[dvt]

> 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.

[pulkitsh1234]

Creation 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.

[bwfan123]

> 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 ?

[pbhjpbhj]

>a "telescope" is not a new creation

It was a new concept, combining lenses to look at things far away as if they are close to. The literal atoms/molecules weren't new, but the form they were arranged in was. The purpose of the arrangement was new too.

[IAmGraydon]

I’ve long been fascinated by this idea. It’s interesting that in religious texts, god is often called the “Creator” and that is what differentiates him from man. To be able to create would be to be a god.

[ulbu]

that’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).

[kenjackson]

"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]

The map is not the territory.

[cthalupa]

I 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]

> 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]

Well, 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.

[grey-area]

I think you're saying a pithy saying proves nothing (Voltaire), which is true; sometimes it summarises a line of argument though.

Math is a mental map which coincides with reality in useful ways. Different maps can also be useful. The models we construct are based on arbitrary axioms which we hold to be true. Different axioms could lead to different theories which are just as useful. So it isn't discovered (i.e. mapping directly to reality and waiting to be discovered), it is created.

To pick one example, adding the concept of zero changed our model/map of reality fundamentally without changing reality.

[cthalupa]

You have a minority view on this argument, though. Scientific and structural realism both reject the idea that math is just a map. You've got company with the instrumentalists and antirealists, but the majority consensus is that math is somewhere between the structure underlying the territory to all the territory.

Zero was already part of the territory. Lack of something is a very normal state in the universe. Once we added it to our understanding of math, we were discovering it, not creating it. Of course people who are scientific or structural realists would agree it didn't change reality - because reality already had it, whether we knew it or not.

[bbor]

Does that correction matter, tho…? Discovered or created, it would be new to us, and is clearly not easy to reach!

[black_knight]

Where does this mathematics exist before we discover it?

I know of no realm where mathematical objects live except human minds.

No, it seems clear to me that mathematics is a creation of our minds.

[hackinthebochs]

If it were merely a creation, there would be no reason for two independent mathematicians to land on the same creation given some directed effort. But of course we do see that. There is an objectivity to mathematics that must be accounted for.

"Where" mathematics exists is in the abstract combinatorical space of an infinite repeating application of logical rules. This space doesn't exist in a substantive sense, but it is accessible/navigable by studying the consequences of logical rules. It is the space of possible structure.

[black_knight]

If this space of possible structure is real, but seemingly immaterial, how does our matter brain access it?

I think we create mathematics as thought structure in our mind. We can agree on things when we create the same structures. But this structure did not exist prior to creation.

[hackinthebochs]

I don't know what real means; I might call it real depending on one's definition. I definitely wouldn't call it immaterial (though it's not material either). We access it by construction: apply relevant rules and discover their consequences. Two people probing this structure are equally constrained by the requirements of consistency. There is no Benacerraf-style access problem.

[bbor]

  math 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]

Math is not based on science!

Maths follows logical (or even mathematical) rigour, not scientific rigour!

[bbor]

You'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]

I 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]

It 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]

What 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]

Sure. 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-a...

[golol]

I think based on the class of problem that RH is an independence result is not something that "really happens".

[Tenobrus]

what basis do you have for assuming an LLM is fundamentally incapable of doing this?

[truncate]

What'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.

[EMM_386]

"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]

>> 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.

[dist-epoch]

LLMs are already making the next great songs. Just check out the Billboard charts.

[truncate]

I'm sorry, I don't consider them "great songs". Obviously, different people have different taste.

[blueone]

> 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]

Good 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]

In 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]

Yeah, 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]

Deep Blue was brute forcing chess in the sense that AlphaGo wasn't brute forcing Go.

[Applejinx]

And today he's got salient observations on politics which hold much of his attention, and Deep Blue is shut off and has done nothing further.

Not a good argument for turning everything over to the Deep Blues. What's Deep Blue done for me lately?

[zardo]

This is something that could be demonstrated rather than just argued.

Train an LLM only on texts dated prior to Newton and see if it can create calculus, derrive the equations of motion, etc.

If you ask it about the nature of light and it directs you to do experiments with a prism I'd say we're really getting somewhere.

[gjm11]

We tried this experiment with humans, back in the 17th century, and only a few[1] out of millions managed it given a whole human lifetime each.

[1] Obviously Newton counts as one. Leibniz like Newton figured out calculus. Other people did important work in dynamics though no one else's was as impressive as Newton's. But the vast majority of human-level intelligences trained on texts prior to Newton did not create calculus or derive the equations of motion or come close to doing either of those things.

[davebren]

Newton did it at 23 and there would have been very few people with mathematical training. The LLM would be trained on the entirety of recorded human knowledge and mathematics up to that point, and would get to use a lot more energy so it still has a massive material advantage over young Isaac. Yet I don't believe calculus would magically appear in its response.

[pickleRick243]

Except this has been said since the 2010's and has been proven wrong again and again. Clearly the theory that LLM's can't "extrapolate" is woefully incomplete at best (and most likely simply incorrect). Before the rise of ChatGPT, the onus was on the labs to show it was plausible. At this point, I think the more epistemologically honest position is to put the burden back on the naysayers. At the least, they need to admit they were wrong and give a satisfactory explanation why their conceptual model was unable to account for the tremendous success of LLM's and why their model is still correct going forward. Realistically, progress on the "anti-LLM" side requires a more nuanced conceptual model to be developed carefully outlining and demonstrating the fundamental deficiencies of LLMs (not just deficiencies in current LLMs, but a theory of why further advancements can't solve the deficiencies).

Incidentally, similar conversations were had about ML writ large vs. classical statistics/methods, and now they've more or less completely died down since it's clear who won (I'm not saying classical methods are useless, but rather that it's obvious the naysayers were wrong). I anticipate the same trajectory here. The main difference is that because of the nature of the domain, everyone has an opinion on LLM's while the ML vs. statistics battle was mostly confined within technical/academic spaces.

[davebren]

> Clearly the theory that LLM's can't "extrapolate" is woefully incomplete at best (and most likely simply incorrect).

What example is there where an LLM has extrapolated? All I've seen is a data set so large and an extra decomposition process making it so interpolation feels like extrapolation if you don't look close enough.

> but a theory of why further advancements can't solve the deficiencies

How about LeCun's?

[dvt]

Because 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.)

[fnordpiglet]

I 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]

People 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]

If 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]

Sorry, 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.

[lukol]

This "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]

> 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.

[jfyi]

Newton and Leibniz were "hand-waving"?

If anything, they were fighting an uphill battle against the perception of hand-waving by their contemporaries.

[dehsge]

It’s not that. Consider the definition of the limit. The idea existed for a long time. Newton/Leibniz had the idea.

That idea wasn’t formally defined until 134 years later with epsilon-delta by Cauchy. That it was accepted. (I know that there were an earlier proofs)

There’s even arguments that the limit existed before newton and lebnitz with Archimedes' Limits to Value of Pi.

Cauchy’s deep understanding of limits also led to the creation of complex function theory.

These forms of creation are hand-wavy not because they are wrong. They are hand wavy because they leverage a deep level of ‘creative-intuition’ in a subject.

An intuition that a later reader may not have and will want to formalize to deepen their own understanding of the topic often leading to deeper understanding and new maths.

[dvt]

> 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.

[baq]

And yet nowadays you can restate all of it using just combinations of sets of sets and some logic operators.

[nh23423fefe]

god of the gaps

[iwontberude]

non overlapping magisteria

[satvikpendem]

What is creativity if not permutation? A brain has some model of the world and recombines concepts to create new concepts.

[KoolKat23]

It pretty much is, otherwise it is randomness or entropy.

[lajamerr]

LLMs 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]

You're proving the GP's argument - LLMs aren't creative you say as much, it's the driving that is the creative force

[lajamerr]

You 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]

Let 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?

[charlie90]

I believe when we have AI Agents "living" 24/7, they will become creative machines. They will test ideas out their own ideas experimentally, come across things accidentally, synthesize new ideas.

We just haven't let AI run wild yet. But its coming.

[awesome_dude]

So are self-driving cars - as they have been for the last... decade or so

AGI has been "just over the horizon" for literal decades now - there have been a number of breakthroughs and AI Winters in the past, and there's no real reason to believe that we've suddenly found the magic potion, when clearly we haven't.

AI right now cannot even manage simple /logic/

[Barbing]

If 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)