Each year, the Niels Bohr International Academy has a series of public talks. Part of Copenhagen’s Folkeuniversitet (“people’s university”), they attract a mix of older people who want to keep up with modern developments and young students looking for inspiration. I gave a talk a few days ago, as part of this year’s program. The last time I participated, back in 2017, I covered a topic that comes up a lot on this blog: “The Quest for Quantum Gravity”. This year, I was asked to cover something more unusual: “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”.

Some of you might notice that title is already taken: it’s a famous lecture by the physicist Wigner, from 1959. Wigner posed an interesting question: why is advanced mathematics so useful in physics? Time and time again, mathematicians develop an idea purely for its own sake, only for physicists to find it absolutely indispensable to describe some part of the physical world. Should we be surprised that this keeps working? Suspicious?

I talked a bit about this: some of the answers people have suggested over the years, and my own opinion. But like most public talks, the premise was mostly a vehicle for cool examples: physicists through history bringing in new math, and surprising mathematical facts like the ones I talked about a few weeks back at Culture Night. Because of that, I was actually a bit unprepared to dive into the philosophical side of the topic (despite it being in principle a very philosophical topic!) When one of the audience members brought up mathematical Platonism, I floundered a bit, not wanting to say something that was too philosophically naive.

Well, if there’s anywhere I can be naive, it’s my own blog. I even have a label for Amateur Philosophy posts. So let’s do one.

Mathematical Platonism is the idea that mathematical truths “exist”: that they’re somewhere “out there” being discovered. On the other side, one might believe that mathematics is not discovered, but invented. For some reason, a lot of people with the latter opinion seem to think this has something to do with describing nature (for example, an essay a few years back by Lee Smolin defines mathematics as “the study of systems of evoked relationships inspired by observations of nature”).

I’m not a mathematical Platonist. I don’t even like to talk about which things do or don’t “exist”. But I also think that describing mathematics in terms of nature is missing the point. Mathematicians aren’t physicists. While there may have been a time when geometers argued over lines in the sand, these days mathematicians’ inspiration isn’t usually the natural world, at least not in the normal sense.

Instead, I think you can’t describe mathematics without describing mathematicians. A mathematical fact is, deep down, something a mathematician can say without other mathematicians shouting them down. It’s an allowed move in what my hazy secondhand memory of Wittgenstein wants to call a “language game”: something that gets its truth from a context of people interpreting and reacting to it, in the same way a move in chess matters only when everyone is playing by its rules.

This makes mathematics sound very subjective, and we’re used to the opposite: the idea that a mathematical fact is as objective as they come. The important thing to remember is that even with this kind of description, mathematics still ends up vastly less subjective than any other field. We care about subjectivity between different people: if a fact is “true” for Brits and “false” for Germans, then it’s a pretty limited fact. Mathematics is special because the “rules of its game” aren’t rules of one group or another. They’re rules that are in some sense our birthright. Any human who can read and write, or even just act and perceive, can act as a Turing Machine, a universal computer. With enough patience and paper, anything that you can prove to one person you can prove to another: you just have to give them the rules and let them follow them. It doesn’t matter how smart you are, or what you care about most: if something is mathematically true for others, it is mathematically true for you.

Some would argue that this is evidence for mathematical Platonism, that if something is a universal truth it should “exist”. Even if it does, though, I don’t think it’s useful to think of it in that way. Once you believe that mathematical truth is “out there”, you want to try to characterize it, to say something about it besides that it’s “out there”. You’ll be tempted to have an opinion on the Axiom of Choice, or the Continuum Hypothesis. And the whole point is that those aren’t sensible things to have opinions on, that having an opinion about them means denying the mathematical proofs that they are, in the “standard” axioms, undecidable. Whatever is “out there”, it has to include everything you can prove with every axiom system, whichever non-standard ones you can cook up, because mathematicians will happily work on any of them. The whole point of mathematics, the thing that makes it as close to objective as anything can be, is that openness: the idea that as long as an argument is good enough, as long as it can convince anyone prepared to wade through the pages, then it *is* mathematics. Nothing, so long as it can convince in the long-run, is excluded.

If we take this definition seriously, there are some awkward consequences. You could imagine a future in which every mind, everyone you might be able to do mathematics with, is crushed under some tyrant, forced to agree to something false. A real philosopher would dig in to this corner case, try to salvage the definition or throw it out. I’m not a real philosopher though. So all I can say is that while I don’t think that tyrant gets to define mathematics, I also don’t think there are good alternatives to my argument. Our only access to mathematics, and to truth in general, is through the people who pursue it. I don’t think we can define one without the other.

vampyriconThis is way above my level but this reports on a paper providing evidence that the cardinality of the continuum is \aleph_2:https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/

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4gravitonsPost authorYeah…there’s a philosophical question of whether this is “evidence that the cardinality is X” vs. “evidence that the assumption that the cardinality is X is nice and useful”. Those may end up the same thing, at least under the perspective I outlined in the post: if every mathematician ends up thinking that the cardinality of the continuum ought to be \aleph_2 then in some sense that’s what it is, much as one could have other axioms for set theory than ZFC but ZFC is the one that gives the thing most mathematicians call set theory.

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