“Unless a grain of wheat falls into the earth and dies, it remains alone; but if it dies, it bears much fruit.” – John 12:24
“The last enemy to be destroyed is death...Death is swallowed up in victory.” – 1 Corinthians 15:26, 54
"The greater the sin, the greater the mercy, the deeper the death and the brighter the rebirth.” - C. S. Lewis
"This story...has the very taste of primary truth." - J. R. R. Tolkien

Monday, May 12, 2008

A Theology of Mathematics: Mathematical Beauty

4. Mathematical Beauty

Further support for a realist interpretation of mathematics can be found in the “beauty” and “depth” perceived in mathematical laws and equations. Anyone who has studied mathematics in some depth perceives, at least to an extent, the elegance of mathematics and of the mathematical description of physical reality. For example, Bertrand Russell wrote that “mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere” (Russell 47). In stating “God made the integers; all else is the work of man” (Bell 477), Leopold Kronecker expressed a similar sense of awe towards numbers. What is it about simple numbers that brings about such striking statements? This perception of beauty and depth is quite remarkable (as is our perception of beauty and goodness more generally).

The natural numbers, and the prime numbers in particular, have been thought to be foundational and central to mathematical truth and beauty. Mathematicians today are still hard at work to discover a deeper structure behind the numerous mysterious patterns formed by prime numbers. In fractals we see how beautiful visual structures and order can be constructed, as it were, out of chaos. The “divine proportion” φ=(√5+1)/2 is the ratio for which the ratio of two quantities equals the ratio of the sum of the two to the larger of the two, but it is also the ratio approached by adjacent terms in the Fibonnaci sequence. This proportion has been found to be the built into the structure of many things in the physical world. These and other instances of mathematical beauty point to an underlying simplicity and interweaving of what appears on the surface to be complexity or randomness.

The so-called “imaginary” number i adds a whole new dimension to mathematics, revealing deeper connections and an elegance that suggests it may not be so imaginary after all. Perhaps the most “beautiful” and “deep” equation in mathematics is Euler’s identity, eiπ+1=0. If any equation touches upon the core of mathematical truth, this is it. In the words of Stanford mathematics professor Keith Devlin, Euler’s equation “reaches down into the very depths of existence” (Nahin 1). Nineteenth century mathematician Benjamin Pierce described it as “absolutely paradoxical; we cannot understand it, and we don’t know what it means, but we have proved it, and therefore we know it must be the truth” (Kasner and Newman 103–104). According to Charles Edward White, professor of Christian thought and history at Spring Arbor University, “a mathematics professor at MIT, an atheist, once wrote this formula on the blackboard, saying, ‘There is no God, but if there were, this formula would be proof of his existence’” (White 4). Although the professor apparently demands God’s existence as a condition for the proof thereof and thus implicitly uses circular reasoning, he nevertheless offers a striking statement. White goes on to explain that Euler’s equation brings together what are probably the five most important numbers in mathematics with the central operations of addition, multiplication, and exponentiation, and goes on to say that:

The idea that these two irrational numbers should combine with an imaginary one to yield so utilitarian a result is breathtaking…That these three strange numbers with such diverse origins should work together to produce a result so basic to mathematics argues that there is a profound elegance or beauty built into the system…It seemed to argue that there was a plan where no plan should be (White 4, emphasis added).
This unity and simplicity of the amazingly diverse and complex areas of mathematics calls to mind Colossians 1:17, “in him all things hold together.”

This perception of mathematical beauty is not limited to pure mathematics, but extends into the physical world, which is saturated with things that can be described mathematically. In fact, “Sophisticated theories, such as relativity or quantum mechanics, can be aptly summarized in just a few small mathematical equations and their logical implications” (Byl 4). Simple mathematical equations such as the Gaussian distribution arise again and again in descriptions of vastly different physical phenomena. John Polkinghorne writes “It is a technique of proven fruitfulness in fundamental physics to seek those theories whose expression is in terms of beautiful equations” (Polkinghorne 72). Indeed, the elegance of string theory is acknowledged as evidence in favor of it, regardless of what experimental evidence there may be. Because of these numerous discoveries of mathematical simplicity and elegance, physicists have “at least unconsciously…abandoned a raw naturalism in favor of a theory formation method that has principles of beauty embedded in its core” (Howell 502). Some mathematicians and physicists may be naturalists, but in giving so much weight to abstract principles like beauty, they run the risk of being philosophically inconsistent. In his celebrated paper, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Eugene Wigner describes and demonstrates the miracle of mathematics in the world, arguing that “the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language” (Wigner). Wigner connects this to man’s search for a deeper, “ultimate” truth.

Having looked at several specific instances of the “diverse unity” of mathematics and of the perception of this beauty among both mathematicians and physicists, let us consider what this may imply for our understanding of the nature of mathematics.

5. Beauty and Epistemology

When we examine the human reactions towards numbers and the intricate yet simple patterns and connections that they form, it becomes clear that these thinkers have indeed perceived something powerfully real and true. In considering the implications of this remarkable perception, and of knowledge in general, it is helpful to conceive of different probabilistic levels of epistemology. Obviously we are more certain of some proposed truths than of others. For example, we perceive a real physical world, yet we must acknowledge that there is a small but finite possibility that our sensory perceptions are incorrect, and there is no world. However, when we carefully consider seemingly more distant realities such as moral values or the beauty of music, intuition tells us that these values are absolute, powerfully real, and foundational to existence in a way that physical perceptions are not. We are more certain of their reality. In the same way, when we perceive the “beauty” and “depth” of mathematics, it is intuitively clear that we are seeing something deeply and absolutely real. That is, the beauty indicates reality in a more powerful way. When we consider mathematics, then, the magnitude of the beauty of number shows us that number is real (after all, only real things are beautiful). One might object that all this relies too much on emotion at the expense of rational thought. However, there is no reason to think that emotion does not indicate something important about reality, or that emotion must necessarily be at odds with rational thought. In the words of Blaise Pascal, “the heart has its reasons of which reason knows nothing.”

This perception is really quite remarkable; it is, in a sense, an observation of some component of reality beyond the physical world. We perceive these “very puzzling and mysterious” abstract realities (Barker 69) which cannot be located in space or time – and not only numbers, but also similarly abstract realities such as beauty and moral values. The human mind seems to touch on a realm of reality beyond the physical.

In summary, unless numbers are real, they cannot be described as elegant. There is, however, no good reason to doubt the reality of this perceived beauty and order any more than there is reason to doubt the reality of the world we perceive. If we hold to the premise that “epistemology models ontology,” which we all assume to be true in our everyday lives (that is, we live as if the world is real), then it would seem that the mainstream realist interpretation of mathematics is correct.

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