by Allan Adler (copyright Allan Adler 2000)
The third meeting of Mathematical Culture took place on June 26, 2000 at Barnes and Noble's cafe on Campbell Lane in Bowling Green, KY and was attended by five people, including myself. This cordial gathering was the final meeting of the discussion group, at least for this year under my leadership.
The books we discussed at this meeting were Eric Temple Bell's book Men of Mathematics and Lynn Osen's book Women in Mathematics. Both are collections of biographies of mathematicians. The books are quite different. Bell's book represents a genre which the French mathematician Jean Dieudonne described as "novelistic biography". The lives of the mathematicians therein are often real adventure stories. Unfortunately, Bell doesn't give many references to support his accounts and in many cases he is just plain wrong about the facts. Bell was a novelist as well as a prolific mathematician. He wrote science fiction under the pseudonym John Taine. He wasn't an historian.
Lynn Osen's book is better in that respect, since she is careful to support her assertions with specific references. In some cases, those references are not completely reliable, but at least one knows what they are and one can examine them critically. On the other hand, although she mentions the mathematics that each of her women did, she doesn't explain any of it. Bell does attempt to explain some of the relevant mathematical ideas to his non-mathematical reader. However, even in such attempts, he sometimes does violence to the mathematics in order to achieve a literary goal.
In spite of its weaknesses, Bell's book is well worth reading. In fact, both books, apart from providing purely factual information about the remarkable lives and accomplishments of certain mathematicians, also provide intellectual role models such as most people never meet. These role models can be quite inspiring, although one might have to be somewhat selective about the qualities one might want to emulate. I myself read Bell's book when I was in high school and it certainly inspired me. Lynn Osen's book was written to provide similar mathematical role models for women.
Since Bell has about forty biographies in his book and Osen has about eight, I couldn't hope to discuss all of them at the meeting, but I did try to touch a few bases.
I began by placing a transparency of a pair of pages from Heiberg's edition of the collected works of Archimedes. This is a bilingual edition with the original ancient Greek on the left and a translation into Latin on the right. Heiberg produced his edition during the 19th century. Most people couldn't be expected to read the Greek without difficulty so the Latin translation was supplied to enable the average educated person to read it and perhaps to compare it with the original. Since then, the study of Latin has gone the way of the dodo and this very useful edition of Heiberg, now fairly expensive, has become accessible only to scholars. Therefore, it is perhaps useful to draw attention to an offhand remark that Bell makes on p.224 of his book, where he mentions that any student can master enough Latin in a few weeks to read the works of Gauss and Euler in that language (except, of course, for the difficulties of the mathematics itself). I want to encourage people to do just that and one of the steps I am taking in that direction is to prepare a bilingual edition (Latin and English) of a memoir of Gauss. To illustrate what I have in mind, I placed a transparency of a page of my work in progress on this project. I will elaborate on that work on another occasion.
The paper of Gauss that I am editing is his memoir, "Theoria residuorum biquadraticorum" (theory of biquadratic residues), written in the early 19th century. Around 1949, Andre Weil (pronounced Vay), one of the most important mathematicians of the 20th century, read Gauss' memoir (in the original Latin, of course) and was inspired to formulate what came to be known as the Weil conjectures. These were deep conjectures about the zeta functions of algebraic varieties over finite fields and stimulated major developments in number theory, algebraic geometry and other fields. In mentioning Weil, I showed the three volumes of his collected works to the group. The phenomenon of Weil's inspiration in this manner illustrates the vitality of old sources of mathematics. Even though Gauss' paper was almost 150 years old when Weil read it, it led Weil to the Weil conjectures.
We should not be surprised at this. By having access to the original sources, one can see the motivation of their authors and sometimes one can see that old approaches, obsolete after being superseded by more powerful methods, nevertheless might be more suitable for another purpose. Also, an old piece of research might be misunderstood until some future time when new points of view make it possible to appreciate it. And sometimes, old research is simply forgotten and rediscovered later. One striking example of this is Weil's 1952 paper, "Jacobi sums as Hecke characters". In 1974, he published another paper with a similar title, "Sommes de Jacobi et caracteres de Hecke", in which he carries his old results further but also has a footnote explaining that since the time he wrote the 1952 paper, he learned that the main result of his earlier paper had already been discovered by Gotthold Eisenstein a century earlier. Eisenstein's work had simply been forgotten.
After reexamining Eisenstein's work, Weil wrote a book entitled Elliptic Functions According to Eisenstein and Kronecker, in which he explained their approach to elliptic functions and used them to point out a direction in contemporary research. I read Weil's book and was motivated to make my own study of the work of Eisenstein. This resulted in my article, "Eisenstein and the Jacobian Varieties of Fermat Curves", which appeared in the Rocky Mountain Journal of Mathematics a few years ago. The paper is concerned with an incidental remark in a paper of Eisenstein, in which he says he can do something further than what is in his paper and hopes to return to the matter on a future occasion. Unfortunately, Eisenstein died soon after without carrying this out. I set myself the task of trying to reconstruct what he might have had in mind. To pursue it, I needed to rely on a lot of original sources. These included not only the works of Andre Weil, but the collected works of Carl Gustav Jacob Jacobi (the eight volumes of which I also showed to the group), the two volumes of Eisenstein's collected works, and dozens of other volumes of collected works that were too heavy and numerous to bring with me to the meeting.
Having mentioned Jacobi and gestured to the eight volumes, I declined to discuss his work in detail but contented myself with showing transparencies of four separate pages of one volume of his collected works. One page was a letter to the French mathematician Legendre, in French. A second page was an article on the solution of some differential equations, in Italian. A third page was an article in German. This page afforded an opportunity to make what I believe is an important pedagogical point, particularly for autodidacts, namely, that a page of mathematics can very nearly be read by any mathematically literate person even if he/she does not know the language it is written in. The fact that one knows in detail what the content of a page is in an unknown language makes it possible to guess or decipher a lot of the page, or even to read it, by relying on context, cognates and recognized loan words. Just by looking at the formulas, one could tell that Jacobi was taking the identity exp(-log(1-x))=1/(1-x) and proposing to verify it by brute force using only the power series expansions of the exponential function and the function log(1-x).
The remaining page was in Latin, taken from Jacobi's article, "De Determinantibus Functionalibus". Even if one knows no Latin, one can make an intelligent guess that this is about functional determinants, which today are called Jacobian determinants because of this work of Jacobi.
Thus, Jacobi wrote in several languages and those who wish to appreciate original sources shouldn't shrink from reading them. Equally important is the availability of those sources. Students at Western Kentucky University, for example, are much less likely to be exposed to them because the library doesn't have them: not Weil's works, not Jacobi's, not Eisenstein's, and I could go on to mention practically all of the great mathematicians whose biographies one finds in Bell's book and in Osen's book.
At this point, I put the transparency of Archimedes' paper back on the projector and asked the two students who happened to attend this meeting if they knew the formula for the area of a circle. Happily, they did, namely pi r-squared (or in TeX notation, $\pi r^2$). I then asked them if they knew who had discovered this formula. One guessed Aristotle, but I explained that it was due to Archimedes and that this was the very paper in which he had done it. I pointed out to them that the ancient Greeks were not in possession of algebra as we know it and that Archimedes therefore formulated the result somewhat differently. According to Archimedes, the area of a circle is equal to the area of a right triangle, of which one of the legs is equal to the radius and the other leg is equal to the circumference of the circle. It is an easy exercise in algebra to verify that this is equivalent to the usual formula.
Just as the work of Gauss reached across centuries to inspire Weil, the work of Archimedes reached across two millennia to inspire the people who developed integral and differential calculus. Today we use calculus to justify the formula for the area of a circle, for example, and Archimedes essentially anticipated some of the ideas needed to do so.
After these brief remarks about the work of Archimedes, I spent a little while giving an account of his life.
According to Bell, Archimedes lived from 287-212 BC. Although he traveled to Alexandria, in Egypt, he spent most of his life in the city of Syracuse, in Sicily. Even though Sicily is part of Italy, where people now speak Italian and where one might think they used to speak Latin, actually they spoke Greek in Syracuse at the time, as did a lot of places in Italy. In fact, there are still villages in Italy where a version of ancient Greek is still spoken.
At this point, I asked the group what they would think if someone who had been sitting quietly suddenly got up and yelled, "Eureka". The concensus was that they would think the person had just discovered something. I asked if they could think of anyone who had actually used that expression and no one could come up with one. I don't really know but I suggested that it was a story about Archimedes that had fixed the word "Eureka" in the popular mind. I explained that, although Archimedes was primarily interested in research in pure mathematics, he sometimes helped out the tyrant Hieron of Syracuse by doing applied work. On one occasion, Hieron sought Archimedes' help in finding out whether a gold crown that had been made for him (Hieron) was pure gold or whether it had been mixed with copper. If you could melt down the crown (which of course you wouldn't want to do after having gone to all the trouble to make it), say to form a block, you could determine its volume. Without melting it down, you could determine its weight, and thereby you could compare its density with the density of pure gold. You can determine the volume of the crown without melting it down by submerging it in a container of water, seeing how much the water level rises and determining the volume of the water that has been raised above the former level. According to the story, this idea occurred to Archimedes while he was taking a bath, whereupon he jumped naked out of the bath and ran to the palace yelling, "Eureka".
Archimedes also demonstrated his skill in applied research when it became necessary to defend the city of Syracuse from the army of the Roman general Marcellus. Archimedes designed a crane, called the "claw of Archimedes", which could reach over the walls of Syracuse, lift the Roman ships out of the water and wreck them. He also developed a super catapult. This persuaded the Romans to give up, at least for a while. Instead, they attacked the city of Megara and, after they took it, they sneaked up on Syracuse from behind while the Syracusans were distracted with a rather festive religious holiday. Thus was Syracuse captured. A Roman soldier entered the dwelling of Archimedes and, according to the story, killed Archimedes when the latter told him not to disturb his geometrical drawings. According to a variant of the story, Marcellus so admired his worthy adversary Archimedes that he sent the soldier to bring him so he could meet him, but the soldier got angry when Archimedes spoke to him in this manner. One comparative historical sociologist I know insists that it would have been completely out of character for the Romans to have behaved so nobly in battle and that instead Archimedes was simply hunted down and executed.
Be that as it may, the story of Archimedes contains elements that capture the imagination and which inspire one by its example. This is only one of the stories in Bell's book and it is one of the reasons why, in some sense, this book is the mathematical equivalent of Lives of the Saints. One finds miracles of the intellect and martyrs to mathematics.
Just as the mathematics of the past remains vital, so do the lives of mathematicians through their example. Just as the mathematics of Archimedes reached out across two millennia, so did the life of Archimedes reach out across two millennia to inspire a young girl named Sophie Germaine, one of the mathematicians whose lives we find in Osen's book. Sophie Germaine was born on April Fools day in 1776. She was a teenager during the French Revolution and had to stay home a lot for her safety. She spent a lot of time in her father's library, reading books. One of the books was a history of mathematics by Montucia. When she read about the death of Archimedes while he was studying geometry, she decided that geometry must be a fascinating subject and wanted to learn it. Her parents, afraid that too much study would be bad for her physical and mental well being, tried to discourage her by removing the candles from her room and confiscating her clothes at night, so that she would have to stay in bed instead of studying. She coped with this ineffectual persecution by hoarding candles and studying by candle light at her table, wrapped in her bed quilts. When her family found her one morning asleep at her table, wrapped in her quilts, with the ink frozen in its container and her slate covered with computations, they relented and let her study mathematics. She went on to become an important mathematician who was closely associated with Gauss.
Osen's hope is that such stories will reach out across centuries and inspire women today.
Next, I displayed a sketch of Evariste Galois, taken from the thin volume of his collected works. I described him as a rebellious teenager who also happened to be a mathematical genius. His work settled questions that had been unsolved for centuries and whose implications settled questions that had been unresolved for thousands of years. He died in a duel in his early twenties. Knowing that he would probably die in the duel, he sat up the night before, writing down as many of his mathematical ideas as he could to send them to his friend Auguste Chevalier. He told him to show them to Gauss and Jacobi, not to ask about their correctness but only about their importance. Working frantically, at one point he wrote, "I don't have time". The next day, he was mortally wounded in the duel.
This poignant tale of young genius, thoroughly misunderstood by those who oppressed him, has a rare appeal to many young intellects and Bell exploits it to the hilt. Among the other authors who wrote about Galois were Fred Hoyle, the astronomer and science fiction writer, and Leopold Infeld, the physicist, whose biography of Galois is entitled "Whom the gods love". In order to put the matter in some perspective, I pulled out an article by Tony Rothman on the fictionalization of Evariste Galois. Rothman looks closely at the available sources and pokes holes in the accounts of Bell, Hoyle and Infeld. It isn't that the story is completely false, only that the slants of all three authors are misleading about some important details. It remains true that Galois was a great genius, that he was killed in a duel, that he wrote a letter to Auguste Chevalier the night before the duel telling him about some of his unpublished ideas and that he wrote in the letter, "I don't have time" as the hours of his life ticked away. At the request of one of the students, I read the relevant line in French and freely translated it: "Mais je n'ai pas le temps, et mes idees ne sont pas encore bien developpes sur ce terrain, qui est immense" (But I haven't the time, and my ideas are still not well developed in that domain, which is vast.)
I displayed a transparency of two pages of Galois' letter At this point, I showed the group a book that had been written by my late thesis advisor, Michio Kuga, when he was young. The book is in Japanese and is entitled "Garowa-no Yume", which means, "Galois' dream". We can take that to mean that he was guessing what else Galois wanted to write in his final letter and presenting it in this book. Kuga wrote his book starting from the rudiments of mathematics, even telling the reader what a set is, and leading the reader to the very frontiers of research. Because of the cute illustrations, Kuga often referred to Garowa-no Yume as his comic book. It is an attractive book, so much so that it became (according to Kuga's account) a fashion article, carried around by young women just because of its appearance.
Kuga selected the title partly because of the subject matter but also because Galois, if my muddled recollection of Kuga's account is accurate, had also become a role model for student political activists, just as Galois himself had been a student activist at the time of the revolution of 1830 in France.
Next I showed the group a picture of Sonia Kovalevsky, one of the mathematicians in Osen's book. She was born Sonia Krukovsky, daughter of General Krukovsky, in 1850 and died in 1891 at the age of 41. Although she was allowed by her parents to study mathematics, in Russia at that time women were not allowed to attend universities. She couldn't simply go abroad to get an education since that would have been scandalous for an unmarried woman. So she did what a number of women did under the circumstances: she got married to someone who was willing to provide her the legitimacy marriage would confer on her trip abroad. Such marriages, which were not intended to be consummated, were called "form marriages" or "nominal marriages". We tend to think of such conditions as the barbaric past we left behind, but I have no doubt that many women in the world today are faced with the same dilemma.
Her marriage enabled her to travel to Germany, where she was still not allowed to take classes officially, but where she was at least allowed to do so unofficially. She then travelled to Berlin to study with Weierstrass, but found the university would not let her attend classes under any circumstances. So Weierstrass tutored her privately. Subsequently, she returned to Russia, but the only job a woman mathematician could get teaching would have been in a girls' school, teaching multiplication tables. Sonia declined this opportunity, with the excuse that she was not good at multiplication. Instead, she spent her time writing novels. Subsequently, the Swedish mathematician Mittag-Leffler got her a job in Sweden and she was finally able to have a career in mathematics.
When one reads that an interesting character like that wrote novels, it is natural to want to read them. Unfortunately, they seem to be unavailable as far as I know, and that points to some important work that needs to be done. The works of the past need to be kept in print and available to all who might want to read them. In the age of the internet, it is probably not difficult to keep such works online. The same applies to the mathematical work of these role models, which is likewise often not available. Even when it is, it is often hard to persuade libraries to acquire them even when they can be obtained for free. There needs to be an inexpensive way to keep this heritage in print and easily accessible.
I concluded my remarks by showing the group some of my other books that bore some relation to the mathematicians whose lives are featured in the books of Bell and Osen. I will just list them here with some brief remarks:
(1) The book Fantasia Mathematica, which we discussed at the first meeting of mathematical culture, contains a satyrical account by Karel Capek of the death of Archimedes. Karel Capek was the author of the play R.U.R. and of the ingenious novel The War With The Newts.
(2) Most undergraduates learn about the philosophy of Plato from some anthology containing just a few of the dialogues. I showed the group the edition by Edith Hamilton and Huntington Cairns of the complete dialogues of Plato and drew the group's attention to the dialogue "Parmenides", which most students never see. The dialogue includes Zeno as a character. Zeno's biography is given briefly in Bell's book. The dialogue itself is noteworthy for a reason I would like to emphasize here. Students are all taught that Plato and Socrates believed in ideal forms. In this dialogue, however, young Socrates tells Parmenides his ideas about ideal forms and Parmenides tears him apart. This raises the question (and let me take this opportunity to emphasize that it is incorrect to use the expression "begs the question" here), "If Plato believed in ideal forms, why did he write a dialogue in which Socrates presents this view and is torn apart?". It is my impression that this question is known as "The Parmenidean Problem" and conferences have been held to discuss it. I've never had the matter explained to my satisfaction.
(3) One of the benefits of reading biograhies is that they help us to understand, at a personal level, other times and places: they give us a glimpse of daily life as others experienced it. Nevertheless, there is a tendency, even in books such as those of Bell and of Osen, to focus only on those aspects of daily life that directly affected the ability of the person to do mathematics. The communications between savants become nearly the only voices heard. That is by no means enough and one must seek further to experience the fabric of society. I am not a historian and I can't recommend the best way to do that. However, I have found the book of Fernand Braudel, The Structures of Everyday Life (the first volume of his Civilization and Capitalism, 1400-1800), to be a rich source of information and I expect the other two volumes of the set to be equally informative. Another book I'm reading now is Paul Faure's La Vie Quotidienne des Colons Grecs de la Mer Noire A l'Atlantique au siecle de Pythagore, 6e siecle avant J.-C. (Daily life of the Greek colonies from the Black Sea to the Atlantic in the century of Pythagoras, 6th century BC). An account of the life of Pythagoras is given briefly in Bell's book and is noted in Osen's book as well because of the fact that some of the Pythagorean's were women.
(4) Euclid's Elements, available in an inexpensive edition in English from Dover (a copy is in stock at Barnes and Noble), contains, in addition to the geometry itself, a lot of historical information about Greek geometry. One of the reasons Euclid wrote his Elements was that the Pythagorean foundations of the theory of ratios was found to be unsound after the discovery of irrationals and Euclid's work placed it on the new foundations of Eudoxus.
(5) Rene Descartes is one of the mathematicians in Bell's book. Descartes' "La Geometrie" is available in a bilingual edition from Dover for just a few dollars, with the French in facsimile.
(6) Bell's book contains biographies of Newton, Leibniz, Pascal and Fermat, who were all contemporaries of Descartes. Although it is far from a history of the times, I showed the group the book Histoire Amoureuse des Gaules (Amorous History of the Gauls, i.e. the French), by Roger Rabutin, count of Bussy, a courtier in the court of Louis XIV. Bussy wrote this as a satyrical account of the love lives of the other courtiers, with their names disguised through the use of nicknames. With Louis XIV was an absolute monarch, one could not simply publish the book, and Bussy contented himself with showing it to his friends privately. One of his friends, Madame de la Baume, told him that she really had to look it over more carefully to get all the nuances and begged him to lend it to her for just 48 hours. It is somewhat astonishing to think that Bussy, after writing with such acuity about the treachery of his fellow courtiers, actually consented to lend her the manuscript. During the 48 hours that it was in her possession, she had a copy made, which she then circulated. Eventually, a copy made its way over the border to Holland, where someone decided that it would be fun to publish it. Eventually, word of this publication reached the court of Louis XIV and poor Bussy was arrested and thrown in the Bastille. Eventually, he was released for reasons of health and after 28 years, he was allowed back in the court and even given a pension. A book such as this brings the people of the court to life, even though it is fiction, and makes it seem suddenly more exciting to read the letters and novels of the courtiers, to study the little civil war in France known as Le Fronde, and so forth. One of the characters in Bussy's novel is given the nickname Bristoe, but the key at the end reveals that this nickname refers to the Earl of Bristol, also known as Sir Kenelm Digby. Actually, Kenelm Digby was one of the correspondents of Pierre de Fermat. I explained Fermat's Last Theorem, noting that Diophantos of Alexandria influenced him across millennia, and then finished by showing the group a letter from Sir Kenelm Digby to John Wallis. The letter is to be found in the collected papers of Fermat. In it, Digby apologizes to Wallis for not having written in such a long time and offers two excuses. One was that he was so touched by Wallis' last letter that he was for a long time at a loss how to respond to it adequately. But the real reason, he says, is that he was hoping to hear something from Fermat about a theorem which Wallis had communicated to him. Eventually, Digby says he received a letter from Fermat saying that he was very busy with a trial (Fermat was a judge, not a professional mathematician) in which he was going to sentence a priest to be burned at the stake. Digby then remarks that he thinks that this is not a very good reason for Fermat to take so long, since a man of Fermat's abilities is capable of doing several things at once. Such are the tidbits one finds lying around in history.
After I concluded my remarks, one of the students asked to see Bell's book. I drew her attention to a place in Cantor's biography where the font mysteriously changed and explained that this was because some changes had quietly been made between the 1937 edition and the present one, notably in some antisemitic remarks which Bell apparently thought were justified at the time as a kind of literary licence.
Another question from that student about languages gave me an excuse to state my opinion that as long as there is something worth reading which was originally written in another language, there is reason enough to learn that language. I gave the example of James Joyce, who learned Danish to read Ibsen.
To see the advertisement that was used for the third meeting, click on: