## Tuesday, April 17, 2007

### Sets and Categories

There are basically two approaches to the foundation of mathematics: set theory and category theory. In his book, 'Infinity and the Mind', Rudy Rucker argues that "if reality is physics, if physics is mathematics, and if mathematics is set theory, then everything is a set." However, category theory is able to embrace objects which are not sets, hence mathematics cannot be identified with set theory, and it may be, therefore, that the physical universe is not a set, but an object in a category.

A category consists of a collection of objects such that any pair of objects has a collection of 'morphisms' between them. The morphisms satisfy a binary operation called composition, which means that you can tack one morphism onto the end of another. In addition, each object has a morphism onto itself called the identity morphism. For example, the category Set contains all sets as objects and the functions between sets as morphisms; the category of topological spaces contains all topological spaces as objects, and has continuous functions as morphisms; and the category of smooth manifolds contains all smooth manifolds as objects and has 'smooth' (infinitely-differentiable) maps as morphisms. However, the definition of a category does not require the morphisms to be special types of functions, and the objects need not be special types of set.

The notion of an object in a category is held by many to provide the best definition of a structure. The philosopher of mathematics, David Corfield argues that "category theory allows you to work on structures without the need first to pulverise them into set theoretic dust. To give an example from the field of architecture, when studying Notre Dame cathedral in Paris, you try to understand how the building relates to other cathedrals of the day, and then to earlier and later cathedrals, and other kinds of ecclesiastical building. What you don't do is begin by imagining it reduced to a pile of mineral fragments."

The crucial distinction between a set and a structure is as follows: two sets are the same if and only if they have the same elements. In contrast, two structures are the same if and only if they are isomorphic. Sets can be equipped with structures, but whilst two different sets can be equipped with isomorphic structures, they are, nevertheless, different sets. Different sets with the same structure are different instances of the same structure. The question, then, is this: is our physical universe a specific structured set, or is it a structural object in a category?