In The Big Questions (pages 18-19) I talk (channeling the physicist Eugene Wigner) about the apparently unreasonable effectiveness of mathematics in revealing truths about the physical world. In Wigner’s words, “It is difficult to avoid the impression that a miracle confronts us here.”
But the physicist Peter Landsberg (no relation!) observes that sometimes the miracle runs in the opposite direction, and offers a curious use of physical reasoning to reveal a purely mathematical truth!
The mathematical truth in question concerns “arithmetic means” and “geometric means”, so let start by telling you what those are. Start with, say, 4 numbers; say 1, 3, 5, and 6. The arithmetic mean is just what you used to call the mean back in high school, and the average back in elementary school: To compute it, you add the numbers and then divide by 4 (or 5 or 6 or 7 if you started with 5 or 6 or 7 numbers). In this case, that gives you (1+3+5+6)/4 = 3.75. To compute the geometric mean, you multiply the numbers and then take the 4th root (or the 5th or 6th or 7th root if you started with 5 or 6 or 7 numbers): The fourth root of 1 x 3 x 5 x 6 is about 3.08.
Now here’s a mathematical truth: the arithmetic mean is always at least as big as the geometric mean. For example, 3.75 is larger than 3.08.
And here’s how you could discover that fact if you knew a little physics: Start with several buckets of water, all at different temperatures. Bring them together and let them sit until they all reach a single new temperature.
The laws of thermodynamics tell us two things: First, energy is conserved. That means the new temperature is the arithmetic mean of the original temperatures.
Second, entropy can only increase. If you write down the formula for the change in entropy, you’ll see that for entropy to increase, that new temperature must exceed the geometric mean of the original temperatures. Voila: A truth of pure mathematics directly accessible from established principles of physical science.
A tip of the hat to my colleague Bob Knox , who put me on to this.