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Physicist Eugene Wigner 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

as it it indicated in his essay “The Unreasonable Effectiveness of Mathematics”. His words have been interpreted by many philosophers and physicists (Like Max Tegmark) as suggesting that the universe IS mathematics.

In this book*: “The Pythagorean World: Why Mathematics Is Unreasonably Effective In Physics”, the author indicates that Wigner’s view of mathematics is not as inclusive as Graham Priest’s, since it does not include paraconsistent logic (Priest is a philosopher who has worked extensively in this area).

But what does this exactly mean? What does it mean that Wigner did not include Paraconsistent logics in this view? Does it mean that he simply did not explicitly mention them in his essay?

*Link to the book

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    Perhaps it means the Wigner is not tolerant of contradictions. And part of the mystery that Wigner has detected is that a powerful method of progressing math and science is to NOT be tolerant of contradictions.
    – puppetsock
    Jan 14, 2020 at 18:06
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    Like puppetsock l see no reason for Wigner to consider paraconsistent logic or accept true contradictions. Dialethism is a minority sport. .
    – user20253
    Mar 31, 2020 at 11:42
  • @PeterJ why do you see no reasons for that?
    – vengaq
    Apr 3, 2020 at 18:39
  • @vengaq - I know of no instance of a true contradiction so see no argument for modifying logic. I see it as a strength that Wigner ignores paraconsistent logic.
    – user20253
    Apr 4, 2020 at 13:21
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    There's a great point that I saw Tim Maudlin make somewhere to effect that Wigner's paper is probably the single most extreme example of people referring to and using a paper based on its title, without having read it. It is short, easy to read, and available for free, there no excuse for this. Not accusing anyone here btw...
    – Rollo Burgess
    Mar 26, 2021 at 13:23

5 Answers 5

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A paraconsistent logic is a logic that does not validate the principle of explosion ("from a contradiction, anything follows"). A paraconsistent plurality of worlds will therefore be open to nontrivial worlds in which there are true contradictions.

Cantor identified both God and "inconsistent multiplicities" as examples of absolute infinity. The latter were structures like "the ordinal of all ordinals," or a naive V, which would be rather "tall" or "large" such that saying there were V-many worlds would be saying that there were as many worlds as could be. A paraconsistent naive V would therefore "seem larger" than a consistent refined V (where the class of all sets doesn't have a commensurable "size," exactly). (See also the SEP article on impossible worlds.)

As far as physics goes, Wigner's eschewal of paraconsistency would be Wigner not applying inconsistent mathematics to physics, or not thinking it applied, or something along these lines.

EDIT: or maybe the idea is that the application of inconsistent mathematics to physics is not as "mysterious" as the application of consistent mathematics supposedly is?

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Wigner might have been a very great physicist, but the mysteriousness of why numbers are important in physics is not at all mysterious. The very word 'geometry' betrays its earthy origins. Physics and number met first in geometry, and they have continued together ever since. Given that this meeting was over ten millenia ago, its not at all strange by now, that we have a great deal to show just how much overlap there is between the two.

The other observation is that both mathematics and physics study what is neccessary, the first in number abd geometry and the latter in the physical world. Given their mutual relation to the neccessary, it's not surprising that they are intimately related.

Shame on Wigner, he should have thought about it more. Then again, he did remark that the Native Americans made way for the colonising Europeans as a weaker race (implied) made way for the 'stronger'. I suppose racial theorising was all the rage back then. Not a word about genocide, concentration camps, broken treaties, venality married to stupidity and ignorance and an almost insatiable greed...

Mr Wigner, you can't make up for all that by thinking your name rather sounds like wig-wam...

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And there is 'the unreasonable ineffectiveness of mathematics in the biological sciences' discussed here: Does reality have axioms?

It's just symmetry. Continuous symmetries, and symmetries under transformation, are what makes physics work, and what make mathematics apply well to the part of science which focuses on the phenomena with relatively simple symmetries. Because number lines are an abstraction of continuous symmetries. And noting symmetries is the most efficient way to drop irrelevant data.

Euclid's geometry was thought to be fundamental, then you get Riemannian geometry, and Anti-deSitter space. Logic like mathematics was thought fully axiomatisable with self-evident axioms, then Godel. Mathematics and physics don't give privileged access to the world, but emerge from it because of it's regularities. And when we understand the world better, they change. And similarly with the laws of physics. There isn't a magic equation out there waiting for us, that solves everything, because the complexity of the universe is emergent, and the laws that govern it are regularities that emerge too. So they will never be complete. And neither will mathematics.

I would nominate the Wigner-VonNeumann interpretation as the single most misleading and problematic of all proposal for the quantum measurement problem. And as evidence of Wigner's choosing an additional time a kind of science-mysticism, that not only went far beyond the evidence but never withstood basic scrutiny.

I see this whole business of math-mysticism in the Analytic Tradition as Pythagoras' shadow, that Plato's Academy was Socratic method, plus Pythagorean math-cult. Tegmark is in the cult, getting misty-eyed over triangles, until some equivalent of discovering irrational numbers makes him choose his fantasy of order over observations. As Hossenfelder has written, searching for beauty is intrinsically problematic and limiting, for good science.

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I suspect the final sentence in your question points at the answer. In his paper, Wigner cites several examples of the affinity between mathematical ideas and the way in which the Universe seems to work, none of them having anything to do with paraconsistent logic.

I think there is an ironic parallel here, between a common refutation of Wigner's claim- namely that only a small subset of mathematics is useful for physics- and Jane McDonnell's suggestion that Wigner had a narrow view of mathematics.

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I consider mathematics as a quantitative language. Thus, I believe that laws of physics when expressed in this language gives us equations used in physics.

For example, "Magnitude of force is directly proportional to rate of change of momentum with time." can be expressed in the "language of mathematics" as:-

|F| = dp/dt

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