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TimS comments on Rationality Quotes March 2012 - Less Wrong
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I'm not sure that's how it was motivated historically. Note that Euclid's proof (Edit: not Euler) doesn't require measuring anything at all.
To use a different example, how would one go about measuring whether there are more real numbers than integers? The proof is pretty easier, but it doesn't require any empirical facts as far as I can tell.
I think you mean Euclid's proof, and he was working centuries after Pythagoras, who was himself working over a thousand years after the Babylonians, who discovered Pythogorean Triples (the ones you notice by measuring).
To restate, I'm fine with saying that a proof for the Pythogorean Theorem exists that does not require measuring physical triangles, but I'm not comfortable with the statement that it cannot be proved by measuring physical triangles, which is what your original comment implied to me.
As discussed in the other subthread, I think that Deutsch's intention was to argue that any instance of a proof, as an object, has to exist in reality somewhere, which is a very different claim.
It depends on what you mean by "proved". The Pythagorean Theorem applies to all possible triangles (on a flat Euclidean plane), and the answer it gives you is infinitely precise. If you are measuring real triangles on Earth, however, the best you could do is get close to the answer, due to the uncertainty inherent in your instruments (among other factors). Still, you could very easily disprove a theorem that way, and you could also use your experimental results to zero in on the analytical solution much faster than if you were operating from pure reason alone.
It's just the problem of induction.
Other subthread? Don't see where anyone made that point. Moreover, I don't think it is a good reading of the original quote.
That's not fairly represented by saying "All actual proofs are on physical paper (or equivalent)."
I was thinking of this comment. If by "knowledge" he means "a piece of memory in reality," then by definition there is no abstract knowledge, and no abstract proofs, because he limited himself to concrete knowledge.
That knowledge can describe concepts that we don't think of as concrete- the Pythogorean Theorem doesn't have a physical manifestation somewhere- but my knowledge of it does have a physical manifestation.
There are all kinds of quantitative ways in which there are more real numbers than integers. On the other hand a tiny minority of us regard Cantor's argument (that I think you're alluding to) as misleading and maybe false.