Alberto Peruzzi

This book reports on cutting-edge concepts related to Bourbaki’s notion of structures mères. It merges perspectives from logic, philosophy, linguistics and cognitive science, suggesting how they can be combined with Bourbaki’s mathematical structuralism in order to solve foundational, ontological and epistemological problems using a novel category-theoretic approach. By offering a comprehensive account of Bourbaki’s structuralism and answers to several important questions that have arisen in connection with it, the book provides readers with a unique source of information and inspiration for future research on this topic.

After a concise description of issues concerning the causal and the deductive-nomological models of explanation, the flaws in the alternative view centred on relevance-to-context are examined. The paper argues for the need of a wider spectrum of options which takes into account both the Local/Global and the Internal/External aspects in order to determine the sense and the adequacy of any explanation. As a test for this argument, some specific problems are considered about the range of causal bonds, the admission of top-down causation, the appeal to emergence, the shift from explanation to explainability, the equivalence classes referred to as “cause” and “effect”. Finally, the paper deals with the comparison between inequivalent explanations and lists three remaining issues to complete the picture.

The paper addresses the main questions to be dealt with by any semantic theory which is committed to provide an explanation of how meaning is possible. On one side the paper argues that the resources provided by the development of mathematical logic, theoretical computer science, cognitive psychology, and general linguistics in the 20th Century, however indispensable to investigate the structure of language, rely on the existence of end products in the morphogenesis of meaning. On the other, the paper argues that philosophy of language, which, either in the analytic or the structuralist or the hermeneutical tradition, made little use of such resources (when they are not simply rejected). Left the main question unanswered. Though phenomenology intended to focus on the constitutive process, it ended up mostly with philology. Cognitive semantics paved the way to focus on patterns of bodily interaction within the natural environment out of which basic schemes emerge and are metaphorically “lifted” to any universe of discourse. The explanatory commitment is thus endorsed through two hypotheses: (1) these schemes, of topological and kinaesthetic structure, determine the range of forms of atomic sentences of any natural language, and (2) the category-theoretic notion of universality allows for a proper analysis of how such schemes are “lifted”.

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 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.

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.