Humans have worked out the amplitudes for integer n up to n = 6 by hand, obtaining very complicated expressions, which correspond to a “Feynman diagram expansion” whose complexity grows superexponentially in n. But no one has been able to greatly reduce the complexity of these expressions, providing much simpler forms. And from these base cases, no one was then able to spot a pattern and posit a formula valid for all n. GPT did that.
Basically, they used GPT to refactor a formula and then generalize it for all n. Then verified it themselves.
I think this was all already figured out in 1986 though: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.56... see also https://en.wikipedia.org/wiki/MHV_amplitudes
Heck, it's hard to get authors to do literature search, period: never mind not thoroughly looking for prior art, even well known disgraced papers get citated continue to get possitive citations all the time...
This result, by itself, does not generalize to open-ended problems, though, whether in business or in research in general. Discovering the specification to build is often the majority of the battle. LLMs aren't bad at this, per se, but they're nowhere near as reliably groundbreaking as they are on verifiable problems.
Our actual software implementation is usually pretty simple; often writing up the design spec takes significantly longer than building the software, because the software isn't the hard part - the requirements are. I suspect the same folks who are terrible at describing their problems are going to need help from expert folks who are somewhere between SWE, product manager, and interaction designer.
Feel like it's a bit what I tried to expressed few weeks ago https://news.ycombinator.com/item?id=46791642 namely that we are just pouring computational resources at verifiable problems then claim that astonishingly sometimes it works. Sure LLMs even have a slight bias, namely they do rely on statistics so it's not purely brute force but still the approach is pretty much the same : throw stuff at the wall, see what sticks, once something finally does report it as grandiose and claim to be "intelligent".
What do we think humans are doing? I think it’s not unfair to say our minds are constantly trying to assemble the pieces available to them in various ways. Whether we’re actively thinking about a problem or in the background as we go about our day.
Every once in a while the pieces fit together in an interesting way and it feels like inspiration.
The techniques we’ve learned likely influence the strategies we attempt, but beyond all this what else could there be but brute force when it comes to “novel” insights?
If it’s just a matter of following a predefined formula, it’s not intelligence.
If it’s a matter of assembling these formulas and strategies in an interesting way, again what else do we have but brute force?
How many people have tried to figure out a new maths, a GUT in physics, a more perfect human language (Esperanto for ex.) or programming language, only to fail in the vast majority of their attempts?
Do we think that anything but the majority of the attempts at a paradigm shift will end in failure?
If the majority end in failure, how is that not the same brute force methodology (brute force doesn’t mean you can’t respond to feedback from your failed experiments or from failures in the prevailing paradigms, I take it to just fundamentally mean trying “new” things with tools and information available to you, with the majority of attempts ending in failure, until something clicks, or doesn’t and you give up).
Method A) 30% speed reduction and 80% precision decrease
Method B) 50% speed reduction and 5% precision increase
Method C) 740% speed reduction and 1% precision increase
and we only publish B. It's not brute force[1], but throw noodles at the wall, see what sticks, like the GP said. We don't throw spoons[1], but everything that looks like a noodle has a high chance of been thrown. It's a mix of experience[1] and not enough time to try everything.
[1] citation needed :)
Instead of brute-forcing with infinite options, reduce the problem space by starting with some hunch about the mechanism. Then the hard part that can take decades: synthesize compounds with the necessary traits to alter the mechanism in a favourable way, while minimizing unintended side-effects.
Then try on a live or lab grown specimen and note effectiveness. Repeat the cycle, and with every success, push to more realistic forms of testing until it reaches human trials.
Many drugs that reach the last stage - human trials - often end up being used for something completely other than what they were designed for! One example of that is minoxidil - designed to regular blood pressure, used for regrowing hair!
RLHF is an attempt to push LLMs pre-trained with a dopey reconstruction loss toward something we actually care about: imagine if we could find a pre-training criterion that actually cared about truth and/or plausibility in the first place!
> I think this was all already figured out in 1986 though
Naively, the n-gluon scattering amplitude involves order n! terms. Famously, for the special case of MHV (maximally helicity violating) tree amplitudes, Parke and Taylor [11] gave a simple and beautiful, closed-form, single-term expression for all n.
I think this is a prime example of where it is easy to think something is solved when looking at things from a high level but making an erroneous conclusion due to lack of domain expertise. Classic "Reviewer 2" move. Though I'm not a domain expert and so if there was no novelty over Parke and Taylor I'm pretty sure this will get thrashed in review.
Sorry but I just have to point out how this field of maths read like Star Trek technobabble too me.
trekify/SKILL.md: https://github.com/SimHacker/moollm/blob/main/skills/trekify...
Slightly OT, but wasn't this supposed to be largely solved with amplituhedrons?
Can humans actually do that? Sometimes it appears as if we have made a completely new discovery. However, if you look more closely, you will find that many events and developments led up to this breakthrough, and that it is actually an improvement on something that already existed. We are always building on the shoulders of giants.
You could nitpick a rebuttal, but no matter how many people you give credit, general relativity was a completely novel idea when it was proposed. I'd argue for special relatively as well.
I'm not sure about GR, but I know that it is built on the foundations of differential geometry, which Einstein definitely didn't invent (I think that's the source of his "I assure you whatever your difficulties in mathematics are, that mine are much greater" quote because he was struggling to understand Hilbert's math).
And really Cauchy, Hilbert, and those kinds of mathematicians I'd put above Einstein in building entirely new worlds of mathematics...
"Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."
:)
Source: https://www.newtonproject.ox.ac.uk/view/texts/normalized/THE...
And Newton was famously interested in dark religous interference in worldly affairs - what today we would call The Occult. When he did finally succeed in finding his force for moving objects at a distance, without need for an intervening body, he gave credit to these supernatural entities - at least that is how this quote was taken in his day. This religious context is not well known today, nor is Newton's difficult character, so today it is easy to take the quote out of context. Newton was (likely) not disputing the validity of his discovery, rather, he was invoking one of his passions (The Occult) in the affairs of one of his successful passions (finding a force to move distant objects).
It should be noted that some of Newton's successful religious work is rarely attributed to him. For a prominent example, it was Newton that calculated Jesus's birth to be 4 BC, not 1 AD as was the intention of the new calendar.
> In 1902, Henri Poincaré published a collection of essays titled Science and Hypothesis, which included: detailed philosophical discussions on the relativity of space and time; the conventionality of distant simultaneity; the conjecture that a violation of the relativity principle can never be detected; the possible non-existence of the aether, together with some arguments supporting the aether; and many remarks on non-Euclidean vs. Euclidean geometry.
https://en.wikipedia.org/wiki/History_of_special_relativity
Now, if I had to pick a major idea that seemed to drop fully-formed from the mind of a genius with little precedent to have guided him, I might personally point to Galois theory (https://en.wikipedia.org/wiki/Galois_theory). (Ironically, though, I'm not as familiar with the mathematical history of that time and I may be totally wrong!)
> The quintic was almost proven to have no general solutions by radicals by Paolo Ruffini in 1799, whose key insight was to use permutation groups, not just a single permutation.
Thing is, I am usually the kind of person who defends the idea of a lone genius. But I also believe there is a continuous spectrum, no gaps, from the village idiot to Einstein and beyond.
Let me introduce, just for fun, not for the sake of any argument, another idea from math which I think it came really out of the blue, to the degree that it's still considered an open problem to write an exposition about it, since you cannot smoothly link it to anything else: forcing.
As for general relativity, he spent several years working to learn differential geometry (which was well developed mathematics at the time, but looked like abstract nonsense to most physicists). I’m not sure how he was turned on to this theory being applicable to gravity, but my guess is that it was motivated by some symmetry ideas. (It always come down to symmetry.)
> Critique of absolute time and space of Newtonian physics was already well underway
> Many of the phenomena that relativity would later explain under a consistent framework already had independent quasi-explanations hinting at the more universal theory.
Your argument really makes the claim that since there are others pursuing similar directions that this means it is in distribution. I'll use a classic statistics style framing. Suppose we have a bag with n red balls and p blue balls. Someone walks over and says "look, I have a green ball" and someone else walks over and says "I have a purple one" and someone else comes over and says "I have a pink one!". None of those balls were from the bag we have. There are still n+p balls in our bag, they are still all red or blue despite there being n+p+3 balls that we know of.
> I am not a [...] physicist
I only believe that (1) if it hadn't been Einstein, it would very soon have been someone else using very similar concepts and evidence, (2) "completely novel idea" is a stricter criterion than "not in distribution," and (3) better examples of completely novel ideas from history exist as a benchmark for this sort of things.
> Without a formal study of physics it is really hard to differentiate these kinds of propositions. It can be very hard even with that education. So be careful to not overly abstract and simplify concepts. It'll only deprive you of a lot of beauty and innovation.
I agree, but with the caveat that I think ancestor worship is also an impediment to understanding our intellectual and cultural heritage. Either all of human creativity deserves to be treated sacredly, or none of it does.
> To be clear, I don't think coming up with relativity was "in distribution" based on the results of the time.
> only believe that (1) if it hadn't been Einstein, it would very soon have been someone else
> (2) "completely novel idea" is a stricter criterion than "not in distribution,"
> (3) better examples of completely novel ideas from history exist
> I think ancestor worship is also an impediment to understanding our intellectual and cultural heritage
I think the best way to avoid the problem is to remember "my understanding is limited" and always will be. At least until we somehow become omniscient, but I'm not counting on that ever happening.
https://en.wikipedia.org/wiki/Prat%C4%ABtyasamutp%C4%81da https://iep.utm.edu/processp/
Edit: but even it likely relied on his prior experience with nondualistic Hinduisms, of course.
```ai-slop
But wait, this equation is too simple, I need to add more terms or it won't model the universe. Let me think about this again. I have 5 equations and I combined them and derived e=mc^2 but this is too simple. The universe is more complicated. Let's try a different derivation. I'll delete the wrong outputs first and then start from the input equations.
<Deletes files with groundbreaking discovery>
Let me think. I need to re-read the original equations and derive a more complex formula that describes the universe.
<Re-reads equation files>
Great, now I have the complete picture of what I need to do. Let me plan my approach. I'm ready. I have a detailed plan. Let me check some things first.
I need to read some extra files to understand what the variables are.
<Reads the lunch menu for the next day>
Perfect. Now I understand the problem fully, let me revise my plan.
<Writes plan file>
Okay I have written the plan. Do you accept?
<Yes>
Let's go. I'll start by creating a To Do list:
- [ ] Derive new equation from first principles making sure it's complex enough to describe reality.
- [ ] Go for lunch. When the server offers tuna, reject it because the notes say I don't like fish.
```
(You know what's really sad? I wrote that slop without using AI and without referring to anything...)
It is absolutely true that someone else would have come up with special relativity very soon after Einstein. All that would be necessary is someone else to have the wherewithal to say "perhaps the aether does not need to exist" for the equations already known at the time by others before Einstein to lead to the general theory.
General relativity is different. Witten contends that it is entirely possible that without Einstein, we may have had to wait for the early string theorists of the 1960s to discover GR as a classical limit of the first string theories in their quest to understand the strong nuclear force.
As opposed to SR, GR is one of the most singular innovative intellectual achievements in human history. It's definitely "out of distribution" in some sense.
General relativity was a completely novel idea. Einstein took a purely mathematical object (now known as the Einstein tensor), and realized that since its coveriant derivative was zero, it could be equated (apart fron a constant factor) to a conserved physical object, the energy momentum tensor (except for a constant factor). It didn't just fall out of Riemannian geometry and what was known about physics at the time.
Special relativity was the work of several scientists as well as Einstein, but it was also a completely novel idea - just not the idea of one person working alone.
I don't know why anyone disputes that people can sometimes come up with completely novel ideas out of the blue. This is how science moves forward. It's very easy to look back on a breakthrough and think it looks obvious (because you know the trick that was used), but it's important to remember that the discoverer didn't have the benefit of hindsight that you have.
The process you’re describing is humans extending our collective distribution through a series of smaller steps. That’s what the “shoulders of giants” means. The result is we are able to do things further and further outside the initial distribution.
So it depends on if you’re comparing individual steps or just the starting/ending distributions.
But yes, it is not yet clear to what degree there can be (non-linear) extrapolation in the learned semantic spaces here.
From my reading yes, but I think I am likely reading the statement differently than you are.
> from first principles
Doing things from first principles is a known strategy, so is guess and check, brute force search, and so on.
For an llm to follow a first principles strategy I would expect it to take in a body of research, come up with some first principles or guess at them, then iteratively construct and tower of reasonings/findings/experiments.
Constructing a solid tower is where things are currently improving for existing models in my mind, but when I try openai or anthropic chat interface neither do a good job for long, not independently at least.
Humans also often have a hard time with this in general it is not a skill that everyone has and I think you can be a successful scientist without ever heavily developing first principles problem solving.
Even the realm of pure mathematics and elegant physic theories, where you are supposed to take a set of axioms ("first principles") and build something with it, has cautionary tales such as the Russel paradox or the non-measure of Feymann path integrals, and let's not talk about string theory.
> Can humans actually do that?
Seriously, think about it for a second...
If that were true then science should have accelerated a lot faster. Science would have happened differently and researchers would have optimized to trying to ingest as many papers as they can.
Dig deep into things and you'll find that there are often leaps of faith that need to be made. Guesses, hunches, and outright conjectures. Remember, there are paradigm shifts that happen. There are plenty of things in physics (including classical) that cannot be determined from observation alone. Or more accurately, cannot be differentiated from alternative hypotheses through observation alone.
I think the problem is when teaching science we generally teach it very linearly. As if things easily follow. But in reality there is generally constant iterative improvements but they more look like a plateau, then there are these leaps. They happen for a variety of reasons but no paradigm shift would be contentious if it was obvious and clearly in distribution. It would always be met with the same response that typical iterative improvements are met with "well that's obvious, is this even novel enough to be published? Everybody already knew this" (hell, look at the response to the top comment and my reply... that's classic "Reviewer #2" behavior). If it was always in distribution progress would be nearly frictionless. Again, with history in how we teach science we make an error in teaching things like Galileo, as if The Church was the only opposition. There were many scientists that objected, and on reasonable grounds. It is also a problem we continually make in how we view the world. If you're sticking with "it works" you'll end up with a geocentric model rather than a heliocentric model. It is true that the geocentric model had limits but so did the original heliocentric model and that's the reason it took time to be adopted.
By viewing things at too high of a level we often fool ourselves. While I'm criticizing how we teach I'll also admit it is a tough thing to balance. It is difficult to get nuanced and in teaching we must be time effective and cover a lot of material. But I think it is important to teach the history of science so that people better understand how it actually evolves and how discoveries were actually made. Without that it is hard to learn how to actually do those things yourself, and this is a frequent problem faced by many who enter PhD programs (and beyond).
> We are always building on the shoulders of giants.
It's probably worth reading The Unreasonable Effectiveness of Mathematics in the Natural Sciences. It might seem obvious now but read carefully. If you truly think it is obvious that you can sit in a room armed with only pen and paper and make accurate predictions about the world, you have fooled yourself. You have not questioned why this is true. You have not questioned when this actually became true. You have not questioned how this could be true.
https://www.hep.upenn.edu/~johnda/Papers/wignerUnreasonableE...
You are greater than the sum of your partsThese have been identified as various things. Eureka moments, strokes of genius, out of the box thinking, lateral thinking.
LLMs have not shown to be capable of this. They might be in the future, but they havent yet
So that's actually 2 different regimes on how to proceed. Both are useful but arguably breaking off of the current paradigm is much harder and thus rare.
There are genuine creative insights that come from connecting two known semantic spaces in a way that wasn't obvious before (e.g, novel isomorphism). It is very conceivable that LLMs could make this kind of connection, but we haven't really seen a dramatic form of this yet. This kind of connection can lead to deep, non-trivial insights, but whether or not it is "out-of-distribution" is harder to answer in this case.
Five years ago we were at Stage 1 with LLMs with regard to knowledge work. A few years later we hit Stage 2. We are currently somewhere between Stage 2 and Stage 3 for an extremely high percentage of knowledge work. Stage 4 will come, and I would wager it's sooner rather than later.
IOW respect the trend line.
And the same practitioners said right after deep blue that go is NEVER gonna happen. Too large. The search space is just not computable. We'll never do it. And yeeeet...
People have been downplaying LLMs since the first AI-generated buzzword garbage scientific paper made its way past peer review and into publication. And yet they keep getting better and better to the point where people are quite literally building projects with shockingly little human supervision.
By all means, keep betting against them.
The LLMs are very fast but the code they generate is low quality. Their comprehension of the code is usually good but sometimes they have a weightfart and miss some obvious detail and need to be put on the right path again. This makes them good for non-experienced humans who want to write code and for experienced humans who want to save time on easy tasks.
I think the latest generation of LLM with claude code is not low quality. It's better than the code that pretty much every dev on our team can do outside of very narrow edge cases.
In chess, there's a clear goal: beat the game according to this set of unambiguous rules.
In science, the goals are much more diffuse, and setting those in the first place is what makes a scientist more or less successful, not so much technical ability. It's a very hierarchical field where permanent researchers direct staff (postdocs, research scientists/engineers), direct grad students. And it's at the bottom of the pyramid where the technical ability is the most relevant/rewarded.
Research is very much a social game, and I think replacing it with something run by LLMs (or other automatic process) is much more than a technical challenge.
We're talking about significant contributions to theoretical physics. You can nitpick but honestly go back to your expectations 4 years ago and think — would I be pretty surprised and impressed if an AI could do this? The answer is obviously yes, I don't really care whether you have a selective memory of that time.
Whoever wrote the prompts and guided ChatGPT made significant contributions to theoretical physics. ChatGPT is just a tool they used to get there. I'm sure AI-bloviators and pelican bike-enjoyers are all quite impressed, but the humans should be getting the research credit for using their tools correctly. Let's not pretend the calculator doing its job as a calculator at the behest of the researcher is actually a researcher as well.
How much precedence is there for machines or tools getting an author credit in research? Genuine question, I don't actually know. Would we give an author credit to e.g. a chimpanzee if it happened to circle the right page of a text book while working with researchers, leading them to a eureka moment?
Well what do you think ? Do the authors (or a single symbolic one) of pytorch or numpy or insert <very useful software> typically get credits on papers that utilize them heavily? Well Clearly these prominent institutions thought GPT's contribution significant enough to warrant an Open AI credit.
>Would we give an author credit to e.g. a chimpanzee if it happened to circle the right page of a text book while working with researchers, leading them to a eureka moment?
Cool Story. Good thing that's not what happened so maybe we can do away with all these pointless non sequiturs yeah ? If you want to have a good faith argument, you're welcome to it, but if you're going to go on these nonsensical tangents, it's best we end this here.
I don't know! That's why I asked.
> Well Clearly these prominent institutions thought GPT's contribution significant enough to warrant an Open AI credit.
Contribution is a fitting word, I think, and well chosen. I'm sure OpenAI's contribution was quite large, quite green and quite full of Benjamins.
> Cool Story. Good thing that's not what happened so maybe we can do away with all these pointless non sequiturs yeah ? If you want to have a good faith argument, you're welcome to it, but if you're going to go on these nonsensical tangents, it's best we end this here.
It was a genuine question. What's the difference between a chimpanzee and a computer? Neither are humans and neither should be credited as authors on a research paper, unless the institution receives a fat stack of cash I guess. But alas Jane Goodall wasn't exactly flush with money and sycophants in the way OpenAI currently is.
If you don't read enough papers to immediately realize it is an extremely rare occurrence then what are you even doing? Why are you making comments like you have the slightest clue of what you're talking about? including insinuating the credit was what...the result of bribery?
You clearly have no idea what you're talking about. You've decided to accuse prominent researchers of essentially academic fraud with no proof because you got butthurt about a credit. You think your opinion on what should and shouldn't get credited matters ? Okay
I've wasted enough time talking to you. Good Day.
That usually comes up with some support usually.
For a datum of one, the mathematician Doron Zeilberger give credit to his computer Shalosh B. Ekhad on select papers.
https://medium.com/@miodragpetkovic_24196/the-computer-a-mys...
https://sites.math.rutgers.edu/~zeilberg/akherim/EkhadCredit...
I have no problem with the former and agree that authors/researchers must note when they use AI in their research.
for this particular paper it seems the humans were stuck, and only AI thinking unblocked them
In your eyes maybe there's no difference. In my eyes, big difference. Tools are not people, let's not further the myth of AGI or the silly marketing trend of anthropomorphizing LLMs.
― C.S. Lewis, The Last Battle
— Carl Sagan
I have no real way to demonstrate that I'm telling the truth, but I am ¯\_(ツ)_/¯
One way I gauge the significance of a theory paper are the measured quantities and physical processes it would contribute to. I see none discussed here which should tell you how deep into math it is. I personally would not have stopped to read it on my arxiv catch-up
https://arxiv.org/list/hep-th/new
Maybe to characterize it better, physicists were not holding their breath waiting for this to get done.
Probably not something that the average GI Joe would be able to prompt their way to...
I am skeptical until they show the chat log leading up to the conjecture and proof.
I know we've been primed by sci-fi movies and comic books, but like pytorch, gpt-5.2 is just a piece of software running on a computer instrumented by humans.
>I know we've been primed by sci-fi movies and comic books, but like pytorch, gpt-5.2 is just a piece of software running on a computer instrumented by humans.
Sure
Do you really want to be treated like an old PC (dismembered, stripped for parts, and discarded) when your boss is done with you (i.e. not treated specially compared to a computer system)?
But I think if you want a fuller answer, you've got a lot of reading to do. It's not like you're the first person in the world to ask that question.
Not an uncommon belief.
Here you are saying you personally value a computer program more than people
It exposes a value that you personally hold and that's it
That is separate from the material reality that all this AI stuff is ultimately just computer software... It's an epistemological tautology in the same way that say, a plane, car and refrigerator are all just machines - they can break, need maintenance, take expertise, can be dangerous...
LLMs haven't broken the categorical constraints - you've just been primed to think such a thing is supposed to be different through movies and entertainment.
I hate to tell you but most movie AIs are just allegories for institutional power. They're narrative devices about how callous and indifferent power structures are to our underlying shared humanity
(In the hands of leading experts.)
The humans put in significant effort and couldn’t do it. They didn’t then crank it out with some search/match algorithm.
They tried a new technology, modeled (literally) on us as reasoners, that is only just being able to reason at their level and it did what they couldn’t.
The fact that the experts were a critical context for the model, doesn’t make the models performance any less significant. Collaborators always provide important context for each other.
Was the initial conjecture based on leading info from the other authors or was it simply the authors presenting all information and asking for a conjecture?
Did the authors know that there was a simpler means of expressing the conjecture and lead GPT to its conclusion, or did it spontaneously do so on its own after seeing the hand-written expressions.
These aren't my personal views, but there is some handwaving about the process in such a way that reads as if this was all spontaneous involvement on GPTs end.
But regardless, a result is a result so I'm content with it.
SpaceX can use an optimization algorithm to hoverslam a rocket booster, but the optimization algorithm didn't really figure it out on its own.
The optimization algorithm was used by human experts to solve the problem.
Is this so different?
LLMs surpassed the average human a long time ago IMO. When LLMs fail to measure up to humans, it's that they fail to measure up against human experts in a given field, not the Average Joe.
We are surrounded by NPCs.
What's the distinction between "first principles" and "existing things"?
I'm sympathetic to the idea that LLMs can't produce path-breaking results, but I think that's true only for a strict definition of path-breaking (that is quite rare for humnans too).
I can claim some knowledge of physics from my degree, typically the easy part is coming up with complex dirty equations that work under special conditions, the hard part is the simplification into something elegant, 'natural' and general.
Also "LLM’s can make new things when they are some linear combination of existing things"
Doesn't really mean much, what is a linear combination of things you first have to define precisely what a thing is?
over long periods of time, checklists are the biggest thing, so the LLM can track whats already done and whats left. after a compact, it can pull the relevant stuff back up and make progress.
having some level or hierarchy is also useful - requirements, high level designs, low level designs, etc
The real question is, what does it cost OpenAI? I'm pretty sure both their plans are well below cost, at least for users who max them out (and if you pay $200 for something then you'll probably do that!). How long before the money runs out? Can they get it cheap enough to be profitable at this price level, or is this going to be "get them addicted then jack it up" kind of strategy?
Compute costs will fall drastically for existing models
But it's likely that frontier models of the future won't be released to the public at all, because they'll be too good
Benchmarks are one thing, real world performance is very different.
Agree with this. I’ve been trying to make LLMs come up with creative and unique word games like Wordle and Uncrossy (uncrossy.com), but so far GPT-5.2 has been disappointing. Comparatively, Opus 4.5 has been doing better on this.
But it’s good to know that it’s breaking new ground in Theoretical Physics!
The search space is much too rich to be explored anything but greedily in timid steps off the trodden path, and the frameworks (arbitrary) set both the highways and the vehicles by which we move along and out of them.
Now, the argument can be made that the "meta-mathematical" (but outright mathematical, really) setting of frameworks follows the same structure, and LLMs could also explore that space.
Even assuming that, a major roadblock remains: mathematics should remain understandable by humans, and yield fast progress in desirable (by whom? until now, by humans) directions, so the constraints on admissible frameworks are not as simple as "yields to coherent results".
Also, to take a step back, I wonder what the pertinence of using numerical math is to derive analytical math when we can already solve a great deal of problems through numerical methods. For instance, is it worth spending however many MWh on LLMs to derive an analytical solution to an optimization problem, which might itself be very expensive to compute (human-derived expressions tend to be particularly cheap to evaluate precisely because we are so limited; machines (formal calculus for instance) will happily give you multi-page formulae with thousands of operations to evaluate), when there's a vast array of algorithms at the ready to provide arbitrarily precise solutions?
What remains is the kind of math that, arguably, is much more precious to understand than to derive.
It seems to me that all “new ideas” are basically linear combinations of existing things with exceeding rare exceptions…
Maybe Godel’s Incompleteness?
Darwinian evolution?
General Relativity?
Buddhist non-duality?
Aren't most new things linear combinations of existing things (up to a point)?
Thanks for the summary; but this is a huge hand-wave. was GPT Pro just spinning for 12 hours and returend 42?!
But it's worth thinking more about this. What gives humans the ability to discover "new things"? I would say it's due to our interaction with the universe via our senses, and not due to some special powers intrinsic to our brains that LLMs lack. And the thing is, we can feed novel measurements to LLMs (or, eventually, hook them up to camera feeds to "give them senses")
[0]: https://slatestarcodex.com/2019/02/19/gpt-2-as-step-toward-g...
But I’ve successfully made it build me a great Poker training app, a specific form that also didn’t exist, but the ingredients are well represented on the internet.
And I’m not trying to imply AI is inherently incapable, it’s just an empirical (and anecdotal) observation for me. Maybe tomorrow it’ll figure it out. I have no dogmatic ideology on the matter.
If all ideas are recombinations of old ideas, where did the first ideas come from? And wouldn't the complexity of ideas be thus limited to the combined complexity of the "seed" ideas?
I think it's more fair to say that recombining ideas is an efficient way to quickly explore a very complex, hyperdimensional space. In some cases that's enough to land on new, useful ideas, but not always. A) the new, useful idea might be _near_ the area you land on, but not exactly at. B) there are whole classes of new, useful ideas that cannot be reached by any combination of existing "idea vectors".
Therefore there is still the necessity to explore the space manually, even if you're using these idea vectors to give you starting points to explore from.
All this to say: Every new thing is a combination of existing things + sweat and tears.
The question everyone has is, are current LLMs capable of the latter component. Historically the answer is _no_, because they had no real capacity to iterate. Without iteration you cannot explore. But now that they can reliably iterate, and to some extent plan their iterations, we are starting to see their first meaningful, fledgling attempts at the "sweat and tears" part of building new ideas.
Any countable group is a quotient of a subgroup of the free group on two elements, iirc.
There’s also the concept of “semantic primes”. Here is a not-quite correct oversimplification of the idea: Suppose you go through the dictionary and one word at a time pick a word whose definition includes only other words that are still in the dictionary, and removing them. You can also rephrase definitions before doing this, as long as it keeps the same meaning. Suppose you do this with the goal of leaving as few words in it as you can. In the end, you should have a small cluster of a bit over 100 words, in terms of which all the other words you removed can be indirectly defined. (The idea of semantic primes also says that there is such a minimal set which translates essentially directly* between different natural languages.)
I don’t think that says that words for complicated ideas aren’t like, more complicated?
Ideas seem to just be our abstractions of neural impulses from deep in evolution.
There are in fact ways to directly quantify this, if you are training e.g. a self-supervised anomaly-detection model.
Even with modern models not trained in that manner, looking at e.g. cosine distances of embeddings of "novel" outputs could conceivably provide objective evidence for "out-of-distribution" results. Generally, the embeddings of out-of-distribution outputs will have a large cosine (or even Euclidean) distance from the typical embedding(s). Just, most "out-of-distribution" outputs will be nonsense / junk, so, searching for weird outputs isn't really helpful, in general, if your goal is useful creativity.