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In mathematical Platonism, a mathematical cousain of Platonism proper numbers are seen as ideas that are outside of the phenomenal world. It's an obscure doctrine

Consider, the mathematical apparatus describing QM; this uses a hilbert space of states and this is an infinite dimensional space; although it is dependent upon spacetime it is not itself within it in anyway; it supervenes on it.

Can this be used as circumstantial evidence for mathematical Platonism? Or should we argue that this is merely consequence of using mathematics to describe QM.

(Though the latter seems plausible, this can't quite be the case; consider Classical Mechanics, this is also described by mathematics; yet the the primary quantities of position, mass and vocify have a direct physical analogue; this is no where near the case for QM).

Mozibur Ullah
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    "Mathematical platonism" concerns existence of mathematical abstractions, you are talking about metaphysical platonism. But I am not sure why single out QM, any mathematized physical theory can be interpreted in Platonist terms. Theory of Everything, if it materializes, would arguably be stronger "evidence" than a partial theory like QM, and Tegmark's "mathematical universe" is just such an interpretation of it. – Conifold May 03 '18 at 19:24
  • @conifold: I've distinguished metaphysical Platonism (I just call it Platonism) from mathematical Platonism; I've explained why I've singled out QM as opposed to Classical Mechanics, Special Relativity and General Relativity in the question. – Mozibur Ullah May 03 '18 at 19:34
  • @Conifold: Tegmarks universe is his highly speculative theory of everything; for sure it has a family resemblance to Platonism; but this isn't what I'm talking about. I'm asking a small and focused question. – Mozibur Ullah May 03 '18 at 19:39
  • @conifold: Metaphysical Platonism doesn't have a direct relationship mathematical Platonism: where are the Ideas of the Good, Beauty & Justice? This is why what I'm asking about, in my judgement, is closer to mathematical Platonism. – Mozibur Ullah May 03 '18 at 19:42
  • ''Metaphysical" Platonism is best seen as a caricature of mathematics; ideologues and theologians have done their best to hide the fact and indeed people still believe that Mathematical Pl. is derived. Recent usage replaced the term "ídeas"by "forms" and made the historical development more evident. – sand1 May 03 '18 at 20:58
  • @sand1: You're entitled to your opinion - because that's all that it is; however you'll note that the SEP has entry on Platonism in Metaphysics. I think your history of the subject and it's development is somewhat naive. – Mozibur Ullah May 03 '18 at 21:39

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Can this be used as circumstantial evidence for mathematical Platonism? Or should we argue that this is merely consequence of using mathematics to describe QM.

Ultimately it would depend on which interpretation of QM you subscribe to. The Copenhagen and decoherence interpretations are staunchly anti-realist ("Shut up and calculate") and so it would be kind of hard to reconcile them with the realism that Platonism or even Mathematical Platonism entails.

Hilbert spaces and wave function are convenient tools for estimating probabilities, assigning any more meaning to them would be foolhardy, would mostly likely be the response of someone from one of those schools.

On the other hand, hidden variable interpretations of QM seem perfectly amenable to Mathematical Platonism.

You might want to check out Penrose, who was a Platonist and worked on QM and Putnam who was a scientific realist and who also wrote about QM.

Alexander S King
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  • I don't see why hidden variables theories are amenable to mathematical Platonism more than any other interpretation - they use the same mathematical architecture; in fact all the main interpretations do (with variants). – Mozibur Ullah May 03 '18 at 18:10
  • @MoziburUllah I was thinking specifically about Bohmian QM - which I think (I will need to double check later) has it's own mathematical formalism which supplements the standard Hilbert space formalism. – Alexander S King May 03 '18 at 18:20
  • I've come across it. It used to be called pilot wave theory. It's takes off from Schrodingers equation; but as Dirac showed this was equivalent to Heisenbergs Matrix Mechanics - aka the Hilbert Space formalism. It's likely that the Bohmian formalism also expressible in Hilbert space formalism too. – Mozibur Ullah May 03 '18 at 18:32