Penrose (2004: 11-2):
Thus, we must be careful, when considering geometrical assertions, whether to trust the ‘axioms’ as being, in any sense, actually true.
But what does ‘true’ mean, in this context? The difficulty was well appreciated by the great ancient Greek philosopher Plato, who lived in Athens from c.429 to 347 BC, about a century after Pythagoras. Plato made it clear that the mathematical propositions — the things that could be regarded as unassailably true — referred not to actual physical objects (like the approximate squares, triangles, circles, spheres, and cubes that might be constructed from marks in the sand, or from wood or stone) but to certain idealised entities. He envisaged that these ideal entities inhabited a different world, distinct from the physical world. Today, we might refer to this world as the Platonic world of mathematical forms. Physical structures, such as squares, circles, or triangles cut from papyrus, or marked on a flat surface, or perhaps cubes, tetrahedra, or spheres carved from marble, might conform to these ideals very closely, but only approximately. The actual mathematical squares, cubes, circles, spheres, triangles, etc., would not be part of the physical world, but would be inhabitants of Plato’s idealised mathematical world of forms.
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From the perspective of Systemic Functional Linguistic Theory, physical structures and mathematical structures differ in terms of orders of experience. Physical structures are first-order meanings, phenomena, whereas mathematical structures are reconstruals of these as second-order meanings, metaphenomena.
Specifically, these metaphenomenal 'inhabitants of Plato's idealised mathematical world of forms' are construals of quantifying Numeratives of Things, and the relations between them, rather than construals of the Things (physical structures) that they quantify. Mathematical structures are construals of the dimensions of physical structures.
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