ABSTRACT
In a nutshell, it concerns the Euclidean paradigm-the paradigm of axiomatization. It is possible, we know, to devise a finite stock of fundamental principles or axioms from which all of the infinitely many truths of Greek geometry can be derived: this is the Euclidean paradigm.1 Prior to 1931 many people had assumed that what was possible in geometry must be possible anywhere else in mathematics (and perhaps in nonmathematical contexts too); the paradigm must represent the very essence of mathematical method.2 One of the reasons for this relates back to our discussion in the last chapter. Suppose we grant that the meaning of a mathematical expression has to be grasped in terms of how it figures in the truths of a formal theory. Then must there not be some way of ‘capturing’ these truths and providing them with a finite characterizationprecisely what an axiomatization (and that alone?) can supply? How else could anyone assimilate the truths and grasp the expression’s meaning? Again, relatedly, do we have any sense of mathematical truth apart from mathematical provability? When we say that a given mathematical
statement is true, do we not mean (or at least imply) that there is a formal and precise proof of it? If so, it seems that there must be a finite specification of what the relevant canons of proof are-what the fundamental principles are to which we may ultimately appeal.
