ANALYSIS-SYNTHESIS: A PATTERN OF EUCLIDEAN HEURISTIC AND ITS CRITICISM

Prologue on analysis and synthesis

Psi: Teacher, I should like to come back to your proof of the Descartes Euler conjecture. It seems to me that you just cheated.

Teacher: Really?

Psi: You claimed that you proved the Descartes-Euler conjecture from subconjectures like ‘all polyhedra are simple’ and ‘all polyhedra have only simply-connected faces’. Though you did not put it in these words, you in fact criticized those who thought they could prove the conjecture, and showed that it cannot be proved, only deduced from certain subconjectures. The theorem, your improved conjecture, was nothing but a disguised inference: ‘From the lemmas the original conjecture follows.’ I admit that you added that this inference may be regarded as invalid if we stretch some of its concepts, but this is a minor issue. You certainly claimed that your ‘proof’ was a deduction of the original conjecture from certain lemmas – not all of which may have been specified.

Alpha: What are you driving at? Come to the point – if you have one at all.

Psi: Your claim is false. You in fact deduced from the main conjecture and from the lemmas that, for a triangle, V − E + F= 1. But this we knew anyway!

Alpha: What?

Psi: First it was assumed (P) that ‘V − E + F = 2 for all polyhedra’. This is the very assertion we set out to prove.