How has mathematics gotten so abstract? • Hacker News Board

Re: How has mathematics gotten so abstract?

#141

Post by coffeeaddict1 » Tue, Sep 30, 2025, 7:46 PM UTC

btilly wrote: post #88

Proposed rule: People writing about the history of mathematics, should learn something about the history of mathematics. Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos -…

I have to disagree with this. Modern (pure) mathematics is abstract and very often completely detached from practical applications because of culture and artistic inspiration. There is no "objectivity" driving modern pure mathematics. It exists mostly because people like thinking about it. Any connection to the real world is often a coincidence or someone outside the field noticing that something (really just a tiny-tiny amount) in pure maths could be useful.

> forced there by a series of crises where the mathematics that they knew fell apart

This can be said to be true of those working in foundations, but the vast majority of mathematicians are completely uninterested in that! In fact, most mathematicians today probably can't cite you the set-theoretic (or any other foundation) axioms that they use every day, if you ask them point-blank.

Re: How has mathematics gotten so abstract?

#142

Post by BrandoElFollito » Tue, Sep 30, 2025, 7:47 PM UTC

I used to be a physicist and I love math for the toolbox it provides (mostly Analysis). It allows to solve a physical model and make predictions.

When I was studying, I always got top marks in Analysis.

Then came Algebra, Topology and similar nightmares. Oh crap, that was difficult. Not really because of the complexity, but rather because of abstraction, an abstraction I could not take to physics (I was not a very good physicist either). This is the moment I realized that I will never be "good in maths" and that will remain a toolbox to me.

Fast forward 30 years, my son has differentials in high school (France, math was one of his "majors").

He comes to me to ask what the fuck it is (we have a unhealthy fascination for maths in France, and teach them the same was as in 1950). It is only when we went from physical models to differentials that it became clear. We did again the trip Newton did - physics rocks :)

Re: How has mathematics gotten so abstract?

#143

Post by trinsic2 » Tue, Sep 30, 2025, 7:47 PM UTC

m_dupont wrote: post #137

trinsic2 wrote: post #136

>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race. Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far i…

you're not, the proof is a famous error known as zenos paradox. Its only an apparent paradox, and indeed it's been disproven by observing that things do in fact move

Wow this is some serious over complication. How can anyone mix Philosophy and Mathematics? They are not even in the same ball park.. Even with infinity. Its just something that cant be understood in the mind, IMHO.

Re: How has mathematics gotten so abstract?

#144

Post by btilly » Tue, Sep 30, 2025, 7:53 PM UTC

zozbot234 wrote: post #133

btilly wrote: post #103

Earlier quoted context omitted.

I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.

The "philosophical questions" dividing formalism from constructivism are greatly overstated. The point of having those degrees of undecidability or uncountability is precisely to be able to say things like "even if you happen to be operating under strong additional assumptions that let you decide/count X, that still doesn't let you decide/count Y in general." That's what formalism is: a handy way of making statements…

Yes, you can show such equiconsistency statements. As Gödel proved, for any set of classical axioms, there is a corresponding set of intuitionistic axioms. And if the classical axioms are inconsistent, then so is the intuitionistic equivalent. (Given that intuitionistic reasoning is classically valid, an inconsistency in the intuitionistic axioms trivially gives you one in the classical axioms.)

So the care that intuitionists take does not lead to any improvement in consistency.

However the two approaches lead to very different notions of what it means for something to mathematically exist. Despite the formal correspondences, they lead to very different concepts of mathematics.

I'm firmly of the belief that constructivism leads to concepts of existence that better fit the lay public than formalism does.

Re: How has mathematics gotten so abstract?

#145

Post by btilly » Tue, Sep 30, 2025, 7:55 PM UTC

edanm wrote: post #114

btilly wrote: post #104

Earlier quoted context omitted.

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

> When you say "given ZFC", you're assuming a lot. Errr, I'm just assuming the axioms of ZFC. That's literally all I'm doing. > In what sense do [numbers that can't be finitely specified] exist? In the sense that we can describe rules that lead to them, and describe how to work with them. I understand that you're trying to tie the notion of "existence" to constructability, and that's fine. That's one way to play the…

My point is that going from a lay understanding of mathematics to "just accept ZFC" means jumping past a variety of debatable philosophical points, and accepting a standard collection of answers to them. Mathematicians gloss over that.

Re: How has mathematics gotten so abstract?

#146

Post by goatlover » Tue, Sep 30, 2025, 7:58 PM UTC

nitwit005 wrote: post #130

pdpi wrote: post #21

Earlier quoted context omitted.

> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

A proof is just an argument that something is true. Ideally, you've made an extremely strong argument, but it's still a human making a claim something is true. Plenty of published proofs have been shown to be false. Math is scientific in the sense that you've proposed a hypothesis, and others can test it.

Difference is mathematical arguments can be shown to be provably true when exhaustively checked (which is straight forward with simpler proofs). Something you don't get with the empirical sciences.

Also the empirical part means natural phenomena needs to be involved. Math can be purely abstract.

Re: How has mathematics gotten so abstract?

#147

Post by btilly » Tue, Sep 30, 2025, 7:59 PM UTC

zozbot234 wrote: post #135

btilly wrote: post #104

Earlier quoted context omitted.

When you say "given ZFC", you're assuming a lot. Including a notion of mathematical existence which bears little relation to any concept that most lay people have of what mathematical existence might mean. In particular, you have made sufficient assumptions to prove that almost all real numbers that exist can never be specified in any possible finite description. In what sense do they exist? You also wind up with wei…

> In what sense do they exist? In the sense that all statements of non-constructive "existence" are made, viz. "you can't prove that they don't exist in the general case", so you are allowed to work under the stronger assumption that they also exist constructively, without any contradiction resulting. That can certainly be useful in some applications.

Sure, we can choose to work in a set of axioms that says that there exists an oracle that can solve the Halting problem.

But the fact that such systems don't create contradictions emphatically *DOES NOT* demonstrate the constructive existence of such an oracle. Doubly not given that in various usual constructivist systems, it is easily provable that nothing that exists can serve as such an oracle.

Re: How has mathematics gotten so abstract?

#148

Post by goatlover » Tue, Sep 30, 2025, 8:02 PM UTC

abdullahkhalids wrote: post #87

pdpi wrote: post #21

Earlier quoted context omitted.

> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

Plenty of mathematical proofs have been proven true with 100% certainty. Complicated proofs that involve a lot of steps and checking can have errors. They can also be proven true if exhaustively checked.

Re: How has mathematics gotten so abstract?

#149

Post by initramfs » Tue, Sep 30, 2025, 8:10 PM UTC

This article can also be written as "The unreasonable effectiveness of abstraction in mathematics."

Re: How has mathematics gotten so abstract?

#150

Post by zozbot234 » Tue, Sep 30, 2025, 8:16 PM UTC

btilly wrote: post #147

zozbot234 wrote: post #135

Earlier quoted context omitted.

> In what sense do they exist? In the sense that all statements of non-constructive "existence" are made, viz. "you can't prove that they don't exist in the general case", so you are allowed to work under the stronger assumption that they also exist constructively, without any contradiction resulting. That can certainly be useful in some applications.

Sure, we can choose to work in a set of axioms that says that there exists an oracle that can solve the Halting problem. But the fact that such systems don't create contradictions emphatically *DOES NOT* demonstrate the constructive existence of such an oracle. Doubly not given that in various usual constructivist systems, it is easily provable that nothing that exists can serve as such an oracle.

> emphatically DOES NOT demonstrate the constructive existence of such an oracle

Of course, but it shows that you can assume that such an oracle exists whenever you are working under additional conditions where the existence of such a "special case" oracle makes sense to you, even though you can't show its existence in the general case. This outlook generalizes to all non-constructive existence statements (and disjunctive statements, as appropriate). It's emphatically not the same as constructive existence, but it can nonetheless be useful.