This post is the first in a series of things that I think would be fun to discuss on LW. Part two is here.

It seems like there are (at least) two kinds of things we make statements about: physical things, like apples or cities, and logical things, like numbers or logical relations. And it's pretty interesting to question how accurate this seeming is. Are numbers really a "kind of thing," and what do we mean by that anyways? Can we unify these multiple kinds of things, or kinds of statements, into one kind, or not?

For a light review of standard answers, see this nice video. For more depth, you might see the SEP on abstract objects or philosophy of mathematics.

Compare the statements "There exists a city larger than Paris" versus "There exists a number greater than 17." It seems like we use much the same thought patterns to evaluate both these statements, and both seem to be true in the same ordinary sense. Yet the statement about cities seems true because of a correspondence to the external world, but there is no "17" object in a parsimonious predictive model of the world.

To this you might say, "What's the big deal? Even if I don't think numbers are physical objects, it's perfectly reasonable to make this tight analogy between cities and numbers in our reasoning. How is making a big issue out of this going to help us do anything practical?"

Well, in logical decision theory, a recent formulation of some ideas from TDT/UDT, the agent wants to make a causal model of the world that includes (in the model) "causal" effects of a fixed mathematical statement (speficially, the output of the agent's own algorithm). First of all, this is pretty novel and we don't really know how to formalize learning such a model. Second, it's pretty philosophically weird - how is a piece of math supposed to have something like a causal effect on trees and rocks? If we want to solve the practical problem, it might help to be less confused about numbers.

Plus, you know, it's interesting! Why do we think there's such a thing as "numbers," how come the same reasoning works for both numbers and cities, and what are the limits to this analogy, if any?

When one wants to outdo an entire branch of philosophy, it's nice to have some sort of advantage. And the sign of such an advant...

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