General Concepts and Reality (Prof. F. William Lawvere)
General Concepts and Reality
Subject: Re: [cm] particulars vs. generals
Date: Tue, 25 Oct 2011 14:29:51 -0400
From: F. William Lawvere
[...] In my 2003 Florence lectures and elsewhere I elaborated the relation between general concepts and reality in terms of two contrasts:
particulars vs general concepts
Within the general, abstract vs concrete.
A system (or “doctrine”) of general concepts involves a fibered category in which the objects in the base are abstract generals (often called “theories”) and the fibers are the concrete generals (categories of models). The fibration structure (Cat-valued contravariant functor # with domain the base) is precisely “functorial semantics”, because a morphism in the base (“interpretation of one theory in another”) induces contravariantly a functor from one fiber to the other. In case there is an initial theory, every fiber has a canonical “underlying” functor to the thus-distinguished fiber K (which is typically taken to be the category of abstract sets).
A major methodological advance exploited over the last 50 years (starting in linear algebra) came from the observation that a FURTHER FUNCTORIZATION of functorial semantics is often possible, because not only the concrete generals (which are “obviously” categories) but also the abstract generals can often be construed as categories, and the objects of the fibers “more concretely” as functors.
A particular by contrast is a category P equipped with a functor to K regarded as a measurement. This category itself may come from reality as a collection of experiments which are behaviourly comparable; that is it need not be construed conceptually as consisting of structures and structure-preserving maps. However such a conceptual theory P* may be derived (depending on the doctrine of generals) by considering the diagrams of natural transformations on K and close relatives (like AK where A is for example an exponential functor on K, or “arity”). That abstract general P* has over it the fiber P*# which of course is a concrete general. Typically there is a functor P->P*# into this double dual giving concrete models for the objects of the particular. However this functor is surely not surjective and typically not faithful. Thus there is no natural place in this account for any “concrete particular”; that would be a “category mistake”.
While in mathematical practice the part the particulars P may also be given as structures (just of a different kind or different doctrine), the possibility is deliberately included that P is derived from planned perception of reality. By contrast both aspects abstract and concrete of a general concept are purely mental (i.e., belong to collective thinking). To cite a notorious example the category of featherless bipeds as a class is equally mental with the theory of fbps. The contrast between the abstract and concrete aspects of a general concept is an important historically developed tool for developing thinking. It is all part of a method for understanding and dealing with reality, but to confuse reality with the concrete developed to model it is the philosophical mistake of objective idealism.