With what accuracy? And with how many intermediate tokens?

We can replace multiplication with any class of problems which should go from 0->100% solution almost immediately if there was actually a concept learned.

There is not. Because they are plain ol' fits. And there are no "emergent" features that pop out without having a sufficient set, where "sufficient" is absolutely gigantic and equivalent to memorizing enough of the space to compress the problem. LLMs are Rain Man.

They interpolate within a known distribution. Search allows places outside of distribution to be explored.

This paper should be required reading [1]. You can explore the curves yourself. You can see exactly what it's doing. And you also have this nagging thing—which you know and I know—that all these models converge and do not diverge upwards. An "emergent" "hyperintelligence"—a characteristic that could be found if something was actually learned and combined with a new concept—would not have this problem.

Exponentials on exponentials added to compute and data and the problem classes still sit at not great places, and require agents, feedback loops, and trial and error to solve. The models are the problem, but more importantly, the people selling things these models could never do are the problem.

[1] https://hai.stanford.edu/news/ais-ostensible-emergent-abilit...

Edit: It should be mentioned, if there's some scary neural architecture that's super-de-duper and doing something beyond the very obvious next string prediction that LLMs clearly do, it can't do what absolutely ancient ML models could do; a network to multiply two floating point numbers should pop out somewhere without symbolic computation, no?

It does not. There's no magic other than the run-of-the-mill SV fake it til you make it magic. And that magic has failed.

> With what accuracy? And with how many intermediate tokens?

So you are arguing here that LLMs should just "know" the result of a multiplication when the operands are in context, ie, that a hidden multiplier circuit should emerge in their weights.

Is this how you do multiplication, if I give you two 12 digit numbers, does the 24 digit multiplication result just pop in your head? Don't you have to follow a learned algorithm through a tedious system 2 effort? Don't you need to write down the results on paper because you can't actually hold in your head the dozen partial results, each with a dozen digits? How many visual, tactile and reasoning tokens does this consume, moving the hundreds of muscles that make up your hand to draw each number under visual feedback, then reading all those numbers back and transforming ocular activation data into numeric symbols?

It seems to me your system 2 is just following a symbolic algorithm for multiplication, and it does roughly the same steps as the LLM trace I showed you previously.

So if you think this is the mark of your intelligence, why wouldn't it apply to the machine too? Why is it implausible that, following a similar algorithm learned from some mathematical paper, the LLMs has reasoned a new solution to a problem in another math field? Why couldn't the machine combine and morph these algorithms for symbolic manipulation, to yield entirely new and original results? How would those results differ from results human mathematicians generate, using recipes they learned in university?

The emergent behavior that we talk about isn't that the machine can do multiplication in its "head" after knowing the multiplication algorithm. What emerges is the ability to follow any other algorithm, even algorithms that were not in the training set, even algorithms to create other algorithms, which it then executes. This is the emergent behavior that matters for AGI; once it can do that, it's a trivial exercise to create a non-AI tool to automate and accelerate the mechanical tasks - just like we humans do it.