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Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves ..."
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Cited by 10 (3 self)
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “manytopoi” view and modalstructuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a
Loss of vision: How mathematics turned blind while it learned to see more clearly
 In B. Löwe and T. Müller (Eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
, 2010
"... The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical ..."
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Cited by 1 (1 self)
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The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical
Category Theory and Structuralism
, 2009
"... The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond ..."
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The term structuralism occurred in several branches of the humanities and the sciences in the period 1929 – 1970: in Linguistics (Ferdinand de Saussure, Roman Jakobson), Anthropology (Claude LéviStrauss), Developmental psychology (Jean Piaget), Literature (Workshop for potential literature, Raymond Queneau) and in Mathematics (Nicolas Bourbaki). To the layman the structuralist movement in mathematics was perhaps most visible the form of New Math, which was strongly influenced by the Bourbaki school. It has been argued in (Aubin 1997) that there were cultural connections between these movements. (See also A. Aczel 2007.) Some of these interactions may be regarded as rather superficial. The epistemologist Piaget however was very much influenced by Bourbaki, and seems to have suggested that those patterns of thought used to explain cognitive development were closely related to the mathematical “mother structures ” found by Bourbaki. On a very general level, structuralism refers to a mode of thinking involving abstraction from specifics and systematic identification and naming of common patterns. It is the relation of objects under study to each other that is of importance rather than their specific appearance, or “nature”. In mathematics, Richard Dedekind may be said to be the first structuralist. He described the positive integers (1, 2, 3,...) as positions in an infinite progression of elements (a socalled simply infinite system) 1 � � 2
Only up to isomorphism? Category theory and the . . .
"... Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can ..."
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Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a categorytheoretical approach will be highly appropriate. But if sets have a richer ‘nature ’ than is preserved under isomorphism, then such an approach will be inadequate.
3.1 Definition and Directionality of Adjoints............... 6
, 704
"... Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to centerstage as ..."
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Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to centerstage as category theory’s primary tool to characterize what is important in mathematics. Our focus here is to present a theory of adjoint functors. The basis for the theory is laid by first showing that the objecttoobject “heteromorphisms ” between the objects of different categories (e.g., insertion of generators as a set to group map) can be rigorously treated within category theory. The heteromorphic theory shows that all adjunctions arise from the birepresentations of the heteromorphisms between the objects of
Structuralism
"... With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the settheoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist ” have become commonplace. ..."
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With the rise of multiple geometries in the nineteenth century, and in the last century the rise of abstract algebra, of the axiomatic method, the settheoretic foundations of mathematics, and the influential work of the Bourbaki, certain views called “structuralist ” have become commonplace.
DOI 10.1007/s105160079008x ORIGINAL PAPER Adjoints and emergence: applications of a new theory
"... increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to centerstage as the principal lens. ..."
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increasingly recognized as having a foundational role in mathematics. It provides the conceptual lens to isolate and characterize the structures with importance and universality in mathematics. The notion of an adjunction (a pair of adjoint functors) has moved to centerstage as the principal lens. The central feature of an adjunction is what might be called ‘‘determination through universals’ ’ based on universal mapping properties. A recently developed ‘‘heteromorphic’ ’ theory about adjoints suggests a conceptual structure, albeit abstract and atemporal, for how new relatively autonomous behavior can emerge within a system obeying certain laws. The focus here is on applications in the life sciences (e.g., selectionist mechanisms) and human sciences (e.g., the generative grammar view of language).
Adjoint Functors and Heteromorphisms
"... Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to centerstage as ..."
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Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades, the notion of adjoint functors has moved to centerstage as category theory’s primary tool to characterize what is important in mathematics. Our focus here is to present a theory of adjoint functors. The basis for the theory is laid by …rst showing that the objecttoobject “heteromorphisms " between the objects of di¤erent categories (e.g., insertion of generators as a set to group map) can be rigorously treated within category theory. The heteromorphic theory shows that all adjunctions arise from the birepresentations of the heteromorphisms between the objects of