David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Synthese 87 (1):125-142 (1991)
Wittgenstein's philosophy of mathematics has long been notorious. Part of the problem is that it has not been recognized that Wittgenstein, in fact, had two chief post-Tractatus conceptions of mathematics. I have labelled these the calculus conception and the language-game conception. The calculus conception forms a distinct middle period. The goal of my article is to provide a new framework for examining Wittgenstein's philosophies of mathematics and the evolution of his career as a whole. I posit the Hardyian Picture, modelled on the Augustinian Picture, to provide a structure for Wittgenstein's work on the philosophy of mathematics. Wittgenstein's calculus period has not been properly recognized, so I give a detailed account of the tenets of that stage in Wittgenstein's career. Wittgenstein's notorious remarks on contradiction are the test case for my theory of his transition. I show that the bizarreness of those remarks is largely due to the calculus conception, but that Wittgenstein's later language-game account of mathematics keeps the rejection of the Hardyian Picture while correcting the calculus conception's mistakes.
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Christian Greiffenhagen & Wes Sharrock (2006). Mathematical Relativism: Logic, Grammar, and Arithmetic in Cultural Comparison. Journal for the Theory of Social Behaviour 36 (2):97–117.
William W. Tait (1993). Some Recent Essays in the History of the Philosophy of Mathematics: A Critical Review. [REVIEW] Synthese 96 (2):293 - 331.
Toshio Irie (2012). On an Important Aspect of Relations Between a Problem and Its Solution in Mathematics and the Concept of Proof. Kagaku Tetsugaku 45 (2):115-129.
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