Are there any recent works (papers, books, etc) in philosophy of mathematics where it is given an account of mathematical truth in terms of a coherence theory of mathematical truth? I am interested more in accounts which don't use the notion of derivability since this notion as some issues due to the Gödel theorems.

Thank you!

  • I don't understand your objection to derivability. Incompleteness isn't an a priori problem here as at least some coherence theories reject bivalence. – Noah Schweber May 22 '19 at 1:50
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    I am following Benacerraf here in his article "Mathematical truth" he says: "On other such accounts, the truth conditions for arithmetic sentences are given as their formal derivability from specified sets of axioms. When coupled with the desire to attribute a truth value to each closed sentence of arithmetic, these views were torpedoed by the incompleteness theorems. They could be restored at least to internal consistency either by the liberalization of what counts as derivability (...) or by abandoning the desire for completeness." Are there any views that try to do this or something else? – aips May 22 '19 at 2:33
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    View that formally derivable from adopted axioms is all there is to "mathematical truth" is commonly called formalism. It is standard to drop completeness and accept multiple realizations with different truth values for undecidable sentences, called set-theoretic pluralism. See e.g. recent Koellner's review for a tame version of pluralism. Another descendant of formalism is fictionalism that drops "truth" in math altogether. – Conifold May 22 '19 at 9:32
  • Yes, I am aware of formalism. Can we say that formalism can be regarded as a version of the coherence theory of truth? They never appear associated but I guess one is the version other. What other versions there are that don't use strict derivability? That is my question. "It is standard to drop completeness and accept multiple realizations with different truth values for undecidable sentences, called set-theoretic pluralism." Can you give me an example in an article? Thank you! – aips May 22 '19 at 15:14
  • Not exactly, although they are close in spirit. Coherence theory is still committed to the "truth", formalism can go further, as indeed fictionalism does, and it can relax formal derivability. But frankly, the term is not really used in relation to math, and people who talk about the relaxing are usually not coherentists but platonists, like Gödel, or empiricists, like Quine. There is little point to fudging coherence, unless one wants to get at something outside the web. I already linked one article on pluralism, see also Warren-Waxman. – Conifold May 22 '19 at 22:35

First things first, think for a second: Do you agree that a formal system must be both (1) complete and (2) sound. If that is the case, then wouldn't every formal system be truth providing?

Bearing that in mind, how do you think such a loose theory of truth be of use? 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.

It is not hard to see why philosophers have not looked into mathematical truths as coherentists.

Now, just to answer your question in a slightly precise way:

We have three dominating theories for now,

(1) Correspondence theory of truth (platonism, symbolism (Empirical), etc.)

(2) Idealist theory of truth (Not sure who holds this, but it is there).

(3) Formalist theory of truth (deflationary [personal opinion])

That said, these are the only theories of truth I have come across, not so much Idealistic. I have never seen coherentism, as it is defined classically, be used in the context of philosophy of mathematics. That said, you might be able to pass formalism as a form of coherentism... But, not really. They are essentially different.

  • why you think they are essentially different? – aips May 23 '19 at 16:39
  • @aips Unless we are taking a semantic theory of truth (Tarskian), Formalism is not even speaking of the same truth spoken of in coherentism. – Bertrand Wittgenstein's Ghost May 23 '19 at 21:47
  • @aips in fact, formalism is not technically a theory of truth. – Bertrand Wittgenstein's Ghost May 23 '19 at 21:48
  • Exactly, formalism is not technically a theory of truth. – aips May 23 '19 at 22:37
  • @aips Let's not get ahead of ourselves, what I said is my personal opinion in this regard. There are a lot of learned, and brilliant philosopher (every one of them better than me) who do believe and argue formalism is a theory of truth. – Bertrand Wittgenstein's Ghost May 24 '19 at 4:21

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