Tao's Solving Mathematical Problems: A Personal Perspective (https://www.amazon.com/Solving-Mathematical-Problems-Persona...) mentioned here is an excellent book that provides keys to mathematical approach to problems and is now a classic ranking there with Polya's How to Solve It.
The front cover of this book gives its author as "Terence Tao, Fields Medal winner 2006". Well ... yes, and no. The thing is, this book was written when Tao was 15 years old. It reflects the precocious skill and insight of an outstandingly gifted 15-year-old, who had won a gold medal at the International Mathematical Olympiad at age 13 (most participants are 17-ish), but not really those of the outstandingly gifted 31-year-old who won the Fields.
It's only about 100 pages long; the problems it discusses are mostly relatively easy (meaning, say, national high-school mathematics competition level or thereabouts, rather than IMO, so not that easy). It doesn't give away any very deep secrets (if there are any) about how to solve such problems. Write down what you know, look for symmetries, simplify step by step, etc.; the real rocket science, as it were, is hidden away in the bits of Tao's brain that instinctively know what symmetries to look for, what steps are likely to lead in the right direction, and so on.
The good news: You wouldn't know it was written by a 15-year-old if the preface didn't tell you. It is a book about mathematics written by someone with a Fields-medal-quality brain, and a book about Olympiad-style problems written by one of the greatest-ever exponents of that art. It contains some nice problems, with solutions by (I repeat myself) one of the finest minds in the business. It's also quite cheap.
If you're interested in this, you should also look at Paul Zeitz's "The art and craft of problem-solving"; it has more pages and more substance to it, but it's twice the price and wasn't written by a Fields medalist.
Perhaps like Erdos, he could be called more of a problem solver than a theory builder. In contrast to a figure like Cantor, Galois, or Grothendieck whose work has more fundamental ramifications. I think this raises interesting questions about the continued viability of the "great man" theory of methematical/scientific progress going forward. As Tao and Gowers exemplify, the trend is towards increasing collaboration on problems as they became ever more complex and individual mathematicians become ever more specialized. Could the age of the great theory builders one day come to an end? Is it already over?
Not that I'm trying to downplay Tao's achievements. How overrated can a Fields medalist really be? They don't hand it out for nothing.
They're not groundbreaking to you because they don't fit the mold of popular science which requires something to be cool and easily digestible in order for most to consider it ground breaking.
Can you define groundbreaking? In a proper historical context, one might say the same about the likes of Newton, Euler etc. Mathematical ideas don't just happen out of nowhere. Everyone is inspired by something that was done before. So truly groundbreaking things are mostly a myth. The closest it gets is when someone takes ideas from one field and unexpectedly applied it to solve ab important problem in another field. Like Poincare's work on three body problem and celestial mechanics.
He received the Fields in 2006 due in no small part to the paper “The primes contain arbitrarily long arithmetic progressions”. I would call that work groundbreaking.
Define ‘overrated.’ He did get awarded a prestigious prize, didn’t he? (On the other hand, there may be little that’s left to be discovered in mathematics which one could call groundbreaking and which the human mind could comprehend without a life-time effort...)
This seems like a thoroughly detestable use of HN: a gratuitous, mean, and almost certainly uninformed (you spelled his name incorrectly) judgement of another human being as "overrated" or "underrated". Why not use your time to do something worthwhile?
What strikes me the most is the amount of support he received and the people he was able to meet when he was a child. I don't want to dismiss his talent, but this level of support is NOT available to 99%+ of population. His parents have done a wonderful job.
To recall my days as an undergraduate at a well-known UK university studying mathematics - large classes (300+) where everyone is disposable, nothing to do but sleep on the table when you work through the problems quickly at a workshop, no interesting projects or opportunities to latch on to develop yourself, nothing to do for days but party, waste talent and occasionally read a math book to catch up. Essentially you are hammered to be average until maybe you enter a PhD course. At that point it is too late.
+1, Kudos to the view. I also believe the support matters. There are many overqualified students who cannot skip grades because of the system, Most probably students graduates at the average age of 22-23 and then PHD and research may well take 6-7 years if not skipped, meanwhile the presence of good mentor is also a luck.
Nothing that has been done for his early education required lots of money.
(Obviously, being in a developed country like Australia helped a lot --- but one can imagine something similar being done in almost any densely populated urban area in the world.)
However, almost everything in his very early age required large amounts of time, thinking, and attention by his parents.
If you have enough time, and your young children are interested in something, don't leave them to figure it out themselves.
Sure, Tao's an outlier among outliers, but in general nurturing a prodigy does not require a lot of parental time or money. I've met a number of such prodigies in the course of my work, and generally their parental involvement amounts to buying them a book every few months, at their kid's insistence. The kids then teach themselves, often by sneaking glances at their books or math problems while hiding out at the back of a standard school classroom. They are not more privileged or supported than the average middle class child in a first world country.
The world's best educational resources are dropping rapidly in price; with some searching, you can even get an mathematical education up to what Tao knew in graduate school completely for free.
I've taught at the U.S. Physics Olympiad's training camp, and also tutor high school students who want to learn a ton of physics quickly. (These kids usually find me themselves, through Google.)
My most obnoxiously smart student was this 12 year old who would interrupt my lectures on classical mechanics with tricky questions I couldn't solve. The next year he came back, doing the same thing but for quantum mechanics, and the year after that he got bored of this and enrolled in an elite college. I want to stress that none of this required great wealth or privilege. The cost for his books was a small fraction of what other middle class parents would pay for, say, music lessons or SAT tutoring.
But a parent has to sign the permission slip for training camp, or drive the kid to tutoring. It's harder (not impossible of course) when the parent is abusive or antagonistic.
When covering Huygen's principle: the wavelets always give a backwards-moving wave in addition to the forwards-moving wave. In other words, Huygen's principle is time symmetric. So in real life, why don't you get the backwards wave?
When covering the principle of least action: often, applying it will give infinitely many solutions, or none at all. An example with no solutions would be the harmonic oscillator with x(0) = 0, x(t) = 0, and t not a multiple of half the period. An example with infinitely many solutions would be the principle of least time applied between the two foci of an ellipse with reflecting walls. So what happens in these cases? Does Lagrangian mechanics just not work?
These days, you use Google. I personally have a list of books for intro physics [0] and help maintain a meta-list of books up to graduate level [1], and there are hundreds more lists out there.
Heck, when I was a kid I learned from just checking out books at random from the local public library, and even that was fine. There's just an avalanche of great free resources out there.
This might sound weird, but, some parents are not at all supportive of their children’s interests. Their relationship can be one of rivalry and antagonism that punishes success. I met a bunch of misfit prodigy hacker types in the military. At some point we realized that we were all there for the same reason.
I very much like that people like Tao are famous. It’s relatively good for the intellectual spirit and hopefully it inspires others.
However this fixation with his “extraordinary capacity”, scores, medals and accolades; I wish we could skip it.
Let’s talk about the awesome work and why it’s awesome. Let’s play through some of it: I’m relatively sure most people know nothing more about Tao than “he’s a genius”. What’s the point of that?
I think we’d be better off if we took the time to understand great work and let our appreciation of great people live through that.
i've said this same thing (on r/math), because it feels like blind hero worship and fetishization, but i got pushback that "it's only human". oh well. most people don't care about the math but only about hero worship and math is just another opportunity.
For my own mental (and moral) health, I always remind myself to focus on what's published, not the publisher. Too often when I see a cool math result I think, "darn, I wish I published that." Then I realize what that attitude does to me.
I agree and think that the drippy accolades are actually counter productive. Why? Because it can become discouraging (as opposed to encouraging) to most (what I would consider) 'normal' smart people. It's the equivalent of walking into perhaps a high school dance and seeing a bunch of tall and good looking men (if you are a man) or beautiful, thin and poised women (if you are a woman). It becomes overwhelming in so many ways and intimidating. Now sure that feeling will spur some people on but just the same it will also turn some people off. And good people. [1]
Think of all the people that might think 'wow this is what you get at Princeton'. Of course logically you know that is not the case and that this is a feature on one very special person. But emotionally (more powerful) that is not what the feeling is.
[1] In reverse if you want proof of this point look at all the people that have entered politics and in particular the Presidential race since Trump was elected. People who in years earlier would have been scared off thinking they were not 'smart' enough. These are people who never realized that a President (regardless of what you think about Trump) does not have to know everything as he relies on others who have expertise (in theory).
> However this fixation with his “extraordinary capacity”, scores, medals and accolades; I wish we could skip it.
I wonder though if such narratives may be necessary (in the world as it is, rather than how it should be) in order to generate fame, specifically the kind of fame that is best able to propagate and reach the eventual endpoints who may be inspired to follow in his footsteps. Annoyances like this may be some sort of social evolutionary adaptation.
This is an unsolvable problem. There already is plenty of expository content about what Tao actually does, e.g. his excellent blog. It's just that it can't get popular because it's technical; that's the iron law of STEM popularization, you can only choose one.
The most press he's gotten recently has been from this "eigenvectors from eigenvalues" thing that he solved three ways in two hours, probably because it's the least technical thing he's done.
> This is an unsolvable problem. [...] it can't get popular because it's technical; that's the iron law of STEM popularization, you can only choose one.
I'm reminded of education research of the form "We tried to teach topic T to students in grade G. We taught it really, really badly. Surprisingly, that didn't work! We draw the obvious conclusion... students in grade G are developmentally unready to understand topic T."
So yes, it might truly be unsolvable. It's certainly difficult. Non-interactive defusing of misconceptions is dauntingly hard - even harder than remedial filling of foundational gaps. Then you add constraints on article length, and the medium may just not be adequate.
But I would be more comfortable with such an argument, if there was wider recognition of how wretched our current science education content is, and how badly it's failing students. How poor the stories we tell about the physical world. If your five-year old wants to know what finger-paint color to use for the Sun, don't ask first-tier astronomy graduate students - there's a wide-spread misconception, so they'll mostly get it wrong. On HN a few days back, there was something like a 'best 2019 astronomy books for children'... and the books were something vaguely like 0 of 10 on getting it right. We're just not set up to teach misconception-free transferable broad understanding.
Seeing research talks, I frequently think "Oh nifty - that concept/description/graphic/video is awesome: accessible and clarifyingly insightful. It should be part of every introduction to this topic. Down to primary school even." ... and won't be any year soon. The pipeline from researcher conversation, to talk, to paper and professional tome, down and down to education and popular content, is regrettably also a gradient from accessible/insightful/transferable/correct to confused/superficial/unusable/nonsense. There just hasn't been the incentives and infrastructure to do better.
In another comment you mention intro physics and olympiads. So take friction. We now know how friction works, down to nanoscale. But last I saw, we don't even try to teach that. Instead it's the decades-old plug-and-chug on Amontons' Laws of large objects sliding on pig fat. Sure, there's some nice training on system decomposition. But if we cared about actual understanding of the physical world, rather than the educational artifact of "Introductory Physics", our teaching focus would need to be different. Similarly, professors complain about PhD candidates lacking a rough quantitative feel for the field, and the current educational focus is far from fixing that. But, perhaps I'm out of date, and things are more-recently improving? Fermi problems are becoming much more common, for instance.
Ending on an upbeat note. An MIT project to create introductory cell-biology VR, obtained domain expertise by pulling in and interviewing researchers. One challenge was apparently... getting them to leave. Such was their enthusiasm. Suggesting that if infrastructure has the right shape, the massive and scarce expertise needed for better content might actually be plausibly obtained. And personalized education via XR might be sufficient, and sufficiently disruptive, to deliver it. So perhaps there's hope to do transformatively better than we have been.
Sorry, but this is idealistic to the extreme. You are completely dismissing the results of education research by saying that every educator is incompetent, despite almost all of these people having far more education experience than both of us combined. Your proposal to fix education is apparently to hyper-focus on a few technical points, like friction and the Sun's color. I don't see how this will help; have you actually spent time teaching intro physics?
In practice, high school and even college introductory classes have a hard time making the basics stick -- stuff as simple as just F = ma. Redirecting half the time you spend on that towards an exquisitely detailed model of friction is not going to produce better physicists. On the average, it's going to produce students that understand F = ma even less, but can recite a couple friction-related buzzwords.
There is no magic curriculum fix that will suddenly make mass education easy. Everybody who thinks about education starts by imagining there is, then reality hits.
Human brains (and bodies) are relentless optimizers. "Use it or lose it" applies to everything from muscle tone and flexibility up to higher cognitive stuff like mathematics. The reason nothing "sticks" is because kids stop using it the moment they stop being tested on it.
We treat education like an assembly line that's supposed to turn every kid into a productive member of society, using a one-size-fits-all curriculum. That's a wrong-headed approach. We should treat it like a mix between olympic trials and career fair. Identify the precocious and the enthusiastic in each subject area. Let teens decide what they want to do.
It won't work, however, until we give teens the right to self-determination. Helicopter parents trying to force their kids to be engineers or doctors, ability be damned, is one of the most pernicious factors in the sorry state of education. The other, perhaps equally pernicious, is the parents who can't or won't help their kids with anything despite their eagerness to learn.
A while back, I had a conversation with a biology professor, about a speculative alternative approach to teaching a topic, which might yield deeper understanding. Their reply was roughly, "Nifty, but... My students will soon be taking the MCAT. It will determine whether their years of effort and dreams will succeed or fail. The MCAT doesn't test for that deeper understanding. It asks for <demonstration of superficial memorization of topic>. I would be doing my students a disservice, to spend our limited time together, on anything else."
Which is completely reasonable. One encounters similar constraints down into middle-school.
Similarly, imagine you are talking with a teacher in a country whose educational system prioritizes memorization and superficial understanding even more than our own. India perhaps.
You emphasize the importance of moving beyond rote whole-classroom call-and-response memorize-and-regurgitate. Of developing skills of system modeling, equational reasoning, and problem solving. More like the USA. And the teacher replies to you, similarly to the biology professor above.
Which is completely reasonable.
But surely, that's not all there is. One can step back, and deal with a larger picture. At least within the education research community of that country, there seems value in awareness and discussion, that there might be different, better ways of doing things. And an exploration of what might become possible under unfamiliar sets of deployment constraints. To inform long-term funding and effort. And system-level tweaks. But which might also turn up bits of overlooked opportunities for nearer-term improvement.
I suggest there is value in similar awareness, discussion, and exploration, within our own country/system.
And that there's value, possibly great value, in doing more of it than we have been. Examples of existent efforts might be life-sciences degree curriculum rewrites, and quantitative introductory biology.
For a concrete illustration (illustrative examples is what those "hyper-focus" points earlier were also intended to be ;) of opportunity, somewhat analogous to F=ma, is a second-year college biology top-of-the-wish-list that incoming students have a firmer grasp of central dogma. Which it turns out, with appropriate tooling, is accessible and useful down into middle-school. And so there's now community and institutional support for spreading that tooling and content.
Another is, there's some work on engineering early science education content, to "immunize" against later misconceptions. Early as in K and even pre-K, for primary/middle-school misconceptions. Some of it almost as side effect - content you might use anyway, but tweaked to have nice long-term properties. Rather than struggling to overwrite naive understandings later, you nudge them in the right direction while they're still being formed. I don't recall whether anyone has explored doing this for motion misconceptions, but they seem the kind of thing it might work for.
When people are doing desperate triage in production hell, they understandably have limited attention for someone saying "you know, if we had resources that we don't, and time that we don't, it might be possible to do much better here, and hmm... what might that look like...?" But a healthy tool ecosystem will have some people discussing and exploring that.
So Tao's celebrity status as a genius does have that positive aspect. I think he disproves many stereotypes of genius and helps people believe that it's possible to brilliant, decent, humble, social and likable. I think that's useful.
So, by lionizing him as a genius I think media actually undermines its own narrative about what genius is. In that sense, maybe the cult of genius is bad but if we have to put a genius on a pedestal, it's great to have it be one who seems so normal.
With respect to his work, I always think of what Richard Feynman said in the start of this book "What do you care what other people think?" (https://www.amazon.com/gp/product/0393355640) He tells a story about an artist who holds up a flower and says "I, as an artist, can see how beautiful a flower is. But you, as a scientist, take it all apart and it becomes dull." Feynman thinks he's nutty. He can see the beauty of the flower and the beauty of the science.
People who don't understand Tao's work probably think it's dull. I can't understand Tao's mathematical work. I'm studying math and I really hope someday I could. But math and science have gone so far now that work at the cutting edge really is incomprehensible to people outside the field. So, I'm inspired by the person who does the work. It's just the surface but it's what I have access to.
Feynman talks about this at the end of the chapter that starts with the flower. His father helped him love science and taught him how to think. Then he sent him to university to learn all the things he hadn't. But when Feynman came home, he couldn't explain quantum mechanics simply. So his father never learned what he didn't know. I hope he still loved his son for understanding it.
It's easy to say you don't need to be a genius to be a mathematician when you're a genius celebrity mathematician.
Personally I gave up on math as a career once I realized all the people who got into top grad schools for math were like international IMO participants, USAMO or equivalent at least, started taking classes beyond calculus in 10th grade, etc. Even though I love it and even have an undergrad degree in it, and maybe could have performed that well if adults in my life nudged me in that direction, the fact that there were so many people who would always have massive advantages turned me off from it.
True, but if you intend to make a career in math, which is what opportune gave up on, it is very much a competition. The odds are better than succeeding in basketball, but only because there are a lot more minor league teams.
I'd (conservatively) speculate most of us here are in the global 1% in terms of "ability to understand math".
Have tried multiple times to follow his work (from the blog). But I'm not math-fluent enough to understand or otherwise appreciate the content. Possibly I could but it requires so much effort.
It's much easier to enjoy Bach's or Messi's genius :)
This (Bach) is a parallel that comes up often in both lay and professional music on the nature of "genius".
I've had some success with the following analogy. Music is not math, but the conversation between research mathematicians is a bit like the conversation between composers. In order for them to understand each other well they must understand the notation, but also the instruments/tools, the history, etc. However we non-composers can still listen to what they wrote if someone performs it for us - we may not catch everything another composer does, but we can enjoy it. The tough thing about mathematics is that there is no orchestra or possibility of one; in order to consume it at all you have to learn some of the details.
its often said around school, that the higher level mathematicians cant sit down and talk to each other about their work much .. things go too far into specialization..
Music is no exception either. Many if not most people (myself included) do not appreciate highly advanced jazz as it's 'spoken' in a language too technical for it to make sense to them.
Can someone explain what the author meant with the following passage?
"... Navier-Stokes equations, which govern the flow of fluids, including air currents. In this case, let us hope that it does not have a real-world application."
One of Terry Tao's recent results shows that an equation that he calls the "averaged Navier-Stokes equation" can have solutions that "blow up" in finite time, starting from a perfectly nice initial condition [0]. Presumably the writer is referring to this, as such solutions would not be very nice to have if they are actually physically realizable, for example by an airfoil.
I believe the author might be trying to say, for example, "let's hope water won't blow up." To make this implication, I believe the author meant to refer to "Tao's most fanciful work" rather than the equations themselves.
> Yitang Zhang, a mathematician at the University of New Hampshire, proved that there are an infinite number of primes that are separated by, at most, 70 million.
> To date, they have managed to prove that there are an infinite number of primes separated by, at most, 246
I'd like to ask a very noob question. What kind of an approach would give us such an exact upper bound, when primes as a concept (in a layman's intuition) seem abstract and unconcerned with such specific figures?
One way to think about this, without any number theory requirements: if we keep throwing a coin with probability of heads p which is around 1/2 at the beginning, but then keeps decreasing on the rate of log n / n, we would expect (using undergraduate probability theory) that two heads indeed come up very close infinitely many times.
As primes up to n have the frequency of log n / n, this would be enough if the primes behaved like coin flip results. However, that is not the case, as primes are leftovers of some sieve, rather than totally independent and random.
The goal of some of the current methods is to establish a good ``probabilistic'' model of the primes, so that this intuition can be transformed to reality.
Analytic number theorist here. I can answer your question.
The method doesn't really yield "exactly 70,000,000". If you traced through his method, and worked out each step in more detail, you'd probably get a bound of (say) 64,189,288 -- or some random number of like that. If you read through it still more carefully, and tried to introduce genuine improvements, you'd improve this further.
Indeed, the bound has been improved to 246. You can see the(long!) proof here:
Zhang stated 70 million because it's easy to remember.
Anyway, the basic strategy of the proof is to attempt to prove the twin prime conjecture, and not fall too short.
Where do these numbers come from? You end up needing to do lots of random computations in the course of the proof. For example, can you compute a constant C such that the bound
holds for all x between 3 and 5? If this appeared in the course of an analytic number theory proof, you probably wouldn't try to compute the absolute best value C. It would be tedious and boring, and nobody would want to read it. You'd just compute something that was good enough, and move on.
Could anyone here recommend me something that actually talks about how genius mathematicians and others go about thinking about problems? I've read Hamming's "You and Your Research" and really liked it, but I would love to see similar things by other thinkers who might have a different take than someone like Hamming.
The author notes "Let us hope there are no real-world applications" when describing the Navier-Stokes regularity and smoothness problem. It looks like he meant to write the opposite and this was an error.
However, in some sense, the statement may also be true. That is, people have been doing fluid experiments and numerical experiments (numerical fluid mechanics is a mature discipline) without relying on whether the conjecture is true. I mean, flight development was not waiting for an answer!
My point is engineering moved on without an answer to this particular question. At the same time, answering questions such as these (and the ones on number theory two of which were mentioned in the article) that occupy pure mathematicians provide more than artistic pleasure -- they provide the giant leaps that spur new technology. The often
quoted example from Tao's research is compressed sensing but even bigger splashes like the idea of computers and the stored program, that can be traced back to Turing's seminal paper, comes from pure mathematics.
Charles Fefferman, mentioned in the article, makes 3 very cogent arguments[1] for the applicability of pure mathematics to the society, and in fact, says very pertinently that the line between pure and applied math is very blurry in the first place:
[1]: https://www.youtube.com/watch?v=3LgjMjVA4sY
1. that unanticipated applications show up from purely theoretical questions
2. a rigourous study of math provides a way of thinking that prepares students to work in a range of fields that require quantitative or analytical thinking.
3. math is capable of revealing those ground breaking discoveries that happen rarely
Celebrating Tao's successes through media, movies and and other pop-culture vehicles, is very good, and I very happy this is happening.
Shining the light into the continent of Mathematics, and its celebrity inhabitants, brings attention and, just may be a bit of the appreciation, of complex and demanding it is to live there...
But, I do think, that to bring more and more people into the field, we have to also celebrate the diversity and range in the abilities, of the people who have, and who will have, contributed to the science of Mathematics.
I am sure that not every accomplished, and well respected mathematician need to have the cognitive brilliancy of Tao or Nash.
I find inspiration and, to a degree, comfort, in this quote by Alexander Grothendieck (specifically, the point where he felt, by far, he was not the most brilliant person in the room)
>"
..Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was.
I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end.
Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects.
..
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians.
Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound.
They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb.
Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era.
To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.”
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[ 262 ms ] story [ 1731 ms ] threadOne thing the article doesn't touch on is Tao's prominence in massively collaborative math research through his blog (latest such work from there was discussed on HN a day ago https://news.ycombinator.com/item?id=21542054). This new approach, first proposed by Tom Gowers (https://gowers.wordpress.com/2009/01/27/is-massively-collabo...) in 2009 has successfully solved a number of open problems in the past ten years (https://polymathprojects.org/2019/02/03/ten-years-of-polymat...).
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First, the bad news.
The front cover of this book gives its author as "Terence Tao, Fields Medal winner 2006". Well ... yes, and no. The thing is, this book was written when Tao was 15 years old. It reflects the precocious skill and insight of an outstandingly gifted 15-year-old, who had won a gold medal at the International Mathematical Olympiad at age 13 (most participants are 17-ish), but not really those of the outstandingly gifted 31-year-old who won the Fields.
It's only about 100 pages long; the problems it discusses are mostly relatively easy (meaning, say, national high-school mathematics competition level or thereabouts, rather than IMO, so not that easy). It doesn't give away any very deep secrets (if there are any) about how to solve such problems. Write down what you know, look for symmetries, simplify step by step, etc.; the real rocket science, as it were, is hidden away in the bits of Tao's brain that instinctively know what symmetries to look for, what steps are likely to lead in the right direction, and so on.
The good news: You wouldn't know it was written by a 15-year-old if the preface didn't tell you. It is a book about mathematics written by someone with a Fields-medal-quality brain, and a book about Olympiad-style problems written by one of the greatest-ever exponents of that art. It contains some nice problems, with solutions by (I repeat myself) one of the finest minds in the business. It's also quite cheap.
If you're interested in this, you should also look at Paul Zeitz's "The art and craft of problem-solving"; it has more pages and more substance to it, but it's twice the price and wasn't written by a Fields medalist.
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Pedantic note: Tim Gowers, not Tom.
Not that I'm trying to downplay Tao's achievements. How overrated can a Fields medalist really be? They don't hand it out for nothing.
To recall my days as an undergraduate at a well-known UK university studying mathematics - large classes (300+) where everyone is disposable, nothing to do but sleep on the table when you work through the problems quickly at a workshop, no interesting projects or opportunities to latch on to develop yourself, nothing to do for days but party, waste talent and occasionally read a math book to catch up. Essentially you are hammered to be average until maybe you enter a PhD course. At that point it is too late.
Nothing that has been done for his early education required lots of money. (Obviously, being in a developed country like Australia helped a lot --- but one can imagine something similar being done in almost any densely populated urban area in the world.)
However, almost everything in his very early age required large amounts of time, thinking, and attention by his parents.
If you have enough time, and your young children are interested in something, don't leave them to figure it out themselves.
The world's best educational resources are dropping rapidly in price; with some searching, you can even get an mathematical education up to what Tao knew in graduate school completely for free.
If you don't mind, can you elaborate more on what kind of work you do? Do you run a school for the gifted (assuming such a thing exists)?
My most obnoxiously smart student was this 12 year old who would interrupt my lectures on classical mechanics with tricky questions I couldn't solve. The next year he came back, doing the same thing but for quantum mechanics, and the year after that he got bored of this and enrolled in an elite college. I want to stress that none of this required great wealth or privilege. The cost for his books was a small fraction of what other middle class parents would pay for, say, music lessons or SAT tutoring.
When covering the principle of least action: often, applying it will give infinitely many solutions, or none at all. An example with no solutions would be the harmonic oscillator with x(0) = 0, x(t) = 0, and t not a multiple of half the period. An example with infinitely many solutions would be the principle of least time applied between the two foci of an ellipse with reflecting walls. So what happens in these cases? Does Lagrangian mechanics just not work?
Heck, when I was a kid I learned from just checking out books at random from the local public library, and even that was fine. There's just an avalanche of great free resources out there.
0: https://knzhou.github.io/#writing 1: https://physics.stackexchange.com/questions/12175/book-recom...
However this fixation with his “extraordinary capacity”, scores, medals and accolades; I wish we could skip it.
Let’s talk about the awesome work and why it’s awesome. Let’s play through some of it: I’m relatively sure most people know nothing more about Tao than “he’s a genius”. What’s the point of that?
I think we’d be better off if we took the time to understand great work and let our appreciation of great people live through that.
Think of all the people that might think 'wow this is what you get at Princeton'. Of course logically you know that is not the case and that this is a feature on one very special person. But emotionally (more powerful) that is not what the feeling is.
[1] In reverse if you want proof of this point look at all the people that have entered politics and in particular the Presidential race since Trump was elected. People who in years earlier would have been scared off thinking they were not 'smart' enough. These are people who never realized that a President (regardless of what you think about Trump) does not have to know everything as he relies on others who have expertise (in theory).
I wonder though if such narratives may be necessary (in the world as it is, rather than how it should be) in order to generate fame, specifically the kind of fame that is best able to propagate and reach the eventual endpoints who may be inspired to follow in his footsteps. Annoyances like this may be some sort of social evolutionary adaptation.
The most press he's gotten recently has been from this "eigenvectors from eigenvalues" thing that he solved three ways in two hours, probably because it's the least technical thing he's done.
I'm reminded of education research of the form "We tried to teach topic T to students in grade G. We taught it really, really badly. Surprisingly, that didn't work! We draw the obvious conclusion... students in grade G are developmentally unready to understand topic T."
So yes, it might truly be unsolvable. It's certainly difficult. Non-interactive defusing of misconceptions is dauntingly hard - even harder than remedial filling of foundational gaps. Then you add constraints on article length, and the medium may just not be adequate.
But I would be more comfortable with such an argument, if there was wider recognition of how wretched our current science education content is, and how badly it's failing students. How poor the stories we tell about the physical world. If your five-year old wants to know what finger-paint color to use for the Sun, don't ask first-tier astronomy graduate students - there's a wide-spread misconception, so they'll mostly get it wrong. On HN a few days back, there was something like a 'best 2019 astronomy books for children'... and the books were something vaguely like 0 of 10 on getting it right. We're just not set up to teach misconception-free transferable broad understanding.
Seeing research talks, I frequently think "Oh nifty - that concept/description/graphic/video is awesome: accessible and clarifyingly insightful. It should be part of every introduction to this topic. Down to primary school even." ... and won't be any year soon. The pipeline from researcher conversation, to talk, to paper and professional tome, down and down to education and popular content, is regrettably also a gradient from accessible/insightful/transferable/correct to confused/superficial/unusable/nonsense. There just hasn't been the incentives and infrastructure to do better.
In another comment you mention intro physics and olympiads. So take friction. We now know how friction works, down to nanoscale. But last I saw, we don't even try to teach that. Instead it's the decades-old plug-and-chug on Amontons' Laws of large objects sliding on pig fat. Sure, there's some nice training on system decomposition. But if we cared about actual understanding of the physical world, rather than the educational artifact of "Introductory Physics", our teaching focus would need to be different. Similarly, professors complain about PhD candidates lacking a rough quantitative feel for the field, and the current educational focus is far from fixing that. But, perhaps I'm out of date, and things are more-recently improving? Fermi problems are becoming much more common, for instance.
Ending on an upbeat note. An MIT project to create introductory cell-biology VR, obtained domain expertise by pulling in and interviewing researchers. One challenge was apparently... getting them to leave. Such was their enthusiasm. Suggesting that if infrastructure has the right shape, the massive and scarce expertise needed for better content might actually be plausibly obtained. And personalized education via XR might be sufficient, and sufficiently disruptive, to deliver it. So perhaps there's hope to do transformatively better than we have been.
In practice, high school and even college introductory classes have a hard time making the basics stick -- stuff as simple as just F = ma. Redirecting half the time you spend on that towards an exquisitely detailed model of friction is not going to produce better physicists. On the average, it's going to produce students that understand F = ma even less, but can recite a couple friction-related buzzwords.
There is no magic curriculum fix that will suddenly make mass education easy. Everybody who thinks about education starts by imagining there is, then reality hits.
We treat education like an assembly line that's supposed to turn every kid into a productive member of society, using a one-size-fits-all curriculum. That's a wrong-headed approach. We should treat it like a mix between olympic trials and career fair. Identify the precocious and the enthusiastic in each subject area. Let teens decide what they want to do.
It won't work, however, until we give teens the right to self-determination. Helicopter parents trying to force their kids to be engineers or doctors, ability be damned, is one of the most pernicious factors in the sorry state of education. The other, perhaps equally pernicious, is the parents who can't or won't help their kids with anything despite their eagerness to learn.
A while back, I had a conversation with a biology professor, about a speculative alternative approach to teaching a topic, which might yield deeper understanding. Their reply was roughly, "Nifty, but... My students will soon be taking the MCAT. It will determine whether their years of effort and dreams will succeed or fail. The MCAT doesn't test for that deeper understanding. It asks for <demonstration of superficial memorization of topic>. I would be doing my students a disservice, to spend our limited time together, on anything else."
Which is completely reasonable. One encounters similar constraints down into middle-school.
Similarly, imagine you are talking with a teacher in a country whose educational system prioritizes memorization and superficial understanding even more than our own. India perhaps.
You emphasize the importance of moving beyond rote whole-classroom call-and-response memorize-and-regurgitate. Of developing skills of system modeling, equational reasoning, and problem solving. More like the USA. And the teacher replies to you, similarly to the biology professor above.
Which is completely reasonable.
But surely, that's not all there is. One can step back, and deal with a larger picture. At least within the education research community of that country, there seems value in awareness and discussion, that there might be different, better ways of doing things. And an exploration of what might become possible under unfamiliar sets of deployment constraints. To inform long-term funding and effort. And system-level tweaks. But which might also turn up bits of overlooked opportunities for nearer-term improvement.
I suggest there is value in similar awareness, discussion, and exploration, within our own country/system.
And that there's value, possibly great value, in doing more of it than we have been. Examples of existent efforts might be life-sciences degree curriculum rewrites, and quantitative introductory biology.
For a concrete illustration (illustrative examples is what those "hyper-focus" points earlier were also intended to be ;) of opportunity, somewhat analogous to F=ma, is a second-year college biology top-of-the-wish-list that incoming students have a firmer grasp of central dogma. Which it turns out, with appropriate tooling, is accessible and useful down into middle-school. And so there's now community and institutional support for spreading that tooling and content.
Another is, there's some work on engineering early science education content, to "immunize" against later misconceptions. Early as in K and even pre-K, for primary/middle-school misconceptions. Some of it almost as side effect - content you might use anyway, but tweaked to have nice long-term properties. Rather than struggling to overwrite naive understandings later, you nudge them in the right direction while they're still being formed. I don't recall whether anyone has explored doing this for motion misconceptions, but they seem the kind of thing it might work for.
When people are doing desperate triage in production hell, they understandably have limited attention for someone saying "you know, if we had resources that we don't, and time that we don't, it might be possible to do much better here, and hmm... what might that look like...?" But a healthy tool ecosystem will have some people discussing and exploring that.
I think I saw it linked along with this article about the warped portrayal of genius Hollywood feeds us (especially in the Imitation Game): https://www.huffpost.com/entry/turing-the-myth-of-the-heroic...
So Tao's celebrity status as a genius does have that positive aspect. I think he disproves many stereotypes of genius and helps people believe that it's possible to brilliant, decent, humble, social and likable. I think that's useful.
He talked about this eloquently in his own blog where he argues against the "cult of genius" and says one needn't be a genius to be a mathematician: https://terrytao.wordpress.com/career-advice/does-one-have-t...
So, by lionizing him as a genius I think media actually undermines its own narrative about what genius is. In that sense, maybe the cult of genius is bad but if we have to put a genius on a pedestal, it's great to have it be one who seems so normal.
With respect to his work, I always think of what Richard Feynman said in the start of this book "What do you care what other people think?" (https://www.amazon.com/gp/product/0393355640) He tells a story about an artist who holds up a flower and says "I, as an artist, can see how beautiful a flower is. But you, as a scientist, take it all apart and it becomes dull." Feynman thinks he's nutty. He can see the beauty of the flower and the beauty of the science.
People who don't understand Tao's work probably think it's dull. I can't understand Tao's mathematical work. I'm studying math and I really hope someday I could. But math and science have gone so far now that work at the cutting edge really is incomprehensible to people outside the field. So, I'm inspired by the person who does the work. It's just the surface but it's what I have access to.
Feynman talks about this at the end of the chapter that starts with the flower. His father helped him love science and taught him how to think. Then he sent him to university to learn all the things he hadn't. But when Feynman came home, he couldn't explain quantum mechanics simply. So his father never learned what he didn't know. I hope he still loved his son for understanding it.
Personally I gave up on math as a career once I realized all the people who got into top grad schools for math were like international IMO participants, USAMO or equivalent at least, started taking classes beyond calculus in 10th grade, etc. Even though I love it and even have an undergrad degree in it, and maybe could have performed that well if adults in my life nudged me in that direction, the fact that there were so many people who would always have massive advantages turned me off from it.
Have tried multiple times to follow his work (from the blog). But I'm not math-fluent enough to understand or otherwise appreciate the content. Possibly I could but it requires so much effort.
It's much easier to enjoy Bach's or Messi's genius :)
I've had some success with the following analogy. Music is not math, but the conversation between research mathematicians is a bit like the conversation between composers. In order for them to understand each other well they must understand the notation, but also the instruments/tools, the history, etc. However we non-composers can still listen to what they wrote if someone performs it for us - we may not catch everything another composer does, but we can enjoy it. The tough thing about mathematics is that there is no orchestra or possibility of one; in order to consume it at all you have to learn some of the details.
I see what they did there.
"... Navier-Stokes equations, which govern the flow of fluids, including air currents. In this case, let us hope that it does not have a real-world application."
[0] https://terrytao.wordpress.com/2014/02/04/finite-time-blowup...
> To date, they have managed to prove that there are an infinite number of primes separated by, at most, 246
I'd like to ask a very noob question. What kind of an approach would give us such an exact upper bound, when primes as a concept (in a layman's intuition) seem abstract and unconcerned with such specific figures?
I hope the question made sense.
One way to think about this, without any number theory requirements: if we keep throwing a coin with probability of heads p which is around 1/2 at the beginning, but then keeps decreasing on the rate of log n / n, we would expect (using undergraduate probability theory) that two heads indeed come up very close infinitely many times.
As primes up to n have the frequency of log n / n, this would be enough if the primes behaved like coin flip results. However, that is not the case, as primes are leftovers of some sieve, rather than totally independent and random.
The goal of some of the current methods is to establish a good ``probabilistic'' model of the primes, so that this intuition can be transformed to reality.
The method doesn't really yield "exactly 70,000,000". If you traced through his method, and worked out each step in more detail, you'd probably get a bound of (say) 64,189,288 -- or some random number of like that. If you read through it still more carefully, and tried to introduce genuine improvements, you'd improve this further.
Indeed, the bound has been improved to 246. You can see the(long!) proof here:
https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-submitted...
Zhang stated 70 million because it's easy to remember.
Anyway, the basic strategy of the proof is to attempt to prove the twin prime conjecture, and not fall too short.
Where do these numbers come from? You end up needing to do lots of random computations in the course of the proof. For example, can you compute a constant C such that the bound
e^(.1x^2 + .4x) + 5cos(pi * x) - sqrt(x^2 + 1) < Cx + 4
holds for all x between 3 and 5? If this appeared in the course of an analytic number theory proof, you probably wouldn't try to compute the absolute best value C. It would be tedious and boring, and nobody would want to read it. You'd just compute something that was good enough, and move on.
https://www.wolframalpha.com/input/?i=maximize+%7B+%28e%5E%2...
-> You can find C by imposing equality for x = 5.
[0] https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem
My point is engineering moved on without an answer to this particular question. At the same time, answering questions such as these (and the ones on number theory two of which were mentioned in the article) that occupy pure mathematicians provide more than artistic pleasure -- they provide the giant leaps that spur new technology. The often quoted example from Tao's research is compressed sensing but even bigger splashes like the idea of computers and the stored program, that can be traced back to Turing's seminal paper, comes from pure mathematics.
Charles Fefferman, mentioned in the article, makes 3 very cogent arguments[1] for the applicability of pure mathematics to the society, and in fact, says very pertinently that the line between pure and applied math is very blurry in the first place: [1]: https://www.youtube.com/watch?v=3LgjMjVA4sY
1. that unanticipated applications show up from purely theoretical questions
2. a rigourous study of math provides a way of thinking that prepares students to work in a range of fields that require quantitative or analytical thinking.
3. math is capable of revealing those ground breaking discoveries that happen rarely
Shining the light into the continent of Mathematics, and its celebrity inhabitants, brings attention and, just may be a bit of the appreciation, of complex and demanding it is to live there...
But, I do think, that to bring more and more people into the field, we have to also celebrate the diversity and range in the abilities, of the people who have, and who will have, contributed to the science of Mathematics.
I am sure that not every accomplished, and well respected mathematician need to have the cognitive brilliancy of Tao or Nash.
I find inspiration and, to a degree, comfort, in this quote by Alexander Grothendieck (specifically, the point where he felt, by far, he was not the most brilliant person in the room)
>" ..Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was.
I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end.
Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects. ..
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians.
Still, from the perspective of thirty or thirty-five years, I can state that their imprint upon the mathematics of our time has not been very profound.
They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb.
Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era.
To have broken these bounds they would have had to rediscover in themselves that capability which was their birthright, as it was mine: the capacity to be alone.”
..."
see also discussion on HN here
https://news.ycombinator.com/item?id=8604814
https://www.goodreads.com/author/quotes/405977.Alexander_Gro...