Alberto Peruzzi

The issue as to whether an atomistic or holistic viewof knowledge and meaning is correct relies on the way part/whole relationships is analysed,exactly as the issue as to whether a constructive or realistic view of knowledge and meaningis correct relies on the way internal/external relationships is analysed. Both theprinciple of compositionality and the context principle depend on how finely the constituents,the nature and the size of the context are identified; both the notion of meaning andthe notion of truth depend on the resources of internalisation/externalisation. Thus thespectrum of semantic and epistemological theories varies from (global) atomismto (global) holism, and from minimal to maximal internalisation. Are compositionaltheories necessarily extensional? Does formal semantics necessarily rely on set theory?Does the domain-specific character of the notions of element, part and whole prevent anygeneral, non-trivial account? The aim of the present paper is to provide a negative answerto these questions by exploring some of the features a theory covering the phenomenologyof part and whole should have. This phenomenology will only be sketched through a fewparadigmatic examples, showing how the reference of notions of part and wholevaries and which are the constraints inherentin such variation. Categorytheory provides the tools for fashioning this framework, since it allows describing any coherentcollection of objects (with actions defined over them) and (action preserving) maps betweenthe objects, as well as the variation of such collections in terms of suitablefunctors, coding the ways parts and wholes undergo co-variation. The main thesis is thatthere are interference patterns between the two pairs Local/global and Internal/external,only in terms of which the above phenomenology can be properly described.Se mai tu diventeraimetá di te stesso, e te l'auguro, ragazzo, capirai cose al di lá della comune intelligenza dei cervelli interi.Avrai perso metá di te e del mondo, ma la realtá rimasta sará mille volte piú profonda e preziosa. E tupure vorrai che tutto sia dimezzato e straziato a tua immagine, perchè bellezza e sapienza e giustizia ci sono soloin ció che é fatto a brani.

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.

Title descriptionWhich causal patterns are involved in mental processes?On what mechanisms does the self-organisation of cognitive structure rest?Can a naturalistic view account for the basic resources of intentionality, while avoiding the objections to reductive materialism?By considering the developmental, phenomenological and biological aspects linking mind and causality, this volume offers a state-of-the art theoretical proposal emphasising the fine-tuning of cognition with the complexity of bodily dynamics. In contrast to the de-coupling of mind from the physical environment in classical information-processing models, growth of brain’s architecture and stabilisation of perception­–action cycles are considered decisive, with no need for an eliminative approach to representations pursued by neural network models. The tools provided by physics and biology for the description of massive causal interactions, on top of which ‘qualitative’ changes occur, are exploited to suggest a model of the mind as a many-layered, co-evolving system. (Series A)

By developing the concept of the geometric space of conceptualization, this paper seeks to establish common ground between the perceptual and logical structures expressed in language. The author’s overall thesis is that the logical status of language is a projection of geometric and topological patterns, and that these can be described. Although highly technical, the essay has several points in common with other contributions in the book. Like those by Violi and Geeraerts, for example, it analyses natural kinds and prototypes, and the problem of conceptual universals. It shares with Violi the idea that, since there is no fixed constant in a word’s field of variation, there is no stable membership in a kind, and no real motivation for type-judgements. The paper also discusses topics addressed by Wildgen’s essay (semantic fields and the relation between concepts and their contextual extension) and Albertazzi’s (the origin of spatio-temporal patterns). In its treatment of specific issues — the impact of the ‘linguistic turn’, the critique of compositionality, of the model-theoretic approach to semantics, and of generative grammar — Peruzzi’s analysis supports the theses developed in the Introduction by Albertazzi. It shares with the latter a phenomenological approach to the analysis of schemes which in turn relates to the theories set out in the essays by Langacker, Croft and Wood, and Kövecses.

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 issue as to whether an atomistic or holistic viewof knowledge and meaning is correct relies on the way part/whole relationships is analysed,exactly as the issue as to whether a constructive or realistic view of knowledge and meaningis correct relies on the way internal/external relationships is analysed. Both theprinciple of compositionality and the context principle depend on how finely the constituents,the nature and the size of the context are identified; both the notion of meaning andthe notion of truth depend on the resources of internalisation/externalisation. Thus thespectrum of semantic and epistemological theories varies from (global) atomismto (global) holism, and from minimal to maximal internalisation. Are compositionaltheories necessarily extensional? Does formal semantics necessarily rely on set theory?Does the domain-specific character of the notions of element, part and whole prevent anygeneral, non-trivial account? The aim of the present paper is to provide a negative answerto these questions by exploring some of the features a theory covering the phenomenologyof part and whole should have. This phenomenology will only be sketched through a fewparadigmatic examples, showing how the reference of notions of part and wholevaries and which are the constraints inherentin such variation. Categorytheory provides the tools for fashioning this framework, since it allows describing any coherentcollection of objects (with actions defined over them) and (action preserving) maps betweenthe objects, as well as the variation of such collections in terms of suitablefunctors, coding the ways parts and wholes undergo co-variation. The main thesis is thatthere are interference patterns between the two pairs Local/Global and Internal/External,only in terms of which the above phenomenology can be properly described.Se mai tu diventeraimetá di te stesso, e te l''auguro, ragazzo, capirai cose al di lá della comune intelligenza dei cervelli interi.Avrai perso metá di te e del mondo, ma la realtá rimasta sará mille volte piú profonda e preziosa. E tupure vorrai che tutto sia dimezzato e straziato a tua immagine, perchè bellezza e sapienza e giustizia ci sono soloin ció che é fatto a brani.

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.