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4

While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David Hilbert was pretty convinced as to the decidability of arithmetic prior to the Incompleteness/Incomputability proofs, but was aware that the problem remained open.


2

I think the title and body questions are subtly different. Here I'm going to address the title question, which I'll paraphrase for clarity as: What sort of "mathematical truth" can a non-Platonist make sense of? I think this is less strange than it may first appear, since there is an existing parallel: "sharp" vs. "fuzzy" referents in natural language. ...


2

Before Frege, axiomatic systems were not a focus of philosophy, and Goedel is pursuing the immediate upshot of Frege's failure. So, in some sense, no. Nobody cared. Mathematics was grounded in some internal, perfect mental reality and not really based on axioms. Axioms just helped keep things clear. Paradoxes abound throughout the history of philosophy. ...


1

Godel himself said that all he did was formalise the Cretan liar paradox into a formal system. So the idea or notion of undecidable statements was already apparent a long time before Godel but obviously not phrased in such terms. A simple parallel is with arithmetic. Its easy enough to notice that putting things into a group has certain properties which is ...


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