Wednesday, November 28, 2012

The entirely reasonable effectiveness of mathematics in all science

(Originally posted by Pat on 9/7/10)

There is a famous philosophical paper entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” which argues that we should be surprised and amazed at how well mathematics—even very abstract mathematics like topology and number theory—works in our most precise scientific theories, like general relativity and quantum mechanics. The paper begins by noticing that pi, the ratio of circumference to diameter in a circle, appears in the formula for a normal distribution, which has innumerable applications in practical science. (Indeed the paper says “natural sciences”, but we use normal distributions most often in social sciences!) It speaks of this as something marvelous and surprising, even suggesting that perhaps this should lead us to a spiritual sort of reverence for mathematics.
Yet, I must ask, what other possibility was there?

I suppose there is indeed something marvelous about the fact that the universe is at all intelligible to us. I am quite certain that the universe is not so intelligible to most other species. I doubt that insects ever reflect upon the universe's causal structure at all—I doubt they even wonder why it rains or why the sun sets at night. Reptiles may reflect upon a few things, perhaps wondering how mating leads to laying eggs or why the sun sets. Most mammals probably think about these things, but don't really understand them. Primates surely reason about many things, and I would guess that they have theories about astronomy, biology and meteorology not too different from our own common-sense folk theories.
We humans—some of us at least—are capable of far deeper understanding of a far wider variety of phenomena; our sharp senses and plastic brains allow us to develop detailed theories of the world. Even our instincts and emotions support us in this endeavor: human folk psychology is by far the most powerfully predictive scientific theory ever devised, and all the great scientific geniuses will report that many of their discoveries started with an intuitive “hunch” that only later could be confirmed by data. Given the huge variety of life forms on Earth and the even vaster variety that in all probability exists elsewhere in the universe, we are indeed fortunate to be among the lucky few with brains powerful enough to grasp, even as poorly as we do, the fundamental laws of nature. Quantum mechanics is baffling to us—but it is quite simply inaccessible to the vast majority of life on this planet. It may be difficult for us to comprehend how the universe could be guided by waves of probability—yet for an ape, let alone an earthworm, it would be impossible to even formulate the notion of “waves” or “probability”. There is in fact something marvelous about that.
Yet for this, we have an explanation in the Weak Anthropic Principle (the only true anthropic principle—saps who endorse SAP should feel a slap that would sound like WAP). Only a species advanced enough to have a fairly deep understanding of reality would ever be concerned about whether and why it had a deep understanding of reality! Ants don't know, but they also don't care.
Moreover, we cannot but presume that the universe is intelligible. A completely unintelligible universe is one in which we could never make any correct predictions at all. Even the theory “the universe is random and unintelligible” would be no more applicable than any other theory, even if was really true in some philosophical sense. So really I don't see why there is anything problematic about the presumption that the universe is at least somewhat intelligible; it wouldn't hurt us in a random universe, and it clearly does help us in this universe. I doubt that everything about the universe is intelligible, for various reasons; but I have no idea how far the reaches of our minds extend, and I would venture a guess that they extend far further than what we know now.
And once we accept that the universe is to some degree within our understanding, I don't see why it's at all problematic to infer that mathematics is likely to be one of the means we use to understand it. The structure of mathematics is based only upon a formalization of logic; it is what happens when logic is made abstract, formal, precise, and consistent. The application of mathematics rests upon the structure of mathematics and the evidence upon which mathematical models were based.
In essence, mathematics is a system of logical arguments. The observed data provide the premises, and the mathematical operations are the logical steps that allow us to infer a conclusion. To ask “why does the universe obey logic?” strikes me as something close to meaningless—how could it not? There is simply no possible world in which noncontradiction and modus ponens fail to hold. One does not need a causal account to explain the nonexistence of phenomena that cannot possibly exist. When you start asking “Why is logic true?” you have surpassed the bounds of questions that it makes sense to ask. A lot of people like to imagine that God is the reason logic is true; but that makes no sense at all. A thing cannot be such that it makes contradictions true. A contradiction simply can't be true. The very concept is vacuous. Even if there is a God, logic is more fundamental than God—God must obey logic, not the other way around. Our minds are evolved to think that all that is true is true because of some thing that makes it so—for so it is, with most of the ideas we have reason to consider, like “Why is the sky blue?” or “Why doesn't she love me?”. If we are not satisfied by the fact that logic is inherently and necessarily true, then it seems to me it must be our brains which are defective—there simply could not be an explanation for what is necessarily so, for what is explained is by definition contingent upon the thing that explains it, and what is necessarily true is by definition not contingent upon anything.
Moreover, to say that it is abstract higher mathematics that we apply to the real world is to completely misunderstand what mathematicians mean by “abstract, higher mathematics”. Topology and number theory seem abstract to most of us, but to mathematicians they are just a step above elementary. Moreover, we really don't use topology and number theory very often in science. Calculus strikes most people as very advanced, but mathematicians regard it as elementary. Calculus is derivable from Peano arithmetic by means well-known to any student of real analysis—which is to say that insofar as 2+2=4 and 3*4=12, the Chain Rule and the Fundamental Theorem of Calculus are true.
I guess it seems a little weird that we use pi in the equation for the normal distribution, but I think this is because we think of pi as “the ratio of circumference to diameter” when on a deeper level it is really more like “-i ln(-1)”. Then we would ask, “why is the ratio of circumference to diameter equal to -i ln(-1)?” but that is something that we can actually derive with calculus. Integrate over the arc length of the unit circle, and you will indeed come out with the value -i ln(-1), which we call pi. (Actually if you use the typical way, you will get something in the form of an arcsin, but that's all right, because the arcsin is fundamentally just a funny way of expressing complex natural logarithms. We can prove that 2 i sin(x) = e^{ix} – e^{-ix}.) It's just that we discovered circles (and hence named pi) long before we understood calculus, complex numbers, or natural logarithms. Once you realize that on a fundamental level pi has nothing to do with circles but instead is a compact representation of “-i ln(-1)”, asking why it appears in the normal distribution is like asking why the square root or the number 2 appears in the normal distribution. We can show you the derivation; it follows logically. If that doesn't satisfy you, well, that's your problem, not ours. It's certainly not math's problem.
Don't get me wrong; I do think that mathematics is beautiful, and that the universe is a marvelous, amazing place. I think we are incredibly fortunate to be the sort of being that has even a tiny sliver of understanding in this vast and unfathomable cosmos. I think it is wonderful that we share kinship not only with the apes and the dogs (who we always suspected were our cousins), but indeed with the pine trees and the jellyfish and the tuberculosis pathogen. I wish more people had a better understanding of all this.
I do in fact think it is incredibly fortunate that our tiny, limited minds are capable of glimpsing the fundamental order of the universe. I just don't see how there's anything unreasonable about that, or why we should be surprised or looking for an explanation for it. I certainly don't think we have any reason to doubt that it will continue to be true.

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