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The "unreasonable effectiveness" of mathematics in describing the universe is often mentioned, but what about the sections of math that aren't applicable anywhere in physics at all? and why?

Could it perhaps be telling us something about which rules/ symmetries must never be broken even if the math "works", in theory ?

If we list out examples of math that doesn't apply to nature, what are the common patterns among them, if any?

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    Even Kant as the critique philosopher didn't claim any math doesn't or won't apply to the real world deducing from his categories theory, he only emphasized 3 ideas of pure reason not applicable to the real world, ie, God, freedom, and immortality. Thus maybe along the line the Cantorian transfinite (large inaccessible) cardinals is hard to be applied in reality while remains to serve as a foundation only. And for the reverse mathematicians such as Friedman and Simpson, the big 5 subsystems from the recursive RCA0 to the impredicative Π1-CA0 of 2nd-order PA are enough instead of set theory... Sep 19, 2022 at 2:35
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    Wigner's quip is dated, it is more common today to talk about unreasonable ineffectiveness of mathematics everywhere outside of physics. So it is more salient to look not for rules that must not be broken, but for those that should be broken, to make it more effective. But if you are looking for mathematics apparently inapplicable even to physics so far, large cardinals of transfinite set theory would be the prime candidates, although who knows what the future holds.
    – Conifold
    Sep 19, 2022 at 3:00
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    Some big wig 20th century philosopher who I can’t find right now said any number over 80^50 (or was it 50^80) will never be necessary for science. I’ll try to find the quote.
    – J Kusin
    Sep 19, 2022 at 15:22
  • @Conifold “… although who knows what the future holds.” Interesting. In my perception propositions about large cardinals are in principle statements about strings made from a finite alphabet, or just integers. Maybe physicists from the next century will find them handy to describe some extremely complicated systems.
    – fantasie
    Sep 19, 2022 at 15:41
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    Define 'applies'. This thread covers the topic: 'Is there any branch of Mathematics which has no applications in any other field or in real world?' math.stackexchange.com/questions/287673/… My suggested candidate is the hyper operation tetration: en.wikipedia.org/wiki/Tetration But Ancient Greek math parlour-tricks like imagining large numbers helped build out math. Even math toys & play, often prepare ground for applications, evening millennia later.
    – CriglCragl
    Sep 20, 2022 at 18:34

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Well-known examples of mathematical constructs that do not describe the real world (though they were intended to) would be the particle model called SU(5), Kaluza-Klein theory, Nordstrom's model of gravity, the Aether Theory, pre-relativistic mechanics, supersymmetry, and possibly string theory.

Note that physicists today construct models all the time that they know do not correctly describe the real world; these are called "toy models" which are supposed to represent special cases of a real theory in which the mathematics can be simplified enough to provide some insight into the likely structure of the real theory or perhaps provide estimates of things they wish to accurately calculate with the real theory.

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    I think that failed models intended to fit the world, and deliberate approximations are not what the question was asking about. It's more like: has anyone ever constructed a machine which never had any function, although it works correctly? A sort of anti-Rube Goldberg machine.
    – Scott Rowe
    Sep 20, 2022 at 10:21
  • If you stumbled across the Difference Engine (seen it) or H3 (seen it too) and you didn't know what they were for, what would you think? Divine Madness?
    – Scott Rowe
    Sep 21, 2022 at 2:46
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    @ScottRowe Yes! That's kind of what I meant. What provoked this question was wondering "why 3+1 spacetime dimensions?". Maybe this is how complex the universe can get without sacrificing some nice properties - such as manifolds being smooth (continuous) & differentiable everywhere? Higher dimensional analysis exists and is useful in several places though (including physics) , but there could be some obscure branches within this domain that are fundamentally inapplicable to nature. Related - en.wikipedia.org/wiki/Exotic_sphere Sep 21, 2022 at 19:38
  • @WillGraham, we have 3 + 1 dimensionality because if we had 4 dimensions of space then there would mathematically exist no stable orbits for planets around a sun, and we wouldn't be here. If we had 2 + 1 dimensionality, then atoms could not exist and we wouldn't be here either. Sep 22, 2022 at 6:50
  • @nielsnielsen that's not really an argument, are you suggesting there could be some 4D or 2D universes out there, out of our reach? because "not possible physically (for whatever reason, the universe hates these)" is very different from "not possible for life to evolve" Sep 24, 2022 at 4:29
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There are several different kinds of thing that could fit that description, though of course as our scientific knowledge advances, matters could change.

There are mathematical theories that apply only approximately to the real world. Euclidean geometry would be an example, since our universe is not perfectly flat. Nevertheless, it works well on a small scale.

There are mathematical accounts of physical theories that have features that don't seem to correspond to any real value for a physical property. For example, in thermodynamics, equations of state sometimes take the form of a cubic equation in the compressibility factor, and such equations may have negative roots, but we have no way to give physical sense to such values.

There are mathematical theories that deploy concepts that don't seem to have any physical analogue at all, as far as we know. Transfinite numbers would fall into this category. We don't know whether the universe is infinite, but even if it is, there is no obvious application of large cardinals. There are other examples where the mathematical concept of infinity yields unreal results. The Banach–Tarski theorem shows that a solid ball can be decomposed into finitely many pieces and then reassembled into two balls of the same size as the original. That wouldn't work in the real world: it holds because the balls consist of an infinite set of points, which are not 'solid' in the physical sense.

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    Incidentally, did you know that the string "banachtarski" is an anagram of "banachtarskibanachtarski"?
    – Bumble
    Sep 20, 2022 at 3:55
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    -1: I think the property you are looking for is 'amenability'. Go look it up. Who are you trying to scare with banachtarski nonsense? Its not even an anagram!!! Sep 20, 2022 at 4:13
  • -1: He said "at all". But of course thermodynamics is eminently applicable even of we toss out a few answers that just don't fit with our physical intuition. Right or wrong? Sep 20, 2022 at 4:20
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    Thermodynamics is a big subject, and some of it can be described exactly by mathematics, and some of it only approximately. Equations of state are only approximate. My point in referring to equations of state is they often have solutions that are completely non-physical and have to be discarded, and hence to that extent the mathematics does not apply. As to the anagram, it amuses me that you don't understand it, so I'm not going to explain it.
    – Bumble
    Sep 20, 2022 at 8:49
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    There ate lots of tjings that don't have physocal solutioms. I have no apples in my hand. Dors that mean that I also have -1 apple and +1 apple in my hand? No, of cpurse not. You fom't need to brimg in massive complications like thermodynamics to undersyand that physicists use physical intuition aboit the world. Anyway, whats that about an anagram above? Sep 20, 2022 at 8:50
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The imaginary quantity i is used extensively in mathematics although it is known there is no square root of -1.

The science of complex arithmetic derives from the assumption that it does exist, and it can simplify computations and produce verifiable results that don't themselves involve i.

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  • And, it fits in with other values that have more visible uses.
    – Scott Rowe
    Sep 21, 2022 at 2:48
  • There is indeed a number called i in the complex number set which is the square root of minus one. This number does not exist in the real number set but it is just as "real" as any number on the real number line. Sep 21, 2022 at 20:02
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    @nielsnielsen the number 1 is both on the real number line, and can be identified in nature. I was originally going to post that e and π arise from mathematics, but aren't values that can found in nature (which doesn't obey our maths). They form a part of our description of nature. But i is a purely mathematical construct. Sep 21, 2022 at 23:33
  • You gotta love that equation that puts them all together. It's enough to make you believe in God.
    – Scott Rowe
    Sep 22, 2022 at 0:26
  • @WeatherVane, both e and pi do indeed exist in nature; e shows up fundamentally in the logarithmic spiral that sunflower seed patterns follow in the flower head and pi shows up all over the place in the real world- for example, in the ratio between the length of the curved path followed by a meandering river and the straight-line distance from the river's origin to its end point. Sep 22, 2022 at 6:13
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some geodesics

Pierre Duhem, citing

gave the example of what came to be called "Duhem's bull", in Aim & Structure of Physical Theory pp. 138-41, § "An Example of Mathematical Deduction That Can Never Be Utilized", as quoted in the film

Duhem's bull

Here is part of Duhem's description of an object sliding on a bull's head:

First, there are geodesics that close on themselves. There are others that, without ever coming back to their starting point never end up infinitely far away from it; some keep turning around the right horn, and others around the left […]; other, more complicated ones, alternate turns around one horn with turns around the other one, following certain rules[…] On our bull’s head […] there will be geodesics that will go to infinity, one by climbing the right horn, others by climbing the left horn,[…]

Duhem's bull

Two geodesics that set off in nearly the same direction can follow quite different paths. Duhem says it like this:

If a point is launched on the surface in question from a position that is given geometrically, with a speed that is given geometrically, then mathematical deduction can determine the trajectory of this point and determine if this trajectory moves away to infinity or not. But, for the physicist, this deduction is forever unusable.

Notice the subtlety : the geodesic can be calculated mathematically, but this is of no use to the physicist.

There is a world of difference between theory and practice!

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