The Mathematics of Neo-Logicism
Time and place
12:30 PM on Thursday, March 17th, 2011; NAC 5/144
Roy T. Cook (University of Minnesota & Minnesota Center for Philosophy of Science)
Abstract
Remark: This is a talk being run by the Philosophy department.
Neo-logicism is the view that mathematics can be reduced to logic plus a special type of implicit definition—abstraction principles. Regardless of whether the neo-logicist is correct in thinking that abstraction principles are analytic, apriori, etc., abstraction principles allow for extremely elegant axiomatizations of centrally important mathematical theories. This in turn suggests that abstraction principles can provide a particularly fruitful framework within which to study the foundations of mathematics. Unfortunately, there has been no sustained examination of the mathematics of abstraction principles - what results do exist have typically been provided piecemeal by critics of the view.
The present project aims to fill this lacuna, fleshing out the initial steps in a mathematical characterization of abstraction principles and sets of abstraction principles. Of particular interest is the fact that (i) the space of abstraction principles forms a Boolean algebra, and (ii) certain classes of abstraction principles that have been highlighted for their philosophical interest turn out to be natural construction (e.g. filters) on this Boolean algebra.
