Exploring Mathematical Universes
For many generations in mathematics new worlds have been put forward, starting perhaps with non-Euclidean geometry and progressing to the current, sophisticated constructions of higher-dimensional manifolds used in discussions of theoretical physics. On other fronts, starting with the early introduction of Boolean algebras, complex numbers, quaternions, and vector spaces, investigations in algebra and analysis have been finding and analyzing more and more structures and "spaces".
In the last century Category Theory introduced a program to better classify "universes" of objects like those just mentioned.With different motivations, logicians showed how to construe rigorously interpretations of "infinitesimal" quantities in the Calculus, and how to introduce "generic" sets into the set- theoretical universes — in the famous independence proofs in Set Theory. From other directions, efforts have been successful in making clear-cut distinctions between "computable" and "non-computable" or "non-constructive" objects and transformations.
From Philosophical Logic we have had a large number of proposals on how to include such concepts as "modality" or "tense" or "probability" in logic. An important question to answer is how "productive" the new universes are in helping solve mathematical problems or craft new explanations of why mathematics is successful. One proposal being investigated by the speaker is a modal version of Zermelo-Fraenkel Set Theory where every formula has a probability. A brief introduction will be given in this lecture, and technical details will be discussed in a seminar series in weeks 5-8.