The Mathematics are Mad
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0:00–0:20Some are calling this the math apocalypse. That's right, math has been solved. But, mathematicians around the world are reacting with disgust. Gross. So, why are they so mad? Why are mathematicians mad that math was solved? I thought that's their entire, like, modus operandi, as as they say.
Structure
Some are calling this the math apocalypse. That's right, math has been solved. But, mathematicians around the world are reacting with disgust. Gross. So, why are they so mad? Why are mathematicians mad that math was solved? I thought that's their entire, like, modus operandi, as as they say. Well, it turns out the reason why mathematicians are upset is because math solved math. Give math the field medal. That's right, the linear algebra solved math, aka LLMs, aka OpenAI. So, we did miss our revenue by $20 billion, but I did solve all of math. You know what? It's because I do it for the love of the game. No, I'm not talking about the millennial prize problems. Of course, if you're unfamiliar with the millennial prize problems, it is like these really difficult problems that face millennials, such as the Riemann hypothesis, Navier-Stokes, ordering avocado toast while affording your mortgage. Because today they dropped 372 open math problems, which is approximately 80% of all major discoveries in math in a single day, which is a little bit concerning because the list itself contains clear numbering counting problems. They were able to solve problems in which decades went by with no progress, but they can't make a list of numbers in order. So, we'll go over what legendary mathematician Terence Tao has to say, why these boxes could potentially mean the end of mathematics for all humans, and it may turn out that OpenAI got a few of them
Watch from 0:00wrong. But, before we do, a quick thank you to the sponsor, Vibe Coders. Silence. Someone who's worked in enterprise is speaking. You know, simple OAuth sign-in's not that bad to program. For real enterprise applications that need single sign-on, passkeys, user management, roles and permission, enterprise readiness, and passwords and recovery, uh that's a little bit more difficult. And that is where WorkOS comes in. WorkOS is trusted by a lot of the top companies, OpenAI, Anthropic, Cursor, and a whole lot more. But not only that, it's free for your first million users. So, that means for you, that is 999,999 free seats available. Pretty much no matter what back end you have chosen, there is an SDK available for you. So, go check out WorkOS at workos.com. Links in the description. To really understand the story, we do need to take a step back and go to September 11th, 20 26, when Terence Tao, superstar mathematician, releases the following letter. Over the last few months, the mathematical abilities of LLMs have improved dramatically to a point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics and to the mathematical community. The goals of the AI companies and the goals of mathematical community are severely misaligned. The letter goes on to explain more of the problems about why having AI company solve math problems exist. We'll get into that in a moment, but this letter was signed by tons of field medalists. And it's fairly
Watch from 1:31universally accepted that after Navier-Stokes, mathematicians generally were pretty unhappy with OpenAI. Now, of course, OpenAI, seeing that everybody was upset, decided that on October 6th, they should just drop 372 solutions on the mathematics community because obviously everybody in mathematics was upset they only solved one problem. That's clearly the problem, isn't it, OpenAI? And it was made even worse when we learned that OpenAI one-shotted these problems. Hey, academics, would you like free tokens? You know, ChatGPT would just like to offer you free tokens, you know? Just Jippity being, you know, good to fellow humans, you know? Just me being a human, no strings attached, nothing you have to worry about. I just want to see you be successful. You, yeah, you. Just type all of your research into the box, and we'll just make sure it's good, you know? Just let us have a second eye for you on your work. Now, some of these problems were the open-ended math problems that if you were to solve, you would effectively instantaneously get a Field Medal in mathematics, which is like the highest, you know, pre-most prestigious award in all of mathematics. And so, the fact that OpenAI acclaimed to solved 372 of them on a single day has people worried. People are shouting, "Mathematics has been solved." Now, there are two big reasons why people were getting really upset in the math world. Terence Tao writes about the first one. In traditional mathematics, a breakthrough proof of a long-standing open conjecture generates a large amount
Watch from 3:01of subsequent activity and excitement in the field. The authors of the proof are invited to give many talks and meet with other experts in the area. Workshops are set up to discuss the proof as well as other recent developments. As a result, new connections and collaborations are formed, follow-up problems are shared, and the other mathematicians, both junior and senior, are attracted to the field, including in some case the authors themselves, if they initially came from different subjects. As a result of all this activity, the proofs naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. And so, one thing he's trying to say here is that as people make progress in mathematics, it excites and generates new buzz, and thus new mathematicians are able to come in, be able to understand this new piece of knowledge, and then they themselves move the field forward. So, it's this constant generation over generational move of mathematics to higher and higher understanding. The problem with AI is in a single day, all these problems at once are just dropped. It's just like, "Hey, we solved everything." It doesn't generate new understanding, it doesn't generate excitement, it's just everything gets solved. So, what's the purpose of learning if everything just is instantly solved by math itself? And more importantly, a lot of these proofs can be literal millions of lines of Lean, the programming language. If you're not familiar with Lean, the programming language, let me give you just a small example. Right here, I'm saying, "Hey, I'm going to create a new theorem." Now, this theorem takes in two variables, A and B, and they're both in natural numbers. And I'm going to say,
Watch from 4:33"Hey, there's a fact that says A equals B." Now, I'm going to prove this fact by the following: A + 1 = B + 1. And what I'm going to do is I'm going to rewrite the rule by taking the left-hand side and assigning it to the right-hand side in the subsequent statement. And now the statement reads B + 1 = B + 1, which means that this statement is in fact correct. So, we have in fact proved that A = B. Now, it's pretty obvious. The fact that we have a language that can be used by machines to do proofs, of course OpenAI would be able to use this. And LLMs love generating huge amounts of code. So, Lean becomes this really great vehicle for OpenAI to be able to solve and prove problems. And that's exactly what they did. They used this language right here, and they solved hundreds of open problems. Now, not all of the solves actually come with the Lean formalization. I believe it was like 100 and some of the 372 problems did come with Lean formalization. But nonetheless, hundreds of problems have been proven. Now, obviously, the software engineering problem and the math problem are pretty similar here. How do you raise up new people in a field in which AI just simply eats their lunch every single day of the week? That is a real problem in both software engineering and now apparently in math, because if everything can be solved, what is the purpose of even doing it? Why should we even learn it? And then really, we get almost a new problem with all of this because of this following proof that the LLM did. It was able to do this bin
Watch from 6:03packing problem right here. Now, you're probably thinking, "What's the big deal with this bin packing problem?" Well, let me explain it for a moment. You effectively are given these squares, which is which they're unit squares. They're 1 by 1 squares. And the goal of this exercise is to fit it in the smallest area. Now, what we are hoping for is that every solution has some sort of like kind of magic to it, some sort of formula, some sort of order to it, and it's not just chaos. It's not just simply compute. But when you look at this bin packing problem, there's no order. It's kind of insane. There's like that small tiny gap right here that you fit just a little bit of the corner and everything perfectly fits within this area, which means that there is an optimal solution. It just requires an insane amount of computation. But the deeper and more scary kind of implication of this problem is that all of mathematics is in fact this. It just is really computationally bound, which means at some point humans are no longer even capable of doing the most optimized math. I think the best way to show this bin packing problem is through this integer multiplication problem. For a long time, the lower bound accepted solution for as fast as we possibly can get is n log n. But it turns out that OpenAI found out that it's not actually quite n log n. It's n log n to the power of 0.99999 54 nines and then some other additional fractions, which means it's actually you can go actually just a little just a little bit faster.
Watch from 7:34Meanwhile, over in physics land, the fundamental unit of length is this monstrosity. The universe does not care about your aesthetics. A physicist dies and goes to heaven, upon which he asked God why he made the speed of light as arbitrary as 186,282 miles per second. "What do you mean?" God replies. "The speed of light is one." Of course, that all those nines is this right here, 2 to the negative 182. In other words, this is math humans can't really do. See, we we kind of look at things in these perfect ways. We kind of have this expectation that in a sense nature is beautiful and it kind of aligns. It may look disorganized from our perspective, but when you really get to it, there's these bounds these natural bounds that exist such as n log n. But the big worry is that no, it's not that. It's just these insane computational tasks that can only be done by machines that will actually show further and better and better proofs, and humans really become more and more irrelevant at creating these proofs. But now I want to go back to this box problem. See, the thing about this box problem is that there's these ideas of perspectives. Meaning that there's a certain set of problems that are really hard to solve if you look at them from our kind of perspective. A great example of that is the Archimedean spiral. Now, if you were to map this equation and attempt to do something in the Cartesian coordinate system, you effectively can't do it and it's really really hard, but the spiral shows up all the time in nature. But if you convert the coordinate system to polar coordinates, the equation becomes as complicated as y
Watch from 9:04equals mx + b. And that's kind of the beautiful thing about math is that some problems, when you change your perspective, can become dramatically simplified. Of course, you know, Fourier transforms are another version of this exact same thing in a signal processing. But now back to these boxes. You're thinking, well, this is just the way it is. It's super complicated. Well, it turns out there's this paper right here that actually breaks down this image and says, "Hey, these two images are actually the same image." Now, I know this may be a little bit difficult to see, but effectively by taking this planar optimum and wrapping it into a torus, you can actually derive the exact same number for the optimal size of the planar size, but you do it in a Fibonacci torus. And it's beautiful. Look at it. It's actually like uniform and repetitive. And that's kind of this cool thing is that OpenAI may have solved it, but it turns out that there was actually a more beautiful and obvious solution. We just simply had the wrong perspective. And going back to this insane number, something also kind of interesting is happening. You know the 4-minute mile? For a long time people didn't think, oh, well the 4-minute mile, that's the you can't cross that. Okay, humans can't do 4-minute mile. But then Roger Bannister on May 6th, 1954 broke the 4-minute mile. And then almost immediately after that people started going faster and faster and faster. What was once thought to be impossible became probable for people to do. So let's jump back to this. It turns out that yes, n log n is in fact not the lower bound,
Watch from 10:34it's just some insane number, right? So let's start here. Integer multiplication is probably the most shocking result. Similar to the matrix multiplication, it's an asymptotic reduction. Also like matrix result, it's not practical. But very few people would have predicted the previous barrier could have been broken, even slightly. But here's the funny thing is, after doing that, this Doug Kalckit guy ends up reviewing the paper. They come up with a solution that's 2 to the 75th fold increase in the algorithm's exponential saving parameter. In other words, they just went from almost just slightly like infinitesimally better. So okay, this is getting better. Now again, it's such a small number it doesn't really matter. A few hours later, once again, it gets 570 million-fold improvement on their previous result. They yet again make the proof even better. Again, a 500,000-fold improvement. Again, even better. bigly better. Even better. Even better. They're continuously improving the runtime. In fact, they got it all the way to this, 2 to the negative 17, which is just under five nines. It went from 54 nines to five nines. A huge win for the fans of the number nine. So what was once thought unable to be improved now has been drastically improved over and over again to the point where we're getting near practical implications where the running time could very well be n square root of log n, which would be an insane improvement. Because here's the thing is, just because we prove it can run this fast doesn't mean that
Watch from 12:07we're stuck in some sort of crazy irrational number forever, but instead it means that there could be a bound that is actually elegant. Or there's a way of looking at the problem once we change our perspective that makes this really elegant. We just simply haven't got there. And again, so this is the problem that everybody is worried about is that OpenAI is generating all these proofs, but a lot of them are these brute force insanity approaches that they're not actually getting to the real heart of the problem. They're not rotating their perspective and coming up with a beautiful and ideal solution. Instead, they're just proofed. They're just saying, "Hey, yes, this does exist. Here we go. We did it." And finally, it kind of looks like mathematicians are starting to wrestle with this idea that I think software engineering has been wrestling with for a while now, which by the way again software engineering fake term, okay? We're not engineering anything, okay, buddy? When was the last time you whipped out an integral? I think not. But as you can see, they don't just want to review proofs. They actually want to understand what they're doing and hoping to open up new possibilities. And I kind of feel like if you would have transported me back to 2018, you were seeing all these people working on software, but there's also a bunch of people working on ways to make working on software better. New libraries, new ways to kind of work on things, new perspective on things. But since the advent of AI, I feel like that, whatever that was, has taken a dramatic downturn in say volume talking about it. Instead, people don't review code, so the idea of
Watch from 13:38going and trying to make new perspectives or better views onto problems that were once hard in computing have kind of largely faded. But of course, computing is the practical science, right? We are the practical art form because at the end of the day, we have to deliver something that is a value to a customer, not to the next generation of programmers. So, ultimately, the use of AI has much different implications compared to math, because at the end of the day, the purpose of math is for us to develop new perspectives and new ways to solve things and really define the optimal beautiful solution, not just simply 0.999999999 going on effectively forever. And just like, "Dude, just like that, much better." I'd say the strangest thing is probably r/math, though. As you can see, do you see Do you see those top stories? Do any of them mention the fact that 372 open problems were solved in a single day and dropped on GitHub? No. It turns out that if you post on r/math, you're going to get banned. They allow one post, and you may only respond in threads if it's going to do anything with AI, which is kind of like, "Dude, that's so silly. Like, what is going on over on Reddit? You fedora-wearing weirdos, my gosh. Why? Like, this is the potentially one of the greatest shattering days of math, and who knows what the implications are going to be from this, and what is actually going to happen over the next couple years due to this day starting now. And you can't even talk about it on math.
Watch from 15:08But, I do think we can take a lot from the math community as software devs, which is that they see AI, and they recognize the fundamental problem of what happens to juniors almost immediately. Like, that is the thing that Terrence Tao jumped on right away, which is that the discoveries and improvement of math should directly excite and get the next generation ready to take on the mantle and go further than their predecessors. I know this is a weird time that we're living through, because there kind of is that lack of excitement in libraries and cool stuff. I still see some of the of the the trad coders, the hand coders out there still building super cool stuff, and they're really pushing forward a lot of things that otherwise would not have been pushed forward. And so, how do you, as you know, as someone who's getting started obtain this knowledge, I do hope that you take the time and the effort to really learn these concepts deep, even if the hands are not on the keyboard as much as they should be, or they could be, or they would have been as they were just a few years ago. Because much like math, there still needs to be the next generation that understands things, and I do think it's a worthwhile pursuit. The name is the prime number regimen.
Watch from 16:33Full transcript
- 0:00
Some are calling this the math apocalypse. That's right, math has been solved. But, mathematicians around the world are reacting with disgust. Gross. So, why are they so mad? Why are mathematicians mad that math was solved? I thought that's their entire, like, modus operandi, as as they say.
- 0:20
Well, it turns out the reason why mathematicians are upset is because math solved math. Give math the field medal. That's right, the linear algebra solved math, aka LLMs, aka OpenAI. So, we did miss our revenue by $20 billion, but I did solve all of math. You know what? It's because I do it for the love of the game. No, I'm not talking about the
- 0:41
millennial prize problems. Of course, if you're unfamiliar with the millennial prize problems, it is like these really difficult problems that face millennials, such as the Riemann hypothesis, Navier-Stokes, ordering avocado toast while affording your mortgage. Because today they dropped 372 open math problems, which is approximately 80% of all major
- 1:03
discoveries in math in a single day, which is a little bit concerning because the list itself contains clear numbering counting problems. They were able to solve problems in which decades went by with no progress, but they can't make a list of numbers in order. So, we'll go over what legendary mathematician Terence Tao has to say, why these boxes
- 1:23
could potentially mean the end of mathematics for all humans, and it may turn out that OpenAI got a few of them wrong. But, before we do, a quick thank you to the sponsor, Vibe Coders. Silence. Someone who's worked in enterprise is speaking. You know, simple OAuth sign-in's not that bad to program. For real enterprise applications that need single sign-on, passkeys, user
- 1:45
management, roles and permission, enterprise readiness, and passwords and recovery, uh that's a little bit more difficult. And that is where WorkOS comes in. WorkOS is trusted by a lot of the top companies, OpenAI, Anthropic, Cursor, and a whole lot more. But not only that, it's free for your first million users. So, that means for you, that is 999,999
- 2:09
free seats available. Pretty much no matter what back end you have chosen, there is an SDK available for you. So, go check out WorkOS at workos.com. Links in the description. To really understand the story, we do need to take a step back and go to September 11th, 20 26, when Terence Tao, superstar mathematician, releases the following letter. Over the last few months, the
- 2:30
mathematical abilities of LLMs have improved dramatically to a point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics and to the mathematical community. The goals of the AI companies and the goals of mathematical community are severely misaligned. The letter goes
- 2:51
on to explain more of the problems about why having AI company solve math problems exist. We'll get into that in a moment, but this letter was signed by tons of field medalists. And it's fairly universally accepted that after Navier-Stokes, mathematicians generally were pretty unhappy with OpenAI. Now, of course, OpenAI, seeing that everybody
- 3:11
was upset, decided that on October 6th, they should just drop 372 solutions on the mathematics community because obviously everybody in mathematics was upset they only solved one problem. That's clearly the problem, isn't it, OpenAI? And it was made even worse when we learned that OpenAI one-shotted these problems. Hey, academics, would you like free tokens?
- 3:32
You know, ChatGPT would just like to offer you free tokens, you know? Just Jippity being, you know, good to fellow humans, you know? Just me being a human, no strings attached, nothing you have to worry about. I just want to see you be successful. You, yeah, you. Just type all of your research into the box, and we'll just make sure it's good, you know? Just let us have a second eye for you on your
- 3:54
work. Now, some of these problems were the open-ended math problems that if you were to solve, you would effectively instantaneously get a Field Medal in mathematics, which is like the highest, you know, pre-most prestigious award in all of mathematics. And so, the fact that OpenAI acclaimed to solved 372 of them on a single day has people
- 4:16
worried. People are shouting, "Mathematics has been solved." Now, there are two big reasons why people were getting really upset in the math world. Terence Tao writes about the first one. In traditional mathematics, a breakthrough proof of a long-standing open conjecture generates a large amount of subsequent activity and excitement in the field. The authors of the proof are
- 4:37
invited to give many talks and meet with other experts in the area. Workshops are set up to discuss the proof as well as other recent developments. As a result, new connections and collaborations are formed, follow-up problems are shared, and the other mathematicians, both junior and senior, are attracted to the field, including in some case the authors themselves, if they initially came from different subjects. As a result of all this activity, the proofs
- 4:57
naturally become digested, streamlined, placed in context with other results in the field, and ultimately become part of the textbooks and lecture notes for the next generation of mathematicians in the field. And so, one thing he's trying to say here is that as people make progress in mathematics, it excites and generates new buzz, and thus new mathematicians are able to come in, be able to
- 5:18
understand this new piece of knowledge, and then they themselves move the field forward. So, it's this constant generation over generational move of mathematics to higher and higher understanding. The problem with AI is in a single day, all these problems at once are just dropped. It's just like, "Hey, we solved everything." It doesn't generate new understanding, it doesn't generate excitement, it's just
- 5:40
everything gets solved. So, what's the purpose of learning if everything just is instantly solved by math itself? And more importantly, a lot of these proofs can be literal millions of lines of Lean, the programming language. If you're not familiar with Lean, the programming language, let me give you just a small example. Right here, I'm saying, "Hey, I'm going to create a new theorem." Now, this theorem takes in two variables, A and B, and they're both in
- 6:01
natural numbers. And I'm going to say, "Hey, there's a fact that says A equals B." Now, I'm going to prove this fact by the following: A + 1 = B + 1. And what I'm going to do is I'm going to rewrite the rule by taking the left-hand side and assigning it to the right-hand side in the subsequent statement. And now the
- 6:21
statement reads B + 1 = B + 1, which means that this statement is in fact correct. So, we have in fact proved that A = B. Now, it's pretty obvious. The fact that we have a language that can be used by machines to do proofs, of course OpenAI would be able to use this. And LLMs love generating huge amounts of
- 6:42
code. So, Lean becomes this really great vehicle for OpenAI to be able to solve and prove problems. And that's exactly what they did. They used this language right here, and they solved hundreds of open problems. Now, not all of the solves actually come with the Lean formalization. I believe it was like 100 and some of the 372 problems did come
- 7:02
with Lean formalization. But nonetheless, hundreds of problems have been proven. Now, obviously, the software engineering problem and the math problem are pretty similar here. How do you raise up new people in a field in which AI just simply eats their lunch every single day of the week? That is a real problem in both software engineering and now apparently in math, because if everything can be solved, what is the
- 7:23
purpose of even doing it? Why should we even learn it? And then really, we get almost a new problem with all of this because of this following proof that the LLM did. It was able to do this bin packing problem right here. Now, you're probably thinking, "What's the big deal with this bin packing problem?" Well, let me explain it for a moment. You effectively are given these squares, which is which they're unit squares.
- 7:43
They're 1 by 1 squares. And the goal of this exercise is to fit it in the smallest area. Now, what we are hoping for is that every solution has some sort of like kind of magic to it, some sort of formula, some sort of order to it, and it's not just chaos. It's not just simply compute. But when you look at this bin packing problem, there's no order. It's kind of insane. There's like
- 8:05
that small tiny gap right here that you fit just a little bit of the corner and everything perfectly fits within this area, which means that there is an optimal solution. It just requires an insane amount of computation. But the deeper and more scary kind of implication of this problem is that all of mathematics is in fact this. It just
- 8:28
is really computationally bound, which means at some point humans are no longer even capable of doing the most optimized math. I think the best way to show this bin packing problem is through this integer multiplication problem. For a long time, the lower bound accepted solution for as fast as we possibly can get is n log n. But it turns out that
- 8:49
OpenAI found out that it's not actually quite n log n. It's n log n to the power of 0.99999 54 nines and then some other additional fractions, which means it's actually you can go actually just a little just a little bit faster. Meanwhile, over in physics land, the fundamental unit of length is this monstrosity. The universe does not care
- 9:09
about your aesthetics. A physicist dies and goes to heaven, upon which he asked God why he made the speed of light as arbitrary as 186,282 miles per second. "What do you mean?" God replies. "The speed of light is one." Of course, that all those nines is this right here, 2 to the negative 182. In other words, this is math humans can't really do. See, we we kind of look
- 9:31
at things in these perfect ways. We kind of have this expectation that in a sense nature is beautiful and it kind of aligns. It may look disorganized from our perspective, but when you really get to it, there's these bounds these natural bounds that exist such as n log n. But the big worry is that no, it's not that. It's just these insane computational tasks that can only be
- 9:51
done by machines that will actually show further and better and better proofs, and humans really become more and more irrelevant at creating these proofs. But now I want to go back to this box problem. See, the thing about this box problem is that there's these ideas of perspectives. Meaning that there's a certain set of problems that are really hard to solve if you look at them from
- 10:12
our kind of perspective. A great example of that is the Archimedean spiral. Now, if you were to map this equation and attempt to do something in the Cartesian coordinate system, you effectively can't do it and it's really really hard, but the spiral shows up all the time in nature. But if you convert the coordinate system to polar coordinates, the equation becomes as complicated as y
- 10:34
equals mx + b. And that's kind of the beautiful thing about math is that some problems, when you change your perspective, can become dramatically simplified. Of course, you know, Fourier transforms are another version of this exact same thing in a signal processing. But now back to these boxes. You're thinking, well, this is just the way it is. It's super complicated. Well, it turns out there's this paper right here
- 10:56
that actually breaks down this image and says, "Hey, these two images are actually the same image." Now, I know this may be a little bit difficult to see, but effectively by taking this planar optimum and wrapping it into a torus, you can actually derive the exact same number for the optimal size of the planar size, but you do it in a Fibonacci torus. And it's beautiful.
- 11:18
Look at it. It's actually like uniform and repetitive. And that's kind of this cool thing is that OpenAI may have solved it, but it turns out that there was actually a more beautiful and obvious solution. We just simply had the wrong perspective. And going back to this insane number, something also kind of interesting is happening. You know the 4-minute mile? For a long time
- 11:40
people didn't think, oh, well the 4-minute mile, that's the you can't cross that. Okay, humans can't do 4-minute mile. But then Roger Bannister on May 6th, 1954 broke the 4-minute mile. And then almost immediately after that people started going faster and faster and faster. What was once thought to be impossible became probable for people to do. So let's jump
- 12:02
back to this. It turns out that yes, n log n is in fact not the lower bound, it's just some insane number, right? So let's start here. Integer multiplication is probably the most shocking result. Similar to the matrix multiplication, it's an asymptotic reduction. Also like matrix result, it's not practical. But very few people would have predicted the previous barrier could have been broken,
- 12:23
even slightly. But here's the funny thing is, after doing that, this Doug Kalckit guy ends up reviewing the paper. They come up with a solution that's 2 to the 75th fold increase in the algorithm's exponential saving parameter. In other words, they just went from almost just slightly like infinitesimally better. So okay, this is getting better. Now again, it's such a
- 12:43
small number it doesn't really matter. A few hours later, once again, it gets 570 million-fold improvement on their previous result. They yet again make the proof even better. Again, a 500,000-fold improvement. Again, even better. bigly better. Even better. Even better.
- 13:04
They're continuously improving the runtime. In fact, they got it all the way to this, 2 to the negative 17, which is just under five nines. It went from 54 nines to five nines. A huge win for the fans of the number nine. So what was once thought unable to be improved now has been drastically improved over and over again to the point where we're
- 13:24
getting near practical implications where the running time could very well be n square root of log n, which would be an insane improvement. Because here's the thing is, just because we prove it can run this fast doesn't mean that we're stuck in some sort of crazy irrational number forever, but instead it means that there could be a bound that is actually elegant. Or there's a
- 13:45
way of looking at the problem once we change our perspective that makes this really elegant. We just simply haven't got there. And again, so this is the problem that everybody is worried about is that OpenAI is generating all these proofs, but a lot of them are these brute force insanity approaches that they're not actually getting to the real heart of the problem. They're not
- 14:06
rotating their perspective and coming up with a beautiful and ideal solution. Instead, they're just proofed. They're just saying, "Hey, yes, this does exist. Here we go. We did it." And finally, it kind of looks like mathematicians are starting to wrestle with this idea that I think software engineering has been wrestling with for a while now, which by the way again software engineering fake term, okay? We're not engineering
- 14:26
anything, okay, buddy? When was the last time you whipped out an integral? I think not. But as you can see, they don't just want to review proofs. They actually want to understand what they're doing and hoping to open up new possibilities. And I kind of feel like if you would have transported me back to 2018, you were seeing all these people working on software, but there's also a bunch of
- 14:48
people working on ways to make working on software better. New libraries, new ways to kind of work on things, new perspective on things. But since the advent of AI, I feel like that, whatever that was, has taken a dramatic downturn in say volume talking about it. Instead, people don't review code, so the idea of
- 15:08
going and trying to make new perspectives or better views onto problems that were once hard in computing have kind of largely faded. But of course, computing is the practical science, right? We are the practical art form because at the end of the day, we have to deliver something that is a value to a customer, not to the next generation of programmers. So, ultimately, the use of AI has much
- 15:30
different implications compared to math, because at the end of the day, the purpose of math is for us to develop new perspectives and new ways to solve things and really define the optimal beautiful solution, not just simply 0.999999999 going on effectively forever. And just like, "Dude, just like that, much better." I'd say the strangest thing is probably r/math, though. As you can see,
- 15:50
do you see Do you see those top stories? Do any of them mention the fact that 372 open problems were solved in a single day and dropped on GitHub? No. It turns out that if you post on r/math, you're going to get banned. They allow one post, and you may only respond in threads if it's going to do anything with AI, which is kind of like, "Dude, that's so silly. Like, what is going on
- 16:11
over on Reddit? You fedora-wearing weirdos, my gosh. Why? Like, this is the potentially one of the greatest shattering days of math, and who knows what the implications are going to be from this, and what is actually going to happen over the next couple years due to this day starting now. And you can't even talk about it on math.
- 16:33
But, I do think we can take a lot from the math community as software devs, which is that they see AI, and they recognize the fundamental problem of what happens to juniors almost immediately. Like, that is the thing that Terrence Tao jumped on right away, which is that the discoveries and improvement of math should directly excite and get the next generation ready
- 16:53
to take on the mantle and go further than their predecessors. I know this is a weird time that we're living through, because there kind of is that lack of excitement in libraries and cool stuff. I still see some of the of the the trad coders, the hand coders out there still building super cool stuff, and they're really pushing forward a lot of things that otherwise would not have been pushed forward. And so, how do you, as you
- 17:15
know, as someone who's getting started obtain this knowledge, I do hope that you take the time and the effort to really learn these concepts deep, even if the hands are not on the keyboard as much as they should be, or they could be, or they would have been as they were just a few years ago. Because much like math, there still needs to be the next generation that understands things, and I do think it's a worthwhile pursuit. The name
- 17:36
is the prime number regimen.
Description
Don’t vibe code your auth. Use WorkOS: https://trm.sh/workos
[Chapters]
00:00:00 - Why are Mathematicians so mad?
00:01:34 - WorkOS
00:02:19 - Mathematical Misalignment
00:03:55 - The Problems
00:04:27 - No more Proof fanfair
00:05:50 - Lean
00:07:08 - Math Humans Can't Do
00:10:00 - Perspectives
00:11:38 - Four-minute mile
00:13:53 - Heart & Understanding
00:15:16 - Purpose
00:15:47 - Reddit Weirdos
00:16:33 - The Next Generation
Sources:
OpenAI announcement: https://x.com/OpenAI/status/210759671...
Faster than n log n (AcerFur): https://x.com/AcerFur/status/21076067...
Improving n log n (Doug Colkitt): https://x.com/0xdoug/status/210765831...
https://x.com/0xdoug/status/210783236...
https://x.com/0xdoug/status/210791769...
https://x.com/0xdoug/status/210799411...
https://x.com/0xdoug/status/210800595...
Fibonacci squares paper (David Budden): https://x.com/davidmbudden/status/210...
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