Symmetry, Invariance, and Objectivity

Posted on 16 May 2009

5


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.

Advertisements
Posted in: Philosophy, Science