---
title: Remark about Finitism
date: 2012-01-15
techne: :done
episteme: :broken
slug: 2012/01/15/si-remark-about-finitism/
---

Finitism is the position that there aren't any infinite things. No infinite energy, no infinite space, no infinite numbers. Finitists get a lot of bad rap, but it's the only sensible position in my mind. In fact, I'm an ultrafinitist - I believe that not just that uncountable numbers don't exist, but that there really is such a thing as a largest integer.

The common criticism of finitism is, look, whatever number you pick, I can construct a larger one by adding 1. How can there be a largest integer?

The problem is that you can't actually *name* these large numbers. You can't specify them. You can only point to an algorithm that would eventually reach infinitely many numbers - if God executed it. You need a miracle 'cause to reach those numbers, simply counting up by 1 would take ages and in fact, you would run out of usable resources in the universe. Your algorithm would eventually stop, even if you improved it. As long as it's computable, you *will* reach an end. (You probably need Quantum Magic to get hypercomputation. Might as well start praying.)

Thanks to [Kolmogorov Complexity][], I can finally point out the underlying intuition that lead me to initially dismiss finitism. My mistake was that I thought *all numbers are equal*. 2^10, 2^10+1, 2^10+2 are all equally real numbers. *Some* description exists for any integer, so why should I say that some of them are real and others aren't? That seems weirdly arbitrary and outright silly.

But the thing is - numbers *aren't* equal. Some numbers can be *compressed*, but some *can't*. Each number has an inherent algorithmic complexity and that complexity is *not* distributed evenly. π looks really chaotic, but it's actually very simple. And just like that, some tremendously huge numbers like 3^^^^^3 compress to very short instructions, but other don't. I looked at the number line and thought that numbers were spread out nice and smoothly, just continuing on forever. But when you see algorithmic complexity, you notice the gaps. There are random numbers and you really can't reach them. You *are* computation, running on finite resources, and some numbers simply can't be computed.

There is a largest integer.
