A Persistent Answer
Ben Orlin is a math teacher who publishes the clever essay blog, Math with Bad Drawings. Last year, Orlin had the opportunity, during a press conference at the Heidelberg Laureate Forum, to ask a question of Andrew Wiles, the Princeton Professor (now at Oxford) who in 1994 finally solved Fermat’s Last Theorem.
As Orlin reports on his blog, he asked the following:
“You’ve been able to speak to an unusually wide audience for a research mathematician. What are some of the themes you’ve tried to emphasize when talking to a broader public?”
Wiles’s answer, according to Orlin, can be summarized in six words: “Accepting the state of being stuck.”
As Wiles elaborated, research mathematics unfolds as follows:
“You absorb everything about the problem. You think about it a great deal—all the techniques that are used for these things. [But] usually, it needs something else. So, you get stuck.”
At this point, he explains, “you have to stop…let your mind relax a bit…[while] your subconscious is making connections.”
Then you “start again.” Day after day. Week after week. Until, one day:
“You find this thing…Suddenly you see the beauty of this landscape…[before,] when it’s still some kind of conjecture, it seems really far away…[but now] it’s like your eyes are open.”
Wiles admitted that the enemy he fights against most is “the kind of message put out by, for example, the film Good Will Hunting.”
And, in particular, the idea that for some people math comes easy (Matt Damon glancing at the chalkboard, and then dashing out the solution to the impossible problem), and for all others it’s hopeless.
The reality, as Wiles knows, is that math is just plain hard. Regardless of who you are. But it’s also amazingly rewarding if you’re used to the feeling of persisting even when you have no idea about how best to move forward.
A Good Response
I liked this answer (and Orlin’s commentary on it) for two reasons.
The first is personal. As a theoretical computer scientist I spend a lot of my professional life stuck on math problems. It’s hard to explain to outsiders what this is like, and Wiles’s response does a good job of capturing the competing forces of frustration and joy that come from tackling such things on a regular basis.
The second has to do with my recent post about how we lack a good vocabulary for describing the varied cognitive efforts that comprise deep work. Wiles’s answer is a good step toward filling in some of those blanks.
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For more on Andrew Wiles’s attack on Fermat’s Last Theorem, see Simon Singh’s popular book, Fermat’s Enigma. For a more raw and technical treatment of what it’s like to do Fields Medal-caliber math, see the more recent Birth of a Theorem. The image above is taken from Wile’s proof, as it appeared in the Annals of Mathematics.
(Hat tip: Amin)
Obviously to some people math is hard while to some it comes very easy. But whichever way one look at it, it is the approach that one takes that will determine the result.
Wiles doesn’t say that everyone is equally capable of doing math. His point is that even if you’re naturally good at math, like the guy in good will hunting, you will not ever come to a chalk board and solve Fermat’s Last Theorem in a matter of minutes. Math is difficult for everyone’s mind, even the naturals. So you shouldn’t feel hopeless at math and wait around for some savior to solve Fermat’s Last Theorem because there were many “average” mathematicians who contributed to the field in profound ways because they took the right approach: Knowing when to move on from and come back to a problem, and “accepting the state of being stuck.” Sometimes the turtle wins the race because he took the shorter path.
But that’s still wrong, because there’s nothing innately special about the people for whom math comes easy. Those people generally happened to have spent more time and effort when they were younger thinking through the building blocks that make up higher-level math problems, eventually attained a deeper understanding, and so are more easily able to see those building blocks to understand complex problems.
Of course there is something innately special. To proclaim that there is nothing innately different between those to whom math comes naturally and those to whom it doesn’t is just naive. It’s seen in young children who are naturally able to understand abstract ideas much better and FASTER then others – that’s where it starts. It’s pure nonsense that we are all equal and everyone can do this and that. Sure, everyone can learn math , but for some it might take an hour to learn something, while for another it will take a month or even longer. It’s seen in sports and physical activities, but when it comes to mental strengths , some try to proclaim that we are all equal and that there is nothing innately different . It’s called genetics and they play a much bigger role then today’s liberal establishment is willing to admit!
Hi Cal,
I think you are referring to this interview https://www.youtube.com/watch?v=KaVytLupxmo
Regards,
Vinay
What’s nice about this–he says accept being stuck, let your mind can relax, then you can go forward.
It feels like this is a great model for many kinds problems. It’s harder to go beyond something that at first can’t be accepted
This week I got stuck several times while writing a poem (this is a common situation for a poet because poems almost invariably have to ripen over time). I’ve spent intervals away from the poem (cooking, sweeping the porch, driving to the post office), and the poem is gradually coming together. It now hums when I pick it up. Mathematicians, make room for poets!
I’m a novelist. It is so true what you say.
Agreed. Many movies destroy the concept of hard work; even Harry Potter teaches predestination or special powers, let alone the plethora of superhero movies.
Well said EA. We fall so easily for tall tales about how people with super powers save the world. Such tales are harmful as Wiles says. How does on fall in love with the normal reality? That would such an amazing thing to do because it sets you free as you don’t seek relief from boredom by surfing the internet. I don’t know how to fall in love reality as it really is.
This post arrived at the perfect time for me. Despite having a productive week and showing up to do my deep work, yesterday I got stuck writing my research results for climate action mobilization. It was one of those disappointing days when you sit staring at the screen and nothing happens.
Thanks for helping me accept those days–oh, how they fowl my mood–and continually adding value to my work life, Cal Newport.
“You absorb everything about the problem. You think about it a great deal—all the techniques that are used for these things. [But] usually, it needs something else. So, you get stuck.”
At this point, he explains, “you have to stop…let your mind relax a bit…[while] your subconscious is making connections.”
Then you “start again.” Day after day. Week after week. Until, one day:
“You find this thing…Suddenly you see the beauty of this landscape…[before,] when it’s still some kind of conjecture, it seems really far away…[but now] it’s like your eyes are open.”
Cal, what are your thoughts about deliberately harnessing this interplay between the conscious and the subconscious? Because there’s a lot of woo-woo out there about this topic.
Almost every professional thinker I know — myself included — integrate “background processing” as a key step in forming original thoughts and solutions. Measuring this effect in a replicatible way, however, has eluded those who have tried to study it more rigorously.
Perhaps, at a future date, you could expand upon any systematized ways you implement this background processing into your working life.
A useful technique I borrowed from Loran Nordgren for engaging the unconscious is as follows.
Formalize two modes of conscious thinking.
The first mode is “information acquisition”. Where you throw the net out into the external world and collect bits of information.
The second mode is “evaluation mode”. Where you trawl through the bits of information, selecting for new insights and lines of thinking.
Once you’ve set up these two modes, Nordgren suggests allocating a clear divide between the two. It’s this divide, he claims, that allows the information collected to be processed by the unconscious. So that when the evaluation process begins, the unconscious has already been busy “background processing” the information collected, allowing for more original and powerful insights.
I think you referenced Nordgren in Deep Work.
That’s interesting. How does one allocate a clear divide between the two modes? Is there a book or an article written by Nordgren on this topic?
I am a Mathematics teacher in College Level but i haven’t seen this type of equation before may be its helpful for University level students.
“The first is personal. As a theoretical computer scientist I spend a lot of my professional life stuck on math problems. It’s hard to explain to outsiders what this is like, and Wiles’s response does a good job of capturing the competing forces of frustration and joy that come from tackling such things on a regular basis.”
I have four kids all under 6. When I explain my life or the trials my wife and I go through, I always get frustrated because the response is usually, “God I never want children,” or something equally dismal. And I’m like, “No! That’s not what I’m saying?!” It’s rewarding, how is so hard! If you can feel this way about your work, how is this hard to understand when talking about the difficulties of raising kids?!”
I have given this state of affairs a name — ‘semignosis’:
“The state… [in which] the seeds of future gnosis are being sown but there is no current, specifiable increase in knowledge.”
https://rulerstothesky.com/2017/01/25/a-taxonomy-of-gnoses/
During the STEMpunk Project, as I was wrestling with circuit diagrams, car engines, algoriths, code, physics and all the rest, I had to spend a lot of time just sort of letting insights incubate. At first I found this frustrating (I’m not quite at Will Hunting’s level), but once I had named it and grasped its importance to the learning process it wasn’t as bad.
I recently read a book title ‘Aha Moments’ by William B Irvine that talks about how these aha moments unfold. He does an excellent job of unpacking the role of the subconscious in all this and even goes on the explain the political battles one has to fight to get the community to accept the insight and results. None of this is easy and that I suppose is what makes it worthwhile.
Is math hard? The questions it tries to answer are trivial: can every even integer greater than 2 can be expressed as the sum of two primes? That, the most famous unsolved problem in number theory, seems like an easier problem than others: what is the best way to live?
I guess what makes it “hard” is that there are clear right and wrong answers; there is no “good enough,” which is what we all use to get by.
I would say the questions are not so much trivial as easy to state. And that’s just for some of the theorems. There are also theorems whose statements you need two graduate level courses to even understand.
As for whether it’s hard, I would say yes, for us. We are very intelligent, but our intelligence isn’t optimized for very abstract things like building rockets or proving theorems. I remember reading somewhere that our vast brain mass had to do with the complexity of our social structure.
As a math student I’m still frequently struck in the way that I misunderstand very “trivial” mathematical things or fail to make some very “trivial” logical connections despite their being, well, mathematically trivial. Sometimes something that directly follows from definition is nontrivial for the human brain. So in a sense math is hard for humans.
Wiles notion of being “stuck” reminded me of Keats “negative capability”:
“…what quality went to form a Man of Achievement especially in Literature & which Shakespeare possessed so enormously – I mean Negative Capability, that is when a man is capable of being in uncertainties, Mysteries, doubts, without any irritable reaching after fact & reason…”
https://www.bl.uk/romantics-and-victorians/articles/john-keats-and-negative-capability
It all depends on the approach one used or one applied but above all determination remains the watch word, once you are determined then you will surmount any obstacles