Raymond L. Wilder on the Foundations of Mathematics

From the outside, the early 20th century obsession with the foundations of mathematics may seem strange. It may seem even stranger if we realise what the end result of this obsession was. According to Raymond L. Wilder, the modern mathematicians with


"his most powerful symbolic tools and his powers of abstraction and generalisation have failed the mathematicians in so far as 'explaining' what mathematics is, or in providing a secure 'foundation' and absolutely rigorous methods". (197)


It's quite remarkable that Wilder claims that the modern mathematician has failed to explain what mathematics is considering the fact that even the layman would have a good go at the job.


The question is: Why can’t they explain what mathematics is?


Why is this task so difficult?


Was it Gödel’s results that stopped mathematics from ‘providing a secure foundation’ as well as ‘absolutely rigorous methods’? Is it really the case that the search for foundations, as well as for absolutely rigorous methods, is well and truly over, let alone when Wilder wrote these words in 1968?


From what Wilder says next, it seems as if mathematics not having any foundations, or not being free from all contradictions, may not be such a bad thing. More precisely, he writes that
 
"perfect rigour and absolute freedom from contradictions in mathematics are no more to be expected than are final and exact explanations of natural or social phenomena". (197)


And, of course, in science we don't have "exact explanations of natural and social phenomena" and nor are such things ‘expected’ in science. Is this really the case in mathematics as well? Surely not. Perhaps this conclusion, on Wilder’s part, is simply a result of his materialist, sociological or even Marxist position on the practice and history of mathematics. Surely even these positions accept different standards from maths – indeed, they do.


Again, it is no surprise that Wilder says what he says if he that "the only reality mathematical concepts have is as cultural elements or artefacts" (197). This position seems to go even further than constructivism; though perhaps not as far as the late Wittgenstein.


More technically, Wilder expresses his constructivist, Marxist or sociological position on mathematics by elaborating on the notion of a ‘completed infinite’ (198). This sounds like a blatant and direct contradiction. How can any infinite be complete or completed? If it is completed, then surely it's not infinite. What, exactly, does Wilder say on this issue of the completed infinite? -


"For example, an infinite decimal is not something that ‘just goes on and on without end’. It is to be conceived as a completed infinite, just as one conceived of the totality of natural numbers as a completed infinity." (198)


Wilder gives us examples of completed infinities, the infinite decimal and ‘the totality of natural numbers’; though he doesn’t say what such things actually are or what the phrase ‘completed infinite’ means. The following hints at an explanation; though it doesn't help the non-mathematicians much. He writes:


"Symbolically, it may be considered a second-order symbolism, in that it is not susceptible to complete perception, but is only conceptually perceivable." (198)


Do you have a vague idea of what Wilder means by the above? Perhaps it's a kind of ‘direct insight’ or intuition into the nature of completed infinities. It can be conceptually perceived; though not seen – literally or even non-literally.