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Wigner's paper "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a well-known paper in the community of the philosophy of mathematics.

The overbearing question in his paper can be summarised as

Why is mathematics so good at explaining physical phenomena?

This, we do not have a set answer for. But I was wondering, does his "unreasonable effectiveness" problem threaten any common philosophies?

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    But mathematics does not explain anything. Prediction is different; you can be effective even if you have no idea how you manage it. His paper was about effectiveness, which is not the same as explanation. – user20153 Jun 1 '16 at 19:44
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    This is a central issue in the philosophy of mathematics, called 'the problem of appllcability'. See iep.utm.edu/math-app, which also mentions Wigner. – Eliran Jun 1 '16 at 20:18
  • @mobileink But what it is effective at is framing explanations, so this is in some measure hair-splitting. Without the math, modern physics also fails to explain anything. But with it, it succeeds. So how is it supposed to be that the math is not doing any of the explaining? – jobermark Jun 2 '16 at 15:39
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    @jobermark: Explanation is not the same as description. Sciences use math to describe (model) natural phenomena, not explain them. Newton's Laws described how things happen; they did not offer an explanation. Newton himself was explicit about this early on; he did not consider gravity to be a force (adducing a concept of "force" to account for the effectiveness of his equations would be a form of explanation, one with no mathematical component.) – user20153 Jun 2 '16 at 17:54
  • @mobileink If the description makes the understanding of the phenomenon clearer, it is an explanation, kind of by definition. If the math makes the description less ambiguous, it is part of the explanation. You have some concept of 'clear' in mind that insists all formal science is implicitly unclear, which is just silly. – jobermark Jun 2 '16 at 18:18
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To my mind, it demands a reason from each philosophy. But each of them pretty much has one, because this astonishing power was noticed long before now. Alongside the early philosophers in Greece were the Pythagoreans, who worshipped that power as a god.

If you admit any kind of idealism you end up assuming mathematics has transcendental roots, in a Platonic sense. So of course, mathematics describes reality because reality is built upon the mind of God, or whatever your essentialist replacement is, and so are we.

And if you assume total naturalism, but take a view of mathematics that makes abstract objects mental constructions, then mathematics describes reality because we are evolved to deal with reality, and mathematics is built upon our natural intuitions, which are honed over hundreds of thousands of years of adaptation.

Every discipline relies upon some evolved experience and intuition. But our logical and spatial intuitions apply to pretty much every action we take, and therefore have a lot more opportunities to fail us, and get improved. In doing math, we are merely extracting the combinatorial power bred into us.

An exception would be philosophies of science that try not to consider mathematics and the other exact sciences as a special case. If you try to adopt a philosophy of mathematics as an abstract version of an experimental science, rather than as an internal exploration of human psychology or mental structure in general, you can have a very hard time with this question.

Various utilitarians and pragmatists (including, to some degree Karl Popper) have proposed such an approach, but they then focus elsewhere and fail to address this question.

None of our other experimental sciences have failed to go through what Kuhn identifies as a 'revolution', where some underlying model is completely replaced by a different one -- we moved from Alchemy's four elements to Chemistry's limitless number, from Substance Theory to Atomism, from Aristotelian notions requiring direct contact to transmit effects and presuming inherent rest to Newton's laws, from Ptolemaic geocentrism to Copernicanism. In each case, we lost a little ground to the new insights. (e.g. Alchemy had reasons why liquids puddle and freezing water expands -- Chemistry didn't, for quite some time. Ptolemy's math was way better than heliocentric models until Kepler realised orbits are elliptical.)

So if mathematics is based on experience, like other scientific endeavors, we should see the same thing happen there, and we don't. No element of mathematics has undergone such an upset. Even ideas that struck people as insane when they arose, like irrational measures, or multiple infinities, have been cleanly folded into mathematics leaving the original model whole.

  • "transcendental roots" -- Is this supposed to be a pun? As it is, I think this is extremely confusing, since transcendental roots are a thing in math and you don't seem to be referring to that thing here. – Era Jun 2 '16 at 23:26
  • Actually, transcendental numbers are the ones that are not roots, right? No, it is unintentional. I mean they have their roots and grounding in things that are transcendental in the philosophical sense of "lying partly in the ideal realm". – jobermark Jun 2 '16 at 23:56
  • No, transcendental numbers are the ones that are not roots of polynomials (the technical definition is a little longer and more specific). They are the roots of transcendental functions. – Era Jun 2 '16 at 23:57
  • Those are traditionally zeroes and not 'roots'. No one discusses the 'roots' of the Riemann zeta function. And of course, yes they can be the roots of polynomials with transcendental coefficients. The point is they are not very much of a thing, and the use of those two words together is not common enough to make this a decent pun. – jobermark Jun 2 '16 at 23:59
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Wigner's paper amazes any reader, notably any philospher or mathematician, see https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

I do not consider the paper a threat for any philosophy but a challenge, pointing to an open problem.

Mathematics is the main tool to formalize physical theories, capturing their content in an unambigous way. Mathematics is the prerequisite to make any quantifiable and precise prediction or retrodiction about the events from the physical world.

Conversely, mathematics is a game with arbitrary concepts and logical rules. Mathematics gets its power and precision from this simpleness as a free creation of the human mind. A game, which is often played without any physical application in mind.

Why does mathematics fit to reality?

This question has never been answered in a satisfactory manner. Wigner was the first pointing out that here is a deep question.

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Mathematics is just the description of patterns. Nature must possess patterns,regularities, if there is to be anything worth calling existence. The concept of randomness, chaos etc requires patterned structures to compare it to. Anything short of infinite randomness involves patterns.There is then no mystery as to why mathematics is useful in describing nature. It is then trivial that mathematics should be so useful. Whatever patterns exist in nature are fair game for mathematical study. The threatened philosophies are those that don't accept that patterns of experience are what science is all about, which is to say metaphysics, if it is taken to involve distinctions that can't be experimentally/experientially settled. http://www.gresham.ac.uk/lectures-and-events/100-essential-things-you-didnt-know-about-maths-and-the-arts

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