Mathematics Beyond Formalisms

Be that as it may, it seems to me that it is a clear consequence of the Gödel argument that the concept of mathematical truth cannot be encapsulated in any formalistic scheme. Mathematical truth is something that goes beyond mere formalism. This is perhaps clear even without Gödel’s theorem. For how are we to decide what axioms or rules of procedure to adopt in any case when trying to set up a formal system? Our guide in deciding on the rules to adopt must always be our intuitive understanding of what is ‘self-evidently true’, given the ‘meanings’ of the symbols of the system. How are we to decide which formal systems are sensible ones to adopt – in accordance, that is, with our intuitive feelings about ‘self-evidence’ and ‘meaning’ – and which are not? The notion of self-consistency is certainly not adequate for this. One can have many self-consistent systems which are not ‘sensible’ in this sense, where the axioms and rules of procedure have meanings that we would reject as false, or perhaps no meaning at all. ‘Self-evidence’ and ‘meaning’ are concepts which would still be needed, even without Gödel’s theorem.

emperors-new-mindch. 4

According to Brouwer’s extreme intuitionism, this also is neither true nor false at the present time, but it might become one or the other at some later date. To me, such subjectiveness and time-dependence of mathematical truth is abhorrent. It is, indeed, a very subjective matter whether, or when, a mathematical result might be accepted as officially ‘proved’. Mathematical truth should not rest on such society-dependent criteria. Also, to have a concept of mathematical truth which changes with time is, to say the least, most awkward and unsatisfactory for a mathematics which one hopes to be able to employ reliably for a description of the physical world.

emperors-new-mindch. 4

(I don't actually understand how "neither true or false at the present time" is incompatible with Platonism? Can an unproven theorem not also be said to be considered neither true or false at the present time, until such time that a proof is furnished? Why would the knowledge of existence of a proof to human beings change the truthiness of the theorem based on the axioms it depends on?)

This expresses the Platonist point of view, whereby

I have made no secret of the fact that my sympathies lie strongly with the Platonistic view that mathematical truth is absolute, external, and eternal, and not based on man-made criteria; and that mathematical objects have a timeless existence of their own, not dependent on human society nor on particular physical objects.

emperors-new-mindch. 4

... and then there is intuitionism, that (?) accepts only proof by construction, not reductio ad absurdum.


emperors-new-mind Penrose, Roger. 1999. The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press. ↩︎ 1 2 3