Naming Infinity

If you were like me you grew up thinking that math stood as the most obvious, least abstract, and therefore most inherently “atheistic” of the disciplines.  Literature involved interpretation, who knows what happened in the past with History, but math was always math, brutal, hard, and cold.

Long after my last math class I found out that in ye olden days, Greek philosophers considered math the most inherently religious of the disciplines.  It involved, after all, abstractions, universals, unchanging reality, and perfection, the very things inherent in the spiritual realms.  After the Greeks, some of the greatest mathematical advances came from deeply religious people like Pascal and Newton.

A strength of Naming Infinity, by Loren Graham and Jean-Michel Kantor, is that it looks at some of the deepest questions about the truths in Math through one obscure group of people at the turn of the 20th century.  In Russia at that time a group of monks arose who engaged in a practice called Name Worshipping.  The roots of this practice go back to the “Jesus Prayer” prevalent in Eastern Orthodox spirituality, where a person repeats the prayer, “Lord Jesus, have mercy on me a sinner” continually to achieve greater spiritual awareness.  Name Worshippers went further, believing that “the Name of God is God.”  The name of God contains His character, so speaking the name of God revealed God Himself.

The church labeled this practice heretical.  God’s name stands for God Himself to be sure, but the church’s  main criticism of this practice stemmed from the seemingly “magical” qualities the monks attributed to the name itself.  And of course, since the name of God gets rendered differently in different languages, it could open up the charge of polytheism.

Naturally the monks accused of heresy had good arguments in their defense, denying that they believed literally in the divinity of the letters or sounds themselves.  Rather, the name of God stood as the most important signifier of divinity in the world (the idea of “signs” would prove to be a link to mathematical innovation). The book frustrates somewhat on this topic, because although this so-called “name worship” will have an indirect link to the story the authors tell, it gets dropped early in the book.  Still, the authors are not Church historians or theologians, and probably rightly step out of the way of an issue they wouldn’t understand well.  This is one weakness of the book — the authors introduce an esoteric and unfamiliar concept and then drop it for the vast majority of the story.

The real root of the book deals with a mathematical and not a theological controversy (though theology remains indirectly involved in the story).  On one side we had the French, who stressed “continuity,” the idea that to get from one mathematical point to another, one must pass through all intermediate points.  Math then, is a closed system, a measurable system, where numbers have a concrete reality that cannot be manipulated.

Russians occupied the other side, the more interesting one for me.  They stressed “discontinuity,” or the idea that reality is not as set in stone as we think, and thus, reality can be manipulated.  We can call reality into being through the creation of numbers, or sets of numbers. In the west at this time determinism held sway over many scientific and mathematical minds.  A Russian priest and mathematician, Pavel Florensky, led the opposition to this school of thought.  He preached discontinuity in all fields, not just in math.  Here the book frustrates yet again, for they back off from going into any real detail linking mysticism and math.  I think the central idea of the Russian school had to do with the concept of naming sets of numbers (linked to set theory), and hence the connection to the heretical school of Name Worshippers.  To quote the authors,

To take a simple example, defining the set of numbers such that their squares are less than 2, and naming it “A,” and analogously the set of numbers such that their squares are larger than 2 and naming it “B,” brought into existence the real number the square root of 2 (emphasis mine).  Similar namings can create highly complex new sets of real numbers.  . . . When a mathematician created a set by naming it, he gave birth to a new mathematical being.

If math dealt with more than finite possibilities, then “real reality” too had to be more than just finite.  The connections of math with religion become obvious, as creation happened in Genesis 1 via the Word, via naming (this idea is present in Egyptian texts as well).

Freed from normal approaches to mathematical questions, the Russian school made key advances in math.  They also taught in new and unusual ways.  One student recalls that,

Luzin would start from the outset by posing to his students who were hardly out of high school problems of the highest level, problems that stymied the most eminent scholars.

The authors add,

One characteristic of the Russian school stood out — the conviction of the best Russian teachers that the most fruitful attack on problems was direct and straightforward, without any preliminary, long, and heavy reading.  In other words, start from scratch.  By doing so one got an almost physical feeling of being directly in contact with mathematical objects and experienced the sensual pleasure of having to fight intellectually with one’s bare hands.  One of the great mathematicians of the time, Israel Moisseivich Gelfand, “We should study this topic before it has been tainted by handling.”*

Here we sense the mystical side of math where one bypasses “matter” to get right in touch with “reality.”  It sounds thrilling, but I don’t understand it.  I never was any good at math, but this sounds appealingly very little like the math I had in high school.  But one also might sense its weaknesses.  The great Luzin (mentioned above) would sometimes brag that he “never solved equations anymore.”  That is, math resided for him not in reality, but perhaps in some gnostic fairy world.  Math need now always have a direct physical application to have value.  The training of the mind itself has great value.  But math must, I think, have some “physical” applications to root us close to the Incarnation.

But though I found the book oddly structured, and though it bounced around too much from topic to topic, the book has great value in exposing western laymen like myself to a whole new way of thinking about math, and about reality itself.


*This whole approach reminds me of Dostoevsky’s theories on reality as it applied to gambling.  In his story The Gambler, it seemed to me that he thought the interaction of the human will could influence the games played.  It was never about mere statistics.  Likewise, I had a friend who swore that he developed a “system” to win on roulette, which seems like a game one must lose if played for any length of time.  Yet he assured me that over the course of more than a year, with 10+ trips to a local casino, that he had come out ahead $880.