Towards a Theory of Mathematical Argument

Chapter
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 30)

Abstract

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

Argument schemes Argumentation Azzouni Dialectic Informal logic Mathematics Proof Rav. 

Notes

Acknowledgements

A prior version of this paper was published in Foundations of Science (2009), 14(1–2):137–152. It received careful and helpful criticism from Andrew Aberdein and David Sherry. I thank them both. Previous versions of this paper were presented in Las Vegas and Amsterdam. I thank the audiences for their helpful questions and comments.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of Nevada, Las VegasLas VegasUSA

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