There are several ways to think about this question.

First, think of a simple situation. Empty Newtonian space has rotational symmetry. To this we add idealized point particles. Suppose these particles exert the Newtonian gravitational force on one another. This force also has spherical symmetry, but the simplest solution - two particles in an elliptic or hyperbolic orbit - does not! The motion selects a plane and is no longer rotationally symmetric.

The basic conclusion is: solutions almost never share the symmetry of the theory they solve. Look about you. To high precision we live in a Lorentz invariant world, and Lorentz transformations include rotations as well as boosts. Yet nothing you see about you shares this symmetry - you rotate a chair and the change is obvious!

From a mathematical point of view, something like the Higgs mechanism is actually quite general. The kinetic term of a scalar field has high symmetry - no direction or distance is preferred. Envision adding an arbitrary, spherically symmetric potential, V(r). If we expand V(r) in a Taylor series for small r, the first term is just a constant, and the linear term may be removed by a translation. We're stuck with the quadratic, r^2 term. Going higher, it's possible to transform away the cubic but not the quartic term. This leaves us with the essentials of the Higgs mechanism: a "Mexican hat" potential ( r^2 and r^4 terms ) which "half" the time has a minimum at some value of r. Low energy solutions will find this minimum, and now there is a preferred distance. The solution has less symmetry than the theory. (The physical Higgs particle represents the *difference* between the actual energy and this minimum).

We can ask the question the other way around: Why are fundamental theories so symmetric? A two-fold answer to this might be (1) they're often not when we include the details of realistic macroscopic systems and (2) even simple, linear differential equations have extremely complicated solutions capable of modeling all we can measure about some systems. One reason, I think, for the importance of "beauty" in physical equations is that beautiful equations are usually simple (in one way or another), and simple equations are the ones we can actually solve. It is remarkable how many times, in all areas of physics, a linear approximation captures the main features of a problem. An interesting example is general relativity, for which the action functional is linear in the curvature. As it happens, curvature is so small that a quadratic term added to the Einstein-Hilbert action can be barely measurable - the quadratic corrections to general relativity coming from string theory come with a miniscule coefficient of order Planck length squared.

If I recall correctly,  the main standard argument  for the symmetry breaking is that the vacua possessing the whole (unbroken) symmetry are usually unstable and thus are destroyed by the fluctuations, so the only physically viable vacua are those with broken symmetry.

There are several ways to think about this question.

First, think of a simple situation. Empty Newtonian space has rotational symmetry. To this we add idealized point particles. Suppose these particles exert the Newtonian gravitational force on one another. This force also has spherical symmetry, but the simplest solution - two particles in an elliptic or hyperbolic orbit - does not! The motion selects a plane and is no longer rotationally symmetric.

The basic conclusion is: solutions almost never share the symmetry of the theory they solve. Look about you. To high precision we live in a Lorentz invariant world, and Lorentz transformations include rotations as well as boosts. Yet nothing you see about you shares this symmetry - you rotate a chair and the change is obvious!

From a mathematical point of view, something like the Higgs mechanism is actually quite general. The kinetic term of a scalar field has high symmetry - no direction or distance is preferred. Envision adding an arbitrary, spherically symmetric potential, V(r). If we expand V(r) in a Taylor series for small r, the first term is just a constant, and the linear term may be removed by a translation. We're stuck with the quadratic, r^2 term. Going higher, it's possible to transform away the cubic but not the quartic term. This leaves us with the essentials of the Higgs mechanism: a "Mexican hat" potential ( r^2 and r^4 terms ) which "half" the time has a minimum at some value of r. Low energy solutions will find this minimum, and now there is a preferred distance. The solution has less symmetry than the theory. (The physical Higgs particle represents the *difference* between the actual energy and this minimum).

We can ask the question the other way around: Why are fundamental theories so symmetric? A two-fold answer to this might be (1) they're often not when we include the details of realistic macroscopic systems and (2) even simple, linear differential equations have extremely complicated solutions capable of modeling all we can measure about some systems. One reason, I think, for the importance of "beauty" in physical equations is that beautiful equations are usually simple (in one way or another), and simple equations are the ones we can actually solve. It is remarkable how many times, in all areas of physics, a linear approximation captures the main features of a problem. An interesting example is general relativity, for which the action functional is linear in the curvature. As it happens, curvature is so small that a quadratic term added to the Einstein-Hilbert action can be barely measurable - the quadratic corrections to general relativity coming from string theory come with a miniscule coefficient of order Planck length squared.

I suppose the question could be asked 'Why do we think there are symmetries in nature (which then are broken)?'  Is this a consequence of our mathematical modeling or do they really pertain to reality?  Since most of what we 'see' are broken symmetries - why is this not a property of nature, rather than the symmetries (which could be just useful mathematical phantoms)?

Does a 'line', as a mathematical concept, actually exist in nature?  I have never heard of one being discovered or invented.  It is a (very useful) mathematical phantom.

As much as other discussions and posts suggest the inherence of mathematics in nature (mathematics as being inherent in nature), most mathematical concepts cannot be found in nature.   Is symmetry such a concept?

Broken symmetry in nature as mentioned in your question, implies a wide  range of fields such as particle physics, embryology, etc however if we restrict ourselves to field theories , breaking symmetry could be spontaneous and explicit, the first seems to be more important than the latter. Generally speaking , symmetries in this context pertain to the governing group and related Lie algebra which leaves invariant  the Lagrangian or ground state or both , then breaking takes places whenever the group reduces to one its subgroup , So the "breaking symmetry" is a necessary part of  any model when the field of model interpretation requires a development, these examples clarify this notion: Goldstone  bosons (spinwave,magnons etc) are produced as broken symmetry of ground state group; finite maximum velocity ( i.e. light velocity) breaks the symmetry of velocities and results in the especial relativity, discovery of complex number is another example. Broken symmetry , consequently is not an innovation ,rather inherent of nature and the reason of dynamic ism and varieties in nature.

As an experimentalist I have only a vague feeling about the symmetries and their breaking in Nature. First of all why the symmetries should be strictly respected if the theoretical tools we use are only an approximation of a deeper, not yet known theory ? I think that there is also an anthropic principle explaining why the symmetry breaking is necessary in order to allow for complexity (and us) to emerge. I have heard also about funny speculations claiming that we live in fact in a big simulation which is not quite perfect as it happens in the MC programs used to describe our experimental data. Just my three cents...

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According to Plato [ Timaeus 57E-C ]:

Motion cannot exist in that which is uniform..... we accept that whatever is it is uniform, motion is the lack of uniformity.......the generation of nonuniormity produces constat motion....

Regards from Athens,

Panagiotis Stefanides

Symmetries are  broken because symmetric states are high energy states and therefore unstable.Take our universe  to be a closed dynamic thermodynamic system.Then it follows that as time flows then energy density reduces thus high energy states cannot  be stable.This leads to the inevitable breaking of symmetric states.