Alberto Peruzzi

The main motivation of the paper lies in an argument which shows the relevance of the topological notion of lifting for semantic theory. After a brief examination of aspects of knowledge which are described by means of concepts of algebraic geometry, the development of a functorial connection between topology and group theory is related to aspects of logical analysis. In categorical terms, lifting is, with extension, a form of division. As such it is investigated here, starting from simple examples in general topology up to fiber spaces and topoi, in order to appreciate the recurrent pattern, which is finally used to explain the understanding of sentences about abstract domains.

The development of Artificial Intelligence has led to many controversies of philosophical interest involving issues of logic, computer science, software engeneering. The paper goes through the main steps of this development, deals with their philosophical relevance and sketches the unsolved problems, with special emphasis on the AI models of semantic competence. Some criticisms raised to such models in AI, either in its “classical” or “neural networks” format, are examined. On the ground of the discussion of these criticisms, a view is proposed which avoids the inherited gap between the project of AI and naturalistic philosophy and suggests that filling the gap also provides a filter in order to select a definite form of naturalism

The paper addresses the following questions: Which aspects of ‘critical ontology’ can provide effective orientation when confronted by the difficulties inherent to the opposition of logic- inspired philosophy vs phenomenology? Given the cyclical swing between the metaphysical realism of all-embracing Systems and the anti-realistic biases of relativism, does Hartmann offer a really satisfactory equilibrium- point? Is it still possible to construct a philosophical cosmology consistent with the natural sciences, while avoiding positivistic reduction of philosophy to analysis of language? How can we assign philosophy a task that goes beyond the meta-theoretical and the epistemological, while re- nouncing the temptation to adopt the view-from-nowhere? What about the project of a ‘new’ realism that refrains from positing Absolutes and yet admits the existence of perennially open problems, on which the advances of scientific knowledge seem to have little or no effect at all? How to anchor categorial analysis, if not by connecting it to the advances of social, cognitive and natural sciences? Starting from Hartmann’s characterisation of the objective spirit, some of the main motivations behind ontological strat- ification are examined. Its basic principles are formulated in a semi-axiomatic way, followed by a brief scholium in which both the consistency and the indeterminacy of the resulting system are discussed. Then, problems inherent in the hierarchy of strata are approached in terms of category theory. (More specifically, it is argued that the notion of adjointness, central to category theory, is effective in formally representing the constrained emergence of one stratum from another.) A few critical remarks about Hartmann’s treatment of modalities close the paper.

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.