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Comment by throw-the-towel | Hacker News Reader

See the longstanding debate on whether new math is "invented" or "discovered". Most mathematicians I knew thought it's discovered.

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This is like saying a sculpture always existed, the sculptor just had to remove the superfluous material.

Or like a musical octave has only 12 semitones, so all music is just a selection from a finite set that already existed.

Sure the insane computation we're throwing at this changes our perspective, but still there is an important distinction.

The difference is that math answers (can answer) specific questions.

Like, "does the Riemann zeta function have zeroes that don't have real part 1/2," or "is there a better solution to the Erdős Unit Distance Problem."

The selection of question is matter of taste, but once selected, there is a definitive precise answer.

Bob Ross would like a word. He frequently talked about objects or features already existing, and using the tools at his disposal to “find” them.

Michelangelo never used references, instead he simply "freed" the sculptures from within the marble.

Isn't that a way of saying working with the grain of the material?

There's more to it than the sculptor example. You could carve anything but the interesting bits of maths are quite specific.

Any design already exists as a possibility, so it could be said to be both invented and discovered, depending on how you look at it.

All inventions are discoveries, though not all discoveries are inventions.

Depending on your point of view? I see what you did there.

Who knew Obi-one was just smoking and pontificating on Wittgenstein.

On the other hand, it is proven that if you need to count things, the only thing you can discover/invent is the natural numbers.

Really?!

Care to cite a reference to that proof?

"The Nature and Meaning of Numbers" (1888) by Dedekind. He proved that any set of mathematical objects conforming to the second order Peano axioms is isomorphic to the natural numbers. Peano axioms basically formalize the notion of well-behaved (that is practically useful) counting numbers.

If you were to dig deeper, you'd get to the murky philosophical depths of foundations of mathematics, but I prefer to not go there. Practically, if you want to reliably count something, you end up with the natural numbers (or, maybe, their subset that ultrafinitists are trying to formalize).

Regardless of which, both Newton and Leibniz imprint in their findings a 'voice' and understanding different from each other and that of an LLM (for now?)

One can argue that mathematical facts are discovered, but the tools that allow us to find, express them and prove them, are mostly invented. This goes up to the axioms, that we can deliberately choose and craft.

Math is an abstraction of reality, it had to be invented, so more inventions or discoveries could be made within it.

The test goes like ‘is our universe, or any other universe, required for the axioms to exist’ and I don’t see how ‘yes’ is a defensible answer.

What is an abstraction? It is something that arises from human thought and human thought arises from the activity of neurons which are a part of reality. You can't escape reality unless you invoke some form of dualism.

abstractions are objects that come into existence via design and iteration to refine its form. This right here is invention not discovery.

...long standing indeed. It can be traced back to Plato's works.

"The European philosophical tradition consists of a series of footnotes to Plato."

The 90% of the Phillosophical tradition it's just bad discrete math.

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