I don't get it. The article talks not just about Zhang, but also about James Maynard, who is a postdoc and looks young, and is part of what you might call standard academia. His result is independent of Zhang's and is orders of magnitude stronger and is much easier to understand. Comes off pretty good for academia, really.
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 talks not just about Zhang, but also about James Maynard, who is a postdoc and looks young, and is part of what you might call standard academia. His result is independent of Zhang's and is orders of magnitude stronger and is much easier to understand.
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.
> Maynard has done some stellar work to be sure, but he built off of Zhang's breakthrough.
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.
Actually at the bottom of page 2 of Maynard's preprint he states:
"We emphasize that the above result does not incorporate any of the technology used by Zhang to establish the existence of bounded gaps between primes."
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.
That is only a tactic move. it is Zhang's work inspired him to look for a quick better alternative. Zhang showed " there is gold! " And then Maynard ran towards that.
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.
The difference between academic teaching and practical teaching is that academic teaching involves "second order knowledge". I.e. knowledge that is not accessible to direct experience. We can only acquire it through study, reflection, and testing and retesting our mental models.
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.
I think it's unfair to place the blame squarely on the student. Even in college, the academic system emphasizes memorization and surface-level application (knowing how to "use a formula") over the kind of "thinking" you're describing.
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.
> I think it's unfair to place the blame squarely on the student. Even in college, the academic system emphasizes memorization and surface-level application (knowing how to "use a formula") over the kind of "thinking" you're describing.
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.
Wait, what? Academia is a pretty broad brush stroke, if you know anything about being an academic, and you appear to be making that brush broader still.
I'm not sure what you're asking me; the whole point of my comment is that academia is a pretty broad brush stroke.
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".
Are you saying that Maynard independently tackled P(2) or that he independently came up with a better sieve? My take from reading the article is the latter.
It is my understanding from reading Maynard's preprint that he claims independence from Terrence Tao and the Polymath8 project. He doesn't claim independence from Zhang and neither should he, even if he uses a different approach. His result may well turn out to be important or more important but Zhang's result is clearly earlier and well-known.
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.
Okay it appears that you and several others have not read the preprint paper. The article's wording is a bit confusing but if you think carefully you'd realize that a later work can not be independent of an earlier well known result.
"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 e8m5.
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.
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.