It's ok to try tackling a big problem, as long as you know when pause for a while and come back to it later.
What if Ramanujan had been told by Hardy to stay home and stop working on big things that he hadn't had sufficient experience in? While there will ever only be one Ramanujan, he wouldn't have developed into what he did had he restricted himself. He made a lot of mistakes, and that didn't kill his career nor did it stop him from producing many great works.
Similarly, Bezos obsessed about a simple online bookstore website. Jobs obsessed over details for a small number of devices. Torvalds obsessed over an operating system he wrote. DHH obsessed over a web application framework open source project for his company. If you obsess over something, it has a much better chance for success.
If you believe in it, and you see a path to it, don't give up.
Ramanujan produced a constant stream of results and wasn't at all obsessed with one big problem to solve.
Terrance Tao is talking about the "hide in the attic for 10 years working on one problem" attitude. And this he warns against.
As for business development, I'm pretty sure Jobs/Bezos etc actually produced something in fairly short order, ie. the MVP came out quickly. This approach of release early, release often seems to be exactly what Tao would endorse.
Yes, Ramanujan just floated around dealing with whatever number theories inspired him, sometimes in a nearly mystical way so he's not a good example.
Andrew Wiles is a better example, since he hit in plain sight working on Fermat's Last Theorem.
"He dedicated all of his research time to this problem for over six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife."
Exactly. He proves the author's very well. Wiles was tenured, had published in the field already, and made sure that he could continue publishing while working on the Big Problem.
Tenured at Princeton, had already solved some fairly big problems in the field, and it helped that FLT had recently been "reduced" to proving a conjecture about elliptic curves which were already very much in Wiles's wheelhouse.
Not to take away from his achievement, but I think it is not stressed enough how much the proof depends on work from previous decades that a priori had nothing to do with Fermat.
Note that with Andrew Wiles all the secrecy and seclusion almost exploded on his face. When he first came public with the Fermat proof someone found a flaw in it within 2 months.
It took another 2 years (he was ready to give up by then) to finally come up with a "fix" for the flaw that had been found.
So in the end he was only successful after exposing it to the public for scrutiny and breaking down the problem into something smaller.
Sure but if Wiles hadn't been secret, he'd be making progress reports all along and plausibly the person who made the fix would be the one who credited with the proof.
It's kind of shame that the quest for credit works this way, secrecy has all sorts of costs but in this instance it clearly had benefits too.
Context is important here I think. This is part of a series of posts containing career advice for mathematicians at various stages ([1]), and not life advice in general. So I don't believe Tao was addressing cases like Torvalds etc.
This post appears in the "Graduate level" section. At this point many open problems become accessible, while at the same time many grad students (emerging from being a big fish in a little pond) are looking to make a name for themselves. So a common trap is for people at this point to obsess over famous problems, to the exclusion of tackling less exciting but more tractable problems, and Tao is warning against this.
I completely agree. Digging deep in the weeds and pulling yourself out over an over again is how most impactful problems get solved. It's time consuming and the truest test of patience. It's very fulfilling to live this way, even if it seems like world is moving at a much faster pace
I think the way I would phrase the advice is; If you are inspired or challenged by the "big" problems keep an eye on them but work on other things that are not directly or obviously related but may be related. What Tao warns against is a direct assault on a big problem or theory as a career path or plan. Like expeditions up Mt. Everest, that path is littered with attempts that reached an impenetrable barrier and had to retreat or perish on the mountain. If you think you've solved a big problem then your method should also explain the failure of all the prior assaults to be credible.
That said, don't give up on your dream of solving big problems, someone has to do it and like Mt. Everest, every field has its Ark of the Convenant or Holy Grail that challenges and inspires its members. If you are a genius all bets are off and don't listen to me or Tao ;-)
Tao follows this advice well himself. He seems to be working on the millennium problem of Navier-Stokes, but works on many other things simultaneously and published partial results.
There is a state of flow when the mind is completely preoccupied with an idea. Where the idea itself takes over the mind at all time and in every place and context.
Although context switches in good measure are useful to creative thought, there's something to be said about a singular focus.
Tao's article reminds me Poincaré's "Science and Method" book (1908) where it is explained how to split/address big tasks/problems over long periods of time.
> While it is true that several major problems have been solved, and several important theories introduced, by precisely such an obsessive approach, this has only worked out well when the mathematician involved had a proven track record of reliably producing significant papers in the area already; and had a secure career (e.g. a tenured position).
Looks like about 4 years after Tao posted this essay, (St.) John the Commenter shows up one Christmas Eve not long ago:
"24 December, 2011 at 8:02 pm
John
I have always “intuitively” known to follow this excellent advice.
I use this kind of advice to know when to “give up” (temporarily) on a given approach to a particular problem and move on to other things. It is a non-obvious battle deciding whether one is wasting time repeating the same attack or being impatient by not pursuing a given approach long enough. Furthermore, an outside objective observer cannot always tell – unless they are experienced in one’s particular area of math. Same applies to any field."
Continuing to scroll downward through the chronological parent comments, John encounters a comment Larry had posted to Tao not long before John had gotten there:
"3 October, 2011 at 4:37 pm
Larry Freeman
Thank you very much for this well-written essay. I am guilty of the very thing that you warn against (I find myself working exclusively on impossible math problems: collatz conjecture, twin primes, Legendre conjecture) and I agree with each of your points.
The only thing that keeps me chasing these unbelievably difficult problems is the humility I feel when I realize:
(1) I’ve made no progress at all
(2) Any sign of progress is more often than not a sign of a mistake in my assumptions.
(3) I am learning number theory and enjoying it.
I am too old to make real progress in mathematics (I’m 40+) but working on the famous unsolved problems gives me a great respect for the brilliant mathematicians who have made progress in the past and helps me to acknowledge my own limitations.
Regards,
-Larry"
This might have inspired John's following parent comment a few minutes after his first comment:
"24 December, 2011 at 8:09 pm
John
I meant to add: one of the great things about math, and one of the reasons I chose this field, is that one can do great math into old age, unlike fields like sports or ballet, where one has a very limited time that one can do those activities, because one’s body ages.
Thus, in math, one can always keep building and expanding on what one already has done and learned and inserting new research that comes along into one’s work.
I actually formally entered the math field relatively “late” (graduate school, after a period of work in my undergraduate major). I entered math because I needed to solve some extremely difficult applied math problems first before returning to work in my undergraduate major. I intended my foray into math to be just “temporary”, because I (in my naivete) expected to “quickly” solve the major applied math problems, say, in 4-5 years, and then “pop out” of math, and then pop back into work, applying all these wonderful results that I had proved for my math PhD. It’s been 23 years since I “entered math”, and I still have not popped out again, because the problem are just so overwhelmingly difficult, far more than in any other field."
There is some chance that John didn't really mean to add all these personal details until after he had read Larry's comment.
Less than 20 minutes later, my good friend Anonymous, surfing the entire internet as he usually does regardless of whether it is Christmas Eve, randomly stumbles upon John's comment, when John is only the most recent among a handful of commenters to participate in years:
"24 December, 2011 at 8:27 pm
Anonymous
@John, would you be kind enough to share the applied math problem that got you sucked into math in the first place? I am at the verge of leaving math and I feel a sense of relief after having solved several of the open problems in my field; one of them is still pending but I believe the ideas are in place. So I would like to put my experience in perspective of others."
Is it just my imagination, or could this be Tao? And could he really have been on the verge of leaving math before 2012?
Anyway, less than 24 hours later, on Christmas Day, John proceeds to completely spill the beans to the anonymous perspective seeker:
"25 December, 2011 at 1:22 pm
John
Yes, Anonymous. Ever since 7th grade, when I openly declared that I wanted to become an “organic synthesist” when I grew up, my dream was to become a mad scientist! Since then, I’ve learned of a new hope for a path to that dream: nanotechnology. And, my years in chemical engineering lab at school, at work in a chemical lab, tutoring others in math applied to science (linear optimization, statistics, probability), and my lab courses in biotechnology all point to one burning conclusion: the technical problems of moving atoms around in nanotechnology to where you want them to be won’t be solved until we have complete solutions and understanding of the nonlinear partial differential equations (e.g. Schrodinger) that govern those atoms. I am convinced now more than ever before in my life that this is true, as a result of my experiences and interations."
Now that's problem-solving ambition.
This is where I could write paragraphs if not thousands of pages on the subject, I'll spare you.
The quotes speak for themselves.
Over a lifetime, having some familiarity with these technologies and what is required to achieve breakthroughs, I would have to say that if I was a capitalist I would find it most worthwhile to invest in "John" to achieve the kind of partnership which could make as much money as anyone would like.
Plus, maybe John himself provided some inspiration for Tao to remain committed to math leadership ever since.
I think the emphasis of the article is on the "obsess" verb, not on the "big" adjective. Big problems and theories are a fundamental driver but obsessing is not productive.
Hardly anyone uses "one" like that in spoken language.
To take the first sentence of the post, "There is a particularly dangerous occupational hazard in this subject: one can become focused, to the exclusion of other mathematical activity (and in extreme cases, on non-mathematical activity also), on a single really difficult problem in a field (or on some grand unifying theory) before one is really ready (both in terms of mathematical preparation, and also in terms of one’s career) to devote so much of one’s research time to such a project."
that might more commonly be rendered as "There is a particularly dangerous occupational hazard in this subject: becoming focused, to the exclusion of other mathematical activity (and in extreme cases, on non-mathematical activity also), on a single really difficult problem in a field (or on some grand unifying theory) before you're really ready (both in terms of mathematical preparation, and also in terms of your career) to devote so much of your research time to such a project." There's no doubt other ways people would put it. Why is such an alternative "atrocious"?
> Hardly anyone uses "one" like that in spoken language.
It's common in written-language [0][1] especially in a piece of writing whose primary aim is didactic.
> There is a particularly dangerous occupational hazard in this subject: becoming focused,
Actually it should be the infinitive not the adjective-form: to become focused. Elsewhere all you've done is replace the third person pronoun with the second-person e.g.
> before one is really ready
with
> before you're really ready
There's nothing wrong with using the second-person - but criticizing the author based on their use of the third-person pronoun one is only reflective of the lack of education on the part of the commenter, the shout out at political correctness and various alternative movements was totally unnecessary as well.
I often feel that 'we' is a better substitute than 'you' anyway, assuming a desire not to personalise. People seem frequently to say ridiculous things such as "you don't - I mean general-you, not you-you - ...", why do we (<-) go to such lengths to avoid the perfectly fine and much more succinct 'one'?
'We' has similar problems, but at least even if taken specifically and personally it includes oneself; thereby it's less likely to be taken as specific criticism or personal insult, etc.
I was talking about how people typically communicate in spoken language.
> There's nothing wrong with using the second-person - but criticizing the author based one
I didn't criticize the author at all.
And I'd ask you the same thing as I asked the person I was originally replying to... do you think there's something wrong with the alternatives to "one"?
What if Ramanujan had been told by Hardy to stay home and stop working on big things that he hadn't had sufficient experience in? While there will ever only be one Ramanujan, he wouldn't have developed into what he did had he restricted himself. He made a lot of mistakes, and that didn't kill his career nor did it stop him from producing many great works.
Similarly, Bezos obsessed about a simple online bookstore website. Jobs obsessed over details for a small number of devices. Torvalds obsessed over an operating system he wrote. DHH obsessed over a web application framework open source project for his company. If you obsess over something, it has a much better chance for success.
If you believe in it, and you see a path to it, don't give up.