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Tuesday, May 2nd, 2017 11:59 am
I've been thinking about numbers a lot recently. As in, what are they, exactly?

It's one of those simple sounding questions that doesn't actually have a very good answer. You have the Platonists, who want numbers to be abstract, metaphysical objects. Pure and perfect, beyond time and space. But if that's the case, how do we interact with them? Where did we even get the idea of them in the first place? Numbers can't be so pure and perfect if they can get caught in simian brains.

On the other side you have the nominalists, who say that numbers are just an abstraction of groups of actual, real-world things. Which maybe sounds a bit more level-headed, but it really falls apart when you start talking about math more complicated than basic arithmetic. It certainly doesn't explain why math has been so phenomenally useful in describing the universe. If it's just a system of symbols we invented, why should something like imaginary numbers end up being useful in electrical engineering? Why aren't music or poetry or card games useful in the same way? It's really hard to say that the universe and mathematics aren't linked in a very fundamental sense.

The whole argument reminds me of something interesting I noticed in Plato's dialogs last year. In Meno, we find Socrates making a point about innate knowledge. He shows how a slave boy is able to understand geometric proofs, despite having no training. He uses this as proof that the soul is immortal, and that we can retain things learned in previous lives. When I was first reading this, I amused myself by flippantly translating it into modern terms: "We do make use of knowledge developed over previous lives. The pattern recognition capabilities of our brains are the product of millions of years of evolution." Now I'm wondering if that doesn't hold the key to understanding what math really is.

Postulate a "deep logic" upon which the universe is founded. This is the set of very basic principles that all of physics is derived from. By evolving in this universe, our brains have been ruthlessly trained to recognize some of these principles. I mean incredibly basic stuff like "x=x" and "x+1 > x". A squirrel needs to know that adding a nut to its cache makes the cache bigger, after all.

Fast forward a couple million years, and homo sapiens sapiens comes on the scene with the ability to create symbolic systems. They get by fine using the intuitive logic they inherited for a long time, but eventually they start to build on it. They like having things, and they really like having more things than anyone else, so they get the counting numbers. They start buying and selling, so they get integers and basic arithmetic. At some point someone asks what half of five is, and they decide that some numbers are the ratio of integers. They want to mark out territory, so they invent geometry. This leads to wondering what the square root of two is, and the proof that not all numbers can be a ratio of integers. It get incredibly abstract very quickly, no longer even slightly connected to physical objects.

And now here we are, using the math that grew from those roots to describe what happens at 0.9c passing by the event horizon of a black hole! Because it's still all based on the same "deep logic" as the universe, it has remained useful. Whenever a mathematician rejects an idea for being "inelegant" or a proof as not making sense, that's the ghost of our evolutionary apprenticeship at work.

So maybe Platonism and nominalism aren't as diametrically opposed as they're commonly thought to be.

Open question: Is there any corner of the universe based on a different deep logic? What would that look like to us, and could we begin to understand it?
Wednesday, May 3rd, 2017 02:12 am (UTC)
It's possible that quantum mechanics is different deep logic.

With that said, I wonder if math isn't an emergent property of intelligence, rather than being something we've discovered. I agree it's deeply tied into reality. But that might also be a reflection of how we model, and we all have the same modeling hardware.