I 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.
A 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.
I come back to something like this idea when I consider the distinction being made that LLMs can only combine and interpolate between points in their training material. I could write a brute-force program that just used an English dictionary to produce every possible one-billion-gazillion word permutation of the words within, with no respect for rules of language, and chances are there would be some provable, testable, novel insight somewhere in the results if you had the time to sift through and validate all of it. LLMs seem like a tool that can search that space more effectively than any we've had before.
If we managed to create very fast monkeys with typewriters and software that can review their output quickly enough that we end up with a result that's worth reading we'd still have people insisting that we've created intelligence. The monkeys however remain monkeys.
I think intelligence is an orthogonal, mostly philosophical question aside from whether these tools can produce novel, useful output vs purely recombinant output.
I think that enough purely recombinant output will eventually produce novel, useful output.
I think of most things you can get to by guess and checking as definitionally inside of the hull; most forms of guess and checking are you take some existing thing, randomize a bunch of its parameters, and see what you get. Whereas with something like relativity, there's not even a starting point that you can randomize and guess/check from the pre-existing knowledge space that will lead you to relativity. That's more like, adding a new dimension to the space entirely.
It's possible LLMs can handle this after all! But at least so far we only have existence proofs of humans doing this, not LLMs yet, and I don't think it's easy to be certain how far away LLMs are from doing this. I should distinguish between LLMS and AI more generally here; I'm skeptical LLMs can do this, I think some other kind of more complete AI almost certainly can.
I supposed you could just, I dunno, randomly combine words into every conceivable sentence possible and treat each new sentence as a theory to somehow test and brute force your way through the infinite possible theories you could come up with. But at that point you're closer to the whole infinite random monkeys producing Shakespeare thing than you are to any useful conclusion about intelligence.
I think your point about “you could randomly generate a sequence of words, which could in principle produce a text interpretable as expressing any particular expressible-as-a-sequence-of-words novel good idea” pretty much refutes the idea that guessing and checking can only result in things inside such a convex hull, unless said hull already contains everything. Of course, there’s a significant role to play by the “checking” part.
Like, “take a random sequence of bits and interpret it as Unicode” is at one end of a scale, and “take a random sequence of words in a language” is just a tad away from it, and the scale continues in that direction for quite a while.
It's also worth noting in that in very high dimension, the convex hull will contain massive volume. It could well be the case that humans established that convex hull millions of years ago, and all of our inventions and innovations sense have fallen inside it.
> There’s no reason to believe that kind of guess-and-check is outside of the reach of LLMs
This doesn't make any sense, by their nature they can't "guess-and-check" things outside their training set.
> You can brute-force trying random things and checking the result and eventually learn something new.
And most of the mathematicians seem to welcome this "brute forcing" by the LLMs. It connects pieces that people didn't realize could be connected. That opens up a lot of avenues for further exploration.
Now, if the LLMs could just do something like ingesting the Mochizuki stuff and give us a decent confirmation or disproof ...
I like this construction, but I don’t think you take it far enough.
If you have a multi dimensional space, and you are trying to compute which points lie “inside” some boundary, there are large areas that will be bounded by some dimensions but not others. This is interesting because it means if you have a section bounded by dimensions A, B, and C but not D, you could still place a point in D, and doing so then changes your overall bounds.
I think this is how much of human knowledge has progressed (maybe all non-observational knowledge). We make observations that create points, and then we derive points within the created space, and that changes the derivable space, and we derive more points.
I don’t see why AI could do the same (other than technical limitations related to learning and memory).
I was a little muddy in my original post on distinguishing between what I think LLMs might be able to do and what AI broadly might be able to do. I'm skeptical LLMs can expand the hull or add dimensions to the space; but I also don't think the reasons for that skepticism necessarily apply to all AI system generally.
I found this thought provoking and just had to see how the new Gemini 3.5 Flash reasoned about this (I find it fun to go meta on modern AI like this), and I'm happy that I did! Also as an opportunity to trial this recent model.
https://g.co/gemini/share/065ffa89698e