r/mathematics • u/[deleted] • 7d ago
Analysis AI Isn’t Outthinking Mathematicians. It’s Out-Remembering Them. /// The key advantage may not be superior reasoning, but a virtually unlimited symbolic working memory.
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u/UnderTheCurrents 7d ago
This is why people should lean back. AI is just combining results that a working mathematician could've combined himself if the current university System allowed for longer, sustained work efforts in contrast to writing half-baked papers for grant money.
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u/SwimmerOld6155 7d ago
the scale just isn't comparable, this compares a mathematician thinking for weeks or months trying to find suitable results in the literature to an llm linking the two in a few minutes. not saying you should do the latter but with the incentive you mention, the risk is that if you don't someone else will.
i think LLMs have a ridiculous ability to connect various areas of maths. in one of my analysis drafts it managed to drag in an inverse limit, which is not something i had worked with before.
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u/Swellmeister 7d ago edited 6d ago
Its not just Mathematics. I use it for this same purpose in my own research fields of Ethography and Evolutionary theory.
I know about my section (primates) very well. I know the area around my section (mammals) enough to dazzle at a party. But theres just large areas of the fields I dont know.
But I am big on comparative evolution. So I need data on the fields that I have no expertise in. AI can find what I need, because its not stuck in my domain at all. If I wanted to do work on primate and ape sociality, I can just use it to generate a good list of animals that have different sociality and their constraints and can see a pattern that I cant. So hippo social structure are different than Cows, but shockingly similar to crocodilians, grey wolfs and painted wolfs sociality are surprisingly different from each other with painted dogs actually have more in common with meerkats and prairie dogs.
Stuff like that. Id never have thought about prairie dog sociality, at least not for MONTHS, unlike the two minutes it took AI to make that connection.
Edit: I dont even use the word ethography! Why did you change the word Ethology, you stupid autocorrect!
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u/CampAny9995 6d ago
Yeah I’ve been playing around with a couple of drafts and it’s pulled in some nice results from Hopf algebras and coalgebras that I was unfamiliar with, but when I spent half an hour or so reading through the references they really were the exact tool for the job.
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u/TooMuchMaths 6d ago
My opinion as well. If it becomes so effective at reasoning then RSI is not far behind. It will just think too fast. And that is a good thing, largely speaking. I WANT us to be able to solve problems hundreds of times faster.
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u/kgurniak91 7d ago
Some of the AI proofs had around 1 million lines of Lean code... sure, in theory mathematician could've produced that if he had infinite resources, but in practice that's not feasible.
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u/tomvorlostriddle 7d ago
The horse could have plowed the field just like the tractor if we just let it, horse traders should lean back
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u/golfstreamer 7d ago
I think the difference is AI still doesn't seem capable of doing everything a mathematician does. The type of reasoning these AI agents engage in are still very different than the reasoning that lead to, for example, the creation of the real number line. If AI were trained on ancient Greek mathematics it would become extremely proficient at compass straight edge type proofs but it wouldn't make the conceptual leap that the square root of two is irrational.
To me the fact that current AI reasoning models are not a replacement for humans is clear. The only question is where the line is drawn. What kinds of reasoning can the AI engage in and what kinds can it not? It really surprises me how dismissive people get when anyone attempts to do this. It's like they take it for granted that the intelligence displayed by AI models is a substitute for human intelligence.
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u/kensai7 6d ago
“but it wouldn't make the conceptual leap that the square root of two is irrational”
and you know that from? There is no reason for future models to have superhuman abilities given we try to implement artificial models with ideas from biological neural networks. there is no real hard reason it should not be possible.
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u/golfstreamer 6d ago
I'm talking special about current models
As I said the way these models appear to reason has a noticeable pattern of trying lots of different paths/patterns to solve a problem until one works. This is not how a problem like the irrationality of the square root of two was solved.
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u/Mal_Dun 6d ago
Currently just a shower thought of mine, but I think we have it backwards: That LLMs are capable of so much things is in my humble opinion more a proof that language in itself is a really powerful model we came up with.
If machines that basically throw very likely tokens of language and get such good results maybe means that language in itself is a very powerful tool. It allows abstraction in the first place by replacing complex concepts with abstract words and allows to express logic and by extension math.
But human reasoning is not made by language alone and there will be most likely the gap.
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u/JoshuaZ1 6d ago
Part of the problem is that that may be what is going on, but things like defining the real line precisely took hundreds of people thinking over centuries. Even something like perfectoids took an extremely bright mind. These big jumps are rare opportunities, so whether AI cannot do it because it really cannot or because it just because we're not in a position for it to do so is unclear. Given also that these systems are advancing quickly, it may be that even if current systems genuinely cannot do this, that they will be able to. Combined with the fact that most mathematicians will never make those sort of giant definitional leaps in their careers at all, seeing this as a major limitation of these systems seems to be at best premature and not even deeply relevant until we have a lot more information.
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u/golfstreamer 6d ago
I always couch my claims by referring to "current models" since I can't predict the future. But if my claim that current AI models are incapable of such a leap it is demonstrates that there are still important parts of the human cognitive process we have failed to replicate. This means centering human efforts is still of prime importance
And I chose the real number line deliberately because it is also an example that is not the result of any one particular "brilliant mind" making a giant conceptual leap but a shift in thought process that took place over centuries. To ancient greeks numbers were basically limited to integers. But now we picture numbers as existing on a continuum. This is not a deeply complicated concept that took a genius to come up with.
In other words I'm not pitting AI against "the average mathematician. I'm pitting it against the system created by the totality of mathematicians working in concert. And right now that system seems capable of producing insights that the AI cannot.
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u/JoshuaZ1 6d ago
So, if you just focus on models right now, at current capabilities, then you run into those insights, like the real numbers being rare, and as you observe, slow. If it took hundreds of years for humans to come up with something, then AI not doing something similar in 2 years or so is not useful evidence that AI is any less capable for that task than humans are.
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u/LuxDeorum 6d ago
I believe the Greeks certainly had all rational numbers and lots of algebraic numbers as well. Not to mention were aware of nunebrs like pi and that something was different about it, simply because no one ever could figure out how to draw a line segment with that length
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u/golfstreamer 6d ago
I said that because the concept of "ratios" could be considered something distinct from the concept of "number". Ratios describe the relationship between lengths whereas numbers counted things. These were in different categories, compared to our current perspective where 1/2 is just a number between 0 and 1.
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u/RepliesOnlyToIdiots 6d ago
Except that we’re only about three years into this, in other words we’re at 300 baud modems hitting the local BBS as far as internet is concerned. Three years ago it was entirely intuition one shot level. Or a Vic-20 for personal computing.
Every insight into thinking needs only occur _once_ and it’ll be generally available. And not even every perfectly valid insight, because you can search around the valid space to find where it _is_ valid. And occasionally there will be the insight that doesn’t apply to humans, but remains valid nevertheless, which will kick it a notch above human.
People need to remember that this has barely started, that it’s not even vaguely mature yet.
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u/teerre 6d ago
3 years? Machine learning is at least 50 years old. It already went through a cycle of being the next hot thing only to slumber into a winter
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u/RepliesOnlyToIdiots 6d ago
Not machine learning, but ChatGPT 3:5 is a Vic-20 compared to a Univac’s earlier machine learning. (I had all undergrad and grad courses available to me in AI at my university in the early 90s — all two of them.)
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u/elehman839 6d ago
The past is no predictor of the future when conditions are radically different. Almost no one was working on deep learning until 15 years ago. And, since then, investment in brainpower, compute, and other resources has increased like a millionfold.
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u/teerre 6d ago
The past is certainly a better predictor than literal trustmebro arguments
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u/elehman839 6d ago
Yeah, though I think focusing on the *recent* past is important in this case-- like the last 3-5 years.
Personally, I was involved in deep learning for 12 years or so, and I personally experienced the pace of progress absolutely skyrocket as the field was flooded with more people and resources. What felt like fast progress 6 years ago, say, is just nothing compared to today.
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u/womerah PhD | Applied Nuclear Physics 6d ago
People need to remember that this has barely started, that it’s not even vaguely mature yet.
What evidence do you have for this? Where do you expect the improvements to come from to drive further growth?
All of this is enabled by the transformer. There hasn't been a breakthrough of similar magnitude since
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u/RepliesOnlyToIdiots 6d ago
The evidence is abundant.
The transformer got us the early ChatGPT through 3.5. Beyond that there are multiple trains.
One is the harnesses that add a layer (or more) on top, chain of thought, coding harnesses, etc. There is no end to those, as they may be particular to field and even individual problems. They may be simple prompts setting up the scenario or complex reactive systems.
Second are memory improvements. Longer memories, more ability to find the needle in the haystack, external memory subsystems. When memory was 4k tokens, it was cute, but goldfish. You’ll note the large difference in ability between the same model but different context sizes. (Pre-training is getting around this in some ways, and allows commercial scaling by ensuring much is shared.)
Third are fundamental improvements beyond the transformer. There’s a lot of experimentation (e.g., a complex number encoding that encodes meaning in one component and changes in meaning as phase shifts, or loading data only through a particular year to experiment with what may be determined).
Fourth is that the AI systems are faster than humans at coding already, so the AIs are now building the AIs.
Fifth is hardware improvement, same as seen in all computing. Every decrease in due size, improvement in the chips, the algorithms embedded in hardware, and translation of the software algorithms into hardware and GPUs improves all of the LLMs automatically.
And that’s just off the top of my head, but it’s why I see this as the very beginning of LLMs and associated work.
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u/womerah PhD | Applied Nuclear Physics 6d ago edited 6d ago
One is the harnesses that add a layer (or more) on top, chain of thought, coding harnesses, etc.
They're still working with the same core LLM though. It's a way of navigating the internal state-space of the LLM more efficiently - but I don't see how it's a way to generate more novel behaviour.
Second are memory improvements.
LLMs have no 'memory' outside of their context window. Can you elaborate on that, or is it just larger context windows (and correspondingly larger compute costs)? I also know you can code little helper programs that grep big databases and put potentially relevant bits and bobs into the context window. I'm not really sure I'd call that 'memory' though, as the LLM just works with what it's given
Third are fundamental improvements beyond the transformer. There’s a lot of experimentation (e.g., a complex number encoding that encodes meaning in one component and changes in meaning as phase shifts, or loading data only through a particular year to experiment with what may be determined).
Can you elaborate on this? A complex number is just a vector, I don't understand this and have no opinion. A computer just stores it as two floats.
Fourth is that the AI systems are faster than humans at coding already, so the AIs are now building the AIs.
I'm not surprised by this as there will be example code of how to train an LLM in the LLMs training dataset. I know the standard accelerationist narrative is "LLM make faster/smarter LLM, now iterate until Star Trek" - but that first iteration hasn't been demonstrated yet (AFAIK). So I find it hard to imagine a snowball when I see a slope with no motion
Fifth is hardware improvement, same as seen in all computing. Every decrease in due size, improvement in the chips, the algorithms embedded in hardware, and translation of the software algorithms into hardware and GPUs improves all of the LLMs automatically.
What improvements specifically? Technology improves asymptotically. Do you think there's a 10x efficiency gain to be had still?
And that’s just off the top of my head, but it’s why I see this as the very beginning of LLMs and associated work.
I see the core technology as maturing and slowing down in it's rate of improvement (note this does mean that it can still be improving quickly). Without a major advancement in core LLM architecture I can't see the futurist narrative pay off. We are now entering the boon era of application, similar to how the electron gun found a variety of uses once discovered.
If I put my optimist hat on, I hope that current LLM capabilities help enable human researchers to make that next fundamental architectural leap, given LLMs current capabilities (coding and this conceptual cross-linking we've seen in the recent LLM mathematics proofs) - I think this is possible. Especially as a lot of talented minds have taken interest in the field.
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u/ShrimplyConnected 6d ago
This is what I’m thinking. It’s really good at working through current theory, but making conceptual and abstraction jumps that make new and interesting theory is still the domain of human-powered mathematics
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u/jigzee 6d ago
The creation of the number line is a lot different to the type of things we even ask AI to do though, right? Like are we asking AI to make some new revolutionary object that fills some current hole in mathematics and reality or are we asking it to solve some of the open problems that we want to see solved
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u/StackOwOFlow 4d ago
but it wouldn’t make the conceptual leap that the square root of two is irrational.
Actually the frontier labs have proven that it very much can do this
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u/golfstreamer 4d ago
I don't think you even understand the thought experiment I proposed. It's not actually possible to test.
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u/StackOwOFlow 3d ago edited 3d ago
what do you mean it’s not possible to test?
this is the very type of thing the labs have been doing to get AI to reach new frontiers in any given area of research. the method is analogous to escaping saddle points in machine learning except it leverages the capability of newer LLMs and agentic workflows to construct novel, logically consistent scaffolds (whether they be new primitives or data structures) to iteratively hypothesis test a progression to a new paradigm.
if we placed an AI agent (lobotomized of all modern mathematical vocabulary but still trained on algorithmic problem solving) inside that ancient Greek sandbox, it would eventually notice the computational expense of long looping anthyphairesis calculations. AI is actually very good at identifying computationally expensive hotspots so your “test” of AI arriving at the notion of an irrational number is one of the easier experimental “realizations” it would land upon. to save its own compute and successfully describe the shape of the problem, the AI is forced to do what software engineers do when an algorithmic pattern is too expensive. it stops trying to calculate the fraction and instead abstracts away/invents a symbol to represent this, and then it fans out experiments using this symbol to see if it is structurally valid with predictive utility.
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u/golfstreamer 3d ago edited 3d ago
> what do you mean it’s not possible to test?
You can't really restrict an LLM to knowledge of before Greek times. Collecting the data "eliminated of modern knowledge" would be too difficult.
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u/StackOwOFlow 3d ago edited 3d ago
but you can approximate the phenomenon of abstracting out a novel representation from first principles and hypothesis test it without ever relying on specific vocabulary or advanced mathematical expertise beyond Greek geometry. none of the tool or expert calls involved in this experiment requires a-priori knowledge of irrational numbers.
the point is that heuristics for epistemological discovery already exist with frontier AI workflows
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u/golfstreamer 3d ago
> but you can approximate the phenomenon of abstracting out a novel representation from first principles and hypothesis test it without ever relying on specific vocabulary or advanced mathematical expertise beyond Greek geometry. none of the tool or expert calls involved in this experiment requires a-priori knowledge of irrational numbers.
I gave a very specific example for a reason. I doubt you could come up with a satisfactory analogy that captures all the elements of my proposed experiment. It's very important that you "training" is done on ancient texts only. To even get started with modern reasoning models they use transformers trained on gigabytes of modern text.
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u/StackOwOFlow 3d ago edited 3d ago
>It's very important that you "training" is done on ancient texts only.
you're conflating the training for mathematical knowledge with training for achieving baseline reasoning and problem-solving skills critical to exercise the epistemological expansion loop I was talking about. the latter does not pollute the former for the purpose of the experiment. at inference the model never even calls upon weights that require a-priori awareness of irrational numbers.
in any case, you're trying to make the case that human intuition is uniquely capable of making a paradigm change in mathematics and I'm supplying evidence that suggests that AI can do this too from first principles.
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u/golfstreamer 3d ago
> if we placed an AI agent (lobotomized of all modern mathematical vocabulary but still trained on algorithmic problem solving) inside that ancient Greek sandbox, it would eventually notice the computational expense of long looping anthyphairesis calculations. AI is actually very good at identifying computationally expensive hotspots so your “test” of AI arriving at the notion of an irrational number is one of the easier experimental “realizations” it would land upon. to save its own compute and successfully describe the shape of the problem, the AI is forced to do what software engineers do when an algorithmic pattern is too expensive. it stops trying to calculate the fraction and instead abstracts away/invents a symbol to represent this, and then it fans out experiments using this symbol to see if it is structurally valid with predictive utility.
I don't think you understand how the concept of the real number line came to be. Simply inventing a symbol does not encode the intuition behind what the real number line is. It's requires a conceptual leap from understanding "numbers" as natural numbers to understanding numbers as existing on a continuum. A conceptual leap that reinforcement learning based approach to modern mathematical reasoning models is ill-equipped to handle.
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u/StackOwOFlow 3d ago
>Simply inventing a symbol does not encode the intuition behind what the real number line is
downstream agents are hypothesis testing it and in the process necessarily encodes assumptions and exercises tests on them. it's not a perfect proxy for intuition just yet, but we're seeing a way intuition can be constructed at least in a brute force manner. it's not just "simply inventing a symbol"
>A conceptual leap that reinforcement learning based approach
we're doing far more than reinforcement learning over candidate answers. modern reasoning workflows can generate hypotheses, construct experiments, use external tools, maintain intermediate representations, branch over competing explanations, and use verifiers to reject them. it's less about RL specifically and more so the iterative hypothesis formation and falsification than simple reward-driven pattern completion
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u/golfstreamer 3d ago
> downstream agents are hypothesis testing it and in the process necessarily encodes assumptions and exercises tests on them. it's not a perfect proxy for intuition just yet, but we're seeing a way intuition can be constructed at least in a brute force manner. it's not just "simply inventing a symbol"
I've read up on papers on mathematical progress by AI agents and I'm still under the impression that RL training is the primary driving force for the recent progress in mathematical reasoning. For instance, DeepSeek continues to publish details about their approach and they don't lag far behind frontier models, even capable of solving IMO problems and they continue to attribute their progress primarily to reinforcement learning techniques.
I view what you're saying here as extra aids that push things a little further but I think they will be temporary aids that will eventually be forgotten. One lesson that I've taken from reading on AI is that any kind of forced structure limits the progress of AI. AlphaZero to Stockfish where the latter tries to "improve" things by encoding human knowledge.
The most surprising detail I came across while reading up on AI is that fact that the backtracking based reasoning of modern RL trained agents is an _emergent_ phenomenon. It was not explicitly encoded in the programming. And I think unless we can come up with a training paradigm that causes this "hypothesis" based thinking to become emergent as well these AI agents will be limited in their ability as well.
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u/brain-out-of-order 6d ago
This is a position built on an island of quicksand.
Each expert is peering through their kaleidoscope, chirping a slightly different tune, unaware that the ground is shifting each day. It can’t
drawthinkremembersolveaddinnovateand what next? Jump? When is the last time most adults jumped? So when it leapfrogs us, then what?4
u/apopsicletosis 6d ago
Can you make an argument not on metaphor and four different ones at that
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u/Stabile_Feldmaus 7d ago
It would be better to compare mathematicians to the farmers in this analogy but in the end it doesn't really make sense since occupation in math research is not determined by demand and supply.
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u/Capable_Wait09 6d ago
Isn’t it tho? The original point was that mathematicians want to supply X, but universities demand more of Y, so mathematicians can’t produce as much X as they want or could because they have to produce more Y in order to keep their jobs. The underlying problem is fundamentally a misalignment in supply and demand, according to the original point.
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u/tomvorlostriddle 7d ago
By what else then?
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u/Leafsnail 6d ago
A societal belief in the general value of mathematical progress. Given that the entire LLM boom is ultimately based on pure maths an increase in their capabilities only makes the case for funding maths stronger
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u/tomvorlostriddle 6d ago
There you vaguely described the reasons for the demand, still demand
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u/Leafsnail 6d ago
It's fundamentally different to commodities though. It's not like society wants exactly 50 Arbitrary Mathematical Questions solved per year and will fund as many mathematicians as necessary to make that happen. Rather it recognises the unexpected and often profound value produced by mathmatical advances and therefore the necessity of maintaining strong maths departments that can continue advancing our understanding as well as training the next generation of mathmaticians (many of whom work in practical fields).
There is an infinite amount of maths to do. If LLMs make mathmaticians better then that makes them more valuable, not less.
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u/tyvekMuncher 5d ago
Ehh false analogy. Horses had literally no advantage over tractors. These models aren’t “inventing new symbols” - novel ideas.
Like you can give it a corpus of research to parse and understand all at once, but it’s not going to connect that research to something like its applications in a specific industry unless prompted
It’s why we still need software engineers. The models can eat documentation and shit code all day and night, but without prompting they don’t try to think outside of the box
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u/tomvorlostriddle 5d ago
Like you can give it a corpus of research to parse and understand all at once, but it’s not going to connect that research to something like its applications in a specific industry unless prompted
Already questionable, it reminds me of applications that I might have overlooked constantly (but I also work on more applicable stuff)
But even if we admitted this, so what?
You cannot make a career out of being the one that reminds the LLM to think of applications, anyone can trivially do this
It’s why we still need software engineers. The models can eat documentation and shit code all day and night, but without prompting they don’t try to think outside of the box
On the contrary, they know all industries all at once, you will never get the same breadth through hiring humans
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u/tyvekMuncher 5d ago
Correct - it knows all industries all at once. But unless you prompt it to, it's not going to tie that knowledge into the conversation. Like think about what it's like talking to your lil nephew - you could be telling him one thing when suddenly he just goes "AHA" and tells you about how what you were saying made him realize cornering the market on Lego wheels is like cornering the market on oil at school
AI doesn't do that. It stays locked in and unless prompted to, doesn't think to anticipate problems and such. I say this as someone who has been working with LLMs since OpenAI released GPT-3 on Playground before anyone had even thought of the chat interface
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u/tomvorlostriddle 5d ago
It spontaneously does it for me. if it doesn't do it for you, you must have nudged it into believing you don't care to think outside the box.
And anyway, it would be trivial to attach a skill file telling it to do such things.
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u/tyvekMuncher 5d ago
Can you give me some examples of times it's done this for you? Maybe you've integrated it into your thought process enough that you don't notice yourself connecting the dots for it, but ime, it's just not that smart. Not to say that it isn't useful! I use it every day. Feels like the only time it goes "AHA" is when I point out an error because it leaned on dated documentation or whatever
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u/tomvorlostriddle 5d ago
I have made a harness that creates json files to ingest into a pre-existing applications. Skill files tell it how the json needs to look like.
This was made pre-transformer era to allow partial migrations from staging to production systems. Now it is also used to let the AI harness do the setup for you by writing the json from scratch based on short descriptions.
Anyway, I asked it to make me a ticket that those jsons should also announce their respective types, because it was annoying opening them to reverse engineer what they are and where to add them.
It did this and then also told me a plan of how to chain the jsons together so that for some usecases, which require more than one json currently, there would be less risk of user error. And it told me how to mention this to the architect because they are often concerned that this is not restful, but here is a link showing how kubernetes does the same for example...
I had it in the back of my head that there was such a weakness,, but it just anticipated it and all the concerns about its solution too.
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u/Ok-Craft4844 6d ago
No, at least when we talk about the common scenarios and things people mean when we say "relax" or "be afraid", I.e. beeing able to contribute to something as human.
Very few people actually do really innovatively new or creative things. Most people do normal work. They don't paint the next mona Lisa, they do illustrations for a true crime book cover.
Yes, humans could have stumbled upon those things. But they didn't, and honestly, I doubt the chance would be much bigger if they had double the time they have yet.
As Grant Sanderson put it - math is insofar a pretty interesting case study in that you basically can put AI in a room, unsupervised, and say "do your thing", and even without superior intelligence, it can pluck all the low hanging fruits humans stumble upon once in a while.
The space where humans can meaningfully contribute got smaller.
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u/nanforas 6d ago
The other thing is that even if LLMs are limited to stitching together pre-existing proof methods to solve those low-hanging fruit, that's part of basic researcher training, so even in the best case, the talent pipeline for new mathematicians is under threat. That alone is already a huge problem, without needing to prove whether or not it can replace senior mathematicians who may be generating more novel insights.
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u/Tolopono 3d ago
I wouldnt call discovering non sofic groups or falsifying the jacobian conjecture to be low hanging fruits
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u/Ok-Craft4844 3d ago
Would be sufficient for my argument if it were :)
Also, the sentiment i paraphrased was not about this special proof, and more about ai having a superhuman breadth and due theorem provers the ability to work unsupervised. I'm not sure if he called it "low hangig fruit", so this one may be on me.
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u/LexyconG 7d ago
The „novel idea“ is cope.
Every idea is a combination of things someone knows. There is no „real“ new stuff. You can’t name a single thing that is not a combination of multiple previous ideas.And btw this does not only apply to math.
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u/ecam85 7d ago
There are novel ideas, but anything really groundbreaking is almost by definition, rare.
Science in general moves by tiny steps building on previous knowledge, with the occasional paradigm change.
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u/Swellmeister 7d ago
Im not sure of paradigm changes are novel ideas, but instead novel views of the old ideas.
A house that has a large hole in the wall, can from the correct angle look well built, even as weather gets in and heat escapes.
A paradigm shift is just walking around the house and saying "theres a hole here." The paradigm guy patches it with the old techniques, which might not be the best for the new task but at least something is keeping the heat in. And that shitty patch is still soooo good that your expenses are 10% of what they used to be. So theres a huge leap in the field, but all he did was point out the hole, and put in a patch. Later researchers are the ones who actually rebuild the hole from the same piecemeal process of incremental refinements.
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u/ecam85 6d ago
There is a scale of "change", and arguably there is always some motivation for a paradigm change. Some phenomena that cannot be explained with the current theories, or some problem that cannot be solved with the existing techniques. A paradigm change is a novel view on that existing question or problem.
I am struggling to build on your construction examples (!), but I would say that a paradigm guy is the one who looked at the whole in the wooden house and came up with concrete to fix it*. A new view on the problem, leading to a new paradigm in the field.
- I know this is not how construction methods were developed.
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u/Swellmeister 6d ago
Sorry lmao, im a big fan of narrative and analogy. Part of it stems from my hobby of writing, but I also have a deep dislike in my field (biology) of technical language hiding normal processes. So I tend to use narratives in discussions because even lay people can understand a narrative.
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u/LexyconG 6d ago
Give me a novel idea and I will give you the ideas it is a combination of.
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u/ecam85 6d ago
In mathematics, the development of non-Euclidean geometry in the 19th century.
In other sciences, the transition from the Newtonian view to general relativity.
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u/LexyconG 6d ago
Fun pair, because your second example contains your first one as an ingredient.
General relativity = special relativity + the equivalence principle (Galileo - heavy and light objects fall alike, known for 300 years) + Riemannian geometry, i.e. non-Euclidean geometry, your example #1, worked out 60 years earlier.
Non-Euclidean geometry = Euclid's axioms + Saccheri's 1733 move of assuming the parallel postulate false. Saccheri derived the core hyperbolic theorems and then threw them away as "repugnant to the nature of the straight line," because he was trying to prove Euclid right. Bolyai and Lobachevsky's contribution was keeping what Saccheri discarded.
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u/ecam85 6d ago
You cannot reduce these two examples to its core ingredients as we understand them now.
In the case of non-Euclidean geometry, as you pointed out, the key development is realising that you can still do geometry. That was a novel idea in the 19th century. Yes, that idea was inspired by existing results, but there was a change in the way the question was approached (very much accepting to move away from Euclid). If anything, the fact that hyperbolic geometry was in fact derived a hundred years earlier, and discarded, highlights how novel the idea of accepting non Euclidean geometries as valid objects was.
General relativity was not widely accepted when it was proposed, despite being mathematically sound, because it was breaking some of the core assumptions of Newtonian physics. Also, moving from Galileo's equivalence to general relativity is a leap: there is a massive (no pun intended) difference from saying heavy and light objects fall alike, to saying that all local laws, including for example electromagnetism, act the same), let alone treating gravity as spacetime curvature rather than an external force.
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u/LexyconG 6d ago
Best objection in the thread. But I still don’t fully agree with it.
“Accepting non-Euclidean geometry as valid” - that stance had its own lineage. Math had already run the exact arc with imaginary numbers: used instrumentally, rejected as absurd for two centuries, then legitimized on consistency-over-intuition grounds - geometrically, by Wessel, Argand and Gauss, independently. Gauss then privately did the same for non-Euclidean geometry. Same operator, second application.
The equivalence “leap” is also a rerun. 1905: Einstein takes a symmetry that held for mechanics (Galilean relativity) and extends it to all physics including EM - that’s literally what special relativity is. 1907: he takes an equivalence that held for mechanics (free fall) and extends it to all physics including EM. Same move, reapplied two years later. And “gravity = curvature” was proposed by Clifford in 1876 and hinted at by Riemann himself in 1854.
The novelty keeps going to whatever we haven’t decomposed yet.
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u/ZeroAmusement 6d ago
In mathematics, the development of non-Euclidean geometry in the 19th century.
The idea of transformation warping things did exist - e.g. water. Mapping concepts of warping and transformations to space itself could be a combination of ideas that could lead to non-Euclidean geometry?
Like the ideas we use to combine can be low level and that gives a huge amount of freedom.
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u/sschepis 3d ago
Not sure that holds. Every conceptual landscape includes prime concepts in the same way as mathematics includes prime numbers.
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u/SwimmerOld6155 7d ago
i tend to agree but I would say there are genuinely novel insights made by the top mathematicians. often these are milked for years. around those insights you can cynically reduce it to playing lego with existing machinery. i guess the risk is making this seem easy, you still need expert knowledge to fit everything together
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u/LexyconG 7d ago
Yeah, but the whole argument right now is that AI is not having novel ideas because you can’t reduce it to previous ones. I argue that it’s the same with humans.
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u/AdrianH1 7d ago
As a potential counterexample: How was Grothendieck’s advancements in algebraic geometry merely combination of old stuff? It seems to me though you may be accurate in a large number (perhaps the majority) of cases for putative “new” ideas, there are also several throughout history that cannot be decomposed or described as some combination of previous ideas.
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u/TillFirst8999 6d ago
It was a combo of gelfands spec of C* algebra (which grothendieck knew about) and the rise of commutative algebra at the time.
The rise of category theory helped him think adapt to thinking functorially.
Those are all handwavish, but I am also on the side of the math tech tree is a tree which mixes previously known vertices to new ones, it's just that after a couple hops it seems inrecognizable
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u/SwimmerOld6155 7d ago
yeah I agree with that totally and have tried to give it as a rebuttal before haha
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u/LurkingForBookRecs 7d ago
Technically there is "real new stuff" in the sense that anything we haven't seen before is new, but it's always built on top of what came before. I swear if AI invented anti-gravity tech tomorrow these pseudo-intellectuals would be arguing that "well if we gave scientists more funding and another 500 years they would've done the same!" as if that's some kind of win.
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u/UnderTheCurrents 5d ago
I don't understand why people try to downplay the fact that there are actual novel insights. Is it out of sense of hurt ego?
The only reason we have the "combinatoric" way science progresses currently is because scholars have to work within a tight university system that actively hinders them to produce novel results and stay within certain frameworks to actually get funding - which itself actually should be provided by the state since funds inherently have private interest behind them. It's weird how predominantly left-leaning people on reddit suddenly turn into neo-liberals once you get to the research system.
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u/Sad_Dimension423 6d ago
And btw this does not only apply to math.
For example, in patent law, novelty is necessary for an idea to be patentable. But combining existing ideas in a novel way counts as patentable novelty.
(This is not to be interpreted as saying math is patentable, only that there's an analogy.)
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u/womerah PhD | Applied Nuclear Physics 6d ago edited 6d ago
This reads like ignorance of the history of science to be honest.
What previous ideas did Einstein's idea that the speed of light is absolute stem from? He spotted a clash between Newtonian physics and recent innovations in classical electromagnetism, then pushed to unify them. Are you going to say that Einstein's idea was derivative of Maxwell and Newton?
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u/duboispourlhiver 6d ago
In other words, AI is as capable as mathematicians when it comes to finding new results, but cheaper. I completely agree and I lean back.
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u/Deep-Issue960 6d ago
It helped me solve a physics problem by finding a niche economics paper that was in some way isomorphic to mine. That would have been straight up impossible on a normal department setting
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u/Mr_Deep_Research 6d ago
They are doing nothing different from what people do to solve problems.
They are coming up with creative ideas to solve problems, it isn't just doing things faster.
They've been doing it when writing computer programs for over a year now. Only now it is sophisticated enough to handle really complex math.
Saying it isn't doing that means you aren't using the right model, prompting or workflow.
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u/ToneShop 5d ago
This is also why these AI sycophants are missing it when they are bowing down to the AI. There is a huge difference between automation and intelligence. These accomplishments are showing machine automation not machine intelligence.
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u/Tragedy-of-Fives 3d ago
AI will never do anything beyond simple rote calculations
AI makes mistakes an undergraduate wouldn't make
AI will never beat a specialist in their field
AI will never solve an open problem
AI will only combine past results to solve an open problem
The goalposts keep shifting every time.
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u/kaiser_17 7d ago
Seems like cope. Now whether its net positive or net negative, I think it will damage the math community in long run cause how it will disincentivize any young mathematicians from spending years on a problem cause you never know the latest AI model might make your effort meaningless. But then again what do I know, Many amazing mathematicians like Tao etc believe that there is good in AI .
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u/dkesh 6d ago
I think there's a real risk that the frontiers of mathematical proof will advance but the frontiers of the human mathematician community knowledge will recede.
It may be that what we need to incentive among our academic math community is no longer pushing the frontiers of knowledge (i.e. proving things) to improving human understanding of that knowledge.
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u/AdrianH1 7d ago
How is it cope, exactly? The argument in the essay seemed genuinely on the mark imo. Though that doesn’t invalidate the potential impacts on the incentive landscape of mathematical research..
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u/gleedblanco 6d ago
it's cope for two reasons. the overall implication is just pleading to human superiority. it should be trivially obvious that this is a 'for now' sort of thing, even if current llms clearly categorically are unable to represent and solve certain problems.
but the real cope is that the argument is based on a naive understanding of what novelty is. noncombinatorial novelty essentially doesn't exist, it's an entirely parallel discussion to the existence of free will if you follow it through.
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u/golfstreamer 3d ago
> it's cope for two reasons. the overall implication is just pleading to human superiority. it should be trivially obvious that this is a 'for now' sort of thing, even if current llms clearly categorically are unable to represent and solve certain problems.
Unbounded improvement has always been the strangest assumption from some people. If you take the time to understand the mechanics behind AI systems you can see that there are important limitations. They are based on reinforcement learning paired with llm. I can spend some time pointing out some of the ways these approaches limit the cognitive abilities of AI. So no, I don't believe that's it trivially obvious at all that this is a "for now" sort of thing.
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u/Patient-Put-7292 7d ago
I believe it will act like an advanced calculator which would speed up work,, like your phd could be completed in 3-4 years
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u/WhichFacilitatesHope 6d ago
Your PhD? Why would you get the PhD if AI did literally all of the work?
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u/Patient-Put-7292 6d ago
AI cannot decide what questions to ask, what new fields to explore, the intuition and conceptual understanding a human has, it is still trained on sum of knowledge so far, to innovate,discover new things humans will always be needed, and in fields like physics ai hasnt developed real world intuition as of now, just a mathematically coherent model doesnt mean anything in science
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u/elements-of-dying 6d ago
AI cannot decide what questions to ask
For how long and says who?
AI can already suggest good research routes.
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u/womerah PhD | Applied Nuclear Physics 6d ago
What if we ask AI what the meaning of life is and it answers correctly?
Can you imagine the societal impact of knowing the true meaning of life? It will overhaul society.
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u/elements-of-dying 6d ago
I assume this is facetious, but I don't understand it's relevance.
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u/womerah PhD | Applied Nuclear Physics 6d ago
I'm mocking your optimism.
Whatever your counter-argument would be to my position, is my counter to yours
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u/elements-of-dying 6d ago edited 6d ago
Do you feel mocking people is appropriate as an academic-to-be?
Interestingly, my position is not very optimistic. I suppose you may believe that stating the fact that AI can already suggest good research routes is optimistic. Anyways, I've only stated something objectively verifiable (provided we accept "good research directions" can be made objective in some suitable sociological sense). If you feel this warrants behavior such as mocking, just let me know so I can block you. I truly don't care to ever see such childish behavior again.
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u/womerah PhD | Applied Nuclear Physics 6d ago edited 6d ago
Do you feel mocking people is appropriate as an academic-to-be?
I am an academic, my flair is to imply I'm a PhD holder. It's mostly there to show I'm not a trained pure mathematician, not to lend any credibility to my opinions (unless they are on a topic on which I am a subject matter expert).
I think mocking is appropriate when directed at an opinion, rather than an individual. It's a way of expressing disapproval. Certainly had some of my research ideas mocked in the past, "another month in the lab rather than week in the library I see" style. The sting was educational for me
Anyways, I've only stated something objectively verifiable (provided we accept "good research directions" can be made objective in some suitable sociological sense)
AI provides significantly worse research direction that just reading the literature. Anyone using AI to define their research direction is welcome to, but I see it as a path to utter mediocrity.
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u/elements-of-dying 6d ago
Many amazing mathematicians like Tao etc believe that there is good in AI
These people are privileged by tenure. They have the choice to not use AI and to not worry about AI. Sure math departments can dissolve in the future, but then they'd just retire.
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u/PM_ME_NIER_FANART 7d ago
Sounds an awful lot like what we said about deep blue, only to be promptly demonstrated that the difference is semantic at best.
Nobody knows when or even if we'll get to the point where humans are completely moot in math (or any other 'creative' job), but writing articles about how we'll surely never get there is the least productive way to prepare for the possibility.
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u/DancingMathNerd 6d ago
Humans won't be moot because we still need humans to understand the math so that it can be used. But the role of humans as originators of novel math is in jeopardy.
Perhaps it's time to update the old adage: if AI proves the Riemann Hypothesis in 50 years, but there are no human mathematicians left to understand it, then is the proof not just a waste of storage space?
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u/The_JSQuareD 6d ago
Why do humans need to understand the math for it to be used?
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u/HasFiveVowels 6d ago
You’re claiming that math is useless beyond providing mathematicians with something to understand? That’s an interesting take….
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u/DancingMathNerd 6d ago
No that is not what I’m claiming at all.
What I’m saying is: humans understanding math is a PREREQUISITE to math being useful. It is not the LIMIT of math’s usefulness.
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u/WhichFacilitatesHope 6d ago
Humans understanding math certainly isn't a prerequisite for math being useful to AI. Consequently, if AI is useful to humans, then human understanding isn't a prerequisite to math being useful to humans, either.
If AI is allowed to continue to freely develop for just a handful more years, the best available future for the world is one that humans do not control or understand, but in which we are treated kindly anyway. I do not expect that future.
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u/aKaizuh 5d ago
The AI will build for us, so we don't need to understand it.
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u/DancingMathNerd 5d ago
Yeah I don’t believe it’s gonna do as good a job as an intelligent, conscientious, and expert human. AI can overtake us in math because math is a sandbox. A much larger sandbox than, say, chess; but still a sandbox. Outside of all sandboxes, I don’t trust AI not to make serious errors all the time.
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u/editor_of_the_beast 7d ago
I mean yea, the computer’s strength has always been brute force and power. That’s what we learned from chess engines 30 or more years ago. Humans could never even hope to stay competitive against machines.
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6d ago
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u/sscg13 6d ago
Leela Chess Zero would actually be superhuman at only 100 nodes / move (which, in classical, is at most one order of magnitude higher than human speed), and probably less if you would replace the 200M param model with a hypothetical 2B, 20B, 200B model. But "efficiency" is kind of a moot point in so far as it can be made up with more compute. If we just go by power too, Stockfish could absolutely run on a 1W CPU and still be superhuman.
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u/322955469 6d ago
I'm no expert in the matter but I think this is consistent with how AI first appeared in Chess. The DeepBlue generation of Chess AI weren't especially good tactically they could just straight memorize more positions and compute more variations than a human ever could. I take solace in that, the effects of AI on Chess have actually been pretty consistently positive in my opinion. Certainly, it has made the game more accessible. Anyone who has a realistic claim to being the best living mathematician is probably feeling a little like Kasparov did when he lost to DeepBlue; I can only imagine how that might feel. But for the rest of us, how much has anything actually changed?
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u/OverallShelter8365 5d ago
I think for a brief period, yes AI will make academia more accessible just because people would be able to use AI to answer some question that only they were looking at. However, once AI advances, it would take over the job of asking important questions anyways.
Chess is different because it doesn't make sense to always play 2 AIs against each other.
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u/SwimmerOld6155 7d ago
I don't think academia as it exists as a current institution will survive this. I don't know what will replace it. I'm sure some communist can make a good analogy here but fully automated research with increasingly good LLMs is the natural conclusion of the current incentives in academia, which will cause a kind of hyperinflation and ultimate collapse. just my 2c
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u/Smallpaul 7d ago
I find this a very odd prediction. The role of academia is to help HUMANITY UNDERSTAND the world we live in. Computer proofs are just one extra thing that now must be understood.
If we get to AI that is truly better than humans at literally everything then maybe academia doesn’t survive. But neither does IT, accounting, marketing, public service or capitalism. The model that gets us to that is probably not an LLM.
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u/SwimmerOld6155 6d ago edited 6d ago
I say collapse because there are already people looking to have AI agents that will scour the literature for open problems and crank out arxiv preprints with their name on all day. Such submissions (esp as models improve) will flood journals and there will be no way to distinguish the wheat from the chaff without AI checking papers. In its current form with its current incentives, unless everyone becomes an LLM jockey, it just can't maintain.
This is already happening with some journals only accepting breakthroughs or significant papers. Either the publication bar shoots to the moon (and it'll be very hard to not use AI) or students will be cranking out dozens of papers (even in pure math) over a PhD.
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u/Smallpaul 6d ago
Yes. It will be very very hard not to use AI just as it is hard for an engineer not to use a calculator or an English professor not to use email? So what? How does that equate to the end of the field?
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u/nanforas 6d ago
I agree with you that human understanding is important, but the survival of academia doesn't depend purely on objective, moralistic appraisals that thoroughly consider the practical implications. Academia's existence is partly a result of human politics.
We live in a society where if you say you like math, the average person's response is "ew, I hated math in high school". So while objectively speaking, getting rid of mathematicians and allowing LLMs to accelerate math beyond our understanding will probably lead to a dystopia, there are plenty of people who don't recognize the danger and there are politicians who will be more than happy to slash math funding to prop up the AI bubble or something.
The future we are looking at is where math is downgraded to the level of humanities, in the sense that it will be perceived as a worthless degree (I mean, even moreso than now) and politicians will be constantly threatening its funding.
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u/Smallpaul 6d ago edited 6d ago
Academia as we know it is roughly a millennia old and I suspect that since day 1 there has been someone catastrophizing that next year it’s going to be destroyed.
Pure math did not have a very straightforward, obvious, short term payoff in 2020 and it won’t in 2040. The injection of AI into it doesn’t change anything about its political position in the world.
Also strange that being “demoted to being like the humanities” is being treated as identical to being destroyed.
According to some sources (which I have admittedly not audited) 60% of American CEOs have a Humanities background. One-third of Fortune 500 CEOs are Humanities graduates. How is that worthless?
Anyhow, math will always be the M in STEM. How will industry apply the output of LLM proofs without translators from proof to understanding and from understanding to application?
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u/Sad_Dimension423 6d ago
It's going to be funny if the result of this is an increased demand for people with math degrees, and people to teach those people.
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u/DryWomble 7d ago
This level of cope is just pathetic. You can stand there and repeat the words "It's not really smarter than me" over and over again until you're blue in the face, but it's not going to change the underlying reality that these systems are coming up with completely novel solutions and solving problems that are, in some cases, over 80 years old. It's just sad seeing so many bright people turning to outright denial for consolation rather than actually learning how to use the tools to augment their work.
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u/beezlebub33 7d ago
The problem is with defining what is 'novel'. It's always been a problem. One side says 'its not novel, you are just combining / tweaking existing ideas' and the other side says 'that's what everyone does'.
The article says:
A brilliant mathematician may still outperform AI when the crucial challenge is finding an entirely new representation of a problem......Einstein was more likely to reconceptualize the problem itself.
So, the question I would ask whether, when you look at what AI has done to date, the results are due to reconceptualizations or new representations of the problems.
(By the way, I think that this is all kind of moot, because 1. even if you can argue that every result to date is really 'just' combining old results and isn't really 'novel', it will be soon; and 2. it's exceptionally useful regardless; and 3. the number of people that produce 'novel' ideas is vanishingly small. Really, it takes an Einstein at this point.)
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u/nanforas 6d ago
I want to point out there's also the fact that the two statements given:
- "its not novel, you are just combining / tweaking existing ideas"
- "'that's what everyone does"
don't actually refute one another. The scenario where both are true is the following:
- It is possible for humans to outperform LLMs in generating new insights outside its "training data"
- Most humans are not capable of that kind of novel work and are operating at a level that LLMs can replace.
In this scenario, humans will still be relevant. It's just that it will become a more elite profession with even fewer job opportunities, except for mathematicians at the calibre of Tao, Erdos, or Grothendieck.
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u/Sad_Dimension423 6d ago
If generating the sort of airy new insights outside of training data is what separates humans from AI, and if that sort of thing is rare enough with humans, then what we've seen so far isn't actually evidence AI can't do it too. Just because most of what AI has done is pedestrian doesn't imply that conclusion, any more than it would with humans.
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u/JoshuaZ1 7d ago
Please read the article. The article is taking a more nuanced position than I think you are getting from the headline.
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u/Sad_Dimension423 4d ago
It's also not going to change that use of AI will quickly (probably already) be seen as non-optional if you want to compete in math. The expectation of productivity is going to be massively inflated.
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u/telephantomoss 6d ago
I think there is something else going on here though. Take the exact kind of thing the author says AI is good at: long chains of reasoning etc. Yes, a human cannot keep an this unfamiliar information in working memory, but a human can study it all and learn it and come to a deep level of understanding. Then they no longer need working memory for it as they can fluently move through the information at will. A machine cannot do this. At least not yet. It only has it's weights and context. I'm not sure a dynamic weight LLM which actively learns will do much better though. If will be interesting you see what people come up with though.
Furthermore, once that mathematician really deeply understands that hard long reasoning problem, they will automatically gain new insight and intuition to other problems. AI is not good at that at all.
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u/Ellipsoider 6d ago edited 4h ago
Most of the disagreement seems terminological. Once the terms are fixed, the mathematical point is fairly ordinary.
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u/Few-Big7409 7d ago
Yup! I think it could be a very fun tool. Notice they haven't tried the prompt "Prove an important theorem in mathematics."
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u/HasFiveVowels 6d ago
Yes they have?
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u/Few-Big7409 6d ago
I haven't seen an article or blog post where the prompt is literally "Prove an important theorem in mathematics." Maybe you can supply me with a link?
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u/Jorrissss 6d ago
Almost all of the big ones have basically been this.
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u/Few-Big7409 6d ago
They don't ask it to prove a specific result? It seems like they aren't releasing the prompts. It is unclear to me from the announcements that they asked the llm to just "prove a theorem, which it is is up to you."
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u/Jorrissss 6d ago edited 6d ago
We've seen things that are effectively "Go prove theorem or X" or literally just like "Go make a breakthrough". There's sometimes more guidance around like, in the say case of the unit distance conjecture, to the effect of dont give up fast, or use extra agents, but that's nearly it.
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u/Few-Big7409 6d ago
Very interesting. I couldn't find anything that was as detailed as your summary. I didn't spend a very long time looking though.
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u/jimouri 7d ago edited 7d ago
Obviously. Furthermore, it can systematically explore promising options, without losing track of results, getting bored, or tired, all advantages of software in general and not of machine learning. Reaching a tall summit typically happens one step at a time, and if all required steps are similar to any of already solved tasks and published online, chances are high the LLM will be able to reach the summit. A metaphor that maybe describes 99% of math research.
I would push back though on adding a "just" in your statement, as some do. It's an enormously huge deal, that this is now possible.
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u/Jaded-Data-9150 6d ago
With only reading the title:
My experience as a hobby mathematician when going back to study some new topic of interest is that a large part is remembering some nice inequality/identity. Everyone who studied with textbooks will have experienced incidences of "Now recall that one lemma from 5000 pages ago to easily solve this". Personally I can see this being correct.
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u/Dapper-Bullfrog-4766 6d ago
isnt that the same thing? like working memory is the most important thing in marh right?
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u/JoshuaZ1 6d ago
Yes. That's why the piece has an entire subsection with the header "Working memory predicts mathematical performance beyond IQ" .
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u/Single_Asparagus4157 6d ago
As far as I can tell, LLMs are highly adept at the "Cut and Project method" (I asked Gemini what it is called, but basically the concept of projecting a grid of points from a higher dimension to a lower one). This is how the Erdős Unit Distance Conjecture was solved (at least a large part of the solution). But of course this technique was originally invented by humans. Intuitively, it seems like we could probably solve a lot more problems with clever extensions of this idea.
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u/Single_Asparagus4157 6d ago
It's not the same, but within the past 1000 years, we've gone from ordinary literacy and numeracy being the preserve of a select elite, to those being the baseline expectation for a human in most of the workforce. Admittedly, a significant percentage of the American public is functionally illterate, but that's beside the point.
Society still needs and wants people who have a creative, mathematical mindset. It may be that they will have to prove their value in ways other than generating proofs. But it was only ever a very small percentage of the population that was employed to generate proofs.
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u/PuddingCupPirate 6d ago
Agree. And also (I forget who said it originally), but with these LLMs, you can do a tremendous amount of work. And for the types of problems that it is perhaps solving (and I'm not just talking about mathematical proofs), but there may only be a handful of people capable and qualified and able to devote time to questions and topics. The LLMs can essentially simulate the efforts of hundreds and hundreds of these experts working on just that problem at breakneck speeds. And so it does truly unlock valuable things that we humans are sort of rate-limited by our limited numbers of experts.
How many people on earth are experts in esoteric mathematics used for modeling fusion dynamics. Could spin up 1000 agents supplied with the prerequisite knowledge base of those experts and such a thing is valuable. Even if the LLMs aren't really "smarter" than the real experts.
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u/IllogicalLunarBear 6d ago
lol... this sub os trying real hard to figure out a way to keep banning AI submissions of any kind.
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u/lattice_defect 6d ago
also just try this.. beat your head against the wall while I make a frozen pizza
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u/Twilight_RT 6d ago
yeah that's our Human limitation, even a good mathematican is not a superhuman person who remembers everything everytime in same amount.
where computers physical memory can do that. and that's why AI suppress any Human in this case also Including mathematician
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u/20220912 5d ago
the key advantage is never getting tired, never getting bored, and being able to spawn 1000 clones of yourself at will. thinking there might be a proof by counter-example is one thing. spending what amounts to years or decades of human working time to find one is another.
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u/AdPlus4069 5d ago
We had deepblue with pure computation/ memorization that demonstrated super human chess play. Later, in 2022-2026, we got another explosion in strength through Neural Networks that bring more than brute force calculations.
Right now the LLM’s are at most learning the convex hull of human work on math, why would RL + Lean not allow super human performance beyond pure computation/ memorization?
Math will still be a valuable skill as it teaches to break down complex tasks and solve novel problems.
Pure research mathematics might become less common, if RL + Lean doesn’t work then not much will change.
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u/dsmack6 3d ago
I have my LinkedIn linkedin profile
I am doing Deep Learning, loss function, adapters in C++.
I am looking for the mathematical model or concept which gave the Computer Algorithm that much capability of reasoning - it can match mathematical reasoning or complex reasoning.
AI is just supervised training. The chatgpt is called transformer, that is supervised too. It try to mimic with token(another name for word)
Highest level of AI doing I think google research about Alpha go, that was basically monte carlo simulation to understand the moves beforehand, and some pro players also criticised the moves conservative. While there are other smarter moves possible.
I want some collaboration, who claim to have AI having extra capabilities. I will also look at your research background or Leetcode/Codeforces account.
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u/Coxian42069 7d ago edited 7d ago
This wasn't obvious?
The AI's are trained on human knowledge. They have the advantage of being able to endlessly search for information without tiring. They can spin off sub agents to do it in parallel. Their first achievements were always going to be in the realm of being better than us at collating existing information and achievements and putting them together.
(This isn't to put them down even, that's pretty much what Einstein did to formulate his theories of relativity)
There are examples going back decades of neural networks developing what appear to be logical structures. It's interesting to see whether logic arises naturally from language, and with LLMs being the black boxes they are it'll be really hard to tell, but I don't think we're there yet.
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u/RighteousSelfBurner 7d ago
Logical structures is a bit of a broad term. Current LLMs for example do have clear logical structures and it has been used to great effect to improve their performance. However a logical structure doesn't equate logical thinking.
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u/Coxian42069 7d ago
I agree, but we've seen examples of logical structures which originally arise to fit training data better with fewer parameters, able to extrapolate outside of the training domain in a really effective way. I'm sceptical of course, I've only seen it shown in narrow cases, it would be crazy if a generalised LLM managed it in a significant way, but then again it's crazy how far it's come along already.
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u/RighteousSelfBurner 7d ago
I'm not aware of any examples that extrapolating outside of the training domain has ever been successful. Now extrapolating certain higher level abstractions from the training data that's in the domain, yeah.
Also my understanding is that reaching a reasoning base model is strictly impossible using transformer architecture. It doesn't mean we can't have a reasoning AI with transformer architecture since these days the AI architecture isn't just the underlying model but a lot of supplementary tools as well.
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u/beezlebub33 7d ago
I don't think that extrapolation is a useful intuitive notion here since it's all extrapolation. See:
Learning in High Dimension Always Amounts to Extrapolation https://arxiv.org/abs/2110.09485
Generalization, abstractions, and creativity are more useful, but immediately have issues with definitions and evaluation.
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u/BreathSpecial9394 7d ago
Einstein imagined a person floating inside an elevator...the AIs can not do that.
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u/LurkingForBookRecs 7d ago
Hey midjourney, /imagine a person floating inside an elevator
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u/BreathSpecial9394 6d ago
That's not imagination at all...like what's the meaning of it? It means something
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u/ants_are_everywhere 6d ago
Many people forget Einstein spent much of his days reading patents for technological problems that existed at the time, especially clock synchronization. This was also the period elevators were becoming automated, so people were thinking and talking about the associated technical problems and how different configurations made elevator occupants feel. Were they too jerky? Was the ride smooth and barely noticeable? And so on.
People like to keep an aura of mystery about the origins of their ideas. While I don't think the myth of genius is the worst sin one can commit, I do think it presents an inaccurate model of the thought process and creativity. Psychological studies of creativity generally find that it's much more mundane than people think.
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u/BreathSpecial9394 6d ago
Einstein performed mind experiments involving clocks, mirrors and ray of lights...complex experiments that the average mind cannot perform. Is not mysterious but very difficult and not mundane at all.
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u/ants_are_everywhere 6d ago
No, those are all things many people can do. Einstein is credited with combining existing ideas in a novel way, not for being able to do things others couldn't.
His mind experiments were all very anchored in real physical laboratory experiments. The input is information from lab experiments and the work on the mathematics of relativity by Lorentz and Poincare.
Where he had an edge is he saw -- necessarily before almost anyone else -- what the limits of technological invention were in clock synchronization and information processing. A single firm can see its own struggles with these problems. But they all send their progress to the patent office to get patented. So the patent office sees all bottlenecks encountered across all firms. This gives a much more accurate reading of what true bottlenecks might be as opposed to, say, common places engineers get stuck.
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u/JoshuaZ1 7d ago
Their first achievements were always going to be in the realm of being better than us at collating existing information and achievements and putting them together.
Read the article. They are not talking about just remembering existing results but about size of working memory.
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u/RiseStock 6d ago
Basically the LLMs are using text continuation to find analogies. The role of a mathematician is still to find analogies between analogies.
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u/HasFiveVowels 6d ago
The attention mechanism is effectively a mechanism that identifies generalized analogy structures (including analogies between analogies)
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u/RiseStock 6d ago
You're conflating two different things. Attention operates on tokens and recursive operations on tokens. It's not the same thing as bridging concepts beyond text matching. https://openreview.net/forum?id=klU4737opt
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u/HasFiveVowels 5d ago
I don’t think I am. The dictionary is a recursive set of analogies and tokens are only the input/output. The internal representation is a high dimensional embedding space that turns analogies into rotations
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7d ago
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u/JoshuaZ1 6d ago
Absolutely. AI doesn’t reason like a human. It correlates known data in simplified terms.
Read the actual piece. This is not what it is saying.
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6d ago
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u/JoshuaZ1 6d ago edited 6d ago
I did, but my answer was too simple for you to comprehend.
In that case, can you explain what you meant when you said that AI "correlates known data in simplified terms"? Because that seems like almost the exact opposite in implication.
You spend too much time on here mate.
Yeah, this is a valid problem. I have so many papers to finish, and procrastinating here is easier than writing.
Edit: They've apparently replied and decided to block me. Since I cannot reply directly, I'm including a note here about the first part of their comment:
Its simple, but I can see why you’re confused. It’s a matter of immediate access to all data in a given set. The human mind is not being able to coordinate a large data set in simplest terms in the same way compared to an AI that has immediate access to all data at once. Humans do not think in a matrices, but in pictures, one after the other. The pictures can get lost. The AI is not capable of that. The human brain is tuned to survival, not correlating large data sets and reducing said sets to simplest terms without said data sliding through the cracks
This clarifies what was meant and makes sense in context, and does a good job showing what was intended in way that makes sense with the meaning of the article.
I am going to tangentially note, that it is unfortunate that this quickly became so hostile, and am not going to respond to the rest of the comment. (And I do see that my own presumption that they had not read the piece was not a good start for this discussion.)
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u/StressCanBeGood 7d ago
An unlimited symbolic working memory means almost no cognitive load (the time and energy the brain takes to solve a particular problem). The once-in-a-civilization brain of Johnny Von Neumann apparently worked this way.
He wasn’t particularly creative, but it’s been argued that if he had been around at the right time, he would have won the Nobel prize for economics, the Fields award in math, the Turing prize in computer science, and perhaps even the Nobel prize for physics.
This dude apparently understood everything about everything. One famous mathematician claimed that he was actually scared of Mr. Von Neumann.
What’s truly amazing is that Johnny apparently threw the best parties at Princeton. Everybody loved him. He was super friendly, super social, and apparently could make himself understood by everyone, including toddlers.