David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Foundations of Science 14 (1-2):136-152 (2009)
In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.
|Keywords||Argumentation Proof Mathematics Argument schemes Dialectic Informal logic Azzouni Rav|
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References found in this work BETA
Irving M. Copi (2008). Introduction to Logic. Pearson/Prentice Hall.
Douglas Walton & Erik C. W. Krabbe (1995). Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning. State University of New York Press.
Imre Lakatos (ed.) (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
Penelope Maddy (1990). Realism in Mathematics. Oxford University Prress.
Douglas N. Walton (2006). Fundamentals of Critical Argumentation. Cambridge University Press.
Citations of this work BETA
Andrew Aberdein (2013). Mathematical Wit and Mathematical Cognition. Topics in Cognitive Science 5 (2):231-250.
Uwe Riss (2011). Objects and Processes in Mathematical Practice. Foundations of Science 16 (4):337-351.
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