From antiquity mathematics and philosophy has been beset by a number of oppositions, such as the Continuous and the Discrete, the One and the Many, the Finite and the Infinite, the Whole and the Part, and the Constant and the Variable. These oppositions have on occasion crystallized into paradox and they continue to haunt fundamental thinking to this day. In my talk I'll analyze some of these and describe their impact on the development of mathematics and philosophy.

In 1969, Quillen introduced the notion of "model category" as an axiomitization of homotopy theory. In the intervening four decades model categories have been proven to be remarkably useful in homotopy theory. In this talk, I will discuss a structure theorem giving a notion of a presentation of a model category, refining a 2001 paper of Dan Dugger; and I will introduce a monoidal structure on the category of model categories that simplifies many "generic" arguments in model category theory.