Mathematicians don't care about foundations (2022) • Hacker News Board

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

#11

Post by zelphirkalt » Sat, Dec 20, 2025, 9:30 PM UTC

soVeryTired wrote: post #4

IMO it's not far off how most python or javascript devs don't care about registers or cache misses. Someone's thought deeply about those things so you don't have to. Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem use…

> Mathematicians do care about how much "black magic" they're invoking, and like to use simple constructions where possible (the field of reverse mathematics makes the central object of study). For example, Wiles' initial proof of Fermat's last theorem used quite exotic machinery called "inaccessible cardinals", which lie outside of ZFC. Subsequent work showed they weren't needed.

In a way mathematicians can afford to do this more readily than people in software development, because if something is actually proven, then you can 100% rely on that. With software not so much. Or rather: Software usually is not proven to be correct, because that's usually expensive. In mathematics they don't have to consider the runtime of an algorithm, when they "merely" need to prove correctness. The time it needs to run is irrelevant for its correctness. And so they can stack and stack and stack, provided that each piece is proven correct, and it won't have negative consequences. Well, almost. There is some negative consequence in that another human being, wanting to understand a proof, needs to know perhaps many concepts and other proofs, in order to be able to do so. But that's probably the only reason to pursue simplicity in mathematics.

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

#12

Post by black_knight » Sat, Dec 20, 2025, 9:35 PM UTC

steppi wrote: post #3

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…

I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure…

Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.

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

#13

Post by black_knight » Sat, Dec 20, 2025, 9:39 PM UTC

AnimalMuppet wrote: post #7

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…

The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets. You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abst…

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!

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

#14

Post by steppi » Sat, Dec 20, 2025, 9:45 PM UTC

black_knight wrote: post #12

steppi wrote: post #3

Earlier quoted context omitted.

I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure…

Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.

I don't think they usually make the distinction in a formal sense, but I think most are aware. The space of explorable mathematics is vastly larger than what the community of mathematicians is capable of collectively thinking about, so a lot of aesthetic judgment goes into deciding what is and what isn't interesting to work on. Mathematicians differ in their tastes too. A sense of sturdiness vs flimsiness is something that might inform this aesthetic judgment, but isn't really something most mathematicians would make part of the mathematics. Often, ones interest isn't the result itself, but some proof technique that brings some sense of insight and understanding, and exploring that often doesn't make much contact with foundational matters.

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

#15

Post by lanstin » Sat, Dec 20, 2025, 9:50 PM UTC

black_knight wrote: post #12

steppi wrote: post #3

Earlier quoted context omitted.

I'd say that I care deeply about the meaning behind theorems, but just find results which swing widely based on foundational quirks to be less interesting from an aesthetic standpoint. I see the most interesting structures as the ones that are preserved across different reasonable foundations. This is speaking as someone who was trained as a pure mathematician, moved on to other things, but tries to keep up with pure…

Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.

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.

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

#16

Post by Tazerenix » Sat, Dec 20, 2025, 9:54 PM UTC

Something the computer scientists of Hackernews might not realise is that most mathematicians are by nature Platonists, even if they would not try to defend that position when pressed.

most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism https://en.wikipedia.org/wiki/Mathematical_Platonism

Mathematicians begrudgingly retreat to formalism and foundations when pressed because its easier to defend, but the day-to-day of contemporary mathematics is much more an explorative process of a "real" mathematical landscape. They aren't concerned with foundations because it "feels" self-evident that the mathematics they are discovering is true (because their means of discovery, rigour and proof, "guarantee" it to be so).

A lot of the comments here are making false assumptions like "but surely mathematicians all know that their field is ultimately justified as a symbol-pushing game from some axiomatic system right?" in the same way one might say "surely all computer scientists know that every language ultimately compiles down to 1s and 0s processed by a CPU" but that is not at all how most mathematicians think about doing mathematics.

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

#17

Post by romangarnett » Sat, Dec 20, 2025, 9:57 PM UTC

AnimalMuppet wrote: post #7

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…

The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets. You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abst…

Do you not care if your vector space has a basis?

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

#18

Post by black_knight » Sat, Dec 20, 2025, 9:59 PM UTC

lanstin wrote: post #15

black_knight wrote: post #12

Earlier quoted context omitted.

Yes, but most mathematicians do not seem to make this distinction between sturdy and flimsy truths. Which puzzles me. Are they unaware? If so, would they care if educated? Or do they fully commit to classical logic and the axiom of choice if pushed? I can see it go either way, depending on the psychology of the individual mathematician.

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?

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

#19

Post by adgjlsfhk1 » Sat, Dec 20, 2025, 10:11 PM UTC

black_knight wrote: post #13

AnimalMuppet wrote: post #7

Earlier quoted context omitted.

The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets. You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abst…

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.

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

#20

Post by qbit42 » Sat, Dec 20, 2025, 10:13 PM UTC

romangarnett wrote: post #17

AnimalMuppet wrote: post #7

Earlier quoted context omitted.

The foundations have real implications on very little of the mathematics. Say I'm working in differential equations in vector spaces. I really do not care whether the axiom of choice is true or false. I'm not building up my functions of multiple real parameters out of sets. You say you have a foundation where that is in fact what I am doing? Great, if that floats your boat. I don't care. That's several layers of abst…

Do you not care if your vector space has a basis?

It is nicer to state theorems that hold for all vector spaces, so mathematicians like to invoke AoC. However, in any applications that are practically relevant, you can obtain a basis without invoking AoC.