Spaces that are symmetrical have a fundamental importance in modern physics. Why is this? It might be thought that completely exact symmetry is something that could arise only exceptionally, or perhaps just as some convenient approximation. Although a symmetrical object, such as a square or a sphere, has a precise existence as an idealised (‘Platonic’; see §1.3) mathematical structure, any physical realisation of such a thing would ordinarily be regarded as merely some kind of approximate representation of this Platonic ideal, therefore possessing no actual symmetry that can be regarded as exact. Yet, remarkably, according to the highly successful physical theories of the 20th century, all physical interactions (including gravity) act in accordance with an idea which, strictly speaking, depends crucially upon certain physical structures possessing a symmetry that, at a fundamental level of description, is indeed necessarily exact!
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From the perspective of Systemic Functional Linguistic Theory, objects are things that participate in processes, physical interactions are processes, and space is a circumstance of things participating in processes. From this perspective, the symmetry of an object, such as a square or a sphere, is not a symmetry of space, but a symmetry of a thing in space, and the symmetry of physical interactions is not a symmetry of space, but a symmetry of processes in space.
This failure to distinguish space from things and processes in space has very important negative consequences, because, as explained elsewhere, it results in the false notion of curved space(-time) in the interpretation of the General Theory of Relativity.
In this view also, the 'Platonic ideal' of a physical square or sphere is a reconstrual of a first-order meaning (phenomenon) as a second-order meaning (metaphenomenon).
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