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Mathematics in the age of AI

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61–70 of 292 posts

Re: Mathematics in the age of AI

#61
post #53
post #35

Earlier quoted context omitted.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

I can almost see two branches of mathematics developing. One which is human-understandable, the other formally verified. I assume the latter is a strict superset of the former?

I suggest "Catching crumbs from the table" by Ted Chiang. Very short piece published in Nature (2000) and well worth a read. Depicts a scenario where modified humans produce science beyond ordinary scientists' comprehension.

Re: Mathematics in the age of AI

#62
post #48

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I believe this rule of thumb will come to fail. The combination of superhuman mathematical reasoning and synthesis in upcoming AI models plus the rapid build-out of scalable formal verification infrastructure means this exponential in math is going to take off quite explosively, and we've barely seen anything yet. Mathematics is going to decisively move beyond human ability fairly soon (within our lifetimes, if not m…

People say similar things about automation of software engineering. Different, but similar.

I'm deeply suspicious. I do not yet have a concise statement for why, but a lot of literature on the sociology of knowledge work sort of points at my thoughts.

Section 5 of the Thurston article cited by Tao touches the elephant. Raduchel's article on the economics of software [2] also touches it.

I've tried to put words to this for a few years. I think I'm just going to start writing versions of it as see if that helps me shape the thought into something more concise.

So, in the spirit of this article's style, here are some postulates:

1. There is a sociological process happening in the production function during knowledge work.

2. That production function and the associated sociological process spans years or even decades, and must outlast many of the artifacts that are produced during the early years of the function.

3. You cannot get the right lines of code or the right theorems proved without running that sociological process alongside the artifact production process.

4. It is impossible to completely separate the sociological process from the artifact construction process. If you just iterate on artifacts then too much of the required hidden state is lost to make progress in the right direction. This is true even if you include distilled artifacts capturing pieces of the sociological process (eg meeting notes, documentation, commit logs, prompts).

5. So you need that sociological process, or something like it, to still happen.

6. For a lot of knowledge work that process plays out in extremely high-fidelity social interactions [3] that we have not yet captured in the datasets that would be required to reproduce those dynamics.

7. And even if we do collect that data, our current architectures and training algorithms and hardware would be useless given the size of the datasets.

So: the technology today gives us the ability to iterate on the production of artifacts. But it does not sufficiently simulate the social process which gives rise to the Right artifacts.

This isn't exactly what I actually think, but it's a version of the thing that I intuit when I watch heavy use of AI in both software projects and formalization projects. And simulating that process feels way harder than people are currently assuming.

[1] https://arxiv.org/pdf/math/9404236 Section 5.

[2] https://www.nationalacademies.org/read/11587/chapter/11 pp 166-168.

[3] there is a reason we still gather in-person around white boards, and why doing so is more crucial for some types of work than others.

Re: Mathematics in the age of AI

#63
post #53
post #35

Earlier quoted context omitted.

The counterpoint to this comes from chess. High level engines "prove" certain lines correct (not in the mathematical sense) but those "engine lines" are really hard to explain to humans, even by GMs. They can sort of explain that something is a good line but not why. Engines crush GMs and are considered ground truth even if noone really understands what is happening. Would it be a nightmare if math was the same, not…

I can almost see two branches of mathematics developing. One which is human-understandable, the other formally verified. I assume the latter is a strict superset of the former?

Mochizuki enters the chat

Re: Mathematics in the age of AI

#64
post #63
post #53

Earlier quoted context omitted.

I can almost see two branches of mathematics developing. One which is human-understandable, the other formally verified. I assume the latter is a strict superset of the former?

Mochizuki enters the chat

[dead]

Re: Mathematics in the age of AI

#65
post #59

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I don't think the mathematicians are going to be able to make that work, because journals are already struggling to keep up with their review load, and AI seems like it will make that harder. So a solution that involves "journals will do a lot more effort to review each paper" doesn't seem practical. It would work better as a bar for hiring, rather than as a bar for publishing.

It will be interesting to see the evolution of journals in the next ten years for sure. Have they outlived their usefulness? Maybe everyone will just upload papers to arXiv, along with a copy of the formal proof.

Re: Mathematics in the age of AI

#66

Tao's Rule of Thumb (which applies very well to software): > My own suggested rule of thumb: if the authors cannot convincingly demonstrate that they are able to give a clear, expert-level talk on their results, one that is correct and properly attributed, then the result should not be published. A proof that no human can properly explain should be viewed as incomplete, even if it has been formally verified.

I saw an analogous argument posted on LinkedIn the other day from one of the opencode guys: the job of a programmer is still to be able to answer questions - from memory - about how the system works and why.

Re: Mathematics in the age of AI

#67

I don't know why anyone should care about understanding the results if the AI is better at math than us. It'd be like demanding that human mathematicians are banned from publishing until their cats understand the theorems. If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed. The human brain is being obsoleted, soon…

If you are free from physical and mental labor, you are in fact, not supplying labor, and are therefore surplus to requirements.

Whose requirements?

Re: Mathematics in the age of AI

#68

I don't know why anyone should care about understanding the results if the AI is better at math than us. It'd be like demanding that human mathematicians are banned from publishing until their cats understand the theorems. If Amazon uses AI math to come up with better routing, the cats can benefit from cheaper delivery fees just as much as humans can. No understanding needed. The human brain is being obsoleted, soon…

If a result has a real-world application, then it can easily be published in an engineering or applied scientific journal in which it is already the norm to present methods that work empirically with little to no understanding of how.

Re: Mathematics in the age of AI

#69
Goal 6.4 reminds me always of the numerous times AI generated n PR's for a feature and I revolted and threw my laptop because it was incomprehensible or unworkable when viewed as a process/workflow.

Re: Mathematics in the age of AI

#70
post #11

If the title have said in the age of "LLMs", I might have given it a try.

Out-of-hand dismissal of Terence Tao is certainly a take. And the term "artificial intelligence (AI)" has been the name of the field for 70 years and counting. If anything, "LLM" is a misnomer that's been lingering around since 2018-19. When the term was coined, these systems were relatively small, experimental, and could only produce impractical facsimiles of the English language. This is obviously no longer the cas…

"artificial intelligence" is a vague and moving target. I'm not sure I would call that "a field".

It's been used to talk about computers playing chess, then machine learning, and now LLM-based systems.

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