Truth through Proof

Truth through Proof aims, according to its subtitle, to provide ‘A Formalist Foundation for Mathematics’. By this, Weir means a metaphysical rather than epistemic or technical foundation, i.e., ‘a general explanation of what makes-true mathematical utterances’ (p. 241). In his view, which he presents as a revival of a pre-Hilbertian game formalism, mathematical utterances are made true by the existence of proofs, by which he means, actual concrete derivation tokens (physical stuff that either is or could be interpreted by mathematicians as a derivation). The identification of mathematical truth with formal derivability (as in, for example, Hilbert's ideal mathematics) faces difficulties with incompleteness. As Gödel's first incompleteness theorem famously showed, in any consistent formal theory powerful enough to include primitive recursive arithmetic, and in which theoremhood is recursively enumerable, there are statements—Weir calls them indeterminables—such that neither they nor their negations are derivable within that theory, some of which we nevertheless would like to call true. In his revival of formalism, the difficulty seems even more deadly. His revival eschews the notion of ‘in principle possibility’ (IPP), and thus refuses to ground truth in derivability in principle

Calculus (dental) · Epistemology · Formalism (music) · Foundations of mathematics · Gödel · Gödel's incompleteness theorems · Linguistics · Mathematical proof · Metaphysics · Negation · Philosophy of Mathematics · Predicate (mathematical logic) · Computability, Logic, AI Algorithms · Computer Science · History and Theory of Mathematics · Mathematics · Philosophy · Philosophy and Theoretical Science