4 ms·

You don’t need to go through every possible element in that infinite uncountable set to prove that relationship, though. You can create an arbitrary demonstrati

You don’t need to go through every possible element in that infinite uncountable set to prove that relationship, though. You can create an arbitrary demonstration that you can then manipulate in your head. Once you see that water demonstration you can inuit how that relationship must persist at different sizes. Again, that doesn’t really count as a “proof” by modern standards, but it’s how the ancient greeks thought (they used more than just visual intuition/they also used more rigorous and formal propositions than that water thing, but they were visual and didn’t involve finite sets)

wikfwikf 4y ago
But your brain is, presumably, doing finitely many cognitive steps, so it seems that a proof you can understand can also be expressed finitely?
- didericis 4y ago
  That’s a really deep question. If our brains are like digital computers, then yes, that’d be true. But they could be like analog computers, quantum computers, or something we don’t yet have the ability to describe.
  - deleted 4y ago
    [deleted]
  - wikfwikf 4y ago
    The problem with this theory is: what is the part of a geometric proof which cannot be described or validated using a classical, discrete computer? There is no such step. All parts of mathematics which mathematicians are able to agree on can be so described. There is no scope for our brains taking in an analog measurement, doing an analog measurement step on it, and using it to confirm the truth of a mathematical statement. Of course, this is not an argument that our brains are absolutely finite and classical. Perhaps analog computations is required for an appreciation of beauty, or quantum physics is necessary for us to fall in love. But we seem to be able to check math proofs without them.
    - didericis 4y ago
      I think mathematical intuition might be some kind of weird analog calculation, but yes, I can’t think of an actual visual proof that cannot also be described in discrete formal terms and validated with a computer. There might be examples out there somewhere, but I don’t know of any.

wikfwikf 4y ago

But your brain is, presumably, doing finitely many cognitive steps, so it seems that a proof you can understand can also be expressed finitely?

didericis 4y ago
That’s a really deep question. If our brains are like digital computers, then yes, that’d be true. But they could be like analog computers, quantum computers, or something we don’t yet have the ability to describe.
- deleted 4y ago
  [deleted]
- wikfwikf 4y ago
  The problem with this theory is: what is the part of a geometric proof which cannot be described or validated using a classical, discrete computer? There is no such step. All parts of mathematics which mathematicians are able to agree on can be so described. There is no scope for our brains taking in an analog measurement, doing an analog measurement step on it, and using it to confirm the truth of a mathematical statement. Of course, this is not an argument that our brains are absolutely finite and classical. Perhaps analog computations is required for an appreciation of beauty, or quantum physics is necessary for us to fall in love. But we seem to be able to check math proofs without them.
  - didericis 4y ago
    I think mathematical intuition might be some kind of weird analog calculation, but yes, I can’t think of an actual visual proof that cannot also be described in discrete formal terms and validated with a computer. There might be examples out there somewhere, but I don’t know of any.