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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 ..."
Abstract

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
PUTTING THE BITE BACK INTO “TWO DOGMAS”
"... Recent Carnap scholarship suggests that the received view of the CarnapQuine analyticity debate is importantly mistaken. It has been suggested that Carnap’s analyticity distinction is immune from Quine’s criticisms. This is either because Quine did not understand Carnap’s use of analyticity, or bec ..."
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Recent Carnap scholarship suggests that the received view of the CarnapQuine analyticity debate is importantly mistaken. It has been suggested that Carnap’s analyticity distinction is immune from Quine’s criticisms. This is either because Quine did not understand Carnap’s use of analyticity, or because Quine did not appreciate that, rather than dispelling dogmas, he was merely offering an alternate framework for philosophy. It has also been suggested that ultimately nothing of substance turns on this dispute. I am sympathetic to these reassessments and their rejection of the received view, but argue that they fail to pay proper attention to Carnap’s metaphysical deflationism. For it is there that Quine’s arguments ultimately make contact with Carnap, undermining his metaphysical deflationism. Moreover, the viability of deflationism is directly related to the viability of Carnap’s view of philosophy as methodologically distinct from science. Hence, Quine’s criticisms make contact with the deepest aspects of Carnap’s views. 1.