Research — the Foundations of Mathematics

Mathematical logic is my field of research. It involves using the methods of modern logic to study the foundations and limits of mathematics. Often this also results in useful plain mathematics or a new perspective. For example the methods of mathematical logic show that in combinatorics the finite Ramsey theorems follow from the infinite version.

Despite these fun math applications, the main motivation for studying mathematical logic is usually either curiosity or concern with issues that can be thought of as impinging on the philosophy of mathematics. For example, suppose we are in dialogue with someone who distrusts complicated sets whose definitions only arise through the use of infinity and other possibly suspect concepts. We might want to know exactly which types of infinity or otherwise suspect concepts are needed to prove particular mathematical theorems. Investigations along these lines are termed “Reverse Mathematics,” a program inaugurated by the work of Harvey Friedman and Stephen Simpson.

Some sample results of Reverse Mathematics are that relatively weak set-existence assumptions allow one to prove the Hahn Banach Theorem for a separable metric space; but much more complicated assumptions are need to prove the Cantor-Bendixson Theorem, which states that a closed set in a Polish Space is the disjoint union of a countable set and a perfect set.

Many of the technical tools of Reverse Mathematics are provided by “Computability Theory,” which is more precisely where the results of my research should be placed. As others have remarked, it could be called “Incomputablity Theory,” because we almost always study objects that are not  computable. Our interest is in a classification according to how uncomputable they are — this often corresponds to results about which assumptions are needed for proofs of certain theorems.

Zeno’s Paradoxes also fascinate me. Strictly speaking, they are to be placed in the philosophy of mathematics, not in the foundations of mathematics. Nevertheless,  the questions raised are similar. In resolving the paradoxes one wrestles with arguments over the validity of infinity as a concept and the nature of a continuum. The approach I like to take is one based on logic: one of my main goals to understand whether a particular perspective on the paradoxes is logically consistent.

Does this research affect my teaching of undergraduates? It does. I talk often about infinity  — arguably it is the most important concept of modern mathematics. Of course it figures prominently, therefore, in my course on the history of mathematics. It cannot be avoided in calculus (I love using Zeno’s Dichotomy Paradox to introduce the study of series). Even in a subject like Statistics it comes up. For example when we investigate discrete random variables and continuous random variables some distinction in cardinalities must be made.

I suppose you might like to see some of my papers?

Here is my PhD dissertation. “On the Elementary Theories of the Muchnik and Medvedev Lattices of Pi01 Classes,” directed by Peter Cholak at the University of Notre Dame, and submitted in 2009.

Mass Problems and Hyperarithmeticity, with Stephen G. Simpson. Journal of Mathematical Logic, 7:125–143, 2008.

I have written two other papers — I will try to provide links soon. Here is the bibliographic information.

“The AE-Theory of Effectively Closed Medvedev Degrees is Decidable,” with Takayuki Kihara. Archive for Mathematical Logic,  49:1-16, 2010. (Although we wrote together, we came independently to the main result of the paper.)

“Embedding FD(w) into P_s Densely.” Archive for Mathematical Logic, 46: 649-664, 2008.