From what I gather, for realists who are especially fond of a good old-fashioned Mathematical Platonism incorporated in their ontology, there seem to be two ways of getting at it. The first seems to be the need for truth-makers for mathematical propositions/truths, hence the positing of mathematical objects as abstract objects which serve as truth-makers. The other route is to marshal the Quine-Putnam Indispensability Argument, or some contemporary variation, and deduce their existence (am I right in thinking this is a deductive argument, or is it an abductive argument? Perhaps their are versions of both) by noting that the existential quantifier is a device for ontological commitment.

First of all, am I mistaken in thinking that these represent some of the motivations for adopting/positing Mathematical Platonism? Secondly, are there any other motivations for adopting/positing Mathematical Platonism?

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    See Mathematical Platonism: "mathematical platonism is the result of adding to Existence the two further claims Abstractness and Independence." – Mauro ALLEGRANZA Nov 9 '17 at 15:09
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    First, Quinean motivation is a particular case of truth-making, Quine's indispensability dictum is that we must commit to existence of all and only entities that make our best theories true (after paraphrase). Since this is an inference to the best explanation it is abductive. Second, such motivations are too philosophical and "cerebral" for most, mathematicians often cite direct "intuition" of mathematical entities, Gödel and others liken it to sense perception of physical objects. Some mathematical realists, like Burgess, even find indispensability arguments distasteful and unconvincing. – Conifold Nov 9 '17 at 19:50

I cannot speak for Platonists who argue along the lines you mentioned above. I personally find the indispensability argument has no more value than the ontological "proof" of God, namely that God could not be the superior being without existing (whereupon Kant answers "equally well a merchant could add some zeros to his account in order to improve his economical situation").

But mathematicians believing in set theory have an indispensable reason to be Platonists, at least when they are consistent: According to set theory there exist uncounbtable sets, i.e., sets with more elements than ever can be described, defined, mentioned, imagined individually by inhabitants of the universe. So, if existing, these elements do not exist in human mathematics (monologue, dialogue, discourse) but at most in God's knowledge. By the way, that is what Cantor believed from the scratch.

  • Even sufficiently large finite sets have the property that their individual elements can never be described, mentioned, etc. by the collective inhabitants of the universe. You are mischaracterizing uncountability. – user4894 Feb 9 '18 at 18:53
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    @user48942: In ideal, classical mathematics we neglect the constraints of reality since otherwise there is not even potential infinity and mathematics would become hard to handle. But the countability of all definitions, names, etc. is a provable feature within ideal mathematics. Your argument is confusing two very different features. – Wilhelm Feb 9 '18 at 21:24
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    Try to think like a mathematician: The set of definitions is countable. Therefore I am right and you are wrong. If you cannot think yourself but don't believe me, try to read the literature. For instance: All possible combinations of finitely many letters belong to a countable set. Since every real number has to be definable by a finite number of words, there can be only countably many real numbers – in contradiction with Cantor‘s theorem and its proof. [Hermann Weyl, Hilbert's successor at Göttingen] – Wilhelm Feb 10 '18 at 10:14
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    If we pursue the thought that each real number is defined by an arithmetical law, the idea of the totality of real numbers is no longer indispensable. [Paul Bernays, Hilbert's co-author] An uncountable set of relation symbols - such a system of notations can not exist. [Wilhelm F. Ackermann, Hilbert's co-author] If we define the real numbers in a strictly formal system, they are countable. [Kurt Schütte, Hilbert's last student] – Wilhelm Feb 10 '18 at 10:14
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    @user4894 Please read more carefully what I wrote about ideal mathematics. To make it yet easier to grasp: In an infinite and eternal universe every natural number can be defined. Most of uncountably many numbers however cannot be defined. Therefore they are not part of mathematics. – Wilhelm Feb 10 '18 at 21:16

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