No. I’m not advocating for some sort of finitism, nor was Brouwer. In fact, I didn’t actually mention computability, that’s just something gjm brought up. It’s irrelevant to my point. Mathematics is a social activity in the same way politics is a social activity. As in, it’s an activity which is social, or at least predicated on some sort of society. Saying that mathematics is incomplete is as meaningful as saying that politics is incomplete in the same way a formal logic might be. It just doesn’t make sense.
Note that the intuitions which justify the usage of a particular axiom is not part of an axiom system, but those intuitions would still be part of mathematics. That’s largely the intuitionist critique of “old” formalism. It was also used as a critique of logical positivism by Gödel.
I get that old formalism isn’t viable, but I don’t see how that obviates the completeness question. “Is it possible that (e.g.) Goldbach’s Conjecture has no counterexamples but cannot be proven using any intuitively satisfying set of axioms?” seems like an interesting* question, and seems to be about the completeness of mathematics-the-social-activity. I can’t cash this out in the politics metaphor because there’s no real political equivalent to theorem proving.
*Interesting if you don’t consider it resolved by Godel, anyway.
Mathematics is a social activity in the same way politics is a social activity. As in, it’s an activity which is social, or at least predicated on some sort of society.
Are you sayig that nothing a hermit would ever do can be called mathematics? That doesn’t seem right.
I’m pretty sure Anthony isn’t claiming that mathematics is social in the sense that every mathematical activity involves multiple people working actively together. But that hermit would be doing mathematics in the context of the mathematical work other people have done. Suppose the hermit works on, say, the Riemann hypothesis: they’ll be building on a ton of work done by earlier mathematicians; the fact that they find RH important is probably strongly influenced by earlier mathematicians’ choices of research topics; the fact that they find other things RH relates to important, too. Suppose they think of what they’re doing in the context of, say, ZFC set theory; there are lots of possible set-theoretic foundations (note: Anthony would probably prefer to avoid this notion of “foundations”) one could use, and the particular choice of ZFC is surely strongly influenced by the foundational choices other mathematicians have made.
(Anthony might perhaps also want to say, though here I don’t think I could agree, that if e.g. the hermit writes proofs then what those look like will be largely determined by what sorts of arguments mathematicians find convincing: that the point of a proof is precisely to convey ideas and their correctness to other mathematicians, which is a social activity even if those other mathematicians happen not to be there at the time. I don’t agree with this because I think our hermit might well pursue proofs simply for the sake of ensuring the correctness of their conclusions, and a sufficiently smart hermit might come up with something like the notion of proof all on their own with only that motivation.)
That’s a pretty low bar. Is wiping your ass a social activity too? Because, presumably, your mom taught you how to do it, and the fact you’re doing it with paper is strongly influenced by earlier ass wiper’s choices.
But never mind that. Suppose the hermit never learned any math, not even addition. Will you say that his math would still be social, because he already knew the words “zero”, “one”, “two”, which hint at the set of naturals? Then suppose that the hermit has not seen a human since the day he was born, was raised by wolves, developed his own language from zero, and then described some theory in that (indeed, this hermit might be the greatest genius who ever lived). Surely that’s not social. But is it not math?
Personally, I’d be perfectly happy to say that our hypothetical hermit is doing mathematics despite the complete absence of social connections; but I wasn’t endorsing the claim that mathematics is a social activity, merely explicating it. (And of course it’s possible that my explication fails to match what Anthony would have said.) I am not confident enough of my understanding of Anthony’s position to guess at his answer to your hypothetical question.
(But, for what it’s worth, if for some reason I were required to defend the mathematics-is-social claim against this argument, I think I would say that it suffices that mathematics as actually practiced is social; making political speeches is fairly uncontroversially a social activity even though one can imagine a supergenius hermit contemplating the possibility of a society that features political speeches and making some for fun.)
No. I’m not advocating for some sort of finitism, nor was Brouwer. In fact, I didn’t actually mention computability, that’s just something gjm brought up. It’s irrelevant to my point. Mathematics is a social activity in the same way politics is a social activity. As in, it’s an activity which is social, or at least predicated on some sort of society. Saying that mathematics is incomplete is as meaningful as saying that politics is incomplete in the same way a formal logic might be. It just doesn’t make sense.
Note that the intuitions which justify the usage of a particular axiom is not part of an axiom system, but those intuitions would still be part of mathematics. That’s largely the intuitionist critique of “old” formalism. It was also used as a critique of logical positivism by Gödel.
I get that old formalism isn’t viable, but I don’t see how that obviates the completeness question. “Is it possible that (e.g.) Goldbach’s Conjecture has no counterexamples but cannot be proven using any intuitively satisfying set of axioms?” seems like an interesting* question, and seems to be about the completeness of mathematics-the-social-activity. I can’t cash this out in the politics metaphor because there’s no real political equivalent to theorem proving.
*Interesting if you don’t consider it resolved by Godel, anyway.
Are you sayig that nothing a hermit would ever do can be called mathematics? That doesn’t seem right.
I’m pretty sure Anthony isn’t claiming that mathematics is social in the sense that every mathematical activity involves multiple people working actively together. But that hermit would be doing mathematics in the context of the mathematical work other people have done. Suppose the hermit works on, say, the Riemann hypothesis: they’ll be building on a ton of work done by earlier mathematicians; the fact that they find RH important is probably strongly influenced by earlier mathematicians’ choices of research topics; the fact that they find other things RH relates to important, too. Suppose they think of what they’re doing in the context of, say, ZFC set theory; there are lots of possible set-theoretic foundations (note: Anthony would probably prefer to avoid this notion of “foundations”) one could use, and the particular choice of ZFC is surely strongly influenced by the foundational choices other mathematicians have made.
(Anthony might perhaps also want to say, though here I don’t think I could agree, that if e.g. the hermit writes proofs then what those look like will be largely determined by what sorts of arguments mathematicians find convincing: that the point of a proof is precisely to convey ideas and their correctness to other mathematicians, which is a social activity even if those other mathematicians happen not to be there at the time. I don’t agree with this because I think our hermit might well pursue proofs simply for the sake of ensuring the correctness of their conclusions, and a sufficiently smart hermit might come up with something like the notion of proof all on their own with only that motivation.)
That’s a pretty low bar. Is wiping your ass a social activity too? Because, presumably, your mom taught you how to do it, and the fact you’re doing it with paper is strongly influenced by earlier ass wiper’s choices.
But never mind that. Suppose the hermit never learned any math, not even addition. Will you say that his math would still be social, because he already knew the words “zero”, “one”, “two”, which hint at the set of naturals? Then suppose that the hermit has not seen a human since the day he was born, was raised by wolves, developed his own language from zero, and then described some theory in that (indeed, this hermit might be the greatest genius who ever lived). Surely that’s not social. But is it not math?
Personally, I’d be perfectly happy to say that our hypothetical hermit is doing mathematics despite the complete absence of social connections; but I wasn’t endorsing the claim that mathematics is a social activity, merely explicating it. (And of course it’s possible that my explication fails to match what Anthony would have said.) I am not confident enough of my understanding of Anthony’s position to guess at his answer to your hypothetical question.
(But, for what it’s worth, if for some reason I were required to defend the mathematics-is-social claim against this argument, I think I would say that it suffices that mathematics as actually practiced is social; making political speeches is fairly uncontroversially a social activity even though one can imagine a supergenius hermit contemplating the possibility of a society that features political speeches and making some for fun.)