Why did Greek mathematicians publish their analyses? (Q2752082)

In an obviously later interpolation to Euclid, Elements XIII,1 we are told about analysis and synthesis: Analysis starts with what is wanted and looks for a way backward to known statements; Synthesis is the reversal of this sequence and thus establishes the truth of what was wanted. Obviously, this method has its grave logical and psychological problems: Even in the ideal case, analysis would reveal the key idea of a proof, and hence would be of no advantage in inventing one. Therefore to regard analysis as a method of discovery, as certain mathematicians around 1600 did, is deeply problematic. But then, the author asks, what was the purpose of the Greek mathematicians, to publish both, analysis together with synthesis? NEWLINENEWLINENEWLINEHe proposes to understand the function of presenting both ones by observing the different roles of theorems and problems. Theorems are considered to be interesting in themselves, the statement of a theorem is an established fact. On the other hand, when solving a problem, the focus is on the solution, and there will usually be different ways to achieve what was asked for. In this context, polemics and discussions about more or less advanced methods, elegance and the like are possible. (We may have a typical case of a `simpler' method in Euclid's proof of the intersecting chords theorem III,35, which actually is more complicated than the proof using proportions, but avoids this advanced tool. This in addition to the authors quotations from Pappus or Eratosthenes.) Here, the author claims, analysis helps: it creates the illusion of necessity in the synthetic solution of the problem. It is a rhetoric device more than a mathematical one. Moreover, the pair analysis/synthesis can have a pedagogical function, analysis can make the synthetic solution of a problem more convincing and can make it better understood, even if it is created after the fact, i.e. after the synthesis had been found.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00010].