chapter  2
35 Pages

Geometry

Holes have been picked in Euclid’s arguments in the last hundred years. Hilbert uncovered a number of assumptions Euclid failed to make explicit.1 A full formulation of the axioms necessary for the rigorous development of Euclidean geometry is now available. It is much more complicated, and much less intelligible, than Euclid’s presentation, and we should ask ourselves what exactly the point of axiomatization is. Euclid high-lighted certain assumptions he needed in order to prove geometrical theorems, but took othersassumptions of order and of continuity-for granted. They are not assumptions we normally question, although undoubtedly they can be questioned. If we want absolute formal rigour, as Hilbert did, then we should make all our assumptions absolutely explicit, as Hilbert did, and produce a formally valid proof-sequence. But we achieve formal rigour at a price. Whereas Euclid’s presentation is intelligible and has immense intellectual appeal, Hilbert’s is unintelligible, except to those who already know their geometry

If not Logicism, then What?