David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Foundations of Science 16 (4):337-351 (2011)
In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is given that rationalises mathematical development as a process of model production. It aims at placing more emphasis on the application aspects of mathematical results. Moreover, it is shown how such production realises subjective capacities as well as objective conditions, where the latter are mediated by mathematical formalism. The approach is further sustained by Polanyi’s theory of problem solving and Stegmaier’s philosophy of orientation. In particular, the tool and application perspective illuminates which role computer-based proofs can play in mathematics
|Keywords||Argumentation Mathematical knowledge Mathematical practice Formal proof Informal proof Dialectic|
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References found in this work BETA
Michael Polanyi (1967). The Tacit Dimension. London, Routledge & K. Paul.
Imre Lakatos (ed.) (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
Michael Polanyi (1974). Personal Knowledge: Towards a Post-Critical Philosophy. University of Chicago Press.
Y. Rav (1999). Why Do We Prove Theorems? Philosophia Mathematica 7 (1):5-41.
Yehuda Rav (2007). A Critique of a Formalist-Mechanist Version of the Justification of Arguments in Mathematicians' Proof Practices. Philosophia Mathematica 15 (3):291-320.
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