Yesterday’s post of a beautiful proof of the Buffon needle problem, reminded me of Paul Erdos. Paul Erdos is famous for many things, and one is his belief in the importance of finding beautiful proofs. He is supposed to have said that such proofs were from God’s book. Perhaps. It occurs to me that it might be well to tell my one story about Paul Erdos that is original, not previously in the literature, as a contribution towards future biographers. There have been a couple of popular biographies of Paul Erdos, but they didn’t impress me. Perhaps there are other biographies, or perhaps a good mathematical one is still to be written, one that both captures Paul’s enthusiasms and adequately represents his mathematics. Anyway, here is an anecdote:

Paul Erdos had come to my university (Western Ontario) to give a talk, and there was a dinner that evening at the Borwein’s, to which I was invited since I was one of David Borwein’s students. The dinner was a buffet, and we sat around the living room on folding chairs, Paul on the chair next to me, our plates of food in our laps. I was tongue-tied, not knowing what to discuss with the famous mathematician. Paul sensed my hesitation, and held up a bun, asking me if I knew how to establish that the bun had a center of gravity, whatever the varying density of the bread. Ok so far, but then I asked Paul a question. His eyes closed, his head dropped, he slumped down in his chair! I looked around in alarm — had Paul had a stroke? What should I do? Someone reassured me — this was how Paul thought. He would shut out the externals to concentrate on the question. And sure enough, after a minute or so, Paul looked up, and answered my question.

So what, you will ask, was the question which Paul Erdos was thinking about? And what was his answer? Frankly, I’ve completely forgotten, as I was so worried about him during his “slump”, and was still worried as he looked up and responded. I wasn’t paying any attention to the math of his reply, just relieved that he was replying. But, in retrospect, I think my question must have been a brief version of something along these lines. Brief would have been sufficient for Paul — he could fill in caveats and details more rapidly than others could say them.

By passing a plane through the bread, imagining integrating the mass times offset on either side, we can find a plane which passes through the center of gravity. That is, we can imagine balancing the bread along the family of planes which are normal to some line, as the line travels through the bread. Given a plane, we are looking at the integral of x*dm, where x is the positive or negative offset of the plane from a slice of bread of mass density dm. Let’s suppose a straight line for simplicity, though a family of planes sweeping out a curved arc could also be interesting. The integral starts most negative at one side of the bread, and goes most positive at the other side of the bread, and hence there is a position of the plane, somewhere in the middle, where the integral is zero and the bread is balanced. Now pass a plane through the bread in another direction, that is moving along another line, and do the same thing. Do this repeatedly, from various directions. Experience shows that the balance planes will all intersect, at the center of gravity of the bread. But, how do we know (prove) that the balance positions of those planes all intersect in a common point?

There you have the question. What would Paul have said? What is the answer from God’s book?

Best wishes,
Ken Roberts
22-Feb-2014