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Mathematical Category Theory and Mathematical Philosophy

December 1, 2013

Explicit concepts and sufficiently precise definitions are the basis for further advance of a science beyond a given level. To move toward a situation where the whole population has access to the authentic results of science requires in particular making explicit some general philosophical principles which can help to guide the learning, development, and use of mathematics, a science which clearly plays a pivotal role with regard to the learning, development and use of all the sciences. Such philosophical principles have not come from speculation but from studying and concentrating the development of actual mathematical subjects such as algebraic geometry, functional analysis, continuum mechanics, combinatorics, etc. This concentration has more than narrow professional interest because the simplifications in the conceptual relations revealed by such philosophical/mathematical advances are of great value, not only in

(1) clarifying and guiding further research in the mathematical sciences but also in

(2) reforming pedagogy and popularization in ways that will not prove deceptive.

I believe that the mathematical results of such a program will have a noticeable impact on philosophy. By philosophy I mean the assessment of the relation between thinking and being, especially of the laws of the development of thinking, as summarized in the characterization of the roles of the contrasts:

Collective Thinking Vs Individual Thinking

Conceptual Aspects Vs Formal Aspects

Quality Vs Quantity

Concepts Vs Properties

Explicit Presumptions Vs Implicit Presumptions

Abstract Vs Concrete

Being Vs Becoming

In the development of specifically mathematical thinking, these oppositions (and more particular ones such as covariance vs. contravariance, intensive vs. extensive, presentation vs. representation) have always played a role; the assessment of their role in mathematical thinking provides new nourishment to the understanding of their role in thinking in general.

My immediate goal is to understand some of the mathematical developments in their proper generality and in their philosophical relationships. I focus here on assimilating the mathematical advances themselves, but with the prospect of advancing philosophy toward a lively useability in the battle for enlightenment. The discussion will revolve around the following topics:

(1) The meta-category of categories

wherein are found adjoint functors, toposes, algebraic theories as algebraic objects etc. A key idea, valid for both the category of categories and for toposes is that the totality of cohesion preserving transformations from one space (or category) to another is again a space that has its own figures (with their incidence relations) and its own quantities (functionals) varying over it. As one resulting example, motions can be studied as “things”, which is a crucial step in science generally and hence requires a philosophical model having all the generality and power which modern mathematics can provide. Other key ideas include the duality of space and quantity; parallel to the space-quantity duality is the duality between the concrete general and the abstract general in model theory or functorial semantics. Using the tool of the meta-category of categories we can distinguish between geometric and algebraic representations of spaces as structures in a less-structured background. The concept of unity and identity of adjoint opposites and its many ramifications can be displayed. Though we assume the existence of a generic object playing the role of the topos of small sets, many results are independent of that generic object. Indeed, following Galois and Cantor, analogues of sets can be defined within categories of spaces.

(2) The extensive and intensive modes of variation of Quantity

over domain spaces and their interdependence by way of densities and ratios and totals. The Riesz-DeRham construction of extensive from intensive, and a converse construction via internal naturality.

(3) The extensive and intensive Qualities

of spaces as respectively expressed in Hurewicz’s homotopy types and in Thom-Mather’s singularity types, with the resulting possibility to quantitatively measure these qualities. Both of these categories of Quality have extremely special algebraic properties that we want to scrutinize philosophically. The geometric morphism assigning Quality in the small. The induced homotopy types of parameterized objects.

(4) Explicit mathematical theories of motion

wherein forces act on states of Becoming, rather than on mere states of Being. The physical models for such theories initiated by Galileo (anticipated by Fibonacci) forced an analysis of the continuum which can be used for a direct mathematical construction on the basis of the motion germ which represents the tangent bundle functor. The geometric morphism yielding new toposes of becoming Being. The positive-geometric fragment of the Leibniz-Robinson principle.

F. William Lawvere

Ramifications of Category Theory

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