Platonic realism takes the view that mathematics does not create or invent its “object, but discovers them as Columbus discovered America. Now, if this is true, the objects must in some sense “exist” prior to their discovery. According to Platonic doctrine, the objects of mathematical study are not found in the spatio-temporal order. They are disembodied eternal Forms or Archetypes, which dwell in a distinctive realm accessible only to the intellect. On this view, the triangular or circular shapes of physical bodies that can be perceived by the senses are not the proper objects of mathematics. These shapes are merely imperfect embodiments of an indivisible “perfect” Triangle or “perfect” Circle, which is uncreated, is never fully manifested by material things, and can be grasped solely by the exploring mind of the mathematician. Godel appears to hold a similar view when he says “Classes and concepts may be conceived as real objects …existing independently of our definitions and constructions. It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence”(Kurt Godel, “Russell’s Mathematical Logic,” in The Philosophy of Bertrand Russell (ed. Paul A. Schilpp, Evanston and Chicago, 1944), p. 137). (As found in ‘Ernest Nagel and James R. Newman: Godel’s Proof, p. 110.)
Godel’s proof should not be construed as an invitation to despair or as an excuse for mystery-mongering. The discovery that there are number-theoretical truths which cannot be demonstrated formally does not mean that there are truths which are forever incapable of becoming know, or that a “mystic” intuition (radically different in kind and authority from what is generally operative in intellectual advances) must replace cogent proof. It does notmean, as recent writer claims, that there are “ineluctable limits to human reason.” It does mean that the resources of the human intellect have not been and cannot be fully formulized and that new principles of demonstration forever await invention and discovery. We have seen that mathematical propositions which cannot be established by formal deduction from a given set of axioms may, nevertheless, be established by “informal” meta-mathematical reasoning. It would be irresponsible to claim that these formally indemonstrable truths established by meta-mathematical arguments are based on nothing better than bare appeals to intuition.
Nor do the inherent limitations of calculating machines imply that we cannot hope to explain living matter and human reason in physical and chemical terms. The possibility of such explanations is neither precluded nor affirmed by Godel”s incompleteness theorem. The theorem does indicate that the structure and power of the human mind are far more complex and subtle than any non-living machine yet envisaged. Godel’s own work is a remarkable example of such complexity and subtlety. It is an occasion, not for dejection, but for a renewed appreciation of the powers of creative reason. (Ernest Nagel, James R. Newman: Godel’s Proof, p. 112. )
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