Concepts – Intensional and Extensional.

Let us start in this fashion: objects to which concepts apply (or not). The first step in arriving at a theory for this situation is, to assume that the objects in question are completely arbitrary (as urelements in set theory). This assumption is evidently wrong in empirical experience as also in mathematics itself, e.g., in function theory. So to admit this assumption forces us to build our own theory of sets to take care of the case of complex objects later on.

Concepts are normally given to us by linguistic expressions, disregarding by abstraction the origin of languages or signals or what have you. Now we can develop a theory of concepts as follows. We idealize our language by fixing a vocabulary together with logical operators and formulate expressions for classes, functions, and relations in the way of the λ-calculus. Here we have actually a theory of concepts, understood intensionally. Note that the extensional point of view is by no means lost, since we read for e.g., λ_x,yR(x,y) as the relation R over a domain of urelements; but either R is in the vocabulary or given by a composed expression in our logical language; equality does not refer to equal extensions but to logical equivalence and reduction processes. By the way, there is no hindrance to apply λ-expressions again to λ-expressions so that hierarchies of concepts can be included.

Another approach to the question of obtaining a theory of concepts is the algebraic one. Here introducing variables for extensions over a domain of urelements, and calling them classes helps develop the axiomatic class calculus. Adding (two-place) relations again with axioms, and we can obtain the relation calculus. One could go a step further to polyadic algebra. These theories do not have a prominent role nowadays, if one compares them with the λ-calculus or set theory. This is probably due to the circumstance that it seems difficult, not to say actually against the proper idea behind these theories, to allow iteration in the sense of classes of classes, etc.

For the mathematical purposes and for the use of logics, the appropriate way is to restrict a theory of concepts to a theory of their extensions. This has a good reason, since in an abstract theory we are interested in being as neutral as possible with respect to a description or factual theory given beforehand. There is a philosophical principle behind this, namely that logical (and in this case set theoretical) assumptions should be as far as possible distinguishable from any factual or descriptive assumption.

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Category Theory of a Sketch. Thought of the Day 50.0

If a sketch can be thought of as an abstract concept, a model of a sketch is not so much an interpretation of a sketch, but a concrete or particular instantiation or realization of it. It is tempting to adopt a Kantian terminology here and say that a sketch is an abstract concept, a functor between a sketch and a category C a schema and the models of a sketch the constructions in the “intuition” of the concept.

The schema is not unique since a sketch can be realized in many different categories by many different functors. What varies from one category to the other is not the basic structure of the realizations, but the types of morphisms of the underlying category, e.g., arbitrary functions, continuous maps, etc. Thus, even though a sketch captures essential structural ingredients, others are given by the “environment” in which this structure will be realized, which can be thought of as being itself another structure. Hence, the “meaning” of some concepts cannot be uniquely given by a sketch, which is not to say that it cannot be given in a structuralist fashion.

We now distinguish the group as a structure, given by the sketch for the theory of groups, from the structure of groups, given by a category of groups, that is the category of models of the sketch for groups in a given category, be it Set or another category, e.g., the category of topological spaces with continuous maps. In the latter case, the structure is given by the exactness properties of the category, e.g., Cartesian closed, etc. This is an important improvement over the traditional framework in which one was unable to say whether we should talk about the structure common to all groups, usually taken to be given by the group axioms, or the structure generated by “all” groups. Indeed, one can now ask in a precise manner whether a category C of structures, e.g., the category of (small) groups, is sketchable, that is, whether there exists a sketch S such that Mod(S, Set) is equivalent as a category to C.

There is another category associated to a sketch, namely the theory of that sketch. The theory of a sketch S, denoted by Th(S), is in a sense “freely” constructed from S : the arrows of the underlying graph are freely composed and the diagrams are imposed as equations, and so are the cones and the cocones. Th(S) is in fact a model of S in the previous sense with the following universal property: for any other model M of S in a category C there is a unique functor F: Th(S) → C such that FU = M, where U: S → Th(S). Thus, for instance, the theory of groups is a category with a group object, the generic group, “freely” constructed from the sketch for groups. It is in a way the “universal” group in the sense that any other group in any category can be constructed from it. This is possible since it contains all possible arrows, i.e., all definable operations, obtained in a purely internal or abstract manner. It is debatable whether this category should be called the theory of the sketch. But that may be more a matter of terminology than anything else, since it is clear that the “free” category called the theory is there to stay in one way or another.