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.
a curious use of physical reasoning to reveal a purely mathematical truth
Hmm … I would rather call it the use of a mathematical truth to obtain a simple and elegant proof of a physical law. The logic that I see is as follows. We start from the following propositions:
a. the arithmetic mean is never smaller than the geometric mean (a mathematical truth);
b. in the system under consideration, neglecting gravitational energy and other factors which we assume constant, the energy is proportional to the average temperature (this can be considered either a physical law or a definition of temperature);
c. again neglecting other constant factors, the entropy of the system is proportional to the geometric mean of the temperatures of the various buckets of water (again, this can be considered a definition of entropy);
d. the energy of a system is a constant (first law of thermodynamics).
Combining all of the above, we can derive
e. the second law of thermodynamics.
OTOH propositions from b to e (all physical laws or physical definitions) do not appear sufficient for a mathematical proof of proposition a, because physical laws are always at risk of being falsified.
Miracles as an explanation of the possibility of successfully using mathematics is empty. Embedded cognition theories at least have a chance at avoiding empty platonism.
Thank you very much for that enlightening comment, Michael. I shall make sure to disinter that hack Wigner and castigate him.
This leads one to wonder where the difference between mathematics and physics lies. Are the laws of mathematics inherent in our universe, and therefore really a product of physics (and not the other way around), or are they supra-universal?
@Barry, Ok read Lakoff’s “Where Mathematics Comes From”, Michael Dummett’s Intuitionism, and then defend naive realism about mathematical entities.
Or you could just start with Simon Blackburn’s observation that Frege showed us that numbers were not objects but rather adverbs.
And it is possible that smart physicists could be very dumb about ontology? Sure, and it may be a requisite for doing science.