Abstract
Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s (Synthese 191:3367–3391, 2014) suggestion that explanations are those sorts of things that (in the right circumstances, and in the right manner) generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue that the explanatory criteria offered by earlier accounts can all be thought of as features that make it more likely that a mathematical proof will generate understanding. On the functional account, features such as characterising properties, unification, and salience correlate with explanatoriness, but they do not define explanatoriness.
Highlights
What are mathematical explanations? This question has generated substantial interest among philosophers
We will argue that various philosophical accounts of mathematical explanation—including those offered by Steiner (1978), Kitcher (1981), and Lange (2014)—are all derivable consequences of a combination of Wilkenfeld’s functional account and a modern understanding of human cognitive architecture
For this reason we too focus our discussion on proofs, we believe both that non-proofs can offer intra-mathematical explanations, and that the functional account we outline can apply in these cases
Summary
An early account of mathematical explanation was offered by Steiner (1978). He suggests that proofs are explanatory if they make critical use of some ‘characterising property’, defined to be those properties unique to “an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property” (p. 143). An early account of mathematical explanation was offered by Steiner (1978) He suggests that proofs are explanatory if they make critical use of some ‘characterising property’, defined to be those properties unique to “an entity or structure mentioned in the theorem, such that from the proof it is evident that the result depends on the property” Hafner and Mancosu offered a proof of Kummer’s test of convergence, noted that it does not obviously use a characterising property, and reported that the proof’s author explicitly described it as being explanatory. If Resnik and Kushner, Hafner and Mancosu, and Lange are right, Steiner’s focus on characterising properties cannot be the whole story behind explanation in mathematics
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