About this topic
Summary Category theory is a branch of mathematics that has played a very important role in twentieth and twenty-first century mathematics. A category is a mathematical structure made up of objects (which can be helpfully thought of as mathematical structures of some sort) and morphisms (which can helpfully be thought of as abstract mappings connecting the objects). A canonical example of a category is the category with sets for objects and functions for morphisms. From the philosophical perspective category theory is important for a variety of reasons, including its role as an alternative foundation for mathematics, because of the development and growth of categorial logic, and for its role in providing a canonical codification of the notion of isomorphism.
Key works The definitions of categories, functors, and natural transformations all appeared for the first time in MacLane & Eilenberg 1945. This paper is difficult for a variety of both historical and mathematical reasons; the standard textbook on category theory is Maclane 1971. Textbooks aimed more at philosophical audiences include Scott 2006, Awodey 2006, and McLarty 1991. For discussion on the role of category theory as an autonomous foundation of mathematics, the conversation contained in the following papers is helpful: Feferman 1977, Hellman 2003, Awodey 2004, Linnebo & Pettigrew 2011, and Logan 2015. The references in these papers will direct the reader in helpful directions for further research.
Introductions Landry & Marquis 2005 and Landry 1999 provide excellent overviews of the area. Mclarty 1990 provides an overview of the history of philosophical uses of category theory focused on Topos theory. 
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History/traditions: Category Theory

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  1. On Adjoint and Brain Functors.David Ellerman - 2016 - Axiomathes 26 (1):41-61.
    There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms that parses an adjunction into two separate parts. Then these separate parts can be recombined in a new way to define a cognate concept, the brain functor, to abstractly model the functions of perception and action of a brain. The treatment uses relatively simple category theory and (...)
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  2. Hyperintensional Category Theory and Indefinite Extensibility.Timothy Bowen - manuscript
    This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, (...)
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  3. Pluralist-Monism. Derived Category Theory as the Grammar of n-Awareness.Shanna Dobson & Robert Prentner - manuscript
    In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the (...)
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  4. On the self-predicative universals of category theory.David Ellerman - manuscript
    This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...)
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  5. How Category Theory Works.David Ellerman - manuscript
    The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be (...)
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  6. Mac Lane, Bourbaki, and Adjoints: A Heteromorphic Retrospective.David Ellerman - manuscript
    Saunders Mac Lane famously remarked that "Bourbaki just missed" formulating adjoints in a 1948 appendix (written no doubt by Pierre Samuel) to an early draft of Algebre--which then had to wait until Daniel Kan's 1958 paper on adjoint functors. But Mac Lane was using the orthodox treatment of adjoints that only contemplates the object-to-object morphisms within a category, i.e., homomorphisms. When Samuel's treatment is reconsidered in view of the treatment of adjoints using heteromorphisms or hets (object-to-object morphisms between objects in (...)
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  7. On the duality between existence and information.David Ellerman - manuscript
    Recent developments in pure mathematics and in mathematical logic have uncovered a fundamental duality between "existence" and "information." In logic, the duality is between the Boolean logic of subsets and the logic of quotient sets, equivalence relations, or partitions. The analogue to an element of a subset is the notion of a distinction of a partition, and that leads to a whole stream of dualities or analogies--including the development of new logical foundations for information theory parallel to Boole's development of (...)
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  8. Lecture Notes On Eric Schmid's "Prospectus to a Homotopic Metatheory of Language".Jack Kahn - manuscript
    Lecture Notes On Eric Schmid's "Prospectus to a Homotopic Metatheory of Language" Presented at the Book Release Event at Triest Gallery (NYC) on January 19, 2024 -/- Prospectus to a Homotopic Metatheory of Language by Eric Schmid proposes that mathematics does not involve the discovery of a synthetic a priori. In other words, mathematics is not a stable transcendent object of knowledge. Instead, Schmid defines math as a language that depends on an infinitely large network topology of inferences. Importantly, this (...)
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  9. A Sketch of a Sirenia: Meros Theory.Dan Kurth - manuscript
    This sketch of a perhaps future 'Elementary Theory of the Category of Mereological Sums (including Mereological Wholes and Parts)' relates to my previous papers "The Topos of Emergence" and "Intelligible Gunk". I assert that for successfully categorizing Mereology one has to start with a specific setting of gunk. In this paper we will give a sketch of a categorically version of particular mereological structures. I.e. we will follow the example of F.W.Lawvere’s “An elementary theory of the category of sets” -/- (...)
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  10. Isbell Conjugacy for Developing Cognitive Science.Venkata Rayudu Posina, Posina Venkata Rayudu & Sisir Roy - manuscript
    What is cognition? Equivalently, what is cognition good for? Or, what is it that would not be but for human cognition? But for human cognition, there would not be science. Based on this kinship between individual cognition and collective science, here we put forward Isbell conjugacy---the adjointness between objective geometry and subjective algebra---as a scientific method for developing cognitive science. We begin with the correspondence between categorical perception and category theory. Next, we show how the Gestalt maxim is subsumed by (...)
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  11. Category Theory: A Gentle Introduction.Peter Smith - manuscript
    This Gentle Introduction is very much still work in progress. Roughly aimed at those who want something a bit more discursive, slower-moving, than Awodey's or Leinster's excellent books. -/- The current [Jan 2018] version is 291pp.
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  12. Levels: descriptive, explanatory, and ontological.Christian List - 2017
    Scientists and philosophers frequently speak about levels of description, levels of explanation, and ontological levels. This paper presents a framework for studying levels. I give a general definition of a system of levels and discuss several applications, some of which refer to descriptive or explanatory levels while others refer to ontological levels. I illustrate the usefulness of this framework by bringing it to bear on some familiar philosophical questions. Is there a hierarchy of levels, with a fundamental level at the (...)
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  13. An outline of algebraic set theory.Steve Awody - manuscript
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  14. Yale Gallery Talk, Language Perception and Representation.PhD Tanya Kelley - forthcoming - Https://Drive.Google.Com/File/D/1YHzX_YR_wOWC3JUvfBcW7KucWX0Wlr_o/View.
    Yale Gallery Talk, Language Perception and Representation Tanya Kelley and James Prosek Linguist and artist Tanya Kelley, Ph.D., and artist, writer, and naturalist James Prosek, B.A. 1997, discuss color manuals used by artist-naturalists and biologists and lead visitors in close looking and drawing. Presented in conjunction with the exhibition James Prosek: Art, Artifact, Artifice. Space is limited. Open to: General Public .
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  15. Categories and the Foundations of Classical Field Theories.James Owen Weatherall - forthcoming - In Elaine Landry (ed.), Categories for the Working Philosopher. Oxford, UK: Oxford University Press.
    I review some recent work on applications of category theory to questions concerning theoretical structure and theoretical equivalence of classical field theories, including Newtonian gravitation, general relativity, and Yang-Mills theories.
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  16. The Origin and Significance of Zero: An Interdisciplinary Perspective.Peter Gobets & Robert Lawrence Kuhn (eds.) - 2024 - Leiden: Brill.
    Zero has been axial in human development, but the origin and discovery of zero has never been satisfactorily addressed by a comprehensive, systematic and above all interdisciplinary research program. In this volume, over 40 international scholars explore zero under four broad themes: history; religion, philosophy & linguistics; arts; and mathematics & the sciences. Some propose that the invention/discovery of zero may have been facilitated by the prior evolution of a sophisticated concept of Nothingness or Emptiness (as it is understood in (...)
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  17. Category Theory and the Ontology of Śūnyatā.Posina Venkata Rayudu & Sisir Roy - 2024 - In Peter Gobets & Robert Lawrence Kuhn (eds.), The Origin and Significance of Zero: An Interdisciplinary Perspective. Leiden: Brill. pp. 450-478.
    Notions such as śūnyatā, catuṣkoṭi, and Indra's net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nāgārjuna considered two levels of reality: one called conventional reality, and the other ultimate reality. Within this framework, śūnyatā refers to the claim that at the ultimate level objects are devoid of essence or "intrinsic properties", but are interdependent by virtue of their relations to other objects. Catuṣkoṭi refers to the claim (...)
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  18. The Trinitarian Doctrine in the Language of Category Theory.Fábio Maia Bertato - 2023 - In Vestrucci Andrea (ed.), Beyond Babel: Religion and Linguistic Pluralism. Springer Verlag. pp. 325-344.
    In this chapter, I use the language of category theory to address a relevant part of the Christian Trinitarian doctrine. Using a categorical conceptual apparatus usual to mathematicians, it is possible to represent important points of Trinitarian theology and show that such discourse can be considered free of contradictions. Problems concerning the Trinity are therefore approached from a category theory perspective.
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  19. Categorical Abstractions of Molecular Structures of Biological Objects: A Case Study of Nucleic Acids.Jinyeong Gim - 2023 - Global Philosophy 33 (5):No.43.
    The type-level abstraction is a formal way to represent molecular structures in biological practice. Graphical representations of molecular structures of biological objects are also used to identify functional processes of things. This paper will reveal that category theory is a formal mathematical language not only to visualize molecular structures of biological objects as type-level abstraction formally but also to understand how to infer biological functions from the molecular structures of biological objects. Category theory is a toolkit to understand biological knowledge (...)
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  20. Duality, Intensionality, and Contextuality: Philosophy of Category Theory and the Categorical Unity of Science in Samson Abramsky.Yoshihiro Maruyama - 2023 - In Alessandra Palmigiano & Mehrnoosh Sadrzadeh (eds.), Samson Abramsky on Logic and Structure in Computer Science and Beyond. Springer Verlag. pp. 41-88.
    Science does not exist in vacuum; it arises and works in context. Ground-breaking achievements transforming the scientific landscape often stem from philosophical thought, just as symbolic logic and computer science were born from the early analytic philosophy, and for the very reason they impact our global worldview as a coherent whole as well as local knowledge production in different specialised domains. Here we take first steps in elucidating rich philosophical contexts in which Samson Abramsky’s far-reaching work centring around categorical science (...)
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  21. Intuitionistic logic versus paraconsistent logic. Categorical approach.Mariusz Kajetan Stopa - 2023 - Dissertation, Jagiellonian University
    The main research goal of the work is to study the notion of co-topos, its correctness, properties and relations with toposes. In particular, the dualization process proposed by proponents of co-toposes has been analyzed, which transforms certain Heyting algebras of toposes into co-Heyting ones, by which a kind of paraconsistent logic may appear in place of intuitionistic logic. It has been shown that if certain two definitions of topos are to be equivalent, then in one of them, in the context (...)
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  22. Hyperdoctrine Semantics: An Invitation.Shay Logan & Graham Leach-Krouse - 2022 - In The Logica Yearbook, 2021. College Publications. pp. 115-134.
    Categorial logic, as its name suggests, applies the techniques and machinery of category theory to topics traditionally classified as part of logic. We claim that these tools deserve attention from a greater range of philosophers than just the mathematical logicians. We support this claim with an example. In this paper we show how one particular tool from categorial logic---hyperdoctrines---suggests interesting metaphysics. Hyperdoctrines can provide semantics for quantified languages, but this account of quantification suggests a metaphysical picture quite different from the (...)
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  23. A methodological note on proving agreement between the Elementary Process Theory and modern interaction theories.Cabbolet Marcoen - 2022 - In Marcoen J. T. F. Cabbolet (ed.), And now for something completely different: the Elementary Process Theory. Revised, updated and extended 2nd edition of the dissertation with almost the same title. Utrecht: Eburon Academic Publishers. pp. 373-382.
    The Elementary Process Theory (EPT) is a collection of seven elementary process-physical principles that describe the individual processes by which interactions have to take place for repulsive gravity to exist. One of the two main problems of the EPT is that there is no proof that the four fundamental interactions (gravitational, electromagnetic, strong, and weak) as we know them can take place in the elementary processes described by the EPT. This paper sets forth the method by which it can be (...)
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  24. Strong Homomorphisms, Category Theory, and Semantic Paradox.Jonathan Wolfgram & Roy T. Cook - 2022 - Review of Symbolic Logic 15 (4):1070-1093.
    In this essay we introduce a new tool for studying the patterns of sentential reference within the framework introduced in [2] and known as the language of paradox$\mathcal {L}_{\mathsf {P}}$: strong$\mathcal {L}_{\mathsf {P}}$-homomorphisms. In particular, we show that (i) strong$\mathcal {L}_{\mathsf {P}}$-homomorphisms between$\mathcal {L}_{\mathsf {P}}$constructions preserve paradoxicality, (ii) many (but not all) earlier results regarding the paradoxicality of$\mathcal {L}_{\mathsf {P}}$constructions can be recast as special cases of our central result regarding strong$\mathcal {L}_{\mathsf {P}}$-homomorphisms, and (iii) that we can use strong$\mathcal (...)
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  25. Abductive Spaces: Modeling Concept Framework Revision with Category Theory.Rocco Gangle, Gianluca Caterina & Fernando Tohmé - 2021 - In John R. Shook & Sami Paavola (eds.), Abduction in Cognition and Action: Logical Reasoning, Scientific Inquiry, and Social Practice. Springer Verlag. pp. 49-73.
    A formal model of abductive inference is provided in which abduction is conceived as expansive and contractive movements through a topological space of theoretical and practical commitments. A pair of presheaves over the space of commitments corresponds to communities sharing commitments on the one hand and possible obstructions to commitments on the other. In this framework, abductive inference is modeled by the dynamics of redistributed communities of commitment made in response to obstructive encounters. This semantic-pragmatic model shows how elementary category (...)
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  26. A Categorical Formalism of Mechanism.Jinyeong Gim - 2021 - Dissertation, Seoul National University
    All biological objects in living systems are composed of their components. They are also involved in biological mechanisms that produce regular phenomena. Mechanistic philosophy is a recent framework for understanding biological sciences. In this dissertation, I will suggest a categorical formalism of mechanism supported by algebraic constraints by virtue of a mathematical language, category theory. First, I formalize the essential groundwork for mechanistic explanations, component parts, and activities by assuming that mechanistic explanations do not allow for infinite decomposition of components, (...)
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  27. Coalgebra And Abstraction.Graham Leach-Krouse - 2021 - Notre Dame Journal of Formal Logic 62 (1):33-66.
    Frege’s Basic Law V and its successor, Boolos’s New V, are axioms postulating abstraction operators: mappings from the power set of the domain into the domain. Basic Law V proved inconsistent. New V, however, naturally interprets large parts of second-order ZFC via a construction discovered by Boolos in 1989. This paper situates these classic findings about abstraction operators within the general theory of F-algebras and coalgebras. In particular, we show how Boolos’s construction amounts to identifying an initial F-algebra in a (...)
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  28. Impurity in Contemporary Mathematics.Ellen Lehet - 2021 - Notre Dame Journal of Formal Logic 62 (1):67-82.
    Purity has been recognized as an ideal of proof. In this paper, I consider whether purity continues to have value in contemporary mathematics. The topics (e.g., algebraic topology, algebraic geometry, category theory) and methods of contemporary mathematics often favor unification and generality, values that are more often associated with impurity rather than purity. I will demonstrate this by discussing several examples of methods and proofs that highlight the epistemic significance of unification and generality. First, I discuss the examples of algebraic (...)
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  29. The category of equivalence relations.Luca San Mauro, Valentino Delle Rose & Andrea Sorbi - 2021 - Algebra and Logic 5 (60):295-307.
    We make some beginning observations about the category Eq of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations R and S is a mapping from the set of R-equivalence classes to that of S-equivalence classes, which is induced by a computable function. We also consider some full subcategories of Eq, such as the category Eq(Σ01) of computably enumerable equivalence relations (called ceers), the category Eq(Π01) of co-computably enumerable equivalence relations, and the category Eq(Dark*) (...)
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  30. Enriched Meanings: Natural Language Semantics with Category Theory.Ash Asudeh & Gianluca Giorgolo - 2020 - New York, NY: Oxford University Press. Edited by Gianluca Giorgolo.
    This book develops a theory of enriched meanings for natural language interpretation that uses the concept of monads and related ideas from category theory. The volume is interdisciplinary in nature, and will appeal to graduate students and researchers from a range of disciplines interested in natural language understanding and representation.
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  31. Choice-free stone duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...)
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  32. Logical Rules as Fractions and Logics as Sketches.Dominique Duval - 2020 - Logica Universalis 14 (3):395-405.
    In this short paper, using category theory, we argue that logical rules can be seen as fractions and logics as limit sketches.
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  33. Review of Alain Badiou, The Pornographic Age. [REVIEW]Ekin Erkan - 2020 - Theory, Culture and Society 37.
    This review of Alain Badiou’s The Pornographic Age—as well of the essays included in the book by William Watkin, A.J. Bartlett and Justin Clemens—illuminates that this is one of the few, if not only, texts where Badiou reverses the operational directionality of the event qua category theory, so as to “dis-image” power. In doing so, Badiou provides a theory of power based on intentionality and relation, rather than the more common Foucauldian genealogic-historical methodologies so often co-opted by contemporary thinkers of (...)
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  34. A Generic Figures Reconstruction of Peirce’s Existential Graphs.Rocco Gangle, Gianluca Caterina & Fernando Tohme - 2020 - Erkenntnis 85:1-34.
    We present a category-theoretical analysis, based on the concept of generic figures, of a diagrammatic system for propositional logic ). The straightforward construction of a presheaf category \ of cuts-only Existential Graphs provides a basis for the further construction of the category \ which introduces variables in a reconstructedly generic, or label-free, mode. Morphisms in these categories represent syntactical embeddings or, equivalently but dually, extensions. Through the example of Peirce’s system, it is shown how the generic figures approach facilitates the (...)
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  35. Arrow’s impossibility theorem as a special case of Nash equilibrium: a cognitive approach to the theory of collective decision-making.Andrea Oliva & Edgardo Bucciarelli - 2020 - Mind and Society 19 (1):15-41.
    Metalogic is an open-ended cognitive, formal methodology pertaining to semantics and information processing. The language that mathematizes metalogic is known as metalanguage and deals with metafunctions purely by extension on patterns. A metalogical process involves an effective enrichment in knowledge as logical statements, and, since human cognition is an inherently logic–based representation of knowledge, a metalogical process will always be aimed at developing the scope of cognition by exploring possible cognitive implications reflected on successive levels of abstraction. Indeed, it is (...)
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  36. Composition of Deductions within the Propositions-As-Types Paradigm.Ivo Pezlar - 2020 - Logica Universalis (4):1-13.
    Kosta Došen argued in his papers Inferential Semantics (in Wansing, H. (ed.) Dag Prawitz on Proofs and Meaning, pp. 147–162. Springer, Berlin 2015) and On the Paths of Categories (in Piecha, T., Schroeder-Heister, P. (eds.) Advances in Proof-Theoretic Semantics, pp. 65–77. Springer, Cham 2016) that the propositions-as-types paradigm is less suited for general proof theory because—unlike proof theory based on category theory—it emphasizes categorical proofs over hypothetical inferences. One specific instance of this, Došen points out, is that the Curry–Howard isomorphism (...)
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  37. Creating new concepts in mathematics: freedom and limitations. The case of Category Theory.Zbigniew Semadeni - 2020 - Philosophical Problems in Science 69:33-65.
    In the paper we discuss the problem of limitations of freedom in mathematics and search for criteria which would differentiate the new concepts stemming from the historical ones from the new concepts that have opened unexpected ways of thinking and reasoning. We also investigate the emergence of category theory and its origins. In particular we explore the origins of the term functor and present the strong evidence that Eilenberg and Carnap could have learned the term from Kotarbiński and Tarski.
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  38. Category Theory in the hands of physicists, mathematicians, and philosophers. [REVIEW]Mariusz Stopa - 2020 - Philosophical Problems in Science 69:283-293.
    Book review: Category Theory in Physics, Mathematics, and Philosophy, Kuś M., Skowron B., Springer Proc. Phys. 235, 2019, pp.xii+134.
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  39. On the validity of the definition of a complement-classifier.Mariusz Stopa - 2020 - Philosophical Problems in Science 69:111-128.
    It is well-established that topos theory is inherently connected with intuitionistic logic. In recent times several works appeared concerning so-called complement-toposes, which are allegedly connected to the dual to intuitionistic logic. In this paper I present this new notion, some of the motivations for it, and some of its consequences. Then, I argue that, assuming equivalence of certain two definitions of a topos, the concept of a complement-classifier is, at least in general and within the conceptual framework of category theory, (...)
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  40. A Model-Theoretic Analysis of Fidel-Structures for mbC.Marcelo E. Coniglio & Aldo Figallo-Orellano - 2019 - In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency. Cham, Switzerland: Springer Verlag. pp. 189-216.
    In this paper, the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N and O satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to analyze the class of mbC-structures. Thus, substructures, union (...)
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  41. MES: A Mathematical Model for the Revival of Natural Philosophy.Andrée Ehresmann & Jean-Paul Vanbremeersch - 2019 - Philosophies 4 (1):9.
    The different kinds of knowledge which were connected in Natural Philosophy (NP) have been later separated. The real separation came when Physics took its individuality and developed specific mathematical models, such as dynamic systems. These models are not adapted to an integral study of _living systems_, by which we mean evolutionary multi-level, multi-agent, and multi-temporality self-organized systems, such as biological, social, or cognitive systems. For them, the physical models can only be applied to the local dynamic of each co-regulator agent, (...)
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  42. An invitation to applied category theory: seven sketches in compositionality.Brendan Fong - 2019 - New York, NY: Cambridge University Press. Edited by David I. Spivak.
    Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
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  43. The Logic in Philosophy of Science.Hans Halvorson - 2019 - Cambridge and New York: Cambridge University Press.
    Major figures of twentieth-century philosophy were enthralled by the revolution in formal logic, and many of their arguments are based on novel mathematical discoveries. Hilary Putnam claimed that the Löwenheim-Skølem theorem refutes the existence of an objective, observer-independent world; Bas van Fraassen claimed that arguments against empiricism in philosophy of science are ineffective against a semantic approach to scientific theories; W. V. O. Quine claimed that the distinction between analytic and synthetic truths is trivialized by the fact that any theory (...)
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  44. Eta-rules in Martin-löf type theory.Ansten Klev - 2019 - Bulletin of Symbolic Logic 25 (3):333-359.
    The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the (...)
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  45. Category Theory in Physics, Mathematics, and Philosophy.Marek Kuś & Bartłomiej Skowron (eds.) - 2019 - Springer Verlag.
    The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology (...)
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  46. Foundations of Mathematics: From Hilbert and Wittgenstein to the Categorical Unity of Science.Yoshihiro Maruyama - 2019 - In Shyam Wuppuluri & Newton da Costa (eds.), Wittgensteinian : Looking at the World From the Viewpoint of Wittgenstein's Philosophy. Springer Verlag. pp. 245-274.
    Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and Kripkenstein’s paradox, yielding (...)
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  47. Foundations of Mathematics: From Hilbert and Wittgenstein to the Categorical Unity of Science.Yoshihiro Maruyama - 2019 - In A. C. Grayling, Shyam Wuppuluri, Christopher Norris, Nikolay Milkov, Oskari Kuusela, Danièle Moyal-Sharrock, Beth Savickey, Jonathan Beale, Duncan Pritchard, Annalisa Coliva, Jakub Mácha, David R. Cerbone, Paul Horwich, Michael Nedo, Gregory Landini, Pascal Zambito, Yoshihiro Maruyama, Chon Tejedor, Susan G. Sterrett, Carlo Penco, Susan Edwards-Mckie, Lars Hertzberg, Edward Witherspoon, Michel ter Hark, Paul F. Snowdon, Rupert Read, Nana Last, Ilse Somavilla & Freeman Dyson (eds.), Wittgensteinian : Looking at the World From the Viewpoint of Wittgenstein’s Philosophy. Springer Verlag. pp. 245-274.
    Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem and Kripkenstein’s paradox, yielding (...)
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  48. Categories with families and first-order logic with dependent sorts.Erik Palmgren - 2019 - Annals of Pure and Applied Logic 170 (12):102715.
    First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved (...)
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  49. A Categorical Characterization of Accessible Domains.Patrick Walsh - 2019 - Dissertation, Carnegie Mellon University
    Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...)
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  50. Seven Sketches in Compositionality: An Invitation to Applied Category Theory.Brendan Fong & David Spivak - 2018
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