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This has been rangling around my head for awhile. With the death of Hilbert's program via Gödel's Incompleteness Theorems (and the prior damage done to Logicism via Russell's Paradox), have mathematicians become pluralists, of some sort, about their discipline?

In 'Varieties of Logic', Stewart Shapiro says, "Pluralism about a given subject, such as truth, logic, ethics or etiquette, is the view that different accounts of the subject are equally correct, or equally good, or equally legitimate, or perhaps even true."

I dont like the truth bit, but that aside, to make a comparison to geometry, it seems as if with the advent of Non-Euclidean geometries, we couldn't lay down a determinate answer to some questions since the answer depends on the geometry being assumed (Parallel Postulate, for example). As a non-mathematician, it seems as if to mathematicians it's not so much that one or another mathematical system is "true" because the question of which is true is some kind of mistake in the way the above question about the Parallel Postulate is.

Whichever mathematical system we assume does not seem to be inherently off the table because what matters is if the structure is interesting in what it allows one to prove (or fail to prove). There's a notion of interestingness that seems to be at play: triviality proves everything so it gets the axe, paraconsistency slides in for avoiding that.

The above, in addition to a sort of practical pluralism (e.g. logicians or mathematicians on occasion apply particular formalisms to particular domains), does this suggest some kind of mathematical pluralism (or even logical pluralism) is implicitly accepted by mathematicians? Or would they, when pressed, revert to the standard maths?

I couldn't find any other questions about this or polls (would be interesting to see one). But as someone who just reads about maths and logic for fun, it interests me.

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    $\begingroup$ Math mostly consists of deducing theorems from axioms. The set of axioms you start with may be different in different contexts, but that does not make mathematicians any more pluralistic than someone who can play both chess and checkers. The theorems are still real, it's just the rules of the game that have changed. $\endgroup$ Commented Jun 12, 2018 at 0:36
  • $\begingroup$ @Jair_Taylor I don't know, something about that seems odd. If you're changing the rules of the game, that seems to suggest one set of rules isn't good enough (in some sense), otherwise why bother? I think the chess-checkers analogy breaks down since maths is a bit more important. :-) $\endgroup$ Commented Jun 12, 2018 at 0:45
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    $\begingroup$ @MindForgedManacle The power of mathematics comes from the flexibility of the axioms we choose to study. There is no reason to favor one over the other a priori. There are also many examples where new axioms thought to be merely curiosities became valuable in application. When you apply some value judgement like "better" or "worse" you're applying a quantitative measure to what is a fundamentally qualitative distinction and I think that's what's misleading you. $\endgroup$ Commented Jun 12, 2018 at 0:52
  • $\begingroup$ @CyclotomicField I agree about the axioms, but I'm really asking if this is a kind of pluralism (which I don't hold as a bad thing!) I mean, constructive mathematics has some interesting uses, no doubt, and database languages (or at least SQL) make use of non-classical logics so their applicability to particular domains seems clear. I just wonders what this might imply on the pluralism question. Also, I think we can quantify better or worse theories using models of theory choice. Somewhat vaguely, perhaps. $\endgroup$ Commented Jun 12, 2018 at 3:43
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    $\begingroup$ What is "the standard maths"? $\endgroup$ Commented Jun 12, 2018 at 5:13

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I can't really speak for mathematicians in general, but from what I've observed I wouldn't say we ascribe to pluralism. Many mathematicians - those not directly involved in logic - don't even think much about Incompleteness, because the undecidable statements aren't relevant to their field.

Among logicians, including myself, the prevailing opinion seems to be this: there's some sort of "overarching" universe, inside which all the possible mathematical universes are situated. So it's not that (for example) it's "equally correct" to say that the Continuum Hypothesis (a famous example of an independent statement in set theory) is true and that the Continuum Hypothesis is false. It's that it is correct to say that in some universes the Continuum Hypothesis is true and in others it's false.

Another way of thinking about it: Euclidean geometry is correct on an infinite plane. Spherical geometry is correct on the surface of a sphere. We don't say either is the "correct" set of axioms; we just say each applies to a different space. That's not pluralism, that's just limiting the scope of the axioms - instead of talking about everything, they're talking about their own respective situations.

Now, some logicians have firm ideas about what is and isn't true in the overarching universe; for example, I read once that Kurt Godel firmly believed that $2^{\aleph_0} = \aleph_4$, which would be one way for the Continuum Hypothesis to be false. But most of them seem to just not really care - they only care what's going on inside all those smaller universes.

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  • $\begingroup$ On reflection, perhaps I mixed together things I was considering about logical and mathematical pluralism. While the bit about "correct axioms" seems to be what I had in mind about geometries, another issue I hinted at was where genuine disagreement exists. It's all well and good to say the axioms of Euclidean and spherical geometry talk about different spaces. But with something like Classical Logic vs Paraconsistent Logic, I wonder if this overarching universe idea works. Would any mathematical theory fall inside it then, even ones which seem firmly in disagreement? $\endgroup$ Commented Jun 12, 2018 at 3:39
  • $\begingroup$ In my experience, non-classical logics (like paraconsistent logics) are usually demonstrated to be "workable" by building them inside set theory - either just talking about a particular proof system (just symbols and rules for manipulating them, not worrying about truth) or statements about a particular structure built in classical logic which has its own internal idea of what "true" means. The mathematician in question is able to fit either of these into whatever overarching universe they care to consider. $\endgroup$ Commented Jun 12, 2018 at 4:04
  • $\begingroup$ @Reese Re: Demonstrating "workability." It seems to me that any formal proof using unambiguous rules of syntax and inference can be verified using a digital computer, a machine that is itself based on classical logic. In this sense, might classical logic not be considered a "universal" logic? $\endgroup$ Commented Jun 12, 2018 at 15:34
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    $\begingroup$ @Dan Christensen A worry I would have about that is that it was a longstanding criticism of non-classical logics that they couldn't represent their own semantics without going classical in the metatheory, so it didn't see like we were legitimately non-classical. But now we now some theories can do so (paraconsistent set theory, etc.). So can it really be an argument for classical logic being a "universal logic" that it can unfaithfully represent an object theory and yet be a criticism of non-classical logics that are still unable to represent their own semantics? $\endgroup$ Commented Jun 12, 2018 at 15:49
  • $\begingroup$ Gödel at one point falsely thought (he was ill at the time) that he proved $2^{\aleph_0} = \aleph_2$, but the paper was quickly retracted. (After all, $\aleph_4$ would be just too arbitrary, wouldn't it ;-) $\endgroup$
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Mathematicians are about as pluralist about math as politicians about politics. Namely, some are, some aren't. For example, Davies is:

Davies, E. B. A defence of mathematical pluralism. Philos. Math. (3) 13 (2005), no. 3, 252–276.

Errett Bishop isn't:

Bishop, Errett A. Schizophrenia in contemporary mathematics. Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), 1–32, Contemp. Math., 39, Amer. Math. Soc., Providence, RI, 1985

(here alleged schizophrenia is attributed to those who don't subscribe to Bishop's constructive views which are the only correct ones).

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