Saturday, December 6, 2014

Wisdom Of The Week

The Messiah of Zooming Out - A Profile of Alexander Grothendieck was one the best pieces I read this year; sadly I never heard of Grothendieck before this week:

Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…

Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.

Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out.

Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.
There are, of course, times when it does pay to examine the inner workings of things. Jean-Pierre Serre, another titan of modern mathematics with whom Grothendieck had an intimate working relationship, was often, in Grothendieck’s words “the yang to my yin”. If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”

In other words, the philosophy was this: If a phenomenon seems hard to explain, it’s because you haven’t fully understood how general it is. Once you figure out how general it is, the explanation will stare you in the face.


It has been the great privilege of my life to understand some small fraction of the mind of Grothendieck. The world is an infinitely more beautiful place because he lived.

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