1/12/03

1. When I was in college I took second quarter calculus from Professor Nelson. He was a cheerful, friendly guy with a permanent smile on his face and a twinkle in his eye. One day, after we had learned how to calculate the surface area of some curved figures he gave us an interesting demonstration. He took the graph of the function y=1/x^3 from one to infinity and rotated it about the x axis. You get a figure that looks like one of those long straight trumpets but with the mouthpiece way off to infinity to the right.

2. Professor Nelson then went through the steps calculating the volume of the figure which turned out to be some number. So far so good. Next, he calculated the surface area of the figure and lo and behold, it turned out to be infinite. When he finished, he put the chalk in the tray, turned to the class with an impish grin and that familiar twinkle and said, "There, you see? You can fill it but you can't paint it!"

3. Without a moment's hesitation, I blurted out, "What happens if you fill it with paint?"

4. Well, I didn't expect the response I got. The smile vanished from his face as he turned and glared at me. His face turned red and he stammered something about this being mathematics and it had nothing to do with anything real like paint. That made me shrink down in my chair and wish I could drop through the floor. I had no intention of ruining his fun or making him look silly.

5. What Professor Nelson didn't understand was that I was asking a real question. And, as with some other similar questions, like "Why doesn't water burn?", I got no satisfactory answer from my teachers but had to figure out the answer for myself. Incidentally, the question about water burning occurred to me in grade school when I first learned that water was made of hydrogen, a very flammable fuel, and oxygen, whose presence is a requirement of fire. I reasoned that with those two ingredients, water should really burn. My teachers gave me nonsense answers like, "the water generates steam which smothers the fire", or, "the water is too cool to burn and absorbs all the heat". After thinking about it for some time (a few years I think) it dawned on me that water didn't burn because it was already burned. Water is the ashes you get when you burn hydrogen and ashes don't burn. Later, of course, I realized that it wasn't really the ashes but instead was the product of combustion. All the same, water doesn't burn because it has already been burned.

6. So, back to my question for Professor Nelson. I was sincerely interested in the relationship between mathematics and reality. There is no doubt that mathematics is useful for describing some real things, and for predicting their behavior. We have learned much about reality by first using mathematics to predict something and then later running experiments to verify the prediction. Since the boundaries of our knowledge of reality tend to be the very small and the very large, the question arises as to how far in these directions does mathematics match reality. In particular, does anything infinite exist in nature? After all, infinity exists in mathematics so why can't we infer that reality follows suit?

7. After giving some thought to my question about filling the trumpet with paint, the answer was obvious. Since paint is paint because of the particular frequencies of light that reflect from its surface, and since those frequencies are determined by the specific electronic configuration of the molecules comprising the paint, it would no longer be paint if that electronic configuration were changed. So, as the diameter of the trumpet tube gets smaller and smaller, you would end up with a stack of triangles, each formed by three paint molecules, with the thin trumpet tube threading through all the triangles, like a string of pearls, once the tube got small enough to fit in the hole in the center of the triangles. This stack would keep going without limit to the right, so, Prof. Nelson was right: It would take an infinite amount of paint.

8. But, if you stood the trumpet on end and poured paint into it, as soon as the tube diameter got smaller than the diameter of a paint molecule, the tube would plug up and allow you to fill it to the brim with a finite amount of paint.

9. I know, I know. We didn't really fill the trumpet with paint this way. But I think the mathematics proves that if we somehow sliced up those molecules so that the pieces tightly fit into the trumpet leaving no space, a finite amount of paint would still do the job.

10. Now, that's a satisfactory "ashes won't burn" type of answer. But it still leaves the more profound question about the real relationship between mathematical structures and real physical structures. We know, for example that we have drawn the hasty but erroneous conclusion that the geometry of space must be Euclidean. Turns out that it doesn't seem to be. But then again, mathematics isn't restricted to Euclidean geometry. There are many possible geometries which are mutually inconsistent. So does one of these match the structure of space exactly? I think the jury is still out.

11. So I think that the same caution should be exercised with respect to infinity. Just because we have the concept of infinity in mathematics, is not a sufficient reason to suppose that infinity exists in reality.

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