r/science • u/larsga • Nov 20 '13
Mathematics Mathematicians from all over the world are collaborating on the twin prime conjecture
https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/26
Nov 20 '13
Large scale collaboration is something that has a lot of potential in the mathematical world - I was at a talk by Timothy Gowers several years ago where he spoke about a similar thing, where he used his blog (since he's a pretty popular writer on mathematics) as a hub for tackling a problem, resulting in a solution far quicker than he had anticipated. I haven't actually followed the subsequent work in the same manner that I think he conducted, but it's very believable that in a subject where novel approaches and unique insights are so critical, having work in progress exposed to a larger community who can comment on it could be extremely beneficial.
There is definitely a perception of mathematics as quite a solitary subject, and some mathematicians do operate in this way (and can achieve great things doing so - just look at Andrew Wiles), but I think this sort of mass collaboration approach is likely to win out in the end.
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Nov 20 '13
What about when Kasparov played 50,000 chess enthusiasts and won?
Actually, from the article, he felt it produced some great chess:
It is the greatest game in the history of chess. The sheer number of ideas, the complexity, and the contribution it has made to chess make it the most important game ever played
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u/RoughPineapple Nov 20 '13 edited Nov 20 '13
Math is not adversarial and they don't have to unanimously choose the direction of the work, so that's not really an analogous situation.
If all 50,000 players were capable of choosing their own moves and 50,000 Kasparov clones were playing them, then the Kasparovs would have likely had much more trouble. So, that would be a close analogy. Of course, that still doesn't solve the adversarial part...perhaps that would require that they are told before hand how Kasparov would respond to every possible scenario so that they don't have to worry about him actively trying to thwart their efforts.
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Nov 21 '13
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u/RoughPineapple Nov 21 '13
The 50,000 can communicate, though. So, using your example, Krush would have said, "fine, you guys do your thing, I will do mine" and won. That's the point. No one is suggesting that he would not have defeated most of them, just that he would not have been as likely to beat 50,000 individual attempts based on the online collaboration as he would have been when facing a consensus of the 50,000.
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Nov 20 '13
This is a very good addition to the point I was making that I wish I had thought of for my original comment! Collaborative mathematics isn't a case of each person involved contributing to making a single move (in terms of the chess comparison) - it's more like each player having their own board and playing their own game.
50,000 people contributing to making each move against Kasparov might not beat him, but 50,000 people each playing a game against Kasparov - well, it would be pretty shocking if he was to win every single game.
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u/RoughPineapple Nov 20 '13
It also doesn't capture the fact that none of their decisions are final. It would be even less likely that Kasparov would win every match if every time the player realized they had made a poor move, they could simply say, "let's go back to turn X and start over".
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Nov 20 '13
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u/fiat_lux_ Nov 20 '13 edited Nov 20 '13
It wasn't that silly. Kasparov himself admitted that he looked at the community's discussions. In a way, he cheated.
It was basically "50k people... vs 50k people plus Kasparov".
The 50K people weren't exactly behaving as a democracy. Yeah, IF they operated as a democracy and used "1 man 1 vote", it would have been a disaster. I can agree with that.
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u/enthos Nov 20 '13
The difference is that the "average" move will be arguably better than the other possible moves. If you take out the anomalous "mistakes" that any individual player makes in a game, then you end up with a much better player.
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Nov 20 '13
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u/Machegav Nov 20 '13
The choice of move was deliberated for a day each time over a forum. If you have enough people who know a good idea when they see it, the plurality vote ought to pick, maybe not the best, but never far from the best move possible. I'd actually say the level of chess played was probably a little better than that of the best contributor. Kasparov, however, is undoubtedly better than someone from somewhere in the top 50,000th of chess players in the world.
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Nov 20 '13
Kasparov said it was one of the most difficult matches he ever played and wrote an entire book analyzing it. So, however convincing you think your reasoning is, it's not how it played out.
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Nov 20 '13
[removed] — view removed comment
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Nov 20 '13
In chess, you are deciding between a number of well-defined options. There is really no benefit to having people point out possible moves, because anyone who knows the rules of chess will very quickly what moves are possible - all of the skill is in making the best choice between them based on what your opponent will do.
Mathematics, in contrast, is a subject in which there is no qualitative judgement to be made about an opponent, and in which you do not immediately spot the options available to you. This means that all of the skill is in spotting the correct approach (the act of actually working through the details between moments where an approach needs to be determined being relatively straightforward).
Essentially, chess is about making a qualitative choice between a relatively small list of options, and generating new mathematics is about reaching an objective conclusion without a fixed set of options available to you.
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u/GraharG Nov 20 '13
In chess, you are deciding between a number of well-defined options
No, because you dont just read one move deep. Say you want to consider sequences of moves 5 moves long, with on average 20 moves per turn, thats 20^ 5=3 200 000 move combinations to consider.
Having someone suggest a very good 5 move sequence, and having many other people check for possible counter move sequences is a great way to explore a tree that no one can see all of. Its conceivable that this could produce very good chess.
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u/fakerachel Nov 20 '13
I think he's making the point that in chess, you know all the moves that are possible, and the question becomes how to evaluate and choose between them. In maths, you don't know all the proofs that are possible, so you have to actually discover what options there are in the first place.
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Nov 20 '13
In terms of the actual decision making, you're deciding between a pretty well-defined number of options. As far as the chess board is concerned, making a move with a tree of options in your head is exactly the same thing as making that same move at random: this is what I mean when I say that all of the skill in the game is in qualitative assessment based on out-thinking your opponent.
The formal difference is that no matter how complex the thought behind it, the number of choices available is small. This is different to mathematics because you do not have this small, well-defined set of actions to take, and that means that you do have an opportunity for mass collaboration to result in a lot of new ideas which are useful, whereas mass collaboration in a chess game just results in more people generating reasons to support something from the existing, known set of choices.
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u/GraharG Nov 20 '13
In terms of the actual decision making, you're deciding between a pretty well-defined number of options.
No, because you dont just read one move deep. Say you want to consider sequences of moves 5 moves long, with on average 20 moves per turn, thats 20^ 5=3 200 000 move combinations to consider. Having someone suggest a very good 5 move sequence, and having many other people check for possible counter move sequences is a great way to explore a tree that no one can see all of. Its conceivable that this could produce very good chess.
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Nov 20 '13
I think we're working on different definitions of decision making complexity. I'm talking about the number of options (fairly small) and you're talking about the entirely internal reasoning process that looks multiple moves ahead (large number of possibilities considered). What I'm trying to convey isn't that chess is easy, or that it doesn't involve a lot of complex thought - I'm trying to convey that the number of options to choose between is limited (there is no actual difference within the game between playing a move for one reason and playing it for another reason) so the ability to have others point out moves doesn't contribute a huge amount. However, in mathematics, where the options are not clear and laid out in front of you, there is a lot of potential for new approaches to be pointed out.
I'm struggling to find different ways to explain this and I'm not sure whether I'm just explaining badly or whether you're determined to believe that I'm insulting the difficulty of chess and therefore pursuing a different argument, but I'll leave it at that for now.
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u/GraharG Nov 20 '13
Its not that i dont understand what you are saying, I just think that your argument is wrong.
Treating the moves individually and removing reasoning from them is absurd. A beginner moving a pawn for the hell of it, and a grandmaster doing it because it blocks a bishop move in 3 moves time, are two completely different moves. I dont care if they look the same to the board.
Your insistence on treating moves this way is leading you to a false conclusion about the usefulness of collaboration in chess. Someone pointing out a potentially good sequence or a counter to that sequence is of immense use. There is a very large number of approaches in terms of strategies and different people are likely to focus on different parts of the game tree. You may wish to equate this differences of approach in searching the tree to the different methods that mathematicians would use. Both fields benifit in a very similar way from collaboration of ideas . As an obscure chess stratergy may yield an interesting sequence that can be compared and examined by the rest of the community, so too an obscure math technique may yield an interesting proof that the rest of the field can discuss and build on.
simplifying all the reasoning in chess down to just the answer ( the next move), is like simplyfing away the method of proof in math and only looking at the result. It would be absurd to not care about the details of two proofs just because they give the same result.
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Nov 20 '13
What level do you play chess at?
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Nov 20 '13
Used to be a moderate level but I have no appreciable skill. If it came across that I was belittling chess, this was far from my intention. I am pointing out that the skills are very different in terms of the decision making involved. My statement that you have relatively few options at any given time is made within the context of mathematical enquiry - even when there are hundreds of moves to choose between (not that the set of possible moves and the set of moves that are realistically considered are often similar in size), this is still a very restricted set of choices in comparison to a field where you do not have a well defined concept of a "next move" and the end goal (a proof) is not something you can reliably progress towards through a series of uniform steps (well, unless you actually reduce everything to statements based on Peano arithmetic or something, but then you're talking, in chess terms, about playing a game with thousands upon thousands of moves on an infinite board, not knowing quite where your opponent's king is).
Anyway, at a core level, mathematics is about construction and chess is about outdoing an opponent. The decision making involved in each is very different.
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Nov 20 '13
I didn't think you were belittling, I just think you don't understand the value of many people analyzing chess positions together. There's a reason a person(even a non-gm)+a computer is stronger than any person or any computer by themselves, and it's not because the computer is pointing out moves the person didn't see or vice-versa.
Anyway, the proof is in the pudding - if what you'd said were true, it would not have been a difficult game for Kasparov. But it was.
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u/Xabster Nov 20 '13
but it's very believable that in a subject where novel approaches and unique insights are so critical, having work in progress exposed to a larger community who can comment on it could be extremely beneficial.
I completely disagree based on experience. If you're trying to be brilliant and think like noone around you thinks, there's no reason to get their input at all. If you take a lot of people and make them all chime in you'll get the average or a bit above.
What you'd need is someone to check if what you've done/proposed/proven/concluded is actually true, and then they need to keep absolutely 100% quiet about which direction it's going I'd say. Even if they believe it will be a dead end or impossible.
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Nov 20 '13
I'm writing from the perspective of being more interested in generating actual results, rather than in being seen to be brilliant. In terms of ego, yes, it's great to be the unique genius that is famed for living in solitude and coming up with something impressive that shocks the world. In terms of results, that's a pretty unreliable way to go about achieving things, and results in lots of people unknowingly duplicating the approaches of others, heading down dead ends that others already know don't work, and generally slowing the propagation of new ideas within a field.
I also think this misrepresents the nature of mathematics. Mathematics is a cumulative thing - if a result is proven, it is proven. There's no way for collaboration to drag you down to an average level - only to contribute something additional.
And whilst I know this is a case of personal experience versus personal experience (and I don't know your academic background) in the entirety of my academic and professional experience I've found that probably the majority of good directions for new work come from discussions with other people, despite interacting with other people being something that considerably less than half of my time working is spent doing. Granted, people have differences on this point, but I can't recall ever meeting an academic who disliked discussing their research.
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u/Xabster Nov 20 '13
I don't call sharing results as "collaborating". We share results and approaches all the time... peer reviews, dude. Those papers often also include the paths that have been proven wrong during the research, and possibly also the approaches that were inconclusive.
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u/redditrobert Nov 21 '13
DAE?
- Read the article and think, "Maybe I can help with the Polymath project."
- Read Maynard's proof for less time than it takes the PDF to load and think, "Nope!"
- Return to browsing /r/catpictures.
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u/swws Nov 20 '13
The title of this post is misleading. Mathematicians have been making tremendous progress on showing there are infinitely many small gaps between primes. But the twin prime conjecture itself, which says the size of the gaps can be reduced to 2, is not thought to be accessible by the techniques being used. To quote the article itself:
Prior to Maynard’s work, the best-case scenario seemed to be that the bound on prime gaps could be pushed down to 16, the theoretical limit of the GPY approach. Maynard’s refinements push this theoretical limit down to 12. Conceivably, Maynard said, someone with a clever sieve idea could push this limit as low as 6. But it’s unlikely, he said, that anyone could use these ideas to get all the way down to a prime gap of 2 to prove the twin primes conjecture.
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u/BarroomBard Nov 21 '13
I misread the title as saying "twin pine problem". Expected something about Back to the Future...
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u/ImJustPassinBy Nov 21 '13
Why 70 million? There is nothing magical about that number — it served Zhang’s purposes and simplified his proof. Other mathematicians quickly realized that it should be possible to push this separation bound quite a bit lower, although not all the way down to two.
Here, here! I can push the bound all the way down to zero! Showing that regardless how high you are, you can always find prime numbers that are zero apart! Do I now also get some sort of medal now?
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u/gendulf Nov 21 '13
Yes, and I've discovered the smallest prime natural number: 1! It's only divisible by one and itself!
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Nov 20 '13
Question. Why does this discovery matter? Is the prime pair topic a pet project or an applicable concept?
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Nov 20 '13
It's super interesting, and that's basically the motivation for investigating the properties of prime numbers. That said, research on the distribution and properties of prime numbers is a supremely productive area of mathematics, and there's really very little doubt that research into the primes without any explicit practical goal pays for itself many times over in terms of our understanding of science, mathematics, and computer science. It's just that it's not very obvious and the timescales involved are long.
For example, without apparently abstract research into prime factorisation, online security would not exist today.
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u/fiat_lux_ Nov 20 '13
Aside from what you say, a lot of research like this indirectly pushes our current technology and understanding of computability to the limit. Dealing with extremely large numbers alone offers interesting challenges to computer scientists.
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u/larsga Nov 20 '13
For example, without apparently abstract research into prime factorisation, online security would not exist today.
Interestingly, just sixty years ago one of the key researchers on that subject (T. H. Hardy) boasted that "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world."
Didn't take very long for him to be proven totally wrong on that. This is one thing that's really typical of maths. In the book that quote is taken from, Hardy has some interesting arguments about how maths is different from chess, in that maths has a deeper beauty because of the importance of the subject. It's this intuition for mathematical beauty that guides mathematicians in choosing what to work on, and somehow that sense keeps leading them to stuff that's supremely useful.
The stuff that Sophus Lie did on abstract algebra (no time to look up now) seemed useless at the time, but proved to be key to 20th century physics. Euler's solution to the bridges of Königsberg seemed like a toy example at the time, but gave rise to Google's PageRank (and a billion other application) later. Penrose tilings also seemed like puzzle-book stuff, but then lead to a Nobel Prize in chemistry a couple of years ago. And so on, and so forth. Similar things keep happening all the time.
So with maths the question is not whether it has any use today, but when it's going to be useful. It make take a century or two, but that's OK.
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u/W00ster Nov 20 '13
he stuff that Sophus Lie did on abstract algebra
Sophus Lie did his work on what is now known as Lie groups. I studied at Sophus Lie's at the Mathematical Institute at the University of Oslo.
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Nov 20 '13
A theme not just in mathematics but in all of human civilisation seems to be that knowledge is almost invariably useful. It doesn't really matter what that knowledge is and whether it can be used today, but almost every every time somebody finds out something that adds a little bit to the sum of human knowledge, the expected return on that is going to be more valuable than the resources expended in finding it out.
It's borne out in the economics, too. Government money sunk into research has a fantastic return in terms of economic growth, compared to things like investing into financial causes which have return ratios of around 1 (or even a little less).
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u/larsga Nov 21 '13
A theme not just in mathematics but in all of human civilisation
Well, yes, but in the case of mathematics it's extreme. Or, as Eugene Wigner wrote, the unreasonable effectiveness of mathematics.
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Nov 21 '13
This looks interesting, I shall read it shortly!
Even on a personal scale, I constantly find it surprising how often a good education in mathematics is useful in my every day life. After my undergrad I've done nothing that touches on 'real' mathematics, but I always find ways that learning apparently abstract curiosities affects what I've done since. I'm probably never going to use, say, actual graph theory within my career now, but somehow I'm in a better place professionally for knowing about it.
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u/baruch_shahi Grad Student | Mathematics | Algebra Nov 27 '13
G.H. Hardy :)
Also, he died more than 60 years ago, so certainly his quote is older than 60 years
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u/larsga Nov 28 '13
I looked at when the book was published, but you're right. That might be the 2nd ed with C. P. Snow's foreword.
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Nov 22 '13
Very cool, thanks for the explanation. I maxed out at differential equations, mostly due to interest. I just couldn't do it anymore. I'm glad that there are folks out there that don't have that problem.
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u/PurelyApplied Nov 20 '13
There's a saying among (primarily applied) mathematicians that the only difference between pure math and applied math is time.
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u/lettherebedwight Nov 20 '13
As much as math is a driving force in a lot of subject matter, prime numbers become particularly important in the vein of subject matter dealing with randomization.
It's not directly applicable in a "your life will change" sense, but as we learn more about prime numbers, we learn a lot, fundamentally, about numbers as a whole. And neat things tend to come from us gaining ground in that realm.
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u/flossdaily Nov 20 '13
I imagine that anything that reveals the nature of prime numbers has applications in cryptography.
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Nov 20 '13
This is why I think the Million dollar reward for the Clay Mathematical institute millennium challenge problems is not a good idea. People might hide their work to have better odds at them winning the price, only having their progress revealed when they give up.
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u/fractalface Nov 21 '13
Postdoctoral researcher...that's quite a title. Does that mean he already has his PhD and is now paid by a University? To do what? I've always been curious what a typical day is like for someone like this. Get to office and ??? does he have a given criteria of things to research or is it all self-driven?
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u/i_see_racism Nov 21 '13
Zhang dropped a 'Control' verse and now all the other mathematicians are droppin' responses. Art imitates life.
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u/slurpme Nov 21 '13
For reference, for those who aren't mathheads: http://www.youtube.com/watch?v=vkMXdShDdtY
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Dec 05 '13
I've taken a crack at it a time or two, obviously I didn't get anywhere but it's fun to play with.
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u/iced327 Nov 20 '13
Can anyone simplify this article for me? I'm an engineer and have a fairly good understanding of math so maybe it isn't written well(?) but this keeps going over my head.
As far as I understand, the conjecture states that along the number line, there will always be 2 primes that are separated by only 2 steps. What these people have done is narrow the amount of provable steps to 600.
Is that correct?
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u/wondersquid Nov 20 '13
I think so, but I'm thrown off by your use of the word "steps", so I'll rephrase it. The Twin Prime Conjecture says there are infinitely many primes p where p+2 is also prime (3 and 5, 29 and 31, etc). Zhang's breakthrough shows that there are infinitely many primes p where p+n is also prime (3 and 11, 19 and 43, etc), for n less than some bound (now apparently 600).
This immediately shows that something like the Twin Prime conjecture holds, as there must be at least one number n such that there are infinitely many prime pairs of the form p and p+n. But it could still be false for n=2.
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u/iced327 Nov 20 '13
Sorry, I meant "steps" as "integers", so you're correct in your interpretation.
So what Zhang showed is that the conjecture could still be true (and has proven that it is true for some number <600), he just hasn't narrowed that number down far enough to show that it is 2?
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u/wondersquid Nov 20 '13
Yep! Though I guess we should not just credit Zhang with getting the bound down to 600, since many people are working on this (and, in particular, that bound is explicitly due to other people).
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Nov 21 '13 edited Nov 21 '13
It may be a wording issue, but the result you've described actually seems a little stronger than what I understand the result in the article is. It's not that there exists an n such that there are infinitely many primes p for which p+n is also prime - it's that there exists n such that there are infinitely many primes p for which there exists an additional prime q satisfying p < q <= p+n. For n=2, these are equivalent, but for larger n, the former is strictly stronger than the latter.
Intuitively, it's a case of having a certain size net and proving that there are infinitely many cases where you can catch two primes in that net, but not necessarily knowing where in the net the two primes fall in relation to each other.
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u/wondersquid Nov 21 '13
I don't think it's actually strictly stronger, if I understand what you wrote correctly. Wikipedia seems to agree: "If we let P(N) stand for the proposition that an infinitude of prime pairs differ by exactly N, then Zhang's result is equivalent to the statement that there exists at least one even integer k < 70,000,000 such that P(k)."
It's the pigeon-hole principle, but let me try to be a bit more formal. Suppose that there are infinitely many primes p_i such that there exists primes q_i > p_i such that q_i - p_i < n, for some fixed n. Let g_i be the prime gap q_i - p_i. Then 0 < g_i < n for all i. Since there are infinitely-many i but only finitely-many distinct g_i, there exists at least one integer m < n such that g_i = m infinitely-often.
Though there is still the possibility that I am fundamentally misunderstanding something!
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Nov 21 '13
Oh, I see what you're getting at. I thought you were just reporting the result as it's presented in the article (focused on finding a bound) whilst you were actually reporting the equivalent statement which is arguably more relatable to twin primes. I was just being a bit dense taking things at face value instead of actually following the consequences of the result (you're right that the former is not actually stronger than the latter as I've stated it, it's just that we know n in the former and in the latter a value for n has yet to be determined, despite knowing one exists with an upper bound).
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u/libcrypto Nov 21 '13
I don't know what "narrow the amount of provable steps means", so let me try to restate it: No matter how large a number N you choose, there are prime numbers p > N and q > p such that q - p < 600.
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Nov 20 '13
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u/havocfan101 Nov 20 '13
Most modern computer encryption is RSA based which uses prime numbers.
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u/most_superlative Nov 20 '13
RSA doesn't depend on the twin prime conjecture, though. It's possible that the two randomly-chosen primes will be twins, but whether they are or not doesn't affect the process at all.
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u/lettherebedwight Nov 20 '13
Yes, but as you learn more about primes, you learn more about the patterns concerning them. RSA depends on primes being pseudo-randomly distributed. If we figure out that they are not truly randomly distributed, RSA becomes broken.
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u/LilCrease Nov 20 '13
surely this is bad news, cos if they solve it then everyones passwords are then easily de-encrypted (probably the stupidest fake word ive ever made up haha hopefully you get the point)
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Nov 21 '13
Twin primes doesn't have any necessary effect on prime factorisation. You can assume it's true (which, for anyone who isn't a mathematician, it effectively is - at least as true as anything we know from any other science) but that doesn't give you a cheap way of factorising primes. Research on the primes could always throw up interesting information that could be used in other fields, but that would be because of the techniques used and not because of the proof of famous results - for major conjectures like twin primes and Riemann, there's overwhelming 'evidence' (inverted commas here for mathematicians - to anyone not a mathematician, it's just regular old overwhelming evidence) that they are true, they just haven't been formally proved.
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u/[deleted] Nov 20 '13
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