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?
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 advantage is often a bunch of philosphers being loudly wrong about some related issue. But this case, I don't see the signs of an easy advantage. Modern philosophy of numbers doesn't seem to have a bunch of sharp divides or false confidence. Instead, most everyone seems pretty aware that they're confused, despite some fairly interesting ideas being available.
But, okay, I do have some ideas.
See, if you ask philosophers about something that might exist, their first instinct is to try to find a necessary-and-sufficient definition of this thing, more or less on its own terms. Over here, we're much more likely to think of how things are represented in people's models of the world, and ask what chain of events led people to have that representation, which I think is some important philosophical technology.
This wouldn't be a proper post without a pile of links. So here are some options we might want to keep in mind: Taboo your words. Focus on origin or function, like in the example of "truth." Imagine what cognitive algorithm you're using. Keep your eye on the reductionist ball.
Since this post is labeled "part 1," you might expect that I'm going to end this without telling you exactly what I think about numbers. You'd be right!
But I do want to prompt you with some questions I think are more key than "what is math, really?", and corresponding things I think might be hints.
- Why do we say that numbers "exist?"
Why do we need a property called "existence" in the first place, even just for trees and rocks? Those Eliezer-posts about truth may hint at one point of view.
- Why would we want to say that certain abstract sentences are "true?"
Do statements about math have the same properties Eliezer outlined as making "true" a useful word? Why would we want talk about labels on mathematical sentences if math is just a bunch of tautologies?
- Does it make sense to evaluate "There exists a city larger than Paris" and "There exists a number greater than 17" the same way?
What cognitive algorithms could we be using? What are their disadvantages?
- Does this line of reasoning actually help us implement LDT?
I got nothing.