The meaning of category theory for 21th century's philosophy

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’s philosophy, i. e., the logical structure of language, the nature of scientific theories, the architecture of the mind. The tools used to face the difficulties inherent to 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 Lawvere made it clear already in 1969. Such universal constructions are expressed at best by means of adjoint functors and representability up to isomorphism. Here is the relevance of a category-theoretic perspective, which has wide-ranging consequences. One of them is the presence of functorial constraints on the syntax-semantics relationships; another one is an intrinsic view of (constructive) logic, as it emerged in topoi and later on 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 is possible. The lack of any satisfactory solution to them in a purely logical and set-theoretic setting shows up as the result of restrictive options, 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 “crisis” in the foundations, 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 of a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; time is ripe to realise that the same holds for classical topics of philosophy.