8 ms·

The Mind of a Mathematician

259 points by lordgrenville 7y ago

Jun8 7y ago
Tao's Solving Mathematical Problems: A Personal Perspective (https://www.amazon.com/Solving-Mathematical-Problems-Personal-Perspective/dp/0199205604 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. One 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 https://news.ycombinator.com/item?id=21542054). This new approach, first proposed by Tom Gowers (https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ 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-polymath/ https://polymathprojects.org/2019/02/03/ten-years-of-polymat...).
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- gjm11 7y ago
  I'll quote my own Amazon review of Tao's book: ---- 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. ---- Pedantic note: Tim Gowers, not Tom.
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corporateslave5 7y ago
I would argue that Terrence Tao is overrated. He has solved tons of problems in mathematics, but none groundbreaking.
- carlob 7y ago
  A lot of people would have said that of Erdős as well...
- lordnacho 7y ago
  Either this comment, or its parent, is a joke.
- fizixer 7y ago
  https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
- aexq36 7y ago
  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.
- WilliamEdward 7y ago
  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.
- 19f191ty 7y ago
  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.
imbusy111 7y ago
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.
- jitendrac 7y ago
  +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.
- srj203 7y ago
  It is important to note that it is indeed about his parents doing a wonderful job, and not about them being very rich or something. The article http://www.davidsongifted.org/search-database/entry/a10116 http://www.davidsongifted.org/search-database/entry/a10116 describes this nicely. 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.
- knzhou 7y ago
  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.
usgroup 7y ago
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.
- throwlaplace 7y ago
  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.
  - lonelappde 7y ago
    I think our world would be in better shape if people like Terry Tao were the heroes more people worshipped.
- abnry 7y ago
  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.
- gist 7y ago
  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).
RickJWagner 7y ago
"The awards and honors have only multiplied...." I see what they did there.
new2628 7y ago
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."
- kkylin 7y ago
  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. [0] https://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/ https://terrytao.wordpress.com/2014/02/04/finite-time-blowup...
- jshevek 7y ago
  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.
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jeswin 7y ago
> 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? I hope the question made sense.
- srj203 7y ago
  Not an expert, and not answer to this exact question, but the video https://www.youtube.com/watch?v=PtsrAw1LR3E https://www.youtube.com/watch?v=PtsrAw1LR3E and these slides https://pdfs.semanticscholar.org/41c5/e01cf7e3975efdc5a27f7484cdbba6e63da6.pdf https://pdfs.semanticscholar.org/41c5/e01cf7e3975efdc5a27f74... could answer some of your questions. 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.
  - jeswin 7y ago
    Thank you. Very informative links too.
- impendia 7y ago
  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: https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-submitted.pdf 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.
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JoeSmithson 7y ago
I always use the Green-Tao Theorem[0] as my go-to example that very understandable problems are still being solved in modern maths research. [0] https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem
threwawasy1228 7y ago
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.
amilein7minutes 7y ago
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 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
75dvtwin 7y ago
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.” ..." see also discussion on HN here https://news.ycombinator.com/item?id=8604814 https://news.ycombinator.com/item?id=8604814 https://www.goodreads.com/author/quotes/405977.Alexander_Grothendieck https://www.goodreads.com/author/quotes/405977.Alexander_Gro...
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259 points by lordgrenville 7y ago

Jun8 7y ago

Tao's Solving Mathematical Problems: A Personal Perspective (https://www.amazon.com/Solving-Mathematical-Problems-Personal-Perspective/dp/0199205604 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. One 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 https://news.ycombinator.com/item?id=21542054). This new approach, first proposed by Tom Gowers (https://gowers.wordpress.com/2009/01/27/is-massively-collaborative-mathematics-possible/ 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-polymath/ https://polymathprojects.org/2019/02/03/ten-years-of-polymat...).

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gjm11 7y ago
I'll quote my own Amazon review of Tao's book: ---- 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. ---- Pedantic note: Tim Gowers, not Tom.

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corporateslave5 7y ago

I would argue that Terrence Tao is overrated. He has solved tons of problems in mathematics, but none groundbreaking.

carlob 7y ago
A lot of people would have said that of Erdős as well...
lordnacho 7y ago
Either this comment, or its parent, is a joke.
fizixer 7y ago
https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
aexq36 7y ago
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.
WilliamEdward 7y ago
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.
19f191ty 7y ago
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.

imbusy111 7y ago

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.

jitendrac 7y ago
+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.
srj203 7y ago
It is important to note that it is indeed about his parents doing a wonderful job, and not about them being very rich or something. The article http://www.davidsongifted.org/search-database/entry/a10116 http://www.davidsongifted.org/search-database/entry/a10116 describes this nicely. 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.
knzhou 7y ago
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.

usgroup 7y ago

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.

throwlaplace 7y ago
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.
- lonelappde 7y ago
  I think our world would be in better shape if people like Terry Tao were the heroes more people worshipped.
abnry 7y ago
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.
gist 7y ago
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).

RickJWagner 7y ago

"The awards and honors have only multiplied...." I see what they did there.

new2628 7y ago

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."

kkylin 7y ago
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. [0] https://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/ https://terrytao.wordpress.com/2014/02/04/finite-time-blowup...
jshevek 7y ago
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.
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jeswin 7y ago

> 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? I hope the question made sense.

srj203 7y ago
Not an expert, and not answer to this exact question, but the video https://www.youtube.com/watch?v=PtsrAw1LR3E https://www.youtube.com/watch?v=PtsrAw1LR3E and these slides https://pdfs.semanticscholar.org/41c5/e01cf7e3975efdc5a27f7484cdbba6e63da6.pdf https://pdfs.semanticscholar.org/41c5/e01cf7e3975efdc5a27f74... could answer some of your questions. 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.
- jeswin 7y ago
  Thank you. Very informative links too.
impendia 7y ago
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: https://www.dropbox.com/s/85pt6mvzf5ghukw/newergap-submitted.pdf 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.

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JoeSmithson 7y ago

I always use the Green-Tao Theorem[0] as my go-to example that very understandable problems are still being solved in modern maths research. [0] https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem

threwawasy1228 7y ago

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.

amilein7minutes 7y ago

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 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

75dvtwin 7y ago

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.” ..." see also discussion on HN here https://news.ycombinator.com/item?id=8604814 https://news.ycombinator.com/item?id=8604814 https://www.goodreads.com/author/quotes/405977.Alexander_Grothendieck https://www.goodreads.com/author/quotes/405977.Alexander_Gro...

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