How can we be sure that asking If Mathematics is Platonic or If it is Our Own Construction is a meaningful thing to ask, and not just abuse of natural language? Can somebody provide a convincing argument for its validity?

We know what Platonism means -something on the lines that mathematics is out there -like sun, stars and moon, and also what Construction means -that mathematics is certainly our own mental construction, and isn't out there (let's assume this to be part of definition). Then are these two cases perfectly mutually exhaustive? If yes, is that enough to conclude that the question is logically valid, and not a misuse of words (out, there, our, own, mental, construction) which we didn't coin for usage in this context.

  • 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 9:29
  • @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 5:10

If you ask it, and you get a meaningful answer, then it is meaningful. This is an operational definition of semantics and is utilized by the Turing Test. The validity of operational definitions is based on correlation to establish dependence. Idealist and realist positions on the origins of math can occur in a nuanced way, depending on your flavor of idealism and realism, so there are hybrids between extreme idealist and realist positions. Platonism is a meaningful idealist philosophy for those who believe the mental and physical are not causally related. Platonism is also meaningful to those who reject the duality, because it embodies a prime example of a category mistake.

Since you can type it, I can read it, and we can have a conversation is strong evidence that the words have meaning. That it has been more than 2,000 years of discussion about the nature of origins of mathematics also gives credence to it being a meaningful interrogative.

Philosophers generally and mathematical philosophers more specifically have built the edifice of conceptualization and argumentation to provide a solid footing for answering the question since the time of Plato. How one answers the question largely centers around one metaphysical presuppositions. Today, Western philosophers largely fall into two broad camps: analytical (Anglo-American tradition) and phenomenological (Continental tradition) philosophers; the former tend towards objective analysis and the latter towards subjective introspection, which roughly speaking, are two broad approaches to rational and empirical thinking.

The answers you seek will depend on epistemic attitude and ontological commitment. One of the most important determinants of the answer you seek is whether you accept supervenience and reject Cartesian duality. Relatively contemporary philosophers of mind and general philosophers (starting with the ordinary language philosophers) like Ryle, Quine, Searle, Kim, Dennett, and others do. In these philosophers' estimation, the physical is the grounding ontology, and if you accept additional ontologies such as the mental (Dennett is eliminative materialist), then in some way they partially or fully reduce to the physical.

What are the implications, then, for the origins of mathematics? In essence, many ontological and epistemological questions are reduced to questions of psychology. This position, known as psychologism, which was is an influential theory that goes back to the 19th-century German mathematicians and philosophers who are largely responsible for pushing the mathematical program of foundationalism forward. See the SEP article early set theory for a brief description of how set theory became the foundational branch of mathematics and led to model theory and proof theory.

Today, in the 21st century, one academic discipline which bears tremendously is cognitive science which includes psycholinguistics which has a direct, scientific bearing on the notion of mathematical conceptualization and language. For philosophers who have hearkened to Quine's naturalized epistemology, the question is largely settled. All information, including mathematical information, is constructed by the brain of which the mind is a mere property, and mathematical thinking is a subset of the broader notion of cognition. Of course, many well-educated philosophers reject the rejection of Cartesian duality and insist mathematics does not depend on the mind. Whether or not one accepts Platonic Ideals, the theory is meaningful, because acceptance and rejection of theories is a matter of truth, which is a selection of which meaningful theory is the correct one.

See these SE posts:
Is mathematics a mental idea.
A comprehensive introduction to relationship between math and experience
Is mathematics truth

  • 1
    Thanks for the answer. Your point on operational definition of semantics utilised by Turing Test seems like a reasonable thing to stick to. However, that does mean that I cannot be sure that you are not a machine claiming to be interested in philosophy of mind, mathematics, computation and hot dogs!
    – Ajax
    Nov 7 '19 at 21:23

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.

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