Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI | Lex Fridman Podcast #472

0:07:56 · watch on youtube.com↗

Our reading

What sets mathematicians apart is caring whether something holds in 100 percent of cases, when 99.99 percent is good enough for everyone else.

Their wordsIf something holds 99.99% of the time, that's good enough for most things. But mathematicians are one of the few people who really care about whether really 100% of all situations are covered by it.

0:55:26 · watch on youtube.com↗

Our reading

He works as a fox, not a hedgehog: moving between fields, chasing analogies, and re-proving results he likes with the tools he favours.

Their wordsI'm much more comfortable with the fox paradigm. Yeah. So yeah, I like looking for analogies, narratives. I spend a lot of time… If there's a result, I see it in one field, and I like the result, it's a cool result, but I don't like the proof, it uses types of mathematics that I'm not super familiar with, I often try to re-prove it myself using the tools that I favor.

1:07:19 · watch on youtube.com↗

Our reading

Physics will be unified again as it has been before; the obstacle is that relativity and quantum mechanics already explain almost everything we can measure.

Their wordsBut I have faith that we've been doing this for centuries and we've made progress before. There's no reason why we should stop.

1:09:25 · watch on youtube.com↗

Our reading

String theory, the leading candidate for decades, is slowly going out of fashion because it is not matching experiment.

Their wordsYeah, that was a leading candidate for many decades. I think it's slowly pulling out of fashion. It's not matching experiment.

1:25:23 · watch on youtube.com↗

Our reading

Formalising a proof in Lean currently takes about ten times the effort of writing it out: doable, but annoying.

Their wordsSo right now I estimate that the time and effort taken to formalize it, proof is about 10 times the amount taken to write it out. So it's doable, but it's annoying.

1:34:39 · watch on youtube.com↗

Our reading

Lean and tools like GitHub will let experimental mathematics scale far beyond what one mathematician's spaghetti code allows today.

Their wordsI think the platform that Lean and other software tools, so GitHub and things like that will allow experimental mathematics to scale up to a much greater degree than we can do now.

1:45:31 · watch on youtube.com↗

Our reading

Solving one Olympiad problem with three days of Google's server time shows what is possible, but the approach does not scale.

Their wordsYeah, these are great work that shows what's possible. The approach doesn't scale currently. Three days of Google's server time can solve one high school math format there. This is not a scalable prospect, especially with the exponential increase as the complexity increases.

1:51:05 · watch on youtube.com↗

Our reading

AI could compete with human mathematicians once it acquires a mathematical sense of smell: knowing which way of splitting a problem makes it easier rather than harder.

Their wordsVery rarely do you transform into a simpler problem. So if they can pick up a sense of smell, then they could maybe start competing with a human level of mathematicians.

1:55:54 · watch on youtube.com↗

Our reading

When formalising a proof costs no more than writing it, mathematics will flip: papers written in Lean first, and journals refereeing only for significance because correctness is certified.

Their wordsAnd that's a phase shift, because suddenly it makes sense when you write a paper to write it in Lean first, or through a conversation with AI, which is generally on the fly with you, and it becomes natural for journals to accept.

1:59:04 · watch on youtube.com↗

Our reading

His prediction that research-level mathematics papers would be written in collaboration with AI by 2026 has already come true.

Their wordsThere are certainly math results which could only have been accomplished because there was a human authentication and an AI involved, but it's hard to disentangle credit. I mean, these tools, they do not replicate all the skills needed to do mathematics, but they can replicate some non-trivial percentage of them, 30, 40%, so they can fill in gaps.

2:02:33 · watch on youtube.com↗

Our reading

This decade, AI could propose a conjecture connecting two things nobody thought were related, with a real chance of it being correct and meaningful.

Their wordsYeah, this decade I can see it making a conjecture between two things that people would thought was unrelated.

2:02:59 · watch on youtube.com↗

Our reading

Current AI struggles even to rediscover old laws of physics from data, and when it does, contamination from training is the first suspect.

Their wordsThe dream is you just feed it all this data, and this is here is a new patent that we didn't see before, but it actually, even the current state of the art even struggles to discover old laws of physics from the data.

2:30:19 · watch on youtube.com↗

Our reading

In ten years there will be many results much closer to twin primes, perhaps not the whole thing; on the Riemann hypothesis he has no clue.

Their wordsSo I think in 10 years we will have many more much closer results, we may not have the whole thing. So twin primes is somewhat close. The Riemann hypothesis I have no clue.

2:33:00 · watch on youtube.com↗

Our reading

The twin prime conjecture is certainly true, the random model gives overwhelming odds of it, and he just cannot prove it.

Their wordsI can tell you with complete certainty the twin prime conjecture is true. The random model gives overwhelming odds it is true, I just can't prove it.

2:43:20 · watch on youtube.com↗

Our reading

On P versus NP, the evidence leans towards no, and the problem is unusual in how many approaches have been proven not to work.

Their wordsCertainly more on the no than on the yes. The funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem.

3:01:25 · watch on youtube.com↗

Our reading

The stable one-profession career is becoming a thing of the past; what will still be needed alongside AI is reasoning with abstractions and problem-solving when things go wrong.

Their wordsIt's becoming much more a thing of the past. So I think you just have to be adaptable and flexible. I think people will have to get skills that are transferable, like learning one specific programing language or one specific subject of mathematics or something. That itself is not a super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve when things go wrong.

3:07:19 · watch on youtube.com↗

Our reading

Faced with a problem none of your techniques fit, try anything, the stupider the better, because the way it fails is the clue.

Their wordsSo the next step then is to try anything no matter how stupid and in fact almost the stupider, the better, which technically is almost guaranteed to fail, but the way it fails is going to be instructive.