Mathematics and Logic

by Mark Kac & Stanislaw Ulam

This interesting little book, published in 1968, is the result of a collaboration between two of the great figures of 20th century mathematics. Although known mostly for his work on the Manhattan Project and later associated applied mathematics efforts (I first learnt of his work after being told to read about the Fermi-Pasta-Ulam problem by my PhD supervisor), Ulam began his career as a pure mathematician, working on problems in general topology. (His autobiography, Adventures of a Mathematician, is well worth a read.) Kac, best known for asking “Can one hear the shape of a drum?”, made major contributions to probability theory. Kac and Ulam got to know each other in Poland, then both made the move to the United States during the Second World War.

Mathematics and Logic was originally commissioned by the Encyclopedia Britannica as a sort of long-form article to appear as an appendix to the encyclopedia. It was only published separately a little later. It’s particularly interesting because here we have two truly great mathematicians trying to answer the question “What is mathematics?”, aiming at the broader scientific community, trying to give an impression of what it is that mathematicians do. Mostly, to be honest, they don’t do a great job of answering their question. The entertaining aspect of the book is that they seem to have realised right from the start that they’re not going to manage to answer the question they pose as the first sentence of the introduction, so they instead embark on a “show-and-tell” voyage through the world of mathematics. This ends up working well, and to be honest, it makes sense. If you want to learn about a new country, where do you turn first? The CIA World Factbook, or something by Paul Theroux? If you want the raw statistics, the former is handy, but if you want a feel for the place, to take in the sights, you want something a bit more picaresque.

What you end up with here is 105 pages of examples of mathematics, then around 60 pages that tries to synthesise these examples into a coherent picture of what mathematics is, to try to tease out the essence of the subject. It’s a hard task! The examples that Ulam and Kac use range from the very traditional (the infinity of the primes, the irrationality of $\sqrt{2}$) through the more “modern” (transcendental numbers & Cantor’s argument, for instance) to topics that would be identified as resolutely modern: groups and transformations; homology groups; transformations, flows and ergodicity. In all, they give a good idea of the vast range of material encompassed by the word “mathematics” at the time. This part of the book is satisfying, although one often feels the limitations of a short form like this–tell us the details, guys! We want the details!

The second part of the book, I find a little less satisfying. We step from the picaresque to the analytic, to try to pin down just what this subject is. There are some general trends that are easy to pick out, but there’s no clear overarching principle (probably because there just isn’t one). The trends:

First, a growth of rigour, rendering the subject more formal and inward-looking.

Then, axiomatisation: a focus on structures, rules, consequences. For example, and as a possible indicator of one path by which new mathematics can arise, consider the field axioms, originally for real numbers, and think about how one might apply these axioms to continuous functions with convolution as the composition rule. Same axioms, different structures, different consequences, and hence new things to think about.

Next, algebraisation, in particular of topology, an approach with enormous ramifications and many applications.

Then, controversially, the liberation from intuition. The very striking example that Kac and Ulam use is the fact that continuous functions differentiable at at least one point form a set of first category in set of continuous functions, i.e. they are infinitesimally rare! In some sense, “most” continuous functions are like the Weierstrass function, differentiable nowhere.

And then we get to the end results of axiomatisation, Gödel’s incompleteness theorem, Turing’s work on computability and an apparent crisis in the fundamentals of the subject. (Of course, the “crisis” wasn’t too critical: Gödel’s work didn’t much difference for most working mathematicians.)

There’s some material about the relationships between computers and mathematics (still only nascent in the 1960s) and relations with other disciplines.

And then we end. Does the book answer the original question? Not as such. It shows the answer, but this might be the only thing that it’s possible to do. Successful in its aim or not, it’s an interesting book, and Kac and Ulam are charming guides to the territory.