propositional writing

Symmetry Matters in Physics and Mathematics and it also Matters in Music Science

Philip Dorrell, 23 July 2012

Symmetry is a basic concept in physics. Every symmetry of the Lagrangian corresponds to a conservation law – this is Noether's Theorem.

Symmetry is also important in mathematics. Whatever mathematical structure is being studied, a mathematicians will ask: "Under what transformations is this structure invariant?" Originally this insight about the importance of symmetry was applied to geometry by the mathematician Felix Klein, who published the Erlangen Program.

In music science, symmetries are also important. For music there are three questions we need to ask about symmetries:

The pattern of "What, how and why" is not unique to the study of musical symmetries – it applies to the study of any biological phenomenon:

("Why" questions are specific to biology, and for those who dislike the implication of unexplained purpose, they can be rephrased as: "What selective pressure has caused this feature of a living organism to evolve?" And if you think that music science is not part of biology, remember that music is something which is performed and enjoyed by human beings, and human beings are living organisms, which is what biology is the study of.)

Probabilistic Theories of Perceptual Symmetries

Relevant to any analysis of perceptual symmetries is the probabilistic approach championed by Dale Purves, as explained in A Primer on Probabilistic Approaches to Visual Perception. (Note: Dale Purves is one of the co-authors of the paper The Statistical Structure of Human Speech Sounds Predicts Musical Universals referred to above.)

Purve's theory can be regarded as an attempt to create a general theory of perceptual "learning" that explains all situations where different percepts imply the perception of the "same" thing. Such a theory would seem to subsume all situations where the equivalence of "different" percepts of the "same thing" can be given an interpretation as invariant under a set of transformations that belong to a formally defined mathematical group.

To Pre-Wire or Not to Pre-Wire?

From what I've read, Purve's research and that of his co-researchers seems to stop short of making or testing any hypotheses about the "neural wiring" that supports this type of perceptual learning. However, even if the relevant learning processes are driven by actual experience, we might expect the brain to be somewhat "pre-wired" with the evolutionarily evolved expectation that certain types of symmetry are likely to occur in practice.

In particular, if we assume the existence of certain special calibration targets, which drive the learning of particular perceptual symmetries, then these targets would have to be "pre-wired" somehow, as being of special interest early on in life, even though the final outcome of the learning process is not itself pre-wired, and even though the perceptual symmetry is eventually going to be applied to the perceptions of things different to the initial calibration targets.

Also, perceptual symmetries can be partly pre-wired in the sense that neurons representing different percepts that are likely to be learned as being the perception of the "same thing" can be laid out in the cortex in a manner such that they are likely to be close to each other, facilitating the formation of the required connections that form when the experiential learning process identifies those different percepts as being the perception of the "same" (or very similar) thing.

My guess is that, even if the probabilistic explanation of perceptual symmetries is largely correct, such "pre-wiring" does exist both for the symmetries of visual perception and for the symmetries of music and speech perception.

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