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The Unreasonable Effectiveness of Mathematics

by Jonathan Witt
(Dr. Witt is a Fellow of Discovery Institute and of Acton Institute)

Summary:Derek Abbott's "Is Mathematics Invented or Discovered?" asks why mathematics is so effective in describing our universe, and ultimately reduces the debate to a simplistic binary of mathematics as wholly created (Abbott's position) versus the neo-Platonic idea that mathematical models can perfectly and exhaustively describe nature. Abbott overlooks the view that drove the founders of modern science: the cosmos is the product of an extraordinary mathematician but one not restricted to the mathematical. Moreover, because the founders of modern science had theological reasons for emphasizing not only the cosmic designer's surpassing intellect and freedom but also human fallibility, they emphasized the need to test their ideas empirically. In these and other ways, Judeo-Christian theism matured Platonism and, in the process, sparked the scientific revolution.

Derek Abbott's recent piece in The Huffington Post, "Is Mathematics Invented or Discovered?", offers a thoughtful taxonomy of views on an issue with important metaphysical implications, but a crucial alternative possibility goes unexplored in the essay. Since Ben Wiker and I explore these issues in our book, A Meaningful World: How the Arts and Sciences Reveal the Genius of Nature, I'd like to summarize what I find useful in Abbott's piece and what I find incomplete.

The Abbott essay boils down to an effort to answer a question that thinkers have wrestled with for centuries and that was nicely expressed by Albert Einstein in this way: "How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" Abbott says there is no consensus among mathematicians and scientists, but highlights four common answers:

"1) Math is innate. The reason mathematics is the natural language of science, is that the universe is underpinned by the same order. The structures of mathematics are intrinsic to nature.... 2) Math is a human construct. The only reason mathematics is admirably suited for describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes.... 3) Math is not so successful. Those that marvel at the ubiquity of mathematical applications have perhaps been seduced by an overstatement of their successes. Analytical mathematical equations only approximately describe the real world, and even then only describe a limited subset of all the phenomena around us.... 4) Keep calm and carry on. What matters is that mathematics produces results. Save the hot air for philosophers. This is called the 'shut up and calculate' position."

One doesn't have to read very hard between the lines to quickly pick up where Derek Abbott's sympathies lie. He calls #3 the realist position. Without the question-begging term appearing in scare quotes and with only a little reflection it will dawn on the reader that Option #3 primarily functions as another way of arguing for the #2 Option (math is a human construct). Option #3 also implies a straw-man characterization of Option #1 (math is innate), suggesting as it does that leading contemporary proponents of #1 necessarily assume...what?--that planets are geometrically perfect spheres?That E=MC2 perfectly describes the movement of bodies? Or that to see nature as possessing inherent mathematical regularities is, willy nilly, to see it as exhaustively algorithmic? But no contemporary proponents of #1 that I'm acquainted with thinks any of these things.

It's true that some version of Option #1 tends to be the position of theistic mathematicians and scientists, but because theists posit a designer who is free to instantiate designs that may or may not manifest mathematical regularity, a theistic proponent of Option #1 is unlikely to insist that the operations of the natural world are wholly mathematical. Modern theists tend to assume that the maker of the universe is a crackerjack Mathematician, but these same theists also assume that this maker is undoubtedly also a pretty fair Author and Artist. The biological information necessary for the first living cell, after all, is not the compressible, mathematically tractable information of the algorithm. It's information more akin to that in a book or in the software and hardware of computer technology (although almost unimaginably more sophisticated).

Abbott mentions the discovery of fractals in nature, "complex patterns, such as the Mandelbrot set ... generated from simple iterative equations," but dismisses their design implications by arguing that "any set of rules has emergent properties. For example, the rules of chess are clearly a human contrivance, yet they result in a set of elegant and sometimes surprising characteristics." Here perhaps more clearly than anywhere else in the essay we are given a glimpse at the root of Abbott's confusion. With the chess example, he's confusing the act of creating a mathematical description (rather than discovering its existence in nature) with the designing act of creating the rules of chess. But a theistic (or even deistic) understanding of Option #1 would consider the invention of chess as an echo of the cosmic designer's invention of various mathematically tractable regularities that manifest themselves as beautiful patterns in the natural world (e.g., spiral galaxies or snowflake patterns). On this view, the parallel to discovering fractals in nature would be a student of chess discovering certain elegant patterns (and perhaps their concomitant strategy and tactics) in the game of chess.

Abbott proceeds to the trope of infinite monkeys creating meaningful prose as they bang away at random on keyboards. "It appears miraculous when an individual monkey types a Shakespeare sonnet," Abbott writes. "But when we see the whole context, we realize all the monkeys are merely typing gibberish. In a similar way, it is easy to be seduced into thinking that mathematics is miraculously innate if we are overly focused on its successes, without viewing the complete picture." No, just the opposite: when we view the complete picture of the universe, what physicists, astronomers, and cosmologists refer to as the fine tuning problem comes sharply into focus. We now know that numerous laws and constants of physics and chemistry appear fine-tuned to an almost unimaginably precise degree to allow for an evolving cosmos where life could exist--such as the strength of gravity and the strong and weak nuclear forces.

So why is the cosmos not instead one of the almost unimaginably more numerous set of theoretically possible configuration--ones that would have made complex chemistry impossible and so, too, complex life? Cosmologists allergic to theism have at least a couple strategies for explaining away the fine tuning problem. One is to shrug and say, "Well, if the universe weren't such as to allow for complex and even intelligent life, we wouldn't be here to wonder about it." That's a bit like a prisoner surviving a rain of bullets from the firing squad unscathed, and when he opens his eyes and finds a perfect bullet pattern around his body, he immediately concludes that it was pure luck rather than the merciful design of the sharpshooters. "Well after all," he tells his skeptical friends, "I wouldn't be here to wonder about my curiously good fortune if the bullets hadn't been fine tuned to miss me, now would I?"

The other common dodge around the fine-tuning problem brings us closer to the rhetorical sleight of hand that Abbott indulges in during this stretch of his article. He speaks of "infinite monkeys" banging away on keyboards. Set aside for the moment that even this picture presupposes the intelligently designed machinery of keyboards and computer screens. The more immediate question is, how did an infinite slip into the party? Aren't we trying to explain something about the physical universe? Well, in order to explain the curiously fine-tuned nature of our universe, some materialists have posited multiple and even infinite universes (unseen and undetectable), ours being one of the lucky ones configured just so to allow for complex and intelligent life.

Here we have the classic gambler's fallacy. A naïve man is at the roulette tables in a shady speakeasy admiring the winning streak of a guy who looks like he walked right out of central casting for The Godfather. The casino worker running the roulette wheel is sweating profusely under the occasionally threatening glare of the gambler, and the roulette wheel keeps landing on the lucky gambler's number. "Gee!" the onlooker comments, "I bet the odds of this are one in a trillion billion billion. Imagine how many gamblers must be playing roulette all over the planet right now, and here I just happen to be beside Mr. Lucky!" (The probability of a single universe just happening to be fine-tuned for intelligent life, incidentally, is astronomically higher than one in a trillion billion billion).

To his credit Abbott rejects the "fuggedaboutit" non-answer that is Option #4; but then he reduces the debate to a simplistic binary of non-Platonist/Platonist--mathematics as wholly created versus the dead-horse position that our mathematical models can perfectly and exhaustively describe nature.

There's another possibility, and it's one that drove the founders of modern science by drawing them beyond an unrealistically tidy Platonism and toward the humble and searching flexibility of theism. It begins with a fundamental question that gets muddled and missed in Abbott's analysis: Why a cosmos rather than a chaos? Why a universe where highly elegant mathematical models (e.g., Kepler's laws of planetary motion, Einstein's theory of relativity) even approximate so many of the regularities of physics and chemistry, regularities that even undergird the cutting edge scientific engineering work that employ other analytical tools in addition to the purely mathematical?

And more than this, why a universe where one quite accurate mathematical model (Newton's law of universal gravitation) can so compactly and elegantly describe the gravity of large bodies, and then be superseded by an even more precise and more elegant model, as if the cosmos were not only mathematical but fashioned in such a way as to allow us to progress in stair-step fashion from one model to the next? It's the question Guillermo Gonzalez and Jay Richards ask in their groundbreaking book The Privileged Planet. Why does the universe appear not only fine tuned for intelligent life but also fine tuned for that intelligent life to discover the underlying order of the cosmos?

The perfect solar eclipse is the everyman's icon of this puzzler. Perfect solar eclipses have proven crucial in helping astronomers test Einstein's theory of relativity as well as unlock the nature of distant stars. But to be useful, the apparent size of the sun and moon in the sky has to be virtually identical. The sun is 400 bigger than the moon, but it just so happens that it's also 400 times further away, meaning it has virtually the same apparent size as the moon in our sky. The whole thing seems rigged to allow humans to make scientific discoveries and to evoke wonder at the elegance and beauty of it all.

Here it's difficult to talk nonsense about perfect eclipses being "a human construct," and the patent absurdity of that notion can help wake us up to the only slightly more subtle absurdity of claiming that the curious effectiveness of mathematics for describing the physical universe is also purely a human construct.
There is a cosmos, a meaningful world, whose existence needs explaining. The answer that the founders of modern science gave was that the cosmos was the product of a reasonable designer, one whose mathematical intelligence far exceeds that of any human, and yet because the human person (including the human mind) is made in the likeness of that designer (one capable of reason, imagination, and discovery), we have some hope of exploring and discovering some of the underlying mathematical order of that grandest of mathematicians.

Moreover, because humans are finite and flawed, the theists who founded modern science--Copernicus, Galileo, Kepler, Boyle, Newton and all the rest--were well primed to realize that our assumptions about how the designer would have done things might be mistaken, thus the strong emphasis on the need to test those ideas, to put our hypotheses "in empirical harm's way," as philosopher Del Ratszch put it. Then, too, theistic scientists had in view a personal being as the source of the cosmos, leaving them more open to finding the kind of design characteristic of a painting or a book--that is, non-repeating, non-algorithmic design--thereby freeing them from the Platonist's box of the geometrical/mathematical (or at least from the box of a certain type of neo-Platonist).

These elements of theism may be why the founders of modern science tended to be more empirically oriented than was characteristic of the proto-science of the ancient Greeks. The theist, you see, expects underlying order, but he expects the underlying order of nature to run deeper still, and so the drive of the founders of modern science to continue digging for deeper and deeper levels of hidden order. In these and other ways, Judeo-Christian theism can be said to have taken up and matured Platonism and, in the process, sparked the scientific revolution--which, after all, did not begin in ancient Greece or Rome, Arabia or India, but in Christian Europe.

You can email brucechapman@discovery.org

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