We have three dominating theories for now, Now, just to answer your question in a slightly precise way: It is not hard to see why philosophers have not looked into mathematical truths as coherentists. Think of it like this, suppose (X) needs a blanket, but instead of saying 'I need a BLANKET.' (X) instead says, 'I need a big thick piece of cloth preferably temperature insulating.' Though both statements are pragmatically 'correct,' but only the former resonates with our intuition. If that is the case, then wouldn't every formal system be truth providing?īearing that in mind, how do you think such a loose theory of truth be of use? Fictionalism is a way of elaborating on the theory of meaning that proceeds from Wittgenstein's approach that 'meaning is usage', which is from some time in the mid 1940's (though this is confused by his reluctance to publish.)ĭo you agree that a formal system must be both (1) complete and (2) sound. One vision of this many-worlds-but-not-too-many approach is represented by the search for 'Ultimate-L', a map of all the relationships between possible set theories that are not too bizarre to use.īoth these ideas are from the first half of the previous century, so I don't know whether that is 'recent' in terms of the question. This is a logic consistent with Fictionalist formalizations, even if practitioners would find the overall framing abrasive. There are still bounds on the pluralism, and they are set by coherent overlaps between locally Platonic 'pictures', which they assume all hang together in the end. You can best characterize the actual behavior of most modern mathematicians as a faith in local Platonism but involving a limited pluralism that directly implies a complete Platonism is false. (But that fussiness does keep you aware that using concepts with known paradoxes in them, like absolute negation, must always be done provisionally.)

Famous Intutionists like Steven Kleene have done classical math. You can also adopt the framing of Intuitionism (of relativistic psychological Platonism, upholding Fictionalism) without fully adopting the fussy conservatism of its founding cadre. In fact, within early Intuitionism, Brouwer expressed great disdain for Heyting's formal derivations. So neither of these approaches is limited to formal derivability. In both cases, all you get is coherence, not grounding in reality or Formalism's sort of transcendental clarity (that is always perfect by virtue of never necessarily meaning anything.) And counter to the thread in the comments, nobody prevents you from including new intuitions or from positing random axioms just to see whether they become appealing. This gets us the same answer without imposing a theory of the human mind. Likewise, if as the latter suggests, Platonism is obviously false, but it is reliable as a limited playground for the comparison of possibilities, then again mathematics is held together entirely by language and shared imagination, not truth. All of our science is couched in it, not because it represents something real, but because it captures what we reliably understand. If, as proposed by the former, mathematics is an art form based on evolved suppositions, it just extends assumptions that are not true, only necessary for humans. Intuitionism and Fictionalism are two very interesting views of math that I think are coherentist at their core. But his approach gets to be the major contribution from the confrontation?)

It relied upon the completeness of arithmetic, which was formally disproved. (Almost aside: Why does everyone discard the intuitionists and constructivists? The history of this is that Hilbert, the Formalist challenged Intuitionism as an alternative to Platonism and Hilbert's program lost.

Coherence theory of truth in philosophy