Symmetry is intimately related to the idea of invariance, the persistence of something amidst change (e.g., changes in time, space, or perspective). This is also one meaning of objectivity. Thus, symmetry is a key to understanding the objective world, as is well known in physics. The basic idea here is not confined to the esoteric realms of theoretical physics. It can be seen in our everyday experience, if we just look at it right.

Consider a child’s visual experience of a wooden block, i.e., a cube. As the child turns it around and looks at it from different perspectives, the visual appearance changes. The shape of the image changes depending on which side is facing the child. Never does the child see the entire block at once (one side is always hidden, facing away). Yet, the child learns to correlate all these different 2D visual images, abstracting from them a 3D object that persists. This object, however, is not itself experienced. It is conceptually constructed and posited as existing “behind” the variety of 2D visual appearances in experience. At the heart of this connection between the multitude of appearances and the invariant reality behind them is the principle of symmetry.

Put simply, symmetry is an invariance amidst change. More precisely, consider something X (e.g., a set of two points in the plane) and a transformation R that changes X to X’ (e.g., a rotation of the plane). Now suppose we have abstracted a feature D that can be defined for both X and X’ (e.g, the Euclidean distance between points). Then we say that D is a symmetry of R if D is invariant with respect to the transformation R (e.g., distance is a symmetry of rotation if distance is unchanged when the plane is rotated). So, although rotations of the plane change the positions of points, the distances between those points are not changed. We can also express this in terms of a rotation of the coordinate system: although the coordinates of points change when the frame of reference (or perspective) is rotated, the distance between two points remains the same. Thus, the distance is an “objective feature” that persists amidst changes in perspective, while the coordinates are “mere appearances” that depend on the perspective and are not objective.

In physics, Noether’s theorem establishes a correspondence between symmetries in the dynamical laws of a system and conserved quantities (i.e., invariant features of the system). For example, if the laws are rotationally symmetric, then angular momentum is conserved. Simply put, Noether’s theorem relates the objective dynamical features D of a system to the transformations R of its dynamical laws that leave the laws unchanged, i.e., that represent mere changes in perspective and not real changes.

In mathematics, Felix Klein’s Erlangen program characterizes different geometries by their corresponding symmetry groups. Euclidean geometry, for example, corresponds to the group of rotations, reflections, and translations because the geometric features that characterize Euclidean geometry (e.g., distances and angles) are invariant under those transformations. Different geometries can then be related to each other by relating their corresponding symmetry groups. For example, projective geometry has a larger group of symmetries, and it has fewer invariant features (e.g, cross-ratio, incidence, tangency, colinearity, but not distances or angles). The symmetry transformations correspond to changes in appearance due to change in perspective rather than change in something objectively real. This is why the larger symmetry group corresponds to fewer objective features. As physics moves towards higher symmetry, objectivity dissolves until it vanishes in the limit of perfect symmetry where everything is seen as a hierarchy of symmetry groups corresponding to successively deeper levels of perspectives.

Consider a set X and the group G of all possible transformations of X to itself (i.e., the group of automorphisms of X). One feature of X that is symmetric under all the transformations of G is the size, or cardinality, of X. Most often, though, the more interesting features of X are not symmetric under G, but are symmetric under some subgroup of G. For example, if X is the real number line then the Euclidean distance between two points is invariant under the Euclidean group E(1) of translations and reflections, but is not invariant under scalar multiplication. E(1) is a subgroup of G that corresponds to the merely apparent transformations of perspective, the transformations that do not change anything real. The factor group G/E(1) represents the transformations that are regarded as objectively real, factoring out all the merely apparent transformations of the symmetry group E(1). These real transformations effectively define what it means for objects to undergo objective changes, i.e., to change from one object to a different object. The identity element of G/E(1) corresponds to the transformations of perspective that do not change the identity of objects.

Let’s reconsider now the example of the child and the block. Although the child does not know abstract mathematics, the different perspectives on the block are understood as transformations of a symmetry group E(3) with corresponding invariants of distances and angles. Because no other transformations are experienced except for those in the group E(3), the child posits a real (i.e., invariant) object existing in a 3D world with a constant identity amidst the changes in perspective. This is how symmetry and invariance is implicitly built into the very roots of our experience of an objective world.

*Philosophy, Science*

Joel Morwood

26 May 2009

I am not sure about the meaning of your conclusion. Are you saying that symmetry is a result of objectivity, or that objectivity results from symmetry? Please clarify.

integralscience

26 May 2009

Symmetry is built into the very notion of objectivity. So, whenever one has something that is objective, there is an implicit symmetry there. One might say that objectivity is symmetry plus something extra, namely, the act of positing existence upon the invariant of the symmetry.

John

8 October 2009

One of the most fascinating aspects of symmetry and invariance is that each symmetry suggests a conservation law. The result of Lorentz invariance is the “conservation of the centre of energy”. This is the most intriguing of the conservation laws. Its pre-relativistic analogue, the “conservation of centre of mass”, applies to a finite rotating system but “conservation of centre of energy” must apply to an entire reference frame. The analogy with the pre-relativistic case is also highly misleading because a boost is not a simple rotation, it is a relative tilting of a hyperplane of simultaneity.

The law is largely ignored for lack of clear interpretation. I think physics might be missing a trick by failing to properly explain the conservation law due to Lorentz invariance.

integralscience

8 October 2009

It is an interesting topic indeed. John Baez has discussed the subtle aspects of applying Noether’s theorem to symmetries in special relativity here http://math.ucr.edu/home/baez/boosts.html

Klinkert

16 July 2010

I like these ideas, and the whole philosophy that is hidden beyond , it vibrate silent and mighty, and it curves and grows in the fresh air, and it unfold like the seed into a magic tree, the trunk grows and branches expand and dissolve in abstract patterns, symbols made out of sense and thoughts, and a web below, made with knots and waves in retention and pretension; a charged texture of correspondences, distinctions that reflect a symmetric picture of the world!!!.

The substance of the world is made up of object, The object is made up of particles and waves symmetric patterns. We are made out of time, space, forces and different forms; ideal Symmetry and broken symmetry; The form in the microscopic space correspond with the form of macroscopic space. The inner with the outer; simultaneous coexist, each one limited and formed for the other: moving and static on the wave;

The in wave and Out-Waves form a Standing Wave around the Wave-Center ‘particle’; Consciousness forms in the inner space; the sacred and solemn dome; It passes as the water in the river, but it stays as the riversong and the silent weblight;

A silent language in symmetry stay.

Heheh those are just some thought, poorly expressed, that sometimes i think when i read your great essays, they are really good and inspiradores ( sorry for my english, im Colombian (-:)

thank you for your essays, they are fantastic!!!