Higher level languages are still formal languages. I think there's a conceptual difference between moving from one formal language to another (machine instructions to asm or asm to C) and moving from a formal language to natural language. So yes, developing, looking at and understanding a formal description of your system has benefits for an engineer compared to handing off this step completely.
As someone who also dabbles in writing fiction (unpublished) I'm less pessimistic. I keep hearing less people read and the end of long form writing is near. To me this signals a great time to write and publish because the smaller the niche, the happier the people who get quality stuff in it.
The "book bubble" of avid readers is the most anti-AI community I am a member of which is a very interesting perspective (I do research in AI as my main job). I don't touch AI for writing/editing etc. because I see it as a creative endeavor and enjoy spending time with a blank piece of paper or an empty latex file :D
The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not sure. Especially for counterexamples LLM solutions seem fine. They stop humans from wasting time on pointless things. For proofs it gets more hairy but I think if it is formally verified a proof is a proof. Attribution is a problem (should the person who wrangled the answer out of an LLM get the credit, I guess so).
I think these are non-trivial epistemology and science theory problems.
I don't think this is a valid counterpoint at all. Math is cooperative, and comprehension is the point: the proof has value exactly because (and only to that extent) it empowers humans to understand an abstract truth. Chess is competitive: the memorized line has value because it makes you incrementally more likely to defeat your opponent.
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.
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.
> 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.
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?
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.
[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.
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.
Oh it would change a lot. It would be an enormous psychological boost for everyone to find a practical algorithm.
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.)
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.
> a practical, albeit incomprehensible, algorithm for solving NP complete problems.
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.
I thought the point of publishing was a matter of dissemination, to make available for people to then try to understand it? This is like saying, I don’t like music, so I’m going to tear down the venue. Then, all music genres suffer as a result, and all of society does, too. This idea is no good.
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?
Both are right. Practically speaking your view is right, and is similar to one of hilberts quest to come up with a proof spitting machine. Just get a computer to enumerate through all proofs and we absorb the results. But you have to agree this is deeply dissatisfying intellectually. This is like if trigonometry was discovered with no relation to circles and triangles but just as a series of look up tables (like in a calculator) and we just know it works for certain scenarios and that's all there is to it.
I don’t recall anything specific off the top of my head but I am confident that such a proof would have immediate actionable implications.
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.
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
A proof that they are the same is of no use either, since it too wouldn't help you find algorithms that are faster.
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.
Well then, just tell the AI that the statement is true and it will find reductions!
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.
Why wouldn't you be able to do that without a proof? I don't see the value of the proof here, just ask the AI to solve the problem you want and the proof isn't needed.
Math that humans don't understand but nonetheless allows AI systems to develop breakthroughs in various fields of science, technology, physics, engineering, medicine, etc., would have great value to humanity even if it doesn't help humans understand abstract truth at all.
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.
But at that point you have full AGI and its not just today's models. Today's models still need humans to understand things since it builds upon human knowledge.
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.
I don't find AGI to be a useful technical term, as nobody can agree on what it means. For instance, you used it at least five times here, but you never defined it, and I could point to intellectually credible people who would say we've already reached AGI.
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.
So then it sounds like you agree that math has additional utility beyond just human comprehension.
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.
The goals of Chess and Math may be different, but they follow the same principle of exploration large search space according to fixed rules. In case of Chess these are chess rules, in case of Math these rules of mathematical logic.
Memorized proof patterns have value because they lead you to a final proof.
I hope I'm remembering this right: a mathematician claims to have a proof for the ABC conjecture, but can't conceive any other mathematician it's right — it's "too weird", so the proof is rejected?
The consensus is that proof is in fact incorrect. People tried really hard (like putting in a year of effort) and most converged to the same place, that proof of 3.12 is incorrect or has a gap. Peter Scholze (who won Fields Medal) and Jakob Stix did a writeup. People seem to think Shinichi Mochizuki correctly reduced ABC conjecture to 3.12, but didn't prove 3.12, and also are doubtful about the whole program because 3.12 doesn't seem any easier than ABC conjecture while complicating everything.
Not quite?
It is more that
1) someone has gone through it, identified a step he thinks isn’t a valid step, and the author hasn’t been willing to work with that person
2) most consider the proof, due to its length combined with those doubts as to its validity, not worth their time and effort to work through and understand (because it would take a lot of time, and they have jobs to do, doing research and teaching, etc.)
Some of the best mathematicians in the world tried to study his work, found flaws he did not address, and somehow there’s someone every week suggesting there’s a conspiracy against this guy. It’s really baffling. AI will probably help him move on by lean verifying his proof is wrong…
By this point, he is very much nutso enough that a Lean certified counterexample to his theories would not dissuade him. His response would be either that the formalization is incorrect (with no coherent insights on how to fix it), or worse, Lean itself is a tool of Western imperialism and incapable of properly explicating his ideas. He has, in the past, ranted against such things as monotheism and English grammar as being the reason for his theories' lack of popularity.
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.
Yes, this. Comprehension is the point. We could map this to something like physics. If a man on a horse can shoot another man with a bow, empirically he makes correct predictions on gravity, wind and relative motion. But he can’t explain it. It’s not any different if your model has some “embodied” or demonstrable understanding; the model is not part of the discourse.
Math isn't "cooperative". Math is about truths. The length of circumference. The area of a triangle. The formulas for these are true in an objective sense irrespective of whether you understand them.
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.
First of all, mathematics is about so much more than the area of a triangle etc. that any analogy based on such simple things is overwhelmingly likely to be too simple to be of value.
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.
> 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?
No, they're saying that what is true in mathematics is contingent, not absolute. It all depends on which set of axioms use, what assumptions you make.
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.
Math is not about truths, at least not by the meaning of "truth" as a word in daily use.
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.
You seem rather aggressive so replying to you feels pointless and uncomfortable.
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.
> There are uncountably many correct mathematical theorems. Almost all of them can't even be written in symbols.
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.
I think Hardy would be very much on the same page with me, as well as Godel and many others. Mathematical truths exist independently from our feelings and processes to discover them. People do and should argue about which truths are interesting to pursue and refine, but all of them are out there to be discovered... or not.
Whatever philosophy you prefer, math is about establishing objectively valid logical results, completely independent of the human process used to arrive at them.
Axioms don't exist independently from our feelings and processes, we pick axioms we feel are good, and axioms defines mathematics.
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.
Is formal (aka "mathematical") logic part of mathematics? Now that's a philosophical question.
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.
this is primitive understanding of math. what does "truth" mean here? usually arguing over definitions is something i hate, but that's the whole point of mathematics.
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.
I don’t think it’s pointless to spend time trying to prove a conjecture which is ultimately false if along the way you figure out a bunch of different true variations on the conjecture, which is how mathematics actually works. This is something I’m a bit worried about with LLMs since it gets you to the end too fast.
LLMs seem to have worse intuition than experts and compensate by being able to cover a much wider surface area of ideas, so we might just need to extract the intermediate progress along the way.
I can almost see two branches of mathematics developing. One which is human-understandable, the other formally verified. I assume the latter is a strict superset of the former?
Presumably there's not much logical obstruction to all human-understandable math eventually being formalized, although the willingness and ability to commit the requisite enormous amount of time will probably be insurmountable. But definitely that hasn't happened already!
I suggest "Catching crumbs from the table" by Ted Chiang. Very short piece published in Nature (2000) and well worth a read. Depicts a scenario where modified humans produce science beyond ordinary scientists' comprehension.
This is a theme in Blindsight by Peter Watts as well.
In that setting, field experts working at the bleeding edge are so advanced that non-experts literally can't understand what they're saying at all. So there's a whole class of specialists, "synthesists", that specialize in gaining approximate understanding of the experts' work for the purpose of communicating it to outsiders—perhaps wrongly, according to the expert at least, but hopefully more productively vs the unmediated version.
What's amusing to me in this context is, summarizing emails and such has for a while been a supposed use case for AI—the LLM serving as the "synthesist" to explain long texts accessibly. But with this math question, a human "synthesist" would be needed to approximately understand the math discovered and programmatically verified by the LLM. So the roles reverse.
If the proof is formally verified but impossible to understand how would anyone be able to be sure the formal verification is correct? Complex software is bound to have bugs, no?
The whole point of Lean is that you don't need to understand the entire proof to be sure that it's correct. You only need to understand the definition of the theorem being proven, and you need to trust that the relatively small core of Lean is correct.
> In July 2026, a disproof of the Collatz conjecture was verified not only by Lean, but another formal verification system Nanoda. However, investigation quickly revealed that the proof exploited bug(s) in these verifiers.
Why should you trust that the relatively small core of Lean is correct?
The core of Lean got a lot less correct when a well-meaning AI system probed Lean for corner cases (bugs) that would "prove" a false conjecture. Corner cases so arcane that no human exploit in a proof. Basically, humans are too stupid to break human-created Lean, but the AI is not.
Lean does have libraries, but since they are also in lean they are subject to the same rules. It's basically a super strong type checker. If it compiles the proof is valid. Unless there is a bug in the type checker.
My time proving things is long in the past and any systems way back when I was studying (some math among other things) certainly were different and usually quite narrow.
My point was rather more motivated by having seen so many weird ways for machines to fail/not work as expected that I wonder how to deal with that if the output were to be incomprehensible to humans.
This was me as well. +HCU and searchlores.org were so much fun. The coolest thing I remember was the hash-maze but I couldn't find it again. I saw Fravia live in 1999 :D
Switching 0x74 to 0x75 and vise versa lead to me ordering the free Intel books and diving into assembly and osdev for a while. There was something zen-like wading through disassembled code with a complete beginner mind and looking for patterns. This and reading Phrack around the "Smashing the Stack for Fun and Profit" times are probably my favorite memories related to computing because I only had a very vague understanding of what was going on mixed with a sense of "wait you can do this" and a lack of readily available information.
Where does this supposed saying come from? All I hear is PostgreSQL is the database intelligent people use, Git or GTFO and I don't even know what a closed source programming language is. Apache and nginx, Linux, DNS infrastructure.
In an agentic world, why do I want closed source software that my agent can't adjust and adapt to my needs for...anything?
To be clear, the saying is about how companies treat open source, not the intrinsic value of open source to end users (individuals or companies). I'm very much a fan. The point is that profit-seeking only push for open source when they're losing.
Agreed. The author in question (Linnaeus) was from Sweden and professor at a Swedish university (Uppsala). Latin was used across the board in "traditional" sciences in northern Europe at that time and in fact the distinction of disciplines was not as clear as it is today. Linnaeus was a professor of medicine formally but focused on botany. He traveled a lot and had many international "contacts", especially in the Netherlands. His correspondence was usually a mix of Swedish and Latin (which I find interesting because I assume Swedish was not widely studied at the time) but his professional publications were in Latin.
But it was an interesting time as slowly native languages became more common. The chair for economic sciences at Uppsala was specifically to be taught in Swedish (1741).
Southern and central Europe were ahead of the curve a bit and it was more common to see French, English and German for example (French and English scientific journals date back to the 1660s).
I'd also recommend "Scientific Babel How Science Was Done Before and After Global English" by Michael D. Gordin [0]. It goes into detail about which languages scientists have used over time.
Even speakers of more common languages like English used Latin for science. Newton wrote the Principia in Latin and William Harvey wrote De Motu Cordis (the book that showed that the heart was a pump that circulated blood) in Latin, for example.
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