Wed Dec 14, 2005
234 Moses, 6–8 PM
|Working Group in the History and Philosophy of Logic, Mathematics, and Science
Daniel Isaacson (Oxford)
What Is a Mathematical Structure?
Structures in mathematics are of two sorts: particular structures (e.g. natural numbers), and general structures (e.g. groups). The difference between particular and general is marked by the definite and the indefinite articles: we speak of the natural numbers and a group. As articulated by Bourbaki, modern mathematics is more concerned with general than with particular structures. At the same time, the boundary between particular and general structures is not absolute (additional structure can be identified in particular structures in virtue of which they exemplify various general structures, e.g. the real numbers are a metric space with respect to the function |x-y|; and a topological space whose basic open sets are the open intervals determined by the < relation, and the reals have many other topologies). Nonetheless, from the point of view of attempting to answer the (philosophical) question “In what does the reality of mathematics consist?”, particular structures are fundamental. The various examples of a general structure ultimately are particular structures. So we need to be able to say what particular structures are.
Stewart Shapiro has stressed the contrast between considering a (particular) structure as existing in virtue of objects out of which it is composed (e.g individual natural numbers) or as existing independently of objects. Insofar as we wish to avoid the (in my view) hopeless idea that we can made sense of mathematical objects existing individually, i.e. independently of a structure in which they occur, the first of these possibilities is ruled out. General structures are given by their axioms, which are stipulative (a group is any set in which the axioms for a group hold). Particular structures are also given by their axioms, but in a very different way, namely by the categoricity of those axioms, i.e. the (mathematical) fact that any two models of those axioms are isomorphic. For a set of axioms to characterize an infinite structure categorically, second-order quantification is required. The status of second-order quantification is controversial. I shall discuss what can be achieved in this way.