Mathematicians Come Closer to Solving Goldbach's Weak Conjecture(scientificamerican.com)
scientificamerican.com
Mathematicians Come Closer to Solving Goldbach's Weak Conjecture
http://www.scientificamerican.com/article.cfm?id=goldbachs-prime-numbers
7 comments
> the youngest ever International Mathematical Olympiad gold medalist to date
I don't see how when he started dating is relevant here...
On a serious note, I find this puzzle far more interesting than Fermat's Last Thereom, even though it doesn't seem to be well known at all. Probably because it hasn't been solved and subsequently been the subject of TV documentaries, admittedly. Are there any other comparably simple/intuitive rules in mathematics that have never been fully proven?
I don't see how when he started dating is relevant here...
On a serious note, I find this puzzle far more interesting than Fermat's Last Thereom, even though it doesn't seem to be well known at all. Probably because it hasn't been solved and subsequently been the subject of TV documentaries, admittedly. Are there any other comparably simple/intuitive rules in mathematics that have never been fully proven?
Number theory is a good source of unsolved problems that don’t require much background to understand. For example: http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_ma...
Don't get it. How can you celebrate someone as the "youngest ever International Math Olympiad gold medalist" while simultaneously denying the importance of genius in math?
Terence Tao is a humble, self-effacing guy. Or at least he knows that society expects him to act like that rather than mimic the chest-thumping, endzone-dancing wide receiver. So obviously he's not going to say that "yeah, to really move math forward you kind of need to be a genius". Though of course other geniuses like Feynman were more up front about that truth.
Very few people have the innate ability to do what Tao or Feynman did. Recognizing that early is a mercy that allows people to move into other areas where they have a comparative advantage, rather than beating their head against the wall for 10,000 hours only not to get anywhere.
Terence Tao is a humble, self-effacing guy. Or at least he knows that society expects him to act like that rather than mimic the chest-thumping, endzone-dancing wide receiver. So obviously he's not going to say that "yeah, to really move math forward you kind of need to be a genius". Though of course other geniuses like Feynman were more up front about that truth.
Very few people have the innate ability to do what Tao or Feynman did. Recognizing that early is a mercy that allows people to move into other areas where they have a comparative advantage, rather than beating their head against the wall for 10,000 hours only not to get anywhere.
I think a point that is often glossed over here is the extreme value of compound interest and self-inforced feedback loops. Calling someone a genius is a cop-out, in my opinion.
If you're a Tao or Feynman, chances are you have been doing "maths" intuitively since a very young age. All these moments add up and reinforce each other. Is that the same as being a genius? Maybe, but then we are dealing with definitions - in my mind genius is a myth created by society to explain "unexplainable" things. That and an excuse for people's relative incompetence - see Hamming in his article "You and Your Research" [0].
The science of experts and deliberate practise [1, 2] is actually quite solid, despite its gladwellification. For example there have to my knowledge not been found a single person who defies the "logic of practise" - oft cited examples are Mozart and Woods, which are more of a myth making than based in any known facts (consider how both their fathers pushed them extremely hard with the right type of practise from a very early age).
Of course, there will always be variations (but these could in my opinion just as well be ascribed to right-time-right-place mechanisms). I suspect a large "problem" is - if the next 11 year old Tao has already had thousands of hours of something akin to deliberate practise, how on earth are other people - who don't share this natural inclination - going to catch up?
0: From http://www.cs.virginia.edu/~robins/YouAndYourResearch.html
Now for the matter of drive. You observe that most great scientists have tremendous drive. I worked for ten years with John Tukey at Bell Labs. He had tremendous drive. One day about three or four years after I joined, I discovered that John Tukey was slightly younger than I was. John was a genius and I clearly was not. Well I went storming into Bode's office and said, ``How can anybody my age know as much as John Tukey does?'' He leaned back in his chair, put his hands behind his head, grinned slightly, and said, ``You would be surprised Hamming, how much you would know if you worked as hard as he did that many years.'' I simply slunk out of the office!
What Bode was saying was this: ``Knowledge and productivity are like compound interest.'' Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime. I took Bode's remark to heart; I spent a good deal more of my time for some years trying to work a bit harder and I found, in fact, I could get more work done. I don't like to say it in front of my wife, but I did sort of neglect her sometimes; I needed to study. You have to neglect things if you intend to get what you want done. There's no question about this.
On this matter of drive Edison says, ``Genius is 99% perspiration and 1% inspiration.'' He may have been exaggerating, but the idea is that solid work, steadily applied, gets you surprisingly far. The steady application of effort with a little bit more work, intelligently applied is what does it. That's the trouble; drive, misapplied, doesn't get you anywhere. I've often wondered why so many of my good friends at Bell Labs who worked as hard or harder than I did, didn't have so much to show for it. The misapplication of effort is a very serious matter. Just hard work is not enough - it must be applied sensibly.
1: http://en.wikipedia.org/wiki/Deliberate_practice#Deliberate_...
2: http://www.amazon.com/Cambridge-Expertise-Performance-Handbo...
If you're a Tao or Feynman, chances are you have been doing "maths" intuitively since a very young age. All these moments add up and reinforce each other. Is that the same as being a genius? Maybe, but then we are dealing with definitions - in my mind genius is a myth created by society to explain "unexplainable" things. That and an excuse for people's relative incompetence - see Hamming in his article "You and Your Research" [0].
The science of experts and deliberate practise [1, 2] is actually quite solid, despite its gladwellification. For example there have to my knowledge not been found a single person who defies the "logic of practise" - oft cited examples are Mozart and Woods, which are more of a myth making than based in any known facts (consider how both their fathers pushed them extremely hard with the right type of practise from a very early age).
Of course, there will always be variations (but these could in my opinion just as well be ascribed to right-time-right-place mechanisms). I suspect a large "problem" is - if the next 11 year old Tao has already had thousands of hours of something akin to deliberate practise, how on earth are other people - who don't share this natural inclination - going to catch up?
0: From http://www.cs.virginia.edu/~robins/YouAndYourResearch.html
Now for the matter of drive. You observe that most great scientists have tremendous drive. I worked for ten years with John Tukey at Bell Labs. He had tremendous drive. One day about three or four years after I joined, I discovered that John Tukey was slightly younger than I was. John was a genius and I clearly was not. Well I went storming into Bode's office and said, ``How can anybody my age know as much as John Tukey does?'' He leaned back in his chair, put his hands behind his head, grinned slightly, and said, ``You would be surprised Hamming, how much you would know if you worked as hard as he did that many years.'' I simply slunk out of the office!
What Bode was saying was this: ``Knowledge and productivity are like compound interest.'' Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime. I took Bode's remark to heart; I spent a good deal more of my time for some years trying to work a bit harder and I found, in fact, I could get more work done. I don't like to say it in front of my wife, but I did sort of neglect her sometimes; I needed to study. You have to neglect things if you intend to get what you want done. There's no question about this.
On this matter of drive Edison says, ``Genius is 99% perspiration and 1% inspiration.'' He may have been exaggerating, but the idea is that solid work, steadily applied, gets you surprisingly far. The steady application of effort with a little bit more work, intelligently applied is what does it. That's the trouble; drive, misapplied, doesn't get you anywhere. I've often wondered why so many of my good friends at Bell Labs who worked as hard or harder than I did, didn't have so much to show for it. The misapplication of effort is a very serious matter. Just hard work is not enough - it must be applied sensibly.
1: http://en.wikipedia.org/wiki/Deliberate_practice#Deliberate_...
2: http://www.amazon.com/Cambridge-Expertise-Performance-Handbo...
Don't get it. How can you celebrate someone as the "youngest ever International Math Olympiad gold medalist" while simultaneously denying the importance of genius in math?
It sounds like you are under the impression that success in mathematical olympiads is connected with success in mathematical research. It’s actually relatively rare for IMO winners to go on to become successful researchers, because the skills involved, while not completely disjoint, are not exactly the same either.
It sounds like you are under the impression that success in mathematical olympiads is connected with success in mathematical research. It’s actually relatively rare for IMO winners to go on to become successful researchers, because the skills involved, while not completely disjoint, are not exactly the same either.
>> rather than beating their head against the wall for 10,000 hours only not to get anywhere
If you ever played a puzzle game (or dwarffortress), you should understand that "beating our head against the wall only not to get anywhere" could even be enjoyable. If you embark in this math adventure, maybe you will not become the next vonNeumann, but you will still have fun.
If you ever played a puzzle game (or dwarffortress), you should understand that "beating our head against the wall only not to get anywhere" could even be enjoyable. If you embark in this math adventure, maybe you will not become the next vonNeumann, but you will still have fun.
But how would you know that you don't have that, supposedly, innate ability if you don't try first? and why does it matter?
If you enjoy it, do it.
If you enjoy it, do it.
You'll never know 'til you try? Sure, but most people get plenty of chance to try mathematics at school. And while school-level mathematics ain't the same as research-level mathematics, enough of it uses the same bits of brain that by the time you hit university you already know whether you're brilliant at mathematics or not.
Terry Tao was clearly brilliant at mathematics as a child, though what nobody knew then was that he'd go on to be super-brilliant at mathematics as an adult, rather than being one of the many disappointing prodigies.
Anyway, I'm pretty smart, but I don't take career advice from anyone as smart as Terry Tao, just as I don't take dating advice from Scarlett Johannsen.
Terry Tao was clearly brilliant at mathematics as a child, though what nobody knew then was that he'd go on to be super-brilliant at mathematics as an adult, rather than being one of the many disappointing prodigies.
Anyway, I'm pretty smart, but I don't take career advice from anyone as smart as Terry Tao, just as I don't take dating advice from Scarlett Johannsen.
>And while school-level mathematics ain't the same as research-level mathematics
School mathematics isn't even math, it's a bad joke. I recommend Lockhart's Lament[1], I fully share his views. School 'math' destroyed every ounce of interest I had in math, and now, as a CS student, I still have a hard time getting rid of this attitude towards the subject even thought I know a lot better now.
[1]: https://www.maa.org/devlin/devlin_03_08.html
School mathematics isn't even math, it's a bad joke. I recommend Lockhart's Lament[1], I fully share his views. School 'math' destroyed every ounce of interest I had in math, and now, as a CS student, I still have a hard time getting rid of this attitude towards the subject even thought I know a lot better now.
[1]: https://www.maa.org/devlin/devlin_03_08.html
Something interesting about the article (and how analytic number theory is done in general).
As mentioned in the article, it was proved over 50 years ago that every odd N can be written as a sum of three primes, provided N is sufficiently large. The proof extends to cover five primes instead, if you like (indeed, write N = 2 + 2 + (N - 4), but in fact the five primes case is in fact easier).
Usually in such proofs the definition of "sufficiently large" is maybe like 10^(10^(10^(10^(10^10000)))) or something similarly absurd, possibly far worse, and it is typically difficult even to calculate such a fixed N, because then you can't use big-O estimates in your calculations. Sometimes you can't even compute such an N, it is "ineffective", see e.g. here for an example:
http://en.wikipedia.org/wiki/Siegel_zero
Tao's accomplishment, which was really cool, was to bring the value of N down to the level where you could settle the small N case by brute force.
As mentioned in the article, it was proved over 50 years ago that every odd N can be written as a sum of three primes, provided N is sufficiently large. The proof extends to cover five primes instead, if you like (indeed, write N = 2 + 2 + (N - 4), but in fact the five primes case is in fact easier).
Usually in such proofs the definition of "sufficiently large" is maybe like 10^(10^(10^(10^(10^10000)))) or something similarly absurd, possibly far worse, and it is typically difficult even to calculate such a fixed N, because then you can't use big-O estimates in your calculations. Sometimes you can't even compute such an N, it is "ineffective", see e.g. here for an example:
http://en.wikipedia.org/wiki/Siegel_zero
Tao's accomplishment, which was really cool, was to bring the value of N down to the level where you could settle the small N case by brute force.
The paper that is referenced in the article http://arxiv.org/pdf/1201.6656.pdf
Is there any practical application to this? I'm not one of those silly people who only cares about math if it can make a better lawnmower or whatever, but I'm just curious as to whether this could make certain algorithms easier or something, or if it's "just" a pure number theory result.
The conjecture has been verified already for large enough n that it can be applied without a real proof; a proof won’t give any new “real world” applications. Like with Fermat’s last theorem, the real interest in a proof is in the ideas that would be developed along the way. (For example, on the way to proving FLT Wiles established the modularity theorem for a class of elliptic curves, and his work was extended to a proof of the full modularity theorem in 2001.)
Tao mentions in the article that there may be applications in cryptography.
Did the commenter named julianpenrod get confused about causality (the article said "the weak version would follow if the strong were true"; he replied that it is "not correct in stating that, if the weak conjecture is proved, then the strong conjecture is prove[d]") or did the article get edited after he commented?
It's hard to say. In either case, the result makes julianpenrod look rather silly. More online publications should, as the New York Times does, note all corrections to an article since its original appearance in a paragraph at the bottom.
>Moreover, the larger the number, the more ways exist to split it into a sum of two other numbers—let alone three.
I don't think that makes sense. Every large odd number has exactly 0 ways to find its sum with two primes^, so they can't have more ways than small numbers (which also have 0, of course). Perhaps he was speaking about the strong conjecture, but then there would be no point to say 'let alone three', because three is only relevant for the weak conjecture.
Am I missing something obvious, or did the author just word that sentence/paragraph poorly?
^I am pretty sure this trivially follows from the fact that two odd numbers always produce an even number.
I don't think that makes sense. Every large odd number has exactly 0 ways to find its sum with two primes^, so they can't have more ways than small numbers (which also have 0, of course). Perhaps he was speaking about the strong conjecture, but then there would be no point to say 'let alone three', because three is only relevant for the weak conjecture.
Am I missing something obvious, or did the author just word that sentence/paragraph poorly?
^I am pretty sure this trivially follows from the fact that two odd numbers always produce an even number.
He says numbers, not primes, so it's trivially correct.
Though that doesn't directly imply the conjecture being true for sufficiently large numbers, since primes become increasingly sparse as one moves up the number line... it seems more like an appeal to intuition than a real argument, to me.
Though that doesn't directly imply the conjecture being true for sufficiently large numbers, since primes become increasingly sparse as one moves up the number line... it seems more like an appeal to intuition than a real argument, to me.
Just semantics. The writer presumably is considering only even numbers, for which it is generally true that you get more solutions as you increase the number. You are right in that it is worded non-rigorously, but in my opinion it is implied from the context.
This is news from february.
Unlike most programming, mathematical theorems usually survive for a bit longer, so a longer time frame is justifiable.
http://terrytao.wordpress.com/
is not to be missed. The "Career Advice" section of the blog
http://terrytao.wordpress.com/career-advice/
includes several classic articles, of which I especially like and recommend "Does one have to be a genius to do maths?"
http://terrytao.wordpress.com/career-advice/does-one-have-to...
(Tao's answer is no.)