I take your question to be whether geometry can be axiomatized into a deductive system based on certain definitions, as some philosophers believe mathematics can be, or whether, because geometry is in some way related to space--unlike mathematics--it cannot so be axiomatized. I begin by noting that there is disagreement among philosophers of mathematics whether mathematics can indeed be axiomatized in this way. Charles Parsons, for example, following Kant, believes that mathematics requires intuition. Since I don't know the details of Parsons's account, presented in his book Mathematical Thought and Its Objects , I draw instead on Kant's view, which inspired Parsons (who is also a great Kant scholar): consideration of Kant's view of mathematics will also lead us back to geometry. According to Kant, both mathematics and geometry yield a body of necessary truths, truths which are, in Kant's terminology, ' a priori '; moreover, according to Kant, the truths of both mathematics and geometry...

The following story is recounted in John Aubry's Life of Thomas Hobbes : "He was forty years old before he looked on geometry; which happened accidentally. Being in a gentleman's library Euclid's Elements lay open, and 'twas the forty-seventh proposition in the first book. He read the proposition. 'By G ,' said he, 'this is impossible!' So he reads the demonstration of it, which referred him back to such a proof; which referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry. I have heard Sir Jonas Moore (and others) say that it was a great pity he had not begun the study of the mathematics sooner, for such a working head would have made great advancement in it. So had he done he would not have lain so open to his learned mathematical antagonists. But one may say of him, as one says of Jos. Scaliger, that where he errs, he errs so ingeniously, that one had rather err with him than hit the...