April 12th, 2007 ontology vs. logic

One of the more interesting thing I have been pondering about recently is the distinction between logic and ontology in math. Mathematicians spend a good deal of time developing the theory (or the structure, if you will) of mathematics, but spend far less time worrying about ontology. The theory, of course, is developed using the rules of logic. It can be even said that all of mathematics is simply a subset of set theory, which is a subset of logic (formalists think that anyways). But ontology is more subtle because it is not certain whether you can prove a physical existence. The traditional empricist approach of experimentation does not work here because the “Existence” mathematicians speak of is more abstract than a mere materialistic existence. For example, do the Natural Numbers exist? Most mathematicians simply chose to postulate that such a set exists. However, it is unclear whether this can be shown by experimentation. There certainly is no clear way of showing it through logic. Hence, ontology seems to escape both empiricism and logicism. Should mathematics be even concerned with it?

I would argue whole-heartedly that mathematicians should be concerned with it. Because if ontology is not established, than the whole affair of mathematics is useless. For example, I recently read something by the famous mathematician Alain Connes that there is a whole branch of non-standard analysis that is logically sound and very beautiful until you realize that none of the things they are talking about actually exist. So it is a useless affair. (Or is it?) Clearly, ontology is important in this case. But the trouble seems to be that there is no satisfying way to investigating the issue.