Illustration Beyond Proof | Hacker News Reader

Illustration Beyond Proof

terrytao.wordpress.com

108 pointssmilelamp19 comments

Suppose you are a maths researcher and you come across a problem, ... most likely, you will feed it to an AI. There are two possibilities: either the AI solves the problem or it does not.

If it solves the problem, you can ask the AI to explain it you. But if it does not solve it, will you really wager 30 years of your life on the possibility that you might be smarter than the AI?

I keep hearing about people spending their entire career on a single problem. And I think that time has ended. I think in Mathematics will be like chess.

> But if it does not solve it, will you really wager 30 years of your life on the possibility that you might be smarter than the AI?

The "AI" is trained on centuries of human theoretical frameworks and innovations. The "AI" can also work on problems full-time without teaching, admin, or mentoring duties. OpenAI spent millions to race human researchers on Navier Stokes, and that doesn't account for any training costs. How many person-years would that pay for? Since the key ideas were developed by humans just a few years prior, it's not unreasonable to surmise that a comparable level of total effort by humans would also have decent chances of breakthrough, with the additional benefit that the insights uncovered along the way would be immediately digestible and easy to communicate to other mathematicians.

> I think in Mathematics will be like chess.

What do you mean?

Do you think more people than ever will start doing math, with millions of people solving daily math puzzles on their phone on math.com[1] while commuting ?

I don't think Mathematics will be like Chess.

[1]: (or its open-source variant limath.org)

Not OP, but I read it as: it will be similar to chess players using computers to enhance their play [1].

[1] https://en.wikipedia.org/wiki/Advanced_chess

Math research isn't all or nothing on a single problem. Rather, the problem would serve as inspiration to explore achievable subproblems, weaker problems, related topics, etc. These would be interesting papers by themselves, but if lucky may also lead to progress on the main problem. So this leaves lots of room for using AI as an assistant.

I dont get this. What is the difference if you replace "AI" by "Tao"? And people still were doing math.

I get HN is a Software site. But all math posts are invaded by people that think Math=Software. Probably based on the way they were taught maths when they were young.

Why would a theoretical physicist spend 30 years developing a theory that could be made false by a new experiment with a new technology? Was Newton gravitation worth if einstein was going to come later?

We are heading directly into a world where people forget about what values, philosophy, poetry, math, physics, or any other "non economically valuable" things are. And we all know what happened any other times in history when that happened.

> I dont get this. What is the difference if you replace "AI" by "Tao"?

1. Tao lacks a fine-tune that accurately follows my instructions. 2. The usage limit on the 100$/month Tao subscription is quite low.

Thanks for telling it better than I could have! But if you could expand on your last sentence, what instances in history are you referring to? Dot com boom? Great depression?

I was thinking more about dictatorships that block the entry of outside knowledge into the population for example, because its not useful to their goals (which is a particular shape of what economic value is for them). But you dont have to go that far away, you can just see the difference between democratic cultures that value at different levels those fields. In the end "economic value" is less objective than what it seams.

> If it solves the problem, you can ask the AI to explain it you.

Will I get an understandable and verifiable explanation? And, if the system cannot solve the problem, will I get an understandable and verifiable explanation of that? As a researcher I'd think that shared understanding is the point.

> But if it does not solve it, will you really wager 30 years of your life on the possibility that you might be smarter than the AI?

Right now it’s not a terrible bet. If the OpenAI results from the past week are representative then the AI success rate for verified proofs on open math problems is somewhere below 10%. There’s a long way to go before you can assume an AI failure to solve means the problem is unsolvable.

> I keep hearing about people spending their entire career on a single problem. And I think that time has ended. I think in Mathematics will be like chess.

Is the Langlands Program a single problem? Much of it is still ill-defined. How will AI win at newfangled chess when they haven’t got the rules yet?

>But if it does not solve it, will you really wager 30 years of your life on the possibility that you might be smarter than the AI?

Sure. Just because current technology can't prove or disprove something, doesn't mean it can't ever be proven or disproven.

But what if Fable 6 solves it next year?

The goal of mathematics is to solve problems the same way the goal of a marathon is to do a 42km travel.

Why are people running in the first place when they could just take their car?!

Wrong analogy. Solving a problem that is already solved is different from running a distance that is already traveled.

i cant be more against that idea, AI is a copy machine, it knows what we feed it to it, it can draw conclussions of course that may be apparently new information but it is not, it is merely connecting dots. math researchers, as well as any other high level professional will be doing much more than that, thats why AI is a tool, it can not, by definition, create new things

Reminds me of things learned from comics professor Nick Sousanis, who enthuses that drawing IS thinking. Sketches are ideas. For most, the pencil is the tightest bond between your imagination and the external world. And they are in dialog as we sketch, externalize, react, and surprise ourself.

I've long been a fan of the way 3b1b illustrates and explains math.

I wonder if something like OpenAI's recent IntelligentUI feature couldn't also be used to creature beautiful, intuitive walkthroughs of otherwise opaque areas of math?

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