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One philosophy question I've wondered about is how pure pure mathematics truly is: questions like whether "the integers" a true abstract concept, or can it only be explained to an intelligence that has a world model that includes the notion of "counting" or something similar. The math definitions seem crafted to be purely abstract, but my thinking about them always ends up grounded in the real world. Can a true abstract intelligence (which an LLM trained on human text isn't, but is perhaps closer than a flesh-and-blood human) derive all of modern mathematics given only the selected axioms? Some of this, I think, comes back to the IMO still-poorly-answered "what is intelligence?" question.
You don't need a notion of "counting" to be able to define the natural numbers. Upward Lowenheim-Skolem means that there are models of Peano arithmetic of every infinite cardinality, so the "rules" that give rise to the naturals also give rise to structures where you have "natural" numbers which are infinite and can never be arrived at by starting from 0 and taking the successor finitely many times. They're called the hypernaturals and are a fascinating object of study, completely divorced from the ordinary "counting" way people think about numbers, and yet they satisfy all the standard rules of arithmetic.
I've never understood why mathematicians say nonsense like this. My 3,4, and 8yo boys regularly get into "who loves daddy more" fights, and as soon as one of them says "I love daddy infinity", the next one immediately says "I love daddy infinity plus one!". Obviously to them infinity plus one is an entirely different and meaningfully bigger quantity than infinity. My experience is that kids universally understand this simple concept, and that it takes a calculus teacher to beat such sensible reasoning out of them.
Don't get me started on the 0.9999... = 1 nonsense, where non-mathematicians are obviously reasoning using hyperreals and the stupid mathematicians insist on limiting themselves to the ordinary reals.
(I have a math phd and teach in a college math dept, so I feel like this is a fair insider criticism.)
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