Naming Infinity

That's the title of a book by Loren Graham and Jean-Michel Kantor that I finished reading recently. It may be one of the most unusual books I've read, as it focuses on two topics--the mathematics of the infinite and heresy in the Eastern Orthodox Church--that are not typically associated with each other. The story is fascinating and covers the origins of set theory and its early development at the start of the 20th century by French mathematicians. According to the authors, the tendency in France toward positivism and Cartesianism contributed to a bias among French mathematicians against theories that could not be connected to physics. Accordingly, French mathematicians became reluctant to pursue the study of the transfinite. Russia, with its mystical orientation, and particularly what became the Moscow School of Mathematics were a more fertile ground for research in this area. Especially interesting is the role played by Pavel Florensky, a Russian Orthodox monk and mathematician. Florensky represents the link between the Moscow mathematicians, several of them faithful churchgoers, and an obscure sect, the Name Worshippers, within the Russian church. While the authors do not get into the details of the theology of the Name Worshippers, it becomes clear that their beliefs had an effect on the mathematical work of Florensky and perhaps the other Moscow mathematicians. After the revolution and with the growing oppression of the Soviet regime, the overt Christianity that informed these mathematicians' outlooks had to go underground. The exception was the fearless Florensky, who continued to wear his clerical robe even in the presence of leading communists like Trotsky. Florensky ended up in prison camps where he was tortured and eventually killed. The stories and personal travails of many of the other mathematicians are also sad and poignant. But in the midst of totalitarianism, a spirit of faith and mysticism continued to color the Moscow school even in the darkest periods.

The authors are right to conclude modestly that religion does not change mathematics. A Christian mathematician should come to the same conclusions as a positivist mathematician. But the spiritual and philosophical spirit that characterizes a culture can allow ideas to develop that may otherwise be neglected or overlooked. It seems that in this case, at least, it was religion rather than rationalism that proved a more fertile ground for the growth of scientific ideas.

Mathematical Truth

As Zach notes below, the topic for the meeting on Monday is the ontology and epistemology of mathematics, and the reading is an influential paper by the Princeton philosopher of mathematics Paul Benacerraf.

Here's some background that will help in understanding Benacerraf's argument. First, the ontology of mathematics is concerned with the nature of the objects that mathematicians study. For example, what are numbers? Do they exist independent of the human mind or are they creations of the mind? The epistemology of mathematics concerns how we come to have mathematical knowledge. Benacerraf's main point is that those theories of math that give a good explanation of how mathematical statements like "2+2=4" connect with the objects they are about fail to account for how we know about them. And those theories that explain how we know math do poorly in accounting for how statements about math can be true.

One influential philosophy of math is platonism. Platonists hold that numbers are real. They are non-physical entities that exist in a separate realm. Platonism explains well why "There exists one even prime number" is true. It's because quite literally there does exist a prime number that is even, namely, 2, just as the existence of a cat lying on a mat makes it true that "There is a cat on the mat." Problem is that it's really hard to give an explanation of how the human mind can come to know about things like numbers, which (unlike cats), cannot be perceived by our senses. The best that the platonist can do is to posit the existence of some kind of mysterious "intuition" that gives us access to the realm of numbers.

On the other hand, formalists or combinatorialists, as Benacerraf refers to them, reject platonism and argue that mathematical statements are true if they can proved from more basic statements. To be true is simply to be provable using certain rules. Formalism explains how we know that 2+2=4 (because it can be proved from basic statements known as the axioms of Peano arithmetic, which define our concept of natural numbers), but it requires denying that "2+2=4" is true in the same way that propositions like "A cat is on the mat" are true. In essence, if truth is about some correspondence between a statement and the world, there is no such correspondence between mathematical statements and the world for the formalist.

So we are left with a dilemma. Or maybe not. Perhaps the problem with Benacerraf's argument is that he has a deficient theory of knowledge. He assumes that knowledge involves some physical, causal relation between the object of knowledge and the knower. But is that true? That's one question among many that we can discuss on Monday.