Here is a video of my talk from the recent 70th birthday conference of Peter Sarnak. During a talk one always forgets to say certain things, so I realized that my blog could be a good place to give some extra context on points I missed. There are three things off the top that I can add before rewatching the talk. The first is that I made a typo in one of my collaborator’s name (oops!). The second is that I didn’t mention the work of Bost-Charles, whose influence on our work is clear. Indeed the \(m = 0\) version of the holonomy theorem (version III) in this talk is a theorem in their monograph. The third is that my presentation of known irrationality results for *explicit* zeta values makes sense in the context of framing of my talk, but it’s good to note that the irrationality results of Rivoal, Ball-Rivoal, and Zudilin (for example, at least (edit: one) of \(\zeta(5), \zeta(7), \zeta(9), \zeta(11)\) is irrational) in a closely related direction are amazing theorems. There’s probably more to say, and I might add some extra comments if I watch the video again).
Some incidental remarks concerning history I thought about when preparing my talk: I know from popular accounts that Apéry’s result came as a complete surprise. Similarly, the result of Gelfond-Schneider was a complete shock as well. (Hilbert was reputed to say that he didn’t think this problem would be solved within his lifetime.) Now these two theorems are “recent enough” so that the memory of their resolution is still within the collective consciousness of mathematicians. In the first case, I still know a bunch of people (Henri Cohen and Frits Beukers) who were actually at Apéry’s infamous lecture. But what about (edit Lindemann’s) proof that \(\pi\) is transcendental? I have no sense as to what was the reaction at the time, in part due to my lack of historical knowledge but also to the lack (as far as I can see) of easily available informal discussions about contemporary mathematics from the 19th century (I assume that personal letters would be the best source). The best (?) I could find was the following (quoted from here):
In fact his [Lindemann’s] proof is based on the proof that \(e\) is transcendental together with the fact that \(e^{i \pi} = -1\). Many historians of science regret that Hermite, despite doing most of the hard work, failed to make the final step to prove the result concerning which would have brought him fame outside the world of mathematics. This fame was instead heaped on Lindemann but many feel that he was a mathematician clearly inferior to Hermite who, by good luck, stumbled on a famous result.
First, this seems pretty brutal towards Lindemann (to be fair, the continuation of the text does give some more grudging praise of Lindemann). Second, which historians are being referred to here? This seems far too judgemental for the historians I have ever spoken to in real life. If this text is at all accurate, it seems to suggest that Lindemann’s result was lauded but perhaps not considered surprising to his contemporaries? I feel that this is recent enough that one should be able to get a fuller idea of what was going on at the time.
Going back in time further, I also wonder what Lambert’s contemporaries thought of his proof (in the 1760s) that \(\pi\) was irrational. When I was giving a public talk on \(\pi\) in Sydney I looked up Lambert’s paper. The introduction is quite amusing, with the following remark that suggests a modern way of thinking not much different to how I think about things today:
Démontrer que le diametre du cercle n’est point à sa circonférence comme un nombre enteir à nombre entrier, c’est là une chose, dont les géometres ne seront gueres sorpris. On connoit les nombres de Ludolph, les rapports trouvés par Archimede, par Metius, etc. de même qu’un grand nombre de suites infinies, qui toures se rapportent à la quadrature du cercle. Et si la somme de ces suites est unq quantité rationelle, on doit assez naturellement conclure, qu’elle sera ou un nombre entier, ou one fraction très simple. Car, s’il y falloit une fraction fort composée, quoi raison y auroit-il, pourquoi plutôt relle que telle autre quelconque?
(Or in translation, errors some combination of mine and google translate):
We prove that the ratio of the diameter of the circle to its circumference is not rational; something that geometers will hardly be surprised by. We know the number \(pi\) of Ludolph, and expressions for this number found by Archimedes, by Metius, etc. in terms of a large number of infinite series of rational numbers, which all relate to the squaring of the circle. If the sum of these sequences was a rational quantity, we must quite naturally conclude that it will be either a whole number, or a very simple fraction. For, if a very complicated fraction were necessary, what reason would there be to be equal to such a number rather than any other real (irrational) number?
I guess Occam was from the 14th century!
I think there’s a typo and you meant to write” But what about Lindemann’s proof that π is transcendental?”
Michel Waldschimdt writes that all the main ideas for showing the irrationality of Pi were present in Hermite’s 1873 memoir, but doesn’t explicitely quote any contemporary reaction de Hermite’s work. https://journals.openedition.org/bibnum/893?lang=fr#bodyftn5
The text of Emile Picard on the work of Hermite doesn’t either. http://www.numdam.org/article/ASENS_1901_3_18__9_0.pdf
Incidentally, in a footnote of a text on the correspondance of Lebesgue, there is mention of the tragic loss all of Hermite’s correspondance in a fire of a storage unit… So all handwritten reactions are probably lost.
It seems that Hermite’s own opinion is captured by a letter to Borchardt from 1873 (from right-after Hermite’s memoir on the exponential function, where he proved the transcendence of e and started the modern theory of functional rational approximation). The English translation could be something like this:
“I will not venture in search of a demonstration of the transcendence of the number π. Let others try to pull it off. No one would be happier than me in their success. But, believe me, my dear friend, it will not fail to cost them some effort.”
In a nice recent popular (and historically well-researched) book “Tales of Impossibility” (David Richeson, Princeton University Press, 2019), there is the following discussion where the “historian” cited is none other than the famed mathematician Hans Freudenthal, but which completely contrasts with how Hermite viewed these things:
”
In 1873, the same year that Hermite proved the transcendence of e, a German mathematics student named Ferdinand von Lindemann earned his PhD from Felix Klein in Erlangen. Shortly afterwards, he headed abroad to visit mathematicians in England and France. It was during his stop in Paris that Lindemann met Hermite and was able to discuss with him his methods.
Nine years later, in 1882, Lindemann proved that π is transcendental. Some mathematicians have expressed disappointment that Hermite, who had devised the key ideas for what would eventually be Lindemann’s proof, did not prove it. Although Lindemann had a good career, producing 60 PhD students, he was not seen as Hermite’s equal. But Hermite was generous, inspiring, and shared his knowledge far and wide through his correspondences. Freudenthal pointed out this was often to Hermite’s detriment, because it allowed others to achieve the important results — such as Lindemann’s proof that it was impossible to square the circle — that were “rightfully” Hermite’s: [and then follows Freudenthal’s strange and scathing quote in the “Dictionary of Scientific Biography”]
Freudenthal: In a sense, [the proof of the transcendence of e] is paradigmatic of all of Hermite’s discoveries. By a slight adaptation of Hermite’s proof, Felix Lindemann, in 1882, obtained the much more exciting transcendence of π. Thus, Lindemann, a mediocre mathematician, became even more famous than Hermite for a discovery for which Hermite had laid all the groundwork and that he had come within a gnat’s eye of making.
”
With all the hindsight, it is all too easy to make strong statements!
Typo: “at least of of” should be “at least one of”.
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