G?del on Conceptual Realism and Mathematical Intuition

摘要

Godel maintains that with mathemalical reason we perceive the most general concepts and their relations, which are separated from space-time reality insofar as the latter is completely determined by the totality of particularities without any reference to the formal concepts. Godel holds that impredicativity presupposes that the totality of all properties exists somehow independently of our knowledge and our definitions, and that our definitions merely serve to pick out certain of these previously existing properties. Godel asserts that the problem of giving a foundation for mathematics, i.e. the totality of methods of proof actually used by mathematicians, can be considered as falling into two different parts: these methods of proof have to be reduced to a minimum number of axioms and primitive rules of inference, and a justification in some sense or other has to be sought for these axioms.