Beautiful Mathematics

Peter Kuchment is here helping to teach us about Radon transforms and medical imaging. He described the way in which Johann Radon originally discovered the Radon transform: it was just a side project, for fun, without any "useful" purpose. We now know that the mathematics Radon did as a side project in 1917 are foundational for computerized tomography, for which Allan Cormack and Godfrey Hounsfield won Nobel Prizes in physiology in 1979.

Kuchment suggested that Radon probably worked on the Radon transform out of sheer curiosity, simply because the idea was beautiful. He said, "If it's beautiful, it must be important." Mathematicians work on problems for their own sake; but the great mathematicians, according to Kuchment, always seem to find problems beautiful which also have a long term impact on the sciences. The lesson to be learned from this is that one ought to pursue mathematics for the sake of its beauty, and that "if it's not beautiful, it must not be that important, anyway."

There seems to be an implicit theology at work here. Truth, beauty, and goodness are unified in a transcendent way. The pursuer of truth needs not only logical skill but also a heightened sense of beauty. He who discovers what is truly beautiful also discovers both what is true and what the world needs. That the universe should really be this way is a pretty remarkable presupposition. I find this presupposition both deeply theological and deeply compelling.

It might take a bit of sheltering to preserve this kind of faith; I don't know for sure. After all, this world often seems fragmented. Rampant consumerism, promising to yield all that the world needs, has often dulled our sense of what is beautiful. The cultivation of "high art" has often been disconnected from anything true or meaningful. And the rigorous pursuit of scientific "facts" has often been done without regard for anything actually worth valuing, right down to scientific progress being used to destroy human life rather than preserve it. Truth, beauty, and the good seem miles apart from one another, and most people actually seem to stop believing in one or more of them. Thus cynics on the one hand accept the hard facts of life while rejecting its inherent beauty, while optimists on the other hand reject facts for the sake of beauty and goodness.

But then, mathematics is a subject that demands patience. It is a pursuit of the beautiful, but it doesn't always appear so. If we want to find beauty, we cannot listen simply to our hearts; we have to take the hard path of contemplation and rigorous logic. And if we seek what is truly beneficial to society, there again we must be patient. Without taking the long journey of pursuing what is beautiful and abstract, we can never lay a foundation for more powerful applications. So in some sense the belief in the unity of truth, beauty, and goodness is hard earned. It doesn't come naturally but must be slowly cultivated over time.

In this sense, I find mathematics one of the most deeply spiritual things we do as human beings.