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Comment by rjrodger | Hacker News Reader

I wrote my final year thesis on Lakatos’s book “Proofs and Refutations”[0]. It’s a short, mind-blowing book that answers the question: “Is mathematics discovered or invented?”

You may think the abstract structures that you learn about are fixed fundamental truths, but just read Lakatos’s account of the multi-decade debugging of the Euler characteristic for polyhedra and you will realise that maths is both created and discovered.

How do ya like them apples for yer epistemology!

Another example would be limits vs infinitesimals - which is the “true” foundation of calculus? [1]

Or have you ever struggled with the Axiom of Choice?

[0]: https://en.m.wikipedia.org/wiki/Proofs_and_Refutations

[1]: And the correct answer, as all you Coventry City fans will know, is: trick question

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So, first off, Model Theory teaches us that there are an infinite number of ways to represent calculus (or any other branch of mathematics). So there is no 'true' way. There are only ways that we have discovered and used.

Second, I don't know anything about the debugging of the Euler characteristic, but I suggest that the issue here is better characterized as taking decades to discover the correct representation. Sort of like Edison finding 1000 lightbulb ideas that didn't work, before finding one that did. (Don't take that analogy too far... I'm not suggesting that physical things are discovered in the same way that mathematical things are discovered. (But I'm also not suggesting that they are not.))

And the interesting question is, for each of those conceptions, did we call it into being, or was it “there already” in some sense. Consider the case where aliens on the other side of the universe got there first (we consider only possibilities, not probabilities!).

With respect to the debugging of the Euler characteristic, one might ask what truth value formulations during those decades of debugging had, from the perspective of those on the ground at the time. Are we so sure in our present age that there no bugs in Andrew Wiles’s work?

Even if you use automated machine proofs (Mathematica and friends) you still can’t be sure that stray cosmic rays don’t flip a bit every damn time.

Fun times.

I already spend more time than I should in considering how to prove that math exists. So if you're talking about physical things in your first paragraph, that's one of those things that I have to prohibit myself from contemplating for reasons of self-preservation.

I will say, though, that acknowledging that math exists independent of its observation makes it more difficult to argue that possible arrangements of atoms do not exist prior to their physical manifestation.

History is, of course, replete with examples of things that we got wrong for thousands of years - the existence of zero, the existence of negative numbers, "real" numbers, as distinct from "imaginary" numbers. These things are now considered trivial, and are taught to children.

I can't imagine Wiles's proof ever being understood by 1% of humanity. I hope, and believe, that there are no catastrophic errors in it, but I wouldn't bet more than $20 that there aren't.

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