I’ll start with a disclaimer that I don’t know for sure (no one really does) what the difference between solving programming problems and solving pure mathematics problems is (and certainly if you stretch the domains you can emulate each within the other, thereby showing their equivalence… if you like). I’m just speculating, as usual. So if you’re confused, maybe that’s just because I’m wrong.
> most coding has no high level ideas and is just boilerplate, and the ones that aren't are the ones LLM's struggle with?
Pretty much, although calling it boilerplate might be going a bit far.
I’m not here to claim something like ‘mathematicians think and programmers do not’ because that is clearly not the case (and sounds like a mathematician with a complex of some kind). But it is empirically the case that so far GPT-4 and the like are much better at programming than maths. Why? I think the reason is that whilst the best programmers have a deep understanding of the tools and concepts they use, it’s not necessary to get things to work. You can probably get an away without it (I have ideas about why, but for now that’s not the point). And given the amount of data available on basic programming questions (much more than there is of mathematics) if you’re an LLM it’s quite possible to fake it.
I guess one could also make the point that the space of possible questions in any given programming situation, however large, is still fairly constrained. At least the questions will always be ‘compute this’ or ‘generate one of these’ or something. Whereas you can pick up any undergraduate maths textbook, choose a topic, and if you know what you’re doing it’s easy to ask a question of the form ‘describe what I get if I do this’ or ‘is it true that xyz’ that will trip ChatGPT up because it just generates something that matches the form implied by the question: ‘a mathematical-looking answer’, but doesn’t seem to actually ask itself the question first. It just writes. In perfect Mathematical English. I guess in programming it turns out that ‘a code-looking answer’ for some reason often gives something quite useful.
Another difference that occurs to me is that what is considered a fixable syntax error in programming when done in the context of maths leads to complete nonsense because the output is supposed to describe rather than do. The answers are somehow much more sensitive to corruption, which perhaps says something about the data itself.
> most coding has no high level ideas and is just boilerplate, and the ones that aren't are the ones LLM's struggle with?
Pretty much, although calling it boilerplate might be going a bit far.
I’m not here to claim something like ‘mathematicians think and programmers do not’ because that is clearly not the case (and sounds like a mathematician with a complex of some kind). But it is empirically the case that so far GPT-4 and the like are much better at programming than maths. Why? I think the reason is that whilst the best programmers have a deep understanding of the tools and concepts they use, it’s not necessary to get things to work. You can probably get an away without it (I have ideas about why, but for now that’s not the point). And given the amount of data available on basic programming questions (much more than there is of mathematics) if you’re an LLM it’s quite possible to fake it.
I guess one could also make the point that the space of possible questions in any given programming situation, however large, is still fairly constrained. At least the questions will always be ‘compute this’ or ‘generate one of these’ or something. Whereas you can pick up any undergraduate maths textbook, choose a topic, and if you know what you’re doing it’s easy to ask a question of the form ‘describe what I get if I do this’ or ‘is it true that xyz’ that will trip ChatGPT up because it just generates something that matches the form implied by the question: ‘a mathematical-looking answer’, but doesn’t seem to actually ask itself the question first. It just writes. In perfect Mathematical English. I guess in programming it turns out that ‘a code-looking answer’ for some reason often gives something quite useful.
Another difference that occurs to me is that what is considered a fixable syntax error in programming when done in the context of maths leads to complete nonsense because the output is supposed to describe rather than do. The answers are somehow much more sensitive to corruption, which perhaps says something about the data itself.