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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 17 (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
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"... PARTIAL DEFENSE ABSTRACT. Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the founder of structuralism in mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the ..."
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PARTIAL DEFENSE ABSTRACT. Various contributors to recent philosophy of mathematics have taken Richard Dedekind to be the founder of structuralism in mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining and reviving. The discussion focusses on Dedekind’s classic essay “Was sind und was sollen die Zahlen?”, supplemented by evidence from “Stetigkeit und irrationale Zahlen”, his scientific correspondence, and his Nachlaß. 1.
A Structuralist Account of Logic
"... The lynchpin of the structuralist account of logic endorsed by Koslow is the definition of logical and modal operators with respect to implication relations, i.e. relative to implication structures. Logical operators are depicted independently of any possible semantic or syntactic limitations. It t ..."
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The lynchpin of the structuralist account of logic endorsed by Koslow is the definition of logical and modal operators with respect to implication relations, i.e. relative to implication structures. Logical operators are depicted independently of any possible semantic or syntactic limitations. It turns out that it is possible to define conjunction as well as other logical operators much more generally than it has usually been, and items on which the logical operators may be applied need not be syntactic objects and need not have truth values. In this paper I analyse Koslow’s structuralist theory and point out certain objectionable aspects to as well as reasons why such a theory does not fulfil the (possibly unjustified) expectation of getting defined a universal logical structure.
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.
Circular Discernment in Completely Extensive Structures and How to Avoid such Circles Generally
, 2012
"... Abstract. In this journal, D. Rizza [2010: 176] expounded a solution of what he called “the indiscernibility problem for ante rem structuralism”, which is the problem to make sense of the presence, in structures, of objects that are indiscernible yet distinct, by only appealing to what that structur ..."
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Abstract. In this journal, D. Rizza [2010: 176] expounded a solution of what he called “the indiscernibility problem for ante rem structuralism”, which is the problem to make sense of the presence, in structures, of objects that are indiscernible yet distinct, by only appealing to what that structure provides. We argue that Rizza’s solution is circular and expound a different solution that not only solves the problem for completely extensive structures, treated by Rizza, but for nearly (but not) all mathematical structures.