Last night, I took part in a debate organised by the students’ Debating Society and Mathematics Society jointly.
The proposition before the house was
This House Believes That Mathematics Is a Human Invention Rather Than a Discovery.
When I was invited to take part, I was offered the choice of side. I think I could have argued for either side, but I also thought it would be more fun to oppose the motion. I judged, correctly as it turned out, that the object was not to win but to provide the audience with an entertaining and thought-provoking evening.
The other three speakers were all, at least to some extent, philosophers. The proposers made arguments which I thought were not very good. One of them was that it is possible for particular bits of mathematics to be discoveries while mathematics as a whole is an invention. I think that, if mathematics is an invention, this would corrupt the whole subject. The other argument was that there were three qualities that inevitably mark an invention, and mathematics shows all these three. Two of these were open-endedness and goal-directedness; I forget the t hird. I disagree with both parts of the proposition. It is easy to point to inventions which lead nowhere; and that part of mathematics which is goal-directed is exactly what people like G. H. Hardy would dismiss as “not real mathematics”.
However, I only had ten minutes, so I didn’t spend any of this on refuting these arguments, beyond saying that if I wanted to learn about the proposition, I would not go to a philosopher; I would look at mathematics itself for internal evidence.
I did mention that we need an understanding of discovery. Certainly mathematics is not discovered the way that America (or an old Viking sword in the basement) can be. The model I proposed was based on the fact that people living in certain valleys in the Himalaya could see a big mountain, and were convinced that the mountains seen from different valleys were different, until it was discoveredd that they were all the same (what is now called Mount Everest). My argument was that there are some things which run like connecting threads throughout mathematics, and are discovered in different areas by people who don’t at first know that people in other areas have also discovered them. The congruences are too impressive for a suggestion of invention to hold up, and the ubiquity of some of these ideas shows that they are not small local parts of the subject.
Of course, explaining two mathematical examples to an audience with a majority of non-mathematicians (indeed, many who were self-confessedly “bad at maths at school”) in ten minutes was a daunting task; but I resolved to try.
My first example was Monstrous Moonshine. I began by saying that symmetry is a real and important thing, both in everyday life (when we look in a mirror) and in fundamental physics (for classifying elementary particles). Mathematicians study symmetry, and call this topic “group theory”. In the nineteenth century we discovered that finite groups are built from atoms called “finite simple groups”; in the twentieth, we discovered all the finite simple groups, although we didn’t know for sure that we had them all until this century. They mostly fall into infinite families, but there are twenty-six which do not fit into a family. The largest of these was originally called the “Friendly Giant”, both because of its remarkable properties but also to commemorate the discoverers, Fischer and Griess. But now John Conway’s term “Monster” has prevailed.
Early on, when the Monster was still surrounded by swirling fog, some features became clear; its order (more than 1050, large but much smaller than the number of elementary particles in the universe), and the order of the smallest matrices that can be used to describe it, namely 196883, For non-mathematicians I explained that a spreadsheet with 196883 columns and the same number of rows could contain one element of the Monster.
When John McKay learned about this, he was probably one of very few people in the world who also knew his way around classical complex analysis. In particular, he knew about the modular function, which I explained as identifying the integers within the plane of complex numbers. Since the integers are shift-invariant, the modular function can be expressed as a Fourier series. The first two coefficients are 1 and 0, not very interesting. But the third is 196884 = 196883+1. When Conway learned of this, he described the coincidence as “mooonshine”. If mathematics were invented, Conway would have been right. But it was not “moonshine”; further discoveries showed a rich correspondence, which has now been formally proved but not really understood. This new branch of mathematics is called “Monstrous Moonshine”, which (like the “Big Bang”) began as a derogatory term but is now used with pride by its exponents.
I introduced my second example by reference to two classic views of the Universe, as discrete or continuous; this goes back to the pre-Socratic Greeks, and was discribed by Alan Watts as “prickles and goo”, and by the mathematical biologist John Maynard Smith as “quality and quantity”. (Biologists care because, although our bodies seem to be made of goo, the genetic code which contains the instructions for building them is definitely prickles, several very long words over a four-letter alphabet.) From the words he used, it is clear which side Maynard Smith was on, and he actually said “Now we have methods of turning quantity into quality”.
One of these methods, popular fifty years ago, had the dramatic name “Catastrophe Theory”, perhaps more soberly described as “singularlity theory”. Its proponents identified seven (a mystic number) “elementary catastrophes”. Then Vladimir Arnold came along and pointed out that there was much more to it; these singularities were described by some discrete diagrams called the ADE diagrams. I will not go on at length about these since there is plenty elsewhere on this blog, but their ubiquity (occurring from general relativity to graph theory, and from Lie algebras to singularity theory) supports my argument. In particular, John McKay (again!) found a bridge connecting the regular polyhedra in 3-dimensional space with topics on the continuous side such as the octonions.
I used the recent book “ADE: Patterns in Mathematics” as a prop, even though the diagrams on the cover are too small for most of the audience to see, and even went on to explain the remarkable discovery I made of the connection between the ADE diagrams and spectral theory of finite graphs, which eventually led to my being invited onto the authorship group of the book.
Several members of the audience pointed out that there is a difference betweem mathematics and the language we use to describe it; the first could be a discovery but the second is surely an invention. I think my first example bore this out. But in my closing comments I was able to mention Conway’s plea for a “mathematicians’ liberation movement”, where instead of overly complicated constructions of mathematical objects such as the real numbers, we could simply say what properties we want a structure to have and guarantee that it exists. This would nicely get around the problem, but I admitted that it is far in the future at present.
Anyway, the debate ended with a clear majority of the House opposing the motion before it, out of what is said to be the largest audience for one of these regular student debates for quite some time (probably at least since the pandemic). People I spoke to afterwards felt that I had done a good job.
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