Mathematical ancestors of Penn math faculty, from October 1999. (This was the department's 100th anniversary.) This lists the advisor's advisor's advisor's... until historical data that was (easily?) available at that time gave out. The longest-ago person listed on this page is Otto Mencke (Ph. D. 1665, 12 generations from Penn professor Stephen Shatz); most chains die out in the 19th century.
As of right now, the math genealogy project claims to know that my advisor's 26th-generation advisor is Elissaeus Judaeus (who was a student in the 1380s). Most mentions of Judaeus on the Internet seem to be by other people who have discovered this (Judaeus has 77000 or so mathematical descendents). But this post from the person who added him to the database gives some background -- he was for the most part a philosopher, it seems. He is described as "a mysterious figure who may or may not have been a Jew". His student Gemistus Pletho seems a little better understood; Wikipedia says "He was one of the chief pioneers of the revival of Greek learning in Western Europe." It seems that in that time a lot more data has been collected for the 14th through 17th centuries.
(As for me, hopefully in a few weeks it'll be possible to add me to the mathematical genealogy project. I defend on April 15.)
Showing posts with label history of mathematics. Show all posts
Showing posts with label history of mathematics. Show all posts
25 March 2010
22 January 2009
The square root of 3 and mock theta functions. (No, they're not connected.)
Mark Dominus writes about why it really isn't so mysterious that Archimedes had the approximation √3 = 265/153 (which is correct to four decimal places). Apparently historians of mathematics have been mystified by this. Dominus points out that tabulating n2 and 3n2 for the first few hundred integers would be enough. And it might even be enough to go up to 100 or so, observing where n2 and 3m2 are close to each other (which gives an approximation √3 ~ n/m), guess the pattern (it comes from the continued fraction of √3, but Archimedes didn't need to know that), and extrapolate. He suggests that's because the historians themselves weren't so good at arithmetic. Many of these historians date from the late 19th and early 20th century, after when mathematics generally turned more abstract and before computers existed, so it's plausible. If I were a historian I'd have something serious and insightful to say about this.
A more general question: if you're trying to work out the history of mathematics by examining the original sources, how important is it to be a good mathematician? I saw a lecture by George Andrews last week on Ramanujan's lost notebook; he and Bruce Berndt are working on an edited version of it (first volume
, second volume
, review of first volume in the October 2006 Bulletin of the AMS). Andrews happened to be looking through some papers at the library of Trinity College, Cambridge, when he came across these papers. The manuscript conventionally called "Ramanujan's lost notebook" consists of many pages of formulas and almost no words and is concerned with mock theta function; Andrews claims that he would not have recognized the significance of what he was looking at had he not wrote a PhD thesis on mock theta functions.
A more general question: if you're trying to work out the history of mathematics by examining the original sources, how important is it to be a good mathematician? I saw a lecture by George Andrews last week on Ramanujan's lost notebook; he and Bruce Berndt are working on an edited version of it (first volume
, second volume, review of first volume in the October 2006 Bulletin of the AMS). Andrews happened to be looking through some papers at the library of Trinity College, Cambridge, when he came across these papers. The manuscript conventionally called "Ramanujan's lost notebook" consists of many pages of formulas and almost no words and is concerned with mock theta function; Andrews claims that he would not have recognized the significance of what he was looking at had he not wrote a PhD thesis on mock theta functions.
01 September 2008
Who's Tremellius?
There's a web site called the Mathematics Genealogy Project, which traces the "family tree" of mathematicians. The relationship of "parent" in traditional family trees is replaced with "doctoral advisor".
You'd think, then, that everybody would have a unique 1st-level, 2nd-level, etc. ancestor, but they don't, since some people, especially before the advisor-advisee relationship formally came into existence, had two advisors. Leibniz is an example, as is Dirichlet.
Anyway, the natural thing to do, I think, is to keep following the links upwards; this process usually seemed to terminate at Erhard Weigel, who was a mathematics professor in the mid-1600s, and who taught Leibniz. But Leibniz had two advisers, Weigel and Huygens; if you follow the Huygens path back up into the 1500s you end up, eventually, at Immanuel Tremellius, who appears to have been known principally as a translator of Bibles. He advised Rudolph Snellius, who in turn advised his son Willebrord Snellius of Snell's law fame.
Tremellius has 56,128 descendants, which is the most anybody has, tied with Valentine Naibod (they both were advisors of the elder Snellius and nobody else, and so have the same set of descendants). I'm pretty sure some of those links weren't there before; the people at the MGP seem to have found some new data. But the database contains 125,583 people; it would not be reasonable to say that Tremellius is the ancestor of all, or even almost all, modern mathematicians.
You'd think, then, that everybody would have a unique 1st-level, 2nd-level, etc. ancestor, but they don't, since some people, especially before the advisor-advisee relationship formally came into existence, had two advisors. Leibniz is an example, as is Dirichlet.
Anyway, the natural thing to do, I think, is to keep following the links upwards; this process usually seemed to terminate at Erhard Weigel, who was a mathematics professor in the mid-1600s, and who taught Leibniz. But Leibniz had two advisers, Weigel and Huygens; if you follow the Huygens path back up into the 1500s you end up, eventually, at Immanuel Tremellius, who appears to have been known principally as a translator of Bibles. He advised Rudolph Snellius, who in turn advised his son Willebrord Snellius of Snell's law fame.
Tremellius has 56,128 descendants, which is the most anybody has, tied with Valentine Naibod (they both were advisors of the elder Snellius and nobody else, and so have the same set of descendants). I'm pretty sure some of those links weren't there before; the people at the MGP seem to have found some new data. But the database contains 125,583 people; it would not be reasonable to say that Tremellius is the ancestor of all, or even almost all, modern mathematicians.
15 April 2008
Fermat on FLT
...not Fermat's last theorem, but Fermat's little theorem. When I took an introductory number theory course I was inordinately amused by this coincidence of abbreviations. Fermat's little theorem is more useful; I came across it most recently because we're explaining RSA cryptography to the Ideas in Mathematics class. For those of you who don't know it, the RSA algorithm works as follows. In order to enable people to encrypt messages to be sent to you, in such a way that only you can decrypt them, calculate three numbers n, e, and d. The number n is the product of two large primes p and q; e is some number relatively prime to φ(n) = (p - 1)(q-1); d is the multiplicative inverse of e modulo φ(n). Then publicize the numbers n and e; keep d secret. Calculating d from these in the obvious way requires factoring φ(n).
A plaintext message T is then encrypted as a ciphertext C, where C = Te mod φ(n); a ciphertext message C is then decrypted by taking Cd mod φ(n). The reason that decryption is the inverse of encryption -- which is of course what one wants -- is Euler's theorem, which states that aφ(n) mod n = 1 whenever n and a are relatively prime positive integers. Assuming T and φ(n) are relatively prime, then, applying the encryption map and then the decryption map to some plaintext message T gives Tde mod n; but de is one more than some multiple of φ(n), so that's just T. (This is only true when T and φ(n) are relatively prime.)
Anyway, the point of this post really wasn't to talk about RSA cryptography, but to point to a letter written from Fermat to Frenicle in 1640. One often hears that Fermat very rarely proved things but just made motions saying "I have a proof", but the only such remark one actually hears repeated is the famous statement of Fermat's Last Theorem:
which translates as something like:
(I don't know Latin, so I take no responsibility for the correctness of this translation.) It turns out that there's a less famous remark regarding the Little Theorem:
That is:
(The "progressions" refer to Fermat's phrasing of the theorem, in which he refers to the p-1 term of a geometric progression a, a2, a3, ... and claims that it is one more than a multiple of p.) The rest of the letter includes other such statements without proofs, but illustrated with copious examples. (The link I provided is to the version of David Zhao and Amanda Bergeron, which includes both the French text of the letter and their English translation; the English translation of Fermat's French I gave above is my own.)
A plaintext message T is then encrypted as a ciphertext C, where C = Te mod φ(n); a ciphertext message C is then decrypted by taking Cd mod φ(n). The reason that decryption is the inverse of encryption -- which is of course what one wants -- is Euler's theorem, which states that aφ(n) mod n = 1 whenever n and a are relatively prime positive integers. Assuming T and φ(n) are relatively prime, then, applying the encryption map and then the decryption map to some plaintext message T gives Tde mod n; but de is one more than some multiple of φ(n), so that's just T. (This is only true when T and φ(n) are relatively prime.)
Anyway, the point of this post really wasn't to talk about RSA cryptography, but to point to a letter written from Fermat to Frenicle in 1640. One often hears that Fermat very rarely proved things but just made motions saying "I have a proof", but the only such remark one actually hears repeated is the famous statement of Fermat's Last Theorem:
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.
which translates as something like:
It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.
(I don't know Latin, so I take no responsibility for the correctness of this translation.) It turns out that there's a less famous remark regarding the Little Theorem:
Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n'appréhendois d'être trop long.
That is:
This proposition [Fermat's Little Theorem] is generally true for all progressions and for all prime numbers; I would send you the demonstration if I did not fear that it were too long.
(The "progressions" refer to Fermat's phrasing of the theorem, in which he refers to the p-1 term of a geometric progression a, a2, a3, ... and claims that it is one more than a multiple of p.) The rest of the letter includes other such statements without proofs, but illustrated with copious examples. (The link I provided is to the version of David Zhao and Amanda Bergeron, which includes both the French text of the letter and their English translation; the English translation of Fermat's French I gave above is my own.)
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