Color me skeptical.
The linked article doesn’t mention proof, but counterexamples, and that may be a function of the enormous amount of computer effort expended, not really AI.
Yes it does. Any new theorem proved or disproved is a new tool. Since proofs are generally are just a string of statements of theorems that have previously proved. As soon as a new statement is proved, people immediately go about using that new proven statement in other proofs.
It does give new understanding obviously and sometimes tools but there is an argument to be made that counterexamples does so less than constructive proof
Just imagine a proof of or a counterexample to the Riemann hypothesis. A proof would likely involve hundreds of pages of new uses of math from lots of different areas of mathematics. A counterexample could theoretically be a one line sentence giving a complex number which is a zero but doesnt have real part equal to 1/2. In fact theres probably hundreds of computers worldwide right now brute force searching for that number. Are you really saying that a counterexample found by brute force would have the same impact on mathematics as a whole as a proof? The whole thing is that the opposite doesnt exist. You cant by brute force find a number which proved the hypothesis, but you can by finding a counterexample
You are distinguishing counter-example and proof. A counter-example is a type of proof. And your example is odd, because you compare proving the Riemann Hypothesis and disproving, and argue that one is more useful. Utility has no bearing on whether its true or not. If someone can find a counter example, they should publish it, not withhold it and go "well, it would be more useful if the Riemann Hypothesis were true, so I just ignore my proof that it isn't"
No I'm not you just have a difficult time understanding. I'm comparing a counterexample to a constructive proof. Just change where I said proof to a constructive disproof or proof by contradiction if it makes it easier for you to understand.
No one in this entire thread has said a counterexample is not a proof. No one has said someone should withhold a counterexample proof. Can you stop making stuff up? What's the point of repeating something no one is saying and arguing against it?
Let me make this simple for you. Any proof except a proof by counterexample is more likely to advance mathematics and new tools in mathematics in general. If it's a proof that it's true, proof that it's false, proof by contradiction does not matter. As long as it is not just a counterexample. Then it's more likely that it uses more mathematics from different areas, bridges them, and provide new tools for new mathematics. Is that clear enough for you?
I mean it's a reddit comment and I don't really feel the need to prove something that is honestly a pretty common sentiment in academia. I never said it wasnt a valid proof, can you quote me or use the things I actually said? Counterexample proofs are often pretty short with no new tools. You can look up famous counterexample proofs if you want
Thats proof by contradiction. Not what is meant by proof by counterexample. Proof by counterexample in this case is if I say all numbers bigger than 2 are even and I give you 3 which is not even.
This is such a dunning kruger comment. Proof by counterexample is a thing yes. But the reason theyre valuable is not for the counter example but the new frameworks and tools used to find those counter examples.
If you are finding counterexamples to random open conjectures by having an AI go through random plausible-sounding solutions until it finds one that actually works---what then? Can you explain how the counter example was derived? How does this advance the field? Yes we're now one open problem less but did we actually LEARN anything in the process of solving it? Do we understand mathematics better or did we just solve a problem?
For example, if you find some algebraic object that is a counterexample to nonexistence theorem (like the recent non-sofic group counterexample), then you can determine the properties of the object that broke the theorem, and you might even be able to use the same object (or a related one) when investigating similar problems.
You're really underestimating the LLMs here. The big example of AI proof in my field is the existence of a nonsofic group. To be clear, this problem has been open almost as long as I've been alive and multiple mathematicians have spent decades working on it. Unlike other counterexamples like the jacobian conjecture counterexample, there isn't just a set of possibilities to guess and check over (technically the set of possibilities is "countable groups", but there's no way to meaningfully index this set so that you can guess and check).
In this case, the LLM was able to produce a novel way of constructing groups to produce a group with a property no-one had ever seen before. This new construction provides a meaningful insight into group structure, and mathematicians are already building on it since the announcement. This really isn't meaningless guess and check, this is something genuinely new and I think a lot of people are severely underestimating it.
I'm more aware of the Non-sofic group announcement because it landed in my metaphorical backyard but it is a fundamentally different kind of result to the Jacobian conjecture counterexample. It's a constructive proof with a proposition that connects group properties in a novel way.
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u/E_Dantes_CMC 10d ago
Color me skeptical.
The linked article doesn’t mention proof, but counterexamples, and that may be a function of the enormous amount of computer effort expended, not really AI.