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God By the Numbers
 

Why is nature constructed along these lines? One could perhaps describe the situation by saying that God is a mathematician of a very high order, and he used very advanced mathematics in constructing the universe.

-- Physicist Paul Dirac

More than 50 years ago I heard Frederick Buechner, the Christian author and minister at my old prep school, preach a sermon in which he posed this question: If God exists, why doesn’t he just show himself? Why didn’t the Lord just spell out a message in the stars that would remove any doubt? As a matter of fact, there is a long tradition among philosophers and theologians who maintain that the Creation is itself proof of God’s existence, assuming you know how to read it. The 18th-century naturalist and Anglican cleric William Paley reasoned that if a watch requires a watchmaker, then an object even more complex than a watch – say, the human eye – would also require a maker. The Roman orator Cicero made a similar argument using sundials and water clocks: “How then can you imagine that the universe as a whole is devoid of purpose and intelligence, when it embraces everything, including those artifacts themselves and their artificers?” These so-called “arguments from design” continue to this day among proponents of intelligent design. However, the scientific community has generally insisted that the patterns and complexity found in nature all have natural causes. Even Buechner is his sermon doubted it would make much difference in the long run if God spelled out his name in lights.

But what if God announces his presence not with letters but with numbers? Start with the fact that the universe operates according to precise physical laws that can be expressed mathematically. Why should this be so? “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” Einstein wondered. He was not alone. Nobel Prize-winning physicist and mathematician Eugene Wigner wrote an influential scientific paper in 1960 noting “the enormous usefulness of mathematics in the natural science is something bordering on the mysterious and there is no rational explanation for it.” Physicist Paul Davies agrees that “there is no logical reason why nature should have a mathematical subtext in the first place, and even if it does, there is no obvious reason why humans should be capable of comprehending it.” He adds, “The evolving cosmos has spawned beings who are able not merely to watch the show, but to unravel the plot.”

Cognitive scientists believe humans have an innate capacity for numbers that causes the mind to recognize mathematical patterns in the world. What Davies and others find so confounding is that the human brain is hardwired to do math that is far more advanced than anything that could have evolved by the rules of natural selection alone. Yes, you need advanced mathematics to design quantum computers or send a rocket to Mars. But the mental circuitry to do the math was already in place when hunter-gatherer tribes were still hunting mastodons with spears. This would appear to be putting the evolutionary cart before the horse. “These higher intellectual functions are a world away from survival ‘in the jungle,’” Davies says “Many of them were manifested explicitly only recently, long after man had become the dominant mammal and had secured a stable ecological niche.”

Then there is the whole question of whether numbers are truly a product of human thought, as Einstein suggested, or whether they somehow exist independently of human involvement. In other words, did we invent them or discover them? Numbers would certainly seem to be deeply embedded in the workings of the universe, even if no one can quite figure out how these abstract entities operate in the physical world. Many mathematicians take it a step further and insist that numbers exist in a kind of Platonic realm outside of time and space, which still leaves open the question of how they are seemingly able to move pieces around on the chessboard of the universe.

For those of us who never got past introductory calculus, the idea that “God is a mathematician of a very high order,” as Paul Dirac expressed it, might appear to do little to clarify things. What does it mean when a mathematician or physicist calls an equation “beautiful,” if it looks to the rest of us like a Mayan hieroglyph? But then I discovered there is abundant photographic evidence of God’s mathematical handiwork in nature; in fact, I had taken a number of such pictures myself without realizing it.

Lately I have been photographing wildflowers I find growing by the side of the road. I shoot them in a dark room under a cone of white light, using a macro lens that allows me to get in close. I suspect I will never appreciate the beauty of an equation in written form, but it is quite another matter when the equation is expressed as a fractal, a geometric configuration that abounds in nature, including the intricate spiral formations in the head of a daisy or the tiny iterative blooms in Queen Anne’s Lace. Once you know what you are looking for, fractals can by found anywhere: in crystals, snowflakes, pineapples, Romanesco broccoli, lightning bolts, tree branches, even coastlines – all of which can be expressed mathematically.

Admittedly, there are gaps in my education, particularly when it comes to higher math. But I have an excuse this time: there wasn't even a name for fractals until I was out of school. They were discovered – if that is the word -- by Benoit Mandelbrot, a French mathematician who spent most of·his career at IBM, because no university worthy of his talents would hire him. (Late in his career he landed at Yale, which awarded him tenure at age 75.) According to Mandelbrot, fractals are natural or man-made objects that are “irregular, rough, porous or fragmented and which possess these properties at any scale. That is to say they have the same shape, whether seen from close or from far.” To me, their significance is not just that they can be described mathematically, but that many of them are quite beautiful. If God is a mathematician, then he is also an artist.

Dirac did not mean to be taken literally when he said God was a mathematician of a very high order; indeed, like many in the scientific community, he was an atheist. As he told a gathering of his peers, “If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality.” What was most real to him – even more real than everyday reality – was numbers. He was another mathematical Platonist, which meant that for him numbers actually existed and mathematics was essentially an act of discovery. By his lights, numbers were eternal and unchanging. They transcended time and space and were entirely independent of human thought, although the mind had access to them.

Harvard mathematician Barry Mazur has observed that mathematical Platonism is at least implicitly a “theistic” position. Indeed, for most of the time people have been counting higher than the number of fingers on their two hands, mathematics has been explicitly so. The ancient Greek philosopher Pythagoras believed numbers were "the principle, the source and the root of all things” and were divine. The 19th-century German mathematician Georg Cantor, who is most closely identified with set theory, came up with a proof for actual infinite numbers. He equated God with the “Absolute Infinite” and believed God had chosen him to communicate his ideas on infinity to the world. According to the German philosopher Ludwig Feurbach, we all tend to make gods in our own image. There is no reason to think mathematicians are any less prone to this than anyone else, even if, like Dirac, they don’t believe God exists.

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