The Meaning of Category Theory for 21st Century Philosophy

Abstract

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its ''received view''. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a ''non-standard'' approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which ''ontological'' interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax-semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century's philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational ''crisis'', as if algebraic geometry and synthetic differential geometry -not to mention algebraic topology -were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as 'the' philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties * Text of a talk given at the

Efectos comunitarios de los regímenes agroalimentarios

“…3 Badiou (2014), Goldblatt (2006), Lawvere y Schanuel (2002), y McLarty (1992 brindan exposiciones claras y accesibles de la teoría de categorías. Por su parte, Ibarra y Mormann (2006), Mormann (2010), Peruzzi (2006) y el reciente texto colectivo editado por Landry (2017) discuten la importancia de la teoría de categorías para la filosofía de la ciencia. En cuanto a modelizaciones en las ciencias especiales (ciencias cognitivas, neurociencias, biología y economía) con base en dicha teoría véase: Ehresmann y Vanbremeersch (2007), Gómez-Ramirez (2014), Rosen (1958) y Sallach (2015.…”

Advances in technology commercialization models

“…At the time of the formulation of the semantic view, model theory was being developed. Looking ahead, category theory is currently being developed, and Lawvere and others (Landry 2007) suggest viewing some theories as categories (with suitable structure) and models for these theories as functors (preserving that structure) (Szabo 1981, Peruzzi 2006). French also points out that a pluralistic, pragmatic stance is possible.…”