Ask almost anyone, and they'll hold up their hands, put them together and say "there you go!" Except, your hands aren't really symmetrical. In fact every "symmetry" I can think of suffers from this 'accuracy' problem, at some physical level of observation 'symmetry' disappears.

So did we discover symmetry, or do we impose it? (like numbers) And why do we need it?

Ideally I would also like some pointers to discussions of the implications of 'symmetry' as an abstract or real concept.

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    That's interesting. I believe the symmetries of physics are assumed to be exact. Symmetries of nature and conservation laws. en.wikipedia.org/wiki/Noether%27s_theorem. And of course mathematical symmetries are exact. The symmetries of a triangle are perfectly exact, just as abstract mathematical triangles are perfectly exact. But they don't exist in the physical world. Ok here's one. How about the symmetry between you and your mirror image? But no actual mirror is perfect. So in the physical world, there's no exact symmetry.
    – user4894
    Mar 2 '19 at 5:08
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    Your question presupposes real=physically-real. Plato would have disagreed (As would most religious persons of any persuasion) Mar 2 '19 at 5:15
  • @user4894 And if that is true, isn't Physics searching in the wrong direction?
    – christo183
    Mar 2 '19 at 5:22
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    So is the symmetry. And physicists do not find anything to more than n decimal places of precision, nor will they. Which is why modern Aristotelians build the approximateness into the picture, see e.g. Franklin's Aristotelian Realism. You are also free to adopt a "mathematical universe" a la Tegmark and have yourself "real symmetry". In short, there is no fact of the matter concerning this question, it is up to your theoretical scheme. Same as asking if irrational numbers are physically real.
    – Conifold
    Mar 2 '19 at 5:59
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    @Conifold Franklin looks interesting, thanks. One wonders at what point while constructing a "theoretical framework", or conversely, would anything change if we didn't have symmetry? I'm thinking this may help in understanding "Unreasonable effectiveness".
    – christo183
    Mar 2 '19 at 9:59

Here is the question:

So did we discover symmetry, or do we impose it? (like numbers) And why do we need it?

David John Baker describes symmetries (page 2):

...symmetries of a theory are transformations that preserve its laws.

These transformations help us distinguish between "fundamental quantities" and "surplus structure" (page 4):

...physical quantities that change under symmetry transformations (i.e., that are not invariant) must not be fundamental quantities. Qualitatively identical objects or worlds cannot disagree about the fundamental quantities.

This provides a partial answer to why we need symmetries: they help identify the fundamental quantities of a physical theory.

They also help ground the idea of "objectivity". Brading, Catellani and Teh while surveying the philosophical significance of symmetry note:

It is widely agreed that there is a close connection between symmetry and objectivity, the starting point once again being provided by spacetime symmetries: the laws by means of which we describe the evolution of physical systems have an objective validity because they are the same for all observers.

A more complete answer to the question why we need symmetry would then be that the idea of transformations preserving the laws of the theory allow one to (1) identify the fundamental quantities of the theory and (2) describe the theory objectively, that is, independently of individual observers.

The other question whether we discover symmetry or impose it can be described from an ontological viewpoint or from an epistemological viewpoint.

Brading, Catellani and Teh describe the ontological viewpoint as follows:

According to an ontological viewpoint, symmetries are seen as a substantial part of the physical world: the symmetries of theories represent properties existing in nature, or characterize the structure of the physical world. It might be claimed that the ontological status of symmetries provides the reason for the methodological success of symmetries in physics.

The epistemological viewpoint gains strength when one doubts if symmetries can "be directly observed". From this epistemological perspective symmetries need only be approximate enough "such that there is sufficient stability and regularity in the events for the laws of nature to be discovered."

So, which one is most likely correct, the ontological viewpoint or the epistemological viewpoint?

Since science is provisional, that is, it doesn't claim to provide absolute answers but only those that are falsifiable, a safe position to take is the epistemological viewpoint.

Another reason to take the epistemological viewpoint is that symmetries also allow for objectivity. They abstract away the subjective part of reality to more conveniently work with a theory that applies to everyone. That makes the theory useful like taking a photograph or looking in the mirror are useful, but because of that they are only approximations for the reality of what was reflected in the mirror or imaged in the photograph.

Baker, D. J. (2010). Symmetry and the Metaphysics of Physics. Philosophy Compass, 5(12), 1157-1166. http://philsci-archive.pitt.edu/5435/1/SymPhysMeta.pdf

Brading, Katherine, Castellani, Elena and Teh, Nicholas, "Symmetry and Symmetry Breaking", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/.

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    @SmootQ I agree. There are many different kinds of symmetries. I think they are more useful as constraints on theories describing reality rather than constraints on reality. Mar 3 '19 at 13:12
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    On the other hand there has to be some kind of asymmetry underlying everything to account for the non-proportionality between matter and antimatter.
    – SmootQ
    Mar 3 '19 at 13:24
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    It seems the different ways in which the term "symmetry" are used are not... symetrical. Baker stands on "transformations", the product of which are called "symmetries"; but in other physical senses perfect symmetry have not, and can not be demonstrated. I'm starting to think there are conceptual differences as well. - Maybe you remember my question about "incompatible ontologies", maybe symmetry is a candidate for linking these? Anyway great answer!
    – christo183
    Mar 3 '19 at 14:03
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    @christo183 Noether's theorem shows us that what we call the dimensions of space and time are directly equivalent to conservation of energy and momentum on transformation. So, then we see charge and parity getting conserved, and associate additional symmetries, linked to fundamental forces. The blackhole information paradox being briadly agreed in the direction of conservation of information points to a further symmetry. Symmetries make descriptions more compact, and entropy is the decrease in that compactness. These uses of symmetry seem at variance, but in maths are the same.
    – CriglCragl
    Mar 10 '19 at 22:21
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    @SmootQ It's like the issue of whether circles exist outside of our imaginations. Semantics. Don't be bewitched by words. Symmetries in the world are always limited, like circles. You can chase the uncertainty below the apparent resolution of the universe, that is all.
    – CriglCragl
    Mar 11 '19 at 15:45

How do you know that your hand is the same hand that you've had yesterday? How do we recognize a river?

As Kant suggests, perception is not reality. Nature changes constantly, a river is not the same that what you saw yesterday. Your hand is not the same hand that you've had yesterday. A rock is equivalent to smoke, within different scales of properties (molecules attract on both entities; atoms prevent us to see on both objects; both objects mutate with time (yes, it takes the rock thousands of years, and milliseconds to smoke), etc.).

But perception allows you to create a model of a river which allows you to understand that it is moreover the same thing that yesterday was. And we are able to say that "this is the river that I saw yesterday". We are identifying two objects that are equal, that are the same. Check the early empiricists, Locke, Berkeley, Hume, and later, Kant for more about the objects of our perception. The capability to create mental models of things and being able to compare them (in absolute terms, or equality, or partial terms, where symmetry would be a part) is essential to our understanding. Although nature changes, we are able to establish thresholds and establish if something has changed or not. That's an essential mental faculty.

That, regarding equality of objects. Symmetry of objects is similar, except that our mind perceives a change in the object and the model that has a coherence in form, that has some correspondence, while the object is different to the model. That's the property we know as symmetry.

If you accept that our shared subjective perception is equivalent to an objective truth, we can say that we discover symmetry (I would not identify myself with such position: for me, perception is not truth). Otherwise, yes, you can say that we impose it.

  • To bring our notion of Truth into it is a sound idea, I think. But then if we discovered symmetry, how long did it take before we discovered that we really didn't? And why are we still using it? On the other hand, why would we impose it? When did we invent it? - Why do we need it? - If we could have these answers, maybe it would say something about our notions of Truth...
    – christo183
    Mar 7 '19 at 7:28

Nothing that is composed of more than two atoms can be perfectly symmetrical, physically. However, it is possible for the existence of two identical -- and apparently symmetrical -- forms in someone's mind; whether they're known in full or just believed to be symmetrical is another matter. It will suffice to say that symmetry can be perceived.

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