The popular view according to which category theory provides a support for mathematical structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies ‘invariant form’ (Awodey) categorical mathematics studies covariant and contravariant transformations which, generally, have no invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics.

There exists a dispute in philosophy, going back at least to Leibniz, whether is it possible to view the world as a network of relations and relations between relations with the role of objects, between which these relations hold, entirely eliminated. Category theory seems to be the correct mathematical theory for clarifying conceptual possibilities in this respect. In this theory, objects acquire their identity either by definition, when in defining category we postulate the existence of objects, or formally by the (...) existence of identity morphisms. We show that it is perfectly possible to get rid of the identity of objects by definition, but the formal identity of objects remains as an essential element of the theory. This can be achieved by defining category exclusively in terms of morphisms and identity morphisms and, analogously, by defining category theory entirely in terms of functors and identity functors. With objects and categories eliminated, we focus on the “philosophy of arrows” and the roles various identities play in it. This perspective elucidates a contrast between “set ontology” and “categorical ontology”. (shrink)

Nominalism in formal ontology is still the thesis that the only acceptable domain of quantification is the first-order domain of particulars. Nominalists may assert that second-order well-formed formulas can be fully and completely interpreted within the first-order domain, thereby avoiding any ontological commitment to second-order entities, by means of an appropriate semantics called “substitutional”. In this paper I argue that the success of this strategy depends on the ability of Nominalists to maintain that identity, and equivalence relations more in general, (...) is first-order and invariant. Firstly, I explain why Nominalists are formally bound to this first-order concept of identity. Secondly, I show that the resources needed to justify identity, a certain conception of identity invariance, are out of the Nominalist’s reach. (shrink)

The aim of this work is to argue for the idea that processes and process-based entities are to be modelled as relational structures. Relational structures are genuine structures, namely entities not committed to the existence of basic objects. My argument moves from the analysis of Whitehead’s original insight about process-based entities that, despite some residual of substance metaphysics, has the merit of grounding the intrinsic dynamism of reality on the holistic and relational characters of process-based entities. The current model of (...) process ontology requires genuine emergence and this, in turn, requires organizations, i.e., emergence in organizations. Another view about processes rely on a structural specification of processes. I suggest that the two views can be made compatible by the help of a specific sort of structures, namely relational structures. The appeal to the mathematical theory of genuine structures, category theory, reveals the formal plausibility of this convergence. According to this formal approach, genuine structures are essentially dynamic entities for they are relational, namely, as well as organizations, they are not existentially committed to particulars. (shrink)

Formal Axiomatic method as exemplified in Hilbert’s Grundlagen der Geometrie is based on a structuralist vision of mathematics and science according to which theories and objects of these theories are to be construed “up to isomorphism”. This structuralist approach is tightly linked with the idea of making Set theory into foundations of mathematics. Category theory suggests a generalisation of Formal Axiomatic method, which amounts to construing objects and theories “up to general morphism” rather than up to isomorphism. It is shown (...) that this category-theoretic method of theorybuilding better fits mathematical and scientific practice. Moreover so since the requirement of being determined up to isomorphism (i.e. categoricity in the usual model-theoretic sense) turns to be unrealistic in many important cases. The category-theoretic approach advocated in this paper suggests an essential revision of the structuralist philosophy of mathematics and science. It is argued that a category should be viewed as a far-reaching generalisation of the notion of structure rather than a particular kind of structure. Finally, I compare formalisation and categorification as two alternative epistemic strategies. (shrink)