Computability, Algorithmic Randomness and Complexity

I think mathematical ability manifests itself in many different ways. In particular, mathematicians can be drawn to space and geometry, can have strong analytic intuition, can be drawn to formalism, they can be drawn to counting arguments, etc. There is definitely no unique type of mathematician. Maybe those mathematicians who are drawn to algorithmic thinking have found a home in computer science. For myself, I have always found myself drawn to thinking algorithmically. As a student, I recall studying algebra. Naturally, we would be prescribed problems to solve and sit exams. Usually, I would find that instead of some short elegant proof I would grind out some longer, but more basic often algorithmic version. Likely this reflected lack of study, as I was pretty lazy as an undergraduate, so I had not read the notes mostly! Even when studying analysis, I saw this as an algorithmic game in that given , somehow I would try to compute δ, viewing this as a game of us versus an opponent. In my honours year, at Queensland University, I recall studying Szmielew’s decision procedure for the elementary theory of abelian groups and the word problem for groups. After this I moved to Monash to work on effective algebra which was very fashionable at the time. In effective or computable algebra, one tries to understand the effective content of mathematics. This is the extent to which mathematics can be made algorithmic. One imagines the data as being presented in some computable fashion, and then asks for the extent to which aspects or processes of the data can be made computable. A nice illustrative example from combinatorics is Dilworth’s decomposition theorem. The classical version says that if the size of the largest antichain in a partially ordered set is k, then the partially ordered set can be expressed as the union of k linearly ordered chains. The computable version asks whether this can be done computably. To wit, given a computable partially ordered set, (P,≤) (meaning that the domain is computable and the relation ≤ is computable), can we decompose this into k computable chains?