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.
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.
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...)"
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.
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.
> 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
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.
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.
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.
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.
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.
I can see it happening if people start depending AI-generated proof libraries for the work they are interested in. Not because the libraries are proprietary, but because they are impenetrable. AI usage wouldn't result in this directly, but it could be a consequence of formal verification as a substitute for canonicalization in some cases.
(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.)
(Disclosure: I'm not a mathematician or any other kind of scientist. Just a toiler in the software mines, and an onlooker)
> 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.
There could be a difference between academic mathematical research and commercial mathematical research, just as in the difference between FOSS and commercial code. Perhaps the difference lie between pure math and applied math with applied math having more incentive for secrecy given that large swaths of computing is just applied math in fancy wrappers.
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.
There are different levels of outcomes, but even Terence Tao essentially foresees this issue: look at his remark footnote 5 in this very talk. It complements the Balkanization point in the parent comment. Not a big leap or unfathomable to connect to broader issues.
Update: see also the later slide where he problematizes corporate secrets in relation to math culture.
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.
> The purpose of knowledge is the sheer power it brings
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" by vast factories then human flourishing isn't even a by-product.
> 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.
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.
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
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.
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?
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.
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.
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.
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...)"
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.
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. ;)
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.
> 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
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.
Open source intellectual power though. The promise of the personal computer.
Whats personal about a datacenter?
The fact is, open source was a solution to a different problem: closed source. Open source is not a panacea that will solve everything.
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.
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.
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.
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.
I can see it happening if people start depending AI-generated proof libraries for the work they are interested in. Not because the libraries are proprietary, but because they are impenetrable. AI usage wouldn't result in this directly, but it could be a consequence of formal verification as a substitute for canonicalization in some cases.
(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.)
(Disclosure: I'm not a mathematician or any other kind of scientist. Just a toiler in the software mines, and an onlooker)
> 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.
There could be a difference between academic mathematical research and commercial mathematical research, just as in the difference between FOSS and commercial code. Perhaps the difference lie between pure math and applied math with applied math having more incentive for secrecy given that large swaths of computing is just applied math in fancy wrappers.
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.
There are different levels of outcomes, but even Terence Tao essentially foresees this issue: look at his remark footnote 5 in this very talk. It complements the Balkanization point in the parent comment. Not a big leap or unfathomable to connect to broader issues.
Update: see also the later slide where he problematizes corporate secrets in relation to math culture.
Everyone should just read the slides, really
Wouldn't AI make validating the research much cheaper?
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.
> The purpose of knowledge is the sheer power it brings
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" by vast factories then human flourishing isn't even a by-product.
> 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.
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.
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
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.
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.