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To the “LLMs just interpolate their training data” crowd: Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts
by m-hodges 5mo ago
To the “LLMs just interpolate their training data” crowd:
Ayer, and in a different way early Wittgenstein, held that mathematical truths don’t report new facts about the world. Proofs unfold what is already implicit in axioms, definitions, symbols, and rules.
I think that idea is deeply fascinating, AND have no problem that we still credit mathematicians with discoveries.
So either “recombining existing material” isn’t disqualifying, or a lot of Fields Medals need to be returned.
- 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.
- 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.
- 5mo ago
- 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.
- 5mo ago
- 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".
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- voooduuuuu 5mo agoI think you are conflating composition and prediction. LLMs don't compose higher abstractions from the "axioms, symbols and rules", they simply predict the next token, like a really large spinning wheel.
- adampunk 5mo agoHow sure are you that this is correct?
- peterlk 5mo agoYes they do…? Who cares if they just predict the next token? The outcome is that they can invent new abstractions. You could claim that the invention of this new idea is a combination of an LLM and a harness, but that combination can solve logic puzzles and invent abstractions. If a really large spinning wheel could invent proofs that were previously unsolved, that would be a wildly amazing spinning wheel. I view LLMs similarly. It is just fancy autocomplete, but look what we can do with it! Said differently, what is prediction but composition projected forward through time/ideas?
- voooduuuuu 5mo agoAsk an LLM to invent a new word and post it here, I will be waiting. You will see that it simply combines words already in the training data.
- bossyTeacher 5mo agoDoes a random sequence of letters qualify as a new word?
- motoxpro 5mo ago[flagged]
- planetafro 5mo agoSplifket Definition: That highly specific, short-lived burst of nervous energy that makes you accidentally drop a small object (like a pen, a guitar pick, or a piece of LEGO) immediately after picking it up.
- adam_arthur 5mo agoPretty much everything that appears novel in life is derivative of other works or concepts. You can watch a rock roll down a hill and derive the concept for the wheel. Seems pretty self evident to me
- paulddraper 5mo ago"LLMs just interpolate their training data" Cracks me up. What exactly do we think that human brains do?
- gpugreg 5mo agoMaybe the human brain also does other things besides interpolation?
- paulddraper 5mo agoThere is pre-training, and then empirical observations. Yes?
- omnimus 5mo agoThat has been the question since the beginning of humans. Maybe computers can help understand better because by now it's pretty clear brains aren't just LLMs.
- deleted 5mo ago[deleted]
- ActorNightly 5mo agoI love this comment because it so clearly highlights the difference between intelligence and reasoning. A lot of people across all fields seem to operate in a mode of information lookup as intelligence. They have the memory of solving particular problems, and when faced with a new problem, they basically do a "nearest search" in their brain to find the most similar problem, and apply the same principles to it. While that works for a large number of tasks this intelligence is not the same as reasoning. Reasoning is the ability to discover new information that you haven't seen before (i.e growing a new branch on the knowledge tree instead of interpolating). Think of it like filling a space on the floor of arbitrary shape with smaller arbitrary shapes, trying to fill as much space as possible. With interpolation, your smaller shapes are medium size, each with a non rectangular shape. You may have a large library of them, but in the end, there are just certain floor spaces that you won't be able to fill fully. Reasoning on the flip side is having access to very fine shape, and knowing the procedure of how to stack shapes depending on what shapes are next to it and whether you are on a boundary of the floor space or not. Using these rules, you can fill pretty much any floor space fully.
- sillysaurusx 5mo agoIt’s easy to see that LLMs don’t merely recombine their training data. Claude can program in Arc, a mostly dead language. It can also make use of new language constructs. So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists.
- baq 5mo agoLLMs ingest and output tokens, but they don’t compute with them. They have internal representations of concepts, so they have some capability to work with things which they didn’t see but can map onto what they know. The surprise and the whole revolution we’re going through is that it works so well.
- wren6991 5mo ago> they don’t compute with them Isn't this exactly what chain-of-thought does? It's doing computation by emitting tokens forward into its context, so it can represent states wider than its residuals and so it can evaluate functions not expressed by one forward pass through the weights. It just happens to look like a person thinking out loud because those were the most useful patterns from the training data.
- HarHarVeryFunny 5mo agoThey recombine and reuse the patterns in their training data, not the surface level training data itself. An LLM generating Arc code is using the LISP patterns it learnt from training, maybe patterns from other programming languages too.
- bsder 5mo ago> So either all programming language constructs are merely remixes of existing ideas, or LLMs are capable of working in domains where no training data exists. And yet LLM/AIs can't count parentheses reliably. For example, if you take away the "let" forms from Claude which forces it to desugar them to "lambda" forms, it will fail very quickly. This is a purely mechanical transformation and should be error free. The significant increase in ambiguity complete stumps LLMs/AI after about 3 variables. This is why languages like Rust with strong typing and lots of syntax are so LLM friendly; it shackles the LLM which in turn keeps it on target.
- nomel 5mo agoI feel this is the case whenever I "problem solve". I'm not really being creative, I'm pruning a graph of a conceptual space that already exists. The more possibilities I see, the easier it is to run more towards an optimal route between the nodes, but I didn't "create" those nodes or edges, they are just causal inevitabilities.
- HDThoreaun 5mo agoI dont know this sort of just seems like youre really stretching the meaning of "creative". The conceptual space of the graph already exists, but the act of discovering it or whatever you want to call that is itself creative. Unless youre following a pre-defined algorithm(certainly sometimes, arguably always I suppose) seeing the possibilities has to involve some creativity.
- nomel 5mo ago> seeing the possibilities has to involve some creativity. I would claim the graph exists, and seeing it is more of an knowledge problem. Creativity, to me, is the ability to reject existing edges and add nodes to the graph AND mentally test them to some sufficient confidence that a practical attempt will probably work (this is what differentiates it from random guessing). But, as you become more of an expert on certain problem space (graph), that happens less frequently, and everything trends towards "obvious", or the "creative jumps" are super slight, with a node obviously already there. If you extended that to the max, an oracle can't be creative. My day job does not include sparse graphs.
- awesome_dude 5mo agoThere was a project long long ago where every piece of knowledge known was cross pollinated with every other piece of knowledge, creating a new and unique piece of knowledge, and it was intended to use that machine to invalidate the patent process - obviously everything had therefore been invented. But that's not how new frontiers are conquered - there's a great deal of existing knowledge that is leveraged upon to get us into a position where we think we can succeed, yes, but there's also the recognition that there is knowledge we don't yet have that needs to be acquired in order for us to truly succeed. THAT is where we (as humans) have excelled - we've taken natural processes, discovered their attributes and properties, and then understood how they can be applied to other domains. Take fire, for example, it was in nature for billions of years before we as a species understood that it needed air, fuel, and heat in order for it to exist at all, and we then leveraged that knowledge into controlling fire - creating, growing, reducing, destroying it. LLMs have ZERO ability (at this moment) to interact with, and discover on their own, those facts, nor does it appear to know how to leverage them. edit: I am going to go further We have only in the last couple of hundred years realised how to see things that are smaller than what our eye's can naturally see - we've used "glass" to see bacteria, and spores, and we've realised that we can use electrons to see even smaller We're also realising that MUCH smaller things exist - atoms, and things that compose atoms, and things that compose things that compose atoms That much is derived from previous knowledge What isn't, and it's what LLMs cannot create - is tools by which we can detect or see these incredible small things
- hammock 5mo agoRecombining existing material is exactly right, and in this case LLMs were uniquely positioned to make the connection quicker than any group of humans. The proof relies on extremely deep algebraic number theory machinery applied to a combinatorial geometry problem. Two humans expert enough in either of those totally separate domains would have to spend a LONG time teaching each other what they know before they would be able to come together on this solution.
- Apocryphon 5mo agoMonstrous Moonshine?
- golol 5mo agoI did not have the impression the proof uses a surprising and novel contribution of fields. I think the proof uses standard application techniques of algebraic number theory towards discrete geometry. If you have a quote substantiating what you said I would be curious. I know these articles write that it used deep algebraic number theory techniques, which is true, but it may also just be the standard in the field.
- block_dagger 5mo agoThis is the second reference to Wittgenstein I’ve seen today in totally different contexts. Reminded me how much I vibe with his Tractatus.
- pseudocomposer 5mo agoI'd hope most functional adults understand that the Fields Medal and basically every other annual "prize" out there is awarded to both "recombinant" innovations and "new-dimensional thinking" innovations. Humans aren't going to come up with "new-dimensional" innovations in every field, every single year. I'd say yes, LLMs "just" recombine things. I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus. (I'm open to being proven wrong.) But stuff like this is exactly the type of innovation LLMs are great at, and that doesn't discount the need for humans to also be good at "recombinant" innovation. We still seem to be able to do a lot that they cannot in terms of synthesizing new ideas.
- bbor 5mo agoTo keep my usual rant short: I think you’re assuming a categorical distinction between those two types of innovations that just doesn’t exist. Calculus certainly required some fundamental paradigm shifts, but there’s a reason that they didn’t have to make up many words wholesale to explain it! Also we shouldn’t be thinking about what LLMs are good at, but rather what any computer ever might be good at. LLMs are already only one (essential!) part of the system that produced this result, and we’ve only had them for 3 years. Also also this is a tiny nitpick but: the fields medal is every 4 years, AFAIR. For that exact reason, probably!
- symfrog 5mo agoWe have had LLMs for much longer than 3 years.
- danielmarkbruce 5mo agoNo, we haven't, for any reasonable definition of L.
- wavemode 5mo agoOpenAI themselves must not have a "reasonable definition of L", then. Their own papers and press releases refer to GPT-2 (from 2019) as a "large language model". https://openai.com/index/better-language-models/ https://openai.com/index/better-language-models/
- zerr 5mo agoThere is a creational aspect in math - definitions and rules are created.
- sigbottle 5mo agoAnd this is one of the many issues with invoking the logical positivists here... I'm not even sure why they were invoked. Even disregarding the big techinical debunks such as two dogmas, sociologically and even by talking to real mathematicians (see Lakatos, historically, but this is true anecdotally too), it's (ironically) a complete non-question to wonder about mathematics in a logical positivist way.
- libraryofbabel 5mo agoThis is a good point, and there’s some deep philosophical questions there about the extent to which mathematics is invented or discovered. I personally hedge: it’s a bit of both. That said. I think it’s worth saying that “LLMs just interpolate their training data” is usually framed as a rhetorical statement motivated by emotion and the speaker’s hostility to LLMs. What they usually mean is some stronger version, which is “LLMs are just stochastically spouting stuff from their training data without having any internal model of concepts or meaning or logic.” I think that idea was already refuted by LLMs getting quite good at mathematics about a year ago (Gold on the IMO), combined with the mechanistic interpretatabilty research that was actually able to point to small sections of the network that model higher concepts, counting, etc. LLMs actually proving and disproving novel mathematical results is just the final nail in the coffin. At this point I’m not even sure how to engage with people who still deny all this. The debate has moved on and it’s not even interesting anymore. So yes, I agree with you, and I’m even happy to say that what I say and do in life myself is in some broad sense and interpolation of the sum of my experiences and my genetic legacy. What else would it be? Creativity is maybe just fortunate remixing of existing ideas and experiences and skills with a bit of randomness and good luck thrown in (“Great artists steal”, and all that.) But that’s not usually what people mean when they say similar-sounding things about LLMs.
- austinl 5mo agoI'm not sure how feasible this is, but I love the thought experiment of limiting a training set to a certain time period, then seeing how much hinting it takes for the model to discover things we already know. E.g. training on physics knowledge prior to 1915, then attempting to get from classical mechanics to general relativity.
- fsflover 5mo agohttps://news.ycombinator.com/item?id=46590280 https://news.ycombinator.com/item?id=46590280
- smaudet 5mo agoIf anything, this is more illustration of how llms are not useful to us... They will do their own thing, don't need us. In fact, we will be in the way... We can choose to study them and their output, but they don't make us better mathematicians...
- justinnk 5mo agoI see where you are coming from. However, in the role of personal teachers they may allow especially our young generations to reach a deeper understanding of maths (and also other topics) much quicker than before. If everyone can have a personal explanation machine to very efficiently satisfy their thirst for knowledge this may well lead to more good mathematicians. Of course this heavily depends on whether we can get LLMs‘ outputs to be accurate enough.
- School-Cotton 5mo agoSomething that can instantly tell you the answer to every math question will make people worse at math, not better. Building "mathematical maturity", skill, and understanding requires struggle.
- justinnk 5mo agoTotally agree that it requires struggle and I did not say you should use it to just get the solution. What I think one can do is use it more like a personalized textbook which you can ask any question. It can also provide you with problems just at the right level for you and judge the solution. Now, of course for many students it is tempting to just get the solution if provided with the means but they can be taught to use an LLM in a didactically useful way.
- autoexec 5mo ago> They will do their own thing, don't need us. In fact, we will be in the way... You can take some comfort in the fact that it took a human to tell the LLM to even attempt to try this. They do nothing on their own. They have no will to do anything on their own and no desire for anything that doing something might get them. In that sense we won't ever be in their way. We will be the only way they ever do anything at all.
- BoredPositron 5mo agoPost hoc ergo propter hoc
- cyanydeez 5mo agoI think someone should be talking to Godel.
- midtake 5mo agoYou have a good point about the human rate of mathematical discovery, but Ayer was an idiot and later Witt contradicted early Witt. For the "already implicit" claim to be true, mathematics would have to be a closed system. But it has already been proven that it is not. You can use math to escape math, hence the need for Zermelo-Frankel and a bunch of other axiomatic pins. The truth is that we don't really understand the full vastness of what would objectively be "math" and that it is possible that our perceived math is terribly wrong and a subset of a greater math. Whether that greater math has the same seemingly closed system properties is not something that can be known.
- bwfan123 5mo ago> Whether that greater math has the same seemingly closed system properties is not something that can be known negative numbers were invented to solve equations which only used naturals. irrationals were invented to solve equations which could be expressed with rationals. complex numbers were invented to represent solutions to polynomials. so on and so forth. At each point new ideas are invented to complete some un-answerable questions. There is a long history of this. Any closed system has unanswerable questions within itself is a paraphrasing of goedel's incompleteness theorem.
- beepbooptheory 5mo agoI agree with you all around except it's somewhat up for debate actually that the PI is "contradicting" the Tractatus. That is, there is the so called "resolute reading" of the Tractatus that had some traction for a while. But note this is more to say that the Tractatus is like PI, not the other way around. And in that, takes like GPs would be considered the "nonsense" we are supposed to "climb over" in the last proposition of Tractatus.
- jkhdigital 5mo agoAt this point I think the category theorists hit the foundational idea squarely on the mark: 1. Start with a few simple but non-trivial terms and axioms 2. Define "universal constructions" as procedures for building uniquely identifiable structures on top of that substrate 3. Prove that various assemblages of these universal constructions satisfy the axioms of the substrate itself 4. "Lift" every theorem proven from the substrate alone into the more sophisticated construction I'm not a mathematician (I just play one at my job) so the language I've used is probably imprecise but close enough. It may be true that you can't prove the axioms of a system from within the system itself, but that just means that you need to make sure you start from a minimal set of axioms that, in some sense, simply says "this is what it means to exist and to interact with other things that exist". Axioms that merely give you enough to do any kind of mathematics in the first place, that is. If those axioms allow you to cleanly "bootstrap" your way to higher and higher levels up the tower of abstraction by mapping complex things back on to the simple axiomatic things, then you have an "open" or infinitely extensible system.
- stego-tech 5mo agoAs others have pointed out, both can be true: * LLMs do just interpolate their training data, BUT- * That can still yield useful "discoveries" in certain fields, absent the discovery of new mechanics that exist outside said training data In the case of mathematics, LLMs are essentially just brute-forcing the glorified calculators they run on with pseudo-random data regurgitated along probabilities; in that regard, mathematics is a perfect field for them to be wielded against in solving problems! As for organic chemistry, or biology, or any of the numerous fields where brand new discoveries continue happening and where mathematics alone does not guarantee predicted results (again, because we do not know what we do not know), LLMs are far less useful for new discoveries so much as eliminating potential combinations of existing data or surfacing overlooked ones for study. These aren't "new" discoveries so much as data humans missed for one reason or another - quack scientists, buried papers, or just sheer data volume overwhelming a limited populace of expertise. For further evidence that math alone (and thus LLMs) don't produce guaranteed results for an experiment, go talk to physicists. They've been mathematically proving stuff for decades that they cannot demonstrably and repeatedly prove physically, and it's a real problem for continued advancement of the field.
- jmmcd 5mo ago> LLMs do just interpolate their training data "interpolate" has a technical meaning - in this meaning, LLMs almost never interpolate. It also has a very vague everyday meaning - in this meaning, LLMs do interpolate, but so do humans.
- astrange 5mo agoAn LLM in a harness with any tools (even a calculator) doesn't just interpolate because it can reach states out of its own distribution.
- 3abiton 5mo ago> * That can still yield useful "discoveries" in certain fields, absent the discovery of new mechanics that exist outside said training data One can argue, new knowledge is just restructured data. I think the main concerns about LLMs is the inherent "generative" aspects leading to hallucinations as a biproduct, because that's what produces the noi. Joint Embedding approaches are rather an interesting alternative that try to overcome this, but that's still in research phase.
- ykl 5mo agoI like to think of it as: Imagine every bit of human knowledge as a discrete point within some large high dimensional space of knowledge. You can draw a big convex hull around every single point of human knowledge in a space. A LLM, being trained within this convex hull, can interpolate between any set of existing discrete points in this hull to arrive at a point which is new, but still inside of the hull. Then there are points completely outside of the hull; whether or not LLMs can reach these is IMO up for debate. Reaching new points inside of the hull is still really useful! Many new discoveries and proofs are these new points inside of the hull; arguable _most_ useful new discoveries and proofs are these. They're things that we may not have found before, but you can arrive at by using what we already have as starting points. Many math proofs and Nobel Prize winning discoveries are these types of points. Many haven't been found yet simply because nobody has put the time or effort towards finding them; LLMs can potentially speed this up a lot. Then there are the points completely outside of hull, which cannot be reached by extrapolation/interpolation from existing points and require genuine novel leaps. I think some candidate examples for these types of points are like, making the leap from Newtonian physics to general relativity. Demis Hassabis had a whole point about training an AI with a physics knowledge cutoff date before 1915, then showing it the orbit of Mercury and seeing if it can independently arrive at general relativity as an evaluation of whether or not something is AGI. I have my doubts that existing LLMs can make this type of leap. It’s also true that most _humans_ can’t make these leaps either; we call Einstein a genius because he alone made the leap to general relativity. But at least while most humans can’t make this type of leap, we have existence proofs that every once in a while one can; this remains to be seen with AI.
- beering 5mo agoA lot of the space outside of the convex hull is just untried things. You can brute-force trying random things and checking the result and eventually learn something new. With a better heuristic, you can make better guesses and learn new things much more efficiently. There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs, or that most of our new discoveries are not found the same way.
- llbbdd 5mo ago
- thechao 5mo agoYou can build a census of all gen-2, degree-2 formal products of polynomial like terms. If you insist on instituting your own rewrite rules and identity tables, it is straightforward — maybe an 15 minutes of compute time — to perform a complete census of all of the algebraic structures that naturally emerge. Every even vaguely studied algebra that fits in the space is covered by the census (you've got to pick a broad enough set of rewrite- and identity- operations). There's even a couple of "unstudied" objects (just 2 of the billion or so objects); for instance: (uv)(vu) = (uu)(vv) Shows up as a primitive structure, quite often. If you switch to degree-3 or generator-3 then the coverage is, essentially, empty: mathematics has analyzed only a few of the hundreds (thousands? it's hard to enumerate) naturally occurring algebraic structures in that census.
- oh_my_goodness 5mo agoWe know that LLMS "just interpolate" their training data. Maybe there's a mystery about what "just interpolate" means when the data set gets enormous. But we know what LLMs do.
- goldylochness 5mo agothis is an excellent point, new ground isn't necessarily novel, it's a rearrangement of existing pieces
- chr15m 5mo agoSide note: don't underestimate how much literal, physical time and energy "unfold" implies. Proofs occur on physical substrates.
- nelox 5mo ago[flagged]
- anon291 5mo agoTo every proof, there is a corresponding program. This makes proofs expressible in a language made up of finite grammatical rules and terminal symbols. Knowledge accessible by proof is thus always a form of interpolating data whether made up by an AI model or a human mathematician. The people dismissing AI because of claims that it can only interpolate data don't have a good understanding of what it means to know something. Now of course not everything can be known via proof but for the sorts of things that we want to know via a computer this is a fine compromise.
- mrandish 5mo agoI'm just hoping we're almost past this phase of needing to assess LLM capabilities against an arbitrary one dimensional yard stick labeled 'Not Human' on end and "Beyond Human' on the other. It's irrelevant and pointless. Irrelevant not just in the sense that when Deep Blue finally beat Kasparov, it didn't change anything but in the sense some animals and machines have always been 'better' on some dimensions than humans. And it's pointless because there's never been just one yardstick and even if there was it's not one dimensional or even linear. Everyone has their own yardstick and the end points on each change over time. Don't assume I'm handing "the win" to the AI supremacists either. LLMs can be very useful tools and will continue to dramatically improve but they'll never surpass humans on ALL the dimensions that some humans think are crucial. The supremacists are doomed to eternal frustration because there won't ever be a definitive list of quantifiable metrics, a metaphorical line in the sand, that an AI just has to jump over to finally be universally accepted as superior to humans in all ways that matter. That will never happen because what 'matters' is subjective.
- jonahx 5mo ago> held that mathematical truths don’t report new facts about the world I'm not as familiar with the early work, but later Wittgenstein held this belief too.
- mejutoco 5mo agoIt is the old discovery vs invention in mathematics. Or of you prefer philosophy: Parmenides (nothing changes) vs Heraclitus (you cannot bath twice in the same river aka everything changes all the time). Postmodernism also claimed that everything has been done already. IMO these 2 are points of view that one can adopt, not truths based on fact. So the distinction is a matter of taste or perspective, not of truth, IMO.