Also, in the usual ZF set theory, it's empty.

Comments URL: https://news.ycombinator.com/item?id=47621251

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That tends to happen when you're the one defining the curve.

As for whether these definitions have a clear meaning that one can relate to 'the world': I think so. To take just one example (I could do more), finite-dimensional means exactly what you think it means, and you certainly understand what I mean when I say our world is finite (or three, or four, or n) dimensional.

Commutative also means something very down to earth: if you understand why a*b = b*a or why putting your socks on and then your shoes and putting your shoes on and then your socks lead to different outcomes, you understand what it means for some set of actions to be commutative.

And so on.

These notions, like all others, have their origin in common sense and everyday intuition. They're not cooked up in a vacuum by some group of pretentious mathematicians, as much as that may seem to be the case.

After all, facts (whatever that means) about the physical world can only be obtained by proxy (through measurement), whereas mathematical facts are just... evident. They're nakedly apparent. Nothing is being modelled. What you call the 'model' is the object of study itself.

A denial of the 'reality' of pure mathematics would imply the claim that an alien civilisation given enough time would not discover the same facts or would even discover different – perhaps contradictory – facts. This seems implausible, excluding very technical foundational issues. And even then it's hard to believe.

> To the best of our knowledge, such cases are basically coincidence.

This couldn't be further from the truth. It's not coincidence at all. The reason that mathematics inevitably ends up being 'useful' (whatever that means; it heavily depends on who you ask!) is because it's very much real. It might be somewhat 'theoretical', but that doesn't mean it's made up. It really shouldn't surprise anyone that an understanding of the most basic principles of reality turns out to be somewhat useful.

Do you think there might be a fairly obvious reason for that?

or perhaps with others (potentially) getting ahead of us.

Your argument fails right here because you're supposing something that isn't true. LLMs are better than search engines for some things, but you're speaking as if they're a replacement for what came before. They're absolutely not. Reading books — going to the original source rather than relying on a stochastic facsimile — is never going to go away, even if some of us are too lazy to ever do so. Their loss.

Put another way: leaving aside non-practical aspects of the experience, the car does a better job of getting you from A to B than a horse does. An LLM does not 'do a better job' than a book. Maybe in some cases it's more useful, but it's simply not a replacement. Perhaps a combination is best: use the LLM to interpolate and find your way around the literature, and then go and hunt down the real source material. The same cannot be said of the car/horse comparison.