A 4-Line Proof of the Isoperimetric Theorem in 3D
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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I think there might be a relevant difference between these two arguments. Premise 2 in your biological argument is clearly not certain – there are all sorts of reasons why animals wouldn’t choose the optimal shape for warmth, like if that shape happened to be extremely complex, or if that shape was very close to many highly suboptimal shapes.
I think the physical argument I gave is clearly sound, because it can be turned into a purely geometric one. The economic argument is probably a bit more worrying, but it seems to follow from the thought that there really is a unique value for every time-series of money. So there’s a hidden premise about the consistency of the economic theory, but it does seem to show us something about the math.
Your biological argument is an interesting one though, but I think an argument based on soap bubbles would be more plausible. Although even there, the argument could only lead to the conclusion that sphere is a local minimum of SA/V.
Right — the distinction has to be between proofs which lead to their conclusions of necessity and those that don’t. And that’s so much the worse for metaphysical determinists — I suppose?
Consider this argument:
1. The sphere is the shape that minimizes SA/V.
2. The warmest shape is the one that minimizes SA/V.
3. If a shape is the warmest, then animals will overwhelmingly assume that shape when they’re cold.
Therefore If an animal assumes a spherical shape, then it is cold.
The issue with our two proofs is that it depends upon the reader to determine what each of us believes is more fundamental. If I think the math is more fundamental, then I would find my proof more intriguing; if I thought the actions of animals when cold more fundamental, then yours is more interesting.
This is to say the premises in a proof need to be somehow more secure than the conclusion at the outset of the proof otherwise we are just exercising our logic gears. In terms of animals being cold and the shape that minimizes SA/V, we need to first assume that our premise about the state of animals being cold is somehow more fundamental than mathematics (regardless of the existence of easy mathematical arguments) in order to have the conclusion about mathematics be meaningful. Otherwise we are using a result of the underlying mathematical truth to prove the mathematical truth, that is, begging the question. Likewise if we assume that the actions of animals when cold are more fundamental than the mathematics, then my proof begs the question.
Therefore, for either of our proofs to have logical force we would first need to show that one of the two, the actions of animals when cold or mathematics, is more fundamental. We may each have some intuitions about which is the more fundamental, but I suspect no proof of this sort to be forthcoming.
orgel’s rule: “evolution is cleverer than you are.” if that’s so, then we stand to learn a thing or two by paying close attention to its fruits.
remember that evolution finds the most effective and efficient means of accomplishing its “ends.” so i suppose that some mathematical theorem implicit in efficient bipedal locomotion would for this reason be likewise bolstered by facts about the human bone-muscle geometry.
it seems like a good argument (albeit with exceptionally loaded premises), but credence in that sort of argument must derive from antecedent knowledge of the link between evolutionary processes and efficiency.
(very enjoyable post, by the way)
Noah: Thank you for pointing out this symmetry in my argument. Wigner would say, your formulation shows the “unreasonable effectiveness of mathematics in science,” while the original argument shows the “unreasonable effectiveness of science in mathematics.”
Kenny & Alex: you’re right that, unlike Kenny’s argument, my biological ‘proof’ has some pretty questionable premises. (Interestingly, different people seem inclined accept/go after different ones.) However, I also suspect that Will is on to something, that a clever biologist could probably keep filling in more details in response to each objection. What we would likely end up with, I think, is something looking more like a proper mathematical proof.
So what I want to suggest here is a deflationary picture: compelling non-mathematical arguments for mathematical claims “hide” good arguments. They are compelling because we recognize the possibility of filling in the details. But they don’t establish anything, because they aren’t good arguments. (As we all know, there can be multiple devils in the details.)
I think the “bubble argument” for this thoerem that Kenny alludes to is also an example of this. The difference is, it’s more obvious that you can fill in the details of the bubble argument.
(Interestingly, I just became aware of a passage by George Pólya (1954, 170) in which he suggests both Kenny’s and my arguments.)
Proofs like yours are a standard trick for formulating conjectures. One of the big ones right now is the Mirror Symmetry Conjecture, which mathematicians in my field (algebraic geometry) are working very hard on trying to prove, but which was handed to us by the String Theory people with some physical arguments.
My understanding of the level of acceptance of such proofs is that they don’t count as real proofs, but often tend to give ideas about how to go about proving things. For instance, this biological proof of the isoperimetric theorem in three dimensions could be saying that there is a PDE based proof of the theorem based on the heat equation, and a clever person might find it worthwhile to look into it.
Very interesting Charles.
I must admit that to some extent, the very idea that an empirical fact be an assumption in a purely mathematical claim is disturbing to me. This worry leads me to try to argue the following:
Any time an empirical fact appears in a valid argument for a purely mathematical claim, you know that either (i) the empirical “fact” is wrong and the argument doesn’t work, or (ii) the empirical “fact” is actually just shorthand for a purely mathematical argument.
So I would expect that no matter what your PDE looks like, if you’re relying on a fact about the world (such as a law of thermodynamics), then the argument is missing something.
What I was saying wasn’t that an empirical fact lays at the bottom of the hypothetical proof. Rather that the empirical fact indicates that we might want to look in a certain direction to see if there’s a proof. Here the claim is that the isoperimetric theorem follows from some stuff about heat loss in animals, so a proof MIGHT (and making this precise and proving the claims would require a better PDE person than me) go like this:
1) For the heat equation on three space with initial conditions in some object (that is, all the heat is inside) the rate of flux along the boundary is minimized by the sphere.
2) The way to minimize flux is to decrease the ratio SA/V.
3) Thus, the sphere minimizes SA/V.
Now, I don’t see a quick proof for either the first or second claim, but I’m not a PDE person. However, this would be a PURELY mathematical argument which is inspired by the biological argument from before, which is what I had meant to suggest. The empirical facts guide us towards the proof, but do not constitute a proof.