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> I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries. Most discoveries are indeed implied from a
by dvt 5mo ago
> I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
Most discoveries are indeed implied from axioms, but every now and then, new mathematics is (for lack of a better word) "created"—and you have people like Descartes, Newton, Leibniz, Gauss, Euler, Ramanujan, Galois, etc. that treat math more like an art than a science.
For example, many belive that to sovle the Riemann Hypothesis, we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it.
- Tenobrus 5mo agowhat basis do you have for assuming an LLM is fundamentally incapable of doing this?
- dvt 5mo agoBecause by definition LLMs are permutation machines, not creativity machines. (My premise, which you may disagree with, is that creativity/imagination/artistry is not merely permutation.)
- KoolKat23 5mo agoIt pretty much is, otherwise it is randomness or entropy.
- nh23423fefe 5mo agogod of the gaps
- iwontberude 5mo agonon overlapping magisteria
- fnordpiglet 5mo agoI prefer to think of it as they’re interpolation machines not extrapolation machines. They can project within the space they’re trained in, and what they produce may not be in their training corpus, but it must be implied by it. I don’t know if this is sufficient to make them too weak to create original “ideas” of this sort, but I think it is sufficient to make them incapable of original thought vs a very complex to evaluate expected thought.
- drdeca 5mo agoPeople keep saying this, but if you try to interpret this at all literally, it just doesn’t work. Like, it’s phrased like it should have a precise meaning, right? Like, people even mention convex hulls when talking about it. But if you actually try to take a convex hull of, some encoding of sentences as vectors? It isn’t true. The outputs are not in the convex hull of the training data. I guess it’s supposed to be a metaphor and not literal, but in that case it’s confusing. Especially seeing as there are contexts in machine learning where literal interpolation vs literal extrapolation, is relevant. So, please, find a better way to say it than saying that “it can only interpolate”?
- Muromec 5mo agoIf it's all just points in the multidimensional space, why would the thing be restricted to some operations and not others. I'm not buying the argument
- drdeca 5mo agoSorry, I don't understand what you mean. Are you agreeing or disagreeing with me? If it can only interpolate in a literal sense, that means that it only produces good outputs on convex combinations of inputs that appear in the training set. That's what interpolation means. But, if you take the embedding vectors of sentences/prompts, and then take the convex hull of these, it is not typical for new sentences not in the training set to have its embedding vectors be in the convex hull of these.
- fnordpiglet 5mo agoI’m not sure I follow your end to end reasoning. In an n dimensional space interpolation along and within the convex hull is pretty much what they’re doing. How can it possibly not be? How would it interpolate a point that’s not within its vector space? Yes, it’s very complex with non linear transformations and a very high dimensionality, and residuals and other features create more complexity in the shape of the hull. But an LLM can not infer a concept to which it has no information channel. That’s clearly nonsense. The fact that they do bounded, learned, nonlinear compositional generalizations over a representational space induced by training -is by nature interpolation- not extrapolation. I’m sorry, but I believe their immense power has you confusing math with magic.
- lajamerr 5mo agoLLMs by themselves are not able to but you are missing a piece here. LLMs are prompted by humans and the right query may make it think/behave in a way to create a novel solution. Then there's a third factor now with Agentic AI system loops with LLMs. Where it can research, try, experiment in its own loop that's tied to the real world for feedback. Agentic + LLM + Initial Human Prompter by definition can have it experiment outside of its domain of expertise. So that's extending the "LLM can't create novel ideas" but I don't think anyone can disagree the three elements above are enough ingredients for an AI to come up with novel ideas.
- awesome_dude 5mo agoYou're proving the GP's argument - LLMs aren't creative you say as much, it's the driving that is the creative force
- Barbing 5mo agoIf that’s a requirement, aren’t LLMs driven by pretraining which was human driven? Who decides at which the last point it’s OK to provide text to the model in order to be able to describe it as creative? (non-rhetorical)
- lajamerr 5mo agoYou can tell an agentic system. "Go and find a novel area of math that has unresolved answers and solve it mathematically with verified properties in LEAN. Verify before you start working on a problem that no one has solved this area of math" That's not creative prompt. That's a driving prompt to get it to start its engine. You could do that nowadays and while it may spend $1,000 to $100,000 worth of tokens. It will create something humans haven't done before as long as you set it up with all its tool calls/permissions.
- awesome_dude 5mo agoLet me know when the Fields medal arrives in the mail. It won't because even though it looks clever to you, people who /do/ understand math and LLMs understand that LLMs /are/ regurgitating Why does your LLM need you to tell it to look in the first place? Why isn't just telling us all the answers to unsolved conjectures known and unknown? Why isn't the LLM just telling us all the answers to all the problems we are facing? Why isn't the LLM telling us, step by step with zero error, how to build the machine that can answer the ultimate question?
- lukol 5mo agoThis "new math" might be a recombination of things that we already know - or an obvious pattern that emerges if you take a look at things from a far enough distance - or something that can be brute-forced into existence. All things LLMs are perfectly capable of. In the end, creativity has always been a combination of chance and the application of known patterns in new contexts.
- dvt 5mo ago> This "new math" might be a recombination of things that we already know If you know anything about the invention of new math (analytic geometry, Calculus, etc.), you'd know how untrue this is. In fact, Calculus was extremely hand-wavy and without rigorous underpinnings until the mid 1800s. Again: more art than science.
- baq 5mo agoAnd yet nowadays you can restate all of it using just combinations of sets of sets and some logic operators.
- jfyi 5mo agoNewton and Leibniz were "hand-waving"? If anything, they were fighting an uphill battle against the perception of hand-waving by their contemporaries.
- dvt 5mo ago> Newton and Leibniz were "hand-waving"? Yes, and it's pretty common knowledge that Calculus was (finally) formalized by Weierstrass in the early 19th century, having spent almost two centuries in mathematical limbo. Calculus was intuitive, solved a great class of problems, but its roots were very much (ironically) vibes-based. This isn't unique to Newton or Leibniz, Euler did all kinds of "illegal" things (like playing with divergent series, treating differentials as actual quantities, etc.) which worked out and solved problems, but were also not formalized until much later.
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- satvikpendem 5mo agoWhat is creativity if not permutation? A brain has some model of the world and recombines concepts to create new concepts.
- d3ffa 5mo ago[flagged]
- truncate 5mo agoWhat's your basis for assuming LLM is capable of doing this? I honestly don't know personally either way. Based on my limited understanding of how LLMs work, I don't see them be making the next great song or next great book and based on that reasoning I'm betting that it probably wont be able to do whatever next "Descartes, Newton, Leibnitz, Gauss, Euler, Ramanujan, Galois" are going to do. Of course AI as a wider field comes up with something more powerful than LLM that would be different.
- redsocksfan45 5mo ago[dead]
- dist-epoch 5mo agoLLMs are already making the next great songs. Just check out the Billboard charts.
- truncate 5mo agoI'm sorry, I don't consider them "great songs". Obviously, different people have different taste.
- EMM_386 5mo ago"I don't see them be making the next great song" Meanwhile, songs are hitting number one on some charts on Spotify that people think are humans and are actually AI. And Spotify has to start labelling them as such. One AI "band" had an entire album of hits. Also - music is a subjective. Mathematics isn't. And in this case, an LLM discovered a new way to reason about a conjecture. I don't know how much proof is needed - since that is literally proof that it can be done.
- truncate 5mo ago>> Meanwhile, songs are hitting number one on some charts on Spotify that people think are humans and are actually AI. And Spotify has to start labelling them as such. One AI "band" had an entire album of hits. There is quite some questions around that. Music is subjective and obviously different people have different taste, but I wouldn't call any of them to be actual good music / real hits. >> LLM discovered a new way to reason about a conjecture I wasn't questioning LLMs ability to prove things. Parent threads were talking about building new kind of maths , or approaching it in a creative/artistic way. Thats' what I was referring to. I can't speak for maths of hard science as I'm not trained in that, but the creativity aspect in code is definitely lacking when it comes to LLMs. May not matter down the line.
- blueone 5mo ago> what basis do you have for assuming an LLM is fundamentally incapable of doing this? because I have no basis for assuming an LLM is fundamentally capable of doing this.
- sswatson 5mo agoGood on you for spelling out this reasoning, but it is manifestly unsound. For a wide variety of values of X, people a few years ago had no reason to expect that LLMs would be capable of X. Yet here we are.
- TheOtherHobbes 5mo agoIn 1989, Gary Kasparov said that it was "ridiculous!" to suggest a computer would ever beat him at chess. "Never shall I be beaten by a machine!” In 1997 he lost to Deep Blue.
- FartyMcFarter 5mo agoYeah, and back then people moved the goal posts too, saying Deep Blue was just "brute-forcing" chess (which isn't even true since it's not a pure minimax search).
- bananaflag 5mo agoDeep Blue was brute forcing chess in the sense that AlphaGo wasn't brute forcing Go.
- FartyMcFarter 5mo agoBoth of them contained a search algorithm that explored some moves from each considered position, usually not all moves. Both of them contained logic (learned or programmed) to evaluate moves and/or positions. The differences between them are many, but brute force doesn't enter into it in either case.
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- pulkitsh1234 5mo agoCreation is done by humans who have been trained on the data of their life experiences. Nothing new is being created, just changing forms. A scientist has to extract the "Creation" from an abstract dimension using the tools of "human knowledge". The creativity is often selecting the best set of tools or recombining tools to access the platonic space. For instance a "telescope" is not a new creation, it is recombination of something which already existed: lenses. How can we truly create something ? Everything is built upon something. You could argue that even "numbers" are a creation, but are they ? Aren't they just a tool to access an abstract concept of counting ? ... Symbols.. abstractions. Another angle to look at it, even in dreams do we really create something new ? or we dream about "things" (i.e. data) we have ingested in our waking life. Someone could argue that dream truly create something as the exact set of events never happened anywhere in the real world... but we all know that dreams are derived.. derived from brain chemistry, experiences and so on. We may not have the reduction of how each and every thing works. Just like energy is conserved, IMO everything we call as "created" is just a changed form of "something". I fully believe LLMs (and humans) both can create tools to change the forms. Nothing new is being "created", just convenient tools which abstract upon some nature of reality.
- ulbu 5mo agothat’s why we say that with such discoveries we receive a new way – of looking, of doing, of thinking… these new paths preexist in the abstract, but they can be taken only when they’ve been opened. and that is as good as anything “new” gets. (and such discoveries are often also inventions, for to open them, a ruse is needed to be applied in a specific way for the way to open).
- bwfan123 5mo ago> Aren't they just a tool to access an abstract concept of counting ? Humans and animals have intuitive notions of space and motion since they can obviously move. But, symbolizing such intuitions into forms and communicating that via language is the creative act. Birds can fly, but can they symbolize that intuitive intelligence to create a theory of flight and then use that to build a plane ?
- wslh 5mo ago
- kenjackson 5mo ago"new kind of math" Well I think the point is there is no "new kind of math". There's just types of math we've discovered and what we haven't. No new math is created, just found.
- grey-area 5mo agoThe map is not the territory.
- cthalupa 5mo agoI don't know what you're even trying to argue here. We're not comparing math to reality (though there's a strong argument to be made that reality has a structure that is mathematical in nature - structural realism didn't die a scientific philosophy just because someone came up with a pithy saying), we're talking about if math is discovered or invented. Most mathematicians would argue both - math is a language, we have created operations, axioms are proposed based on human creativity, etc., but the actual laws, patterns, etc. are discovered. Pi is going to be pi no matter if you're a human or someone else - we might represent it differently with some other number system or whatever, but that's a matter of representation, not mathematical truth.
- Koshkin 5mo ago> we have created operations It seems that addition (for instance) was "created" long before us. On the other hand, it seems highly unlikely that a civilization similar to ours could "invent" an essentially different kind of mathematics (or physics, etc.)
- cthalupa 5mo agoWell, I was thinking more along the lines of, say, multiplication and division - you can handle every single equation humanity has ever come up with without either of them. It might be messy and awful and annoying, but I would say in particular these operations are invented more than discovered. So, more properly phrased, we created some operations.
- bbor 5mo agomath more like an art than a science. That’s a fun turn of phrase, but hopefully we can all agree that math without scientific rigor is no math at all. we likely need some new kind of math. Imo, it's unlikely that an LLM will somehow invent it. Do you think it’s possible/likely that any AI system could? I encourage us to join Yudkowsky in anticipating the knock-on results of this exponential improvement that we’re living through, rather than just expecting chatbots that hallucinate a bit less. In concrete terms: could a thousand LLMs-driven agents running on supercomputers—500 of which are dedicated to building software for the other 500-come up with new math?
- black_knight 5mo agoMath is not based on science! Maths follows logical (or even mathematical) rigour, not scientific rigour!
- bbor 5mo agoYou're just using a different definition of "scientific". If Math isn't a science, then it cannot follow logical rigor. Math and philosophy are both sciences, even if they differ qualitatively from the natural and human empirical sciences.
- Someone 5mo agoI think “new math” is ‘just’ humans creating new terminology that helps keep proofs short (similar to how programmers write functions to keep the logic of the main program understandable), and I agree that is something LLMs are bad at. However, if that idea about new math is correct, we, in theory, don’t need new math to (dis)prove the Riemann hypotheses (assuming it is provable or disprovable in the current system). In practice we may still need new math because a proof of the Riemann hypotheses using our current arsenal of mathematical ‘objects’ may be enormously large, making it hard to find.
- black_knight 5mo agoIt could be that RH is independent of current mathematical axiom systems. We might even prove that it is some day. But that means we are free to give it different truth values depending on the circumstances! This is also true for established theorems! We can can imagine mathematical universes (toposes) where every (total) function on the reals is continuous! Even though it is an established theorems that there are discontinuous functions! We just need to replace a few axioms (chuck out law of the excluded middle, and throw in some continuity axioms).
- necovek 5mo agoWhat frequently happens when we recombine axioms like that is that they end up leading to inconsistencies or contradictions. Do you know if this topos with every total function on real numbers is continuous has been constructed and proven to be a viable set of axioms? If so, I am curious about the source. My go to example still remains the one of hyperbolic geometry and axiom of parallel lines, so the more approachable examples I can get, the better.
- black_knight 5mo agoSure. These toposes are well known, and proven to be consistent (relative to set theory). For instance Hyland’s effective topos, or Johnstone’s topological topos. The ideas are that these toposes either require everything to be computable, or continuous in some greater sense. There is also this blogpost by Amdrej Bauer, which can be seems as exploring how it is to be such such a topos: https://math.andrej.com/2006/03/27/sometimes-all-functions-are-continuous/ https://math.andrej.com/2006/03/27/sometimes-all-functions-a...
- golol 5mo agoI think based on the class of problem that RH is an independence result is not something that "really happens".