David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Synthese 179 (3):435 - 454 (2011)
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, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.
|Keywords||Mathematical structuralism Category theory Algebraic structuralism Philosophy of mathematics Hilbert Frege Shapiro McLarty Marquis Hellman Mac Lane|
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References found in this work BETA
Stewart Shapiro (1997). Philosophy of Mathematics: Structure and Ontology. Oxford University Press.
Stewart Shapiro (1991). Foundations Without Foundationalism: A Case for Second-Order Logic. Oxford University Press.
Steve Awodey (2004). An Answer to Hellman's Question: ‘Does Category Theory Provide a Framework for Mathematical Structuralism?’. Philosophia Mathematica 12 (1):54-64.
William Bragg Ewald (ed.) (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford University Press.
J. Lambek & P. J. Scott (1989). Introduction to Higher Order Categorical Logic. Journal of Symbolic Logic 54 (3):1113-1114.
Citations of this work BETA
Elaine Landry (forthcoming). Mind the Gap. Metascience:1-6.
Julian C. Cole (2010). Mathematical Structuralism Today. Philosophy Compass 5 (8):689-699.
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Elaine Landry (1999). Category Theory: The Language of Mathematics. Philosophy of Science 66 (3):27.
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