  Since contributions in analytic philosophy to the study of mathematical explanations date back only to Steiner 1978a, one might suspect that the topic was a byproduct of the Quinean conception of scientific theories (see Resnik &amp; Kushner, 1987, 154). Once mathematics and natural science were placed on the same footing, it became possible to apply a unified methodology to both areas. Thus, it made sense to look for explanations in mathematics just as in natural science. However, this historical reconstruction would be mistaken. Mathematical explanations of mathematical facts have been part of philosophical reflection since Aristotle. We have already seen the distinction Aristotle drew between demonstrations “of the fact” and demonstrations “of the reasoned fact”. Both are logically rigorous but only the latter provide explanations for their results. Aristotle had also claimed that demonstrations “of the reasoned fact” occur in mathematics. On account of what we said in section 2, these demonstrations can be called “explanatory” demonstrations. Aristotle’s position on explanatory proofs in mathematics was already challenged in ancient times. Proclus, in his “Commentary on the first book of Euclid’s Elements”, informs us on this point. He reports: “Many persons have thought that geometry does not investigate the cause, that is, does not ask the question ‘Why?’” (Proclus 1970, 158–159; for more on Proclus on mathematical explanation see Harari 2008). Proclus himself singles out certain propositions in Euclid’s “Elements”, such as I.32, as not being demonstrations “of the reasoned fact”. Euclid I.32 states that the sum of the internal angles of a triangle is equal to two right angles. If the demonstration were given by a scientific syllogism in the Aristotelean sense, the middle of the syllogism would have to provide the ‘cause’ of the fact. But Proclus argues that Euclid’s proof does not satisfy these Aristotelian constraints, for the appeal to the auxiliary lines and exterior angles is not ‘causal’: