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Which parts of mathematics are objectively true or false like finite arithmetic and which are only true, false or undecidable relative to a particular axiom system such as Euclidean geometry?

Background

This is a philosophical question that affects the practice of mathematics. For example it led to the development of alternatives to Euclidean geometry. There are well documented classical approaches including Platonism, formalism and various forms of constructivism and intuitionism. I am posting this as a community Wiki to collect less well known or alternative views and to see how they relate to my own thoughts. This is a question that can never be finally resolved given the necessary incompleteness of any sufficiently powerful mathematical system. Debatable extensions to the foundations of mathematics will always be proposed. However, it might be possible to reach consensus on an imprecise philosophical criteria for objective mathematics.

The question is taking on increasing importance today. Solomon Feferman near the end of Does mathematics need new axioms? wrote as follows.

I am convinced that the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way. Moreover, I think the Platonistic philosophy of mathematics that is currently claimed to justify set theory and mathematics more generally is thoroughly unsatisfactory and that some other philosophy grounded in inter-subjective human conceptions will have to be sought to explain the apparent objectivity of mathematics.

He adds in a footnote:

CH is just the most prominent example of many set-theoretical statements that I consider to be inherently vague. Of course, one may reason confidently within set theory (e. g., in ZFC) about such statements as if they had a definite meaning.

In contrast Harvey Friedman has created a substantial body of work in which he proves certain arithmetical questions are decidable in ZFC plus certain large cardinal axioms and not decidable in ZFC or even ZFC plus smaller large cardinal axioms. Feferman's paper led to a symposium, described by Feferman in which Friedman participated.

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I like this topic a lot, but it's more of a philosophical question than a mathematical question, and I don't see it as admitting a clear-cut answer. (For example, it seems to presuppose that arithmetic is objectively true, which I'd agree with but which is already staking out some degree of Platonism. I'm not convinced MO is the right place to debate these sorts of questions.) –  Henry Cohn Mar 29 '12 at 21:26
    
Just to emphasize the non-mathematical nature of this issue, as one of the few, lonely, mathematical anti-Platonists, I would not assert the objective truth of finite arithmetic, but might instead entertain statements such as that our brains find it to be a satisfactory model of what we observe out there in the universe. –  Lee Mosher Mar 29 '12 at 21:44
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I'm partial to the multiverse view, as advocated for instance in arxiv.org/abs/1108.4223 . Within the multiverse one can designate, if one wishes, a single universe as the "objective" one (and in particular, designate one of the possible models of arithmetic as being "true arithmetic"), and it is often convenient to do so for both psychological and philosophical reasons, but one should not be too insistent on claiming that such a choice of objective universe is canonical. –  Terry Tao Mar 29 '12 at 22:03
    
The designation of the standard model of PA is far from arbitrary, since for example it's the only computable model. So this feels very different from the variety of models of ZFC serving different philosophical purposes. I really like the multiverse viewpoint, and I think it's a beautiful perspective on set theory, but for example large cardinal axioms tend to point in the opposite direction (of describing a single universe rather than a multitude of possibilities on an equal footing). The fact that this question leads so readily to debate is a reason why the MO format is suboptimal. –  Henry Cohn Mar 30 '12 at 3:16
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closed as not a real question by Andres Caicedo, Gerald Edgar, Henry Cohn, Simon Thomas, Chris Godsil Mar 29 '12 at 21:53

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