When they assert that there are synthetic judgments a priori, Kant and his followers point in the first instance to mathematics. Our inquiries in the earlier sections, however, have already yielded considerable clarity about mathematical judgments. There is no doubt that mathematics contains strictly valid truth and that mathematical judgments are to that extent a priori. But the absolute exactness of mathematics, as we showed in § 7, may be regarded as guaranteed only in so far as it is a science of mere concepts. We saw in the case of geometry, for example, that it is possible to abstract from all intuitive content of mathematical concepts by defining them through implicit definitions. And modern mathematics not only has acknowledged that it is possible to introduce and determine concepts in this fashion; it has found itself compelled to follow this path because in no other way could it ensure the rigor of its propositions. Geometrical concepts must thus be considered without regard to the intuitive content with which they may be filled and are usually thought of as being filled.