Godel Incompleteness and the Metaphysics of Arithmetic
Godel has claimed that it was his philosophical stance that made his revolutionary discoveries possible and that his Platonism had begun in his youth. However, an examination of the record shows something quite different, namely a gradual and initially reluctant embrace of Platonism as Godel considered the philosophical implications of his mathematical work . It is at least as true that Godel’s philosophy was the result of his mathematics as that the latter derived from the former.
In 1887, in an article surveying transfinite numbers from mathematical, philosophical, and theological viewpoints, Georg Cantor made a point of attacking a little pamphlet on counting and measuring written by the great scientist Hermann von Helmholtz. Cantor complained that the pamphlet expressed an “extreme empirical- psychological point of view with a dogmatism one would not have thought possible ...” He continued:
Thus, in today’s Germany we see, as a reaction against the overblown Kant-Fichte-Hegel- Schelling Idealism, an academic-positivistic skepticism that powerfully dominates the scene. This skepticism has inevitably extended its reach even to arithmetic, in which domain it has led to its most fateful conclusions. Ultimately, this may turn out most damaging to this positivistic skepticism itself.
In reviewing a collection of Cantor’s papers dealing with the transflnite, Frege chose to emphasize the remark just quoted, writing :
Yes indeed! This is the very reef on which this doctrine will founder. For ultimately, the role of the infinite in arithmetic is not to be denied; yet, on the other hand, there is no way it can coexist with this epistemological tendency. Thus we can foresee that this issue will provide the setting for a momentous and decisive battle.
In a 1933 lecture, Godel, considering the consequences of his incompleteness theorems, and perhaps not having entirely shaken off the positivism of the Vienna Circle, showed that the “battle” Frege had predicted was taking place in his own mind:
The result of our previous discussion is that our axioms, if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent.
The axioms to which Godel referred were an unending sequence produced by permitting variables for ever higher “types” (in contemporary terminology, sets of ever higher rank) and including axioms appropriate to each level. He pointed out that to each of these levels there corresponds an assertion of a particularly simple arithmetic form, what we now would call a П0 sentence, which is not provable from the axioms of that level, but which becomes provable at the next level. In the light of later work, a П0 sentence can be seen as simply asserting that some particular equation
where p is a polynomial with integer coefficients, has no solutions in natural numbers. To say that such a proposition is true is just to say that for each choice of natural number values a1; a2,...,a„ for the unknowns,
Moreover a proof for each such special case consists of nothing more than the sequence of additions and multiplications needed to compute the value of the polynomial together with the observation that that value is not 0. So in the situation to which Godel is calling attention, at a given level there is no single proof that subsumes this infinite collection of special cases, while at the next level there is such a proof.
This powerful way of expressing Godel incompleteness is not available to one who holds to a purely formalist foundation for mathematics. For a formalist, there is no “truth” above and beyond provability in a particular formal system. Post had reacted to this situation by insisting that Godel’s work requires “at least a partial reversal of the entire axiomatic trend of the late nineteenth and early twentieth centuries, with a return to meaning and truth as being of the essence of mathematics”. Frege’s reference to the “role of the infinite in arithmetic” is very much to the point here. It is the infinitude of the natural numbers, the infinitude of the sequence of formal systems, and finally, the infinitude of the special cases implied by a П0 proposition that point to some form of Platonism.