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I think it is important to divine what currently AI is good for and what it is not even in such verifiable environments like math. Current hyped announcements about breaking conjectures are notable and are a marker of how much improvement was made. But as I read them, and maybe I am wrong, I see it as a large model+ harness executing a broad brute force search and trying solutions until something sticks. There are lots of problems like that and they should be solved, as often they are perhaps less important or overlooked, or just a slog, any field of research has these, math even more so.

However, this is very different from inventing new mathematical machinery that allows to break old problems, I think it will be a while until AI will be able to do it if at all. For now I think we will be moving to a symbiosis where an AI cracking a problem and giving a solution, inspires a human to invent new techniques.

Agreed, AI is not capable at the moment of coming up with radical ideas to solve the tough problems. Sadly, I would argue many problems in math are likely to be found to be not actually tough in this sense, and those working in "comfortable" areas with fewer tough problems are having real crises of their own right now.

But even for the tough problems, it is good at executing on a particular idea with reasonable competency. It's also quite decent at verification now. That can radically speed up proof development overall, since those aspects can become quite tedious otherwise.

finally someone is saying it our loud lol. mathematicans are just finding lower and upper bounds and having a computer input them into a theorem prover more times than a human is able to on their own. ai imo isnt solving anything.
I loved the appearance of Bill Thurston. He proved enough of what he saw to be considered one of our greatest mathematicians, but his brilliance was what he saw.

The obvious question Terry Tao seems not to address: AI mops up our unsolved problems? Whose unsolved problems? The architecture of mathematics will remain a human endeavor long after we replace human construction workers with machines.

However, the writing often dwells at length on trivialities, while passing very briefly through (or even obscuring) the most interesting and novel portions of the argument.

This is consistently my experience chatting with LLMs. It's an very interesting anti-feature, what does it mean about them fundamentally, could more sophisticated machines one day surpass this, etc.

Chomsky for example took the position that LLMs will never be able to explain things and he thought this anti-property was fundamental to their model of computation. But that was years ago.

Recording of a talk with the same title. I’m not certain it’s the same content as the linked PDF though

https://youtu.be/mS9Lr43cIB4

Note that the SAIR lecture is from 6 months ago, this Beamer presentation is from 2 days ago.
> Beamer

Spotted the German guy. ;)

the software that generated the slides is called Beamer https://en.wikipedia.org/wiki/Beamer_(LaTeX)
This is the taste question applied to Mathematics in a similar way it's been applied to code. In an era of AI abundance, the question becomes what the goals are and what gets verified and digested (adopted by users). Goodhart’s law: the goal of producing code is not just about maximizing the number of tokens used, but the productivity gains and economic surplus.

This could serve as the template for any field in the age of AI: "We are not trying to meet some abstract production quota. The measure of our success is whether what we do enables people to understand and think more clearly and effectively about math (or products, or science, or hardware...)"

> but the productivity gains and economic surplus.

I think you might be missing out vast swathes of human experience here. Some things are about joy, or god help me - fun.

(For the record I think AI has a role to play here as well)

I find AI a joy to use to build software. I was never the type of engineer who loved the code itself. I was always the type who wanted to see how users would react, how metrics change based on the feature, etc. The code was the most boring part for me generally.
This is the template any field should have in the face of AI: screw AI, forget it. If you're using it, it's a disrespect to the field. Let's continue without it. I think that's the only sane approach.
William Thurston's argument (from the quote) was that the point of math isn't to produce more math but to better understand the nature of reality, just like the point of code is not to generate more code but to increase user value.

AI is a tool that reshapes knowledge work but the ultimate goal remains about value.

I posit that one cannot decouple understanding from a healthy social environment in which to practise math. And AI destroys that.
This is the thought process that I fail to see many people take (both here and other places). AI is changing many fields, ok what does that look like and how do we adapt? What can we do now that we couldn't before? What skills should I learn to adapt to this changing world? I think this is how we move forward. If you can get past the scary aspects of AI (not talking about datacenters or billionaires, that is a different subject) then you might be able to see how exciting this can be.
Even as somebody who uses AI every day and has integrated it in useful ways, I don't see how the rise of the AI industry can be anything other than terrifying. It signals the rise of a new feudal system. Couple that with the rolling back of democracy across the globe and growing trend of authoritarianism, even in Europe, and I just can't be excited. In Germany particularly, it feels like every week there's some new idea to increase surveillance and control over the populace while the AfD is knocking on the door.
Compared to what though? When was this golden age when things were not scary or turbulent?

If you read about life in the 19th century, it was more turbulent than today with change.

We seem to frame these debates against some random week in spring 2017 that if it ever existed at all, was extremely fleeting.

I think the bigger issue is that social media doesn't exactly provide historical context. If anything it provides a type of ahistoric , hysterical context to the present moment.

Then if you have no social media, everyone feels more and more like a hysterical lunatic living in a reality that doesn't exist if you just go outside and put the phone away for 5 minutes.

Open source intellectual power though. The promise of the personal computer.
The fact is, open source was a solution to a different problem: closed source. Open source is not a panacea that will solve everything.
Whats personal about a datacenter?
> fail to see many people take (both here and other places)

A lot of very vocal people have already taken a position (as in ideology) on this, and are very reluctant to change it, even when faced with obvious evidence to the contrary. It's like anonymous gary marcus clones. Every time something cool happens, they move the goal posts. Even in this thread there's someone saying that recent math advances are "models just brute forcing until something sticks". shrug

As a (sort of former) mathematician myself, I can say two things. First, one of the most exciting parts of mathematics is to see what other people can do with it and the results they obtain, and learning from them (and sharing with them in return).

And second, I am completely uninterested in what AI has to offer. To me (and most normal people) knowledge pursuit is about inspiring others with discovery and the intellectual joy of doing something yourself. Like it or not, that's a necessary requirement for any field to healthy. To put it more plainly, one cannot decouple the fun from the work. I suspect many are jealous of this, but whatever.

Therefore, I see no benefit to AI in math, none whatsoever. Senior mathematicians like Terry Tao might have a different approach because I can imagine: who can resist an oracle that can help them out with their toughest problems. But Dr. Tao IMO is too blinded by that to see the destructive nature of AI, and to be frank, a lot of top mathematicians have too much hubris to understand that they can't control AI.

I don't see how AI in maths stops anyone from pursuing any sort of intellectual learning themselves.

In the same way children still learn their times tables even though calculators can do the job better.

To me (and most normal people) knowledge pursuit is about inspiring others with discovery and the intellectual joy of doing something yourself.

I disagree, for a lot of people knowledge pursuit is about the knowledge and the results of that knowledge. IE the end result is what matters.

As a simple example, Pythagoras's theorum is useful to lots of people all over the world as in the actual equation. I'm not sure most for most people that the knowledge that Pythagoras himself went through an intellectual journey to discover the theorum is more important than the equation itself.

If we use a thought experiment to say that instead of Pythagoras discovering the theorum it was actually discovered by artificial intelligence do you think most people would care? I would say not.

I think a lot of people like the idea of being at the forefront of their field and their intellectual contributions being able to push the frontier foreward. If AI simply gets good enough to largely absorb the frontier they don't find as much fun in rediscovering what AI has already discovered. People forget however, this has already happened to many fields, chess being one example.

Whether AI gets there with maths is an open question but to say AI has no value in maths whatsoever is to say that the practical implications of a better understanding of maths that AI could potentially give us is worth nothing.

> And second, I am completely uninterested in what AI has to offer. To me (and most normal people) knowledge pursuit is about inspiring others with discovery and the intellectual joy of doing something yourself.

I often hear that people are "uninterested in what AI has to offer" but I don't really understand it. I understand this is a mix of ethics and engineering that I don't often engage with but I do not understand why someone would explicitly state that they are not interested in improvements to the world based on the source of the improvement.

The comment I am replying to did not say that LLMs cannot create improvements. That would be a different discussion! It instead says that it does not care about fruits born from the LLM tree.

It seems highly likely to me that through a combination of LLMs and traditional optimization software like vizier that manufacturing, industrial processes, scientific research, or even bakeries will see some improved output through the mixing of domain specific expertise and enablement from LLMs.

I think right now, in the extreme hype environment we live in where people say AI will do everything and take over the world and take your job and give you nothing, people are burned out. Once we get past this point through I am hoping people start talking about what LLMs can do that will actually improve our lives and which can actually work. Much of what people tell me LLMs can do, don't actually work reliably. But there are many cases where LLMs have been a step function in capabilities.

Examples: OCR that really works, fuzzy search, fixing scoped bugs in software, building prototypes of software, etc.

People here don't want fun for others. The site is full of wannabe entrepreneurs whom daddy bought a startup and got VC friends to fund it.

They think they are better than you and me and everyone is just a human resource who has no right to enjoy anything.

That is why they hope for the industrialization of the sciences. The C students can feel equal to the A students.

Why Tao goes along? He is Chinese and they have an insatiable social pressure to be the first and best at "new" things. Since Yau tried to rip off Perelman, my view of the Chinese pressure cooker influence on Western academics has changed.

Especially since the results aren't the same as in the times when everything was more relaxed in academia and the stopwatch idiots hadn't taken over yet.

My first impressions and thoughts on the presentation speak to the need for human mathematicians to accept, absorb, and understand new AI-generated proofs—and to the fact that it takes time to do so, as well as the increasing burden of sorting through essentially unsolicited material, which robs that same group of the time to work. Software developers are seeing this same issue with AI-generated contributions of software changes in the open-source arena. What they are seeing is a very high number of trivial submissions, which Dr. Tao points out is a propensity of AI generation. The second problem is that submissions are often not good, forcing reviewers to read hundreds if not thousands of lines of intricate yet incorrect code—and they waste their productive time doing so, making them less amenable to future consideration of such contributions. This is a real problem. As to the issue of human mathematicians needing to accept work as useful before it can enter wider consideration—now that is an interesting problem if humans are indeed unable to understand work AI creates that is useful and correct, but not understood to be so (perhaps due to the above issues), and yet we have to consider a point in time when models become more capable. Let me put it another way: how long would you spend trying to get your dog to understand calculus? AI has inexhaustible patience (unlike humans), and it will keep trying to explain its ideas and theorems to us (or our dogs) far past the point humans would be willing to listen—though our dogs, at least, are amenable to treats and used to us talking a lot while handing them cookies.
I feel like at a certain point, there's is not necessarily a reason to go through the peer review publication process for some AI proofs.

Not because "ai bad," but because at some point AI outputs should probably just be treated like public knowledge. Specifically stuff that's provable by "AI please output lean showing X is true," where anyone could kind of reach the same conclusion by asking ai.

You don't want people to have to rerun the prompt though. Collaborating on a registry for proven results makes sense.

This probably looks like contributing Lean code to an open source project and writing docs.

Yeah, I don't disagree with a registry. Especially for problems that are important to humans or are expensive for AI to prove/disprove. I partly just want to avoid a situation where the volume of hard-to-search-for results that are easily-ai-proved/disproved skyrockets...

Basically, I think that for some stuff running a prompt is (or could become) easier than trying to do a search for an existing result. Partly why I think if something is fairly easily ai-proven, it should kind of be treated like it was already known, even if it wasn't actually known. :shrug:

It matters a lot what kind of tasks AI would be able to solve. Two completely different worlds arise if AI can do what I call a modern PhD grunt work vs it being able to produce genius of Nash/Einstein/Lovelace etc. And the way to test it is trying to produce modern math and science results with those historical LLM models or something similar: https://news.ycombinator.com/item?id=46319826 https://news.ycombinator.com/item?id=47927903 https://news.ycombinator.com/item?id=46319826
A few predictions:

1. I fear that AI is about to change from being a human art to becoming something like a bulk-extruded, industrial product.

2. I also suspect that mathematics is about to become substantially less open. As the number of entities on the planet that are capable of top-level maths explodes, and because most of those entiries will have no interest in the validation that publishing papers brings, we will see balkanisation and hoarding of "secret maths".

3. Areas of science that are downstream of maths -- basically everything -- will, in time, be degrated by lack of openness, before suffering the same fate.

Well, that's what AI does. It makes everything about the final product and nothing about the enjoyment of the process. Now a lot of people here will say "you can still read the AI results and enjoy math on your own time", but that is completely at odds with how most people do things: a large part of most people's enjoyment comes from sharing results with others and AI devalues that. People can't help but feel less impressed when you can come up with a result but AI can also.

People on here have also argued with me that enjoyment and fun should be decoupled from work but I'd argue that jobs that are enjoyable are important because I mean, what really counts in learning is inspiration. If no one is having fun, no one will be inspired to learn and do math in the first place. The best professors I've had are clearly having fun.

And sharing that fun on a human level is exceptionally important.

I just don't see the tradeoff (more math at the expense of people being able to have fun expressing themselves), as being worth it. The whole point of discovering knowledge is to enjoy the process and share that with the world to create hope an optimisim in it, not to product like a factory.

Unfortunately, this website tends to bring in the rare type of person who loves to believe that they can enjoy things all by themselves, and maybe they can, but the problem is that they (not necessarily you) foist their unusual view of the world on the rest of the world by developing and supporting AI and get defensive when someone points out that it actually contributes to a horribly destructive social structure - a cognitive dissonance most of them can't handle.

> The whole point of discovering knowledge is to enjoy the process and share that with the world to create hope an optimisim in it, not to product like a factory.

This is the crux of everything you've written and I strongly disagree. The purpose of knowledge is the sheer power it brings, and if there's factory scale churning out of it, I'll gladly take it.

As a closer analogy, consider this probability (I'll be stealing from the game Stellaris here): let's say a few hundred years later, when we are spacefaring, we discover a vast cache/library of knowledge of an alien civilization, we won't just pass up on exploiting it because we need to "enjoy the process of acquiring it on our own" or "derive the satisfaction of sharing the discovery with others". No, I and I daresay most others will take as much as we can without restraint in a quest to make all our lives better/more enjoyable.

And if that's ok to do, using machine intelligence to mine/grab any new knowledge or new math as Tao is suggesting here should also be acceptable and welcomed.

I'm an academic mathematician, and could you explain your second prediction?

It goes entirely against the culture of research mathematics to hoard secrets. If you've done something interesting, you want to share it, and it's in your best interest to share it.

Of course, this culture could change under external pressure from AI tools. But, aside from cryptography and the applied fields, I'm finding it difficult to imagine any possible motivation for spending substantial effort on top-level math and then keeping it secret.

That's the current culture, but culture changes. Gauss, Onsager and Grothendieck hid their work. "Pauca sed matura".

Your best interest only means your current best interest. What makes for the best form of personal interest is evolutionary, not static.

Dresden Codak as a story has a concept of "Dark Science" where the world has the benefit doesn't come from publishing science, but from utilizing it. If you understand a piece of science when nobody else does, and integrate it into your technology, you have a strategic non-academic physical advantage. Trade secrets basically. And trade secrets used to be the norm prior to patent law. We could be in the pre-"AI patent" era where the government created incentive to share your AI findings isn't made strong enough yet.

Science is today built on the shoulders of giants, but what if an agent swarm makes your own tower of giants in a day? Why do you need an ecosystem of peer review, if you can discover your own science from first principles without meaningful effort? Yes, it's duplicated effort, but why does that matter if the monetary cost of effort trends to zero? The value of peer review is going to have to contend deeply with itself just like how open source developers have had to contend with the value and mechanisms of inbound AI generated PRs the last six months.

Wouldn't AI make validating the research much cheaper?
It was an excellent talk. If nothing else, it really helps to establish some much needed vocabulary for us to constructively talk about the future of math
A pleasure to read such a presentation which carefully lays out its argument brick by brick.
Is there a video for this?
To be released imminently per this toot from 2 days ago (saying forthcoming in a day or two): https://mathstodon.xyz/@tao/116977934921819775
It really is sad that supposed leaders of technical fields have been reduced to TED tier clowns for profit
In times when software eng teams push for maximizing token usage, Tao's public view is to encourage verification, limit AI use for tasks that have real-life consequences and prevent low-effort papers that exhaust the review pipeline?

He also focuses on training the next generation of mathematicians and prioritizing the process of slow digestion of results, which to me sounds very.. reasonable?

There's no real point in engaging with these bot accounts. They'll keep making points that sound superficially correct but allow an argument against.

TT has real results which is why his opinion is worth listening to even when you disagree. These random accounts are dismissable immediately.

Except to make jokes about of course.

Taos public view is to engagement farm and he'll swap to the new nfts or crypto when this is done because once you go down this road theres no going back with the lifestyle creep
The man just cannot stop. First, a lengthy introduction with caveats which can be used to hedge later.

Then he goes all in on AI again. He is sponsored by the AI for Math Fund (Renaissance Technologies) and I'd really like a yes/no disclosure about OpenAI stock options.

People give him the benefit of the doubt because he always has been unable to stay off the Internet for more than a day. But this is really unprecedented.