A couple of weeks ago, I had an interesting but frustrating conversation with a fellow about the nature of mathematics. He insisted that we “made up” mathematical rules rather than discovering them. I tried to show him that the order of mathematics is inherent in reason itself. He suggested that we can create alternate mathematical systems. I responded that we could devise special rules, but the playing out of those rules would follow a mathematical structure that we do not create. Such special rules, then, become mere functions that we set up within the mathematical order.
I freely admitted that certain ways of expressing these mathematical relationships were conventional. The words that we use to denote numbers and their relations, the symbols employed for digits and operations, and even the base number system all could be quite different, but the mathematical structure that they concern is known, in Kantian terms, through universally accessible a priori reasoning. Obviously, no one sees this order in its entirety; the discipline of mathematics is always progressing. From mathematical developments achieved in ancient Sumer, Egypt, India, China, America, and Greece to contemporary work accomplished in Boston and Oxford, human beings, individually and collectively in their intellectual cultures, discover these mathematical relationships inherent in reason. Far from being a matter of custom, mathematics is the most universal language that we have.
Nonetheless, the fellow persisted in his opinion. What can you do with someone who is committed to relativism even with respect to mathematics? Is there any hope for him? Surely, reason has ceased to function for someone like him, which leads me to believe that such folks retain their intellectual commitments to nonsense for irrational reasons. When a nominalist or a materialist begins to consider the nature of mathematics, he endangers his understanding of the world. So, instead of entertaining heresy, he turns himself into a fool. I have witnessed it among Christian fundamentalists with respect to their religious views, and I have likewise seen it occur with modernists of various stripes. The mind often prefers delusions to the harsh prospect of reassessing its convictions.