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Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected.

One feature of the standard model of particle physics is that the weak force is not symmetric with respect to a parity transformation (to be more specific only left handed fermions interact with the weak force).

This is an example of a special kind of counter-intuitive result that defies a symmetry that looks extremely reasonable given our intuition about the world. This implies that the world we see when we glance at the mirror is "unphysical" (or at least obeys a different law for weak interaction).

I'm looking for results in maths where an obvious symmetry was suggested by a certain problem yet careful consideration revealed that the symmetry is in fact broken for certain cases.

Although there are many threads in SE about counter-intuitive results in maths I don't think my question is a duplicate. If it is indeed a duplicate or not at all appropriate I'm sorry.

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3 Answers

For finite dimensional real vector spaces $V$ we have an "obvious" and canonical isomorphism with its bidual $V^{**}$. This "symmetry" breaks for infinitedimensional spaces.


Also in the finite case, ordinals and cardinals are "the same", but this breaks for infinites

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All the paradoxical decomposition might count. We have the famous Banach-Tarski (decompose a sphere into 2 despite using only rotation and translation), and the related von Neumann (decompose a square using only area preserving affine transformation). It's intuitive that somehow these transformation should not change the "size" of the shape, and thus very much a symmetry, and we should be able to measure all shape, but as it turns out, nope.

Another one might be the notion of length/area defined by approximation. For a rectifiable curve, the length can be defined by approximating it with piecewise linear curve, in any dimension. Doing the analogous thing for area of surface and you will end up with even simple surface (such as cylinder) having infinite area. It seems intuitive that a cylinder surface is just an upgrade of a circle when you upgrade the space from 2D to 3D, so if it work for curve, it should work for surface. Apparently not.

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In example in probablity theory would be the monty hall problem (http://en.wikipedia.org/wiki/Monty_Hall_problem). Most people seem, at least initially, to assume that there's a symmetry between the two unopened doors, and hence believe that the probability of winning is the same regardless of whether you switch doors or not. Yet it turns out that the probability of winning is higher if you switch.

Another problem, although one which deals with an extrapolation rather than a symmetry in a strict sense, is solving polynomial equations. After being presented the formula for finding roots of univariate polynomials of degrees $1$, $2$, $3$ and $4$, you'd naturally assume that there's a formula for every degree. Yet it turns out that there's no universal formula for degree $5$ and higher.

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