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I have read that Eugene Paul Wigner thought that all mathematical structures had physical existence.

Does that mean that he believed in a multiverse containing all mathematical structures as separate universes?

And if yes, did he consider all types of mathematics (for example mathematics based on other kinds of logic like paraconsistent logic)?

  • The answer to all questions of the form "Did X think that all mathematical structures have physical existence" is no, so you do not have to ask them one at a time. Mathematical structures being physical is specific to Tegmark, even he can not explain what that means, and even he does not extend it to all of them. – Conifold Aug 27 at 3:48
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Max Tegmark references Eugene Wigner's "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" from a collection of Wigner's essays, Symmetries and Reflections in "The Multiverse Hierarchy" (page 12):

In a famous essay, Wigner (1967) argued that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious”, and that “there is no rational explanation for it”. This argument can be taken as support for assumption 1: here the utility of mathematics for describing the physical world is a natural consequence of the fact that the latter is a mathematical structure, and we are simply uncovering this bit by bit.

Tegmark's Assumption 1 is

That the physical world (specifically our level III multiverse) is a mathematical structure

It seems odd that Tegmark should cite an assertion that "there is no rational explanation" for the enormous usefulness of mathematics as support of his own rational explanation for it. If Wigner agreed with Tegmark surely Tegmark would be able to find something relevant to quote.

Rather than thinking mathematical structures are the universe, Wigler believes, in the article Tegmark quoted, that mathematics is the correct language (page 230):

However, it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language.

To counter somewhat the correctness of the language Wigler also notes that some theories are not as correct as others such as Ptolomy's epicycles and Bohr's early ideas of the atom. In the end, "we do not know why our theories work so well". (pages 236-7)

Having the correct language is not the same as believing that such a language is reality. That the language accurately describes reality is surprising for Wigler is as close as one may get to his accepting Tegmark's Assumption 1.


Tegmark, M. (2009). The multiverse hierarchy. arXiv preprint arXiv:0905.1283. Retrieved on August 24, 2019 from arXiv.org at https://arxiv.org/pdf/0905.1283.pdf

Wigler, E. P. Symmetries and Reflections. Retrieved on August 27, 2019 from Internet Archive at https://archive.org/details/SymmetriesAndReflections/page/n1

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