Just for fun, I’ll take a stab at your final questions (apart from the last one, where I too have nothing).
Why do we say that numbers “exist”? I think there are two separate questions here. (1) Why do we treat numbers as real? (2) Why do we bother with terminology like “exist”, (2a) at all and (2b) for numbers in particular? I think at least part of our answer to (1) is that our best explanations for our observations have numbers in them, and this seems to be robust in the sense that it applies across a wide range of explanations of a wide range of things at a wide range of sophistication levels. Dunno whether it’s the whole reason. For (2a) … well, actually mostly we don’t talk about things “existing” except in marginal cases where there’s disagreement or doubt (only when doing philosophy or drugs does anyone ask whether tables exist). Our language has nouns, and our thinking has things, because the most convenient approximations to how the world works (at most levels—this one does break down a bit in some of our very best models) treat it as made out of things of various kinds; we not infrequently make mistakes about what there is, and to talk about such questions—which we need to do given our fallibility—we need a term like “exists”. We need something similar to state generalizations: “for all x, …” is the same as “there doesn’t exist an x such that not-...”. For (2b), our best models of the world involve mathematical structures in which propositions like “there exist infinitely many prime numbers” make sense. It’s not clear (to me, at least) how much of all this is dependent on details of human brains and minds—to what extent we should expect intelligent aliens, AIs, archangels, etc., to have notions of “things” and to use them in their mathematics. My guess is that we should expect both.
Why would we want to say that certain abstract sentences are “true”? Of course “true” is like “exists”. Most of the time we don’t say that X is true, we just say X; just as most of the time we don’t say Y exists, we just talk about Y. We need the term “true” (and its opposite “false” and other terms like “meaningless” and “unclear”) mostly because sometimes we err, and we need ways to think and talk about whether we’re erring. (E.g., we have some logical argument; it starts with innocuous premises and ends somewhere surprising; it’s useful to be able to go through step by step and ask at each step “well, is this true?”.) We can err just as easily when doing mathematics as when doing anything else.
Does it make sense to evaluate “There exists a city larger than Paris” and “There exists a number greater than 17″ the same way? I’m not sure what “the same way” really means here. The steps I would go through to try to convince a thoroughgoing skeptic of those two propositions would diverge quite quickly. It’s clear that there’s something in common between my mental representations of those propositions, but I don’t think I’d want to describe it by saying that I evaluate them the same way. Does it make sense for them to have as much in common as they do? Yup. Numbers have this comparability-structure that resembles that of city-sizes; why? partly because one thing we use numbers for—one thing that picks the number systems we use out of the vast universe of possible abstract structures—is comparing things. And then we use parallel language to describe these things with parallel logical structure. (I don’t want to overstate the extent to which we get to choose our abstract structures. E.g., if we want a minimal structure in which we can count indefinitely then we automatically get an ordering. This isn’t a coincidence because you can think of the ordering as arising from the counting process. In some cases it may be better to say “we can compare city-sizes because we can compare numbers” than “we use numbers rather than something else because we can compare them as we can city-sizes”. But whichever comes first, there are correspondences between things in the world and the abstractions we use, and this suffices to make it appropriate to use similar language and similar thoughts to handle both.)
Note 1: In the above, I’ve repeatedly claimed things whose rough form is “we do X because doing X is useful”. That shouldn’t be taken as meaning that someone sat down, figured out that doing X would be useful, and convinced everyone else to do it. The processes by which we ended up doing X are mysterious and complicated, and probably include biological evolution (perhaps our brains have structure in them that predisposes us to use nouns and numbers because our candidate-ancestors who didn’t were thereby disadvantaged in figuring out the world) and memetic kinda-sorta-evolution (if there were ever languages without anything nounlike, I guess they weren’t helpful to their users) and who knows what else.
Note 2: Nouns and things are so deeply built into our language and cognition that there’s no getting away from them. So there’s some circularity here.
Just for fun, I’ll take a stab at your final questions (apart from the last one, where I too have nothing).
Why do we say that numbers “exist”? I think there are two separate questions here. (1) Why do we treat numbers as real? (2) Why do we bother with terminology like “exist”, (2a) at all and (2b) for numbers in particular? I think at least part of our answer to (1) is that our best explanations for our observations have numbers in them, and this seems to be robust in the sense that it applies across a wide range of explanations of a wide range of things at a wide range of sophistication levels. Dunno whether it’s the whole reason. For (2a) … well, actually mostly we don’t talk about things “existing” except in marginal cases where there’s disagreement or doubt (only when doing philosophy or drugs does anyone ask whether tables exist). Our language has nouns, and our thinking has things, because the most convenient approximations to how the world works (at most levels—this one does break down a bit in some of our very best models) treat it as made out of things of various kinds; we not infrequently make mistakes about what there is, and to talk about such questions—which we need to do given our fallibility—we need a term like “exists”. We need something similar to state generalizations: “for all x, …” is the same as “there doesn’t exist an x such that not-...”. For (2b), our best models of the world involve mathematical structures in which propositions like “there exist infinitely many prime numbers” make sense. It’s not clear (to me, at least) how much of all this is dependent on details of human brains and minds—to what extent we should expect intelligent aliens, AIs, archangels, etc., to have notions of “things” and to use them in their mathematics. My guess is that we should expect both.
Why would we want to say that certain abstract sentences are “true”? Of course “true” is like “exists”. Most of the time we don’t say that X is true, we just say X; just as most of the time we don’t say Y exists, we just talk about Y. We need the term “true” (and its opposite “false” and other terms like “meaningless” and “unclear”) mostly because sometimes we err, and we need ways to think and talk about whether we’re erring. (E.g., we have some logical argument; it starts with innocuous premises and ends somewhere surprising; it’s useful to be able to go through step by step and ask at each step “well, is this true?”.) We can err just as easily when doing mathematics as when doing anything else.
Does it make sense to evaluate “There exists a city larger than Paris” and “There exists a number greater than 17″ the same way? I’m not sure what “the same way” really means here. The steps I would go through to try to convince a thoroughgoing skeptic of those two propositions would diverge quite quickly. It’s clear that there’s something in common between my mental representations of those propositions, but I don’t think I’d want to describe it by saying that I evaluate them the same way. Does it make sense for them to have as much in common as they do? Yup. Numbers have this comparability-structure that resembles that of city-sizes; why? partly because one thing we use numbers for—one thing that picks the number systems we use out of the vast universe of possible abstract structures—is comparing things. And then we use parallel language to describe these things with parallel logical structure. (I don’t want to overstate the extent to which we get to choose our abstract structures. E.g., if we want a minimal structure in which we can count indefinitely then we automatically get an ordering. This isn’t a coincidence because you can think of the ordering as arising from the counting process. In some cases it may be better to say “we can compare city-sizes because we can compare numbers” than “we use numbers rather than something else because we can compare them as we can city-sizes”. But whichever comes first, there are correspondences between things in the world and the abstractions we use, and this suffices to make it appropriate to use similar language and similar thoughts to handle both.)
Note 1: In the above, I’ve repeatedly claimed things whose rough form is “we do X because doing X is useful”. That shouldn’t be taken as meaning that someone sat down, figured out that doing X would be useful, and convinced everyone else to do it. The processes by which we ended up doing X are mysterious and complicated, and probably include biological evolution (perhaps our brains have structure in them that predisposes us to use nouns and numbers because our candidate-ancestors who didn’t were thereby disadvantaged in figuring out the world) and memetic kinda-sorta-evolution (if there were ever languages without anything nounlike, I guess they weren’t helpful to their users) and who knows what else.
Note 2: Nouns and things are so deeply built into our language and cognition that there’s no getting away from them. So there’s some circularity here.
Thanks! I think this is pretty darn good.