The Unsolvability of The Quintic: A Case Study in Abstract Mathematical Explanation

Pincock, Christopher

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The Unsolvability of The Quintic: A Case Study in Abstract Mathematical Explanation

Christopher Pincock

January 2015, Volume 15, No. 3, pp. 1-19

Permalink: http://hdl.handle.net/2027/spo.3521354.0015.003

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Abstract

This paper identifies one way that a mathematical proof can be more explanatory than another proof. This is by invoking a more abstract kind of entity than the topic of the theorem. These abstract mathematical explanations are identified via an investigation of a canonical instance of modern mathematics: the Galois theory proof that there is no general solution in radicals for fifth-degree polynomial equations. I claim that abstract explanations are best seen as describing a special sort of dependence relation between distinct mathematical domains. This case study highlights the importance of the conceptual, as opposed to computational, turn of much of modern mathematics, as recently emphasized by Tappenden and Avigad. The approach adopted here is contrasted with alternative proposals by Steiner and Kitcher.

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