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What is the best argument against the Platonic idea that mathematical objects are concrete things with causal powers?

But what is a Platonic Form or Idea? Take for example a perfect triangle, as it might be described by a mathematician. This would be a description of the Form or Idea of (a) Triangle. Plato says such Forms exist in an abstract state but independent of minds in their own realm.

So if someone asserted that logic had some form of concrete existence in the realm of ideas, what would be a good counter argument against this assertion? The tricky thing is that it seems logic can only be proven by logic itself, so we cannot know anything about logic itself, is that the rebuttal or is there a better rebuttal for this?

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    I am not sure how anything Plato says about Ideas can be interpreted as them being "material things". Objective - yes, concrete - perhaps (at least, in the sense of having causal powers unlike modern abstract objects), but they are as ideal as it gets. As for objections, Plato himself discusses some in Parmenides. Aristotle's objection that Ideas in their own realm are pointless because they fail to explain the workings of this world is still influential.
    – Conifold
    Commented Aug 3, 2022 at 2:49

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If one accepts Conifold's criticism that Plato was arguing for the objectivity of numbers, then one simply needs to challenge the existence or nature of objectivity. From WP:

Plato considered geometry a condition of idealism concerned with universal truth.

So, one needs to argue that mathematics is inherently a subjective enterprise. One famous mathematical enterprise that aligns itself with such an argument might be intuitionism:

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.

And that enterprise isn't that difficult to defend. Do rocks count? No. Did theorems exist before people? No. Are there eternal, universal facts about numbers? Certainly not. What there is are humans with a penchant for thinking. That they think largely in the same way is not because truths are objective, but rather that subjectivity has dispositions derived from biological similarities; this might be considered a characterization of intersubjectivity.

This conflict in philosophy was characterized by Dummett as a disagreement between mathematical realists and anti-realists. Mathematicians, such as Linnebo, are generally realists of one shape or another, but the view isn't as appealing to those outside of mathematics. For instance, cognitive semanticists, who have specific beliefs about where mathematical meaning derives from, reject what they called disembodied, objective mathematics. See Where Mathematics Comes From that bases its arguments on cognitive science.

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    This implies that there was no number 11 until the first human counted that high. And if that human died without sharing his counting method, the number popped out of existence again. It implies that 11 was not odd until a mathematician proved it was, and that it was not prime until a mathematician proved it. It implies that most numbers between 10^40 and 10^50 don't exist because no one has ever thought of them. Commented Aug 3, 2022 at 8:40
  • @DavidGudeman Of course. Ideas don't exist unless there are minds to construct and apprehend them. Since you have a hungry mind, Springer-Verlag has a fantastic biography on Brouwer. The philosophy he is famous for is Intuitionism in the Philosophy of Mathematics (SEP). It's an anti-realist doctrine.
    – J D
    Commented Aug 3, 2022 at 16:25
  • thanks for the citation. I read Brouwer many decades ago and never found anything of value in his ideas--in part because of the problems I pointed out in my comment, in part because his rejection of the notion of truth is self-defeating. Commented Aug 4, 2022 at 7:04

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