Saturday, July 16, 2011

The improbable effectiveness of mathematics -- a Saturday reprise

(Following the posts of the last two days, here is a Saturday reprise from April 2009.)

It dawned on me like a hammer blow to the head, at about chapter ten of my high school physics book. For weeks we had been solving problems. The problems involved frictionless pulleys, massless strings, perfectly smooth inclined planes, and other idealizations of the real world. And I thought: Physics isn't about the real world at all; physics is about a world that exists only in the physics text. So why were we studying this imaginary world?

It was because only in that idealized world were there problems that could be solved with high-school mathematics. And solving the problems was fun. I loved solving the problems. I loved doing my physics homework.

And I loved that imaginary world of the physics text that seemed to be built out of pure mathematics -- number, algebra, geometry, trig.

Years passed. College physics. Grad-school physics. The pulleys acquired friction, the strings acquired weight. The problems became ever more difficult to solve, but solving them was still fun. More and more fun, actually. I especially loved solving problems involving electromagnetic fields around charged objects.

But the world we were playing with was still an imaginary world. When I put down the pencil and turned away from the text, it was a very different world that attracted my attention. My real body was bathed, no doubt, in real electromagnetic fields, but there was no way I could describe them with the analytical elegance of the fields in the text. The text world had a simple beauty that appealed to my esthetic sense. The real world was messy.

What is the connection between the mathematically elegant world of the physics text and the real world? Why is physics, which is an exercise of the mind, so fabulously successful in practical application? Einstein once said: "I am convinced that we can discover by means of purely mathematical constructions...the key to understanding natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. The creative principle lies in mathematics."

We are in the face here of one of the deepest mysteries of philosophy -- the uncanny resonance of mind and world. And all these years later I can still remember the precise moment when this mystery of mysteries leapt off the page of the high-school physics text.