No country for mediocre mathematicians
garvvee.substack.com
garvvee.substack.com
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.
⸻
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"
⸻
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.
There are still things that will remain difficult (or illegal) in domestic settings for DIY, of course, but that's no model for full employment.
I haven't got any experience with them but there's even vendors for house assembly kits that cost a fraction of a finished one. Imagine you idiot-proof house planning and building: just do exactly what the app says.
I wonder just how far we have to push this until we finally get the 10h work week.
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..
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.
Could you share more about that?
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.
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.
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.
Mathematics is the thing you try to understand, don’t,
get frustrated about, and then do.
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.
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.
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.
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.
https://terrytao.wordpress.com/career-advice/does-one-have-t...
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.
Nevertheless you don't have to lie to kids in any field, science, art or otherwise.
- it’s not about lying, that’s the wrong way to say it. We explain too simply. We lie by omission…
The point of Pratchett is to make fun of how the university humbles the students, trading their self-assurance in their knowledge for actual knowledge that is dried inside its books
...if and only if the kid in question has the mental capacity to take on the whole truth without confusion.
Some kids have the potential to go hog wild on multivariable calculus, and they should be given the chance to handle it all. But they shan't be the benchmark that others with differing capabilities must hit to understand a concept.
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...
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 or appreciate basic set logic is to know all about infinite sets and ZF axioms... Most people, even mathematicians, will not understand all of that and have only heard about it in the most basics if at all.
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
There we have it, one of the many secrets to happiness.
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 them up to some shallow level quite fast, which suffices to create some video about the respective topic, and then move on.
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.
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.
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...
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.
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 him maybe another year compared to pretty much all the just really good / great mathematicians.