 Hilbert’s approach thrived because he had come to realise that there was a mathematics of axioms, a study of different but inter-related axiom schemes and their implications, and of the axioms and their possible interpretations (models). He employed this new method to solve questions about independence and consistency, hoping to find also new insights into simplicity, the solubility of mathematical problems, and decidability. Poincaré, in his review (1902) of Hilbert’s book, accepted the new geometries as valid, but regretted that they were, as he put it, incomplete, because they lacked a psychological component. By this he meant that they could not be accommodated in his explanation of how we develop our knowledge of the geometry of physical space on the basis of experience (see section 6.2 below). 