This may not be a timely commentary, but I only recently read the book Naming Infinity (Harvard University Press 2009). It was a gift from my husband who rightly expected that I would be interested in a book purported to be about how mathematicians were supported through a conceptual crisis by the bold work of believers in the mystical tradition of name worship. The authors can never fully display the correspondence between the belief in name worshipping and the mathematics itself, but they do successfully tell the story of the way the religious, emotional, mathematical and political lives of a group of Russian mathematicians converged. This is certainly a story worth telling. The passion and devotion we are often lead to see in stories about artists or saints is seen here in these mathematicians’ lives. Just this glimpse of their dedication is enough to tell us that something more is happening in mathematics than mere problem solving.

There have been a number of disputes among mathematicians, particularly through the late 19th century and early 20th centuries, about which math ideas were legitimate and which ones should not be allowed in the discipline. Interestingly enough, along with set theory notions and characterizations of the infinite, there was even objection to discontinuous functions. Reading the different positions in these arguments shows us something about the nature of mathematics itself and the intellectual and psychological struggles it can create. This particular book highlights an important question, namely – what does it mean for a mathematical object to exist? How *does naming* something contribute to or even produce its existence? These are beautiful questions and the answers are not obvious. I hope to continue to discuss them in upcoming blogs.

The Grahm/Kantor book does a very nice job of revealing history that will surprise us and it brings us into the world of some Russian mathematicians we may know little about.

Interesting — I’d like to read the book. But I’d like to hear what you have to say about the questions you raise. What DOES it mean for a mathematical object to exist?

It seems that the existence of a mathematical object rests entirely in its relationship to other mathematical objects. As Courant says in What is Mathematics?: “…a ‘point’ is not a ‘thing in itself,’ but is completely described by the totality of statements by which it is related to other objects.” He goes on to say that the existence of something like points at infinity (in projective geometry) is assured once we state clearly their relation to ordinary points and to each other. But because the ancestors of so many math ideas seem to come from sensory experience (like counting objects or drawing lines) it’s easy to be skeptical of the existence of a notion that appears to have no relationship to the senses or the physical world.

In my opinion, it is through the body that we define existence, our own that of everything else. And I think the trick is in recognizing that the body builds abstractions on its own while, in a more conscious way, we sort of pull mathematics out of that activity. Then the existence of a mathematical object relies on whether it can actually live consistently within the bodies images, which are a mix of sensory images and conceptual ones. And the mathematician can only use the mathematics itself to determine that.