It's interesting that this piece stands well enough if you replace mathematics with probably any other intellectual profession, including software development or pretty much anything else.
And, I lie to anyone who asks me why I’m a mathematician.
It is much easier to claim “I love learning the laws of life,”
while literally handwaving, than it is for me to flashback to
the twenty or so pivotal moments that lead to me walking out
of Gainesville with a PhD in Arithmetic Geometry.
Since “normal” people mostly don't understand what the software development job is about, handwaving in response to regular questions: “what do you do at work?”, “what do you like about your job?” – is pretty normal. I think that most of us have some prepared answers ready to use.
Mathematics is the thing you try to understand, don’t,
get frustrated about, and then do.
Just like the software development. I truly believe that the only people who can survive a software job are those who can tolerate the constant feeling of frustration caused by things not working or breaking for random reasons, and persevere in this environment to do the things you need to do.
I feel like the number is higher for software work where I'm at, and lower for every other profession. Scary when you think that your doctor is in it for the money
You can be a software developer without any formal training and without any licensure. Not so for an MD.
It's going to attract more people who have the mentality of artists or musicians, i.e. people who do it for the love of the craft and as a creative outlet.
True. I like math and CS theory, but despise coding (pays well though). AI is finally taking over this soul-sucking occupation and all I can say - good riddance.
It's an awesome profession for those of us who love it.
Tbf coding with ai is still super fun though. I am hoping that engs who hate it like you will finally get kicked out as productivity increases from ai and it will finally go back to just us nerds.
It's kinda soul-sucking being around all you guys that just hate this work, please get out and go do farming or something lol.
A PhD in Arithmetic Geometry and a publication in a top tier journal is hardly mediocre in and of itself, as far the author is concerned. The equivalent for a software engineer is probably a leading AI engineer, with a strong publication count. I think a lot of people who are not mediocre are unaware of what mediocrity actually is. Yeah if you're only in the top .1% and comparing yourself to the literal best in the world, you will feel mediocre in that sense . A mid mathematician maybe publication in worse journals, teaching community college.
> We're all frustration addicts. We just want to bang our heads against problems we don't yet know how to solve.
I've been tapering off AI lately. I think I've realized that conquering the struggle is the fun part, and accomplishments just don't hit the same if AI is smoothing over every friction and cordoning off all the pitfalls and rabbit-holes.
> My physicist friend once asked me what the point of doing research was if someone like Terence Tao could have figured out everything in my dissertation in a tenth of the time. I answered by pointing out that Terence Tao didn’t. Terence Tao did not find a small open problem posited by my advisor and publish a bite sized result making incremental progress. He has only so much time and so many other fish to fry.
This reminds me of a post I saw recently, although I can't remember the platform. It said something along the lines of assessing the limits of AI by finding the dumbest questions it can't solve. I think that pairs well as an additional way to view meaning through one's work.
The linked post points out constrained attention as a way to bring meaning to novel work that no one else took on. With AI, this can still be applied to compute.
I'm just wondering if there are a class of problems that humans, at least in the short-term, where humans need to be in the loop to solve more efficiently.
> My physicist friend once asked me what the point of doing research was if someone like Terence Tao could have figured out everything in my dissertation in a tenth of the time.
> what the point of doing research was if someone like Terence Tao could have figured out everything
A similar question is now being asked: what is the point of doing research, etc. if something like AI can figure out everything?
The question betrays the parochial way in which many people think about knowledge. For them, knowledge is merely an instrument or an effect. It does not occur to them that knowing is a valuable thing in itself, that understanding is valuable and desirable. Yes, some knowledge has merely practical value, but theoretical knowledge is primarily sought for its own sake, because we desire to know reality.
So, even if Terrence Tao, an AI agent, or who or whatever arrives at some bit of new knowledge, it doesn't benefit you as a knowing subject unless you understand it yourself and make it your own.
This is why I still like solving software problems on my own. Because those solutions now live in my head, rather than being spit out by some agent and then disappearing from the Dixie Flatline's memory once the session shuts down. And they deepen and enrich my life, and my life deepens and enriches them.
A brief example: When I was a teenager I had the most profound crush on a girl, as teenagers do. Gorgeous and gregarious, she was often surrounded by a circle of friends and acquaintances, and I noticed the peculiar way in which she would give attention to each in turn. She would exchange a few sentences with them, and then maybe her head would turn a certain way or her eyes would glance elsewhere, and that's how you knew your time was up and she had moved on to the next. To continue the conversation you had to hold onto the state in your head and wait for the next go around.
From her I learned a lot about how multitasking works, and how task schedulers distribute little quanta of time for each task to do some work before moving onto the next, and how this was achieved in cooperative multitasking by mutual communication between the task and the scheduler.
Would a vibe coder be able to have that insight? Maybe, but would they have been able to elaborate it into a working implementation? Perhaps, but I suspect with more time and difficulty than I did, because both the initial insight and the elaboration of detail that let me show that it worked lived in my head, not in some ephemeral AI context.
Hobbies aren’t as fun when you have an overbearing friend who constantly shows off how much more they know and how quickly they can switch to talking about anything you want but in greater depth than you.
I do not have a single hobby where I'm very far beyond the median hobby-haver in skill (that is to say: if you took all other humans who share my hobby, I'm likely somewhere near median for all of them). This may put me in top whatever percentile among all humans, since most humans don't share my hobbies and so are terrible at those things, but there are enough humans out there, and enough humans who share my hobbies, that I have always known that there are people who are vastly better at them than I am. Now yes, if I was constantly being followed around by one of the top 10 hobby-havers, pointing out to me all the mistakes or sub-optimal decisions I was making, that would indeed be annoying and reduce my enjoyment in the hobby. But why do you expect AI to be like this? I would love if I had one of those top 10 hobby-havers on call to answer every one of my (often inane) questions with infinite patience (and who would only talk about the hobby when I specifically initiated the topic). I'm already under no illusions that I'm the best, but it's now easier than it has ever been (for some hobbies, I expect others to join them over time) to get better at them....if one so desires.
> mathematicians or physicists fall into this category of knowing a lot about many thing . Sabine videos for example . she knows everything it seems
Sabine Hossenfelder, for obvious reasons, knows quite a bit about physics, though on some physics topics she has opinions that are outside the mainstream. For other areas, I am rather certain that she has a talent to learn about it up to some shallow level quite fast, which suffices to create some video about that topic, and then move on.
She is just arrogant. And asserts a lot of silly opinion as fact, her stuff on philosophy and theology is generally terrible.The sort of atheist that never actually looks into theology but knows the thing that first came to her head is a definitely the most amazing point against classical theology that has never been considered before.
> Lying is a core part of communicating mathematics. We lie to kindergarteners when explaining fractions. We lie to fourth graders when approaching limits...
Hard disagree.
Lying is with intention to deceive.
Teaching is simplifying with the intention that they understand and get the correct intuition.
> As humans, we have invented lots of useful kinds of lie. As well as lies-to-children ('as much as they can understand') there are lies-to-bosses ('as much as they need to know') lies-to-patients ('they won't worry about what they don't know') and, for all sorts of reasons, lies-to-ourselves.
> Lies-to-children is simply a prevalent and necessary kind of lie. Universities are very familiar with bright, qualified school-leavers who arrive and then go into shock on finding that biology or physics isn't quite what they've been taught so far. 'Yes, but you needed to understand that,' they are told, 'so that now we can tell you why it isn't exactly true.'
> Discworld teachers know this, and use it to demonstrate why universities are truly storehouses of knowledge: students arrive from school confident that they know very nearly everything, and they leave years later certain that they know practically nothing. Where did the knowledge go in the meantime? Into the university, of course, where it is carefully dried and stored.
Yes: this is about building the quotient field (field of fractions) [1] for some integral domain, or more generally, building the localization ([2], [3]) of a commutative ring with respect to some given set that is closed under multiplication (the special case of the quotient field for a ring R is obtained when one chooses R\{0} as such a set).
Really enjoyed reading your perspective. I am a recent math PhD graduate, and I also find it both exciting and terrifying to see what AI is doing to the field of math.
That was long-winded, but I appreciate that it seems to have actually been written by a human being.
For every landmark theory, theorem, or conjecture, there have been incremental, partial results supporting intuition and inching towards the white whale. When I attended BARD, a small computational number theory conference, one of the organizers preached of the outsized impact we could have just by being willing to program the numerical experiments that other mathematicians only theorized about. The small ball player can completely change the approach and intuition of the leading names without ever joining their ranks. The mediocre mathematician has always had purpose.
Yes, yes, YES! F*cking yes.
The greatest challenge of the AI Age (which is also the Climate Change Age and the Demographic Trap Age and a lot of other ages) is going to be finding an appreciation of the mediocre and mundane, when so many things are going very right, and so many things are going very wrong. Most of the time, the top of the bell and an SD in either direction can overwhelm either end, for better or worse. So respect for the unremarkable is warranted, if you want good things to happen and bad things not to.
I have some bad news for the non-mediocre mathematics. Given it another year or two or so and there won't be much need for non-mediocre mathematics either. Instead everyone will have on call a near magic mathematician who can push the state of the art for their needs.
Math is actually a perfect fit for AI because it is possible to express everything in terms of written language and you can write formal verifications of things. It is just a set of abstract rules, perfect for a computer.
And remember computer science was initially a sub-discipline of mathematics. So after Claude/Codex conquer writing code, it makes sense to move on to mathematics.
One can tell how much you hate all human skill and beauty. I also think your one of these AI booster people who don't know anything about formal verification and methods or it prerequisites, but are 100% sure it's going to get rid of human talent, beauty, only brutish concerns with what the market demands. May this reality come but only for you and you can occupy your place in the bowels universe as a contemptible slave.
Whenever there is a breaking AI-generated proof, it's the job of actual leading mathematicians to formalize/check it . Laypeople are not checking or writing these AI-assisted proofs. Even when Lean is used, it's mathematicians writing these proofs and checking if the formalization was done right. Terrance Tao's career trajectory has reached new highs due to AI. He's more relevant than ever. This is the exact opposite of Ai making mathematicians obsolete.
> Terrance Tao's career trajectory has reached new highs due to AI.
Given he is uniquely brilliant, he is likely one of the very last mathematicians to be rendered obsolete for his skills. But AI is pretty unstoppable here, so I would give me maybe another year compared to pretty much all the just really good / great mathematicians.
I’m not sure about this. Anthropic’s AI constructed complex structures on S^6 and wrote a 108 page paper about it, and a few days later there was already a 250k line lean program claiming to verify it.
an obvious question would be if 250k loc is what is required for the proof or if it can be shortened massively, is this essentially going to be AI trying to search for a smaller proof or is it that a human being would be beneficial in that loop.
> It’s highly nontrivial to verify that a 250k loc Lean program actually represents that which it claims.
Generally you only need to look at 10-100 lines (unless you have a highly novel theorem that essentially invents a new field of math or builds on a field that has never been worked on in Lean before) of the 250k to verify what it claims. This is why there is excitement around formal verification. The rest of it is perhaps useful to read to figure out why the proof works, but is not necessary for checking.
Human verification of the Lean program only requires verifying that the theorem itself is represented correctly. The theorem will only make up a very small part of the entire Lean program.
To the credit of the original commenter, that is why they said "give it one or two years". _Right now_ we need the experts to formalize/check. They're saying they think LLMs will reach a point in the near future where that won't be necessary.
honestly curious question: do you expect this to remain true? If so, for how long? I can think of two potential reasons why it might not stay true.
1. The very best humans remain able to understand/check the proofs, but we go for so long with every proof checking out that society more broadly just decides to trust. We are already doing that with human mathematicians. I can't verify what Terence Tao tells me is correct, I just trust that it is because he (and other human mathematicians) tell me it is. How many proofs/years of them checking out before we reach this point? I don't know, but history suggests that eventually, humans might keep checking, but they will do so only as a hobby. For any purpose that actually matters, we will just start to trust and use it.
2. The proofs that AI comes up with become too difficult/complex for even the very best human mathematicians to understand, and our options become to either trust or to not use at all.
Obviously it's possible that neither of these happens if AI capabilities stall out not too far beyond where we are now, but if they keep progressing at the current rates for another few years, I expect at least one, and maybe both, to eventually come to pass.
"Throughout the process, I felt that my only role was to teach the AI how to write things in a way that I could understand. Its initial language was extremely condensed—so compressed that I could barely follow it—but somehow the AI agents themselves seemed to understand it perfectly well."
It won't take much longer before AI is consistently better at validation than humans, and at that point, why continue to have humans do the validation? I think we're being naive about the end game - admittedly I don't know what it is though.
> do you expect this to remain true? If so, for how long?
For the foreseeable future. Left to their own devices current LLMs kinda wander off into outsider art territory. They aren’t grounded in the real world and they need that feedback loop to stay within the category of relevant ideas. I haven’t seen anyone working on fixing that.
Regarding 1, the same is true of every other scientific field. Verifying some tidbit of knowledge for yourself as an individual isn’t optimally useful in all circumstances.
Regarding 2, if the proof isn’t understandable then it probably isn’t useful. Many people today work in the hypothetical world where the Riemann Hypothesis is true, and many work in the hypothetical world where it is false. If it takes decades to validate that some horrifically complex AI proof of either fork is true, people will probably continue working on the other fork just in case.
> For the foreseeable future. Left to their own devices current LLMs kinda wander off into outsider art territory. They aren’t grounded in the real world and they need that feedback loop to stay within the category of relevant ideas. I haven’t seen anyone working on fixing that.
I have. DataAnnotation and these other AI-training piecework companies are pretty much the backstop now against total navel-gazing model collapse. With the Dead Internet Theory now pretty much reality, it's not like there is, or is going to be, gobs of untainted human-generated data out there ripe for the harvesting so it's going to take active human effort to keep the models grounded. That is, of course, until they start inhabiting robot bodies so they can live and move around in the real world, and thereby achieve their grounding, as in GitS or Ex Machina...
For software development, normal people will just assume it's money and probably not even ask...
It's going to attract more people who have the mentality of artists or musicians, i.e. people who do it for the love of the craft and as a creative outlet.
Don't jinx it. We are extremely lucky in this regard, and it actually looks like a rare exception.
but as one of the nearby professors is famous for saying: "C students gotta go somewhere."
( and since this is HN - he didn't mean the programming language :D )
Tbf coding with ai is still super fun though. I am hoping that engs who hate it like you will finally get kicked out as productivity increases from ai and it will finally go back to just us nerds.
It's kinda soul-sucking being around all you guys that just hate this work, please get out and go do farming or something lol.
Yes, by definition of "love" and "awesome".
> Tbf coding with ai is still super fun though.
Agree, it could be entertaining.
> I am hoping that engs who hate it like you will finally get kicked out
Ain't gonna happen, as I am pretty good at it.
> please get out and go do farming or something lol.
I thought about, but it is not well paid. I make money mostly from investments though, still do occasional coding stuff - for money.
I prefer to say "I liked a girl" (because it's the truth)
It kinda is (sadly) because unlike engineering there aren't thousands of postdoc jobs in arithmetic geometry.
And ofc in a year because of AI all math PhDs will be mediocre by definition.
I've been tapering off AI lately. I think I've realized that conquering the struggle is the fun part, and accomplishments just don't hit the same if AI is smoothing over every friction and cordoning off all the pitfalls and rabbit-holes.
This reminds me of a post I saw recently, although I can't remember the platform. It said something along the lines of assessing the limits of AI by finding the dumbest questions it can't solve. I think that pairs well as an additional way to view meaning through one's work.
The linked post points out constrained attention as a way to bring meaning to novel work that no one else took on. With AI, this can still be applied to compute.
I'm just wondering if there are a class of problems that humans, at least in the short-term, where humans need to be in the loop to solve more efficiently.
That's really mean thing to say
https://www.gatesnotes.com/a-turbulent-ai-era-and-critical-c...
A similar question is now being asked: what is the point of doing research, etc. if something like AI can figure out everything?
The question betrays the parochial way in which many people think about knowledge. For them, knowledge is merely an instrument or an effect. It does not occur to them that knowing is a valuable thing in itself, that understanding is valuable and desirable. Yes, some knowledge has merely practical value, but theoretical knowledge is primarily sought for its own sake, because we desire to know reality.
So, even if Terrence Tao, an AI agent, or who or whatever arrives at some bit of new knowledge, it doesn't benefit you as a knowing subject unless you understand it yourself and make it your own.
A brief example: When I was a teenager I had the most profound crush on a girl, as teenagers do. Gorgeous and gregarious, she was often surrounded by a circle of friends and acquaintances, and I noticed the peculiar way in which she would give attention to each in turn. She would exchange a few sentences with them, and then maybe her head would turn a certain way or her eyes would glance elsewhere, and that's how you knew your time was up and she had moved on to the next. To continue the conversation you had to hold onto the state in your head and wait for the next go around.
From her I learned a lot about how multitasking works, and how task schedulers distribute little quanta of time for each task to do some work before moving onto the next, and how this was achieved in cooperative multitasking by mutual communication between the task and the scheduler.
Would a vibe coder be able to have that insight? Maybe, but would they have been able to elaborate it into a working implementation? Perhaps, but I suspect with more time and difficulty than I did, because both the initial insight and the elaboration of detail that let me show that it worked lived in my head, not in some ephemeral AI context.
Sabine Hossenfelder, for obvious reasons, knows quite a bit about physics, though on some physics topics she has opinions that are outside the mainstream. For other areas, I am rather certain that she has a talent to learn about it up to some shallow level quite fast, which suffices to create some video about that topic, and then move on.
Hard disagree.
Lying is with intention to deceive.
Teaching is simplifying with the intention that they understand and get the correct intuition.
Math is not about lying, that's just silly.
> As humans, we have invented lots of useful kinds of lie. As well as lies-to-children ('as much as they can understand') there are lies-to-bosses ('as much as they need to know') lies-to-patients ('they won't worry about what they don't know') and, for all sorts of reasons, lies-to-ourselves.
> Lies-to-children is simply a prevalent and necessary kind of lie. Universities are very familiar with bright, qualified school-leavers who arrive and then go into shock on finding that biology or physics isn't quite what they've been taught so far. 'Yes, but you needed to understand that,' they are told, 'so that now we can tell you why it isn't exactly true.'
> Discworld teachers know this, and use it to demonstrate why universities are truly storehouses of knowledge: students arrive from school confident that they know very nearly everything, and they leave years later certain that they know practically nothing. Where did the knowledge go in the meantime? Into the university, of course, where it is carefully dried and stored.
What exactly is the lie? 1/4 and 3/8 equals 5/8. Is there’s something more to that? Is that fundamentally wrong?
Yes: this is about building the quotient field (field of fractions) [1] for some integral domain, or more generally, building the localization ([2], [3]) of a commutative ring with respect to some given set that is closed under multiplication (the special case of the quotient field for a ring R is obtained when one chooses R\{0} as such a set).
--
[1] https://en.wikipedia.org/w/index.php?title=Field_of_fraction...
[2] https://en.wikipedia.org/w/index.php?title=Field_of_fraction...
[3] https://en.wikipedia.org/w/index.php?title=Localization_(com...
https://terrytao.wordpress.com/career-advice/does-one-have-t...
The greatest challenge of the AI Age (which is also the Climate Change Age and the Demographic Trap Age and a lot of other ages) is going to be finding an appreciation of the mediocre and mundane, when so many things are going very right, and so many things are going very wrong. Most of the time, the top of the bell and an SD in either direction can overwhelm either end, for better or worse. So respect for the unremarkable is warranted, if you want good things to happen and bad things not to.
Math is actually a perfect fit for AI because it is possible to express everything in terms of written language and you can write formal verifications of things. It is just a set of abstract rules, perfect for a computer.
And remember computer science was initially a sub-discipline of mathematics. So after Claude/Codex conquer writing code, it makes sense to move on to mathematics.
Given he is uniquely brilliant, he is likely one of the very last mathematicians to be rendered obsolete for his skills. But AI is pretty unstoppable here, so I would give me maybe another year compared to pretty much all the just really good / great mathematicians.
I guess it could be AI turtles checking and summarizing all the way down, but is that any more credible than a single AI checking it? I doubt it.
Generally you only need to look at 10-100 lines (unless you have a highly novel theorem that essentially invents a new field of math or builds on a field that has never been worked on in Lean before) of the 250k to verify what it claims. This is why there is excitement around formal verification. The rest of it is perhaps useful to read to figure out why the proof works, but is not necessary for checking.
1. The very best humans remain able to understand/check the proofs, but we go for so long with every proof checking out that society more broadly just decides to trust. We are already doing that with human mathematicians. I can't verify what Terence Tao tells me is correct, I just trust that it is because he (and other human mathematicians) tell me it is. How many proofs/years of them checking out before we reach this point? I don't know, but history suggests that eventually, humans might keep checking, but they will do so only as a hobby. For any purpose that actually matters, we will just start to trust and use it.
2. The proofs that AI comes up with become too difficult/complex for even the very best human mathematicians to understand, and our options become to either trust or to not use at all.
Obviously it's possible that neither of these happens if AI capabilities stall out not too far beyond where we are now, but if they keep progressing at the current rates for another few years, I expect at least one, and maybe both, to eventually come to pass.
It's already the case that it's becoming not true. For example see this post from Lin Yang: https://x.com/lyang36/status/2092092709251293611
"Throughout the process, I felt that my only role was to teach the AI how to write things in a way that I could understand. Its initial language was extremely condensed—so compressed that I could barely follow it—but somehow the AI agents themselves seemed to understand it perfectly well."
It won't take much longer before AI is consistently better at validation than humans, and at that point, why continue to have humans do the validation? I think we're being naive about the end game - admittedly I don't know what it is though.
For the foreseeable future. Left to their own devices current LLMs kinda wander off into outsider art territory. They aren’t grounded in the real world and they need that feedback loop to stay within the category of relevant ideas. I haven’t seen anyone working on fixing that.
Regarding 1, the same is true of every other scientific field. Verifying some tidbit of knowledge for yourself as an individual isn’t optimally useful in all circumstances.
Regarding 2, if the proof isn’t understandable then it probably isn’t useful. Many people today work in the hypothetical world where the Riemann Hypothesis is true, and many work in the hypothetical world where it is false. If it takes decades to validate that some horrifically complex AI proof of either fork is true, people will probably continue working on the other fork just in case.
I have. DataAnnotation and these other AI-training piecework companies are pretty much the backstop now against total navel-gazing model collapse. With the Dead Internet Theory now pretty much reality, it's not like there is, or is going to be, gobs of untainted human-generated data out there ripe for the harvesting so it's going to take active human effort to keep the models grounded. That is, of course, until they start inhabiting robot bodies so they can live and move around in the real world, and thereby achieve their grounding, as in GitS or Ex Machina...