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Why do you have to say the math is "outside" the brain? I do understand that the model of the natural numbers is particularly useful in making elegant predictions about our physical universe, but why does that say something about the numbers or the math? The integers are an example of a formal system, but we can construct other formal systems where the formula 2+3=6 holds (I don't know of any interesting such formal systems, though). I can easily see that we have these formal systems, and we also have inductive arguments that they describe the world well. I get the sense Eliezer that you posit a third thing "exists". But, wouldn't this be a case of the "mind-projection fallacy"? Why do we need a third thing exist when the formal system and the inductive argument account for everything (or, perhaps they don't, and I'm missing the point...).