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Mathematicians don't care about foundations (2022)

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Re: Mathematicians don't care about foundations (2022)

#21

Post by fpoling »

black_knight wrote: post #18
lanstin wrote: post #15

Earlier quoted context omitted.

No one not working on foundations has any problem with axiom of choice. It has weird implications but so what? Banach Tarski just means physical shapes aren't arbitrarily subdividable.

Banach Tarski is not about physical shapes. The thing is, the foundations negating axiom of choice are just as consistent as those with. So, how do mathematicians justify their faith in AC?

AC makes things much easier as it allows to play God powers. Negating AC is not significantly different from constructing mathematics that avoids AC (no assumption about validity of AC). And that makes things way harder with longer proofs and only in sub-cases of classical theorems.

Re: Mathematicians don't care about foundations (2022)

#22

Post by black_knight »

fpoling wrote: post #21
black_knight wrote: post #18

Earlier quoted context omitted.

Banach Tarski is not about physical shapes. The thing is, the foundations negating axiom of choice are just as consistent as those with. So, how do mathematicians justify their faith in AC?

AC makes things much easier as it allows to play God powers. Negating AC is not significantly different from constructing mathematics that avoids AC (no assumption about validity of AC). And that makes things way harder with longer proofs and only in sub-cases of classical theorems.

Simply assuming the negation of AC is boring, as negations often are. But there are stronger statements, implying the negation of AC which might be as useful. I think for instance one could assume all subsets of the plane to be measurable. Seems convenient to me.

Same with law of the excluded middle. Tossing it out we can assume all functions are computable and all total functions in the real are continuous. Seems nice and convenient too!

Re: Mathematicians don't care about foundations (2022)

#23

Post by black_knight »

adgjlsfhk1 wrote: post #19
black_knight wrote: post #13

Earlier quoted context omitted.

Very little of mathematics, like analysis? I am sure the analyst will care about all functions on the reals suddenly turning continuous. (Or rather losing the discontinuous ones) Or what of commutative algebra and their beloved existence of maximal ideals!

you're kind of coming at this backwards. it's not that someone doing analysis doesn't care about whether all functions on reals is continuous, it's that if you hand them a foundation where that's true, they'll disagree with whether your foundation is correctly modeling functions/real numbers.

At which point we would have an interesting debate! I could tell them all about how this foundation will give them a more nuanced view on continuity!

Re: Mathematicians don't care about foundations (2022)

#24

Post by johngossman »

black_knight wrote: post #2

This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game. I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. But choosing foundation has real implications on the mathematics. You…

Newton and Gauss and Euler did just fine without such solid foundations. If you get a PhD, very likely even a undergraduate degree in mathematics you cover this stuff, then (unless you choose foundations as your field) you go about doing statistics, or algebra (the higher kind), or analysis knowing you're working on solid fundamentals. It would be crazy if every time you proved something in one of those fields you had to state which derivation of real number you were using. And I guarantee at least 90% of PhD mathematicians could do so if they really needed to.

Re: Mathematicians don't care about foundations (2022)

#25

Post by johngossman »

black_knight wrote: post #23
adgjlsfhk1 wrote: post #19

Earlier quoted context omitted.

you're kind of coming at this backwards. it's not that someone doing analysis doesn't care about whether all functions on reals is continuous, it's that if you hand them a foundation where that's true, they'll disagree with whether your foundation is correctly modeling functions/real numbers.

At which point we would have an interesting debate! I could tell them all about how this foundation will give them a more nuanced view on continuity!

I suggest you go meet some PhD mathematicians and have that discussion.

Re: Mathematicians don't care about foundations (2022)

#26

Post by black_knight »

johngossman wrote: post #25
black_knight wrote: post #23

Earlier quoted context omitted.

At which point we would have an interesting debate! I could tell them all about how this foundation will give them a more nuanced view on continuity!

I suggest you go meet some PhD mathematicians and have that discussion.

Having a PhD in mathematics myself, I have been surrounded by such and had this discussion a few times. Some even like the ideas suggested!

I would say the most common counter argument is cultural: Classical mathematics is the norm in the field, hence one must use it to participate in research in this field.

But that seems to me a rather intellectually unsatisfying argument, if one cares about the meaning of the work.

Re: Mathematicians don't care about foundations (2022)

#27

Post by black_knight »

johngossman wrote: post #24
black_knight wrote: post #2

This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game. I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. But choosing foundation has real implications on the mathematics. You…

Newton and Gauss and Euler did just fine without such solid foundations. If you get a PhD, very likely even a undergraduate degree in mathematics you cover this stuff, then (unless you choose foundations as your field) you go about doing statistics, or algebra (the higher kind), or analysis knowing you're working on solid fundamentals. It would be crazy if every time you proved something in one of those fields you ha…

We are not talking about having to return to foundational axioms in every argument! Just that what axioms one chooses has an impact on which arguments are valid, and thus in turn what truths there are.

Re: Mathematicians don't care about foundations (2022)

#29

Post by Sniffnoy »

black_knight wrote: post #2

This seems to me to be the same as saying that mathematicians do not care about the meaning of their theorems. That they are only playing a game. They care about consistency only because inconsistency means one can cheat in their game. I know TFA says that the purpose of foundations is to find a happy home (frame) for the mathematicians intuition. But choosing foundation has real implications on the mathematics. You…

> But choosing foundation has real implications on the mathematics. You can have a foundation where every total function on the real numbers is continuous. Or one where Banach–Tarski is just false.

I mean, mathematicians do care about the part of the foundations that affect what they do! Classical vs constructive matters, yes. But material vs structural is not something most mathematicians think about. (They don't think about classical vs constructive either, but that's because they don't really know about constructive and it's not what they're trying to do, rather than because it's irrelevant to them like material vs structural.)

Re: Mathematicians don't care about foundations (2022)

#30

Post by jesuslop »

black_knight wrote: post #18
lanstin wrote: post #15

Earlier quoted context omitted.

No one not working on foundations has any problem with axiom of choice. It has weird implications but so what? Banach Tarski just means physical shapes aren't arbitrarily subdividable.

Banach Tarski is not about physical shapes. The thing is, the foundations negating axiom of choice are just as consistent as those with. So, how do mathematicians justify their faith in AC?

My 2 cents is they do justify it by the interest of the consequences, as Tychonoff or Nullstellensatz. I wouldn't call that faith: Best practices is to state Tychonoff as "AC implies Tychonoff" and that last is logically valid. Sometimes the "AC implies..." is missing, buried in the proof or used unawaredly or predates ZFC, and is a bad thing. But very ofen one now see asterisks on theorems needing it.
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