Comment by msteffen | Hacker News Reader

Comment by msteffen | Hacker News Reader

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> what happens is that, as in software, certain ideas get ossified. That’s why, for example, every OS has a POSIX layer even though technically the process/namespace/security model could be radically reimagined possibly to create more easily engineered, correct software.

Total amateur here, but it strikes me that one important difference is that performance matters in software in a way that it doesn’t in mathematics—that is, all proofs are equally valid modulo elegance. That means that abstractions in software are leaky in a way that abstractions in mathematics aren’t.

In other words, in software, the same systems get reused in large part because they’ve been heavily refined, in terms of performance, unexpected corner-case behavior and performance pitfalls, documentation of the above, and general familiarity to and acceptance by the community. In math, if you lay new foundations, build some new abstraction, and prove that it’s at least as powerful to the old one, I’d think that you’d be “done” with replacing it. (Maybe downstream proofs would need some new import statements?)

Is this not the case? Where are people getting stuck that they shouldn’t be?

I know what you're saying but elegance is not simply an aesthetic concern.

The value of a proof is not only its conclusion but also the insight that it provides through its method.

The goal of mathematics is not to prove as many theorems as possible but rather to gain an ever deeper understanding of why certain statements are true. The way that something is proved can be more or less useful to advancing that goal.

As an example the elementary proof(s) of the prime number theorem are just about as famous as the original proof. Sometimes the second bite of the cherry is even juicier than the first.

Exactly. The reason mathematicians and physicists care about elegance is because they care about understanding things. Elegance, like you said, isn't about aesthetics, even though people seem to think they're synonymous. But the elegance is that you've reduced things to simple components. That not only makes it easier for us humans to understand but it means you're closer to the minimal structure. Meaning you know what matters and more importantly, what doesn't.

Tbh, elegance is something programmers should strive for too. Elegant code is easier to build upon, easier to read/understand, easier to modify, easier to adapt. For all the same reasons mathematicians want elegance. Though it's true for many domains. People love to throw around the term "first principles" but that's not something you (usually) start at, that's something you derive. And it's usually not very easy to figure out

Agreed; e.g. if you prove something about the real numbers, the matter of how R is constructed out of your axiomatic system doesn't matter

there are questions where the abstraction of real numbers becomes leaky, and some axioms (or their lack) poke through.

https://en.wikipedia.org/wiki/Axiom_of_choice#Real_numbers

The picture isn't quite so clean in the constructive context, which is what many of these proof systems are rooted in, e.g., https://mathoverflow.net/questions/236483/difference-between...

Some proofs have become extremely long, and the raw size has created worries about correctness. It's easy to make a mistake in hundreds of pages.

Ultimately, a proof is an argument that something is true. The simpler "more elegant" proof is generally going to be more convincing.

Proof irrelevance I don't think is accepted in constructivist situations. Those are, however, not that relevant to the recent wave of AI math which uses Lean, whose type system includes classical mathematics.

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