Terence Tao: Mathematics in the Age of AI [pdf]
teorth.github.io
teorth.github.io
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.
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.
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.
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.
Spotted the German guy. ;)
- he works in Germany (https://vedran.miletic.net/files/cv-vedran-miletic.pdf)
- the original author is Till Tantau (https://www.beamer.plus/Introduction.html) who is German (https://www.tcs.uni-luebeck.de/en/mitarbeiter/tantau/cv/)
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...)"
AI is a tool that reshapes knowledge work but the ultimate goal remains about value.
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)
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.
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.
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.
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.
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.
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
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.
This probably looks like contributing Lean code to an open source project and writing docs.
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:
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.
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.
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.
Actually, this is the fundamental divide. There are a group of people (like you) who want the world to be one way, and there are a group of people (like me) who think that world is abhorrent. I don't think we should go after that power.
But, I'm glad we got to this point in the discussion because it shows this irreconcilable divide. The truth that most people will never say is that these two groups are enemies and will forever be. They fit into this world because we currently participate in the same global system, but because of this situation, it's an unstable one. And AI is exactly the system that is distilling this antagonism to its limit.
A dismal prospect.
When knowledge is acquired at a human scale, and under the influence of a human culture, then extraordinary things can happen. Agriculture, the transistor, Gödel's theorems.
When knowledge is "churned out" at "factory scale" then human flourishing isn't even a by-product.
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.
If you can derive some special formula that prints money from accurate prediction of asset valuations if and only if you're the only one with the efficiency, you're incentivized to keep it quiet.
If you're a nation state that's cracked a simple method to decrypt modern "pre-quantum" encryption schemes with less compute/wall time using a development in mathematics, it is in your interest to keep this quiet so that the parties you surveil to not switch or adapt. Similarly if you've figured out a handy new form of encryption, you're incentivized to prevent leaking of details as it is harder to reverse such operations without details of the methodology.
Or you've designed some special optimizer that you then bundle in a subscription service. Your business now only profits while the algorithm is secret. e.g. The DVD Copy Control Association's CSS encryption was only a valuable thing that the company could charge for or entice clients with before DeCSS was published/became popular.
Update: see also the later slide where he problematizes corporate secrets in relation to math culture.
Everyone should just read the slides, really
(Disclaimer: I had quite a bit of formal training in mathematics at one point, but have worked in software for more than two decades now. I wasn't on the applied side, but I recall that MATLAB evoked similar concerns back in the day.)
> 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.
You're describing an enviable human culture with a reward system suited to people. People publish papers etc. to enrich the human domain and to get recognition. AIs don't care about things - and neither, I'm asserting, will their owners. That culture will become a vestige.
Mathematics as the underpinning of science and technology that can be exploited for inter-state competition, including cryptography but not limited to that (image analysis, targeting algorithms, conflict modelling, whatever) won't see the light of day. The AIs that discover the mathematics will be paid for by the warfighting budgets of states, and by the corporations that feed on them.
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.
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?
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.
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.