Tag: mathematics

One of the more interesting random questions that I was asked in class recently was about whether it makes sense to “do different things”.

By this, the student who asked the question meant the following: should one focus on doing one thing, and one thing only in terms of studies, in terms of a career and in terms of hobbies, or does it make sense to do a lot of different things in each case.

The obvious answer to give these days is that one should read Range, by David Epstein – and it certainly is a good way to start thinking about the subject. And I agree with the central idea in David’s book – variety is good! You can take a similar lesson from statistics, if you like, or you can focus on the English phrase “variety is the spice of life“… or you can read and reflect on a wonderful profile of one of the winners of the Fields medal this year.

One might say the same of his path into mathematics itself: that it was characterized by much wandering and a series of small miracles. When he was younger, Huh had no desire to be a mathematician. He was indifferent to the subject, and he dropped out of high school to become a poet. It would take a chance encounter during his university years — and many moments of feeling lost — for him to find that mathematics held what he’d been looking for all along.

https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/

The entire essay is, of course, well worth your time, and I would strongly recommend that you read it. Your eyes might glaze over some of the mathematics in the profile (mine certainly did), but there is a lot to be learnt by reading the essay carefully, especially if you want to think about whether “doing different things” is a good idea or not.

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The profile is much about June Huh as it is about ruminations on productivity, daily routines, the connection between beauty and mathematics and the role that luck plays in our lives. Consider for example, the fact that Huh never works for more than three hours in a day. Folks who work in the corporate world, especially in India, take an almost perverse pride in the number of hours that they spend in office, but I would argue (from personal experience, more than anything else) that the actual work done doesn’t take more than four to five hours at best.

It gets even better, because there is mention later on in the article about an inversion of the Pomodoro technique:

Huh can still muster only enough energy to work for a few hours each day. “Other people work one hour and just take a five-minute rest,” Kim said. “He is like, one hour do something else, and just focus for five minutes, 10 minutes.”

https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/

Me, I’m very good at the doing-something-else-for-one-hour bit. It’s the five minutes of deep and focused work that I struggle with!

But the bit that I enjoyed the most was the part about somehow, just knowing:

For Huh, when he is working, there’s something almost subconscious going on. In fact, he usually can’t trace how or when his ideas come to him. He doesn’t have sudden flashes of insight. Instead, “at some point, you just realize, oh, I know this,” he said. Maybe last week, he didn’t understand something, but now, without any additional input, the pieces have clicked into place without his realizing it. He likens it to the way your mind can surprise you and create unexpected connections when you’re dreaming. “It’s just amazing what human minds are capable of,” he said. “And it’s nice to admit that we don’t know what’s going on.”

https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/

Two references that this reminded me of:

Poincaré then hypothesized that this selection is made by what he called the “subliminal self,” an entity that corresponds exactly with what Phaedrus called preintellectual awareness. The subliminal self, Poincaré said, looks at a large number of solutions to a problem, but only the interesting ones break into the domain of consciousness. Mathematical solutions are selected by the subliminal self on the basis of “mathematical beauty,” of the harmony of numbers and forms, of geometric elegance. “This is a true esthetic feeling which all mathematicians know,” Poincaré said, “but of which the profane are so ignorant as often to be tempted to smile.” But it is this harmony, this beauty, that is at the center of it all.

Pirsig, Robert M.. Zen and the Art of Motorcycle Maintenance (p. 240). HarperCollins e-books. Kindle Edition.

From further on in the same book:

Then Poincaré illustrated how a fact is discovered. He had described generally how scientists arrive at facts and theories but now he penetrated narrowly into his own personal experience with the mathematical functions that established his early fame. For fifteen days, he said, he strove to prove that there couldn’t be any such functions. Every day he seated himself at his work-table, stayed an hour or two, tried a great number of combinations and reached no results. Then one evening, contrary to his custom, he drank black coffee and couldn’t sleep. Ideas arose in crowds. He felt them collide until pairs interlocked, so to speak, making a stable combination. The next morning he had only to write out the results. A wave of crystallization had taken place. He described how a second wave of crystallization, guided by analogies to established mathematics, produced what he later named the “Theta-Fuchsian Series.” He left Caen, where he was living, to go on a geologic excursion. The changes of travel made him forget mathematics. He was about to enter a bus, and at the moment when he put his foot on the step, the idea came to him, without anything in his former thoughts having paved the way for it, that the transformations he had used to define the Fuchsian functions were identical with those of non-Euclidian geometry. He didn’t verify the idea, he said, he just went on with a conversation on the bus; but he felt a perfect certainty. Later he verified the result at his leisure.

Pirsig, Robert M.. Zen and the Art of Motorcycle Maintenance (pp. 239-240). HarperCollins e-books. Kindle Edition.

And finally, a fascinating paper, of which I learnt via a blogpost on MR, by Alex Tabbarok:

Many people have claimed that sleep has helped them solve a difficult problem, but empirical support for this assertion remains tentative. The current experiment tested whether manipulating information processing during sleep impacts problem incubation and solving. In memory studies, delivering learning-associated sound cues during sleep can reactivate memories. We therefore predicted that reactivating previously unsolved problems could help people solve them. In the evening, we presented 57 participants with puzzles, each arbitrarily associated with a different sound. While participants slept overnight, half of the sounds associated with the puzzles they had not solved were surreptitiously presented. The next morning, participants solved 31.7% of cued puzzles, compared with 20.5% of uncued puzzles (a 55% improvement). Moreover, cued-puzzle solving correlated with cued-puzzle memory. Overall, these results demonstrate that cuing puzzle information during sleep can facilitate solving, thus supporting sleep’s role in problem incubation and establishing a new technique to advance understanding of problem solving and sleep cognition.

https://journals.sagepub.com/doi/abs/10.1177/0956797619873344?journalCode=pssa

And what better way to end then, than with this perfectly appropriate final extract:

The office is spare, practically empty. There’s a large desk, a couch for sleeping — Huh typically takes a nap later in the morning — and a yoga mat rolled out on the floor (just for lying down, he said; he doesn’t actually know how to do yoga).

https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/