Here’s an example of what might be called “biological proof” of a mathematical claim.
Proposition (The Isoperimetric Inequality). The solid that minimizes the ratio of surface area to volume (SA/V) in Euclidean 3-space is the sphere.
A Biological Argument. Consider the large class of animals capable of changing their ratio of surface area to volume. (And note that these animals live — approximately — in Euclidean 3-space.) What do these animals do when it’s cold? They curl up into a ball! More precisely, they assume the closest approximation to a ball that they can manage. This is because any exposed surface area is a place where heat is lost, and curling up into a ball minimizes that surface area. So a spherical or “ball” shape keeps animals warmer. Now, here’s how these ideas can be turned into a 4-line “biological proof” of the above proposition.
- Animals overwhelmingly assume a spherical shape when they are cold.
- If animals overwhelmingly assume some shape when they’re cold, then it is because that shape is the warmest.
- The warmest shape is the one that minimizes SA/V.
- Therefore, the sphere is the shape that minimizes SA/V.
It’s a pretty easy argument, and you can write it down in four lines. At the same time, there simply are no easy mathematical arguments for this proposition in 3-Dimensions (for a nice survey, see Osserman (1978).)
Now, here come the tricky questions: what’s the status of an argument like the one that I’ve given here? What does it allow me to infer? And especially: how does it compare to an argument in which, say, the conclusion is produced by way of the calculus of variations?
And just to ratchet the trickiness up another notch: there are apparently many of these non-mathematical arguments for mathematical claims. For example, a lovely economic argument for a mathematical conclusion was recently described about by Kenny, which he calls part of the “unreasonable effectiveness of the sciences in mathematics.”
But let me encourage a little caution. Arguments with correct conclusions are easy to come by. An example: Bibble, babble, blat, blit: therefore, the primes are infinite. We must not be mislead by the fact that this argument for the infinitude of the primes (which is drivel) ends with a correct conclusion.
The biological argument above is analogous, to an extent (though I hope not quite drivel). Suppose that premise (1) turned out to be wrong — for example, our empirical data may have been poorly gathered, or we may have interpreted it incorrectly. Then this little “proof” would be just another bad argument for a correct conclusion.
This doesn’t mean that non-mathematical arguments for mathematical claims can’t lead to new knowledge. I think that they can, and in many cases they do (is this mathematical intuition?). But my suspicion is this: these non-mathematical tricks are instructive only insofar as they lead to good arguments.
Update, 2 July 08. I just found out that George Pólya (1954, 170) has already suggested this kind of argument could be made. But the difficulty of the exercise remains, as I suggest above, in how you work out the details.
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