It needs a concept of objective reality, but it doesn't require belief that concept is true, or anything more than a foundational element of a pragmatically useful contingent model.
What? No, it's not. Your attempt to justify this with the statement that "arithmetic truth cannot be defined in arithmetic" fails, because that's not the same thing as a there being "no absolute truth" in mathematics.
what else is this going to mean? "absolute truth" in mathematics should be a mathematical concept. Is that not the definition of "in mathematics"? I don't think you should accuse me of moving the goalposts when they aren't clearly... posted.
> For example, there are no round squares. There are also no square circles. The angles of a triangle add up to 180 degrees.
When Richard Feynman talks about cargo cult science [1], everyone agrees that he has made a deep point about the process by which science is done and is clearly in favor of science. All the same, it is possible to criticize science as a contextual, social construct in order to find the ways it can fail while still being in favor of science. A great deal of postmodern philosophy falls into this camp, but you won't learn that by cherry-picking summaries from wikipedia that say less than you think they say.
Aside below
In mathematics, the fact that there is no absolute truth is merely a technical fact. Tarski's undefinability theorem [2] says that arithmetic truth cannot be defined in arithmetic. This is easy to see: Arithmetic makes use of arbitrarily nested quantifiers. If arithmetic truth were an arithmetic statement, then it would necessarily have some finite number of alternating quantifiers (forall A. exists B. forall C. ...), but the truth of a formula with a greater number of alternations can't be defined by such a truth formula of lesser complexity. Like all results of this sort, they are just an application of the diagonalization argument.
This result is the same for all theories of non-trivial expressivity. There is no way around this, you cannot formalize truth. You can only push the notion of truth out into a meta-level, which again has an unformalizable notion of truth. This is not a deep philosophical failure, it's just a technical aspect of mathematics.
If mathematics cannot pin down absolute truth, what hope does the concept have in far more complicated situations? It's a useful heuristic but when you look at it closely enough, you see its incoherence. I'm glad that there are philosophers out there who critically engage with the social implications of this incoherence.
[1] http://calteches.library.caltech.edu/51/2/CargoCult.htm
[2] https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theo...