[pseudocomposer]

I'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.

[godelski]

  > Humans aren't going to come up with "new-dimensional" innovations in every field, every single year.

In fact, they are more rare. Specifically because they harder to produce. This is also why it is much harder to get LLMs to be really innovative. Human intelligence is a lot of things, it is deeply multifaceted.

Also, I'm not sure why CS people act like axioms are where you start. Finding them is very very difficult. It can take some real innovation because you're trying to get rid of things, not build on top of. True for a lot of science too. You don't just build up. You tear down. You translate. You go sideways. You zoom in. You zoom out. There are so many tools at your disposal. There's so much math that has no algorithmic process to it. If you think it all is, your image is too ideal (pun(s) intended).

But at the same time I get it, it is a level of math (and science) people never even come into contact with. People think they're good at math because they can do calculus. You're leagues ahead of most others around you, yes, and be proud of that. But don't let that distance deceive you into believing you're anywhere near the experts. There's true for much more than just math, but it's easy to demonstrate to people that they don't understand math. Granted, most people don't want to learn, which is perfectly okay too

[bbor]

To 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]

We have had LLMs for much longer than 3 years.

[Nevermark]

I took humans thousands of years, then hundreds of years, to come to terms with very basic concepts about numbers.

Its amazing to me when people talk about recombining things, or following up on things as somehow lesser work.

People can't separate the perspective they were given when they learned the concepts, that those who developed the concepts didn't have because they didn't exist.

Simple things are hard, or everything simple would have been done hundreds of years ago, and that is certainly not the case. Seeing something others have not noticed is very hard, when we don't have the concepts that the "invisible" things right in front of us will teach us.

[adi_kurian]

Anyone in the arts is aware that creativity is not the new, it is the repackaging of what already exists into something that is itself new.

[RajT88]

Except for "Being John Malkovich". That movie was way out there on its own.

[fragmede]

It's "just" a Man-vs-Self story, of the ~7 story archetypes out there.

[godelski]

It's why the invention of teaching has been so important. Took a long time for humans to develop calculus. A long time to then refine it and make it much more useful. But then in a year or two an average person can learn what took hundreds of years to invent. It's crazy to equate these tasks as being the same. Even incremental innovation is difficult. You have to see something billions of people haven't. But there's also paradigm shifts and well... if you're not considered crazy at first then did you really shift a paradigm?

[Nevermark]

And yet it is still taught in less than optimal form, lacking algebraic closure in ways that are completely unnecessary.

It isn't a secret, but the percentage of people who don't know that, plus the percentage of mathematicians who vaguely or more directly know that, but habitually use the broken, more difficult (i.e. less algebraic) notation is ... virtually everyone.

I am not trying to pick on calculus, this is everywhere. Important and useful concepts are right in front of all of us, that we don't see even in the context of what we are relatively fluent with.

Because we learn quickly, where we have (almost always inherited) the right preparatory perspectives (earned over lifetimes by others), we vastly overrate our ability to reason independently.

[bananaflag]

What is that algebraic calculus you are hinting at?

[danielmarkbruce]

No, we haven't, for any reasonable definition of L.

[wavemode]

OpenAI 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/

[danielmarkbruce]

Yes, and 1.5 billion parameters meets no reasonable current definition of large. It would be considered a tiny language model. OpenAI themselves refer to their small/fast models as small models all over their documentation.

[wavemode]

The term doesn't change its meaning because something new comes along.

The point of the term "large" is to highlight the massive parameter count (compared to traditional statistical models, where having 1.5 billion parameters was basically unheard of). It leads to the "double decent" phenomenon that allows them to generalize in ways traditional statistical models can't.

The idea that the "large" descriptor was just a subjective exclamation, like "oh wow this model is pretty large ain't it", is revisionism.

[Yizahi]

Sure we do, since Fei-Fei Li and team created that annotated dataset, which allowed to train first LLMs. So LLMs are here for more than a decade already.

[danielmarkbruce]

You are confused by what the L and L mean in LLM, or which data set she created, or both, or in general.

[Yizahi]

Or it is you who are confused. And I want to remind you that you can't retcon historical word use.

[asdfasgasdgasdg]

When people say this what they mean is that we've had plausibly useful LLMs for around three years, and I would say that is basically true. The stuff before 2023 could barely be classified above the level of an interesting toy.

[asdfasgasdgasdg]

When people say this what they mean is that we've had plausibly useful LLMs for around three years, and I would say that is basically true.

[nextaccountic]

Fine, 8 years? That's not a long time

[m4x]

I think your comment about inventing new words is an interesting one. One of the things that I believe limits our ability to discover new ideas is our ability to describe related concepts. For example, the reason we still can't have clear discussions on consciousness is probably partly due to the fact that the necessary concepts haven't been cemented in language. We need new language before we can describe consciousness.

I would guess LLMs are limited in their ability to be genuinely novel because they are trained on a fixed language. It makes research into the internal languages developed by LLMs during training all the more interesting.

[pegasus]

The fundamental paradigm shift is the categorical distinction. And what would constitute many new words for you? It introduced a bunch of concepts and terms which we take for granted today, including "derivative", "integral", "infinitesimal", "limit" and even "function", the latter two not a new words, but what does it matter? – the associated meanings were new.

[azakai]

There was a lot new in calculus, but it also didn't come out of nowhere.

That Newton and Leibniz came up with similar ideas in parallel, independently, around the same time (what are the odds?), supports that.

https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculu...

[JonathanMerklin]

I agree with almost all of what you have stated, save for a minor nitpick: I frankly don't think most functional adults think about the Fields Medal, similar annual prizes, or the qualities of the innovations of their candidate pools. I also think that that's totally okay. I think among a certain learned cohort of adults it's okay to hope that, and I think it's okay to imagine an idealized world where having an opinion on this sort of matter is a baseline, but I don't think it's realistic or fair to imply that (what I believe handwavily to be a majority of) adults are nonfunctional for not sharing this understanding.

[amelius]

> I still don't think if you trained an LLM with every pre-Newton/Liebniz algebra/geometry/trig text available, it could create calculus.

Yes but that is because there was not enough text available to create an intelligent LLM to begin with.

[hgoel]

I think an LLM trained on pre-calculus material would easily stumble into reinventing at least early calculus. It's already pretty easy for students to stumble into calculus from solid enough fundamentals.

We even think that the Babylonian astronomers figured out they could integrate over velocity to predict the position of Jupiter.

[bananaflag]

Something like this would be interesting:

https://en.wikipedia.org/wiki/Adequality

[kelseyfrog]

> 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.)

The experiment is feasible. If it were performed and produced a positive result, what would it imply/change about how you see LLMs?

[pegasus]

GP was stating that they don't believe this would happen (I don't either), but also to make the point that it's a falsifiable view. (At least in theory. In practice, there probably won't even be enough historical text to train an LLM on). No, I don't think it would be falsified. Asking what if I'm wrong is kind of redundant. If I'm wrong, I'm wrong, duh.

[sumeno]

How are you going to train a frontier level llm with no references to post 1700 mathematics?

[bjt]

"frontier level" is doing a lot of work there, but the idea would be to only feed it earlier sources.

There are people working on this.

e.g. https://github.com/haykgrigo3/TimeCapsuleLLM

[lovecg]

The problem is the amount of data with that cutoff is really minuscule to produce anything powerful. You might be able to generate a lot of 1700s sounding data, you’d have to be careful not to introduce newer concepts or ways of thinking in that synthetic data though. A lot of modern texts talk about rates of change and the like in ways that are probably influenced by preexisting knowledge of calculus.

[NewJazz]

Doesn't it prove GP's point then, that LLMs themselves simply aren't capable of creating/proving new theories and axioms?

[codebje]

Without passing opinion on GP's point, I think that just proves it's hard to establish a data set that doesn't bias toward the result you're hoping to find.

[kelseyfrog]

Time cutoff LLMs are regularly posted to HN. It takes just one success to prove feasibility.

Besides, we can forecast our thoughts and actions to imagined scenarios unconditioned on their possibility. Something doesn't have to be possible for us to imagine our reactions.

[anthk]

Archimede was close.

[nathan_compton]

I don't think its really feasible - there just isn't enough training data before calculus. I would guess all the mathematical and philosophical texts available to Newton and Leibniz would fit on a CD-ROM with loads of space to spare.