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.

Contradiction

Well, having become interested in the theological views of certain famous mathematicians recently, I decided to read Father Florensky's The Pillar and Ground of the Truth.

I realize this is not the kind of everyday pleasure reading that most people would engage in. But it appeals not only to my tendency towards abstraction (hello, I'm a mathematician), but also my deep personal desire to answer those scary questions, like what is truth?

This book has had a profound impact on me. Not that everything it says is thoroughly compelling to me, but its approach is unlike anything I've seen before. I love how Florensky starts from such an intensely personal point of view, and then delves into philosophical abstraction not only with intellect, but with passion.

The (Eastern) Orthodox way of thinking is fiercely Trinitarian. This is the Ground of the Truth. There simply is no Truth without Trinity.

Already in this we see Florensky powerfully seek to dismantle Western rationalism. For him any attempt to search for the Truth based on "givens," the axioms of the human mind, is deeply flawed. Thus he strongly condemns the Law of Self-Identity:

"A = A. That is the final answer. But this tautological formula, this lifeless, thought-less, and therefore meaningless equality A = A, is, in fact, only a generalization of the self-identity that is inherent in every given. But by no means is this formula an answer to our question 'Why?'

...

"'I = I' turns out to be nothing more than a cry of naked egotism: 'I!'"

The Truth, on the other hand, is absolute precisely because it transcends this egotistical self-assertion. It finds its self-identity through the denial of self, receiving itself back through love.

"Instead of an empty, dead, formal self-identity A = A, in virtue of which A should selfishly, self-assertively, egotistically exclude every not-A, we get a real self-identity of A, full of content and life, a self-identity that eternally rejects itself and that eternally receives itself in its self-rejection.

...

"Truth is the contemplation of Oneself through Another in a Third: Father, Son, and Spirit."

So in this view, Truth is found not (as dear old Schaeffer would have it) in the Principle of Non-Contradiction, but rather in the Principle of Self-Contradiction, the principle of denying oneself in order to find self-identity through love, rather than rationality.

This view of the Truth also leads to a similarly styled view of the truth (with a lower-case "t"). If the nature of Truth is the denial of this self-identity "I = I," then it's not surprising that Florensky comes to a view of "truth" (meaning knowledge about the Truth) that relishes antinomy, or contradiction:

"A rational formula can be above the attacks of life if and only if it gathers all of life into itself, with all of life's diversity and all of its present and possible future contradictions. ... It follows that truth is a judgment that also contains the limit of all its refutations, or (in other words) that truth is a self-contradictory judgment."

But if you really hate all this abstraction, you'll thank Florensky for pointing out how concretely this idea appears in scripture. He actually gives a little list of antinomies in scripture (one or two examples omitted):

Divinity: both One and Three
Christ: both human and divine
Relation of Man to God: both predistination (Rom 9) and free will (Rom 9:30 - 10:21)
Sin: both through the fall of Adam (Rom 5:12-21) and inherent in the flesh (1 Cor 15:50)
Retribution: both retribution according to works (Rom 2:6-10, 2 Cor 5:10) and free forgiveness (Rom 4:4, 9:11, 11:6)
Final Fate: both universal restoration (Rom 8:19-23, 11:30-36) and the "double end" (Rom 2:5-12)
Works: both the necessity of works (1 Cor 9:24) and the lack thereof (Rom 9:16). See also Phil 2:12 and then 2:13 for a related antinomy
Faith: both free and depending on free will (John 3:16-28) and God's gift not found in human will but in the will of God (John 6:44)
Judgment: "For judgment I come into the world" (John 9:39) "I came not to judge the world" (John 12:47)

Florensky isn't giving skeptics a cheat-sheet for how to argue against Christianity. He's stating powerfully what he feels to be the very strength of Orthodoxy.

"Contradiction! It is always a mystery of the soul, a mystery of prayer and love. The closer one is to God, the more distinct are the contradictions. In Heavenly Jerusalem, there are no contradictions. Here, on earth, there are contradictions in everything; and they can be removed neither by social reorganization nor by philosophical argument."

In this way he disparages all rationalistic worldviews. (No doubt the Soviets didn't approve of this, as their own Marxism depended on the strength of "social reorganization" to solve the contradictions of life.)

It is with these things in mind that I read a recent article by Al Mohler disparaging the postmodernism of "emergent church" types. He writes:

"The problem is this -- [the] argument that truth is plural means that the church should both embrace and celebrate different and even contradictory understandings of these doctrinal statements and core truths."

I wonder what Florensky would have to say about this classical Protestant attitude?

"A heresy, even a mystical one, is a rational one-sidedness that claims to be everything. ... Orthodoxy has a universal nature, but heresy essentially has a sectarian nature. The spirit of a sect is the egoism that emanates from it, spiritual separateness. A one-sided proposition takes the place of absolute Truth..."

Under this view, I suppose that Protestantism is, by its very nature, heretical. At its very core is the desire to get the Bible right, to boldly affirm one interpretation to the exclusion of other contradictory interpretations.

I wonder if Florensky has a point. The empirical evidence from the Bible is almost compelling enough. Can anyone actually resolve these passages together logically? Somehow I think those who do inevitably suck the life out of one passage or another.

But if that's not enough, can't you just take a look around at the sheer number of Protestant denominations and ask, didn't these all come from a desire to be "right"? Is there maybe a hint of egoism in our Protestant tendency to split, not only from the historic Church, but even from each other?

On the other hand, Protestantism is a reaction against idolatry. Thus it always seeks to remind humanity that its conceptions of God are always too small. If only Protestants would continually apply this principle to our own rationalistic interpretations of scripture!

There are many reasons to think about this conception of truth other than to critique Protestantism, for sure. This just happens to be on my mind at the moment, as it often is.

In any case, The Pillar and Ground of the Truth is shaping my thinking in lots of ways, and this won't be the last time I mention it.