openpolitics.com

Somewhere, you describe your approach to mathematics, in which one does not attack a problem head-on, but one envelops and dissolves it in a rising tide of general theories.

.. My friend Bob Thomason once told me that the reason Grothendieck succeeded so often where others had failed was that while everyone else was out to prove a theorem, Grothendieck was out tounderstand geometry. 

.. But Grothendieck lived by the conviction thateverything is easy if you look at it right — which means there have got to be enough points. And if we think there aren’t, it must be because we haven’t yet figured out what a point is.

.. There are even points where every function is equal to some expression like (3x²+1)/(7x³+4). You might object that that’s not a constant — but it is, because the x in that expression is not a variable; it’s just a symbol, and that symbol always remains just x.

..

When you look at, say, the ordinary Euclidean plane, the points you see — the points that stretch out to infinity in all directions, the ones you familiarized yourself with in high school — are just the real points. But from a Grothendieckian perspective, that’s not the whole plane. There are also plenty of (invisible) complex points, (and those points, incidentally, can “spin in place”, ultimately because the complex numbers contain two square roots of minus one, which can be interchanged.) And there are plenty of far more complicated points besides. The plane is teeming with points you never learned about in high school.

It turns out that when you have all those extra points to work with, a lot of technical problems melt away, and you can solve a lot of problems you couldn’t solve before. Generalize sufficiently — allow the possibility that your notion of a point was always too specific and too cramped — and hard problems suddenly get easy.

.. What, then, is a curve? A curve is a place where you can move around, and look at things from many points of view. Is 7 a prime number? It might depend on where you’re standing. So we can identify a curve with a different mathematical universe, a universe that admits a certain amount of ambiguity — not at all the same as the classical universe we’re used to, but still a perfectly valid object of study.

.. First, it turns out, miraculously enough, that when we see points and curves and surfaces as the homes for entire mathematical universes, we are able to use that insight to solve hard problems in classical geometry and arithmetic. We’re still studying the same old points and curves, but by recognizing that each of these points and curves supports an entire Universe, and by making good use of that insight, we can (rather incredibly) learn new things about the ordinary geometry of those points and curves.

.. Grothendieck’s lifelong insistence that mathematical objects are intrinsically uninteresting — instead it’s the relationsbetween mathematical objects that matter. The internal structure of a line or a circle is boring; the fact that you can wrap a line around a circle is fundamental.