Proclus' Account of Explanatory Demonstrations in Mathematics and its Context

Explanation in mathematics: Proofs and practice

“…Another reason to study ME, then, is because doing so promises to enrich (and perhaps correct) our ideas about mathematical practice. See for instance Detlefsen (1988), Mancosu (1999), Tatzel (2002), Harari (2008), Mancosu and Hafner (2008), and D'Alessandro (2018) for more on the role of explanatory concerns in historical and contemporary mathematics.…”

“……For the interior angles are equal to two right angles even if there are no exterior angles, for there is a triangle even if its side is not extended. (Friedlein, 1873, 206.12‐26, quoted and translated in Harari, 2008, p. 138‐139.…”

Explanatory Proofs and Beautiful Proofs

“…3 Its neglect is remarkable. Mathematical proofs that explain why some theorem holds were distinguished by ancient Greek mathematicians from proofs that merely establish that some theorem holds [17], and this distinction has been invoked in various ways throughout the history of mathematics. 4 Fortunately, mathematical explanation has now begun to receive greater philosophical attention.…”

“…17 An especially simple result typically cries out for a proof that exploits some similar, simple feature of the setup. In contrast, a proof where the result, in all of its simplicity, appears suddenly out of a welter of complexity -through some fortuitous cancellation or clever manipulation -tends merely to heighten our curiosity about why the result holds.…”

“…Such a proof leaves us wanting to know where such a simple result came from. 17 Poincaré says that a brute-force proof is unsatisfying when the result exhibits some noteworthy feature such a symmetry or simplicity: "[W]hen a rather long calculation has led to some simple and striking result, we are not satisfied until we have shown that we should have been able to foresee, if not this entire result, at least its most characteristic traits. .…”