Category: Language

If you had to write a circular novel, that can be started at any point of the narrative and that would circle back to itself, how long would it be? Probably, most mathematicians would say that either 63, 628, or 6283, pages would be the most appropriate.

And so it’s probably not surprising that the one circular novel I know, Joyce’s “Finnegans wake“, is indeed (in all editions I’ve seen) 628 pages long. Amusingly, this is not mentioned in the introduction to the one I have (which is, also, about the only thing I’ve read of the book, with the exception of the first few and last paragraphs, and isolated snatches here and there).

The reason for this post is really that today is Bloomsday, the day when, fictionally, the action of the earlier “Ulysses” evolves, and that Zürich is well-connected with Joyce. Not only is he buried there, very close to the zoo, so I can use one appropriate quotation

As the lion in our teargarten remembers the nenuphars of his Nile…

from “Finnegans wake” (one of those few snatches I have actually read), but he also wrote parts of “Ulysses” in a house on Universitätstrasse, quite close to the main building of ETH Zürich where I work, as memorialized by the plaque below:

(I’ve heard that there are a few other such places in Zürich, the reason being that, unable to pay the rent, Joyce had to move frequently…)

For years, I have been convinced that the proper French translation of “equidistribution” was not the faux ami (false friend) “équidistribution”, but rather the word “équirépartition”. The latter is for instance used by Serre (and Bourbaki).

But then  I realized recently that Deligne uses “équidistribution” in his great paper containing his second proof of the Riemann Hypothesis over finite fields, which contains in particular his famous equidistribution theorem (see Section 3.5, entitled “Application: théorèmes d’équidistribution”).

Since, in fact, neither word appears in the French dictionaries I have available (unsurprisingly: “equidistribution” is not in the OED), and since moreover “distribution” and “répartition” do appear and are identified as synonyms, it seems now that in fact both words should be acceptable…

Every mathematician has heard of the Peano axioms of arithmetic. Here is a lesser known contribution of Giuseppe Peano: the “Peano paragraphing method”. This is a numbering system for sections/subsections/etc in books where the different items are identified by a decimal number (e.g., 9.132), where the integral part is the chapter number, and the decimal part is arranged in increasing order within each chapter. So for instance 9.301 is a subsubsection lying between 9.3 and 9.31.

I had noticed this system in Titchmarsh’s book “The theory of functions”, from 1932, without understanding it (it is not explained, nor attributed to Peano). Then I saw it again just recently as I was looking up a reference in Whittaker and Watson’s “A course of modern analysis” from 1927, where the explanation and attribution are given in a remark at the beginning. This greatly clarified my previous perplexity in navigating the book of Titchmarsh, which I had found extremely confusing; for instance in Chapter 9, we have

9.1, 9.11 up to 9.15, 9.2, 9.3, 9.31, 9.32, 9.4, 9.41 to 9.45, 9.5, 9.51 to 9.55, 9.6, 9.61, 9.62, 9.621 to 9.623, 9.7…

Looking into other classical books, I can see this system in Watson’s treatise on Bessel functions, but it is not used in either Hardy and Wright’s “Introduction to the theory of numbers”, nor in Titchmarsh’s “The theory of the Riemann zeta function”. It is also absent from Zygmund’s “Trigonometric series” (which, on the other hand, uses a continuous numbering scheme X.Y (Chapter.Item) both for equations, theorems, etc), and from Hardy and Rogosinsky’s “Fourier series”.

Note finally that it seems rather euphemistic to say that this is “lesser known”: neither Google nor Wikipedia seem to be able to give a reference or explanation!

A quip of S. Lang states that “analysis is number theory at the place infinity”.  (Which implies, correctly, that analytic number theory is some particularly exalted form of number theory).

The equally quipful E. Witten goes rather further in reducing mathematics to its essentials: during the conference organized by the Mathematics Department of Princeton University in honor of the 250th anniversary of Princeton University, he said something like: “Most of 20th century mathematics is the study of the harmonic oscillator”. (This can be seen, in a slightly different and weakened form, on page 120 of the Google Book preview linked above; my memory is that he did state, during his lecture, something closer to what I wrote; but that was a while ago, so I may be over-reacting in hindsight…)

P.S. For the obligatory etymological epilogue: the word “one-upmanship” is quite recent (1952), but “quip” goes back to the early 16th century. I didn’t know about the charming derivative “quipful” before looking in the OED.