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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
Pluralism and the Foundations of Mathematics
, 2006
"... A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose ..."
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A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that asufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of ontological multiplicity and relativity encountered in the natural sciences as well. 1
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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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
FOUNDATIONS OF UNLIMITED CATEGORY THEORY: WHAT REMAINS TO BE DONE
"... Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unli ..."
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Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited ” or “naive ” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. From the very beginnings of the subject of category theory as introduced by Eilenberg & Mac Lane (1945) it was recognized that the notion of category lends itself naturally to
Russell’s Absolutism vs.(?)
"... Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And ..."
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Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modalstructuralism and a category theoretic approach as remaining nonabsolutist
Sets, Categories and
"... First we introduce some basic theoretical issues that set the stage for subsequent accounts. Secondly, we touch upon some important issues: Settheory vs. category theory, various conceptions of sets, ..."
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First we introduce some basic theoretical issues that set the stage for subsequent accounts. Secondly, we touch upon some important issues: Settheory vs. category theory, various conceptions of sets,
THE CENTRAL INSIGHT OF CATEGORICAL MATHEMATICS
, 2008
"... ``The central insight of categorical mathematics,'' ..."
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Reconstructing Hilbert to Construct Category Theoretic Structuralism
"... This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, its relation to the “algebraic”1 approach to mathematical structuralism. I first consider the FregeHilbert debate with the aim of distinguishing b ..."
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This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, its relation to the “algebraic”1 approach to mathematical structuralism. I first consider the FregeHilbert debate with the aim of distinguishing between axioms as assertions, i.e., as statements that are used to express or assert truths about a unique subject matter, and an axiom system as a schema that is used to provide “a system of conditions for what might be called a relational structure ” (Bernays [1967], p. 497) so that axioms, as implicit definitions, are about whatever satisfies the conditions set forth. I then use this inquiry to reevaluate arguments against using category theory to frame an algebraic structuralist philosophy of mathematics. Hellman has argued that category theory cannot stand on its own as a “foundation ” for a structuralist interpretation of mathematics because “the problem of the home address remains ” (Hellman [2003], pgs. 8 & 15). That is, since the axioms for a category “merely tell us what it is to be a structure of a certain kind ” and because “its axioms are not assertory ” (Ibid. 7), we need a background mathematical theory whose axioms are
Philosophia Mathematica (III) (2005) 13, 44–60. doi:10.1093/philmat/nki006 Learning from Questions on Categorical Foundations
"... We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman. There are two ways to take the question ‘Does category theory provi ..."
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We can learn from questions as well as from their answers. This paper urges some things to learn from questions about categorical foundations for mathematics raised by Geoffrey Hellman and from ones he invokes from Solomon Feferman. There are two ways to take the question ‘Does category theory provide a