Are Mathematicians Pluralists About Math? 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.
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.
Answering in a completely personal, psychological and somewhat emotive fashion:

After having worked with very productive researchers in signal processing and artificial intelligence, I saw that everyone has their own understanding about the mathematical objects they are using. Some thought in terms of methafors, subjective images, others in a more formal, proof/rational/logic way. Both types, and I am not trying to be exhaustive here, yielded great researchers in this practical but highly mathematically-suported area.

Going a little to the phylosophic side, people like Nietzsche and Jung have a very poignant view of "truth". Nietzsche claims that "God is dead" in a way that seems to be the same as "absolutism is dead". Jung says that truth is "Whatever works". Darwin says something similar. Einstein says everything is relative. He really means that: Nothing is able to exist, let alone be described, by only itself. 

My point is that we are subject to our minds when we do maths. We have our own cognitive biases and everyone thinks of the "same" objects differently, although we are frequently still able to understand ourselves. Furthermore, it is virtually impossible for a mathematician to grasp every pertinent theory within his lifetime. Each one has its specific axioms, propositions based entirely on intuition. These intuitions allow us to handle specific problems the best way.

So yes, I think mathematics is completely plural these days. Use the theory that works best for you! The goal is to solve something specific, after all!