Quote unquote

In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist when the last of their radiant host shall have fallen from heaven.

E. Everett, Orations and speeches on various occasions, Volume 3 (Boston 1970), p. 514.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.

Bertrand Russell, Mysticism and Logic (1918).

Two very contrasting views of our subject! Everett was a mid-nineteenth century American orator who was much in demand in an age that valued public speaking more highly than ours does. He spoke at Gettysburg, though his address has been completely overshadowed by Abraham Lincoln’s much briefer and more memorable speech. (In fact, I came across this quote as an undergraduate, I don’t now remember where, and was under the impression that he had actually said it at Gettysburg; it was Clark Kimberling who put me right. I have not managed to get hold of the volume in which this speech is published, and I don’t know exactly where he gave it. If anyone knows, please let me know!)

Bertrand Russell was, at that time of his life, a professional logician, and worked very hard on removing mysticism from mathematics. His statement sounds mysterious, even mystical, but it is not. He is saying that our symbols have no meaning in themselves, and are manipulated according to the rules of logic with no reference to truth in the external world; but then, of course, they will give us information about the external world, of the type “If our assumptions hold in a given situation, then so do our conclusions”. The wide applicability of mathematics stems in large part from this refusal to assign meaning to its formulae.

I have been collecting mathematical quotes since I was a student; the two above are among my first examples. I became a bit more systematic about it when I wrote a textbook on combinatorics in the early 1990s. Combinatorics is a subject which has traditionally been held in low regard by “real” mathematicians, typefied by the quote attributed to Henry Whitehead, “Combinatorics is the slums of topology”. I wanted to illustrate this strange phenomenon and give people a chance to make up their own minds. The Preface to the book begins with two quotes:

Ive got to work the E qwations and the low cations
Ive got to comb the nations of it.

Russell Hoban, Riddley Walker, Jonathan Cape, London, 1980.

We have not begun to understand the relationship between combinatorics and conceptual mathematics.

Jean Dieudonné, A Panorama of Pure Mathematics: As seen by
N. Bourbaki, Academic Press, New York, 1982.

The first of these is from what I regard as one of the best novels of the last century, set in a post-apocalyptic world in which the English language is much changed and a few surviving ideas from physics and theology hang on as myth. In the second, Dieudonné is presumably speaking with the voice of Bourbaki; you can here the puzzlement and impatience in his voice as he says it.

Later on in the book I quote from Stanslaw Lem, English traditional folksong, von Neumann and Morgenstern, an essay on T. S. Eliot, Lao Tse, Ecclesiastes, the Guinness Book of Records, a little rant by Thomas Kirkman, William Wharton, Motzkin’s definition of Ramsey Theory (“Complete disorder is impossible”), E. M. Forster (“Only connect!”), Russell Hoban a couple more times, Richard Brautigan (the quote that forms the tagline for this blog), Eliot again, The Mabinogion, N. Ya. Vilenkin, Kurt Singer, and Ursula K. LeGuin. This is not to prove that I am widely read – I am a compulsive reader, and note down anything interesting I find – but simply to try to add something to the book for its readers. Maybe some people will go out and read Russell Hoban’s classic novels from the 1970s and 1980s; they are well worth the trouble!

I have made considerable use of my collection in writing things for this blog. The lazy way perhaps, finding someone who said what I want to say better than I can say it; but I look at it differently. If I tell you how I work as a mathematician, you have no incentive to think that this is more than just my idiosyncracy. But if I can quote to you from the writings of Gauss, Poincaré, Hadamard, and other greats, it might be more convincing to you that this is how we do it.

I find other uses too. In 2005, I gave a short course at Groups St Andrews. One of the other lecturers was Slava Grigorchuk. After a while, a competition developed between us to illustrate our lectures with more and more outlandish quotes. There wasn’t a winner, but the audience was entertained!

Quotes offer an interesting, though not very accurate, view of history. In Dorothy L. Sayers’ translation of Dante’s Inferno (part I of the Divine Comedy), two lines caught my eye:

For [Luck] your science finds no measuring-rods; . . .
Her permutations never know truce nor pause

Not too seriously, I interpret the first line as saying that there cannot be a theory of probability; and the second as saying that a random permutation has no fixed points. Neither of these assertions is correct, but Dante was prescient in raising the questions! (Of course, he probably meant something much different).More seriously, Karl Petersen, in his book Ergodic Theory, quotes two sentences by the fourteenth-century scientist Nicole Oresme, one of my heroes:

It is probable that two proposed unknown rations are incommensurable because if many unknown rations are proposed it is most probable that any [one] would be incommensurable to any [other].

Nicole Oresme, De proportionibus proportionum (1351)

No sector of a circle is so small that two such [bodies bodies moving with uniform but incommensurable velocities] could not conjunct in it at some future time, and could not have conjuncted in it sometime [in the past].

Nicole Oresme, Tractatus de commensurabilitate vel incommensurabilitate motuum cell

As Petersen points out, this is Oresme’s argument against astrology. It is based on his clear understanding that (in modern language) rational numbers form a null set, while multiples of an irrational number mod 1 are dense in the unit interval.

Anyway, the whole collection is available online here. Browse it, use it if you wish, and most important, send me more interesting quotes to add!.