Mathematics is a part of what should be considered beauty. Centuries ago, the great astronomer Johannes Kepler (d. 1630) found that math enabled him to predict the relative distances between the various planets and the sun. This finding made him feel “possessed by an unutterable rapture,” bedazzled by “God [thereby] signing his likeness into the world,” symbolizing “all nature and the graceful sky” by “the art of geometry.” Modern scientists, such as the pioneer quantum physicist Paul Dirac, were similarly struck. “The fundamental physical laws are described in terms of a mathematical theory of great beauty and power.... 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.”
If mathematics has impressed many of its greatest theorists with its beauty, many scientists have been equally struck by the peculiarity of its ability to map nature with acute accuracy. The Nobel laureate physicist Eugene Wigner wrote a celebrated essay about how the ability of math to tell us about nature is “unreasonable.” Einstein said that our ability to make sense of the universe with mathematics is a “miracle,” for a universe arising from randomness would not be expected to be so rational as to correspond with such precision to the precise system that mathematics is. In his book The Mind of God Paul Davies registered his surprise at this correspondence. “Much of the mathematics that is so spectacularly effective in physical theory was worked out as an abstract exercise by pure mathematicians long before it was applied to the real world . . . and yet we discover, often years afterward, that nature is playing by the very same mathematical rules that these pure mathematicians have already formulated.”
Is there a link between the beauty of mathematics and its description of nature? There is—in music. Millennia ago, the Greek philosopher Pythagoras noticed that musical tones are in harmony if their tensions arc ratios of the squares of small whole numbers. These rules, translated into frequency, state that notes sound good together if their frequencies arc in ratios of small whole numbers. He was the first major thinker to see the connection between nature and music through the medium of mathematics. This “music of the spheres,” which Pythagoras thought was created by the harmonies among the sun and moon and planets, is the same “heavenly harmony” that Kepler described. The physicist Wilczek said they were on to something profoundly true. Atoms, he wrote, are tiny musical instruments. “In their interplay with light, they realize a mathematical Music of the Spheres that surpasses the visions of Pythagoras, Plato, and Kepler. In molecules and ordered materials, those atomic instruments play together as harmonious ensembles and ordered materials.”
So, mathematics links the world of nature and our minds by a kind of music. But do we comprehend it because we have created mathematics? Is it just another artificial human construct that we have imposed on the world? We have already seen that the fit between mathematics and the world is so close and precise that artificial imposition is unlikely. In fact, there is something about mathematics itself that suggests it has a life of its own completely apart from its use to describe nature. In other words, it seems to have its own existence and nature, quite removed from the physical world.
As particle physicist Peter Bussey argued, mathematics is not physics, even though physics depends on math. “Mathematics seems to have a life of its own.” Musician and science writer Kitty Ferguson told of the day when it dawned on her that mathematics has an objective reality apart from its use in science. “I remember clearly when it first dawned on me those human beings might have discovered mathematics, not invented it; that it might lie waiting in nature; that mathematical truth might be a part of independent reality. It wasn’t in mathematics class, but in music theory, when I studied the harmonic series. It seemed to me that this pattern could not be a human way of sorting things out. It would have existed even if human beings had never existed.”
Astrophysicist Roger Penrose also noted that mathematics has a transcendent, objective character. “There often does appear to be some profound reality about these mathematical concepts, going quite beyond the mental deliberations of any particular mathematician. It is as though human thought is, instead, being guided towards some external truth—a truth which has a reality of its own, and which is revealed only partially to any one of us.”
Once again, then, we are faced with oddities. This world seems to have been fine-tuned not just with one or two parameters but with many. If a few or perhaps even one of them were different, we would not be here. These corroborations of fine-tuning are strange enough, but the fact that there is beauty at the deepest physical levels of the universe, and that it has been noticed by agnostic and believing scientists alike, is also odd. Why would a randomly formed universe be beautiful? Furthermore, why would the beautifully rational system of mathematics so perfectly describe the way this world works? If this world is finally irrational, which is what one would expect from a cosmic accident (as it is said to be by atheists such as Richard Dawkins), then one would never expect a rational system like mathematics to fit it so closely and usefully.
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