While applying, I looked at the current SoTA, (briefly) read through some of the papers, and realized that I am very far away from understanding them.
Understanding one of these proofs is the work of several months, years or lifetimes depending on whether or not something clicks. It requires a kind of stamina that I quite frankly don't have, but I would like to develop.
If the mathaton's organizers accept my team, I realized that I would spend the next few years working through the result.
So why apply to the Mathaton?
"Many years ago the great British explorer George Mallory, who was to die on Mount Everest, was asked why did he want to climb it. He said, 'Because it is there.'
Well, [theoretical math] is there, and we're going to climb it, and [topology] and [number theory] are there, and new hopes for knowledge and peace are there. And, therefore, as we set sail we ask God's blessing on the [~~most hazardous and dangerous and greatest adventure~~] on which [we have] ever embarked."
More seriously, I applied because I was hoping to get access to the models without the veil. I don't think people realize just how big the gap is between what exists behind the scenes at these entities, and what we get out here.
And it's frustrating. Because I think it's within the rights of frontier labs to decide whether or not to sell access to a product, but the labs aren't just doing that. They're trying to thumb the scale to make sure that none of us ever get access to these models at peak performance. Ever.
And I think humanity is worse off for that. I am worse off for that.
I have studied the shape and structure of historical technological revolutions (and I've written about it), and usually the world doesn't realize how big of a big deal the big deal is because the big deal is often flawed, broken, and under-delivers. In the short term.
In the long term...? The world changed in the past few months. I think mathematics is one small part of that.
For most of human history, higher mathematics would have been inaccessible to me, and other outsiders, no matter how well heeled. Mathematics is, or rather was, a living discipline that existed piecemeal in a handful of minds across the world. These people's time was finite and valuable. To just meet them, you'd have to jump through hoops, and spend years proving yourself.
There is no price for an hour of tutoring from Terence Tao. But now, with AI? You can have an entity with the capabilities of Terry Tao help you understand the subtleties of math.
AI has changed what mathematics is. And every prominent mathematician seems to know it. They feel like mathematics has been devalued, and in some ways it has. Mathematics has gone from being a living discipline kept alive by a chosen few to a wellspring everyone can sip from. For the first time in human existence, learning and accessing higher mathematics doesn't involve jumping through hoops and knowing the right people. You can just ask.
I can just ask.
Except I can't. Because that capability is being gate kept. And I want to know. I want to climb the mountain.
"There is no royal road to Geometry" - Euclid. Mathematical knowledge isn't from tutoring but doing. And no one is against the AI helping explain the tricky parts to help you practice. Rather it's about dumping a giant bunch of low quality text with some Lean claiming a big result is done.
Yes, it's why I love math. You can't buy fluency.
There are subtleties to mathematics that aren't easy to understand from the written page alone. It's why it's a living medium.
For example, as we're talking about LLMs... why not, there are ways to reason about vector spaces that weren't intuitive for me to understand. It's something that required talking things out with a friend who is a practising mathematician (albeit in training).
I am not smart enough to reconstruct all of mathematics on my own from scratches on paper alone. That back and forth is necessary. And it's something that you couldn't have "bought" for cutting edge math at any price a few months before this point in time. Because it exists in the minds of people and it needs lots of back and forths with those people.
It's why LLM proofs can be slop on paper. A proof that no one can check or understand is not but scratches on paper. BUT LLMs are also the solution to the problem they create. The machines that can generate proofs are also machines that can help us understand them.
The living medium can now be represented and scaled inside of a machine. I can now sit down at an airport and have that discussion. I think that's transformative for our species.
Just wait until BCI and/or fast Pavlovian conditioning force fed by agents into human learners.
I've been vibe coding my own SRS software that is vastly superior to my learning style than Anki, and I know I'm just scratching the surface of accelerated learning. Who knows where this goes.
MRI and glucose injections with agent-tutor steering and millisecond feedback to learning?
We might be able to Matrix "I Know Kung-Fu" things into brains one day.
Can you, I, or any mathematician who isn't well connected (let's say someone who is a young Maryam Mirzakhani or just someone who is in grad school) learn from the system that produced the solution to the unit distance problem? <a href="https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf" rel="nofollow">https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...
You will notice that it says on the first page,
"first mathematically generated in one shot by an internal model at OpenAI"
Mathematicians want to talk to the exact model variant whose summarized chain of thought is,
<a href="https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf" rel="nofollow">https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...
And I want to talk to models of similar aptitude and capability to help me understand nuances of the proof. Mathematicians will happy to pay for this. I've heard that people and non-profits are putting together $$$ for this to get access to these systems so that they can all interrogate them.
But the issue is that we can't. And I'm using the royal we here.
The paper says that the labs shouldn't release proofs from models that mathematicians can't interrogate. It's very clear that the models we get as users aren't the models used to produce the breakthroughs. And as LLMs display emergent capabilities, it's uncertain whether or not the model actually understands what it's explaining.
Because if I don't understand it. Professional mathematicians who are subject experts don't understand it. Then how do we know the model does? How do we know that it's correctly representing the proof produced by a more capable model? It's not logical to take any random model at its word, unless we can verify. Or, if it's the same model that produced the proof.
And that's what the mathematicians want. Access to the actual models.
areoform · · focus · HN ↗
While applying, I looked at the current SoTA, (briefly) read through some of the papers, and realized that I am very far away from understanding them.
Understanding one of these proofs is the work of several months, years or lifetimes depending on whether or not something clicks. It requires a kind of stamina that I quite frankly don't have, but I would like to develop.
If the mathaton's organizers accept my team, I realized that I would spend the next few years working through the result.
So why apply to the Mathaton?
"Many years ago the great British explorer George Mallory, who was to die on Mount Everest, was asked why did he want to climb it. He said, 'Because it is there.'
Well, [theoretical math] is there, and we're going to climb it, and [topology] and [number theory] are there, and new hopes for knowledge and peace are there. And, therefore, as we set sail we ask God's blessing on the [~~most hazardous and dangerous and greatest adventure~~] on which [we have] ever embarked."
More seriously, I applied because I was hoping to get access to the models without the veil. I don't think people realize just how big the gap is between what exists behind the scenes at these entities, and what we get out here.
And it's frustrating. Because I think it's within the rights of frontier labs to decide whether or not to sell access to a product, but the labs aren't just doing that. They're trying to thumb the scale to make sure that none of us ever get access to these models at peak performance. Ever.
And I think humanity is worse off for that. I am worse off for that.
I have studied the shape and structure of historical technological revolutions (and I've written about it), and usually the world doesn't realize how big of a big deal the big deal is because the big deal is often flawed, broken, and under-delivers. In the short term.
In the long term...? The world changed in the past few months. I think mathematics is one small part of that.
For most of human history, higher mathematics would have been inaccessible to me, and other outsiders, no matter how well heeled. Mathematics is, or rather was, a living discipline that existed piecemeal in a handful of minds across the world. These people's time was finite and valuable. To just meet them, you'd have to jump through hoops, and spend years proving yourself.
There is no price for an hour of tutoring from Terence Tao. But now, with AI? You can have an entity with the capabilities of Terry Tao help you understand the subtleties of math.
AI has changed what mathematics is. And every prominent mathematician seems to know it. They feel like mathematics has been devalued, and in some ways it has. Mathematics has gone from being a living discipline kept alive by a chosen few to a wellspring everyone can sip from. For the first time in human existence, learning and accessing higher mathematics doesn't involve jumping through hoops and knowing the right people. You can just ask.
I can just ask.
Except I can't. Because that capability is being gate kept. And I want to know. I want to climb the mountain.
wbl · · focus · HN ↗
areoform · · focus · HN ↗
There are subtleties to mathematics that aren't easy to understand from the written page alone. It's why it's a living medium.
For example, as we're talking about LLMs... why not, there are ways to reason about vector spaces that weren't intuitive for me to understand. It's something that required talking things out with a friend who is a practising mathematician (albeit in training).
I am not smart enough to reconstruct all of mathematics on my own from scratches on paper alone. That back and forth is necessary. And it's something that you couldn't have "bought" for cutting edge math at any price a few months before this point in time. Because it exists in the minds of people and it needs lots of back and forths with those people.
It's why LLM proofs can be slop on paper. A proof that no one can check or understand is not but scratches on paper. BUT LLMs are also the solution to the problem they create. The machines that can generate proofs are also machines that can help us understand them.
The living medium can now be represented and scaled inside of a machine. I can now sit down at an airport and have that discussion. I think that's transformative for our species.
echelon · · focus · HN ↗
Just wait until BCI and/or fast Pavlovian conditioning force fed by agents into human learners.
I've been vibe coding my own SRS software that is vastly superior to my learning style than Anki, and I know I'm just scratching the surface of accelerated learning. Who knows where this goes.
MRI and glucose injections with agent-tutor steering and millisecond feedback to learning?
We might be able to Matrix "I Know Kung-Fu" things into brains one day.
fragmede · · focus · HN ↗
omnicognate · · focus · HN ↗
In what way are these "gatekeepers" stopping you asking an LLM questions about maths?
areoform · · focus · HN ↗
Can you, I, or any mathematician who isn't well connected (let's say someone who is a young Maryam Mirzakhani or just someone who is in grad school) learn from the system that produced the solution to the unit distance problem? <a href="https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-remarks.pdf" rel="nofollow">https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29a...
You will notice that it says on the first page,
Mathematicians want to talk to the exact model variant whose summarized chain of thought is, <a href="https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf" rel="nofollow">https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925d...And I want to talk to models of similar aptitude and capability to help me understand nuances of the proof. Mathematicians will happy to pay for this. I've heard that people and non-profits are putting together $$$ for this to get access to these systems so that they can all interrogate them.
But the issue is that we can't. And I'm using the royal we here.
The paper says that the labs shouldn't release proofs from models that mathematicians can't interrogate. It's very clear that the models we get as users aren't the models used to produce the breakthroughs. And as LLMs display emergent capabilities, it's uncertain whether or not the model actually understands what it's explaining.
Because if I don't understand it. Professional mathematicians who are subject experts don't understand it. Then how do we know the model does? How do we know that it's correctly representing the proof produced by a more capable model? It's not logical to take any random model at its word, unless we can verify. Or, if it's the same model that produced the proof.
And that's what the mathematicians want. Access to the actual models.