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.
We all believe it. It's a magic oracle. Now what?
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.
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.
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.
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.
For what? 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.
[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.
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.
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.
In any case, I think it's better to read PP as somebody would find a practical, albeit incomprehensible, algorithm for solving NP complete problems.
Although I probably disagree with PP, because even a candidate algorithm that mysteriously works without proof would have practical value, so this case is not predicated on proving.
I think a better example of genuinely practical but rather uninteresting (YMMV) mathematical proofs are proofs of convergence of numerical methods, FEM for example. (I have been through it in school, it was a torture.)
> Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm.
OK, I tell you that P=NP, and that I am a magic oracle. So, you have you psychological boost for finding a practical algorithm for free. :-)
a) I am already convinced that P=NP
b) You have to convince many other people as well (that you're a magic oracle), because for the effect to work, lot of people would have to work on the problem (or at least spend tokens)
Nevertheless, a plausible magic oracle (such as Lean-verified proof, even if non-constructive and incomprehensible for humans) would convince many to take a 2nd look.
It would not not necessarily be practical, even if it ran in polynomial time. It may have cost O(n^c), with a totally out of order exponent like c=A(5,5) or whatever.
Somebody, eventually, somewhere would understand it, or at least aspire to understand it. And even if he doesn’t, what have they learned in the process? About themselves, about their environment? About failure? I would bet a lot. How useful then, can we say that it is, not because we can understand it, but because we can try? That is useful. This is about the journey. Sometimes the journey is the point.
This is like if math was fascist, this is what would happen. When you start controlling the flow of knowledge like this, it will be bad news all around. And who is to say whether or not something can be understood?Aside from the math nazis.
If we are going to dictate what gets published like this, why bother publishing anything? This feels like a gatekeeping…that’s exactly what it is. Ya’ll getting nervous?
We only compute with two kinds of things:
- small data; or,
- extremely lower power and coefficient algorithms
We lack the power to, eg, use a quintic algorithm in anything but nearly trivial cases.
Furthermore, careful analysis of the latter would as likely as not yield further understanding and, actually /would/ help finding such algorithms.
Finally, it has been observed time and time again that often (again, nothing comes up and i don’t want to ask AI) the certainty that something is possible and has been done is motivation and inspiration enough for people to independently solve a problem. Sometimes it is even enough for someone new to simply not know that something is “hard” to solve.
It even “motivates” llms, it seems (eg https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98...)
Of course this is all pure speculation concerning a hypothetical proof that most likely doesn’t exist, or indeed might be so complicated as to not be approachable even after hundreds of lifetimes of study.
Nevertheless your conclusion does not follow from the premise
You would need an algorithm that finds solutions, not just a proof they exist. So the value here would almost entirely come from how you proved p = np, since that proof will probably be the first step towards finding the polynomial solutions. But if humans don't understand it good luck finding any.
Is Amazon still delivering food to your cat?
Humans don't need to understand what AI generates. We still can get the rewards.
In reality, it wouldn't depend on the truth value of the statement, but on the AI understanding the proof. If it understands it then it might be able to use it to find reductions.
So the point remains, knowing that P=NP isn't what's important, it's the proof that matters.
Imagine if humans couldn't understand multivariable calculus, but we had access to an AI system that developed it, it initially seemed useless, then another AI system found a predictive model of electromagnetism using it.
When you have full AGI of course you no longer need humans to understand math.
> Imagine if humans couldn't understand multivariable calculus, but we had access to an AI system that developed it
Developing multivariable calculus requires much more than just solving problems though, it requires defining an entirely new system and space. That is not the situation mathematicians face today, modern AI cannot do that.
When talking about mathematicians and AI don't use fictive examples, we can look at what AI can do today and extrapolate that they can do more of that tomorrow, that is what we have to work with.
In the case you posit where AGI exists there is no reason to even discuss what is left for humans to do, since AGI is defined as when humans are no longer needed for anything, the AGI can do every bit of thinking humans can.
Anyway, if we put the AGI framing aside, I think the main point you're making is that AI mathematics hasn't yet demonstrated the ability to theory-build in the way that the great human mathematicians have (Grothendieck, Scholze, etc.). And I'd agree with you on that. Where we disagree, I suppose, is I think that capability is coming -- I don't see anything that would prevent its development.
The idea of AI stepping from a graph theory/combinatorics innovation to some new and useful algorithm isn't crazy.
If I understand you correctly, you're just qualifying that that will only be the case when AGI exists. To be clear, I actually disagree with you here because I think it's very plausible to find a use case for human-incomprehensible math proofs before AGI exists. I'm just saying it sounds like you're agreeing with the parent comment that math is not purely about human comprehension.
Memorized proof patterns have value because they lead you to a final proof.
https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture...
IIRC he has expressed support in the past for attempts to formalize IUT in Lean, but we'll see where that really goes, because he's absolutely not clearheaded enough to lead such a project himself.
In history, we made much more use of hitting things with bows than abstractly comprehending arrow flight.
That said, without understanding, Math can't evolve. Comprehension of a proof is very important, but not what Math is fundamentally about.
Computer programs are Math. You can use them without understanding how they work.
Secondly, there is no truly objective truth to the area of a triangle. At bottom, this “truth” is simply “everyone is convinced, and for good reason”.
Without persuading other people of the “truths” that you discover, there is no real mathematics.
Why? If I sat around and studied math by myself and discovered something true yet not yet known but didn't share it, it's still true. Are you saying I didn't "do math" because I didn't share the result? Math exists on another plane and it has 'truths' that we haven't discovered, yet are still 'true', no?
The area of a triangle doesn't have 1 unique formula, it has many, depending on the system you use. A triangle in plane geometry has a different area than a triangle in spherical geometry, and different again in hyperbolic geometry.
When you get to studying the geometry of manifolds, you realize the area of a triangle can be any damn thing you want, depending on how you construct the manifold you embed it in.
A proof can just be "assuming these axioms.....the area of a triangle is X"
Math has been almost purely arbitrary since ~ late 19th/early 20th century. There are uncountably many correct mathematical theorems. Almost all of them can't even be written in symbols. Even if you have a tape with infinite length (which is already longer than the whole physical universe!) filled with theorems, they are still only 0% of all correct theorems. That's how arbitrary math is.
> The area of a triangle
Yes, even this is arbitrary. The rigorous definition of triangle is arbitrary. People just subconsciously choose something that vaguely approximates their physical intuition.
You seem to be vaguely waving in the general direction of a point, without making any concrete claims or bothering to engage with the GP’s argument.
The tastes and interests of humans are absolutely not arbitrary. They are dictated by fate, the sun and the moon gods. Or maybe by the unitary evolution of the universe’s quantum state. Or by the probability distribution of finding ourselves in a particular branch of the universe.
What bearing does this have on whether math is a collaborative endeavor?
And what is unique to math, that your argument wouldn’t apply equally to physics, sociology or financial markets? All, “truth seeking” disciplines.
Nonetheless, the person writes, “ Math has been almost purely arbitrary”.
This is simply a misusage of the word arbitrary, which is a word with a specific meaning you can look up if you are unaware, since mathematics is (obviously) not arbitrary in the sense this person wants to convey, in part for the reasons I state. Humans are not choosing arbitrary logical statements to prove true or false.
What are you talking about? A theorem is a statement that has been proved from some axioms. A statement itself is a finite sequence of symbols satisfying some syntactical rules. The set of symbols for set theory, arithmetic, etc is finite, so the set of statements, and a fortiori the set of theorems, is at most countable. Where do you get the uncountability?
I suspect you are confusing theorems and theories. Assuming the set theory ZF (for instance) is consistent, then a consequence of Gödel's incompleteness theorem is that there are indeed uncountably many inequivalent extensions of ZF that are complete and consistent. Also, not a single one of these extensions can be described in symbols in the sense that there does not exist a computer program that enumerates a possible set of axioms for the extension.
As for your general point about arbitrariness, the late 19th century was a period when mathematicians started being concerned with the rigorous formalization of mathematics. Sure, there are some arbitrariness in the particular choice of formalization in the same way that the particular form of a programming language like C is arbitrary. However, the Gödel stuff has nothing to do with that arbitrariness, it is about the limit of formalization itself. The programming analogue is the undecidability of the halting problem. Saying that mathematics are arbitrary sounds to me a bit like saying that an algorithm like Quicksort is arbitrary because you saw an implementation in C and the particular form of the C language is arbitrary. Obviously, if you don't like the C language, you can implement Quicksort in another language. The same is true for mathematics. If some day, somebody finds a contradiction in ZF, or simply a new formalization that people find more convenient, then most mathematics will simply get translated and very little will change.
Whatever philosophy you prefer, math is about establishing objectively valid logical results, completely independent of the human process used to arrive at them.
Mathematicians even argue which axioms we should have, it isn't objective in the slightest, mathematics is therefore very closely linked to our feelings and intuition. Remove that and you just have formal logic, a very different field.
From my perspective, I feel you restated what I said with the opposite conclusion. You say that axioms "defines" mathematics. If I were Claude, I'd say that the word "define" is doing a lot of work, is load bearing or something like that.
"Define" is where we turn these axioms into consequences - what I call "truth". As opposed to all the other stuff people could say that don't follow from these axioms. These are nonsense and, most certainly, un-mathematical.
it starts as a tool for humans, then evolves into a set of interesting properties of those tools, then grows into an art form, a set of "games" where cooperation is half of the point. the other half is discovering beauty in this weird parallel world of our reasoning and imagination. once tools become autonomous and start making up their own games we can't even play then mathematics loses it's meaning as a discipline. the only retort you can come up with is that "it's going to be useful". how would you know? because your autonomous tool that's too smart for you told you so? they could be as useful as morning orange juice to Claude Shannon was in terms of inventing information theory. I.e. you drinking it won't make you any closer to inventing anything of the sort anytime in your lifetime.
do triangles exist IRL? is the world discrete or continuous? can you prove it? If you have an answer to all of those I know you're wrong.
also in your computer program example just shows you don't understand it at all. those programs ARE NOT understood by you, but someone else who built them did. someone who bothered to read and architect it did. The whole Google codebase might be incomprehensible in its totality if you go bottom up but it is comprehensible by construction by us. Same with math. You don't understand all the bits of it, but someone built every brick and so you know it is "true". once the bricks become black boxes you're screwed.