Alberto Peruzzi

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 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; the time is ripe to realise that the same holds for classical topics of philosophy.

The paper deals with three sources of holistic arguments: confirmation of scientific theories, translation between languages, and the relationships of meaning with belief. Certain difficulties of each of the involved versions of holism are pointed out: such difficulties concern the lack of evidence in support of holistic theses, as well as the presence of slippery slope arguments, and finally inconsistencies, in holism. We argue that the dualism of atomism and holism is just a polarization of a much richer spectrum of cohesiveness levels, and that their consideration is needed in order to provide explanatory power to naturalistic epistemology. In the end, the features of a "local" alternative to holism are sketched, taking into account the role of gestaltic schemata in making stable reference possible.

The principle of compositionality (PC) claims that the meaning of a compound expression is determined by the meanings of its constituent expressions and the way they compose. Is it true or false? Does it apply to both natural and formalised languages? In order to answer, we must examine various formal versions of PC, the notion of meaning and the patterns of composition. Moreover, further principles are called for to determine its import and, in particular, its relationships with the Context Principle, which seems to be inconsistent with PC. The paper deals with some aspects of the issues involved, by considering both empirical and model-theoretic results on compositionality obtained in recent years. The main thesis is that only if the parametric form of PC is acknowledged, the above questions can receive a definite answer. To this aim, the paper makes the conditions for the consistency of PC with context-dependence explicit. Such conditions allow for the stability of a schematic conceptual/epistemic core, in contrast with the slippery slope leading to holistic pragmatism.

An excursus is carried out through the principal steps in the development of the theory of descriptions (TD) from B. Russell until now, and its most important advantages and disadvantages are sketched. TD is studied in the context of model theory (in A. Robinson's style), taking preservation and classification theorems based on normal forms into consideration. Finally, the categorical formulation of TD in topos theory, starting from M. Fourman and D. Scott, is presented with reference to sheaves.