The fol lowing is immediate.
quence.
paration of I unit
10
The braid diagrams for I1 , h and
THEOREM 8 (g(O), g(l),
0
paration of 3 2 unit
I
I
•
throv n at
0
I
1=
thrown at
Figure 5.
0
0
g( n - l l) is a juggling se
In other words, the sites b( a) and h(b) in which the balls land will swap positions. Indeed, the term site swap is used because of this property. In our context, when attempting to construct the gener ators of the braid group, the creation of a crossing is sim ply an extension of this idea of swapping sites. Starting with the identity braid In, the strands to be crossed are chosen and two of their throws are manipulated (swapped), cre ating a crossing in an otherwise trivial braid. Figure 6 shows a part of the braid diagram of 1,/ t), where only the k and k + 1 strands are depicted. The length of heats between Z two hands is scaled by a factor j k- 2
Over-crossing:
Let a);
==
h�
a k+ l and
_
a'k = (33
h� = _
a k + 32k+ l . Then
1) .
3z k- 2 < 3z( k+ l)- 1
=
dln(ak"J ,
satisfying the conditions of Theorem 8. We take a funda mental chamber of In( t) and swap the two sites d!,(a%) and d!,lb'k ) , resulting in
{
+ ( h); - a);) if t = a); mod 2 C/e,Jt) = J,(aJ:) - ( b'k - aJ:) if t = b'k mod 2 J,( t) otherwise. I,( b);)
Notice the length of its fundamental chamber is
·
2
·
32 "- 1 32 " - 1
·
32 "- 1 .
LEMMA 9 j(Ck,n(t))
is the solid torus n-braid with its k-strand crossing over its (k + 1)-strand.
Proof Theorems 7 and 8 guarantee that Ck, n( t) is a juggling pattern. We need to show the crossing information is as claimed. The throw value of ball k + 1 at a'k increases from 32( k+ ] )- l to y k- 1
+ Cb'k - aiD = (33 + 3 - 1 l .
3zk- z .
© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3 , 2007
19
I I
Figure 6.
rati n
f 3 2k - 2 unit
..
•
Y.
O O O O
O O O
;�z
Braid diagram only of the k and k + 1 strands of I,z(t).
,.
:
(
__
Figure 7.
0 0
Braid diagram showing only the k and k + 1 strands of Ck,n(t) .
0 Figure 8.
Braid diagram only of the k and k + 1 strands of Ck,�(t).
Crossings will not interfere with other }strands for which j > k + 1 , because 32j- l
2:
32(k+ 2 )- 1
=
35 . 32k-2 > (33 + 3
.
- 1 ) 3 2 k- 2 .
Similarly, the throw value of ball k starting at position b'k decreases from 32k- l to
So crossings will not interfere with other }-strands for which j < k. Thus, the only crossings that can occur are between the k and k + 1 strands around the swap location. Figure 7 depicts the details: the k-strand at position b'k + y k- z has a throw value of 32k+ 1 , thus crossing over the (k + 1 )-strand thrown from b'k + Yk- l with a value of 32k- l . D
Under-crossing:
This is identical to the situation above, except for different choices of sites to swap. Let a� = a k+ 1 and b� = a k + 32k- l . Then b� - a� = 2 32k- z < 32C k+ l ) - l dln(a�), satisfying the conditions of Theorem 8 above. We take a fundamen tal chamber of In(t) and swap the two sites dlnCa�) and dln(b�), resulting in ·
20
THE MATHEMATICAL INTELLIGENCER
=
{
InCh�) + ( b� - a�) if t == a� mod 2 32 n- l � Ck, ( t) = In C a�) - ( b� - a�) i f t = h� mod 2 32"- 1 otherwise. In( t) � LEMMA 1 0 :J(Ck, (t)) is the solid torus n-hraid with its k-strand crossing under its (k+ 1)-strand. ·
·
Proof Again, Theorems 7 and 8 guarantee Ck, n(t) to be a juggling pattern. The throw value of ball k + 1 at a� de creases from 32Ck+ l) - l to 3zk- 1
+ ( h�
_
a�) =
y k- l + 2
.
3zk- z.
Thus crossings will not interfere with other j-strands where j < k. Similarly, the throw value of ball k starting at posi tion b� increases from 32k- l to 3zck+ D - l
_
( b�
_
a�)
=
3zk+ l
_
-.
2 . 3zk z
So crossings will not interfere with other }-strands, where j > k + 1 . Thus, the only crossings that can occur are be tween the k and k + 1 strands around the swap location; see Figure 8 for details. Notice the k-strand at position a� 3 2k- z with throw value of 32k- l crosses under the (k + 1)-
-
.·
.·
Figure 9.
:f 2(3) and :J3(3) mapping to the figure-eight knot and the Borromean rings.
strand at position a% having a throw value of 3 2k- J 2
.
3zk-z.
+
D
To finish the proof of Theorem 4, we construct a juggling pattern for every solid torus braid. Let w = aT, a;, · · a7, be a word describing a solid torus n-braid. Let ) I"(t n
=
{
111C
·
bp + < h71 - a:)
) � ) - ( b'11· - n' In ( n ""'�t '" "'1r. . . . 1,/t)
if t = a;, + Z( j - 1 ) · 32"- 1 if t = b'11· + 2( ,,. _ 1 ) . otherwise.
·
32"- 1 ,
·
mod (Zr · Y"-1)
1 (_2 r · L221 mod J - 1 _)
air then bT = b� and a7 a�; similarly, if a7 = aj 1, then b� and a7 a�'. Theorem 7 along with the lemmas above guarantee I�((t) to he a juggling pattern; we leave it to the reader to provide details. We claim that :fWD maps to w. In Iif(t) , r copies of the fundamental chambers of I,l t) are used, one for each element in w; the length of its funda mental chamber is 2 r · 3 2 n - l . Each copy is altered by an ap propriate swapping of sites corresponding to the generating element in w. This alteration provides the appropriate cross ing needed. This completes the proof of Theorem 4. If a7
b;
=
=
=
=
Note that choosing the starting position of the fundamen tal chamber of /�\ t) yields an ordering of the position of the strands, determining w. This ordering is not necessar ily the ordering of the n balls, which are used to label the strands, since a site swap switches the strand formed by the k-th ball with the strand formed by the ( k + 1 )-st ball. However, since our elements are solid torus braids, we have the ability to slide our strands in the solid torus to the ap propriate order (due to conjugation in Markov's theorem).
Looking Forward Although our construction allows us to prove all links can he juggled, it is far from realistic or efficient. The throw values were chosen in powers of 3 (a prime) in order to
make transparent the construction of a valid juggling se quence, along with isolating crossings of two strands. This requires a juggling sequence with n balls to have throw values up to 3 2 rz- l . As juggling sequences with values of 9 are near impossible to perform, the method above is cer tainly not realistic. Let us now look at how realism can be introduced and measured. We begin with the map j from juggling sequences to solid torus braids. This map is based on taking the closure of the fundamental chamber. Thus, the classic (3) cascade sequence, having fundamental chamber of length two, is allowed to be "active" for only two beats, resulting in the unknot. But what if our juggler wishes to juggle longer, for more beats? Be cause we are going to dose the resulting braid, juggling mul tiple copies of the fundamental chamber can be allowed. De fine 3'"\./') to he the closure of k adjacent copies of the fundamental chamber associated to the juggling pattern] Fig ure 9 shows j2(3) and j3(3); the first maps to the figure-eight knot, the latter to the Borromean rings. Given a link, we ask to find the best juggling pattern which maps to it. To try to measure what "best" means, we need to look at a few factors. Given a link /, let j ( l ) he the set of juggling patternsfsuch that j k( j' ) maps3 to l for some k E N .
DEFINITION 1 1 The ball index of a link l is the minimum number of halls needed for juggling pattern/ for allfin j( /).4 The throw index of a link I is the minimum over all the max imum throw values of a juggling pattern f for all .f in j( l ). It is straightforward to show that for non-trivial links, the ball index must be greater than one and the throw index must be greater than two. Consider the trefoil as an ex ample. Two possible ways to construct it are by j 3(4 0) ,
3Strictly, :Jk maps a juggling sequence to a solid torus braid; we abuse terminology and sometimes refer to the composition of this map with stabilization. 4Ciearly, the braid index of a link is less than or equal to the ball index.
© 2007 Springer Science+Business Media, Inc., Volume 29. Number 3. 2007
21
and :}(5, 5, 5, 1), as shown in Figure 4. Since (4, 0) is a ball pattern, the ball index of the trefoil must be two.
2-
PROBLEM Study the properties of the ball index and throw index of links. An underlying issue to this problem is understanding :J . We have shown that :J i s surjective, but we d o not know much more about the map itself. Based on the figures above, up to conjugation, :J maps (7, 1 , 1)
� �
(5, 5 , 5 , 1) (3, 4, 5) � (4) �
u2u1 u:Z 1 <TI 1 num ber and various generalizations in cluding generalized Gauss sums; the theory of Kummer fields, class num ber, class fields, power characters and laws of reciprocity in the theory of algebraic fields; Fermat's quotients and other arithmetic quotient forms; congruence theories as applied to power series; abstract algebra includ ing, particularly, group theory and semi-groups; and many types of Dio phantine equations aside from the Fermat relation itself. It seems Vandiver was rather carried away with enthusiasm. Only some of these topics have substantial, direct con nections with FLT. On the other hand, Vandiver himself was led to explore many of these potential payoffs, partly because of his interest in questions that arose from, or were thought to be use ful for solving FLT. In fact, in 1 952-53 he published a two-part article on as sociative algebras and the algebraic the ory of numbers, a paper he regarded as more important and innovative than any of his work directly connected with FLT. He was disappointed that this article was seldom cited: As far as I know, only one person has studied thoroughly this paper, and he is Alonzo Church.41 In private correspondence Church raised some interesting criticisms re garding the axiomatic debate devel oped by Vandiver, and Vandiver was quick to include Church's comments in a follow-up to this article.
40[Vandiver 1 946]. 41Vandiver to Mientka: March 1 3. 1 964 (HSV). See also, Vandiver to Herstein, April 2, 1 960.
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THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+Business Media, Inc.
As late as the early 1 960s Vandiver was still publishing new results related to FLT. He also continued to work on a book about FLT and related topics in number theory, a project he pursued for many years. His archive contains hun dreds of typewritten pages with whole chapters nearly ready for publication, but for some reason this book was never published.
Correspondence on FLT As Vandiver came to know, having your name publicly attached to Fermat's problem could impose a considerable burden on a mathematician. Yet some experts found efficient ways to duck the unwelcome task of reading the steady stream of faulty proofs submitted by rank amateurs. In the early twentieth century, Edmund Landau came up with a nearly ideal solution to this corollary to Fermat's problem. As Gottingen's leading authority on number theory, Landau was officially entrusted with handling all correspondence related to the Wolfskehl Prize, which offered 100,000 Marks to anyone who could solve Fermat's conjecture. Landau had little interest in the problem, so he took on this duty with little enthusiasm. When the flow of incoming correspon dence from amateurs eventually be came unbearable, he became openly disgusted. So he prepared a form letter that looked something like this: Dear . . . . . . . . . , Thank you for your manu script on the proof of Fermat's Last Theorem. The first mistake is on: Page . . . . . . Line . . . This invalidates the proof. Professor E. M. Landau An assistant read through the manu scripts and filled in the missing details in the form letter. Fortunately for Vandiver, no rich oil man came along to establish a similar prize fund at the University of Texas for a successful proof of FLT. So this cir cumstance surely diminished the num ber of would-be problem solvers who might have written to him. Nevertheless he did receive enormous quantities of mail that only grew from year to year.
In fact, many American mathematicians (perhaps all of them?) saw in Vandiver the default address to which any letter on the topic should be redirected. Van diver's attitude toward these corre spondents was essentially positive, per haps because he had been something of an amateur mathematician himself. In response to attempted proofs by rank amateurs, he sent a pre-written, but rather polite reply. His archives contain no fewer than 225 such answers, sent between 1 934 and 1966. To those he considered qualified mathematicians he usually answered in some detail, though even in these cases the task became in creasingly onerous with time. An interesting letter from 1949 attests to this problem in the case of a math ematician, Taro Morishima, whose con tribution Vandiver truly appreciated. He was forewarned that Morishima was about to submit an article on FLT to an American journal. This prompted him to take preemptive action by contacting several editors (Aurel Wintner of the American journal of Mathematics. Rudolph E . Langer, Saunders Mac Lane, and Leonard Carlitz) to request that the article not be sent to him. He would certainly like to read this paper, he said, but at his leisure and not under pres sure to finish within some given period of time, however reasonable. Number theory, he added, "seems to be getting popular," but Vandiver felt he was drowning under the enormous corre spondence he now had to handle.42 A few months later, he again complained bitterly about this to another colleague, while requesting that no further letters be sent to him. Only if he received a manuscript from Siegel, Hasse, or Rademacher would he he willing to ex amine the work in detail. To which he added: "After nearly forty years of look ing at such manuscripts, good and bad, don't I deserve a respite?"43 One revealing interchange took place in 1961 around a proposed proof of FLT by Lucien Hibbert, then Execu tive Director of the Inter-American Bank for Development in Washington, D.C. Hibbert had been directed to Vandiver by Israel Herstein and Marshall Stone.
Born in 1899 in Haiti, he had received in 1937 a Ph.D. at the Sorbonne, work ing with Arnaud Denjoy (1 884-1974). In Haiti he had been professor of mathe matics and physics, and director of the Haitian Statistical Institute, and had en joyed a very successful career in the fi nance sector. He was Ambassadorial Representative of Haiti to the Organi zation of American States, and later served in the Ministry of External Af fairs. As usual, Vandiver read the man uscript fairly carefully and replied po litely and in some detail. In his answer, he summarized his general attitude to ward this matter: Next October 21 I shall start my 80th year of age. Beginning in the year 1 9 1 4 I published several articles on the Fermat problem which received attention from readers to the extent that many of them wrote letters to me, generally containing their opin ions . . . about the problem, and also what they regarded as proof of Fer mat's statement or contributions to that encl . For some years I made a practice of replying to such letters and giving my estimates of the value of their work. However, as I con tinued to publish from time to time through the years articles pertaining to the Fermat problem, my corre spondence along that line became so heavy that if I had continued to do this it would have taken most of my time. . . As an example of this, l have received four letters within the last few weeks and about seven since the first of the year pertaining to the Fermat problem . . . one of them . said the full proof of the theorem coven.:d about 50 pages. In his resume of the paper given me in his letter, he made a number of statements I could not understand at all: . . . so I told him I was sorry I had to refuse to help him . . . In my 50 years' experience with the problem I have often been con vinced for a time that I had a proof of the theorem using only the tools of elementary number theory and al gebra, but I found in every such case
that I had an error in my argument. After a time I became more and more skeptical of any apparent proof that I found using such ele mentary means, as I felt that if an elementary proof existed, it could hardly have escaped the attention of such great mathematicians as Euler, Legendre, Lame, Abel, Gauss, Cauchy, and Kummer, all of whom worked at the problem! . . . I have looked over the general character of your argument . . . and as far as I can see . . . you have used nothing but elementary alge bra therein, hence I cannot help be ing skeptical as to the accuracy of your work. if you regard this as a disparagement of your work, please note that I have just di!>paraged above all my own efforts ofthis char acter.14 Vandiver further advised Hibbert to write up full proofs for various specific cases, to see if they worked. And he added: "In giving you this advice, I am assuming that you would prefer to find the error yourself, if one exists, than to have someone else find it." Then, in a letter to Stone he explained what he really feared about cases like this one: For many years, in connection with "proofs" of FLT sent to me and which I examined it turned out in nearly every case that if I called the author's attention to an error in his work, soon after I would receive an other ms. which he assumed was a correction of his original paper. Also, if instead of pointing out an error I would merely state to him that there was a step in his argu ment which I did not understand, then the author would reply that I did not understand his [entire] argu ment. These things would be the be ginning of a long correspondence that I would have with him.45 Some years prior to Hibbert, Vandiver was involved in another notewor thy exchange with a Pakistani air force officer named Quazi Abdul Moktader Mohd Yahya, who was formerly "Pro fessor of Mathematics at Brajali Acad emy, East Pakistan." In various letters
42Vandiver to various: December 8, 1 949 (HSV}. 43Vandiver to J. R. Kline: January 1 2 , 1 950 (HSV}. 44Vandiver to Hibbert: June 23, 1 96 1 (HSV}. Emphasis in the original. 45Vandiver to Stone: June 26, 1 96 1 (HSV}.
© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007
37
written to colleagues about this man, Vandiver referred to him as X, noting that "I do not wish to be sued for libel, in case the information in this letter somehow reaches him." He received a manuscript from him on FLT that the au thor wished to submit to the Proceed ings of the National Academy of Sci ences. As in other cases, Vandiver answered the initial letter politely, but this led to a lengthy and futile series of interchanges. Vandiver tried to put an end to this by suggesting that Yahya send his manuscript to a "regular math ematical journal," one "preferably in Switzerland or Germany where they seem to have more interest in number theory than in the U.S." He feared that this "dangerous character" might "write me a threatening letter, as some of these birds have done in the past. "46 Eventu ally Yahya was able to publish his (ob viously flawed) proof in a Portuguese journal in 1976.47 A third interesting correspondence over FLT took place in 1960-61 when Vandiver was contacted by a junior high school pupil from Baltimore named Joel Weiss. After learning the names of the three persons in the US who had re cently done work on the Fermat prob lem, Joel wrote to Vandiver (and also to the Lehmers) for advice on this topic, which he had chosen for a school term paper. He was willing to work hard, and so he explained his choice as follows: This theorem, which originally was a curiosity to me, turned out to be a stimulating research project well worth the 45 hours of work neces sary to complete it. I hope that my conclusion will start a new train of thought leading to an eventual proof of Fermat's Last Theorem. Joel later indicated at the end of his fin ished paper what this desired train of thought might be: I conclude that Fermat's Last Theo rem has been proven all this time, and that its entire proof is that of n 3. I have reached this conclu sion from an analyzation of a suc cession of cases of the theorem with exponents 3 through 9. After study=
ing these cases, it is apparent that the deviation between the sum of the terms in the left-hand member of the equation and that of the right hand member increases steadily with higher exponent value. There fore, I feel that it is only necessary to prove n = 3 because this is the point of lowest deviation. Any ex ponent value above this is immedi ately ruled out as a result of the fact that the deviation is greater than that of the third power thus making it impossible to suit the equation.48 Vandiver, who had written several po lite and possibly helpful letters to Joel along the way, also reacted politely to Joel's conclusion: most mathematicians, he kindly remarked, would not agree with the closing statement of his paper.
Figure l. Harry S. Vandiver (Creator: Walter Barnes Studio (HSV).
Recognition and Oblivion In the currently available literature, Van diver's name is barely mentioned in connection with FLT. For instance, in the popular "MacTutor History of Math ematics Archive" website, Vandiver barely rates a very short entry of his own. His name appears only in passing in the site's article on FLT, and he is not mentioned at all in the article on Dick Lehmer. From the point of view of current mathematical research asso ciated with the problem, especially fol lowing Wiles's dramatic breakthrough, this may be understandable. But from the point of view of the history of the problem, this lack of recognition is com pletely unjustified, though the reasons for this are not difficult to find. Although Vandiver was the undis puted world's leading expert on FLT during his lifetime, contemporaries of ten took an ambivalent attitude toward him and his passionate quest. Certainly he was well-known and respected both within the American mathematical com munity and abroad, but his interests were also viewed as exotic, and evi dence abounds that he was viewed as more bizarre than brilliantly original. Thus, it is not surprising that when his friend G. D. Birkhoff prepared a list of the 10 most prominent American math-
ematicians in 1926 for the Rockefeller foundation, Vandiver, then 44 years old, was not on his list. 49 Even during his most creative phase as a researcher he seems to have received less recognition than he probably deserved. Yet Vandiver received several high honors, including the Cole Prize and an honorary doctorate from the University of Pennsylvania; and, of course, he was the recipient of many research grants. Harry Vandiver was the only American mathematician whose work received mention in Edmund Landau's 1927 clas sic textbook on number theory. He was elected vice-president of the AMS for the term 1933-1935, and in 1935 he was an AMS Colloquium Lecturer. He served as assistant editor of the Annals of Math ematics from 1926 to 1939, and in 1934 he was elected to the National Academy of Sciences. Still, he always remained part of a small and rather marginal sub-community within the larger Amer ican mathematical research enterprise. Strongly fixated on his own work, he was certainly not a shaker and mover. He would not manage to attract large numbers of young researchers to his chosen field; he did not establish a re search school, nor did he develop an influential network of contacts with like-
46Vandiver to Hayman: April 3, 1 958 (HSV). 47Mathematica/ Reviews
lists a "private edition" by the author [Yahya 1 958], and three additional articles in
Portugaliae Mathematica
(1 973, 1 976 and 1 977).
48The entire correspondence appears in HSV: File 1 6-3. 49See [Siegmund-Schultze 2001 , 51]. Birkhoff's list included only mathematicians from three leading centers: Cambridge (Birkhoff, Morse, Osgood, Wiener, Whitehead); Chicago (Bliss, Dickson, E. H. Moore, Moulton); and Princeton (Alexander, Eisenhart, Lefschetz, Veblen).
38
THE MATHEMATICAL INTELUGENCER © 2007 Springer Science+ Business Media, Inc.
minded mathematicians. Nor was he an organizational talent who excelled when it came to promoting journals or organizing professional meetings. The honors conferred on Vandiver occasionally betray ambivalence. For example, only after Vandiver himself applied some direct pressure on uni versity authorities was he named Dis tinguished Professor at TU in Austin, in 1947. But his title, "Distinguished Pro fessor of Applied Mathematics and As tronomy," was certainly odd given his research expertise. More telling still is the context sur rounding a Festschrift published in his honor. In 1966 Bellman's journal of Mathematical Analysis and Applica tions brought forth the special issue dedicated to Vandiver on his eighty third birthday. The editors wished to honor him not only for his contributions to FLT and algebraic number theory but also because "he has profoundly influ enced the development of American mathematics for a period of over sixty years." And yet the American contribu tions to this volume were all written by his former students and close collabo rators. Side by side with these papers one finds a score of others written by leading number-theorists from abroad, figures such as Mordell, Hasse, Erdos, Szemeredi, Gel'fond and Morishima. It's odd that such a collection appeared in a journal far removed from Vandiver's own fields of interest. Evidently the de cision to publish such a Festschrift came from close friends who wanted to pay long-overdue tribute to the man and his work, yet sensed that no one outside Vandiver's inner circle would ever un dertake it. The honoree, then in deli cate health after undergoing surgery, was deeply touched by this gesture.so Vandiver's lifetime endeavor was characterized by remarkable indepen dence and a willingness to pursue self styled, original research programs. As a researcher, his style was marked by an indefatigable appetite for endless cal culations, by a peculiar style of collab oration with small groups of people who were dose to him, and by his pi oneering use of electronic computers in his fields of expertise. While Vandiver's contributions played no direct role in shaping the train of ideas that eventu-
Figure 4. AM S MAA meeting in Washington D.C. (HSV). Source: Capi tol Photo Services, Inc. -
Figure 5. joel Weiss with
a poster presentation of his work on FLT (HSV).
ally led to the general proof of FLT, and while opinions may vary as to the in trinsic mathematical significance of the ideas deve lo ped in his work, one can not make sense of the history of FLT without giving prominence to the story of this man, the only one ever to de vote his entire professional life to solv ing the problem. ACKNOWLEDGMENTS
Albert C. Lewis and David Rowe read earlier versions of this article. I thank them for the critical remarks which led to significant improvement. I have used archival material found in several institutions. I thank the archivists for assistance in locating and copying the originals, and for granting rermission to
quote. Pictures are reproduced and sources are quoted with pennission, us ing the following abbreviations: HSV: Vandiver Collection, Archives of American Mathematics, Center for American History, The University of Texas at Austin. MOHP: Oral History Project, The Legacy of R.L. Moore, Archives of American Mathematics, Center for American History, The University of Texas at Austin. HUG: George David Birkhoff Papers, Harvard University Archives: Call Num ber HUG 4213.2, Box 3, Folder "T-V " . APS: American Philosophical Society Archive. GFA: The John Simon Guggenheim Memorial Foundation Archive.
50Dorothy W. Baker to Bellman: September 29, 1 965 (HSV).
© 2007 Springer Science+Business Media, Inc., Volume 29, Number 3, 2007
39
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and Minutes of the General Faculty, The Uni versity of Texas at Austin, 1 97 4, 1 0926Herstein, Israel. (1 950), "A Proof of a Conjec ture of Vandiver," Proc. AMS 1 , 370-371 . lwasawa, Kenkichi, and Charles Sims (1 965), "Computation of Invariants in the Theory of
569-584.
Fermat's last theorem (second paper) , " Duke Math. J. 3, 4 1 8-427. --
(1 946), "Fermat's Last Theorem, " Am.
Math. Mo. 53 (1 946), pp. 555-578. -- (1 954), "Examination of methods of attack
on the second case of Fermat's last theorem," Proc. Nat!. Acad. Sci. USA 40, 732-735. (1 963), "Some of my recollections of
George David Birkhoff," Jour. Math. Analysis and Applications 7, 271 -283.
Vandiver, Harry S., Derrick H. Lehmer, and
Siegmund-Schultze, Reinhard (2001 ), Rocke
Emma Lehmer (1 954), "An application of
feller and the Internationalization of Mathe
high-speed computing to Fermat's last
matics between the Two World Wars, Basel
theorem," Proc. Nat!. Acad. Sci. USA 40, 25-33.
and Boston, Birkhauser. Vandiver, Harry S. (1 9 1 4), "Extensions of the
Vandiver, Harry S., John L. Selfridge, and
criteria of Wieferich and Mirimanoff in Con
Charles A. Nicol (1 955), "Proof of Fermat's
nection with Fermat's Last Theorem, " Jour.
last theorem for all prime exponents less than
reine u. angew. Math. 1 1 4, 3 1 4-31 8.
4002," Proc. Nat!. Acad. Sci. USA 41 , 970-
--
(1 920), "On Kummer's Memoir of 1 857
Concerning Fermat's Last Theorem," Proc. Nat!. Acad. Sci. USA 6, 266-269. --
(1 922), "On Kummer's memoir of 1 857,
concerning Fermat's last theorem (second paper)," Bull AMS. 28, 400-407. --
(1 929), "On Fermat's Last Theorem, "
--
973.
Vandiver, Harry S., and George E. Wahlin ( 1 928), Algebraic Numbers -II. Report of the Committee on Algebraic Numbers, Wash
ington, DC, National Research Council. Wagstaff, Samuel S. (1 978), "The irregular primes to 1 25000," Math. Camp. 32 (1 42), 583-591 .
Trans. AMS 31 , 61 3-642.
1 0940.
(1 937), "On Bernoulli Numbers and Fer
mat's Last Theorem, " Duke Math. J. 3,
--
(2008), "Number Crunching vs. Number
Science (Forthcoming).
ber," Bull AMS 40, 1 1 8-1 26.
-- (1 937a), "On Bernoulli numbers and
Buhler, J . P . , R. Crandall, R. Ernvall, T. Met
diver, the Lehmers, Computers and Number
(1 934), "Fermat's last theorem and the
second factor in the cyclotomic class num --
York, Springer-Verlag.
Buhler, J.P. , R. E. Crandall, and R . W. Sam
Corry, Leo (2007), "FLT Meets SWAC: Van
--
(1 930), "Summary of results and proofs
Yahya, 0. A. M . M . (1 958), Complete proof
on Fermat's last theorem (fifth paper) , " Proc.
of Fermat's last theorem. With a foreword
Nat!. Acad. Sci. 1 6, 298-305.
by Dr. Razi-Ud-Din Siddiqui. Available from
--
(1 930a), "Summary of results and proofs
the author, Pakistan Air Force, Kohat, West
Cyclotomic Fields," J. Math. Soc. Japan 1 8,
on Fermat's last theorem (sixth paper)," Proc.
Pakistan (1 4 pp. Mimeographed appendix,
86-96.
Nat!. Acad. Sci. USA 1 7, 661 -673.
3 pp.).
LEO CORRY is head of the Cohn lnstrtute. His latest book,
and the Axiomatization of Physics, 1 898- 1 9 1 8
David Hilbert
(Kiuwer), was published
in 2004. His current research interests include the history of FLT and of computational approaches to number theory.
Cohn Institute for History and Philosophy of Science and Ideas
Tei-Aviv University 69978 Tei-Aviv Israel
e-mail:
[email protected] 40
THE MATHEMATICAL INTELLIGENCER © 2007 Springer Science+ Business Media, Inc.
Spectral Variation, N ormal M atrices, and Finsler G eometry
RAJENDRA BHATIA
ow did two matrix-theorists who had never worked together before come to prove a theorem which has had consequences throughout the field and beyond? I will try to put together the personal and the mathemati cal sides of the Hoffman-Wielandt Theorem, its prehistory, and attempts (both successful and unsuccessful) to gener alise it. Wielandt was really trying to do the thing for operator nonns and the Frobenius nann was his second choice. Thus begins Alan Hoffman's commentary on his joint paper with Helmut Wielandt [HW] , one of the best known in linear algebra. The paper is less than three pages long and, of a piece with that brevity, Hoffman's commentary con sists of just one paragraph. It continues, Infact, he had a proof ofHW with a constant bigger than 1 in front. It was quite lovely, involving a path in matrix space, and I hope someone else has found a usefor that method. Since linear programming was in the air at the National Bureau of Standards in those days, it was nat ural for us to discover the proof that appeared in the pa per. The most difficult task was convincing each other that something this short and simple was worth publish ing. In fact, we padded it with a new proqf qf the Birk ho.ff theorem on doubly stochastic matrices. I think the reason for the theorem 's popularity is the publici�y given it by Wilkinson in his hook on the algebraic eigenvalue
problem U. H. Wilkinson, The Algebraic Eigenvalue Prob lem, Clarendon Press, Oxford, 1965). In this article I will explain what it was that Wielandt was really trying to do, why he wanted to do it for oper ator norms, and what some others had done before him and have done since. Wielandt's mathematical works [Wie l] straddle two dif ferent fields: group theory and matrix analysis. He began with the first, was pulled into the second, and then hap pily continued with both. The circumstances are best de scribed in his own words: The group-theoretic work was interruptedfor severalyears while, during the second half of the war, at the G6ttin gen Aerodynamics Research Institute, I bad to work on vibration problems. I am indebted to that time for valu able discoveries: on the one band the applicability of ab stract tools to the solution of concrete problems, on the other band, the-for a pure mathematician-unexpected dif.ficul�y and unaccustomed responsibility of numerical evaluation . It was a matter of estimating eigenvalues of non-se{fadjoint differential equations and matrices. I at tacked the more general problem of developing a metric spectral theory, to begin with forfinite complex matrices. The links between all parts of our story are contained in the two paragraphs I have quoted from Hoffman and from Wielandt.
© 2007 Springer Science+ Business Media, Inc, Volume 29, Number 3, 2007
41
By the time Wielandt came to Gi:ittingen in 1 942, Her mann Weyl had left. Thirty years earlier Weyl had pub lished a fundamental paper [We] on asymptotics of eigen values of partial differential operators. Among the several things Weyl accomplished in that paper are many interest ing inequalities relating the eigenvalues of Hermitian ma trices A, B, and A + B. One of them can be translated into the following perturbation theorem: If A and B are n X n
Hennitian matrices, and their eigenvalues are enumerated as a1 ::::: a2 ::::: a n, and {31 ::::: {32 ::::: ::::: f3n, respectively, then •
•
•
•
•
•
(1) Here IIAI I stands for the norm o f A a s a linear operator o n the Euclidean space e n; i.e., (2)
ll ll ii ll I IAI I = max l Ax : X E e n, x = 1 ) .
Apart from the intrinsic mathematical interest that Weyl's inequality (1) has, it soothes the analyst's anxiety about "the unaccustomed responsibility of numerical evaluation. " If one replaces a Hermitian matrix A by a nearby Hermitian matrix B, then the eigenvalues are changed by no more than the change in the matrix. Almost the first question that arises now is whether the inequality remains true for a wider class of matrices, and for a mathematician interested in "estimating eigenvalues of non-selfadjoint differential equations and matrices" this would be more than mere curiosity. The first wider class to be considered is that of normal matrices. (An operator A is normal if AA* = A*A. This is equivalent to the condi tion that in some orthonormal basis the matrix of A is di agonal. The diagonal entries are the eigenvalues of A, and A is Hermitian if and only if these are all real.) The eigenvalues of a normal matrix, now being com plex, cannot be ordered in any natural way, and we have to define an appropriate distance to replace the left-hand side of (1). If Eig A = {a�, . . . , aJ and Eig B = {{3�, . . . , f3 J are the unordered n-tuples whose elements are the eigen values of A and B, respectively, then we define the opti
mal matching distance
(3)
d(Eig A, Eig B) = min max laJ - f3cr(J)I , u l $.j$. n
RAJENDRA BHATIA has been associated with
where a varies over all permutations of the indices { 1 , 2, . . . , n}. The question raised by Weyl's inequality is: if A and B are any two normal matrices, then do we have
ll d(Eig A, Eig B) :::; II A - E ?
(4)
This is what Wielandt, and several others over nearly four decades, attempted to prove. We will return to that story later. The operator norm (2) is the one that every student of functional analysis first learns about. Its definition carries over to all bounded linear operators on an infinite-dimen sional Hilbert space. That explains why this norm would have been Wielandt's first choice. There are other possible choices. The Frobenius nann of an n X n matrix A is defined as
IIAIIF = (tr A* A)1 12 =
(5)
(6)
dF(Eig A, Eig B) = m�n
42
THE MATHEMATICAL INTELLIGENCER
] 112 .
Hoffman credits ]. H. Wilkinson [Will with the publicity responsible for the theorem's popularity. Wilkinson writes
Tbe Wielandt-Hoffman theorem does not seem to have attracted as much attention as those arising from the di rect application of nonns. In my experience it is the most useful resultfor the error analysis of techniques based on orthogonal tranifonnations in floating-point arithmetic.
He also gives an elementary proof for the (most interesting) special case when A and B are Hermitian. In spite of Wilkinson's reversal of the order of names of its authors, the theorem is known as the Hoffman-Wielandt theorem. Unknown, it would seem, to Hoffman and Wielandt, and
151 Delhi most of the
Rao,
KR
whom we thank) shows him on his anival at University of Califomia
India
a1 - f3crcJ)I 2
(7)
Parthasarathy. The photograph of him here (taken by George Bergman,
e-mail:
[email protected] l
THEOREM 1 Let A and B be any two nonnal matrices. Tben
nealogy Project, he is scientifically a direct descendant of Arthur Cayley,
Indian Statistical Institute Delhi
L�
=
Instead of ( 4), Hoffman and Wielandt proved the following.
via A. Forsyth, E. Whittaker, James Jeans, RA. Fisher, C.R
New Delhi, I I 00 I 6
1.]
This norm arises from the inner product (A, B) tr A* B, and, for this reason, it has pleasant geometric features. It can be easily computed from the entries of A. If we replace the norm (2) with (5), then we must make a similar change in the distance (3) and define
time since his graduate student days. According to the Mathematics Ge
Berkeley on a post-doctoral fellowship, 1 979.
(L l a;1 12 r2 .
to Wilkinson, the Hermitian special case of (7) had been announced several years earlier, by Karl Lowner in 1 934 [Lo]. This paper is very well-known for its deep analysis of operator monotone functions. Somewhat surprisingly, there is no reference to it in most of the papers and books where the inequality (7) is discussed. (Incidentally, Lowner was at the University of Berlin between 1922 and 1928. Wielandt came to study there in 1929 and obtained a Ph.D. in 1935. Lowner's original Czech name was Karel but, because his education was in German, he was known as Karl. Later, when he had to move to the United States, he adopted the name Charles Loewner.) Lowner does not offer a proof and says that the inequality can be established via a simple vari ational consideration. One such consideration might go as follows. When x = (x1 , . . . , x,J is any vector with real coordinates, let x t = ext, . . . ' xf) and xi = cxl, . . . x �) he the decreasing and increasing rearrangements of x. This means that the numbers x1, . . . , Xn are rearranged as xf ::::: ::::: x� and as x I :5 :5 x �· Then for any two vectors x and y, we have '
·
·
·
·
for every differentiable curve U(t) with U(O) = equivalent to saying
_!]_I dt
PROPOSITION 2 Let A and B ces. Then (9) (Eig l (A) , Eig i (B))
:5
tr
be
n
X
n Hermitian matri
AB :5 (Eig l (A),
Eig 1 (B)).
OUs
n
X
n unitary matrices, and let
= { U BU* : U E U(n)},
be the unitary orbit of B. If we replace B by any element of OUs, then the eigenvalues of B are not changed, and hence neither are the two inner products in (9). Consider the function j(X) tr AX defined on the compact set OUs. The two inequalities in (9) are lower and upper bounds for j(X). Both will follow if we show that every extreme point Xo for f commutes with A. A point Xo on OUs is an extreme point if and only if =
_!]_I dt
tr A U( t)Xo U(t)*
t=O
=0
=
0
=
0.
The trace of a product being invariant under cyclic per mutation of the factors, this is the same as saying tr K(XoA - AXo)
=
0.
Since (K, L) - trKL is an inner product on the space of skew-Hermitian matrices, this is possible if and only if X0A - AXo 0. 0 =
=
Using the second inequality in (9) we see that
I lA - Ell} = IIAII} + II BII} ::::: IIAII} + II BII}
-
n
2trAB 2(Eig t (A), Eig t (B))
= '\' L lA ;t (A) - A .!t cB)I2
( 10)
j= l
This proves the inequality (7) for Hermitian matrices. The same argument, using the first inequality in (9), shows that
II A - B lli :::; I
(11)
j= l
}
}
I A CA) - A CB)I 2.
There is another way of proving Proposition 2 that Lowner would have known. In 1923, Issai Schur, the ad viser for Wielandt's Ph.D. thesis at Berlin, proved a very in teresting relation between the diagonal of a Hermitian ma trix and its eigenvalues. This says that if d = (d1, . . . , d,J and A = (A1 , . . . , A,J are, respectively, the diagonal en tries and the eigenvalues of a Hermitian matrix A, then d is majorised by A. This, by definition, means that k k '\' d.1l :::; L '\' A .Jl ' for 1 :5 k :5 n (12) L j= l
f= l
and
Proof If A and B are commuting Hermitian matrices, this reduces to (8). The general case can be reduced to this special one as follows. Let U(n) be the set of all
(AKXo - AXoK)
tr
·
To see this, first note that the general case can be reduced to the special case n = 2. This amounts to showing that whenever x1 ::::: x2 and Y1 ::::: Yz, then XtYI + XzYz ::::: XtYz + XzY1 . The latter inequality can be written as (x1 - x2) (y1 - y2) ::::: 0 and is obviously true. A matrix analogue of this inequality is given in the fol lowing proposition. If A is a Hermitian matrix we denote by Eig l (A) = (A iCA), . . . , A �(A)) the vector whose coor dinates are the eigenvalues of A arranged in decreasing or der. Similarly Eig i (A) = (A I (A), . . . , A �(A)) is the vector whose coordinates are the same numbers arranged in in creasing order. The bracket (x, y) stands for the usual scalar product IJ�1 XJYI
tr Ae 1KX0 e- tK
This is
for every skew-Hermitian matrix K. Expanding the expo nentials into series, this condition reduces to
·
(8)
t=O
I.
n
n
j= l
f= l
I df = I A f .
(13)
The notation d < A is used to express that all o f the rela tions ( 1 2) and (13) hold. Schur's theorem has been gener alized in various directions (see, e.g., the work of Kostant [K] and Atiyah [A]), and it provided a strong stimulus for the theory of majorisation [MO, p4]. A good part of this theory had been developed by the time Hardy, Littlewood, and P6lya wrote their famous book [HLP] in 1934, the same year as that of Lbwner's paper. The condition d < A is equivalent to the condition that the vec tor d is in the convex hull of the vectors A 1 . If this is to be in Theon's ladder,
2b - a is a- b the appropriate entry. Moreover, algebra shows that 2 b - a is positive, as is a - h, and that a - b is a denominator smaller than b. (For example, to establish that 2 b > a, square both sides and make use of the relation a2 - 2 b2 = 2:: 1 .)
what must the term before it be? One checks that
Repeated use of this backwards approach must eventually produce an expression with denominator equal to 1 . As this entry belongs to Theon's ladder, so too then must the expression
a --;;·
0
FURTHER. Start Theon 's ladder instead with the fraction 5 + 12 17
= - . Show that every second term corre�ponds to 13 13 a Pythagorean triple with legs that differ by 7. (Do we get every such triple?) W'hat do the even terms of the sequence give? Can we generate all Pythagorean triples via Theon 's method? W'hat are the appropriate "seed" terms? W'hat do we obtain if we run Theon 's ladder backwards, infinitelyJar to the left? What can be garneredfrom the equa tions x2 - 3.l = :t 1 ? (Approximations to the square root of 3? Variants of Pythagorean triples?)
---
Of course a tremendous wealth of information lies hid den in Fell's equations and its simple variants. Students of the Boston Math Circle have spent many an hour delving into its riches. 3.
Is Figure 4 enough? Have we, in fact, established al-Khazin's algebraic identity
(a2 + h2)(c2 + d2) = (ac + bdi + (ad - bc)2?
FURTIIER. J.fN and kN are both sums of two squares, must
.
.
Figure 4. Multiplying areas.
a >0 and b > 0 satisfying a2 - 2 b2 =
k he too?
A1·ea 50
4. One can read a braid from top down as a sequence of crossings. Let L denote the crossing of a left strand over or under the middle strand, and R the crossing of the right two strands.
© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3 , 2007
57
LLRLR
Observe that it does not matter if an L is an under or an over crossing. Quick experimentation shows that one can convert any L under-crossing to an over-crossing, and vice versa, by pushing the wooden spoon between the right two strands just below the crossing. (Try it!) The same is true for any R crossing. Thus any braid is encoded in a well defined manner as a string of Ls and Rs.
Note, moreover, that a braid possessing two consecutive L-crossings is equivalent to a braid with those two crossings removed: convert one to an under-crossing and the other to an over-crossing. Similarly, we can remove any two consecu tive Rs from a braid. Thus, any braid with fixed ends is equiv alent to a braid encoded by an alternating string of Ls and Rs.
and to an L if it was an R. (For example, the code LRLL RLRLL becomes LLLRRRRRL .) Count the number ofLs in the modified code and, from it, subtract the number of Rs. If this difference is a mul tiple of three, then the original braid can be untangled. Ifthe difference is congruent to 1 modulo 3, then the braid is equivalent to a single L, and equivalent to R if it is con gruent to - 1 .
This provides a wonderfully swift technique for analyzing three-braids!
FuRTHER. Take a rectangularpiece offelt, cut in it two slits,
and make the braid RLRLRL with fixed ends. Notice that, with the appropriate under and over crossings, the strands of this braid can be made 'Jlat" (that is, there are no internal twists in any of the individual strands).
Classify all flat three-braids with fixed ends that can be so produced. (In 2005, high-school students attending the St. Mark's Institute of Mathematics research group made sig nificant progress with this tough question.) 5. (My thanks to Elizabeth Synge, seventh-grade home schooler, for her help in writing the details of this proof.) Let T(n) denote the number of triangles with integer side lengths ("integer triangles") of perimeter n. The first 20 val ues of T(n) are
0, 0, 1 , 0, 1 , 1 , 2 , 1 , 3 , 2 , 4, 3 , 5 , 4,
Also, if a braid ends in either sequence LRL or RLR, then a 180° rotation of the spoon deletes the sequence from the
braid.
=
Thus every braid with fixed ends reduces to one of five possibilities: LR, RL, R, L, or the "empty braid," the untan gled state. But the braid LR is equivalent to just R (via LR = RRLR = R), and the braid RL to L. Thus any braid reduces to either R, L, or the untangled state. Neither of the first two options has yellow strand returning to the middle position. Thus, the only permissible option for a braid with the mid dle strand ending in the middle position is for the braid to be equivalent to the untangled state! High-school students from 2005 St. Mark's Institute of Mathematics went further and established:
Given a string ofLs and Rsfor the code ofa braid, change every second entry of the code-to an R if it was an L,
58
One striking pattern of note is that the values of the se quence seem to repeat after a shift of three places. Specif ically:
CONJECTURE 1. T(2n) LRL
THE MATHEMATICAL INTELLIGENCER
7, 5, 8, 7, 10, 8
=
T(2 n - 3) for n > 1 .
Thus the sequence {T } appears to be two intertwined copies of the sequence 0, 0, 1 , 1 , 2, 3, 4, 5, 7, 8, . . . If this conjecture is indeed true, we need only focus on this se quence of even terms in our analysis. We will need the following result: LEMMA 2. Given a positive integer n, three positive integers a, b, and c with a 2: b 2: c and a + b + c = n are the side
lengths of an integer triangle ofperimeter n if, and only if, a is strictly less than half of n. This quickly follows from the triangle inequality.
CoROllARY 3. No integer triangle of even perimeter pos
sesses a side of length 1 .
We can now establish the conjecture:
PRooF OF CONJECI1JRE 1. Let (a, b, c), with a 2: b 2: c, be
a triple of integers representing the side-lengths of a triangle of perimeter 2 n. (Notice that c > 1 .) Then (a - 1 , b 1 , c 1) are the side-lengths of an integer triangle of perimeter 2n - 3. This correspondence is one-to-one and onto. D -
-
In 2005 students of the St. Mark's School Institute of Math ematics came to this point very quickly (within two hours of playing with the problem!) and were excited by the cor respondence of simply "adding one" to each side length. In their musings they explored the option of "adding two" and "adding four" (so as to keep within the class of triangles of even perimeter). This led to the following key result: LEMMA 4. For n
even
with n > 1 2 , T( n) - T( n - 12)
=
.!!. - 3.
2
PROOF OF LEMMA 4. Let (a, b, c), with a 2: b 2: c, be a triple
of integers representing the side-lengths of a triangle of perimeter n - 1 2 . Then (a + 4, b + 4, c + 4) is a triple representing a triangle of perimeter n, with a ::S .!!. - 3, and
2
every triangle of even perimeter n > 1 2 with longest side at most this length arises this way. The correspondence to this subset is one-to-one and onto. Our correspondence "misses" the triangles of perimeter n with longest sides of lengths .!!._ - 1 and .!!._ - 2 . A simple counting argument
2
2
n shows that there are precisely - - 3 of these triangles. D 2 If we set T(O) = 0, then this formula is valid for all n even with n 2: 1 2 . Set n 1 2k + a for k 2: 0 and a = 0, 2 , 4, 6, 8, or 10. We have : =
T(l 2 k + a)
=
=
I ( T(12r a) - T(l2(r I (6r + 2 3) + T(a) k
+
r= l
1) + a)) + T(a)
1 = 3k2 + - ka + T(a) 2
(12k + a)2 48
_
a2
48
48
.l. Thus it is precisely the fractional amount needed 2 (12k + a)Z .
to round
up or down to the nearest mteger,
48
namely T(l2k + a). Thus, for n even, we have:
T(n)
=
� :; )
=
T(n + 3)
/ ( n + 3)2 = \ 48
=
·
Given integers n and k, the quantity
).
l �J
counts the num
ber of times k appears in a list of all the factors of all
l �J +
the numbers 1 through n and the (finite) sum
l
l �J ;J + +
·
·
·
counts the total number of times a
square number appears in the list of all factors of num bers 1 through n. We have:
l l + ; + � (l � J � J J . . ) n - I -n l k2 J 1 k k2 - --;; ( : IL : JI) l IAve(n) - I k 1 1 -n I (-nk - -nk J ) I ::::; Ave( n) =
_
·
+
l v;; l
1
k= l
=
tv;; J 1
2
l
-
2
·
Notice that
IVn l
1
2 -
=
1
IVnl
k= l
1 [v;;j
limn-->oc Ave( n)
=
I -k21 00
k= l
-n k=l I1=
=
lVnl � o -
n
1f2 . 6
CoMMENT. Advanced students at the Boston Math Circle were oc
aware of the value of I
1
2
(and could recount Euler's k clever approach to finding it) . They were able to develop essentially the same limit argument presented here. k= l
One can go quite far with the general idea of this ap proach and prove, for instance, that if a series of positive terms
where angled brackets indicate rounding to the nearest in teger. For n odd:
T( n)
r
. This turns out to be correct and 6 9 we can make the argument rigorous as follows:
4
as n � oo, from which it follows that
+ T( a) .
Since T(a) = 0, 0, 0, 1 , 1 , 2 for a = 0, 2 , 4, 6, 8, 10, notice a2 that T(a) is a fractional quantity with modulus less
than
1 + ..!._ + ..!._ + . .
k= l
!!:_ -
r= J
6. Every number possesses 1 as a factor and, speaking loosely, one quarter of the numbers possess 4 as a factor, one-ninth the factor 9, and so on. This suggests that the average count of square factors among all numbers is
I _!__ converges to a finite value L then, on average, ak
k=J
a number possesses L factors from the set {a1 , a , a3 , . . . ) . 2 Thus, for example, a number possesses, o n average, two factors that are a power of two, two factors that are trian gular numbers, and e - 1 factors that are factorials.
FuRTIIER. Let S(n) be the number a.(scalene integer trian
FURTIIER. Show that, on average, a number possesses ln2
COMMENT. In 2002, students of the Boston Math Circle in
St. Mark's Institute of Mathematics 25 Marlborough Road Southborough MA 01 772
gles ofperimeter n. What do you notice about the sequence of these numbers? dependently discovered this result, as well as some nice con nections to partitions of integers. They published their work in FOCUS ( Vol. 22, no. 5, 4-6).
more oddfactors than even factors.
USA e-mail: JamesTanton@stmarksschool. org
© 2007 Springer Science+ Business Media, Inc., Volume 29, Number 3, 2007
59
i;i§iil§i,'tJ
Osmo Pekonen , Ed itor
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by Terence Tao
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103 PP. , PAPERBACK, US $24.95,
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[email protected] 60
BY JOHN J. WATKINS
ike many other professional math ematicians I have spent a good portion of my mathematical life actively involved with the sort of prob lems posed in mathematical competi tions such as the various international Mathematical Olympiads and the William Lowell Putnam Mathematical Competition, and so I have devoted a great deal of effort, time, and thought not only to solving many of these prob lems, but also to the far more elusive task of helping students learn how to do so for themselves. Thus, it was with a great deal of anticipation that I picked up Terence Tao's new book Solving Mathematical Problems. After all, Tao had just been awarded the Fields Medal earlier in the summer at the International Congress in Madrid, and here he was taking the time to write a book for young people on how to solve competitive math problems. I knew his perspective would be a tremendously interesting and useful one for me and my students. It turned out I was right, though not for exactly the reason I thought. In fact, I became utterly confused by the very first sentence in Tao's preface to this second edition of his book: "This book was written 1 5 years ago; literally half a lifetime ago, for me. " But, as I mentioned, I am a professional mathe matician and am perfectly capable of digging myself out of a hole when nec essary. I quickly recalled that Terence was 3 1 when he received the Fields Medal in Madrid in the summer of 2006
THE MATHEMATICAL INTELLIGENCER © 2007 Spnnger Science+Business Mecia, Inc.
and since I could see that the preface had been written from his home insti tution at UCLA in December 2005, he must have been 30 at the time; so it sud denly all made sense. The book in my hands had not been written, as I had thought, by a brilliant 30 year-old Fields Medal winner, but instead by a rather extraordinary 1 5-year-old prodigy. What I most hope to do in this re view is convince you that this remark able work is the fine book it is precisely because it was written by the 1 5-year old prodigy and not by the mature math ematician the author would later be come. Tao himself seems to be fully aware of this, and indeed resisted the urge to bring to bear his formidable later experiences and current level of insight in order to eliminate what he now de scribes as "a certain innocence, or even naivety" from the original exposition. He wisely recognized that "my younger self was almost certainly more attuned to the world of the high-school problem solver than I am now." It is the voice of the younger Tao speaking directly to today's would-be problem-solvers. By the time Terry was 1 1 , in Adelaide, Australia, he was already participating in international competition. By 1989, he became the youngest gold medal win ner ever in the International Mathemati cal Olympiad-he had previously won the silver medal in 1988 and the bronze medal in 1987. Perhaps a completely nat ural next step for this highly precocious teenager was then to write a "How-To" book on problem solving. What is quite surprising, however, is that at that young age he had the maturity to do such a good job of it. In this book Terry-and I'll call him Terry as a way to remind you that the author is 15 and not yet a Fields Medal ist-shows us how to solve only 25 problems, which works out to about four pages per problem; he spends a lot of time discussing his thoughts about each of them. He begins with the fol lowing problem: A triangle has its lengths in an arith metic progression, with difference d. The area of the triangle is t. Find the lengths and angles of the triangle. (See Figure 1 .)
a rea t
b+d
Figure
He uses this first problem to lay down a few basic fundamentals of problem solving such as understanding the problem, the data, and the objective, as well as selecting good notation and writing down what you know in the selected notation and drawing a dia gram. He offers three reasons why it is helpful to put everything down on pa per: a) you have an easy reference for later on; b) the paper is a good thing to stare at when you are stuck; and c) the physical act of writing down what you know can trigger new inspirations and connections. My first reaction when I read (b) was to laugh, but then-and this was the moment it re ally hit me that this 1 5-year-old had some important things to teach me about problem solving and about teaching-! realized how many times in my life I have stared at a diagram and bits of notation on a piece of pa per while waiting to get unstuck on a problem (and because I trained myself to do it on paper I now without much effort can do it in my "mind's eye" and can work on problems at almost anytime and almost anywhere) . For the next page or two he dis cusses ways to modify this problem, a theme he returns to again and again, and always there is a sense of motion, never linear, but always searching, cast ing a net of connecting ideas and facts, looking for a promising route towards a solution. Eventually he settles on an early idea, Heron's formula, that was
previously jotted down and looked es pecially promising. In this problem the semiperimeter s simplified to s = 3 b/2 (applying his axiom on good notation, Terry had chosen b to represent the middle length among the three lengths b - d, b, and b + d in the triangle) . The solution then falls out routinely. There are details to be worked out, to be sure; one has to use the quadratic formula on a quartic polynomial and then know how to use the law of cosines to eval uate the angles in the triangle once the sides are known. But this happens quickly and matter-of-factly in the text. The real action and attention is in the exploratory stage of solving the prob lem. In many ways the entire book is contained in this one example. Tao re peats the procedure for us on one beautiful problem after another, grad ually allowing his thought processes to sink in for the reader. I couldn't help but notice that Terry began his book with a geometry prob lem that was based in a fundamental way on arithmetic progressions. Al though Terence Tao is now widely re spected for his work in many areas of mathematics including the n-dimen sional Kakeya problem, wave maps in general relativity, Horn's conjecture, and nonlinear Schroedinger equations (with a group of four other mathemati cians known as the "I-team"), he is most celebrated at the moment for settling in 2004, with Ben Green of the University of Bristol in England, one of the most
famous conjectures in number theory, a conjecture concerning arithmetic pro gressions! It had long been conjectured that there exist arbitrarily long, but finite, arithmetic progressions of prime num bers. (It is a fairly easy exercise to see that any infinite arithmetic progression contains infinitely many composite numbers.) For example, 47, 53, 59 is an arithmetic progression of length 3 consisting entirely of primes, and 25 1 , 257, 263, 269 is an arithmetic progres sion of length 4 consisting entirely of primes. But longer sequences are quite difficult to find. A recently discovered sequence of 10 consecutive primes be gins with a prime number having 93 digits and the numbers in the sequence have a common difference of 2 1 0 . The text I used when I last taught number theory, an excellent 2002 edition, claimed, with seemingly justifiable con fidence: "Finding an arithmetic pro gression consisting of 1 1 consecutive primes is likely to be out of reach for some time." The author clearly had not anticipated a math prodigy from Aus tralia who had honed his skills in in ternational mathematics competitions, for in 2004 Tao and Green proved that the prime numbers contain arithmetic progressions of any finite length what soever, thus putting to rest a centuries old conjecture. One of my favorite problems in the book begins with a rectangle, con structs the intersection of its diagonals, and then extends two of its sides to create two more points and then asks the problem-solver to show that three ratios, respectively, of six of the given or constructed line segments are equal. What I enjoyed most about this prob lem was watching Terry solve it, much in the way I recently watched in awe a virtuoso performance by the young violinist Joshua Bell. As always, Terry looks for ways to reformulate the prob lem. He first plays a bit with the ratios to get them as simple and symmetric as possible, but he then considers other rearrangements including one which multiplies them out to get products in stead. This doesn't seem to help much but gives him a slight opening since it looks a little familiar to him (though not to me). Here is a formal statement of the problem (see Figure 2):
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c Figure 2
Let ABFE be a rectangle and D the intersection of the diagonals AF and BE. A straight line through E meets the extended line AB at G and the extended line FE at C so that DC = DG. Show that AB!FC = FC!GA = GA!AE. And after awhile here is what he is looking at: FC X BC = AG X BG. And here is what it reminds him of. If a point P is outside of a circle and a line from P cuts the circle at two points Q and R, then the product PQ X PR is called the power of the point P. This terminology was not used by Terry but was introduced by Jacob Steiner in 1 826. The astonishing thing-and I urge you to draw a diagram-is that this product is independent of the line itself; that is, the product depends only on P and not on Q and R. So, in par ticular, if the line is chosen to be a tan gent to the circle, then Q and R coin cide at T, the point of tangency, and PQ X PR = PT2, and in turn PT2 can easily be expressed in terms of the radius of the circle and the distance from P to the center of the circle. (See Figure 3.) So, that's how Terry, with the flour ish of a virtuoso, could polish off this problem. There is not a circle in sight in the statement of the problem, but along the way he is vaguely reminded of a beautiful result from high-school geometry (not that I recall ever seeing it), and so then he notices that a circle
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THE MATHEMATICAL INTELLIGENCER
with center in the middle of the rectan gle conveniently passes through points A, B, and F, and suddenly he is done. Besides the fun of watching Terry solve problems such as these, there is much to learn from his book. One of the lessons of course is that it helps to know things. It helps to know Heron's formula or to know about the "power of the point". But the more important lesson that Terry is so adept at show ing repeatedly is that it is absolutely vi tal to be on alert for some subtle clue to turn up in the problem that is telling us that one of these marvelous facts lodged somewhere deep within our brains might actually be relevant to the problem at hand.
I think this book is destined to be come a classic, to find its place on our bookshelves alongside P6lya's How To Solve It. But I also think this book is likely to be especially effective for its target audience, the young problem solver. In his preface-his first preface, that is-Terry tells us why the Greek philosopher Proclus believed we should like mathematics, and then Terry tells us "but I just like mathe matics because it is fun . " The entire book is filled with this youthful exu berance and it runs through every problem and solution like a splashing river. He speaks of "gung-ho algebraic attacks", "the first sneaky thing to be done" , "our equation is a mess", "hack and-slash coordinate geometry", "we should just play around with it", "the general proof smells heavily of induc tion", "geometry is full of things like this". The reader is just swept along by the sheer joy of it all. Terence Tao had an amazing year in 2006. In addition to the Fields Medal, he was also awarded several other im portant prizes, but most notably he re ceived one of the year's MacArthur Fel lowships which comes with a "no strings attached" $500,000 stipend. The year 2006 crop of MacArthur Fellows not surprisingly included only one mathematician; however, it also hap pened to include an author of chil dren's books, David Macauley. I won der if anyone on the MacArthur selection committee realized that they also selected among their Fellows for the year 2006 a truly remarkable child author, Terry Tao.
Figure 3
REFERENCES
1 . D. M . Burton, Elementary Number Theory,
fifth edition, McGraw-Hill, 2002. 2. G. P61ya, How To Solve It, new edition,
Princeton University Press, 2004. Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail:
[email protected] Introduction to Circle Packing: The Theory of Discrete Analytic Functions by Kenneth Stephenson NEW YORK, CAMBRIDGE U NIVERSITY PRESS, 2005, 356 PP. $60.00, HARDCOVER, ISBN-10: 0-521-82356-0 REVIEWED BY J. W. CANNON, W. J. FLOYD, AND W. R. PARRY
his book describes the rich math ematics associated with patterns of tangent circles in two-dimensional surfaces; the theory is a full discrete ver sion of the classical theory of one com plex variable. Classical theorems have discrete analogues that are visually strik ing. The size of an individual circle is generally proportional to the absolute value of the derivative of some analytic mapping, so that the distortions of a con fo rmal mapping are visible. The au thor, Ken Stephenson, describes the the ory, gives applications, and illustrates
T
Figure 2.
The Apollonian gasket.
Figure I . The owl and its uniformization (p. 1 2). A page number accompa nying a figure caption refers to a figure in the book, reprinted with permis sion from Cambridge University Press.
everything with explicit packings cre ated by his marvelous suite of computer programs titled CirclePack (available at no cost from the website http://www. math.utk.edu/�kens/). Ken's trademark is the owl as shown in the accompany ing illustration. (Figure 1 ) The most famous classical circle packing is the Apollonian gasket: insert a fourth tangent circle in the triangu lar gap formed by three mutually tan gent circles and, iteratively, continue to insert tangent circles into the trian gular gaps formed, ad infinitum. (Fig ure 2) In 1 934, Paul Koebe [Koebel, whose 1907 uniformization theorem is a cen tral result in the theory of conformal mappings, employed circle packings as a tool in conformal mapping. In 1938, Lester R. Ford [Ford] described the Farey fractions as the tangency points on the real line in a version of the Apollonian gasket. E. M. Andreev [Andreev] stud ied circle packings in the construction of convex polyhedra in Lobachevskii
space. William P. Thurston rediscov ered Andreev's theorems in the late 1970s, gave a new proof, and employed circle packings in his famous orbifold trick, which he used in the proof of his Geometrization Conjecture for Haken
Figure 3.
manifolds. (See [Thurs1), [Morgan] , and [Kapovich].) In 1985 Thurston awakened general interest in the theory of circle packing by discussing his elementary approach to Andreev's theorem at the de Branges Symposium, and using the theorem to give a constructive, geometric approach to the Riemann Mapping Theorem [Thurs2] . Thurston's talk has spawned a literature with well over a hundred pa pers, and now Stephenson's book of 356 pages. A circle packing, in its simplest form, is a collection of circles in the plane or 2-sphere such that each gap between circles is triangular, abutting three pair wise tangent circles. Every child is fa miliar with the fundamental packing of the plane by pennies, in which circle centers form the vertices of the planar tiling by equilateral triangles. We would call this penny-packing the equilateral packing, but it is more commonly known as the standard hexagonal pack ing. In the last section of this review, we will consider a sequence H; of hexagonal penny-packings, where the size of the pennies in packing H; is 1/ i. (Figure 3) Stephenson's book considers the rigidity and flexibility of circle packings,
The penny-packing and its dual triangulation.
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63
Figure 4.
vertex).
The Dual Triangulation (outward pointing arrows all go to the same
• • •
•
•
•
•
• • •
Figure 5.
Schwarz Reflection (p. 1 93).
Figure 6.
Convergence to Riemann mapping (p. 278).
64
THE MATHEMATICAL INTELLIGENCER
•
•
and the approximation of conformal maps by circle packings. Rigidity de scribes the discrete version of the clas sical uniformization theorem for Rie mann surfaces. Flexibility mirrors, in discrete terms, the huge variety of con formal and analytic mappings used in mathematics and other sciences. Ap proximation joins the discrete with the classical by showing that many of the most important classical mappings can be approximated by circle packings. Thurston once carefully constructed a pattern of tangent circles for the cover illustration of one of his publications and sent the layout to a graphic artist for professional rendering. The artist came to him in frustration. "I can't get the circles to match the pattern, " he said. "I used my plastic template with circles of all sizes and started to lay out the pattern, but I couldn't make the cir cles fit!" "I had forgotten to tell him," said Thurston, "that, when the outer cir cle of the pattern has been chosen, and when its tangency points with three of the inner circles have been fixed, then the combinatorial tangency pattern de termines the exact size and position of every circle. The entire configuration is unique up to linear fractional transfor mation of the plane. " The reader might try constructing some packings by hand in order to appreciate the difficulties. Stephenson has managed, by means of his program CirclePack, to construct circle packings with prescribed patterns of tangency that involve more than 50,000 circles. His program is modelled on Thurston's algorithms. The existence of circle packings with any plausible pat tern of tangencies is assured by the fun damental theorem of the book, the Dis crete Uniformization Theorem. This is an exact analogue of the classical uniformi zation theorem for Riemann surfaces. Plausible tangencies are described by triangles. If three circles are to be mu tually tangent, with no other circles in the triangular gap, then one assigns to the three circles an abstract triangle whose vertices correspond to the cen ters of the three circles. An edge of such a triangle represents a path from one center to another, the edge passing through the point of tangency. Such an edge is said to be dual to the tangency; it might be an edge of another such tri angle as well. We require that these tri angles triangulate a surface S. The re-
suiting configuration of triangles is called a complex triangulating S. In the illus trated example, the complex triangulates the 2-dimensional sphere. (Figure 4) There is great freedom and flexibil ity in choosing an appropriate complex. The Discrete Uniformization Theorem says that each corresponds to an es sentially unique circle packing .
(" •
•
• , D
• •
./
II • ("
THEOREM (DISCRETE UNIFORMI ZATION THEOREM) Let K be a com
Figure 7.
plex that triangulates a topological sur face S. Then there exist a Riemann surface SK homeom01phic to S and a cir cle packing P for K in the associated in trinsic spherical, Euclidean, or h;perbolic metric on SK such that P is univalent and fills SK. The Riemann surface SK is unique up to conformal equivalence and P is unique up to conformal autom01phisms ofSK
Dessin d'enfants (p. 289).
I
llo
F igure 8.
Convergence to the Riemann mapping (p. 27).
The proof is leisurely and instructive, though it occupies about eighty pages and comprises Chapters 4 through 9. Despite this rigidity, circle packings are flexible. The rigid universal cover ings can be mapped with great freedom if one allows exotic branching, covering, and boundary behavior. The results be come essentially as flexible as classical analytic functions. Stephenson shows how to model Blaschke products, the functions of the disc algebra of analytic functions, all sorts of boundary behavior, discrete entire functions, discrete poly nomials, discrete rational functions, and discrete exponentials. He describes at tempts at a discrete error function. A few of his examples are shown in the three accompanying graphics. (Figures 5-7) Among the classical notions that have interesting circle packing ana logues (as theorems, techniques, con jectures, or problems) are extremal length, the type problem, Schwarz re flection, the Schwarz-Pick Lemma, Liou ville's Theorem on bounded entire func tions, Koebe's 1/4-Theorem, normal families and convergence, the maxi mum principle, and the monodromy theorem. As the author says, "Discrete analytic functions not only mimic their classical counterparts, . . . but actually approxi mate them. . . . It turns out that given the slightest chance, circle packings will almost trip over themselves in their rush to converge" (p. 247).
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Thurston's approach to the Riemann Mapping Theorem via circle packings was completed by Rodin and Sullivan [Rodin-Sulll. Given a bounded, simply connected domain [! in the complex plane, the goal is to approximate a Rie mann mapping that takes the open unit disk D onto [! by circle packing map pings. One chooses a penny size, say 1/ i, and fills [! as nearly as possible with a connected portion P; of the hexago nal penny packing H;, where the un derlying triangulation K; defined by P, is a closed topological disk. The Dis crete Uniformization Theorem yields a packing PK; of the open unit disk D by circles whose tangencies correspond ex actly to the edges in K; and whose outer, or boundary, circles are all tangent to the boundary circle of D. The corre spondence between the circles of PK; and those of P; can be used to define a mapping from most of D to most of f!. The Radio-Sullivan Theorem claims that, after a minor normalization, these par tial mappings converge to a Riemann mapping from D onto f!. (Figure 8) The original proof made strong use of the combinatorics of the hexagonal penny-packing. Other authors have re moved many of the restrictions in volved. The best result to date seems to be that of He and Schramm (see the references in this book). REFERENCES
Press, Orlando, San Diego, San Francisco, New York, London, Toronto, Montreal, Syd ney, Tokyo, Sao Paulo (1 984): 37-1 25. [Rodin-Sull] Rodin, Burt, and Sullivan, Dennis, "The Convergence of Circle Packings to the Riemann Mapping," J. Differential Geometry 26 (1 987): 349-360. [Thurs1] Thurston, William P., "The Geometry and Topology of 3-Manifolds," Princeton Uni versity Notes, preprint.
[Thurs2] Thurston, William P . , "The Finite g Mapping Theorem," invited talk (An Interna tional Symposium at Purdue University in Celebration of de Branges' Proof of the Bieberbach Conjecture, March 1 985). J. W. Cannon Department of Mathematics Brigham Young University Provo, Utah 84602 USA e-mail:
[email protected] W. J. Floyd Department of Mathematics Virginia Tech Blacksburg, VA 24061 USA e-mail:
[email protected] W. R. Parry Department of Mathematics Eastern Michigan University Ypsilanti, Ml 481 97 USA e-mail:
[email protected] [Andreev] Andreev, E. M . , "Convex Polyhedra in Lobacevskii Space," Mat. Sbornik 81 , No. 1 23 (1 970a). [Russian] "Convex Polyhedra in Lobacevskii Space," Math. USSR Sbornik 1 0 (1 970b): 41 3-440
[English]. "Convex Polyhedra in Lobacevskii Space," Mat. Sbornik 83 (1 970c): 256-260 [Russian] .
"Convex Polyhedra of Finite Volume in Lobacevskii Space," Math. USSR Sbornik 1 2 (1 970d): 255-259 [English] . [Ford] Ford, Lester R . , "Fractions," American Mathematical Monthly 45 (1 938): 586-601 .
[Kapovich] Kapovich, Michael, Hyperbolic Man ifolds and Discrete Groups, Birkhauser,
Boston (200 1 ) : 467 pages. [Koebel Koebe, P., "Kontaktprobleme der kon formen Abbildung , " Ber. Sachs. Akad. Wiss.
Introduction to Cryptography with Coding Theory, Second Edition by Wade Trappe and Lawrence Washington SADDLE RIVER, NJ, PRENTICE-HALL, 2006, 592 PP., US$ 90.20, HARDCOVER, ISBN 0-13-186239-1 REVIEWED BY MICHAEL ANSHEL AND KENT D. BOKLAN
Leipzig Math.-Phys. Kl. 88 (1 936): 1 41 -1 64.
[Morgan] Morgan, John W., "On Thurston's Uni formization Theorem for Three-Dimensional Manifolds, " in The Smith Conjecture, Morgan, John W., and Bass, Hyman, eds., Academic
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THE MATHEMATICAL INTELLIGENCER
ost ten-year-old boys and girls run around a lot. Many play video games. Some accidentally download computer viruses. And quite
a few invent secret codes, their very own means of disguising their communica tions from parents and peers. Children quickly learn the rules of cryptography: their techniques must be efficient and their methods must be able to be un done, too. (Budding cryptanalysts, who spend their efforts breaking the systems of their classmates, are scarcer than young cryptographers.) It's the bread and butter of cryptography, the en crypting, and there's a popular mythol ogy to Top Secret ciphers and spy in trigue-with the television shows with the strong encryption that somehow al ways manages to get broken. Today we are inundated with media pronounce ments of strong (or strongest!) protec tions with such ubiquitous phrases as, " 1 28 bit encryption." It seems that every one does it or claims to do it. Even I can do it, with the Captain Midnight de coder badge that I bought on e-bay. But exactly how does it all work? Cryptog raphy is not just the latest trend, like the hula hoop, Betamax, and the Spice Girls. It's here, it's not going away, and some one needs to know how it really works-and if it's really strong.
Ah, but a man 's reach should exceed his grasp, or what's an SSL for? An excellent first step toward the un derstanding of the black boxes of (com mercial) encryption is to work through
Introduction to Cryptography with Cod ing Tbeory, second edition, by Wade
Trappe and Lawrence Washington (which we dub WaTr for purely metri cal purposes). Read it and you'll learn the answer to that mysterious question, "What's [in] that SSL thing?" You may still fumble, though, when your friends ask you, "Should I really trust ama zon.com with my credit card number?" An introduction to cryptology, the sum of cryptography and the cryptanalysis, usually starts with a fundamentals class at the elective undergraduate or early graduate level. WaTr fills about two of these courses, two semesters worth, and it's aimed at an audience of computer science, engineering and mathematics students. But this text is not just replete with the classical ciphers-the Vi generes and the Enigma's-but is full of the flotsam and jetsam that fill the ether about them, those cryptographic prim itives and applications (like the key dis tribution protocols and the digital sig-
natures) that are the backbone of the few high-profile protocols upon which so much of today's data security rests. WaTr provides pedagogy in two dis tinct voices. Unfortunately, this duality is often distinguished by the strengths of the expositions. As we are told in the preface, WaTr plans to "cover a broad selection of topics from a math ematical point of view." To be com prehensive is a near Herculean labor. Some volumes, like [2] and [6], do very well to touch upon almost all of the notable features of cryptography today, but they are not texts for a first lesson in the mechanics of how and why and the mathematics behind it all. WaTr fea tures a sound balance of methods and attacks; it is a pleasure to read. There are the occasional proofs of the math ematical statements, when the proofs are elementary, but WaTr is about a wider introduction. Certainly, a lot lurks hidden beneath the surface, including the hidden Markov models. Trappe and Washington fittingly point to many of the deeper ideas, especially in the the ory of elliptic curves, and they keep the reader both aware and enticed for further study. On the down side, the privation of implementation details and implementation issues in WaTr is a real loss for the student who wants to run with the encryption ball; the devil, after all, is in the cryptographic small print. There are exceptions, though, and WaTr does include the very clever work of [4]. But this is a first text and, as such, serves well. This second edition of WaTr features several important additions to the first edition, including identity-based public key cryptogra phy, an elegant construc tion which holds substantial promise in future applications. Significant cryptan alytic advances in the theory of hash functions are included. The study of hash functions (and collision-finding) is experiencing a revival due to exciting work beginning with [3] and culminat ing in [7]. Thanks to these efforts, we're now looking for new hash algorithms because our faith in the old ones has been ruffled. Also new to the second edition of WaTr is a chapter on lattice methods (including the Lenstra-Lenstra Lovasz method for finding short vec tors). Notable by its absence, though, in both the first and second editions of WaTr is the Merkle-Hellman Knapsack
scheme, the Icarus of public key cryp tosystems. Knapsack was all the rage in the late 1970s: it was elegant and based upon a known NP-hard problem (unlike the RSA system) . Shamir, in 1982, shook the foundation and cer tainly the confidence of the young field of modern cryptography by cracking it-and in so doing changed the face of (public key) cryptography and crypt analysis to this day. The story of Knap sack is part of the history. It makes for great reading, it's high drama, and it provides a strong lesson. But it's not in WaTr and it should be. What is in WaTr and is a highlight of the text is the gentle introduction to DES, (which was) the Data Encryption Standard. WaTr presents it slowly, a few rounds at a time. Block ciphers, like DES, 3DES ("triple DES") and Rijndael (the new standard, the Advanced En cryption Standard) are the load-bearers of data encryption. They are each com posed of rounds, and a single round is a structured shaking of the input. Start ing with the plain text input to the first round and repeated on the output of the previous round, a block cipher is designed with enough rounds so that the result, the cipher text, is jumbled enough-meaning that the influence of the input on the output (and vice versa) is fully diffused. The mixing steps are usually a combination of permutations and substitutions and some non-linear lookups (DES has some famous S-boxes that do this) all the while being de signed to be invertible so plain text can be recovered. WaTr explains the work ings of DES in parallel with the crypt analytic method of differential crypt analysis. By so doing, it becomes clear(er) why DES has 16 rounds. The discourse does get a bit technical. How ever, the parallel presentation is well worth the effort of careful study. The best cryptographic algorithm designs are structured around what attacks are known and then laid out to be resistant to them. For symmetric key protocols, like block ciphers, it's all about the mud dling and the repetition of the
The paradig m of "easy to do but hard to undo " lies at the heart of e1yptography.
processes. For asymmetric schema, as in public key cryptography, designs are predicated upon mathematical prob lems that are "easy" (computationally efficient) to perform but believed to be "hard" (computationally infeasible) to invert-without some extra piece of knowledge, a key. These problems are called trapdoor one-way functions and are not to be confused with one-way functions for which there is no key to undo them. (Hash functions are one way functions.) There is no real proof that trapdoor one-way functions exist since obtaining lower bounds for these kinds of complexities seems near im possible. There is faith, based upon many years of very limited success, in a few select problems: the discrete log arithm problem and that of finding, for some e, an e-th root modulo a number of unknown factorization. Discrete logarithm problems (dip's), as involved in, for example, the classi cal Diffie-Hellman key agreement pro tocol, take a form such as: find x if
7x � 2434711235764822669040730
(mod 4083497104378553871 280549).
WaTr's treatment of the discrete log problem and approaches to solve it, the Pohlig-Hellman algorithm and the index calculus, are exemplary for first-year stu dents. The hard (but nevertheless toy sized) dip above can be solved on your laptop. When the modulus has several hundred digits, things get very tough. (The index calculus approach, while sub-exponential complexity, does not scale well enough to be efficient.) The RSA architecture involves raising a message, m, to a fixed known power, e, modulo a number n whose factor ization is a secret (and e and n are rel atively prime). A result may look some thing like this (for e = 31): m3 1
�
1970517852344637324142632145 5642097240677633038639787310457 022491789 (mod 495960937377360 604920383605744987602701 1013993 99359259262820733407167).
Breaking RSA is about finding m. If n can be factored, this is easily ac complished. (Raise both sides of the equation above to the power d where
d-1
�
e (mod rp(n)).
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That recovers m. The proof is a simple application of Euler's generalization of Fermat's Little Theorem.) It is unknown if there is a way to find m that is more efficient than factoring the modulus. Since factoring special types of large numbers is believed to be hard, the RSA system, with a sufficiently large, hard modulus is currently considered secure. (The example presented here is, hope fully, a small step towards dispelling the common misconception that an RSA modulus need be the product of two large primes. For efficiency, it ought to be-or close. It need not be, though.) WaTr gives a quick overview of some factoring techniques; enough to convince the would-be RSA code breaker that things can be tremen dously challenging. The questions of the (computational) equivalence of the RSA problem and of factoring-and of the discrete logarithm problem and the Diffie-Hellman proto col-are amongst the most important open issues in cryptology today. Progress has been made on the latter question (see [5]) in the affirmative di rection. On the former, the recent work is less than convincing. WaTr really shines in its initiation into the world of elliptic curves and el liptic curve cryptography. From a sim ple introduction to the group law to H. Lenstra's beautiful elliptic curve factor ing method (which can solve the RSA question in this review) to the elliptic curve analogue of the discrete logarithm problem and the Diffie-Hellman proto col, WaTr provides a fine rendering for the initiate. The elliptic curve one-way trapdoor function is simple: given an el liptic curve E defined over a finite field, a point P on E and k some positive integer, find k given kP (where kP = P + P + P + · · + P, k times). Again, it seems (computationally) difficult-or infeasible-to do this for large enough carefully chosen examples. There's a lot of data security (and commercial prod ucts) banking on that. There are unfortunate omissions in WaTr. There's no real discussion of (al gorithmic) complexity where it may have been well-placed to provide the reader with a sense of appropriate key sizes and protocol (and attack) strength. And there's the sporadic lack of the sense of largeness, the why's of why are things like this?. More generally, ·
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THE MATHEMATICAL INTELLIGENCER
what is incidental and what is of real import in the digital world is not always clear. These gaps, though, are alleviated in part by an assortment of excellent (and detailed) end-of-chapter exercises and computer problems that allow and encourage the reader to identify some of the subtleties and gain a deeper ap preciation of the why's. WaTr skips almost the whole field of stream ciphers. That's a shame. Stream ciphers are a major component of encryption technology today. And WaTr features only a cursory look at linear feedback shift registers, the pri mary constituent of most stream ciphers over the past century and many very good random-number generators. Lin ear feedback shift registers (and their associated tap polynomials) are rich in mathematical theory and can be designed and combined to provide very satisfactory output. Bad random number generators, at least for crypto graphic purposes, are based upon sim ple linear congruential generators of the form X11
=
AXn- l + B (mod m)
where m is fixed and A and B (and �) are unknowns (but chosen so that the period of the generator is large). One can easily deduce the next "randomly" generated number from knowledge of the previous three-and this predictabil ity makes for a very bad random-num ber generator. (Yet this is how many rand functions work!) Stream ciphers are needed for real time encryption when you can't wait for a whole block of plain text to arrive before you use your block cipher. Stream ciphers aren't just for voice communication anymore. Cryptography sells, from the great propaganda of "the only provably se cure system" (one-time pads) to the in troduction of quantum cryptography. Using principles of quantum mechanics for cryptographic applications is an idea now a few decades old-and remains ever intriguing. It also makes for great press. Most notable among quantum methods is the key exchange protocol introduced by Bennett and Brassard which allows legitimate participants to (probabilistically) recognize the exis tence of an eavesdropper on their communication. It's a lovely idea that requires substantial overhead (a chan nel so clean that a photon in transit is
undisturbed). WaTr presents the ideas, this glimpse of a possible future with quantum cryptography and with quan tum computers (if substantial ones can ever be built). Then, with Shor's algo rithm, the pre-eminent quantum com putational cryptanalytic tool, most everything would change-for security, for the Internet, and for cryptology leading us to wonder, in the words of Buffy, "Where do we go from here?" A world of post-quantum cryptogra phy is being studied in anticipation of one plausible future. Non-abelian ap proaches have been suggested which do not seem to succumb to quantum attacks. Though such ideas are not in WaTr, for they are still in their early de velopment, these considerations are providing new avenues of investigation. ([1] offers an analogue of the Diffie-Hell man key establishment protocol wherein, instead of a discrete logarithm problem, the restricted conjugacy search problem serves as the trapdoor one-way function.) WaTr tries to cover a lot: the past, the present, and the (uncertain) future. It is occasionally uneven in its mathe matical level, the knowledge expected of the reader. The Information Theory and the Error Correcting Codes chap ters are not as carefully composed as much of the rest of the book and do not have the same (encouraging) instructional rhythm. The latter part could benefit from some compression and reordering, and the Information Theory section could afford some ex panded coverage of language recogni tion. (How does your computer know an acceptable decryption when it finds one?) WaTr is the best book of its kind. Appendices of Matlab, Maple, and Mathematica exercises support the rhetoric of the individual chapters be cause in cryptology small examples can give a false sense of security. We can quibble with what's not in WaTr, but you can't do it all at once. And what WaTr does is almost always done well. To do it all-that would be as daunt ing as the task of breaking " 1 28 bit en cryption," whatever that is. REFERENCES
[1] I. Anshel, M. Anshel, D. Goldfeld, An alge
braic method for public-key cryptography. Math. Res. Lett. 6 (1 999), 287-291 .
[2] Menezies, van Oorschot and Vanstone, Handbook of Applied Cryptography, CRC
Press 1 997.
t was an uninteresting assignment, except for the tenth problem: Evalu ate
[3] A. Joux, Multicollisons in iterated hash func tions. Application to cascaded construc tions, Advances in Cryptology-CRYPTO 2004, Lecture Notes in Computer Science
31 52, Springer-Verlag, 2004, 306-31 6. [4] P. Kocher, Timing attacks on implementa tions of Diffie-Hellman, RSA, DSS, and other systems, Advances in Cryptology-CRYPTO 96, Lecture Notes in Computer Science
1 1 09, Springer-Verlag, 1 996, 1 04-1 1 3 . (5] U. Maurer, Towards the Equivalence of Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms, Advances in Cryptology- Crypto '94, Lecture Notes in Computer Science 839, Springer-Verlag,
1 994, 271-281 . (6] B. Schneier, Applied Cryptography, 2nd edition, John Wiley, 1 996. [7] X. Wang, Y. Yin, H . Yu, Finding collisions in the Full SHA-1 , Advances in Cryptology Crypto 2005, Lecture Notes in Computer Science 362 1 , Springer-Verlag, 1 7-36.
Michael Anshel Department of Computer Sciences The City College of New York, CUNY 1 38th Street and Convent Avenue New York, NY 1 0031 USA e-mail:
[email protected] Kent D. Boklan Department of Computer Science Queens College, CUNY 65-30 Kissena Boulevard Flushing, NY 1 1 367-1 597 USA e-mail:
[email protected] Dr. Euler's Fabu lous Formu la: Cures Many Mathematical I l ls by Paul]. Nahin PRINCETON, NJ, PRINCETON UNIVERSITY PRESS, 2006. xxii + 380 PP. $29.95 ISBN: 978-0-69111822-2; 0-691-11822-1 REVIEWED BY PAMELA GORKIN
remember thinking, "You can't do that." Then I figured it out. It was one of those mathematical moments that makes you say "wow." Paul ]. Nahin's hook, Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills, is filled with such moments. The book, like its title and cover, is clever, creative, and unique. It contains every short story Nahin can think of that uses e7Ti + 1 0, plus a few that don't. You might know that Euler's formula was one of the most frequently cited "great equations," according to a poll con ducted by Physics World in 2004. In 1990, readers of The Mathematical In telligencer voted it the most beautiful of 24 formulas, with a score of 7.7/10. Why did this equation receive such a high score? Many people cite its sim plicity and brevity, as well as the con nection between five important con stants in mathematics. Though there are dissenters, most mathematicians agree: this formula needs no introduction. Nahin has a gift for recognizing good stories and has put together a collection of mathematical "tales" about Euler's formula that would make a fine addi tion to a differential equations or com plex analysis class. The reader should, however, be forewarned: although the back cover informs us that the book is ''accessible to any reader with the equivalent of the first two years of col lege mathematics," to read and enjoy this book, most readers will need more mathematical maturity. In addition, though Euler's formula may need no in troduction, applications of Euler's for mula need motivation-and they don't always get it in this book. Many of the formulas and computa tions incluclecl are among the highlights of a typical complex analysis course (Wallis's formula, for example). There are also many stories that will be new to readers. There is an account of the Gibbs phenomenon, which is a story with a fas cinating history. (A longer version of this history, without Nahin's biography of the overlooked Henry Wilbraham, appeared in an article by Edwin Hewitt and Robert E. Hewitt in 1 979.) A wonderful =
chapter titled "Vector Trips" features R. Bruce Crofoot's story about his clog Rover. Crofoot runs a pretty compli cated path, which he sketches for the reader, each morning. He is the proud owner of a well-trained clog who always runs exactly one foot to his owner's right. Given the path, the owner, and the clog, it turns out that Crofoot runs farther than Rover. The question is: How much farther did Crofoot run? I liked the article when I read it in Math ematics Magazine and I liked it here too. It's not really an application of Euler's formula, but it is a nice use of complex numbers and vectors. On the other hand, the discussion of the vibrating string problem (as well as a development of a solution to the wave equation) really does use the fact that ffx = cos x + i sin x in an essential way. This serves as the introduction to the story of what "was probably (almost certainly) the first 'Fourier series' . " When you think o f Fourier series you probably don't think of funny stories, but in Nahin's hands they become amusing. He presents Euler's "remark able claim" that 7T -
- �
2
=
.
f'
_
t
----
n=l
SID
sin( nt)
-----
n
•\ sin(2 t) sin(3 t) + ( t; + --- + 2 3 ---
·
·
·
.
As Nahin points out, this is indeed re markable, in part because it is not true (check out what happens at t 0). But now Nahin has your attention; now you should want to know the story behind Euler's claim. Other stories would have benefitted =
from a little motivation. Nahin presents
the "beautiful formula" 00
L
n=l
C nn+11/ n2 -
=
�/1 2
and Euler's result "which made him world famous":
These are followed by more sums, in cluding one "dazzling result," a "spec tacular application of Parseval's for mula," a "pretty result, " and "an even more beautiful generalization" of it that will appear in the succeeding chapter. Now, "excited" is not the first word that comes to mind to describe my students
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when I cover series, particularly one af ter the other (as Nahin does here), no matter how enthusiastic I am when I present it. So, I would imagine that young readers would need more than a list of beautiful infinite sums to keep their attention. Consider the way Nahin motivates finding X
I m= l
( - l) m cos(mx) ( m + l)(m + 2)
·
After a very brief discussion in which we learn, primarily, that Ramanujan was a self-taught genius who was in terested in this question, we read: " this problem will be interesting because at one time it 'interested' a genius. " Often the motivation i s there but it follows the result. For example, we learn that "writing cos(wt) and sin(wt) in terms of complex exponentials is the key to solving difficult problems," which appear three chapters later. Then, in a discussion on Fourier series, we find out that we'll get examples of valuable results . . . soon. On p. 214, Nahin promises that we'll see how forming the product of two time func tions is essential to the operation of speech scramblers and radios. He ful fills this promise on p. 289. The wait for motivation is, occasionally, too long and the phrase ''I'll tell you later" ap pears far too often. Nahin's strength is his ability to draw the reader in. For this reason, it is dis appointing when he claims that his ex planation or proof is simply the result of cleverness. This happens frequently: in a discussion of how to evaluate "Dirichlet's discontinuous integral"
we learn that the solution will depend on an auxiliary integral. This idea, we are told, came from someone who was "very, very clever. " This is followed by the "clever trick" of evaluating a dou ble integral over a triangle by chang ing the order of integration. Then, to aid the discussion of Fourier's integral theorem, Nahin presents the "devilishly clever trick" of thinking of a function defined on the real line as periodic with infinite period. Many of these are not tricks at all, but rather insights, meth ods, or techniques. The author is an electrical engineer writing a book for a general audience.
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THE MATHEMATICAL INTELLIGENCER
His approach is one that m1mm1zes technicalities and rigorous justification. The reader should be aware, therefore, that Nahin is a man who will, in his own words, "reverse the order of inte gration on a double integral as fast as you can snap your fingers," who never hesitates to interchange integrals and in finite sums, can't resist pulling deriva tives through integrals, rearranges series without comment, and, again in his own words, manipulates two "Fourier series equations in a pretty rough-and-ready way with little (if any) regard to justi fying the manipulations. " The author points out that, sometimes, being overly concerned with technicalities can "par alyze" a mathematician into inaction. In addition, Nahin's approach makes the book readable and helps to maintain the reader's interest (though there are some things that will be pretty hard for a pure mathematician to swallow, in cluding one derivation that Nahin him self admits, "many 'pure' analysts are truly aghast [at]"). While Nahin's style will appeal to a wide audience, a problem arises when the author holds others to a different standard. Nahin criticizes Philip Davis and Reuben Hersh for using the unique ness of factorization of the integers without mentioning it in their proof that Vz is irrational. In an early (and oth erwise beautiful) chapter, titled Tbe Ir rationality of r, Nahin tells us that he is following Carl Siegel's book, Tran scendental Numbers, which was written for graduate students at Princeton. He adds that Siegel often "leaped" over steps and some of the leaps are of "Olympic size." In fact, the holes Siegel left are standard fare for a good grad uate student, and the treatment here is quite close to that of Siegel except for the attention to relatively small details. At times it appears that Nahin is unable to decide whether Dr. Euler's Fabulous Formula is a rigorous, detailed treat ment of mathematical formulas or an ef fort to convey the beauty of the for mulas to a wide audience without regard to technicalities. Nahin offers us enthusiasm. He tries to surprise and, sometimes, startle the reader. When he offers his opinion on various subjects outside the main thrust of the book we are reminded of Strunk and White's advice: "Do not inject opin ion. . . . Opinions scattered indiscrimi-
nately about leave the mark of egotism on a work." Consider the following: Nahin begins his book with a discus sion of beauty, mathematical and oth erwise. This leads him to a discussion of Jackson Pollock, of whom Nahin says, "anybody who can observe the re sult of simply throwing paint on a can vas-what two-year-olds routinely do in ten thousand day-care centers every day . . . and call the outcome art . . . is delu sional or at least deeply confused (in my humble opinion). " Fortunately for the reader, the number of opinions ex pressed is a decreasing function of the page number. The book ends with a short biogra phy of Euler. Nahin treats Euler's life in stages; Euler's years in Switzerland, the years in St. Petersburg, the years in Berlin, and the return to St. Petersburg. Nahin's writing is entertaining, marred only by the curious statement that "while there is a steady stream of bi ographies treating famous persons . . . there is not even one book-length bi ography, in English, of Euler." At this point it would have been more appro priate for Nahin to focus on the large amount of work about Euler's mathe matics and life that does exist: There is William Dunham's book Euler, Tbe Mas ter of Us All and Varadarajan's recently published book, Euler Through Time: a new look at old themes. There are pa pers on Euler's years in St. Petersburg, a history of analysis beginning with his work, and a history of the logarithm ending with his work. Birkhauser Ver lag and the Euler Commission of Switzerland have published Leonbardi Euleri Opera Omnia, an enormous work containing Euler's mathematics and correspondences. There's an Euler society, an Euler archive, Sandifer's on line column How Euler Did It, and a big celebration planned in Basel for Euler's 300th birthday. Euler was even featured on the old Swiss 10-franc note, not a claim most "famous persons" can make. Frequent lack of motivation, strongly stated opinions, and overzealous at tempts to get the reader's attention de tract from an otherwise well-written, well-researched, and interesting idea. In other words, despite its flaws, Dr. Euler's Fabulous Formula is an exciting mathematical read. This book is ideal for readers who see themselves in Nahin's description of G. H. Hardy: "dis-
playing an unevaluated definite integral to Hardy was very much like waving a red flag in front of a bull . . . " But this is a mathematics book with a sense of humor and a lot of opinions. If that's not how you like your mathematics books, you probably won't be curling up by the fire with this one. In any case, one thing is certain: if you present the short stories here to a classroom full of students who have successfully com pleted their first course of differential equations and who still have a mathe matical twinkle in their eyes, and if you present them with Nahin's energy and gusto, you are surely going to be a very popular teacher. Department of Mathematics Bucknell University Lewisburg, PA 1 7837 USA e-mail:
[email protected] Tribute to a Mathe magician Edited by Barry Cipra, Erik D. Demaine, Martin
L.
Demaine,
and Tom Rodgers WELLESLEY, MA, A. K. PETERS, HARDCOVER, 350 PP., 2004, US$ 38.00, ISBN: 1568812043 REVIEWED BY CLIFF PICKOVER
ach year, as I begin to write my next popular mathematics book, I gaze at my bookshelves filled
with books by Martin Gardner, and I
chant to myself, "What has Martin not already done? What hasn't he done'" Many consider Gardner to be the father of recreational mathematics. He has brought mathematics to the general pub lic, and many mathematicians began their lifelong love of mathematics as a result of Gardner's influence. His mind has roamed far and wide. He is also an avid debunker of pseudoscience, and his "Mathematical Games" column in Scien tific American was key to introducing important and fascinating mathematical subjects to a wide audience. Some of his hottest and most memorable columns in cluded topics on flexagons, john Con way's Game of Life, polyominoes, the
soma cube, Penrose tiling, and fractals. Yes, Gardner is my hero and my inspi ration, and his articles, books, and kind and humble approach to life will leave a mark upon the world forever. Tribute to a Mathemagician is the third book in a series written to honor the mind, writings, and works of Mar tin Gardner. Each book is a collection of articles by seasoned and amateur mathematics and puzzle aficionados all of whom Gardner has inspired. The current book is based on dozens of ar ticles, many of which were presented at a "Gathering for Gardner" conference held in 2004. Sample puzzles and games in this book include blackjack, Chinese ceramic puzzle vessels, paper folding, Mongolian interlocking puzzles, rolling block puzzles, and sliding puzzles. The articles range from one-page teasers to full-length articles. The topics are organized in six parts: Braintreasures, Brainticklers, Brainteasers, Braintem plers, Braintaunters, and Braintools. Each part contains a variety of chal lenges, which vary in difficulty so that both students and veteran mathemati cians will find something to delight. For example, the chapter titled ·'Chinese Ce ramic Puzzle Vessels" contains a valu able history of a peculiar set of attrac tive puzzles, many of which are based on historically known laws of physics, such as the use of siphons. Another de lightful chapter describes a three-legged hourglass that lets users measure frac tions of the total time. The hourglass designed by M. Oskar van Deventer and illustrated proudly in the book-con tains eight minutes' worth of sand alto gether. The three lobes of the hourglass
Dissections," and "Rolling Block Mazes." Martin Gardner, now in his 90s, at tended only the first two Gatherings for Gardner meetings. But as a writer who has encouraged the field of recreational mathematics to blossom, and inspired thousands of careers, he remains the guiding spirit of both the book and the conference. Many readers of The Math ematical lntelligencer have been molded and remolded by Gardner, and we know that he has left a mark on us all.
are situated at the corners of an equi
about them-what should I read?
lateral triangle. When it is turned, the sand in the top lobe flows equally into the lower two lobes. Operating instruc tions involving rotations are given to achieve various timings. Other chapters deal with polyomino number theory and Godelian puzzles. A brief sample of some favorite chapter titles gives a flavor of the diverse content and in cludes: "Configuration Games, " "Five Algorithmic Puzzles," "Mongolian Inter locking Puzzles," "Fold-and-Cut Magic," "The Three-Legged Hourglass," "The In credible Swimmer Puzzle," "Sliding Coin Puzzles," "Underspecified Puz zles," ''The Complexity of Sliding-Block Puzzles and Plank Puzzles," "Hinged
P.O. Box 549 Millwood, New York 1 0546-0549 USA e-mail:
[email protected] Elementary Set Theory with a U niversal Set by Melvin Randall Holmes LOUVAIN·LA·NEUVE, BRUYLANT-ACADEMIA, 1998, 241 PP. HARDCOVER, 1150 BF, ISBN 2-87209488-1 REVIEWED BY ROBERT JONES
c: A: s:
he following conversation was overheard during a stroll across the campus. Persons: Professor Calculus Professor Algebra one of their students.
s: In calculus this morning you were talking about sets. I don't know much c: You might begin by looking at Holmes's book on the universal set. s: The universal set? What's that? c : It's the complement of the null set. s: But that's not a set, that's everything! e : That's a popular myth. Of course the universal set is a set. The null set has a complement, every set does. A: You can't get along without alge bra! s: It still sounds paradoxical. c: No it's not. Holmes shows you how to avoid the paradoxes. A: Most standard introductions to set theory use a so-called relative comple ment notation to form set complements.
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c:
Take a look at Halmos's book
Naive Set Tbeory.
Why bother, if it's naive? It's anything but naive! s : Sophisticated naive set theory? A: Now you've got it. s : But why should you do away with relative set complements? A : They're a throwback to your fears about sets that are too big. With a uni versal set, we can use absolute set com plements. They're simpler to write down. Don't worry, there's no danger when using them. s : How did all this come about; what led up to it? c: Well, before you read Holmes, take a look at Grattan-Guinness. He ex plains why you need sets in a calculus course. s : Okay, I will, but tell me anyway, who thought up the universal set? c : Holmes reviews its history. It be gan with Quine. Quine showed how to have a universal set, yet avoid the set theory paradoxes. He first presented the material in a course in mathematical logic that he gave in his first year of teaching at Harvard. His system is known as "New Foundations," or just NF, after the title of his 1 937 paper. s: Wow, I suppose that mathemati cians got excited! c : Well, NF has a dramatic history. Ernst Specker, a mathematician in Zurich, derived the first really deep re sult about NF: he showed that it ex cludes, or is inconsistent with, the un restricted Axiom of Choice. A: That disappointed some re searchers, but it led to a better under standing of NF. c : Yes, his result showed everyone that they should not try to append the Axiom of Choice to the axioms of NF. s : And then? c: Rosser wrote a foundation for all of mathematics using NF. Jensen pro duced a set theory, NFU, that is a ver sion of NF, but which was proven, with out the AC, to be consistent. This gave mathematicians a choice of using either NF or, with similar proof methods but known consistency, NFU. A: Perhaps the best assessment of NF, as a foundation for mathematics, is in the book by Fraenkel, Bar-Hillel, and Levy. c : That was "the great non-special ized beginning of set theories with a s:
A:
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THE MATHEMATICAL INTELLIGENCER
universal set," the time of the great am ateurs. s: But what's happened lately? Or did it just stop there? c: No, indeed. Holmes covers that in just the first 14 chapters. s: So it moved on? c: Yes, after that, the mathematicians claimed the field as their own. But the rest of the story is more specialized and too complex to summarize. In 1987 there was a seminar to honor Quine and report on the work he had brought about. By then he had become the grand old man of the subject. Oswald, who spoke in the seminar, said "W.V. Quine confessed to be amazed at the work that has been done on the sub ject he had fathered, and at seeing what a 'Pandoras's box' he had invented in
1937."
Maybe I should read the seminar proceedings? c: Holmes's book is a better intro duction; I think it is the best introduc tion now available. s: What does he say about this latter period? c: He makes his own choice of top ics. I think the most interesting part is the next-to-last chapter, which he wrote with Robert Solovay. It includes un published work. The last chapter is an excursion into the prospects for found ing mathematics upon functions with out appealing to sets to define them. A: Don't you have any criticisms? No book is perfect. c : One small point and a complaint. s: And those are? c: My small point is this: Holmes sug gests an alternative name for the theory customarily called "Morse-Kelley set theory." But why rewrite history? s: And your complaint? c: Holmes should have taken ad vantage of his collaboration with Robert Solovay to describe, at least briefly, Solovay's beautiful model of the real numbers, in which every subset of the real numbers is Lebesgue measurable. A: Not many calculus students know about Lebesgue measure. s: I don't. c: Mine do. You'll learn it next se mester. But if you want to know more about Solovay's model nowA: What person with at least a mod icum of interest in mathematics would not? s:
c: -then take a look at Kanamori's book on large cardinals. Anyway, I wish Holmes had discussed how Solovay's model fits in with NF and NFU. None of them requires the Axiom of Choice. s: I 'd like to buy Holmes's book. c: It's out of print. But Holmes promises a second edition on his web site. You might also read Forster's book-it's called Set Tbeory with a Uni versal Set-and Holmes's review of it in the journal of Symbolic Logic.
REFERENCES
Boffa, M . , and E. Specker (cochairs). Mathe matisches Forschungsinstitut Oberwolfach, Tagungsbericht, Number 9, 1 987: New Foundations, March 1 -3, 1 987.
Forster, T. E. Set Theory with a Universal Set. Exploring an Untyped Universe. Oxford logic guides, no. 20, Clarendon Press, Oxford Uni versity Press, Oxford and New York, 1 992, viii + 1 52 pp. Second edition, Oxford logic guides, no. 31 , Oxford University Press, 1 995. Fraenkel, A A., Y. Bar-Hillel, and A Levy. Foun dations of Set Theory, North-Holland, sec ond edition, 1 984, Studies in logic and the foundations of mathematics, 67. Grattan-Guinness, lvor. From Calculus to Set Theory, London, 1 980. Halmos, Paul. Naive Set Theory, van Nostrand, Princeton, 1 960. Holmes, M. Randall. review of Forster's book above. The Journal of Symbolic Logic, 58 (1 993), 725-728. Jensen, Ronald Bjorn. On the consistency of a slight (?) modification of Quine's 'New Foun
dations' , " Synthese, 1 9 (1 969), 250-263. Kanamori, Akihiro. The Higher Infinite. Large Cardinals in Set Theory from their Begin nings, Springer-Verlag, Berlin and New York, 1 994, Section 1 1 . 1 . Kelley, John L. General Topology, Van Nos trand, New York, 1 955. Quine, W.V.O. "New foundations for mathe matical
logic,"
American
Mathematical
Monthly, 44 (1 937), 70-80.
Quine, W. V. 0. "The inception of 'New Foun dations' , " Bull. Soc. Math. Belg. -Tijschr. Be/g. Wisk. Gen 45 (1 993), 3, Ser. B,
325-327. Rosser, J. Barkley. Logic for Mathematicians, Second Edition, Chelsea, New York, 1 973. Specker, E. P.: "The axiom of choice in Quine's 'New Foundations for Mathematical Logic' ", Proceedings of the National Academy of Sci ences of the U. S.A., 39 (1 953), 972-5.
Rurweg 3 D-41 844 Wegberg Germany e-mail: ivanhoe491
[email protected] l
