All of the fields of mathematics have been experiencing exponential growth since the 1950's; this impetus is not likely to flag any time soon. Research mathematicians are not inclined to spend much time re-visiting the past. Surprisingly, when they do, they are often quite satisfied to absorb the dreadful pulp being fed to the public by such authors as Sylvia Nasar, Amir Aczel, Marc de Sautoy, Douglas Hofstadter, etc. ( I do not deny that there are some good writers in this field, starting with Martin Gardner: Ian Stewart, Reuben Hersh, James Gleick and others) .
As a defensive reaction, the despised historians (in some quarters the adjective is not too strong ), may tend to meticulous pedantry. They resent being treated as if they don't know anything worth knowing, or as if they were impotent in reasoning power. These reactions led to an interesting interchange in the question period of one of the morning lectures.
I walked into the IHP around 9. The secretaries, who know me from 5 years of enrollment for the Fall Trimestre, had my badge and conference bag ready for me. Bag it was: a handsome black plastic handbag secured by a lock that broke almost immediately, filled with all kinds of small goodies: the conference schedule; a porcelain coffee cup with a portrait of Galois on the outside; a little "automorphism" game (pushing numbered tiles around with thumbs), courtesy of the Societé des Mathématiques de France , a small tablet and pen for taking notes; and the conference schedule.
Coffee, small pastries and fruit juices were available from fancy catering carts in the dining lounge. It is standard for these things to be provided at any IHP conference, though rarely at so high a level of catering. Illumination on this phenomenon came to me from a casual glance at the list of sponsors.
The bureaucracy of support organizations for mathematics in France is truly stupendous. The list of intellectual, educational and professional societies that contributed to honor the brief career of the 19th century's paradigm of romantic genius, Evariste Galois, included :
Inside the handsome old IHP building, the corridors were packed with high intelligence, all in various degrees of abstraction (sometimes distraction). This turnout had to include the crème de la crème of the mathematics community of France! I welcomed conversations from everyone I met.
Saint-Jean Patrick was standing against the right wall, waiting for the auditorium doors to be opened. He is, by his own description, a mathematical biologist. There are several schools of mathematical biology in France, some of them inspired by the research of René Thom, whose essays I've translated. Patrick is friendly, happy to engage in conversation, rotund with a sparse, ambiguous beard. (Since a recurrent theme of the conference was 'the theory of ambiguity'I'd begun seeing ambiguity everywhere!). It turns out that he was also interested in the kind of mathematical research associated with music that one finds in places like IRCAM the research facility at the Centre Pompidou established by Pierre Boulez, where I've given a few lectures, and a colorful permanent floating seminar called MaMuPhi (Math, Music , Philosophy).
He does research in pure mathematics as well. His field is Non-Transitive Topology. I'm not sure of what that means, but the title is suggestive. No matter. He told me that the great Alexander Grothendieck has gone on record for saying that "Topology is Transitive". Because of Grothendieck's standing, no journal in France will publish Patrick's work if he uses the term Non-Transitive Topology! He must disguise it as Extended Topology, or Modified Topology.
This kind of carping appears to be typical of the French mathematics community. A few days later I met a group theorist who, within less than a minute of shaking hands, began complaining that no journal in France will publish articles on finite group theory anymore; they have to be on Galois groups, or Lie groups.
Patrick joined me later in the Salle Hermite, an auditorium with a steeply sloping floor that can hold about 300 people, plus standing room (which attained its maximum (+50 to 70?) at the final lecture of the conference, given by Alain Connes). I was directed to a seat by the IHP concierge, Madame Bonnet. A good-hearted lady, Madame Bonnet sometimes feels the need to take charge when she doesn't have to. This morning she was instructing everyone as they came in where they should sit.
I rebelled: not for political reasons but because she'd place me far too close to the screen and the light was hurting my eyes.I spontaneously climbed (the benches are low) out of the seat and moved back up into the auditorium a good 6 rows. As far as I could see, I was the only person with the audacity to make up my own mind about where I wanted to sit. Madame Bonnet is used to this kind of behavior on my part, but she always seems to be shocked; for about 2 minutes.
I found myself sitting next to a dour,learned and wearyingly polite British savant stationed at Oxford University,Dr.Peter Neumann. Dr. Neumann asked me for my name. That he'd never heard of "Roy Lisker" seemed to discountenance him: he usually knows the names of most people he meets at conferences. Neumann perked up when I told him about Ferment Magazine, and I passed him a business card. He seemed to have thought that I manage an On-Line research journal, so he gave me some interesting information: On-Line Journals in England have to print up 5 copies of each issue, to be sent to the 5 "Libraries of Reference": Cambridge University, the Bodleian Library at Oxford, the British Library in London, Trinity University in Dublin, and the University of St. Andrews in Edinburg. If or when he looks up Ferment Magazine, I'm not sure he will consider it an On-Line Journal in his sense, but I'd better start making my trip to the printers as soon as I return to Middletown.
The conference was opened by Yves André, mathematics research director at the Ecole Normale Superièure. He seemed suited (even to the appearance of his blue suit!) for his role, an effective speaker with a somewhat bureaucratic manner. I later discovered that he also works on projects with MaMuPhi and IRCAM. The cross-fertilization of mathematics with music does not seem to have the stigma in France that it has acquired in the US. Here, if you reveal that you are interested in this subject, the assumption is that either:
(1) You are an amateur mathematician disguising your incompetence by claiming to apply it to music;
(2) You are an amateur composer disguising your incompetence with high flown mathematical language;
(3) Both.
It is a matter of historical record that Evariste Galois was kicked out of the ENS in 1830 for insubordination. He shares this distinction with Léon Blum, expelled from the ENS around 1909 for his activities as a left-wing journalist. Galois had sent a letter to a radical journal, La Gazette des Ecoles condemning the actions of the principal of the ENS (then called the Ecole Preparatoire) , a M. Guigniault, in the revolution of July, 1830: Guigniault had locked up the student body to prevent it from going out to join the street battles.
Guigniault's personality was in fact tyrannical, but he also saved Galois' life. Had Galois been killed on the barricades of July, 1830 we would not have one scrap of the research that makes him famous.It was a perilous time for the sciences in general. In a famous letter written by the mathematician Sophie Germain to Guillaume Libri in April 1831, she talks about the "misfortunes falling on all of mathematics" before going on to describe the wretched mental and physical state of Galois. (Libri (even the name was prophetic!) was later to win fame as the most notorious book thief in European history. In 1848, under cover from the fog of war, Libri smuggled 30,000 stolen books out of France to England. To this day they are still being recovered.)
In the question period after the talk on Tuesday by Nobert Verdier (author of Galois, le mathématician maudit) I asked him about the meaning of her comment; he didn't have any ideas. I have my own hypotheses:
One of the good things the French Revolution accomplished was the recognition of the importance of science. Following the revolution, major institutions were set up that protected and advanced the sciences, including mathematics for which France had already become internationally renowned.
When, in 1824, Charles X replaced his relatively liberal brother, Louis XVIII as king, he inaugurated a broadly conceived ultra-royalist reaction. This included restoring public education once more into the hands of the Jesuits. To be a teacher it was more important to don a black soutane than it was to know anything. The resultant chaos between pre- and post- Enlightenment conceptions of education and the sciences must have lasted long past the revolutions of 1830 and 1832.
The Ecole Normale Superièure has come a long way since Galois' day, and it was perfectly suitable that Dr. André should preside over the Bicentenary Conference. Indeed one should, with more justice, associate Professor André with Louis-Paul-Emile Richard, the teacher of mathematics at the Ecole Preparatoire who was the first to recognize Galois' amazing abilities.
André praised Galois for inventing the concept of a "group". He quoted Alexander Grothendieck (the name will re-occur constantly) as saying that the two most important inventions in the history of mathematics are the "concept of zero" and the "group concept". Galois certainly did not act in a vacuum, though he supplied the word "group": the essentials of the concept are already present in the works of Abel, Cauchy, Lagrange, Cayley, Kirkman, Hermite and Jordan, some of the most famous names in 19th century mathematics. They were sometimes called "substitutions" or "permutations". However, Galois' work was unpublished until 1847, and Group Theory per se, as we learned from the distinguished talk by Caroline Eherhardt on Tuesday, did not emerge until the beginning of the 20th century.
M. André then introduced the first speaker: Massimo Galuzzi, historian of mathematics from the University of Milan. Galuzzi spoke of the two most direct influences on Galois' work: Auguste Cauchy(1789- 1857) and Joseph Lagrange (1736-1813). It is clear from his writings that Galois had set himself the task of resolving certain problems and questions posed by Lagrange in his famous textbook: Reflexions sur la Resolution Algebrique des Equations (1771).
.. Parenthetically, when was the last time in our own day when the advance of science was prompted by a famous textbook? Hawking and Ellis' Large Scale Structure of Space-Time(1973)? Perhaps. Bourbaki has been given far too much credit vis-a-vis its' real influence. Indeed, for the most part this is considered to have been negative. Today's textbooks, at least in the United States, tend to be big, glossy over-priced bricks with large pictures and sparse text, and little impact on anything beyond the pocketbooks of impecunious students ..
However, Lagrange is not mentioned in any of Galois' memoirs, yet Cauchy is referenced in numerous places. Galuzzi argued that this is so because one doesn't find any method behind the ingenious techniques devised by Lagrange for the solution of algebraic equations. Both Cauchy and Galois exhibit a congenital tendency to elaborate methodical structures: Theorems, Lemmas, Scholia, Corollaries, Propositions. (Can this obsession with formal structure in the presentation of ideas can be correlated with Galois' concern with the inherent "structure" of algebraic polynomial equations? Possibly.) As one finds in modern algebra, concerns with structure in Galois' writings predominate over detailed arithmetic computations or geometric pictures.
An interesting idea. After the lecture I asked Galuzzi if Lagrange had been inclined to investigate the symmetries of the Euler-Lagrange equations central to all the branches of theoretical physics. Clearly a language barrier was involved because he didn't understand the question: Galuzzi has written about this and related topics. Other speakers did in fact make allusion to the parallels between the symmetries of differential and algebraic equations, notably in the related area of the Picard-Vessiot theory of differential equations. Understanding the symmetries of the Hamilton-Lagrange equations, and their relationship to the conservation laws of physics was the great achievement of Emmy Noether in 1918.
Galuzzi's French was too rapid, at least for me (and I suspect even for French auditors). No doubt aware of this problem, his entire paper was flashed onto a screen and read off from the text on his computer. Evariste Galois' indifference to explicit computations was undeniably one of the reasons why the first two copies of his memoir to the Académie des Sciences were lost, and the third rejected. Galois had invented a new way of doing mathematics for the times, which did not involve explicit computations but was based on the relationship between symbols.
One is talking about a procedure that was, of course,standard practice in logic; however this field would not come out of its 2000 year-long deep freeze until the end of the 19th century, when it would join forces with the many streams converging to modern algebra. His judges at the Académie des Sciences (Lacroix, Poisson, Poinsot) did not understand how such methods could lead to real understanding of the solvability of equations.
Harold Edwards:
Galois' version of Galois Theory
"The symmetries that offered the way out of the problems of elementary particle physics in the 1950s were not the symmetries of objects, not even objects as important as atoms, but the symmetries of laws."
This is a clear statement of the way in which Galois' ability to "think outside the box" has overtaken all of science.
However, Galois Theory per se underwent its own paradigm shift when it was recast in the language and concepts of modern algebra by Emil Artin in the 30's! One therefore finds two "types" or shall we put it, two "styles" of Galois Theory floating about in the mathematics education community. There are teachers and researchers who know their "Galois Theory" very well, but can't make head or tale of the Artin reformulation; and those (for example the textbook of I. N. Herstein, "Topics in Algebra") which seem to owe nothing to Galois himself.
One is talking of two paradigm shifts in other words: (a) pre-Galois to post-Galois; (b) pre-Artin to post-Artin (and, for those who really understand the subject,(c )pre-Grothendieck to post-Grothendieck!).
Edwards clearly thinks that pre-Artin Galois Theory has been neglected and he's set himself the task of correcting this. He described Galois as a "genius of constructive mathematics". He supplied quotes from the Premier Memoire to show that Galois truly understood that it was possible to make purely algebraic computations.
To illustrate this point Edwards analyzed Lemma 2 of the Premier Memoire. Given a monic polynomial f(x) with integer coefficients, Galois constructs the Resolvent, a new polynomial whose degree may up in the thousands; but he isn't interested in evaluating it . Instead he constructs an abstract form V= Ax+By +..+Zz, where x,y, ..,z are the roots, and A,.., Z are integers to be determined. One calculates all possible values of V under all permutations of the roots, then multiplies them together. V is called the "primitive element" (Emil Artin dispenses with this construction, apparently on aesthetic grounds).
An ingenious argument to show that A,B,..,Z can be chosen so that all permutations of the roots give distinct values, is taken by Galois as sufficient "proof" that it is possible to find those values! No explicit computations are made.It was surely this that perplexed his judges.
In the question period afterward,the pedantry of the historians was pitted against the rigor of the mathematicians. Most notable was an exchange between Harold Edwards and Peter Neumann. They argued for some time over whether Galois used the words "permutation" and "substitution" independently, or interchangeably. Edwards argued that Galois thought of the two notions as interchangeable right from the start. But Neumann reminded Edwards that on page such-and-such of the Premier Memoire, Galois had written "permutation", then crossed it out and wrote "substitution" above it! Neumann has translated all the texts I'd looked at in the library of the Institut de France, and published a book with the French on one page, English on the facing page; not surprisingly, he's all but memorized everything in them.
Not to disparage the issue altogether: this minor point does have something to do with Galois' fame as the inventor of group theory. The group has the property that there is no real distinction between a "function" and a "variable", as one finds in the calculus. In a finite group, a "permutation" (an arrangement of numbers that can be represented as a permutation matrix), and "substitution" (scheme for moving the numbers about), do become identified.
Edwards' presentation was followed by a talk by Christian Houzel, its' title translated as "Transcendental aspects of the works of E. Galois". Houzel is frequently cited for his survey essay Fonctions Abeliennes et Elliptiques in Jean Dieudonné's Abrigé d'Histoire des Mathétiques .
What might Houzel mean by the word "transcendental" (transcendant in French) I wondered. The word has a very specific meaning in mathematics: a transcendental number is any real number that is not algebraic. Yet it could also be used in the general sense to mean "transcending the norm"; or, in a religious context, "heavenly", "aspiring to the beyond."
It turns out that what Houzel means has to do with his investigations into the discoveries in the papers remaining after Galois' death that do not relate directly to the solutions of algebraic equations. These have been loosely described as the "Troisième Memoire" and were not published in his life-time. Houzel may also be attempting a far-fetched pun, combining the notion of "non-algebraic functions", with Galois' "transcendence" over all his contemporaries!
Houzel is a lucid, compelling lecturer. Though he spoke in French I could easily understand everything he said. The evidence in these unedited, unpublished pages shows that Galois was at least 20 years ahead of his time in the study of the properties of elliptic functions, and elliptic and Abelian integrals. According to Houzel, very little has been done, even after 140 years, in terms of bringing to light Galois' research in these areas. For one thing, everything he did was eventually discovered later. Also, the history of mathematics is a somewhat neglected area. If some past mathematician's research doesn't make a direct contribution to contemporary research there is not much incentive to pick it up and take a second look.
That ends this coverage of the Galois conference for Monday, October 24th. I probably spent the evening at a reading/book-signing at the Shakespeare & Company bookstore. Unfortunately I did not get a chance to visit with its founder, George Whitman, during this visit to Paris. He is one of my oldest friends in the city. He's over 97 and spends much of his time in bed. An emergency developed while I was there and he had to be taken to the hospital. Just before I left I checked back with the bookstore, and learned that he was back at home and recovering.
Continued at Galois3