Foundations for Mathematical Structuralism

We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoret...

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Bibliographic Details
Published inMind Vol. 123; no. 489; pp. 39 - 78
Main Authors Nodelman, Uri, Zalta, Edward N.
Format Journal Article
LanguageEnglish
Published Oxford Oxford University Press 01.01.2014
Oxford Publishing Limited (England)
Subjects
Abstract objects
Axioms
Definitions
Identity theory
Mathematical objects
Mathematical problems
Mathematical relations
Mathematical set theory
Mathematical sets
Mathematical structuralism
Mathematical theorems
Ontology
Philosophical object
Theoretical mathematics
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Summary:We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts.
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ISSN:0026-4423
1460-2113
DOI:10.1093/mind/fzu003