I’ve submitted an entry for the Foundational Questions Institute Essay Contest: “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”. You can check out my entry here.
The backstory to this is that I was participating in a weekly mathematical physics seminar back at Florida State (although I use the word “seminar” pretty loosely – it was regularly attended by only myself and one other individual!), and in the process of working on presenting on some NCG topic, I came across “The Bost-Connes System”. This is a particular -algbera, on which you can define some dynamics. What makes it special is that if you calculate the partition function for this dynamical system, you get the Riemann Zeta function! Since the partition function can be used to generate predictions for a statistical mechanical system, I wondered how possible it was to construct a real physical system with the same symmetry as the Bost-Connes system. Then you would have experimental access to (at least some features of) the Riemann Zeta. There is a great deal of mathematical important to Zeta, including a $1 million dollar prize for finding the zeros!
I wasn’t exactly thinking about which Benz to buy with my prize money yet, but I thought it was an interesting idea – experimental verification of a mathematical theorem. I wasn’t aware that anything like this had been done before. Normally the “flow of ideas” works the other way – constructions in mathematics find usefulness in physics, or theoretical models become interesting mathematical systems. It stuck in my head for a while, I did a few calculations to determine what the zeros of a partition function might look like, but nothing really came of it.
When this essay contest came around, I thought it might be an opportunity to share this idea. I figured that if it was going to be taken seriously, you needed to raise experimental verification to the level of mathematics – after all, if I prove a conjecture is true outside of the field in which the conjecture is stated, we should not take the proof very seriously! I needed to make experimental physics a subfield of mathematics. It turns out that this is pretty easy, and so that’s what the essay is about. If you take your physical model as a set of formal axioms, and add in an additional axiom which can be used to experimentally verify a theorem (I call this “an axiom of measurement”), you can formulate physics as a complete formal system. As a bonus, the axiom can be used to add a little more structure to the Platonist viewpoint on universal versus physical forms.
Now, the FQXi Essay Contests are Contests; the community and the public can vote on the quality of the essays, and the quality of the essays vary widely, since nearly anyone is allowed to submit an entry. I actually think my essay represents a pretty mainstream viewpoint about physics – that we are not really studying “nature” or “the universe” when we do physics, we are really studying a “model for the universe”, which is confirmed by our everyday observations as well as carefully constructed experiments. Since it’s not a new, dramatic viewpoint on any particular aspect of the relationship between the two fields, I don’t expect to be winning any awards. But, I had an opinion with an interesting idea behind it, and an essay seemed like the ideal place to explore it.
Anyway, if you’re so inclined go over and check out my entry as well as all the others.