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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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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
Typical ambiguity: trying to have your cake and eat it too
 the proceedings of the conference Russell 2001
"... Would ye both eat your cake and have your cake? ..."
Subspaces in abstract Stone duality
 Theory and Applications of Categories
, 2002
"... ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idemp ..."
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Cited by 4 (3 self)
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ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a selfadjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Paré showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales. The paper is largely concerned with the construction of such a category out of one that merely has powers of some fixed object Σ. It builds on Sober Spaces and Continuations, where the related but weaker notion of abstract sobriety was considered. The construction is done first by formally adjoining certain equalisers that Σ (−) takes to coequalisers, then using Eilenberg–Moore algebras, and finally presented as a lambda calculus similar to the axiom of comprehension in set theory. The comprehension calculus has a normalisation theorem, by which every type can
Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
A Theory of Adjoint Functors —with some Thoughts about their Philosophical Significance
, 2005
"... The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal map ..."
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The question “What is category theory ” is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to centerstage as category theory’s primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of “chimeras ” or “heteromorphisms ” between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical
Three Conceptual Problems That Bug Me
, 1996
"... Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought ..."
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Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought it would be worthwhile on this occasion to bring them to your attention side by side. In this talk I will explain the problems, together with some things that have been tried in the past and some new ideas for their solution. Types of conceptual problems. A conceptual problem is not one which is formulated in precise technical terms and which calls for a definite answer. For this reason, there are no clearcut criteria for their solution, but one can bring some criteria to bear. These will vary from case to case. There are three general types of conceptual problems in mathematics of which the ones that I will discuss today are examples. These are: 1 ffi<F
A POSSIBLE MODAL FORMULATION OF COMPREHENSION SCHEME by JAN KRAJÍÈEK in Prague (Czechoslovakia) 1)
"... We will propase a set theory MST formalized in modallogic and we will try to show that its consequences are relatively powerful in relation to its simple axiomatization. The result was announced in [10]. ..."
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We will propase a set theory MST formalized in modallogic and we will try to show that its consequences are relatively powerful in relation to its simple axiomatization. The result was announced in [10].
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
Gödel’s incompleteness theorems, free will and mathematical thought
"... Abstract. Some have claimed that Gödel’s incompleteness theorems on the formal axiomatic model of mathematical thought can be used to demonstrate that mind is not mechanical, in opposition to a FormalistMechanist Thesis. Following an explanation of the incompleteness theorems and their relationship ..."
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Abstract. Some have claimed that Gödel’s incompleteness theorems on the formal axiomatic model of mathematical thought can be used to demonstrate that mind is not mechanical, in opposition to a FormalistMechanist Thesis. Following an explanation of the incompleteness theorems and their relationship with Turing machines, we here concentrate on the arguments of Gödel (with some caveats) and Lucas among others for such claims; in addition, Lucas brings out the relevance to the free will debate. Both arguments are subject to a number of critiques. The article concludes with the formulation of a modified FormalistMechanist Thesis which prima facie guarantees partial freedom of the will in the development of mathematical thought. 1. Logic, determinism and free will. The determinismfree will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological and logical character; my concern here is to limit attention to two arguments from logic. To begin with, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no