Bolzano and Cournot on mathematical explanation (Q2714827)

The author compares and contrasts the views of Bolzano and Cournot on the distinction between mathematical proofs which just prove the theorem at hand, and those which also cast some light on the reasons for its truth (relative to its assumptions). In Bolzano's case this distinction related to his theory of logical consequence between propositions. Cournot, who does not seem to have known the contributions of his predecessor, also stressed the superior merit of explanatory proofs, and considered the related distinction between the logical order of theorems (as propositions) and their rational order within a theory: the further relationship to historical order seem not to have occurred to him. NEWLINENEWLINENEWLINEWhile the basic distinction is interesting, it is not obviously sharp. How would one appraise, for example, the proof by mathematical induction of the binomial theorem, or that of the Pythagoras theorem by similar triangles?