He dropped out to become a poet – now he’s won a Fields Medal
quantamagazine.org
quantamagazine.org
https://www.quantamagazine.org/a-path-less-taken-to-the-peak...
Are you sure about that? I know at least some number theory can help with quick mental calculations. For example, I remember that if the digits of a number add to 9 the number itself is divisible by 9. I'm sure there are lots of other relationships like that.
Personally, my first introduction into the more abstract mathematical concepts was a trigonometry class in high-school and later was fortunate enough to take more theoretical classes at university. It would be wonderful to introduce some of the more accessible, elegant concepts to kids at an earlier age.
However he took a course high in mathematical content and it sounds like he must have switched to mathematics for grad school, I.e. after his undergraduate or masters. In other words, he did the preparation. If it is anything like physics courses I have seen they feature a lot of mathematics courses anyway.
Some people have a lot of natural ability in mathematics but this is not enough to do good research. There's no way around hard work.
Also, dropping out of high school in many places significantly reduces your chances of going to university because you do the qualifications you need at the high school equivalent.
Yeah, you don't "happen into" a course on algebraic geometry. He must have dedicated a large amount of undergraduate courses to math, or demonstrated aptitude in another way.
Yes, in the United States, you can take a GED to graduate but that's usually reserved for adult high school students. I've never heard of it being used to skip high school and go straight to college admissions.
The closest parallel that I know of is Early Bird/Junior Admission. Essentially, one must first obtain all the credits required to graduate, and then as a high school junior apply to a college/university (although that only seems to be an available option for a few colleges) but it isn't exactly the same as what you're describing.
He proceeds just as deliberately when doing mathematics. Wang was shocked when he first witnessed it. “I have this math competition experience, that as a mathematician you have to be clever, you have to be fast,” he said. “But June is the opposite. … If you talk to him for five minutes about some calculus problem, you’d think this guy wouldn’t pass a qualifying exam. He’s very slow.” So slow, in fact, that at first Wang thought they were wasting a lot of time on easy problems they already understood. But then he realized that Huh was learning even seemingly simple concepts in a much deeper way — and in precisely the way that would later prove useful.
Having gone through the American education system, I felt like I was in a factory being evaluated as a potential end product. Steamroll through as much material as quickly as possible. Optimize behavioral patterns for the next performance metric. Rinse and repeat. Is this really the best system we can come up with? Can the pace of mathematical and scientific advancement be optimized by steamrollers? I wonder how much potential we've missed as a society because the education system has been trying to force everyone through the same hole, and destroying intellectual curiosity in the process.
Personally, my experience with this educational system turned me off from "math." It was only much later in life when I learnt that the mindless algorithms that I had to personally execute for years on end to get into university could barely be considered math at all. Only then could I begin to appreciate how elegant real math could be. And I doubt I'm alone in my experience.
"My specificity today among French mathematicians is not to know more mathematics than others, I can even admit that I know less mathematics than most French and foreign mathematicians. It's not about being 'great' either, I'm not great at all, on the contrary I have a rather slow mind. When I listen to a seminar, I'm probably one of those who understand the least what is being said: an idea in an hour, for me, that's already a lot. No, the only explanation for the successes that I have been able to obtain in mathematics is my being imbued by the reading of the great classics."
https://www.biennale-lf.org/b22/index.php?p=https://www.bien...
As for your experience, you are indeed very far from alone.
"In my parents' house, there were a lot of books: all the great authors of the 17th, 18th and 19th centuries, translated foreign authors. I spent my youth, and still today I spend most of my time, reading literary works, and particularly those of French literature."
If this doesn't make you think about math, nothing will.
But he is slow! He did a bunch of cognitive tests where he was testing 3rd std deviation on most tasks. Except processing speed, where he was in the 2nd percentile. You'd present him with his times tables, and he would take 15 seconds to get "8x7"... but he didn't slow down as as the psychiatrist progressed through the test. He'd get to some 6th grade problem, and it'd be the same 15 seconds and blank look before the solution came out. He was an O(1) machine with a huge constant.
And school utterly killed his interest in math. From grade 2 onwards it was speed competitions and public humiliation in front of his class. After grade 3 he refused to do anything related to math outside of what's required because he was "way to stupid for that".
My school had this thing called mental math, learning to do arithmetic really fast as a large portion of my 4th or 5th grade material. I absolutely sucked at it and I lost interest in any kind of math. My dad who teaches college math for a living was very disappointed when that happened.
Around 8th grade, I started doing some plane geometry and it sort of opened up my mind in a way that's hard to explain. I remember spending a lot of evenings working my way through several geometry books and problem sets. I sort of branched out into other areas and developed enough skill to make it to national olympiads and other such competitions. I don't do math now, but I think it had a reasonable amount of influence of my eventual path through life.
Who knows, maybe he'll pick it back up again in a few years.
The bar for SATs and similar college admissions tests is set remarkably low, though, such that one may never run into this problem without going into a mathy field. But it does seem like a slow-and-steady approach taken by a greater number of people could encourage more interest and deep thinking, and possibly a greater contribution to the field and society overall.
There is a known (but mostly ignored) educational method called https://en.wikipedia.org/wiki/Mastery_learning -- you should teach kids until they actually get it, and only then move to the next lesson. This method may be slower at the start, but later gains speed because kids with solid background understand the new concepts faster, and do not need to relearn the old concepts all the time.
If you think about the problem, discuss it, and then implement the solution properly, it's only one ticket. But if you make a half-assed solution, change it twice, and fix seven related bugs, that is ten tickets, with a potential to make even more tickets in the future.
In practice, if I write my own ticket, some manager will give it a priority 2. And the tickets with priority smaller than 1 never get done, because there are always enough tickets with priority 1, i.e. those written by the management. A few years later, when the project is decommissioned, the tickets with priority lower than 1 are still in the backlog. Of course, if having your ticket in the backlog makes you happy, just go ahead and make one.
It is not how it should be done, in theory. Knowing that does not help much.
There's tickets I know we'll never get to, but things we do actually have time for get real attention, with enough focus and resources to do whatever solution is actually correct. There's often tradeoffs for time, but they're ones we have real input in.
When I started my undergrad in maths (in the UK) they gave us a day 1 test to check we could recall certain pre-requisites from school. I did poorly because I had forgotten all the tricks for those particular exam questions and was put in an additional "catch up" class. It was a real gut punch and I questioned whether I was cut out for maths at all. Fast-forward six years and I ended up with a PhD in pure maths from the same university. Turned out that when given the time to read and think at my own pace and allowed reference to books and my own notes I was pretty good at abstract math.
Fast-forward another ~15 years and I'm a machine learning researcher for a SV company. The same issue rears it's head now as an inability to contribute usefully in meetings. I can't think fast enough to keep up and don't have anything useful to say when put on the spot. I much prefer to read written communication and take the time to formulate my responses. This is not the company culture though, so I do suffer problems with effective collaboration.
In school I never had a problem with speed or understanding, nor at university. But what hit me was the realization that "actual math" is probably not a load of little riddles. I often wondered how I'd ever solve something that someone hadn't contrived to have an elegant solution, something completely new and insightful that nobody had proven before. There's quite a leap from your tricky olympiad problem that you can find a good video for, and proofs of things in higher math.
The other thing that struck me was that our current system destroys all appreciation for beauty. There's so many beautiful things in math, but we're forced to calculate, quickly. Especially in the engineering side I went into, it ended up becoming very mechanical.
Currently I'm doing an ML research internship at a SV company and this problem of "slowness" at meetings is really resonating with me. I have a hard time following quick verbal communication, I just need the time to think in order to contribute. I don't know it's like I just phase out or forget what we were talking about.
Have you found any useful techniques to do better in this type of environment? Is there any advice you would give your younger self?
I think I know some good strategies, but I often fail to live by them still. Firstly, encourage written/async communication by setting a good example:
- Keep up-to-date written descriptions of projects and progress in a wiki that is available to all colleagues.
- Deliver high quality presentations on your research periodically. These don't have to be reserved for big public conferences.
- Send regular but concise status update e-mails to a list of likely interested colleagues. Advertise your wiki here.
- Send out edited meeting notes within a day or two of working meetings.
General advice I'd give my younger self is pretty cliche, but to trust in my abilities and the value I can provide by working in ways that are comfortable to me. It's more important to take care of my long term mental health than it is to be in the spotlight or maximizing output all the time. Quality work speaks for itself and doesn't need to be continually delivered. Don't get sucked in to false urgency. Delivering a couple of genuinely useful things per year (and publicizing those appropriately) is enough to keep you in people's minds but give you the space to think deeply about the real work.
I say this from a place of privilege though where, somehow, I've managed to accrue a large amount of experience and ended up in a secure research position. The company (and team) I work for give me a lot of autonomy and I have a very supportive manager.
Applying computational algebraic geometry to robotics sounds fascinating. What are some highlight papers in that area?
This is why the conceit of "intellectual property" has always bothered me. I relate to what he describes as "finding things". Those things are are eternally "out there" for all to find. I can put up a signpost or build a road to show you the way, but how can a mere man own what is part of the infinite and eternal?
In principle, a patent should cover specific machinery/process/material designed for a task rather than the underlying more ephemeral/indefinite concept powering the invention.
The interesting thing is that, of those, machine and process seem like the least conceited. Composition of matter seems more like sticking a flag in infinite ground. But in practice it’s the former two that power most of what I consider an abuse of the spirit of patent law. But perhaps that’s just my personal bias and the bio or materials people will disagree.
What can we say? Profit motive is a strong incentive and lawyers/judges are clever.
Anyways, with 7B people and counting, a term limited right to one idea in the infinite space of ideas seems less harmful than perpetual rights to a piece of land.
We live in this complexity nightmare partly because all the simple and obvious solutions are easily patented.
Interestingly, society agrees with you; for the most part, we don’t grant perpetual rights to a piece of land. Instead we charge property tax to ensure you are doing something productive with it. You have to generate enough free cash flow to pay the taxes or you lose the land.
I have a theory that the misnomer "intellectual property" is most ardently championed by (perhaps older) intelligent/creative people who fear losing their power. Fear that this may be their last great idea fosters withholding and subtracts from the opportunities of the many.
This comes up in other contexts. In cybersecurity, if one discovers a new exploit the natural instinct for most people is responsible disclosure, even without a bounty. However, the temptation to privatise that knowledge as a sellable "zero-day" is powerful. But what we see is that it's the mediocre who do that more. Top 0.1% hackers are so confident in their ability to find more and deadlier exploits at will, they don't hoard as it's beneath them.
Or they probably see it as leverage for a job with a company at some point in the future. Or an achievement they can put on their resume or show the world.
I might be wrong in assuming this, but I take it that you believe the disclosures of these top tier hackers are a result of an altruistic benevolence. If so, I think you're describing a soul rarer than the Loch Ness Monster. The best hackers do zero-day work because they love it and it pays (in cash, achievement, fame, etc.). If it doesn't pay, they'll probably find some other way of getting their kick upto and including trading exploits or data for cash.
Even the hacker who saved the net from Mirai all those years ago sold exploits.
I'm not sure I believe in that. Or rather, I think the idea of altruistic benevolence as a simple exclusive categorical is naive. Humans have super complex motives, often quite unconscious, multi-faceted and even in contradiction.
What I would say is that very high performing people think and behave in different ways. In political science (I'm thinking of Nicomachean Ethics but it's a recurring theme in modernity too) is the conceit of the "Great/Noble Man", whose competence places him "above" everyday perspectives and rituals. They're rare not because apparent altruism is rare, but because people of that calibre are rare.
Also, the question whether math is discovered or invented is an old one, with arguments going either way. It’s not like people just stumble upon certain proofs, or that Newton was just lucky.
https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Math..., for example, holds that math would still exist if there were no life anywhere in the universe (maybe even if there were no universe)
The idea to incentivize public sharing of knowledge for a short term rent is not so stupid.
This is…basic philosophical knowledge about Plato’s ideas. It’s not up for debate.
And there is nothing wrong with this concept; it encourages people to search and create, and prevents people from being abused by the strong and getting their findings stolen. Generally, this is beneficial for everyone and lets society moving forward and to higher levels of quality.
That's the reason I deleted my personal blog. I felt that when I was writing publicly about my life/thoughts I was focusing too much on my ego.
Why do you want to avoid that?
I still write sometimes when I want to clear things for myself in a notebook nobody will see. I think it can be therapeutic at times.
But I'm in a point in my life where I think I need to get myself out of the way, be humble and enjoy everything else out there.
Our ego is not our friend.
Or maybe I’m just lazy.
Hit me up if you figure this one out. I can't be bothered to get my personal website back up and it's just an nginx configuration away for about a year now.
I absolutely love this distinction.
In Polish there's a saying "Chcę, ale mi się nie chce." which is hard to translate but it's something along the lines:
I want ... but I don't want to.
I want to win the World Cup, but I'm not getting out of bed to go to the gym.
I want to be rich and famous, but I don't want to do the work.
I want to be a bride, but I don't want to build a relationship.
I buy books but don't read them.
I buy tools and supplies but don't build.
I read books but don't do exercise.
I want to be a person who does great things, but in each moment I don't do the work.
That I'm exactly the person that I want to be
- Amanda Palmer, "In My Mind"
Also don’t want to diminish his quite remarkable achievements in any way, either!
[Upon hearing that one of his students had dropped out to study poetry]”
― David Hilbert
https://www.quantamagazine.org/tag/2022-fields-and-abacus-me...
https://en.wikipedia.org/wiki/Kim_Ung-yong
The other problem being that "high IQ explains everything" people don't actually go around personally going around giving people IQ tests. They just read somewhere that X guy has IQ Y and then decide that's evidence for something.
From the older article [0]: "Soon after he posted his proof of Read’s conjecture, the University of Michigan invited Huh to give a talk on his result. On December 3, 2010, he addressed a room full of many of the same mathematicians who had rejected his graduate school application a year earlier. By this point Huh’s talent was becoming evident to other mathematicians. Jesse Kass was a postdoctoral fellow in mathematics at Michigan at the time. Just before Huh’s visit, a senior faculty member encouraged Kass to watch the talk because '30 years from now you can tell your grandchildren you saw Huh speak before he got famous,' recalled Kass, who’s now a professor at the University of South Carolina.
"Huh’s lecture did not disappoint.
"'The talk was somehow very polished and very clear; it just went to the right points. It’s a bit unusual for a beginning graduate student to give such clean talks,' said Mircea Mustaţă, a mathematician at Michigan.
"Following his talk, the Michigan faculty invited Huh to transfer, which he did in 2011. By that point he’d learned that Read’s conjecture was a special case of a larger and more significant problem — the Rota conjecture.""
[0] https://www.quantamagazine.org/a-path-less-taken-to-the-peak...
Traditionally this would be the midwife making him go out to fetch hot water repeatedly.
> he dropped out of high school to become a poet. It would take a chance encounter during his university years — and many moments of feeling lost — for him to find that mathematics held what he’d been looking for all along.
Having taken a similar path myself -- I dropped out of high school due to a combination of factors, but was still able to get into university 6 months later -- I can empathize with mandatory education NOT being something people fit into, and where making the right career choice isn't always clear or easy.
Not everyone who 'dropped out' did it because of unlimited mom-and-dad-money and boundless professional and social opportunities, nor wears it like a badge of honor.
I get tired of the trope and survivorship bias as well, but in this case it seems like the author did still put in the hard work of learning his niche the traditional way, if in a round-about manner: through a lot of hours of study, personal investment, natural curiosity, and using the abilities he has to the best of his abilities.
How does that even work?