  In addition, Proclus also held that proofs by contradiction were not demonstrations “of the reasoned fact”. The rediscovery of Proclus in the Renaissance was to spark a far-reaching debate on the causality of mathematical demonstrations referred to above as the Quaestio de Certitudine Mathematicarum. The first shot was fired by Alessandro Piccolomini in 1547. Piccolomini’s aim was to disarm a traditional claim to the effect that mathematics derives its certainty on account of its use of “scientific demonstrations” in the Aristotelean sense (such proofs were known as “potissimae” in the Renaissance). Since “potissimae” demonstrations had to be causal, Piccolomini attacked the argument by arguing that mathematical demonstrations are not causal. This led to one of the most interesting epistemological debates of the Renaissance and the seventeenth century. Those denying the “causality” of mathematical demonstrations (Piccolomini, Pereyra, Gassendi etc.)  argued by providing specific examples of demonstrations from mathematical practice (usually from Euclid’s Elements) which, they claimed, could not be reconstructed as causal reasonings in the Aristotelian sense. By contrast, those hoping to restore “causality” to mathematics aimed at showing that the alleged counterexamples could easily be accommodated within the realm of “causal” demonstrations (Clavius, Barrow, etc.). The historical developments have been presented in detail in Mancosu 1996 and Mancosu 2000. What is more important here is to appreciate that the basic intuition—the contraposition between explanatory and non-explanatory demonstrations—had a long and successful history and influenced both mathematical and philosophical developments well beyond the seventeenth century. For instance Mancosu 1999 shows that Bolzano and Cournot, two major philosophers of mathematics in the nineteenth century, construe the central problem of philosophy of mathematics as that of accounting for the distinction between between explanatory and non-explanatory demonstrations. In the case of Bolzano this takes the form of a theory of Grund (ground) and Folge (consequence). Kitcher 1975 was the first to read Bolzano as propounding a theory of mathematical explanations (see Betti 2010 and Roski 2017 for recent contributions). In the case of Cournot this is spelled out in terms of the opposition between “ordre logique” and “ordre rationnel” (see Cournot 1851). In Bolzano’s case, the aim of providing a reconstruction of parts of analysis and geometry, so that the exposition would use only “explanatory” proofs, also led to major mathematical results, such as his purely analytic proof of the intermediate value theorem.