Categories in Context: Historical, Foundational, and Philosophical
Identifieur interne : 001610 ( Istex/Curation ); précédent : 001609; suivant : 001611Categories in Context: Historical, Foundational, and Philosophical
Auteurs : Elaine Landry [Canada] ; Jean-Pierre Marquis [Canada]Source :
- Philosophia Mathematica [ 0031-8019 ] ; 2005-02.
Descripteurs français
- Wicri :
- topic : Mathématiques.
English descriptors
- KwdEn :
- Abelian, Abelian categories, Abelian groups, Abstract kind, Abstract kinds, Abstract level, Abstract levels, Abstract structure, Abstract structures, Abstract system, Abstract systems, Adjoint, Adjoint functor, Adjoint functors, Algebra, Algebraic, Algebraic geometry, Algebraic theories, Algebraic theory, Algebraic topology, Ante, Appropriate type, Awodey, Axiom, Axiom system, Axiomatic method, Background theory, Bourbaki, Cambridge university press, Cartan, Categorical, Categorical context, Categorical foundation, Categorical framework, Categorical logic, Categorical logicians, Categorical perspective, Category, Category theorists, Category theory, Columbia university, Completeness theorems, Concrete kind, Concrete kinds, Concrete level, Concrete system, Concrete systems, Constitutive, Context principle, Continuous mappings, Dordrecht, Eilenberg, Elementary topos, Essential features, Foundational, Foundational framework, Foundational research, Free topos, Fregean, Fregean demand, Functor, Functors, General theory, Generalized sketches, Grothendieck, Group theory, Hellman, Hilbertian, Homological, Homological algebra, Independent existence, Isomorphic, Isomorphism, Italic, Lambek, Landry, Lawvere, Lecture notes, Local mathematics, Makkai, Mathematica, Mathematical concept, Mathematical concepts, Mathematical knowledge, Mathematical object, Mathematical structuralism, Mathematical structures, Mathematical systems, Mathematical theory, Mathematics, Mathematics studies structure, Mclarty, Modalized version, Morphisms, Natural isomorphisms, Natural numbers, Natural transformation, Other words, Oxford university press, Philma, Philosophia, Philosophia mathematica, Philosophical position, Possible types, Princeton university press, Pure intuitionistic type theory, Resnik, Same kind, Same type, Schematic type, Schematic types, Shapiro, Sheaf theory, Structuralism, Structuralist, Structure theory, Subject matter, Such theories, Symbolic logic, Theorist, Topological, Topological spaces, Topology, Topos, Topos theory, Type theory, Weak categories.
- Teeft :
- Abelian, Abelian categories, Abelian groups, Abstract kind, Abstract kinds, Abstract level, Abstract levels, Abstract structure, Abstract structures, Abstract system, Abstract systems, Adjoint, Adjoint functor, Adjoint functors, Algebra, Algebraic, Algebraic geometry, Algebraic theories, Algebraic theory, Algebraic topology, Ante, Appropriate type, Awodey, Axiom, Axiom system, Axiomatic method, Background theory, Bourbaki, Cambridge university press, Cartan, Categorical, Categorical context, Categorical foundation, Categorical framework, Categorical logic, Categorical logicians, Categorical perspective, Category, Category theorists, Category theory, Columbia university, Completeness theorems, Concrete kind, Concrete kinds, Concrete level, Concrete system, Concrete systems, Constitutive, Context principle, Continuous mappings, Dordrecht, Eilenberg, Elementary topos, Essential features, Foundational, Foundational framework, Foundational research, Free topos, Fregean, Fregean demand, Functor, Functors, General theory, Generalized sketches, Grothendieck, Group theory, Hellman, Hilbertian, Homological, Homological algebra, Independent existence, Isomorphic, Isomorphism, Italic, Lambek, Landry, Lawvere, Lecture notes, Local mathematics, Makkai, Mathematica, Mathematical concept, Mathematical concepts, Mathematical knowledge, Mathematical object, Mathematical structuralism, Mathematical structures, Mathematical systems, Mathematical theory, Mathematics, Mathematics studies structure, Mclarty, Modalized version, Morphisms, Natural isomorphisms, Natural numbers, Natural transformation, Other words, Oxford university press, Philma, Philosophia, Philosophia mathematica, Philosophical position, Possible types, Princeton university press, Pure intuitionistic type theory, Resnik, Same kind, Same type, Schematic type, Schematic types, Shapiro, Sheaf theory, Structuralism, Structuralist, Structure theory, Subject matter, Such theories, Symbolic logic, Theorist, Topological, Topological spaces, Topology, Topos, Topos theory, Type theory, Weak categories.
Abstract
The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning.
Url:
DOI: 10.1093/philmat/nki005
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<term>Natural isomorphisms</term>
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<term>Philosophia</term>
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<term>Abstract kinds</term>
<term>Abstract level</term>
<term>Abstract levels</term>
<term>Abstract structure</term>
<term>Abstract structures</term>
<term>Abstract system</term>
<term>Abstract systems</term>
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<term>Adjoint functor</term>
<term>Adjoint functors</term>
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<term>Algebraic geometry</term>
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<term>Appropriate type</term>
<term>Awodey</term>
<term>Axiom</term>
<term>Axiom system</term>
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<term>Categorical framework</term>
<term>Categorical logic</term>
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<term>Category theorists</term>
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<term>Concrete system</term>
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<term>Context principle</term>
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<term>Foundational framework</term>
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<term>Generalized sketches</term>
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<term>Group theory</term>
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<term>Homological</term>
<term>Homological algebra</term>
<term>Independent existence</term>
<term>Isomorphic</term>
<term>Isomorphism</term>
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<term>Lambek</term>
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<term>Lawvere</term>
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<term>Makkai</term>
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<term>Mathematical concept</term>
<term>Mathematical concepts</term>
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<term>Mathematical structures</term>
<term>Mathematical systems</term>
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<term>Mathematics</term>
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<term>Modalized version</term>
<term>Morphisms</term>
<term>Natural isomorphisms</term>
<term>Natural numbers</term>
<term>Natural transformation</term>
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<term>Oxford university press</term>
<term>Philma</term>
<term>Philosophia</term>
<term>Philosophia mathematica</term>
<term>Philosophical position</term>
<term>Possible types</term>
<term>Princeton university press</term>
<term>Pure intuitionistic type theory</term>
<term>Resnik</term>
<term>Same kind</term>
<term>Same type</term>
<term>Schematic type</term>
<term>Schematic types</term>
<term>Shapiro</term>
<term>Sheaf theory</term>
<term>Structuralism</term>
<term>Structuralist</term>
<term>Structure theory</term>
<term>Subject matter</term>
<term>Such theories</term>
<term>Symbolic logic</term>
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<term>Topological</term>
<term>Topological spaces</term>
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<term>Topos</term>
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<front><div type="abstract" xml:lang="en">The aim of this paper is to put into context the historical, foundational and philosophical significance of category theory. We use our historical investigation to inform the various category-theoretic foundational debates and to point to some common elements found among those who advocate adopting a foundational stance. We then use these elements to argue for the philosophical position that category theory provides a framework for an algebraic in re interpretation of mathematical structuralism. In each context, what we aim to show is that, whatever the significance of category theory, it need not rely upon any set-theoretic underpinning.</div>
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