In the case of pure mathematics, I think “the reason X is true” is often ill-defined; there will be different ways of thinking about the things involved in X, and they will lead to different answers to “why is X true?”.
(Is the prime number theorem true because the zeta function has no zeros on Re(s)=1, or does the zeta function have no zeros there because of how the prime numbers are distributed? Or is PNT perhaps true because of Selberg’s “fundamental formula”, which can be used to give a proof of PNT that makes no mention of complex analysis or the zeta function—or vice versa? My inclination is to say that those high-powered theorems we use to prove PNT are true because the prime numbers are distributed as they are—but we are then in the uncomfortable situation that all known ways of proving that they’re distributed that way go via these high-powered theorems, so X is true because Y is true but we can only prove X by proving Y first.)
It seems to me that “because” language in pure mathematics is co-opting a bunch of intuitions that were formed by considering cause-and-effect in the physical world. Causation is a tricksy notion even in the physical world, and it’s worse still in pure mathematics.
I agree (I think; it’s easy to go astray in this stuff) that your “upward” induction principle is equivalent to the formulation in terms of well-orderings. For what it’s worth, I think your principle works “because” we can take R = { (A, smallest thing not in A) } rather than thinking that the formulation in terms of well-orderings works “because” it guarantees we can think of making new elements from old; it’s interesting how different things are more-intuitive to different people :-).
I’m not assuming that my poset is totally ordered, so there isn’t always a “smallest thing not in A”. For example think of the binary trees you mentioned above, with ancestry as the order relation.
I get the impression we do still have a substantive disagreement, so I’ll taboo “because” and instead ask the question: how are the natural numbers built? Or in other words, what does it mean to say that something is a natural number? I would say that it just means that it is one of the objects in the collection you get if you start with 0, and each time you have n then you add n+1 to your collection. A book such as the one discussed in the OP, however, will give a different answer: a natural number is an element of the intersection of all sets that contain 0 and are closed under the operation n↦n+1. These two definitions are arguably equivalent—but such an equivalence cannot be proven in a language like ZFC, since the only way ZFC has of formalizing the first definition is to beg the question and encode it according to the second definition.
In a sense all properties of a mathematical object are “because” of the way that it was built. So perhaps our different intuitions about “because” are based on different ideas of what it means to be a natural number? (Or perhaps not...)
Regarding the prime number theorem: I’m not an expert so I don’t have an opinion on which way we should say that the causality goes in that particular case, but I do think a lot of the time it is useful to ask such questions, and that giving good answers (even if they aren’t the type of thing that can be rigorous) can help one understand a subject better.
I think it’s highly debatable whether the natural numbers are built at all. Arguably they’re just there (in some sense). One can construct particular “implementations” of the natural numbers, and there are many ways to do it; for instance, the usual way to do it in NF is a Frege-like construction: natural numbers are equivalence classes of finite sets under the relation “can be put in bijection with”; “finite” means “can’t be put in bijection with any finite subset”. (You can’t do that in ZF(C) because there are too many finite sets, but perhaps you can do it in a system that adds proper classes, like NBG.)
I don’t have strong feelings about how the natural numbers “should” be built, or what they “really” are. I’m happy thinking of them as “sizes of finite sets”, or as the things you get if you start at 0 and add 1 as often as you like (though there’s a certain circularity about that definition), or even as finite sequences of bits (though, again, there’s some circularity there). I don’t think it’s coincidence that these all lead to the same theorems, but I don’t feel any particular need to pick one definition and say that the others are all somehow parasitic on it.
Incidentally, when Frege came to define the natural numbers (this was in 1884, a few years before the usual Peano axioms were formulated, and I think he was the first person to do anything of the kind) he did it by (1) defining cardinal numbers as equivalence classes of sets under the same-size relation, and then (2) saying that a natural number is anything you can get to 0 from by counting downwards. Make of that what you will.
In the case of pure mathematics, I think “the reason X is true” is often ill-defined; there will be different ways of thinking about the things involved in X, and they will lead to different answers to “why is X true?”.
(Is the prime number theorem true because the zeta function has no zeros on Re(s)=1, or does the zeta function have no zeros there because of how the prime numbers are distributed? Or is PNT perhaps true because of Selberg’s “fundamental formula”, which can be used to give a proof of PNT that makes no mention of complex analysis or the zeta function—or vice versa? My inclination is to say that those high-powered theorems we use to prove PNT are true because the prime numbers are distributed as they are—but we are then in the uncomfortable situation that all known ways of proving that they’re distributed that way go via these high-powered theorems, so X is true because Y is true but we can only prove X by proving Y first.)
It seems to me that “because” language in pure mathematics is co-opting a bunch of intuitions that were formed by considering cause-and-effect in the physical world. Causation is a tricksy notion even in the physical world, and it’s worse still in pure mathematics.
I agree (I think; it’s easy to go astray in this stuff) that your “upward” induction principle is equivalent to the formulation in terms of well-orderings. For what it’s worth, I think your principle works “because” we can take R = { (A, smallest thing not in A) } rather than thinking that the formulation in terms of well-orderings works “because” it guarantees we can think of making new elements from old; it’s interesting how different things are more-intuitive to different people :-).
I’m not assuming that my poset is totally ordered, so there isn’t always a “smallest thing not in A”. For example think of the binary trees you mentioned above, with ancestry as the order relation.
I get the impression we do still have a substantive disagreement, so I’ll taboo “because” and instead ask the question: how are the natural numbers built? Or in other words, what does it mean to say that something is a natural number? I would say that it just means that it is one of the objects in the collection you get if you start with 0, and each time you have n then you add n+1 to your collection. A book such as the one discussed in the OP, however, will give a different answer: a natural number is an element of the intersection of all sets that contain 0 and are closed under the operation n↦n+1. These two definitions are arguably equivalent—but such an equivalence cannot be proven in a language like ZFC, since the only way ZFC has of formalizing the first definition is to beg the question and encode it according to the second definition.
In a sense all properties of a mathematical object are “because” of the way that it was built. So perhaps our different intuitions about “because” are based on different ideas of what it means to be a natural number? (Or perhaps not...)
Regarding the prime number theorem: I’m not an expert so I don’t have an opinion on which way we should say that the causality goes in that particular case, but I do think a lot of the time it is useful to ask such questions, and that giving good answers (even if they aren’t the type of thing that can be rigorous) can help one understand a subject better.
I think it’s highly debatable whether the natural numbers are built at all. Arguably they’re just there (in some sense). One can construct particular “implementations” of the natural numbers, and there are many ways to do it; for instance, the usual way to do it in NF is a Frege-like construction: natural numbers are equivalence classes of finite sets under the relation “can be put in bijection with”; “finite” means “can’t be put in bijection with any finite subset”. (You can’t do that in ZF(C) because there are too many finite sets, but perhaps you can do it in a system that adds proper classes, like NBG.)
I don’t have strong feelings about how the natural numbers “should” be built, or what they “really” are. I’m happy thinking of them as “sizes of finite sets”, or as the things you get if you start at 0 and add 1 as often as you like (though there’s a certain circularity about that definition), or even as finite sequences of bits (though, again, there’s some circularity there). I don’t think it’s coincidence that these all lead to the same theorems, but I don’t feel any particular need to pick one definition and say that the others are all somehow parasitic on it.
Incidentally, when Frege came to define the natural numbers (this was in 1884, a few years before the usual Peano axioms were formulated, and I think he was the first person to do anything of the kind) he did it by (1) defining cardinal numbers as equivalence classes of sets under the same-size relation, and then (2) saying that a natural number is anything you can get to 0 from by counting downwards. Make of that what you will.