[crystal_revenge]

> but I would like to understand the problem, too

But why should it be the case that this is always possible?

It's entirely reasonable that the set of useful mathematical proofs is a proper superset of human intelligible useful proofs.

In fact, to argue the contrary would imply there is something incredibly remarkable about human cognition.

[crote]

> It's entirely reasonable that the set of useful mathematical proofs is a proper superset of human intelligible useful proofs.

If you can't explain something in a way that a child could understands it, you don't fully understand it either.

[zmgsabst]

No, it doesn’t imply that.

Just that the set of proofs a human can interpret and the set of statements a human can understand overlap; conversely, you require that the statements/theorems humans can understand be a larger class than the proofs they can understand.

To me, it’s not obvious which of those should be true:

- can we only understand theorems for which we comprehend their proof?

- or can we understand theorems despite not comprehending the proof structure?

Within the mathematics community, opinions differ. But you’re elevating your perspective on that question into a law, without any evidence.

[crystal_revenge]

> understand theorems for which we comprehend

I don't know what your distinction between "understand" and "comprehend" but my point was not about these words, but about being "useful" and being "understandable".

I'm saying there's no relationship between a mathematical statement being useful and it being understandable.

If it is true that "understanding is a prerequisite for usefulness" (where "understanding" means that a statement can be proven in a way that is intelligible to humans) was a property of mathematical expressions, then this fact would certainly be useful (we could exclude any statements that no human understand from the world of useful mathematical expression). But, by that definition, we would need to understand that statement, so you would need to be able to prove that "understanding is a prerequisite for usefulness" in a human intelligible way.

Now what I just wrote is in itself not a proof that we can't know, but proving the above statement would involve expressing the claim in a mathematically verifiable way that was also understandable by humans, which does imply something remarkable about human cognition (something that would be intelligible no less!)

[poetaster]

Well, there is something remarkable in human intelligence. We have yet to find anything like it in the known universe. As for the rest, the wise mathematicians are leaning, sorry, hard to lean. TT and co.