Does Category Theory Provide a Framework for Mathematical Structuralism

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis a 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 “many-topoi” view and modal-structuralism. 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 uniform way, by relativization, sustaining insights of Zermelo.

[1]  Geoffrey Hellman,et al.  Three Varieties of Mathematical Structuralism , 2001 .

[2]  A. Pitts.  INTRODUCTION TO HIGHER ORDER CATEGORICAL LOGIC (Cambridge Studies in Advanced Mathematics 7) , 1987 .

[3]  Hellman Geoffrey,et al.  Structuralism without Structures , 1996, Mathematics and Its Logics.

[4]  Geoffrey Hellman,et al.  Mathematics without Numbers: Towards a Modal-Structural Interpretation , 1989 .

[5]  Solomon Feferman,et al.  Set-Theoretical foundations of category theory , 1969 .

[6]  S. Lane.  Mathematics, Form and Function , 1985 .

[7]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[8]  Solomon Feferman,et al.  Categorical Foundations and Foundations of Category Theory , 1977 .

[9]  s Awodey.  Structure in Mathematics and Logic: A Categorical Perspective , 1996 .

[10]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[11]  Gerhard Osius,et al.  Categorical set theory: A characterization of the category of sets , 1974 .

[12]  Samuel Eilenberg,et al.  Algebra, topology, and category theory : a collection of papers in honor of Samuel Eilenberg , 1976 .

[13]  M. Beeson.  Foundations of Constructive Mathematics , 1985 .

[14]  A. Fraenkel.  Untersuchungen über die Grundlagen der Mengenlehre , 1925 .

[15]  Michael D. Resnik,et al.  Mathematics as a science of patterns , 1997 .

[16]  David Lewis,et al.  Parts Of Classes , 1991 .

[17]  W. Quine,et al.  Philosophy of mathematics: Truth by convention , 1984 .

[18]  R. Diaconescu.  Axiom of choice and complementation , 1975 .

[19]  Geo.,et al.  Maximality vs . Extendability : Re fl ections on Structuralism and Set Theory , 2005 .

[20]  J. Neumann.  Eine Axiomatisierung der Mengenlehre. , 1925 .

[21]  Harvey M. Friedman,et al.  The lack of definable witnesses and provably recursive functions in intuitionistic set theories , 1985 .

[22]  F. W. Lawvere.  Variable Quantities and Variable Structures in Topoi , 1976 .

[23]  John L. Bell,et al.  Category Theory and the Foundations of Mathematics* , 1981, The British Journal for the Philosophy of Science.

[24]  Ieke Moerdijk,et al.  Algebraic set theory , 1995 .

[25]  M. Fourman.  The Logic of Topoi , 1977 .

[26]  D. Lewis,et al.  Mathematics in Megethology , 1993 .

[27]  F. W. Lawvere,et al.  The Category of Categories as a Foundation for Mathematics , 1966 .

[28]  David Lewis,et al.  Papers in Philosophical Logic: Mathematics is megethology , 1997 .

[29]  Gianluigi Oliveri,et al.  Mathematics. A Science of Patterns? , 1997, Synthese.

[30]  Richard E. Olson.  On Truth by Convention , 1975 .

[31]  S. Shapiro.  Philosophy of mathematics : structure and ontology , 1997 .

[32]  J. Bell,et al.  From absolute to local mathematics , 1986, Synthese.