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2.2.2 Modern Geometry

In the past two hundred years, geometry has evolved to accommodate new concepts, in particular, that space itself has shape, can move, bend and expand - properties which were previously attributed to objects. This led to the idea that space itself can be treated as some kind of object; consequently, there has arisen a confusion between the objects that occupy space and the space that they occupy.

These recent advances in geometry make it clear that geometry is pure mathematics. It is important, therefore, to understand that while geometry may be used to describe physical reality, physical reality is not a geometric system.

In the original geometry described in Euclid's 'Elements', objects are categorized as lines, surfaces and solid objects existing in empty space. To this, Plato added his theory of 'ideal forms', circles, triangles, spheres etc, which are perfect shapes abstracted from the shapes of real objects. In the modern era Newton summarized the situation neatly when he wrote "Practical geometry teaches us to draw straight lines and circles, whereas, Mathematical geometry requires only that they be drawn".

This simple concept of objects existing in empty space is essentially the 'atoms and the void' philosophy of Democritus (400 BC). This idea was rejected by Aristotle (350 BC), who believed that there is an invisible 'Ether' which fills the space above the Earth and between the stars. The Ether theory was adapted by Galileo and Newton to explain the action at a distance of gravity. It was still widely accepted in 1900, by which time it had also become the medium which supported Maxwell's theory of Electromagnetism. After Einstein dismissed the Ether as being 'superfluous' in 1905, it re-emerged in 1908 in the guise of a 'space-time' continuum. In this form it has become the 'Ether' of Cosmology.

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