Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity by Loren R. Graham
My rating: 1 of 5 stars

A Badly Hidden Agenda

Professional mathematical competence has no obvious link to religious conviction; nor does religious conviction often depend on scientific analyses. Nonetheless, at the turn of the twentieth century, a crisis in worldwide Christian theology and in mathematics was brought about, at least in part, by advances in mathematical set theory. These advances shed new light on what had been a persistent mathematical surd and a traditional theological meme - the concept of infinity.

Naming Infinity purports to tell a story about the interaction of theology and mathematics at the turn of the twentieth century. In fact, it is a subtle polemic masquerading as a history of science. The implicit purpose of the book appears to be the promotion of a rather nasty intellectual and political prejudice. Graham and Kantor are pros in providing extraneous and excessive biographical detail, but fairly light on comprehensible mathematical explanation. Worse, they have very little understanding of theology and no comprehension of the sociology of organizations that is relevant to their principle narrative.

In the eleventh century, a theologian and philosopher, Anselm of Canterbury, inspired by the pagan philosopher Plotinus, formulated a somewhat novel argument for the existence of God. Instead of making assertions about reality and then trying to demonstrate that these assertions imply a divine being, Anselm took a different approach. He simply defined God as “that which nothing greater can be thought.” That’s it, a neatly concise statement. This came to be called the Ontological Argument because it has to do with the necessary existence of God as a concept.

The Ontological Argument has fascinated philosophers ever since Anselm’s formulation. The range of opinions about its validity is extreme - from those who find it completely meaningless to those who believe it to be brilliantly definitive. But not one of Anselm’s opponents or defenders would ever seriously contend that Anselm believed he was creating God through his definition; he was merely providing a positive specification. Anselm was a philosophical Realist who considered the connection between words and things - metaphysics in modern language - were problematic but subject to analysis and improvement.

The authors of Naming Infinity make a brief derogatory reference to Anselm using a jocular comment by a nineteenth century philosopher. But they simply ignore the fact that the Russian mathematicians at the centre of their story, also inspired by Plotinus, followed Anselm in his philosophical realism. Rather Graham and Kantor choose to make the absurd claim that these respected scientists were devotees of a mystical cult which sanctioned ‘naming’ as a virtually magical act which conjured up the reality from the name. The Russian mathematicians, they contend, considered they were making God as well as numbers real by their linguistic representation. At best this is a pathetic misinterpretation; at worst it is a purposeful slander of a group of mathematical realists, as well as a group of mystically-minded monks with whom they were associated

Graham and Kantor supply no credible evidence for their contentions. They claim, for example, that the Russians were motivated by an offhand quip of the mathematician Georg Cantor, the inventor of set theory. Cantor had remarked that by naming mathematical sets he created them. However it’s highly unlikely that Cantor intended the remark in the way presented by the authors, and certain that the Russians, through their own denials, didn’t interpret it to mean what the authors suggest.

Graham and Kantor also use the Russians’ religious links to the so-called Name Worshippers in the Orthodox Church, whom they think believe in the mystically creative power of the divine name, to accuse the mathematicians of some sort of spiritual black magic. The implication made by the authors is that the mathematicians had compromised their science through their religious beliefs This is simply untrue and a fundamental distortion of the mystical practices involved, which are very similar to other Christian as well as Jewish, Islamic and Buddhist forms of prayer. These forms, although clearly mystical, are not considered by any faith-group of which I am aware to be generative of reality in the sense used by the authors. Even at their most radical, for example in the Zohar of medieval Judaism, they are merely poetic interpretations of what is considered a pre-existing metaphysical reality.

Finally, the authors dwell on the condemnation of the so-called Name Worshippers by the Orthodox hierarchy as evidence of the superstitious character of both the mystical practitioners and the mathematicians. This is simply absurd and demonstrates the authors’ lack of understanding of both theology and hierarchical organizations like the Orthodox Church. What the members of this sect, largely Russian monks resident on Mt. Athos in Greece, threatened was not the introduction of magic into Christian practice, but like all mystics, the authority of the Orthodox hierarchy and, eventually, the Soviet state. Because mysticism is an entirely subjective experience which is usually independent of credal or doctrinal assertions, it is literally beyond the reach of authority seeking to maintain orthodoxy. All mystical practices are therefore considered suspect by religious hierarchies.

The only plausible reason I can surmise for Graham and Kantor’s contentions is that the authors are themselves hard and fast nominalists. For nominalists, all use of language, including mathematical and religious language, is a kind of creative fiction. The elements of nominalist fictions are presumed to be ‘real’ to the extent they are useful in inquiry. But they have have no more ontological status than any other fiction. Nominalists solve the metaphysical issues of the connection between words and things by simply ignoring these issues. The authors obviously have a gripe with philosophical realism that they don’t make clear at any point in their exposition. But it is an opposition to philosophical realism which constitutes their main agenda, and fatally undermines the credibility and wider usefulness of their narrative. Their conflation of philosophical realism with a sort of primitive theological idolatry through their nominalistic presumptions is inexcusable.

The debate between nominalism and realism is perennial and has taken place for centuries with no clear intellectual outcome, in either theology or mathematics. Who knows, we each may be born with a dominate gene that determines our preferences. But what is certain is that nominalism is the option of choice for those with dictatorial ambitions from Pericles to Trump. Nominalism implies that, lacking any reliable connection between words and things, language means what authority says it means. Orwell’s novel 1984 represents a nominalist paradise in which meaning is a coercive tool employed by those in power who want to stay in power. When Margaret Thatcher claimed that there was no such thing as society, when Trump denies any knowledge of fascist support, they are using rhetorical tactics typical of philosophical nominalism.

So, the Orthodox Church, and the Soviet state, condemned the Name Worshippers for precisely the same reason that the Roman Catholic Church was simultaneously condemning the so-called heresy of Modernism, and during the same period when the American Fundamentalists were criticizing the loss of ‘correct’ belief among liberal Protestant churches. Yet Graham and Kantor fail to see these wider connections. In every one of these cases the maintenance of hierarchical, nominalistic control over language was the primary objective. The issue in these cases is not the literalness of the interpretation of creeds and doctrinal pronouncements - subjective meaning of such things is not public - but the formulae of words themselves and their ‘correct’ use in public ritual and liturgy. The exercise of church authority is a tribal rather than a spiritual event which re-establishes both tribal membership and the submission of members to the leaders of the religious group.

By turning infinity into an analyzable component of the mathematical repertoire, set theory undermined mathematical certainty about its own foundations. By demonstrating that there were many different kinds of infinity, and in a sense therefore ‘taming’ the traditionally dominant concept of the divine, set theory also compromised an implicit presumption of Anselm’s Ontological Argument - perhaps there is no limit to that which could be thought by the human mind; perhaps there were an infinity of conceivable infinities, and therefore deities; or none at all. Over a century later, neither mathematics nor theology has recovered from the paradoxical shocks to their traditional foundations. And the problematic ontological status of both God and numbers shows no prospect of being settled.

The intellectual and spiritual drama sparked by set theory is, therefore, a worthwhile subject of historical and philosophical research. And there has been little written about the mutual implications of crises in mathematics and theology. The mere simultaneity of their crises is sufficient to justify inquiry. But Naming Infinity does not make a contribution to documenting this drama. It is tendentious, mis-guided, and simply wrong. One can only hope for further effort to fill the considerable hole left unfilled by this book.

A Digression on Systems Theory

There is a sense in which naming is a creative act, but which doesn’t involve any magical bringing into existence. For example, the selection of any arbitrary segment of the infinite Continuum (that is the set of all numbers) results in an infinite set which can be named. That set is identifiable by the name. The selection, of course, does not bring the set into existence, it merely distinguishes it from the rest of the Continuum. The selection, nevertheless, is aesthetically creative; it creates a ‘form’. It also establishes the selected set as a newly discovered whole which can be described and analysed to determine its unique properties. This kind of selection is the basis of what would eventually be called Systems Theory, the analytic/aesthetic discipline of studying parts and wholes simultaneously.


Postscript: for an interesting fictional account of the relevance of advances in mathematics to American fundamentalism see: https://www.goodreads.com/review/show.... For an analysis of the metaphysical import of the Jewish mystical language of Kabbalah, see https://www.goodreads.com/review/show...

View all my reviews