I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part. I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them. Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it. And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
I have a hatred for people who think they can use this method.
If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless.
Ask anyone unfortunate enough to ask for help on IRC
The math field is having to speedrun something that the 'thought work' field has been dealing with for a few years now. I remember when "writing code was never the point" became a mantra. There was truth in it, but removing the coding has certainly taken away a lot of texture, and we find ourselves just tech-leading teams of agents now, which will suit some more than others.
The moat for coders now seems to be that AI can automate tasks but not a full job (unclear how long that will hold). But in math, doing the math really was _the_ job, my PhD certainly was. Since this is academically funded, they now have to attempt to pivot that to save their profession. I am not optimistic myself.
We are all staring at the same existential dread, just seeing it unfold a bit slower. We're being told that utopia is being obsolete, and that is difficult to accept.
Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.
Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.
Likewise all other technology [ and culture ].
LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.
Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
A moratorium on AI development might be the only way to achieve this preservation of human culture.
One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.
I do neither maths nor science with AI but in my experience most models are perfectly willing to burn tokens on a ton of sidequests at the earliest opportunity.
While I understand and emphatically with Tao's concern I'm afraid it's missing the forest from the trees. Unless you can make a claim that AI will never be able to perform intellectually at the same level as any human at a much lower cost, there's an outstanding utility problem that remains unaddressed.
Sure enough, the AI may not have taste or goals, or many human traits, but that's irrelevant to the much thornier (and much broader than mathematics or even academia) question related to who's getting paid how much and for what.
I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).
I'd be interested in hearing a field report on this! For example, I can easily imagine that they're great at walking through the proof step by step, explaining background as necessary; but as TFA notes, one of the most important questions is "why is this definition the way it is?", and my bet would be that the Lean is not enough to help the LLMs meaningfully in answering that.
I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.
Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".
The theorem thing is invented by human to help other people better understand Math structure in a easier way.
It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.
A majority? I doubt it, simply because the majority doesn't know what math is at all.
At university level introductory calculus, the person teaching class had to reassure students that math wasn't entirely arithmetic or adding up numbers. He did this because it's a common misunderstanding.
The people building AI claim it will surpass human intelligence in all
respects and prerhaps kill
us all. Should we just cease all human activity on the basis of what AI might do in future?
Personally, I doubt AI can surpass a good human explainer because explanation requires empathy, which benefits from being an instance of the kind of entity you are explaining the thing to. That gives you a way of exploring and evaluating the space of possible explanations that isn't available to an LLM.
It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.
Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability. But of course mathematics is all about proof, and for that reason I was wary of it for a very long time.
>Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability
not really. You can consider positive proof as an experiment confirming your theory and the negative proof and counter examples as an experiment falsifying your theory.
Yes, really. "Positive proof" opposes the concept of falsification. You can only have it within a system formal logic, and science can contain those, but isn't one.
I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part. I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them. Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it. And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
[delayed]
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
a good teacher remembers the journey, not just the destination.
socratic method exists. almost none follows it.
>socratic method exists. almost none follows it.
I have a hatred for people who think they can use this method.
If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless.
Ask anyone unfortunate enough to ask for help on IRC
The math field is having to speedrun something that the 'thought work' field has been dealing with for a few years now. I remember when "writing code was never the point" became a mantra. There was truth in it, but removing the coding has certainly taken away a lot of texture, and we find ourselves just tech-leading teams of agents now, which will suit some more than others.
The moat for coders now seems to be that AI can automate tasks but not a full job (unclear how long that will hold). But in math, doing the math really was _the_ job, my PhD certainly was. Since this is academically funded, they now have to attempt to pivot that to save their profession. I am not optimistic myself.
We are all staring at the same existential dread, just seeing it unfold a bit slower. We're being told that utopia is being obsolete, and that is difficult to accept.
Its a reasonable view to take that "human math" [ math residing in human minds ] is the only math that counts.
Math that only resides in the weights of models, or arcane forms such as a long lean proof or even an unread textbook .. is not the math that we should be striving for.
Likewise all other technology [ and culture ].
LLMs and AI / AGI / ASI could lead to a new renaissance of math discussion and expansion of human math and science. Or the opposite, where we outsource all our thinking to the AI, and no new generation of artisans is trained by doing hard problems, and in a generation we have killed off human math.
Likewise all of the fields of human intellect. We need to make sure we protect future generations of doctors, biologists, software developers, architects, engineers, librarians, musicians, artists ...
A moratorium on AI development might be the only way to achieve this preservation of human culture.
I want to agree with this, but I have a hard time seeing how it can be done.
Tao is speaking of a very particular kind of mathematics, that done out of pure curiosity.
But maths, even at the highest levels, often finds applications sooner or later.
It will be economically impossible to justify boycotting correct mathematics that no humans understand on grounds only of purity.
This may happen very soon: one of the obvious applications of novel mathematical results is in building stronger AI models.
This goes in a necessary direction, from my personal take away of Gower's recent post on the subject.
Mathematics is suffering from Goodhart's Law:
"When a measure becomes a target, it ceases to be a good measure."
Doing something difficult was a signal that you:
a- understood it and all the background information it requires
b- internalized techniques and methods that are helpful in problem solving in general
Now it just means nothing
Timothy Gower
One thing that concerns me from all this is "understanding" is very important to human progress. The fact it took 400 years to crack Fermat's theorem resulted in a lot of "Side Quests". These side quests helped grow other fields (for instance elliptical cryptography). Im concerned with AI that we will loose these side quests.
I do neither maths nor science with AI but in my experience most models are perfectly willing to burn tokens on a ton of sidequests at the earliest opportunity.
Yeah but do you read through and find if one of those side quests is useful?
While I understand and emphatically with Tao's concern I'm afraid it's missing the forest from the trees. Unless you can make a claim that AI will never be able to perform intellectually at the same level as any human at a much lower cost, there's an outstanding utility problem that remains unaddressed. Sure enough, the AI may not have taste or goals, or many human traits, but that's irrelevant to the much thornier (and much broader than mathematics or even academia) question related to who's getting paid how much and for what.
I enjoy learning math from LLM proofs with the help of LLMs https://github.com/htzh/flt_for_human . It is amazing how well models do when they are well grounded by formalized proof traces (even if created by other models).
I'd be interested in hearing a field report on this! For example, I can easily imagine that they're great at walking through the proof step by step, explaining background as necessary; but as TFA notes, one of the most important questions is "why is this definition the way it is?", and my bet would be that the Lean is not enough to help the LLMs meaningfully in answering that.
I can’t help but feel a little schadenfreude. STEM folks may soon find themselves masters of skills as esoteric as translating Ancient Greek poetry or analyzing 18th century novels. The ability to construct complex mathematical proofs will become a party trick, rather like the ability to mentally multiply 10 digit numbers. The arguments that STEM snobs dismissed in favor of the study of the humanities will be the very same arguments that they now turn to. We will hear about how math and science make you a better rounded person, have inherent as well as instrumental value, etc. etc.
Even if we can proof/disproof any statement in Math (not possible due to halting problem), Human still need to decide which statement to be called "theorem".
The theorem thing is invented by human to help other people better understand Math structure in a easier way.
Interesting headline.
It's interesting because, as far as I'm aware, the vast majority of people already believe that math is more than proof. A slightly smaller but still very large majority don't even include proofs in their mental concept of what math involves.
> don't even include proofs in their mental concept of what math involves
Technically, the largest majority are the people who go: "What are proofs?" :P
A majority? I doubt it, simply because the majority doesn't know what math is at all.
At university level introductory calculus, the person teaching class had to reassure students that math wasn't entirely arithmetic or adding up numbers. He did this because it's a common misunderstanding.
Math academia 2025
> Sorry, only epic problem solvers allowed here
Math academia 2026
> We were more than just problem solvers
I think people are overblowing this though. Wake me up when GPT-whatever writes gcc from scratch, then by the Curry-Howard I'd be impressed
Jacob Tsimerman claims [1] we might have superhuman expositors by April, so then what?
[1] https://youtu.be/H7_d_sgui6o?t=4436 (timestamped url)
The people building AI claim it will surpass human intelligence in all respects and prerhaps kill us all. Should we just cease all human activity on the basis of what AI might do in future?
Personally, I doubt AI can surpass a good human explainer because explanation requires empathy, which benefits from being an instance of the kind of entity you are explaining the thing to. That gives you a way of exploring and evaluating the space of possible explanations that isn't available to an LLM.
It's all good until we have superhuman appreciators :)
It starts to sound like medieval science - "understanding" instead of proofs. And like a medieval army loosing a battle in the open field tries to retreat back into the fortress, people, facing the prospects of machine doing intelligent tasks better than humans, start to retreat into areas like intuition which supposedly aren't reachable by the machine. Some go even further starting to talk about religion. It is very Hegelian that the crown jewel achievement of our civilization starts to drive people away from the foundational principles of that civilization.
Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability. But of course mathematics is all about proof, and for that reason I was wary of it for a very long time.
>Basing science on proof (or anyway believing that you can) is Logical Positivism, a mindset that opposes the method of falsifiability
not really. You can consider positive proof as an experiment confirming your theory and the negative proof and counter examples as an experiment falsifying your theory.
Yes, really. "Positive proof" opposes the concept of falsification. You can only have it within a system formal logic, and science can contain those, but isn't one.
Mr. Tao is an excellent politician. Lots of awards and texts, yet no major problem solved.
It seems now that NS is solved he is mobilizing the community to convince taxpayers continue to pay even though AI may do a better job in his work.
Also, his opinion of AI has continually changed in the past years, after the capabilities were demonstrated.
Gr8 b8 m8