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
They tend to be higher risk though. If you've got the capacity for memorizing all the stuff you need to become a doctor, there's not much risk involved in attaining your doctor salary.
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.
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.
> constant feeling of frustration caused by things not working or breaking for random reasons
That’s not a feature of software development, but a feature of working higher up in the stack and/or interfacing with lower-quality external systems. I am lucky to have been working on relatively self-contained systems for most of my career, where that experience hasn’t been a constant.
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.
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.
> 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.
I don't think she is untalented as a researcher in physics. Where she did not have talent at (and which is why she became a science YouTuber) is in "playing the career game" in physics research. Being a great researcher in some science requires very different talents than climbing the career ladder in this area of academic research.
> 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.
> 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.
'All truths are easy to understand once they are discovered; the point is to discover them'
If knowing was the valuable part, then nobody would need a PhD. You could know more by just reading textbooks. Research mathematicians research, everybody else just learns.
> 'All truths are easy to understand once they are discovered; the point is to discover them'
There are plenty of things that are difficult to understand that have been known by others for a long time. The point is that you don't understand those things. Your understanding of something doesn't benefit from someone else understanding it, per se. I will agree that having a guide does make it easier.
> If knowing was the valuable part, then nobody would need a PhD. You could know more by just reading textbooks. Research mathematicians research, everybody else just learns.
Knowing is the most valuable part! It's the whole point of research!
And where knowledge is concerned, there isn't a sharp line between what constitutes "learning" and what constitutes "research". How do you know a claim in a book is correct? You can take it on authority and just believe it. Or, you can seek to verify it yourself. As far as your own mind is concerned, you've discovered something for yourself. And what is a researcher doing? He's inferring things and reasoning and verifying his inferences. These are activities you use both during learning and during research.
Who knows. I hope we figure out how to align the AI so that there's a sustainable economy for the majority of humanity, but I'm not in a position to influence that. I can just do my best to make myself less likely to be crushed.
> 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.
It's actually a really dumb thing to say and OP's response is the best one because it's a microcosm about how the world works. For every Terence Tao there's probably 20 more people cranking out high-quality work that's just a little less inspired. Their work is valuable and important, and asking that question is devaluing the entire endeavor of human knowledge. It's essentially positing that nobody else can contribute anything if it's not on that same level. It's like asking "why bother competing in marathons if you haven't won any?" Well that's not the point.
Very true but you can go even further than that. Advancing knowledge is a community endeavor. Consider where the field of mathematics would be if everyone except Terrence Tao and a handful of other luminaries stopped doing math research. It would die.
He's right, of course, for those people who already have a mathematical gift, but the problem with that is that Tao has never experienced not being a mathematical genius, and is no more capable of understanding what it's like to not be a genius as an ordinary person is capable of understanding what it's like to be one.
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.
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.
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.
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.
> 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.
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.
> 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.
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.
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.
> 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.
I hate struggling. I hate the realization that comes after the struggle about how easy was the problem I tried to solve. I hate realizing that I only reduced the number of problems from infinity + 1 to infinity, and have to do it again, forever.
Yet I also hate when I get away from solving problems and feel like I'm wasting my life with nothing to show for it.
> I hate the realization that comes after the struggle about how easy was the problem I tried to solve.
You're far from the only one to feel this way, but I want to point out that this attitude is a choice, not an intrinsic feature of the problem. An equally valid perspective is that learning turns difficult problems into easy ones.
One advantage of the latter perspective is that it makes solving a problem a moment to enjoy and celebrate, whereas your perspective turns it (almost definitionally) into a moment of self-recrimination. "Hooray, I understand it!" vs "Why didn't I understand it sooner? (I'm so stupid!...)"
I actually suspect that there is natural selection for people with the more upbeat perspective to succeed at becoming mathematicians.
> 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.
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).
It's mighty pretentious to say that one needs all that theory to simply answer the question lol. For many questions, only the most rudimentary theory is plenty to get an answer, that is exactly the same answer as a more elaborate theory would yield.
If you just want to do some stupid computations: sure.
But this is not what mathematics is centrally about. The central point is the kind of thinking about the respective topics (and understanding it) which these more abstract definitions encode.
Understanding the topic just enough to do some elementary computations does not give you the kind of thinking that is often near a transcendental experience.
Just to give one example: the reason why the localization of a commutative ring (a generalization of the field of fractions) is introduced is that many properties of ring hold if and only if they hold for all of its local rings; see for example [1]. This means to understand some property of a commutative ring R, we "just" have to understand its (simpler) local rings.
This is an example why one wants to study such ideas; on the other hand, I can imagine sooo many more exciting things to do with my life than dividing numbers by each others to form fractions. :-)
Those "stupid computations" comprise the bulk of useful work in the world. You're proving my point about the pretentiousness of insisting on the theory when one doesn't need it.
Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician.
Imagine arguing that the only way to understand basic set logic is to know all about infinite sets and ZF axioms... Most people, even mathematicians, will not understand all of that.
A similar phenomenon happens with philosophy. Imagine arguing that simple logic is "stupid" and that one can only reason well if they have a total understanding of epistemology. I happen to think epistemology matters, and that people can benefit from at least being aware of it, but it is really a separate topic from actual mechanical logic and argumentation.
>
Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician.
You are free to ignore mathematics that is not completely trivial. I prefer (and would rather recommend) to understand it, and use this understanding to build a >1-billion-USD/EUR application out of it. :-)
I often ignore mathematics that is not completely useful to me these days lol. I get the appeal of learning neat theories and feeling like you got it all figured out, but I'd rather limit my studies to topics that are likely to bear fruit in my life. Complication and abstraction does not necessarily deter me, but it has been my observation that overly complicated or abstract ideas rarely pay off in my endeavors.
> Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician.
What a coincidence this came up today.
I'm not a mathematician. I minored in math, but even the undergrad work was honestly difficult for me.
For most people, this might be nonsense. But it doesnt have to be.
I'm trying to learn about Fast Fourier Transforms because they're relevant for an embedded device system I'm investigating. I'm also not an Electrical Engineer so it is mostly new to me.
To understand the language of FFTs, linear bases and the like, I've started working through Axler's Linear Algebra Done Right.
First, just learning more theory shows me we can learn and grow in our old age.
This week I worked through linear spaces. I'm actively asking myself questions and working with other fields besides the reals and complex numbers so I can understand coding theory in general more.
And the parent's comment about quotient fields and rings is directly related to a question I asked myself about the
> Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician.
What a coincidence this came up today.
I'm not a mathematician. I minored in math, but even the undergrad work was honestly difficult for me.
For most people, this might be nonsense. But it doesnt have to be.
I'm trying to learn about Fast Fourier Transforms because they're relevant for an embedded device system I'm investigating. I'm also not an Electrical Engineer so it is mostly new to me.
To understand the language of FFTs, linear bases and the like, I've started working through Axler's Linear Algebra Done Right.
First, just learning more theory shows me we can learn and grow in our old age.
This week I worked through linear spaces. I'm actively asking myself questions and working with other fields besides the reals and complex numbers so I can understand coding theory in general more.
And the parent's comment about quotient rings is directly related to an active learning question I asked myself about whether the set with only the zero element is a linear space. I don't think it is a field if 0 is the multiplicative identity, 0 can't be 1, so it can't be a linear space, right?
But it works. I guess the set of the field for the scalar in a linear space is always assumed to contain more elements. It's a different set than the linear space. It seems like, duh, of course it is. But you don't see it until you work through it. And I'm guessing my experience can inform teaching others.
It's confusing to me, maybe because the notation is sparse in explicitly defining the set of the linear space and the set of the associated linear space.
But this helps me truly understand linear codes down the line, and Linear Feedback Shift Registers and FFTs. It's not just theory to me, I can now understand what my peers are saying and contribute my own thoughts.
OK now you're getting into some more useful topics. Linear algebra is a great one with tons of applications. But if you want my advice, prioritize useful stuff and spend less time on theoretical baubles or bedrock-level boilerplate. This is unless, of course, you enjoy puzzles and hard work with little application.
> This is unless, of course, you enjoy puzzles and hard work with little application.
Wouldn't that be great? If we could teach our kids that learning for the sake of learning is awesome? If only that was all this world was.
But you're right, you and the thousands of others posting, it's not. You have that voice, and the hundreds of online stories about college losing value have that political voice.
In the post-AI future I can imagine recreational mathematics being a socially approved past time. It helps with age-related cognitive decline, etc.
But I can also understand people in that post-AI future who had smart political ideas that were more important to them than smart math ideas.
Impactful ideas.
Just getting snapshots and puzzling out that post-AI future I don't know if we'll have the right space to encourage positive care for our mental and physical landscapes.
I'm not making any political statement here, or at least that was not my intention. Life is short and there are infinite things we could be learning about. The advanced math I'm talking about avoiding here is not recreational math, but you could potentially apply the same logic to recreational math. Personally I see a big difference between "routine" abstract/advanced math and recreational math, though there can be some overlap. People have written books about what makes a pleasant puzzle, for example. Some puzzles are deceptively simple to state and impossible to solve without elite level theories. I think one of the characteristics of a good puzzle is that it can be solved with reasonable effort and not consume your whole life. Another is that it not be so contrived and abstract, that you need to be heavily initiated to even understand it. The worst are the ones that require a ton of mechanical computations even after you figure out the key insight. I don't think those are very fun either. But then again, there are people who don't get tired of Sudoku lol...
Don't we still teach kids that e.g. 3/4=6/8, that they need to make a common denominator to add, and that they should cross multiply to check equality? I suppose we don't teach zero divisors, but otherwise, jargon aside, I'd be hard pressed to explain how we don't teach kids that fractions are members of ZxZ* mod (ac-bd).
Lies to children are like... time-reversal symmetry.
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.
Really enjoyable read. I escaped a career in Mathematics by being sufficiently bad at it that I only have a Masters. Fortunately, I escaped to a field immune to AI: CS and then a software engineering career. Well, not really, but I did get a good 15 years out of it so thank god for that.
About half of my friends are founders of various startups and the rest are executives of and almost all of them have the view that it’s better for everything to be a failure than to be in the “it could make it” category for half a decade or more.
In that way, I am glad I found that it wasn’t for me. I had the curiosity, but not the doggedness to face difficulty (not enough curiosity perhaps?) or the ability to not encounter such difficulty. And fortunately that meant I was never in the “I could make it” category. God bless clear and present boundaries and may the devil take the grey zone.
Yeah, I’m pretty much on a trajectory of building up as much retirement savings as possible before AI completely kills my career. The sweet spot seems to be about three years out. I honestly don’t know how to guide my 12-year-old son who loves math¹ and coding.
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1. Which at his age essentially means solving computation problems, although he figured out how to take something like 0.3535… and turn that back into a rational number without any guidance at all.²
2. I want to see how close to the general solution he’s gotten on his own, but given that he’s not had any formal algebra, it’s damned impressive and bodes well for his future development.
Im going with this... scifi stuff still requires engineers. Its laughable and insulting to think these billionaire owners are going to make robots and llm loops to get real shit done
So far, every "AI will never be able to do X" is aging like fine milk. Or do you think that engineering is somehow more special than software development or math?
You seem to have one goal in every thread and that’s “rage boost” LLMs, often to the point of getting flagged multiple times. I’m honestly wondering if you aren’t just trolling HN.
From what I’ve seen of what AI can do for coding, the part of software engineering that I like is going to be automated out of existence, much like the ability to do serious calculations by hand got automated out of existence with the development of computing. Writing code by hand will be as marketable as factoring 5-digit numbers in your head.¹
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1. I enjoy doing both, but I doubt that anyone will pay me what I’ve been paid in the past for this five years from now.
If you want AI-proof careers it seems risky to go into knowledge work at all. Robotics lag behind mere computation, so something where you do awkward work in places designed around human bodies seems a lot safer. Plumbing, landscaping or something like that perhaps.
This is a romantic and very appealing idea that flies in the face of reality. The market for skilled trade workers like plumbers, electricians, landscapers and so on is more or less in equilibrium, and the demand side of it is relatively inelastic. Dumping hundreds of millions of new workers into these trades will not increase demand sufficiently to let anyone earn a viable living from them.
There is nothing that can absorb millions of new workers. If AI destroys jobs to that degree pitchfork manufacturers will be the only ones left in business.
Where I live at least right now it is quite hard to find contractors for anything and many are close to retirement age and have a hard time finding replacements.
There's also the thing where trade work is in the process of pricing itself out of what normal folks are willing to afford. Combined with Youtube tutorials and AI help that is getting better, it wouldn't surprise me if more and more people would just decide to do all this stuff themselves at a fraction of the cost.
As someone who's also thinking about their child's future, I've come to the conclusion that building their capacity for precise language by expanding their vocabulary and grammar in an array of domains and subdomains is the best way forward. Not only will they have a more detailed understanding of what's out there, but using specialized, narrow terminology with AI provides vastly better solutions than general prompt language.
Being an artist has always been a pretty difficult career path. But lots of people do it because they love creating art. If AI actually does what people fear it will to programming, people like your son and mine can choose to be an artist as a hobby or make a go of it as a profession.
Funded positions are much less than PhD number. However, people can fund themselves with a job. In a country, if it's easy to to get a part-time job with enough payment, mathematicians can continue work.
As an ex-academic (not mathematician), this really resonates. Every generation of researchers has to outperform the researcher generation before them - there are fully tenured profs out there who, with their track record, wouldn't get a postdoc nowadays. It's just a ratchet where every generation has to be more outstanding than the previous one, so yeah, you suddenly need 'triple the conferences'. Eventually that ratcheting reflects on your self-worth and you start calling yourself mediocre, even though you vastly outperform the previous generation. It's an extremely unfair game designed by careless people.
Throw AI into the mix and your self-worth crashes. Just today I saw a Claude Science set of results that made my own work of the past 2 months completely superfluous, and I sit here and wonder what's the point.
Sure! I was building a workflow that assesses human gene model accuracy. Which part is supported by what evidence? Important for drug development; if we want to change $THING it better exist in the first place.
I came up with a set of rules, collated external databases, and then slowly (Claude-code assisted) built a Nextflow pipeline that gives me an automated report, which I then manually expand by 'human' assessment of the evidence.
Claude Science prompted with 'assess this gene' came up pretty much with the same rules, built a report, and did the manual assessment of evidence pitfalls for about 10% of a Claude Max subscription's tokens in about 15 minutes. Some details differ from my report - a different tool here or there - but overall, what we needed out of these reports is in the Claude Science report.
Consider that if this was ran from your machine - it would have all of your memories and might be relying on the past 2 months. I'd be curious what would happen if it was run independently.
(I am assuming they didn't train on your prompts, which is always a worry)
All research and progress boils down to Brownian Loop Soup.
Someone/something having a result showing a connection does not mean they have explored "the way" to do it.
Not even the most elegant mathematically perfect solution is guaranteed to be the best way to crack a problem, or provide a definite answer.
Meandering paths through whatever we set our minds to do and serendipity is the way of human beings for the past few hundred millennia.
As a middle-aged mathematician, AI is provoking a serious crisis among the better mathematicians I know. Many are only half jokingly talking about retiring. We use AI a lot in research already and it's obvious that whoever doesn't will fall behind fast. At the very least it gives access to the full library in a few minutes, serving as a kind of knowledge oracle generally more useful than the expert who ought to be in the office next door but isn't. And we see what can happen when it is effectively coupled with lean to automate theorem proving and discovery.
We aren't dumb enough to see this as the end of the profesión - Esther it's clearly a shift in how we will work - but we like doing computations and playing around with examples and how one does that just changed a lot. The other problem is we know we don't have the energy of youth to learn to use AI as effectively as the kids, although we are wiser and have better judgment and do know some things.
Mathematicians who are not taking seriously how to adapt to AI are deluding themselves.
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[ 0.23 ms ] story [ 33.2 ms ] threadFor software development, normal people will just assume it's money and probably not even ask...
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 )
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.
Saying most tenured faculty at research 1 institutions are mediocre seems to be stretching the definition, though.
I prefer to say "I liked a girl" (because it's the truth)
That’s not a feature of software development, but a feature of working higher up in the stack and/or interfacing with lower-quality external systems. I am lucky to have been working on relatively self-contained systems for most of my career, where that experience hasn’t been a constant.
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.
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.
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.
If knowing was the valuable part, then nobody would need a PhD. You could know more by just reading textbooks. Research mathematicians research, everybody else just learns.
There are plenty of things that are difficult to understand that have been known by others for a long time. The point is that you don't understand those things. Your understanding of something doesn't benefit from someone else understanding it, per se. I will agree that having a guide does make it easier.
> If knowing was the valuable part, then nobody would need a PhD. You could know more by just reading textbooks. Research mathematicians research, everybody else just learns.
Knowing is the most valuable part! It's the whole point of research!
And where knowledge is concerned, there isn't a sharp line between what constitutes "learning" and what constitutes "research". How do you know a claim in a book is correct? You can take it on authority and just believe it. Or, you can seek to verify it yourself. As far as your own mind is concerned, you've discovered something for yourself. And what is a researcher doing? He's inferring things and reasoning and verifying his inferences. These are activities you use both during learning and during research.
That's really mean thing to say
It still gets this person to somewhere they weren't.
https://terrytao.wordpress.com/career-advice/does-one-have-t...
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.
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'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.
Yet I also hate when I get away from solving problems and feel like I'm wasting my life with nothing to show for it.
You're far from the only one to feel this way, but I want to point out that this attitude is a choice, not an intrinsic feature of the problem. An equally valid perspective is that learning turns difficult problems into easy ones.
One advantage of the latter perspective is that it makes solving a problem a moment to enjoy and celebrate, whereas your perspective turns it (almost definitionally) into a moment of self-recrimination. "Hooray, I understand it!" vs "Why didn't I understand it sooner? (I'm so stupid!...)"
I actually suspect that there is natural selection for people with the more upbeat perspective to succeed at becoming mathematicians.
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.
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).
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[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...
But this is not what mathematics is centrally about. The central point is the kind of thinking about the respective topics (and understanding it) which these more abstract definitions encode.
Understanding the topic just enough to do some elementary computations does not give you the kind of thinking that is often near a transcendental experience.
Just to give one example: the reason why the localization of a commutative ring (a generalization of the field of fractions) is introduced is that many properties of ring hold if and only if they hold for all of its local rings; see for example [1]. This means to understand some property of a commutative ring R, we "just" have to understand its (simpler) local rings.
This is an example why one wants to study such ideas; on the other hand, I can imagine sooo many more exciting things to do with my life than dividing numbers by each others to form fractions. :-)
[1] https://en.wikipedia.org/w/index.php?title=Localization_(com...
Nothing in your comment makes me want to go learn more theory, and I would argue that it's nonsense to anyone who is not a mathematician.
Imagine arguing that the only way to understand basic set logic is to know all about infinite sets and ZF axioms... Most people, even mathematicians, will not understand all of that.
A similar phenomenon happens with philosophy. Imagine arguing that simple logic is "stupid" and that one can only reason well if they have a total understanding of epistemology. I happen to think epistemology matters, and that people can benefit from at least being aware of it, but it is really a separate topic from actual mechanical logic and argumentation.
You are free to ignore mathematics that is not completely trivial. I prefer (and would rather recommend) to understand it, and use this understanding to build a >1-billion-USD/EUR application out of it. :-)
What a coincidence this came up today.
I'm not a mathematician. I minored in math, but even the undergrad work was honestly difficult for me.
For most people, this might be nonsense. But it doesnt have to be.
I'm trying to learn about Fast Fourier Transforms because they're relevant for an embedded device system I'm investigating. I'm also not an Electrical Engineer so it is mostly new to me.
To understand the language of FFTs, linear bases and the like, I've started working through Axler's Linear Algebra Done Right.
First, just learning more theory shows me we can learn and grow in our old age.
This week I worked through linear spaces. I'm actively asking myself questions and working with other fields besides the reals and complex numbers so I can understand coding theory in general more.
And the parent's comment about quotient fields and rings is directly related to a question I asked myself about the
What a coincidence this came up today.
I'm not a mathematician. I minored in math, but even the undergrad work was honestly difficult for me.
For most people, this might be nonsense. But it doesnt have to be.
I'm trying to learn about Fast Fourier Transforms because they're relevant for an embedded device system I'm investigating. I'm also not an Electrical Engineer so it is mostly new to me.
To understand the language of FFTs, linear bases and the like, I've started working through Axler's Linear Algebra Done Right.
First, just learning more theory shows me we can learn and grow in our old age.
This week I worked through linear spaces. I'm actively asking myself questions and working with other fields besides the reals and complex numbers so I can understand coding theory in general more.
And the parent's comment about quotient rings is directly related to an active learning question I asked myself about whether the set with only the zero element is a linear space. I don't think it is a field if 0 is the multiplicative identity, 0 can't be 1, so it can't be a linear space, right?
But it works. I guess the set of the field for the scalar in a linear space is always assumed to contain more elements. It's a different set than the linear space. It seems like, duh, of course it is. But you don't see it until you work through it. And I'm guessing my experience can inform teaching others.
It's confusing to me, maybe because the notation is sparse in explicitly defining the set of the linear space and the set of the associated linear space.
But this helps me truly understand linear codes down the line, and Linear Feedback Shift Registers and FFTs. It's not just theory to me, I can now understand what my peers are saying and contribute my own thoughts.
Wouldn't that be great? If we could teach our kids that learning for the sake of learning is awesome? If only that was all this world was.
But you're right, you and the thousands of others posting, it's not. You have that voice, and the hundreds of online stories about college losing value have that political voice.
In the post-AI future I can imagine recreational mathematics being a socially approved past time. It helps with age-related cognitive decline, etc.
But I can also understand people in that post-AI future who had smart political ideas that were more important to them than smart math ideas.
Impactful ideas.
Just getting snapshots and puzzling out that post-AI future I don't know if we'll have the right space to encourage positive care for our mental and physical landscapes.
Lies to children are like... time-reversal symmetry.
OK there’s still no intent to deceive but almost all of the “rules” you learn have giant exceptions
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.
About half of my friends are founders of various startups and the rest are executives of and almost all of them have the view that it’s better for everything to be a failure than to be in the “it could make it” category for half a decade or more.
In that way, I am glad I found that it wasn’t for me. I had the curiosity, but not the doggedness to face difficulty (not enough curiosity perhaps?) or the ability to not encounter such difficulty. And fortunately that meant I was never in the “I could make it” category. God bless clear and present boundaries and may the devil take the grey zone.
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1. Which at his age essentially means solving computation problems, although he figured out how to take something like 0.3535… and turn that back into a rational number without any guidance at all.²
2. I want to see how close to the general solution he’s gotten on his own, but given that he’s not had any formal algebra, it’s damned impressive and bodes well for his future development.
https://rcsnyder.github.io/open-frontier-curriculum/
Ya "claude can you build me the next cern Thanks"
So far, every "AI will never be able to do X" is aging like fine milk. Or do you think that engineering is somehow more special than software development or math?
AI can be trained on that body of work. But then AI has enough issues in output that it needs to be verified.
Otherwise we'll live in a world of titan submersibles ordered by CEOs running on hopes&dreams.
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1. I enjoy doing both, but I doubt that anyone will pay me what I’ve been paid in the past for this five years from now.
Where I live at least right now it is quite hard to find contractors for anything and many are close to retirement age and have a hard time finding replacements.
many schools have shuttered since ~2022
artist gigs are at an all time low
if you practice artistry as a craftsman in niches like carpentry, maybe you could make a living..
There we have it, one of the many secrets to happiness.
Throw AI into the mix and your self-worth crashes. Just today I saw a Claude Science set of results that made my own work of the past 2 months completely superfluous, and I sit here and wonder what's the point.
I came up with a set of rules, collated external databases, and then slowly (Claude-code assisted) built a Nextflow pipeline that gives me an automated report, which I then manually expand by 'human' assessment of the evidence.
Claude Science prompted with 'assess this gene' came up pretty much with the same rules, built a report, and did the manual assessment of evidence pitfalls for about 10% of a Claude Max subscription's tokens in about 15 minutes. Some details differ from my report - a different tool here or there - but overall, what we needed out of these reports is in the Claude Science report.
(I am assuming they didn't train on your prompts, which is always a worry)
Not even the most elegant mathematically perfect solution is guaranteed to be the best way to crack a problem, or provide a definite answer.
Meandering paths through whatever we set our minds to do and serendipity is the way of human beings for the past few hundred millennia.
We aren't dumb enough to see this as the end of the profesión - Esther it's clearly a shift in how we will work - but we like doing computations and playing around with examples and how one does that just changed a lot. The other problem is we know we don't have the energy of youth to learn to use AI as effectively as the kids, although we are wiser and have better judgment and do know some things.
Mathematicians who are not taking seriously how to adapt to AI are deluding themselves.
Man if you were learning about limits in 4th grade I must've been lagging behind HARD because they didn't even bring that up for me until college.