Showing papers in "The Mathematical Intelligencer in 1984"

Counting and recounting: The aftermath

Abstract: This is a recount of the letters that I received from readers who continued where I left off by offering solutions to the problem The problem is not that new P Erd6s, on reading the article, was quick to point out to me that Hungarian mathemat ic ians tackled it in the thirties: P Veress proposing and G Hajos solving it In his letter to me I Gessel (MIT) has given a survey of the more recent history of the problem Proofs were published by D Kleitman (Studies in Applied Mathematics 54 (1975), also by his student D J Kwiatowski (PhD Thesis, MIT, 1975) It also found its way into texts (Feller, Mohanty) In addition, I received solutions by A Bondesen (Royal Danish School of Educational Studies, Copenhagen), K Gr~nbaum (Roskilde Universitetscenter, Denmark), J Hofbauer, jointly with N Fulwick (Universit/it, Wien, Austria), D Zeilberger (Drexel University, Philadelphia), and verbally from C Pearce (Adelaide University), directly after reading the article All solutions are based, with some variations, on the count of lattice paths, or equivalently (1,0) sequences Figure 1 is used to illustrate the simplest version It represents a two-dimensional coordinate lattice, or a network of streets running East and North We consider paths of length 2n, beginning at O, proceeding in unit steps, heading East or North It is clear that there are 22n ways in which a lattice point on the boundary AB can be reached This gives the left-hand side of the identity Counting in a different way, assume that the last crossing of a path with the NE line (OM on the diagram) is at K(k,k), which of course may coincide with O or M It is easy to see that there are (2~) possible paths from O to K Assuming for the moment that the number of ways the remaining 2n 2k steps, (avoiding OM) may be taken, is similarly (2n_-k2k), we obtain the desired right-hand side: ~=ot2k) (2n-~k) 9

The Proof of the Mordell Conjecture

Abstract: Andre Weil has frequently criticized the use of “conjecture” in mathematics: Sans cesse le mathematicien se dit: “Ce serait bien beau” (ou: “Ce serait bien commode”) si telle ou telle chose etait vrai. Parfois il le verifie sans trop de pein; d’autres fois il ne tarde pas a se detromper. Si son intuition a resiste quelque temps a ses efforts, il tend a parler de “conjecture”, meme si la chose a peu d’importance en soi. Le plus souvent c’est premature. [17]

Kurt Gödel in Sharper Focus

Abstract: The lives of great thinkers are sometimes overshadowed by their achievements—a phenomenon perhaps no better exemplified than by the life and work of Kurt Godel, a reclusive genius whose incompleteness theorems and set-theoretic consistency proofs are among the most celebrated results of twentieth-century mathematics, yet whose life history has until recently remained almost unknown.

Integer programming and cryptography

Abstract: Not long ago it was reported m the press that Adi Shamir, from the Weizmann Institute of Science m Israel, had broken one of the first public key cryptosystems, the Merkle-Hellman knapsack System Saentific American (August 1982, p 79) reported maccurately that He did so by provmg that a mathematical problem called the knap sack problem, which had been considered e^ceedmgly dtfficult, am be solved rapidly by a simple Computer algonthm

On the Steps of Moscow University

Abstract: MOSCOW, Aug. 26—A University of California mathematics professor was taken for a fast and unscheduled automobile ride through the streets of Moscow, questioned and then released today after he had criticized both the Soviet Union and the United States at an informal news conference.

Spurious mathematical modelling

Abstract: The main object of the present paper has been to say something about the role of mathematics in the exploration of reality. Without underestimating this role, I have advocated a non-aprioristic point of view.

Lost greek mathematical works in arabic translation

Abstract: The works of the ancient Greek mathematicians which we possess represent only a few fragments from the wreck of the great treasure ship of Hellenistic mathematics. What has come down to us is little more than a reflection of the pedagogical interests of the schoolmen of late antiquity and Byzantine times, who caused to be copied only those works which were of some use in the curricula of their institutions of higher education at Alexandria, Antioch, Athens, Constantinople, and a very few other places. Their choice illustrates the impoverished intellectual climate of the Greek world in the millennium from the third to the thirteenth century a.d. Thus the compendium of elementary geometry which goes under the name of Euclid was transmitted through the schoolrooms, but none of the works on higher geometry which Euclid wrote (probably in the early third century b.c.) has been preserved; and only the first four books of Apollonius’ Conics, which treat the elements of the theory, continued to be copied in Byzantine times: The last four books, which deal with more advanced topics, are lost in Greek.

Sofia kovalevskaia and the mathematical community

Abstract: Many mathematicians know at least a little bit about Sofia Kovalevskaia. She was, after all, a glamorous figure, and had a life packed full of exciting events. But those mathematicians tend to think of her as somewhat divorced from the mathematical wor ld of her time. They see her as a talented amateur, as a "weakness" of Karl Weierstrass, or as the flighty, slightly hysterical personal i ty that comes across in the accounts of Eric Temple Bell, Anna Carlotta Leffler (G6sta Mittag-Leffier's sister), and others. These impressions are misleading. Kovalevskaia was a dedicated professional. She was one of the most active of the early editors of Acta Mathematica, and with Mittag-Leffier took responsibility for propagandizing the journal in Europe. She was on cordial terms with the best mathematicians in the world (indeed, in her time, she was considered one of them), and was included in their discussions of academic politics, promotions, appointments, and so on. Kovalevskaia numbered Charles Hermite, Emile Picard, P. L. Chebyshev, Carl Runge, Weierstrass, and Mittag-Leffler among her frequent correspondents. These people were particularly assiduous in their praise of one another 's achievements. The eminent astronomer Hugo Gyld6n referred ironically to the pentad of Hermite, Weierstrass, Picard, Koval-

Human activity: The soft underbelly of mathematics?

Abstract: Human activity and experience are an integral part of mathematics. They not only play a crucial role in the creative process of research but are also basic to the way human beings learn. I have suggested three lines of attack which would give students a better understanding of what mathematics is really about. First, we have to show them how (at least some of) the material they are studying originated and evolved into its present form through the combined efforts of generations of mathematicians. Second, we must encourage them to take an active part in their own learning, not just by practising routine exercises but by developing their own creative talents—thereby sharpening their insight into the way mathematics is created. Third, we have to find some way of relating higher mathematics to students’ previous experience. “The more abstract the truth is that you would teach, the more you have to seduce the senses to it” (F. Nietzsche,Beyond Good and Evil).

Gerard Debreu wins the Nobel Prize

Abstract: I was at the office early Monday, October 17, when my wife Clara called with the news that our friend Gerard Debreu had won the Nobel Prize in economics Although I had anticipated this event, it was exciting and a pleasure to hear that it had actually happened Gerard and I have been close friends since we met fifteen years ago I still remember our first encounter He came to my office to ask about some mathematics needed for his work (he was trying to show that a general econorrry could have only a finite number of equilibria) I found Debreu friendly; we could communicate easily about mathematics and economics I was especially impressed with his ability as a mathematician, his clarity and rigor His questions were the beginning of my own work in economic theory In the following years we spent long hours in discussions (Our day-hikes at Point Reyes have been especially memorable) We exchanged questions, his more mathematical and mine about economics Eventually, Gerard joined the mathematics department; I joined the economics department Debreu's great contribution is his profound use of mathematics in the central theme of economic theory, consolidating an insight of Adam Smith more than 200 years ago Debreu has given the foundations of general equilibrium theory in his classic work "Theory of Value" The award of the Nobel prize to Debreu gives a valuable impetus to basic research in mathematical economics In 1776, "The Wealth of Nations," Adam Smith suggested an "invisible h a n d " promoted economic activity He wrote:

A bit of gossip: Koebe

Abstract: Paul Koebe, known from uniformisation theory, was born a century ago (1882) in Luckenwalde (50 km from Berlin, 25,000 inhabitants), which was my birthplace too. His family ran a renowned fire-engine plant. I was born in the same year when he took his Ph.D. at Berlin University. I do not believe I saw him more than once -a t a dis tance-in our common birthplace. When on the first day of my study I was inscribed at the Mathematical Institute of Berlin University, it happened that Bieberbach was around and heard that I came from Luckenwalde. He turned and asked me: "So you are one of the Luckenwalde streetboys who run after Koebe to call 'there goes the famous function theoretician,' are you?" I was flabbergasted and did not even deny it, though afterwards people explained the heart of the matter to me. K o e b e was so candid that he never concealed that he was a famous man. "It is the talk of all Europe: Koebe mailing reprints." In hotels he never registered as Koebe. He travelled incognito because he could not stand waiters and chambermaids asking him whether he was a relative of the famous function theoretician. Among colleagues he was given the nickname of "the greatest Luckenwalde function theoretician." Younger people, when introduced to Koebe, almost automatically reacted: "Ah, the great function theoretician." This earned one of my friends an assistantship with Koebe. In Brouwer's works II, p. 575, I told a story about Koebe and Brouwer. There is a famous footnote in Brouwer's paper on uniformisation which appeared in G6ttingen Nachrichten, 1912. The note was added without Brouwer's knowledge after he had seen the last proof; it acknowledges some priority of Koebe. There is a cloak-and-dagger story here: On one dark af ternoon in March 1912 an unident i f ied person wear ing a large hat, a tu rned up collar, and blue glasses, called at the printing office of GOttingen Nachrichten, and asked for the printer's proof of the next issue. He got it and after a while gave it back. The identity of this person has never been determined, nor is it known whether he made any change in Brouwer's reading proof, which of course d i sappeared after printing. Years later, when Koebe was asked about this incident, he explained it as a trick that somebody had played on him. This is not as impossible as it looksG6ttingen was a paradise of practical jokers. In later times poor Bessel-Hagen was the most popular target. Once a night a wel l t imed bat tery of alarms were hidden at various places in his bedroom; every hour on the hour one of them went off. But let us spare Bessel-Hagen for another note. The Battle of the Frog and the Mouse (from The Fables of Aleph) by John Hays