On the Alleged Simplicity of Impure Proof

  • Andrew AranaEmail author
Part of the Mathematics, Culture, and the Arts book series (MACUAR)


Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self-evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof-theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim.



Thanks to Walter Dean, Michael Detlefsen, Sébastien Maronne, Mitsuhiro Okada, Marco Panza, and Sean Walsh for helpful discussions on these subjects.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Unité de Formation et de Recherche de PhilosophieUniversité de Paris-Sorbonne and Institut d’Histoire et de Philosophie des Sciences et des TechniquesParisFrance

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