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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!

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  • 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
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(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. It relied upon the completeness of arithmetic, which was formally disproved. But his approach gets to be the major contribution from the confrontation?)

Intuitionism and Fictionalism are two very interesting views of math that I think are coherentist at their core.

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. All of our science is couched in it, not because it represents something real, but because it captures what we reliably understand.

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. This gets us the same answer without imposing a theory of the human mind.

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. So neither of these approaches is limited to formal derivability. In fact, within early Intuitionism, Brouwer expressed great disdain for Heyting's formal derivations.

You can also adopt the framing of Intuitionism (of relativistic psychological Platonism, upholding Fictionalism) without fully adopting the fussy conservatism of its founding cadre. Famous Intutionists like Steven Kleene have done classical math... (But that fussiness does keep you aware that using concepts with known paradoxes in them, like absolute negation, must always be done provisionally.)

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. 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. This is a logic consistent with Fictionalist formalizations, even if practitioners would find the overall framing abrasive.

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.

Both 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. Intuitionism arose in the 1920's. 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.)

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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.

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  • 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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