**Bound Gaps Solved**

Last year, Yitang (Tom) Zhang published a paper in the *Annals of Mathematics* titled “Bounded Gaps Between Primes.” The abstract for the paper is simple enough for a non-mathematician to understand. It states that there are* infinitely many* pairs of consecutive prime numbers that are no more than 70,000,000 apart.

Don’t let the simplicity of the claim fool you: people have being trying to prove something like this for over 150 years.

At the time when Zhang submitted his result he held a “tenuous” temporary position in the mathematics department at the University of New Hampshire. As reported in Alec Wilkinson’s elegant *New Yorker* profile, before a friend set Zhang up with the New Hampshire position, he bounced around odd jobs, including a stint keeping the books at a Subway franchise.

Soon after his result was published, everything changed. His employer (wisely and with haste) made him a professor. He was invited to spend six months at the Institute for Advanced Study and accepted lecture invitations across the country. That same year, he was awarded a MacArthur “Genius” grant.

*What caught my attention about Zhang, however, was not the elegance of his result (which, as a lowly applied mathematician, I cannot come close to understanding) but the elegance of his work habits.*

**A Deep Life**

Zhang began to work on bound gaps (as it’s often called) in 2010. He didn’t find his “door” into the problem until the summer of 2012, when, standing in a friend’s backyard in Colorado, smoking a cigarette and watching for deer, he finally saw a way to penetrate its knotty interior.

In that two year period, between when Zhang started on the problem and had his first breakthrough, he spent much of his waking life simply thinking. Here’s Wilkinson’s description of Zhang’s “method”:

A few years ago, Zhang sold his car, because he didn’t really use it. He rents an apartment about four miles from campus and rides to and from his office with students on a school shuttle. He says that he sits on the bus and thinks. Seven days a week, he arrives at his office around eight or nine and stays until six or seven. The longest he has taken off from thinking is two weeks. Sometimes he wakes in the morning thinking of a math problem he had been considering when he fell asleep. Outside his office is a long corridor that he likes to walk up and down. Otherwise, he walks outside.

I don’t have a piece of pragmatic advice to extract from this story. Most people cannot spend two years thinking about the same thing for ten hours a day, and it wouldn’t help their professional life much if they did.

* But I think Zhang’s story highlights the beauty and potential of a mind left to do nothing but think without interruption at its highest capacity…*

(Photo from the John D. & Catherine T. MacArthur Foundation)

2 YEARS OF DEEP THINKING!!? :shocked: That’s a lot man! I even find it hard to concentrate for 2 hours straight….. My mind is always wandering while studying (that’s my work for now as am still a student)… Any tip or link for me on how to stay focused for long periods, on how to get into the “flow state”?

Right now, I am stuck on which career path to follow 🙁

Nice post BTW. Thanks for sharing (as always).

Kay. 🙂

Perseverance and Patience beats Intellect

So you’re saying that Zhang has no or little intellect?

He’s not saying that, he’s saying, “Genius is one percent inspiration, ninety nine percent perspiration.” Zhang has the intellect, but that isn’t why he succeded.

I’d put it differently: “Perseverance, patience and intellect beat intellect.”

But that’s not all. Zhang picked a good problem to work on, a problem that was important, and for which he felt he could develop a solution.

In the article it is mentioned that Zhang’s memory is enormously retentive, and that is a HUGE convenience. If you are struggling with a problem, and you can easily recall previous work that is related to your problem, the likelihood to develop a creative breakthrough is much greater.

Take me, for example. I know nothing about number theory, my knowledge is exhausted by citing that there are prime numbers and things like greatest common divisor and least common product.

If I worked on a proof of that theorem, the likelihood of finding one would be less than zero (defying Kolmogoroff’s axioms 😉 ) in the sense that not only would I not find one, but produce garbage instead.

Being able to recall theorems and reconstruct proofs many years after you read them is a giant advantage when trying to find solutions to new problems.

But, as Binnig and Rohrer (also Nobel Prize laureates) emphasize, you must not become “contaminated” by knowledge. I.e. you still have to be able to question your knowledge, no matter how certain it seems.

That’s a fine line to walk on. Just my 2 cents, as I’m most certainly not a genius.

I agree with what you say Brendon, and I’d also say that genius includes many more factors than just perspiration and inspiration. This is a great example though!

i don’t think he thinks for 10 hours a day, at least its not obvious from the excerpt, which only states that he thinks on the bus.

Perhaps the pragmatic advise would be to think how to solve your most pernicious problem everyday and focus on the biggest problem until you solve it.

Or focus everyday on your profession’s biggest problem. e.g. For a marketer it might be how to make the most sales with the least effort etc.

Btw Cal, are you still on the 9 to 5 fixed schedule you wrote about so many years ago? Where you totally shut off at 5 or 6 but focus like a laser when working ?

I get the impression from the full profile that he does really do little else but think. On the other hand, he’s quite eccentric, “even for a mathematician” (to use a quote from the piece).

I still do use fixed schedule productivity. My main exceptions, however, is that I write my weekly blog post at night after the kids go to bed (I lump blog post writing in the same category as reading an interesting book), and I will often load up a problem to think deeply about on and off during days off and weekends (though not always).

Kay, I’m like you. In fact, the ability to concentrate over long periods is a phenomenon which I do not understand, but I have learned that the ability to extend that period comes with practice and dedication. I had the same problem you described when I was a young student. But my stamina has grown with work, discipline, and dedication. Cal didn’t respond to you, but I suspect he may have additional advice that will help you learn to concentrate for longer periods. Also, have you noticed it is easier to concentrate longer on things you are deeply interested in compared to those things you are only marginally interested in? Wonder what Cal would say about that too?

Actually, the result is quite counter-intuitive.

One would imagine that as the numbers become infinitely large, the gaps between primes should also become infinitely large, because between them there are products of infinitely many numbers below, which should increase the chance that you “hit” many numbers between the primes.

Intuitive or not, the statement of the result remains quite simple…

Yes, it does… I was merely expressing surprise about the result, rather than bewilderment 🙂

For one reason or another I decided to go through the Little Schemer without pen and paper, contrary to the books advice, and I observed that understanding something consisted of reading at what I call ‘the semantic level’ where you’ve read the material at least 4 times within a day for example. At the 5th reading you find yourself just making sense and no longer parsing syntax.

What counts for all the time spent arriving at comprehension seems to be in my case repetition until familiarity is attained. Is there a way of reducing the syntax parsing to say 2 counts?

In Mr. Yitang Zhang’s case though it seems to be a matter of finding the right room in a long corridor of doors; it will take as long as it takes!

Cal, your interpretation of the result is actually not quite right. And A.I.’s intuition is right. Primes do become more sparse. What the result actually days is there are infinitely many primes that are less than 70,000,000 apart.

Without getting too technical, it is because you seem to confusing lim inf with the limit.

Your main point however stands.

I did confuse lim with lim inf! Good catch…

Thanks for clarifying! So while you can always find pairs that are less than 70,000,000 apart, there can be pairs that are 70,000,000,000 apart or any greater number.

Still, it is a remarkable theorem…

e.g. 70,000,000,000 of course

I’ve always thought his story wasn’t so much about his incredible work ethic, focus, and perseverance, but the real problem that this economy/society has with smart/well trained/highly educated people.

Einstein’s fate was quite similar. He couldn’t get a PhD position after finishing his Master’s degree. So he worked at the patent office, and only later was he awarded a PhD for his breakthrough papers.

The only difference was that this occured at a much younger age.

Hamming thought Einstein wasted his creativity during the second half of his life because he didn’t work on what Hamming called “good problems”.

Do not take this offensive. But he is quite an opposite example what you’re suggesting for years. See his passionate comments about math, see how he follows his passion ALTHOUGH he lost too much doing this across his career: Otherwise such a mathematician would not have to be working as a delivery worker or accountant. Surely he would be able to do some “low-hanging fruits” work, he has the ability. But he followed his passion, and here he is, in this blog. This is not a result of the persistency. Passion results in persistency. This is the result.

I agree with you, Mr. D. In order to do such a thing, the guy has to really be passionate about math. And gifted, tell by the way. Otherwise, he would not produce such impressive work. A lot of people are into math as their day job. But it requires more than that to stand out. Nature does not share our values about fairness. That’s why some individuals are more prominent at particular skills/tasks/activities. Whatever. Besides, passion indeed plays a fundamental role in performance. Apparently, people who love what they do are more likely to make more effort compared with those who don’t.

You haven’t understood his philosophy, passion builds on previous success. He chose a discipline, and chose to hone his skills to work on a good problem. The various skills he gained along the way build his “passion”, and confidence to solve bigger problems.

Easier said than done, Prof Zhang’s commitment to deep work, and getting good is what stands out.

Not quite. He has almost no papers before this one. So there is no previous success that he can build his “passion”. He was an ordinary calculus teacher. You all see this “deep” thing in hindsight, after he solved the problem (Heard hindsight bias?), not before. Before, he would seem you as a loser just because he followed his passion…

You are looking at publishing papers as the only way to have ‘success’ or evidence of honing his skills.

You ignore that started his interest in math at a very early age, and took the initiative to learn the basics himself, studied math/science/history to a point of mastery for China’s entrance exam to Peking University (the Ivy league of China), did very well there to be considered for a study abroad program at Purdue (was also introduced to Number theory, but was forced to pick another area of focus-forbidden fruit?), was very intense about his work till he graduated with a PhD. All the hours spent, DEEP hours, would surely produce some passion for his work, no?

Afterwards, he kept honing his skills by keeping up with latest advancements in his field (spent DEEP hours in the library), and kept attacking the same problem for a long time. His passion, I think, started long before he started teaching calculus – probably when he was scavenging for math books so he could be one of ‘great mathematicians’ he read about and the cultural revolution disapproved of; and so spent hours learning the basics without the guidance of a teacher.

It’s not impossible to be so good they can’t ignore you at something you are very passionate about.

The problem is the converse of the statement, namely that you just have to be passionate to be so good they can’t ignore you.

I see really no point in trying to argue that Zhang wasn’t passionate about number theory at first, or whether his passion or his competence developed first.

It really doesn’t matter what is developed first, as long as the process results in developing competence.

Just reread the article – notice he learned the basics of math himself, since it was hard to get a education during the cultural revolution in China. Math might have even been the ‘forbidden fruit’ that he couldn’t help but learn

https://www.youtube.com/watch?t=137&v=M_TOW1Cjb8Q

He says he was first inspired by “great mathematicians”

I read about Zhang in “Spektrum der Wissenschaft”, January 2014 edition. Interestingly, there is more detail about why he and his doctoral advisor parted ways.

Apparently, Zhang based his doctoral dissertation on previous work by Moh. But then Zhang discovered irreperable flaws in Moh’s work, and had to start over with an entirely new topic.

The article implies that the troubled relationship between the two contributed to Zhang’s inability to find an academic position, as the advisor can help the applicant or decide not to help.

Cal, maybe you know Hans Engler, who is also professor at Georgetown. He wrote the article for Spektrum I’m talking about. So maybe you can verify in person with him whether the story’s true that Zhang found errors in Moh’s previous work.

Update: Zhang’s original figure of 70,000,000 has been reduced to 252.

I find walking and thinking a great help when struggling with a knotty problem. IIRC, some genius did this routinely and walk with his hand on the wall so he didn’t have to think about routes. He would wander in and out of rooms if the door was open, etc.

While at the university, I would ride home on my bike. Talking with another professor who rode home along the same route with a different starting place, we found that about the same time/distance from work an insight would frequently occur.

As a programmer and writer, “Step away from the keyboard” is often good advice.

People severely underestimate the importance of focused thought.

I feel there’s an interesting connection between this story and the last (Asimov’s essay). Both suggest that aspiring to become a “professional creator” – like a tenured professor, or a corporate R&D scientist – might actually be an

obstacleto true creativity. From Asimov’s essay:“Probably more inhibiting than anything else is a feeling of responsibility. The great ideas of the ages have come from people who weren’t paid to have great ideas, but were paid to be teachers or patent clerks or petty officials, or were not paid at all. The great ideas came as side issues.To feel guilty because one has not earned one’s salary because one has not had a great idea is the surest way, it seems to me, of making it certain that no great idea will come in the next time either.”Academics frequently complain that the “publish or perish” system discourages truly innovative research, because it creates powerful incentives to pursue quantity over quality – that is, to churn out lots of modestly interesting publications for your C.V. instead of one or two brilliant ones.

Which is why Zhang’s story is interesting. He seems to have succeeded not simply

in spiteof lacking a professorship, butbecauseof it. By accepting a life of unprestigious and poorly-paid “grunt work” instead of chasing a more prestigious mathematical job, he was able to focus his energies on problems that truly mattered (like Einstein in the patent office).The lesson here seems to be that linking your creative work to your livelihood can actually be counterproductive, or at least highly dangerous. I’m curious what Cal thinks, as a professor himself.

Tom,

this is quite an intriguing point. It reminds me of Feynman, who did write about this topic.

After the war, he was hired by Cornell first, I think, and he felt burnt out and couldn’t produce any results. Feynman felt extremely guilty about that and thought: “Those people think I am a genius, and they don’t know that they hired a burnt-out man.”

Later, his supervisor talked to him and said they were happy with his work. And he added: “Whether you will make great contributions to physics or not, no one can know – it is our risk to hire someone as a professor who turns out to be a man of few ideas.”

(It’s debatable whether this attitude still exists in today’s academia.)

Feynman went on to think that he was not responsible of the expectations others had of him.

When he was offered a position at Princeton’s Institute for Advanced Studies – he declined, and he even thought it would have been crazy to accept it.

He argued that in times of intellectual stagnation, teaching students physics would still give him the feeling of being useful, of making meaningful contributions.

So this point may be worth considering.

Cal,

When working on a problem, especially a large one with multiple aspects and branches that requires deep thought over many weeks/months, how do you keep thinking contained to certain hours of the day and prevent the intrusion of thoughts about that problem during hours dedicated to other work?

I find it especially difficult when I have a thought about the problem that I’m afraid I’ll lose if I don’t immediately jot it down, but then I don’t get the work I had planned to do during that time done which has other negative repercussions.

Any thoughts on this subject would be very appreciated.

Thanks.

Cal,

On this same subject you should check Ruslan Medzhitov (one prominent immunologist-arguably a must Nobel winner by his peers) that said:

“I was basically spending all of my time reading and thinking,” (The university was broke, and Medzhitov didn’t have the equipment he needed to run experiments.)

https://mosaicscience.com/story/why-do-we-have-allergies

That is interesting, we generally are always in a rush to figure things out. The more I read about things like this, things that point to us taking the time to understand something deeply and work on it, the more I understand that outside of blind luck, there is no other way the big problems are handled.

The hard part of me is making myself think :-). We live in a world where we are rewarded for engaging in interruption. It has boiled down into my routine. Lately I have put it into my calendar and taking myself to a coffee shop, but there is a part of me that thinks this isn’t deep enough. I will find out more with experimentation though, and that is the first step.

Two observations from this story:

(1) the author of the New Yorker piece describes Zhang as speaking from “diffidence” during their interview, while a Harvard Professor said Zhang’s seemed “brave and independent” when lecturing on his result. This reminds of me of certain leaders whose sense of self resides not in generic egotism, but flows from an innate mastery and commitment to a problem or effort.

(2) after a couple of years of intense thinking on bounded gaps, a door to the problem opened to Zhang only when he was on break in the Colorado mountains. This appears to be an instance of how breakthroughs occur as explained by Fritjof Capra in The Tao of Physics:

“Rational knowledge and rational activities certainly constitute the major part of scientific research, but are not all there is to it. The rational part of research would, in fact, be useless if it were not complemented by the intuition that gives scientists new insights and makes them creative. These insights tend to come suddenly and, characteristically, not when sitting at a desk working out the equations, but when relaxing, in the bath, during a walk in the woods, on the beach, etc. During these periods of relaxation after concentrated intellectual activity, the intuitive mind seems to take over and can produce the sudden clarifying insights which give so much joy and delight to scientific research.”

Hi,

Do you think everybody has the ability to do well in mathematics? You focus a lot on habits (which is important, I think, especially for somebody like me who has had to seriously work at improving his study behaviours), but this emphasis has left me wondering whether talent and intelligence (measured by IQ or whatever) are real things that affect achievement. If so, do you think a person is basically doomed if they don’t possess the aptitude needed to be successful at some of these more technical disciplines?