Picking up from here…
Jordan Ellenberg writes:
[Gödel] believed that mathematical objects, like numbers, were not human constructions but real things, as real as peanut butter sandwiches. …
If numbers are real things, independent of our minds, they don’t care whether or not we can define them; we apprehend them through some intuitive faculty whose nature remains a mystery. From this point of view, it’s not at all strange that the mathematics we do today is very much like the mathematics we’d be doing if Gödel had never knocked out the possibility of axiomatic foundations. For Gödel, axiomatic foundations, however useful, were never truly necessary in the first place.
Gödel didn’t prove numbers are “real things,” but by proving that there are true statements about numbers that cannot be proven, did he not imply that numbers (and thus, possibly other things) exist apart from the human mind?
- Logic will eventually gather all the knowledge there is to gain, and
- If it cannot be attained through logic, then the so-called ‘knowledge’ is not ‘real’, e.g. ‘knowledge’ gained through ‘faith’.
But if “no system of logical axioms can produce all truths about numbers because no system of logical axioms can pin down exactly what numbers are,” [Ellenberg] how, then, are those truths to be produced and pinned down, except by a rudimentary kind of faith? It is easy to forget that Pythagoreans were mystics who used mathematical ideas for religious purposes.
I found help in an essay by Princeton mathematician, Edward Nelson–
The notion of truth in mathematics is irrelevant to what mathematicians do, it is vague unless abstractly formalized, and it varies according to philosophical opinion. In short, it is formal abstraction masquerading as reality.
[‘Truth’] is a correspondence between a linguistic formulation and reality.
Abstract ideas [about truth] have concrete consequences– this is their power.
[Nelson, Mathematics and faith, in “The Human Search for Truth: Philosophy, Science, Theology – The Outlook for the Third Millennium,” International Conference on Science and Faith, The Vatican, 23-25 May 2000, St. Joseph’s University Press, Philadelphia, 2002.]
Perhaps the truth of mathematics is metaphorical. Perhaps the only way we can perceive truth is through metaphor. Perhaps the only way we can make the connection from metaphor to truth is to employ a kind of faith.