Mathematical Explanation beyond Explanatory Proof

Abstract

Much recent work on mathematical explanation has presupposed that the phenomenon involves explanatory proofs in an essential way. I argue that this view, ‘proof chauvinism’, is false. I then look in some detail at the explanation of the solvability of polynomial equations provided by Galois theory, which has often been thought to revolve around an explanatory proof. The article concludes with some general worries about the effects of chauvinism on the theory of mathematical explanation. 1Introduction2Why I Am Not a Proof Chauvinist 2.1Proof chauvinism and mathematical practice2.2Proof chauvinism and philosophy3An Example: Galois Theory and Explanatory Proof4Conclusion

The explanatory and heuristic power of mathematics

“…Further recent developments in this area are due to Pincock (2015), Lange (2017, Ch. 7 and 8), Colyvan et al (2018) andd'Alessandro (2020).…”

Counter Countermathematical Explanations

“…Extramathematical explanations explain empirical facts, in part, by mathematical facts (Baker, 2005;Lange, 2013;Lyon and Colyvan, 2008). Intra-mathematical explanations explain mathematical facts such as an explanatory proof for why a mathematical theorem should be accepted (Mancosu, 2008;D'Alessandro, 2017;Lange, 2018). An extra-or intra-mathematical explanation can be expressed in the form of a counterfactual.…”

“…The scope and validity of these accounts remains a matter of criticism. For instance, drawing on mathematical practice, (Lehet, 2019;D'Alessandro, 2020) discuss examples of intra-mathematical explanations that go beyond explanatory proofs; Mancosu and Hafner (2008) show that Kitcher's model makes predictions about explanatoriness that go against specific cases in mathematical practice.…”

“…This gives too many countermathematical explanations. On the other hand, non-vacuists such asNolan (1997),Berto et al (2017), and BCR (2017, 2020 argue that some counterpossibles are false and some are true. On this view, a mathematical impossibility may or may not explain another fact depending on whether the corresponding countermathematical comes out true or false.7 (β) also explains (Γ).…”