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Certain propositions can be meaningless. How do we know if "Are there abstract mathematical entities (platonism)?" a meaningful question, and not an abuse of language?

Ajax
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  • It is unclear what you are asking, I am afraid. We do not need two possibilities A and B to be mutually exclusive or exhaustive to ask if one of them holds, even asking if a single one non-exhaustive possibility holds is a "logically valid" question. And it needs no "argument for validity". Of course, it could happen that neither holds, or both in part, and that would be the answer. And there are plenty of alternatives to both Platonism and constructivism, as well as blends of the two. – Conifold Oct 29 '19 at 20:36
  • Platonism is at his roots a generalization of mathematics, imho, For historical reasons it became more widely known and that created the inverted illusion of mathematics being platonic. – sand1 Oct 30 '19 at 09:29
  • @Conifold Let me rephrase in terms of *if, then* argument. I ask **IF** Platonism and Constructivism are *mutually exhaustive, exclusive, and meaningful concepts* (in our World, i think this to be the case -since either something exists independently of us, or we create it), **THEN**, is it logically valid to apply this distinction to anything (and in this case, to foundations of mathematics)? – Ajax Oct 30 '19 at 14:14
  • Your "if" is rather obviously false, just look over what SEP has on philosophy of mathematics. Or think of a hammer or any other artifact, it exists independently of us, yet we created it. But if we did have a valid dichotomy we could obviously apply it. So the question seems doubly moot. – Conifold Oct 30 '19 at 17:53
  • @Conifold As for hammer, it exists independently of us, but it was created by us, so in this case, it strictly belongs to Constructivism. As for a valid dichotomy, my query resolves to *IF* existence of a 'valid dichotomy' can serve as a concrete (necessary and sufficient) argument in favor of asserting that Platonism vs Constructivism is *not* a nonsensical thing to discuss in context of foundation of mathematics. – Ajax Oct 31 '19 at 05:10
  • To add to Conifold's point, the world 'out there' may be our construction so your dichotomy can be challenged. A thing can be 'out there' and also a mental construction. –  Nov 04 '19 at 12:01
  • Platonism can be supported by the [indispensability of math](https://en.wikipedia.org/wiki/The_Indispensability_of_Mathematics) thesis proposed by Quine and Putnam: *This thesis is based on the premise that mathematical entities are placed on the same ontological foundation as other theoretical entities indispensable to our best scientific theories*. Field famously argued against such thesis in his *Science Without Numbers* to some degree but not completely and convincingly. If something is indispensable then its ontic existence can be argued for, and it's not exclusive with constructivism... – Double Knot Dec 23 '21 at 23:27

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I think mathematics has both platonic and constructive aspects. If we view mathematics as a language for describing patterns evident in numbers, sets, etc., then these patterns might exist in some platonic sense, but the language used to name and describe them has been purposely constructed.

sfmiller940
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