Also, remember that most people on Earth aren't in academia, but most people still think. So the null hypothesis is that most innovations won't come from academia. But a huge amount of them do anyway, including even your counterexample, who--despite his economic struggles--was still a lecturer at a university. You could make a similar argument for young people.
Our university system and our cultural obsession with youth deserve plenty of criticism, but the accusation that they're not making important discoveries isn't one of them, and it's totally unwarranted from this article.
The article does have a link to Maynard's preprint: http://arxiv.org/pdf/1311.4600v1.pdf
Here is an expert overview of the situation: http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf
"In April 2013, Yitang Zhang proved the existence of a nite bound B such that there are in nitely many pairs of distinct primes which di er by no more than B. This is a massive breakthrough, makes the twin prime conjecture look highly plausible (which can be re-interpreted as the conjecture that one can take B 2) and his work helps us to better understand other delicate questions about prime numbers that had previously seemed intractable. The original purpose of this talk was to discuss Zhang's extraordinary work, putting it in its context in analytic number theory, and to sketch a proof of his theorem. Zhang had even proved the result with B 70 000 000. Moreover, a co-operative team, polymath8, collaborating only on-line, had been able to lower the value of B to 4680. Not only had they been more careful in several dicult arguments in Zhang's original paper, they had also developed Zhang's techniques to be both more powerful and to allow a much simpler proof. Indeed the proof of Zhang's Theorem, that will be given in the write-up of this talk, is based on these developments. In November, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to 600, by a substantially easier method. Both Maynard, and Terry Tao who had independently developed the same idea, were able to extend their proofs to show that for any given integer m ¥ 1 there exists a bound Bm such that there are in nitely many intervals of length Bm containing at least m distinct primes. We will also prove this much stronger result herein, even showing that one can take Bm e8m􀀀5. If these techniques could be pushed to their limit then we would obtain B( B2) 12, so new ideas are still needed to have a feasible plan for proving the twin prime conjecture. The article will be split into two parts. The rst half, which appears here, we will introduce the work of Zhang, Polymath8, Maynard and Tao, and explain their arguments that allow them to prove their spectacular results. As we will discuss, Zhang's main novel contribution is an estimate for primes in relatively short arithmetic progressions. The second half of this article sketches a proof of this result; the Bulletin article will contain full details of this extraordinary work."
EDIT: it appears that copying and pasting from the PDF is problematic and please read the linked PDF if interested.
Maynard has done some stellar work to be sure, but he built off of Zhang's breakthrough. Maynard's result is one of a sequence of very quick findings all happening in just months after Zhang's work became public, all make possible by that work, and relatively easy, as can be seen from their number and rapid pace.
I have not read all the papers here, but my guess is that Zhang's result is hugely more impressive.
That was actually not my understand from the article (I haven't read the papers either). What I got was that there was an earlier paper ("GPY") that Zhang based his work on. Maynard based his work on a flawed predecessor of GPY. Thus he did not base his work on Zhang's. In fact, the article seems to indicate that further progress could be made by combining their approaches, but it certainly seems from my reading that they're independent. Of course, that could all be wrong, but it is what the article indicates.
So it seems like it is actually a coincidence, though they were both building off of the same previous work (GPY) so maybe it's not that surprising.
The article makes it very clear that his work is completely independent from Zhang's, which is why there is expectation that they can be used together to get bounds into the 10s, and maybe even below that limit.
I take issue with that.
People think, but they don't think far beyond what is directly accessible to experience, so I wouldn't call it academic thinking.
How sure are you that academia is different from the general run of mankind in this regard?
Time and again, phenomenographic research shows that students simply don't do this. This is the bane of any academic teacher's life. Students are demonstrably smart enough to understand the material, but they just don't engage with the material at a deep enough level for it to make any real impression. They don't want to think.
For instance, Physics students become adept at manipulating kinematics equations, but on further questioning it becomes clear that they still think like Aristotle did. Or Computer Science students struggle mightily to memorize all the syntax and semantics of Java, but their learning is so shallow they still can't solve the FizzBuzz problem.
Why don't people think? For the same reason they don't exercise, if they don't have to. It's hard!
So, yes, I claim that the majority of people don't think. Are there academics who don't think much? A few don't. Many don't think much about matters outside their specialism. And academics receive precious little training in overcoming the cognitive biases that hinder clear thinking for most people. But I still think that academics are more likely to be deep thinkers than most, just like sportspeople are more likely to be enthusiastic exercisers than most.
Also, many professionals are called upon to solve problems that require this same kind of "thinking". Academics are not the only ones wrestling with difficult problems.
You're quite right. I can't possibly give a fair, nuanced description of the issue in a couple of paragraphs, and I'm nowhere near qualified to give the definitive account of the problem. Still, you're right that I do place more blame on the learners than most. Maybe people shy away from these ideas because they're dangerously close to some rather sinister Brave New World-style educational theories.
In teaching, I think we have a chicken-and-egg problem. We want students to learn both the underlying abstraction and the surface details. But the weaker students are resistant to learning the underlying abstraction, so we drop that and drill them harder on the surface details instead.
I see no easy solutions to this problem.
> Also, many professionals are called upon to solve problems that require this same kind of "thinking". Academics are not the only ones wrestling with difficult problems.
Sure. Computer programmers are an obvious case. But a perusal of the Daily WTF's archives (http://thedailywtf.com/), or the fact that many professional programmers with long careers still can't pass the FizzBuzz test, shows you that there is still a problem.
Care to elaborate?
My parent comment says that, outside academia, most people's thoughts aren't significant enough to merit the term "thinking". I'm saying that, if you want to consider that that's the case outside academia, you should realize that it's also the case inside academia.
Personally, I would be happy to dignify most people's thoughts with the term "thinking".
My differentiation is that number theorists are looking for better sieves all the time. Zhang's work was a breakthrough in bringing the prime gap from infinity to a finite number. It is not surprising that better sieves could bring the gap a lot closer.
What I'm trying to get at is that there's two ways you can think of academia: 1) roughly the set of people in traditional institutions, and 2) a basically unorganized set of people that happen to come together through working on related problems, in related ways.
2) is arguably the purer view of academia, and in that sense, it's as strong as ever.
I agree with you that there's deep, serious problems with 1), and they have to do with Pournelle's law, (in the US) with Bayh-Dole, and a litany of other hard, nasty problems of social life. But, even in that case: is that any worse than other cultures?
The only example of out-of-academia, in the real sense, was Ramanujan[1] who was a sort of pure genius out of nowhere and was able to pull out formulas like this [2]. He wasn't constrict by the rigorous need for proof that Europeans were looking in order to accept any discovery for being true.
De Sautoy[3] states that Ramanujan non-formal (western-style) education kept him away from European mathematical achievements and but also away from European fears of inability to attack specific problems (i.e. like Riemann's Hypothesis for example[4]) and that was a good thing.
[1] http://en.wikipedia.org/wiki/Srinivasa_Ramanujan [2] http://bit.ly/1dh0E5w (Google image) [3] http://plus.maths.org/content/music-primes [4] http://en.wikipedia.org/wiki/Riemann_hypothesis
This depends heavily on what you mean. Of course there are areas of math requiring ten (or more) years of study before you can work in them. However...
- In theory, all high school graduates have studied math for at least twelve years.
- There are many, many problems on "the bleeding edge" of mathematics, in the sense that no one can solve them, but which can be stated in terms an algebra student would have no trouble understanding. Further, there are many mathematical techniques which are "advanced" in the sense that they are never taught in school, but are "simple" in the sense that the algebra student I mentioned earlier would have no trouble following a proof using them. If you're interested, try reading through the archives of Tanya Khovanova's blog; she has plenty of examples of IMO problems and special Soviet math exam problems for Jews.
The only times I can think of where this is not the case is after a major, yet easy to understand, discovery makes a lot of formerly hard problem easy, or a field of mathematics that is either young enough, or uninteresting enough that it has not yet had significant time devoted to it.
It's stories like these that make me jump with joy inside every time I think about the wonder that is the internet and the possibilities that creates for us as a human race.
The possibility that exists by having things like free 2,000 book encylcopedias (see http://en.wikipedia.org/wiki/Wikipedia%3ASize_of_Wikipedia#H...) at the fingertips of nearly every human on earth is just mind boggling.
I won't be surprised if Joe Average discovers the cure for Cancer tomorrow. Or AIDS. Or our fossil fuel addiction.
In fact, I expect those things to happen sooner than most of us might think.
While access to freely available knowledge is increasing, this knowledge is mostly broad, not deep. So we can have information on a lot of topics, but these topics aren't covered in very much depth.
Also keep in mind that Zhang was very much within the fold of academia when he discovered the 70 million upper bound on the separation of primes.
This has bothered me quite a bit. I want to go to wikipedia and be able to find, to use Feynman's term, a map of the cat. I don't understand why it's not there.
Tenure exists to escape people from the "3 papers a year" BS which means you get a neverending stream of minor refinements and very little thorough investigation.
It seems like more and more of the really big discoveries are being made by people who go off and work really hard on their own.
I vaguely recall a claim that the greatest discoveries aren't being made by people in their twenties any more just because math is so huge you need ten years of study to get to the bleeding edge.
Anyway, it seems to say something about academia.
Edit: Just to be clear I neither deny that academia can achieve great things nor do I claim that young people can't do similarly. Certainly, anyone halfway near academia probably see many flaws but that's for another post.
[1] http://en.wikipedia.org/wiki/Yitang_Zhang