It has struck me that axiomatic treatments of mathematics resemble the creation myths of primitive religion. In both cases the attempt is being made to provide an ordered account of something that already exists. In the one case the ordering is by deduction; in the other by the temporal sequencing of narrative. By way of illustration I offer a paraphrase of the opening pages of Hubert's Grundlagen in the style of a creation myth.
Of course, I do not use ‘myth’ with any derogatory intention. Nor do I wish to disparage historical attempts from Euclid onwards to present mathematics as a deductive system; indeed I will join the current chorus of regret that this aspect of the subject is now absent from school mathematics. The trouble is to know how best to present axiomatics. It is no longer possible, as it was for Euclid and even for Frege, to suppose that deduction leads back to ‘primitive truths’. One cannot pretend that the mathematics lies wholly in the deduction itself; as Wittgenstein observes, this is like saying that cabinet-making consists in gluing. To say that an axiomatic system is a ‘free creation of the human mind’ does not begin to address the question of purpose and motivation. Perhaps the geometrical deductive systems of Euclid and Hilbert should be seen as conceptual models, designed to make sense of geometrical phenomena, rather as a creation myth is designed to make sense of biophysical phenomena.