On the Alleged Simplicity of Impure Proof
| Autor: | Andrew Arana |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Computer science
media_common.quotation_subject 010102 general mathematics 06 humanities and the arts Mathematical proof 01 natural sciences Measure (mathematics) Philosophy of mathematics Analytic geometry 060105 history of science technology & medicine Proof theory Calculus 0601 history and archaeology Reverse mathematics Simplicity Analytic number theory 0101 mathematics media_common |
| Zdroj: | Simplicity: Ideals of Practice in Mathematics and the Arts ISBN: 9783319533834 |
| DOI: | 10.1007/978-3-319-53385-8_16 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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