Characterizing Properties And Explanation In Mathematics

Josephine Salverda

Abstract


Mark Steiner proposes one of the earliest contemporary accounts of mathematical explanation, which appeals to characterizing properties of entities referred to in proofs. Unfortunately Steiner’s remarks are often quite vague, sometimes described as ‘very puzzling indeed’, and this lack of clarity has led to a lack of understanding and a tendency to reject Steiner’s account in the philosophical literature.

I argue that Steiner’s account repays deeper analysis by providing a sympathetic reading that makes sense of his puzzling remarks and draws out some important questions.

I focus on a simple mathematical example involving sums of number sequences and identify three key conditions that the proof must meet to count as explanatory for Steiner. I propose a suitable characterizing property and show that on my suggestion, the proof indeed  ts Steiner’s account. Subsequently, I present a few potential problems relating to Steiner’s focus on the generalizability of proofs, and show how my reading of generalizability helps to avoid these worries.

Finally, I show how (my extension of) Steiner’s proposal can account for what I take to be the primary epistemic function of an explanation, namely, to help us see why the fact to be explained is true.


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