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The mathematician and mathematical physicist Alain Connes has expressed in many occasions that he is a Platonist and he thinks that mathematics itself does exist in the same level (or even in a "stronger" level) as physical reality.

This is very similar to Max Tegmark's hypothesis of the Mathematical Universe (MUH), which basically says that every mathematically possible structure exists as its own universe.

Since both approaches are almost identical, I was wondering if Connes has ever commented anything on Tegmark's views. I cannot find anything on the Internet, but since both ways of thinking are almost the same, it seems strange to me that Connes has never said anything about it. Does anyone know if Connes has commented anything on this?

J D
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vengaq
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  • see https://philosophy.stackexchange.com/questions/22448/what-are-the-historic-stances-on-the-epistemological-status-of-mathematics/22484#22484 – Swami Vishwananda Dec 13 '21 at 10:10
  • Suggested links and tag. – J D Dec 13 '21 at 17:22
  • MUH is incoherent, and it's really obvious based on mathematical logic. Connes' views seem to be too vague rather than incoherent. You should read up about [reverse mathematics](https://en.wikipedia.org/wiki/Reverse_mathematics). Mathematicians generally do not naturally come up with mathematical constructions beyond a certain proof-theoretic strength, roughly around bounded ZFC, but there is no evidence for platonic reality of mathematical entities beyond ATR0, which is in turn way stronger than what is needed for real-world applications of mathematics (roughly ACA). – user21820 Dec 14 '21 at 19:24
  • @user21820 Been I while since I read it, but MUH is based on the assertion nothing beyond mathematical objects are needed to describe/predict/experience all empirical facts. I would not call *that* mathematical logic unless you accept the MUH. A "person" is making that metaphysical judgment; it is not a proof arrived at after mathematical axioms and deductions. Now given MUH, a "person" is just a mathematical object, which would make everything mathematical logic. But you deny the MUH, so you can't have your cake and eat it too; MUH is not mathematical logic to the non-MUH'er. – J Kusin Dec 14 '21 at 20:28
  • @JKusin: Sorry, but your comment is as incoherent as MUH. There is simply no such thing as a coherent definition of "mathematical object" or "mathematical structure" that does not depend on a **fixed foundational system**. – user21820 Dec 14 '21 at 20:40
  • @user21820 I'm not sure why you think there can't be a fixed foundational system that leads to a ToE though? Self-reference and incompleteness? – J Kusin Dec 14 '21 at 21:44
  • @JKusin: It's not that there cannot be a fixed foundational system underlying reality, but once you specify it then it automatically makes the idea of "every mathematical structure exists" completely incoherent. **That** is why MUH is incoherent. You can easily check that proponents of MUH can never properly define "mathematical structure". It's as incoherent as claiming that "every fantasy exists". – user21820 Dec 15 '21 at 00:07
  • @user21820 Everything is mathematical structure, not every mathematical structure necessarily exists. Mathematical structures are entities defined purely by their relations according to Tegmark. – J Kusin Dec 15 '21 at 02:29
  • @JKusin: Your comment just goes to show that you do not know basic logic at all. I have nothing else to say. – user21820 Dec 15 '21 at 19:32
  • @user21820 You’re levying that at Tegmark, those are his words directly from OMU. – J Kusin Dec 15 '21 at 19:54
  • I do not care what people who are ignorant about logic say. If their statements are incoherent, no amount of blur followers can make them coherent. – user21820 Dec 15 '21 at 19:56
  • @user21820 I guess I'm especially dense and ignorant because I don't see what's incoherent (hard to believe sure) about finite mathematics/computability as a foundation either. Tegmark prefers this option (or related). He writes "if the [Computable Universe Hypothesis] turns out to be correct, it will instead be because the rest of the mathematical landscape was a mere illusion, fundamentally undefined...". – J Kusin Dec 15 '21 at 20:37
  • I'll say for the last time, go and actually **learn** basic logic (i.e. FOL at least up to a rigorous proof of the completeness and compactness theorems). People who do not understand basic logic can never grasp what they do not know, nor can they distinguish meaningful ideas from crankery. – user21820 Dec 15 '21 at 20:41
  • @user21820 A red-herring and/or already incorporated by Tegmark. "Tegmark goes on to note that although conventional theories in physics are Gödel-undecidable, the actual mathematical structure describing our world could still be Gödel-complete, and "could in principle contain observers capable of thinking about Gödel-incomplete mathematics," https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis – J Kusin Dec 15 '21 at 20:57
  • @JD: You may be interested in reading [my recent post regarding CUH](https://philosophy.stackexchange.com/a/88482) and the related comments. – user21820 Dec 18 '21 at 20:08
  • @user21820 Already did, and you got my vote. ; ) For me, Tegmark fails at the idea that all computation has to have, ultimately physical embodiment in the empirical sense and until empirical evidence proves the physical is actually abstract, it's non-scientific – J D Dec 18 '21 at 20:59

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I've thought of Tegmark as a neo-Pythagorean but I've recently come to reject that that characterisation. Both Platonism and Pythagoreanism and their variants are quite clear that the cosmos has a moral dimension. This is conspicuously lacking in Tegmark's theory.

He's more akin to mathematical Platonism which as a severely truncated form of Platonism is not Platonism at all. Hence it's better described as mathematical realism. Mazur, a mathematician, has described Conne's as a definite mathematical realist. Although I haven't come across any remarks of Conne's on Tegmark's theories, I think it's fair to say that Tegmark is also a mathematical realist on the basis of his theories.

Of course this says nothing about how Connes views Tegmark's theories. There is nothing that I can see on the net. As the book has been well publicised and that we know Conne's is interested in mathematical physics, I think the safe answer would be - not very much.

Mozibur Ullah
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