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Comment by czgov | Hacker News Reader

If it’s an oracle and we know it’s an oracle then it’s not useless. Humans make mistake and there are examples of published results that were widely believed to be correct by experts that later proved to be wrong. Why do you think human verified proofs are better than machine verified proofs?

Suppose an oracle tells us the Riemann Hypothesis is correct. There are a vast number of results of the form:

If RH is correct then A.

It would be very useful to have an oracle tells us whether or not RH is correct.

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I asked how would it change anything. What's the next step if an oracle were to tell you p=np that changes anything about the world?

We all believe it. It's a magic oracle. Now what?

If it is known that A is provably true then one can study the consequences of A being true. It changes things becuase the body of knowledge has expanded.

In this case won't this oracle also tell you what is the consequences as soon as it tells you RH is true and also much more? At this point what is the point of you knowing what is true and what is not?

There is no actual oracle. In this discussion oracle means Lean.

> If it is known that A is provably true then one can study the consequences of A being true

But one can already study the consequences of P=NP right now. You don't need to know that it's provably true in order to do that.

Knowing an actual proof would be useful, but an oracle revealing merely that it's true (or even provable) without telling you the proof does not let you do anything you couldn't do before.

Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement. In the present discussion it’s not about letting me do something I can’t do now but about whether or not the endeavor is worthwhile.

A lot of people spent a lot of time and effort to prove or disprove the Jacobian Conjecture. AI solved it easily. It is increasingly becoming the case that humans are not as good at mathematics as computers. You are free to ignore computer generated proofs but I don’t think this position will win out in the long run.

> Some people (almost all mathematicians) wouldn’t want to spend time on consequences of a false statement.

No, people constantly prove statements of the form "if P=NP, then strange implication X". They do not consider it wasted effort at all, because of the contrapositive: if X is indeed very strange, they might be able to prove that it is false, and then they've settled P!=NP.

If a counterexample to a conjecture is found then all work toward proving consequences of the conjecture will cease. No one is trying to discover consequences of the Jacobian Conjecture now.

At some point an AI will prove a result that is so long and complicated that no human will understand it. This should not preclude people from using that result. In general, whenever the body of knowledge is increased it is a good thing. Even if it isn’t increased by humans.

> It would be very useful to have an oracle tells us whether or not RH is correct.

For what? Which product becomes better if it is correct?

You have an unfortunate view of the value of knowledge. The frontiers of science would be static if everyone believed as you do.

The whole point why anyone cares about these proofs is that the things we learn as we make the proof might add value, proving p = np itself isn't interesting, that knowledge has no application and therefore no value in itself.

I have published mathematics so I do value knowledge, but for most of mathematics the value of the knowledge isn't the thing you try to prove it is all the things you learn as you try to prove it. p = np is one such thing.

So the whole interesting bit about it is the proof, not the fact.

You are wrong as far as most mathematicians believe. The fact is important. The proof of the fundamental theorem of algebra is interesting and important but the theorem itself is also important.

For what? Which product becomes better if it is correct?

This sentiment is anti-thetical to the whole point of pure math and theoretical science. No product became better when Euler proved the fundamental theorem of algebra.

Someone claimed it would be "useful", without saying what for. Hence the questions "what for?". To try to shame people for that question in the name of science of all things is wild.

Which product becomes better if it is correct?

Tying the worthiness of theoretical knowledge to the whether or not it improves a product is asinine in my opinion.

I generally[0] agree, but IMO that should be taken up with https://news.ycombinator.com/item?id=49369458 more than with people who ask for examples for that claim.

[0] and am only adding that "generally" because I can think of examples where I'd disagree, e.g. a kid that wants to count all stars in the night sky before it has dinner would just starve and then not be able to count stars, either.

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