But, as powerful mathematicians like Minkowski and Hilbert found striking harmony between their pure mathematical results and the workings of the physical world, many found the claims for such a harmony hard to resist. Thus, in the early years of the twentieth century, we begin to see why Minkowski's application of complex numbers to the description of space and time was hailed by one physicist as 'one of the greatest revolutions in our accepted views'.

Aristotle draws a sharp dividing-line between the activities of the physicist and those of the mathematician. The mathematician limits his enquiry to the quantifiable aspects of the world and so dramatically restricts what is describable in mathematical terms. Physics, for Aristotle, was far wider in scope and encompassed the earthly reality of sensible things. Whereas Plato had maintained that mathematics was the true and deep reality of which the physical world was but a pale reflection, Aristotle claimed mathematics to be but a superficial representation of a piece of physical reality. Such is the contrast between idealism and realism in the ancient world.

If the Universe possesses intrinsically random elements in their make up, inherited from its quantum origins or from random symmetry-breakings during its early evolution, then we must take our own existence into account when evaluating the correspondence between reality and the cosmological predictions of any Theory of Everything. Moreover, if these random cosmological elements lead to a Universe which differs significantly from place to place over the very large distances, then our local observations of a possibly infinite Universe will inevitably leave our knowledge of its global structure seriously incomplete.

The fact that simultaneous discovery occurs in mathematics, as well as the sciences, points toward some objective element within their subject matter that is independent of the psyche of the investigator.

The ease with which collaboration occurs in mathematical research and the essential similarity of the fruits of such collaboration to that of individual work points suggestively towards a powerful objective element behind the scenes that is discovered rather than invented.

Plato argued that the material world of visible things was but a shadow of the true reality of eternal forms. He proceeds to explain the nether world of eternal blueprints most completely in the case of the elements of matter: earth, air, fire, and water. These he represents by geometrical solids: the earth by a cube, water by an icosahedron, air by an octahedron, and fire by a tetrahedron. His position is that ultimately the elements are just these solid geometrical shapes not simply that they possess geometrical shapes as one of their properties. The transmutation of elements one into the other is then explained by the merger and dissolution of triangles. This strictly mathematical description characterizes Plato's discussion of many other physical problems, For him, mathematics is a pointer to the ultimate reality of the world of forms that overshadows the visible world of sense data. The better we can grasp it, the closer we can come to true knowledge. Thus, for Plato, mathematics is more fundamental, truer, closer to the eternal forms of which the visible world is an imperfect reflection, than the objects of physical science. Because the world is mathematical at its deepest level, all visible phenomena will have mathematical aspects and be describable by mathematics to a greater or lesser extent, determined by their closeness to their underlying forms.