> In either case I believe people who can put AI to the most value are the mathematicians themselves
The net output of math will increase, and mathematicians have more work now to unravel all this, and make it useful. AI plays the role of a monkey in the infinite monkey theorem [1]. We now need an LLM corollary - Something like: A finite number of LLM agents will almost surely find all theorems given an infinite token budget.
It's impossible for finite number of LLMs to solve all theorems. This would imply that the busy beaver sequence is computable which implies the halting problem is decidable.
For any finite program (eg some LLMs), there is a true math theorem which they cannot prove or disprove (given fixed input of the statement with no other information sources). If that weren’t true, BB would be computable.
Math is beyond computation. Since AI is just bits in bits out, it has this fundamental limitation.
Any magic of AI systems comes from the transformed meaning of its input data. With fixed weights any LLM is just an artifact. For example a human prompting an LLM constitutes an extra information source, which removes the above limitations. In theory any input from the natural world would remove the limitations too. The natural world is a black box and we don't know what kind of meaning or intelligence could underly it.
We are talking about the same thing, but I would actually put this the other way around.
Computation and computability is "the final frontier". Math is a "subset" of that. Doesn't matter if we choose ZFC or in the future discover some "better" subset of core axioms, we will always hit limits where BB will trivially skip over whatever we could prove (let alone Gödel's theorems).
> given fixed input of the statement with no other information sources
Also, this is just trivially avoidable, so not sure if we really should be concerned about this limitation. An LLM in a loop where it can write on a tape can be Turing complete, ergo it can compute anything computable and is "bigger" than math at that point.
Because then you would actually learn something. What else?
I don’t know what correction to make because you are not even wrong. You are _kinda_ wrong in comparing math and computing via a subset of relation, but it also doesn’t make much sense. If you consider computing as constructive math it should be a subset, but those are not sets, so it doesn’t really help.
bwfan123 · · focus · HN ↗
The net output of math will increase, and mathematicians have more work now to unravel all this, and make it useful. AI plays the role of a monkey in the infinite monkey theorem [1]. We now need an LLM corollary - Something like: A finite number of LLM agents will almost surely find all theorems given an infinite token budget.
[1] <a href="https://en.wikipedia.org/wiki/Infinite_monkey_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Infinite_monkey_theorem
srcreigh · · focus · HN ↗
For any finite program (eg some LLMs), there is a true math theorem which they cannot prove or disprove (given fixed input of the statement with no other information sources). If that weren’t true, BB would be computable.
Math is beyond computation. Since AI is just bits in bits out, it has this fundamental limitation.
Any magic of AI systems comes from the transformed meaning of its input data. With fixed weights any LLM is just an artifact. For example a human prompting an LLM constitutes an extra information source, which removes the above limitations. In theory any input from the natural world would remove the limitations too. The natural world is a black box and we don't know what kind of meaning or intelligence could underly it.
gf000 · · focus · HN ↗
We are talking about the same thing, but I would actually put this the other way around.
Computation and computability is "the final frontier". Math is a "subset" of that. Doesn't matter if we choose ZFC or in the future discover some "better" subset of core axioms, we will always hit limits where BB will trivially skip over whatever we could prove (let alone Gödel's theorems).
> given fixed input of the statement with no other information sources
Also, this is just trivially avoidable, so not sure if we really should be concerned about this limitation. An LLM in a loop where it can write on a tape can be Turing complete, ergo it can compute anything computable and is "bigger" than math at that point.
rsrsrs86 · · focus · HN ↗
gf000 · · focus · HN ↗
rsrsrs86 · · focus · HN ↗
I don’t know what correction to make because you are not even wrong. You are _kinda_ wrong in comparing math and computing via a subset of relation, but it also doesn’t make much sense. If you consider computing as constructive math it should be a subset, but those are not sets, so it doesn’t really help.