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Platonism received its first major set-back from one of its arch proponents: Plato's predecessor Pythagoras [6'th century BC]. Obviously Pythagoras did not use the term `Platonism' - a term that was not used until the twentieth century -- but his approach to mathematics conformed to the philosophy just outlined. In Pythagoras' time numbers and arithmetic were considered the field where absolute truth was most certain to be found. Geometry also, to a lesser extent, provided one of the foundations of objective truth. At around the time of Pythagoras, there was the discovery of irrational numbers. (There is some doubt as to whether it was Pythagoras who actually made this discovery. See [#!Sza78!#].) An irrational number, like or , cannot be written as a fraction. Though is a good approximation, it can be shown that no fraction can be exactly equal to . Here the problem is explained using modern terminology but the Greeks did not know about irrational numbers nor even fractions, the numbers they used were the ordinary counting numbers 1, 2, 3, ... They were interested in similar triangles and could prove that two triangles were similar by showing that the ratio of the sides (a:b:c) was the same for both triangles. However for a right angled, isosceles triangle it is not possible to find the ratio of the sides as the ratio of three integers. The sides were called incommensurable which means that they had no common measure.

Numbers like this were considered as irrational, and probably not numbers at all. Thereafter it was geometry that became the more solid base for certain knowledge. Also, the unexpected and shocking discovery of irrational numbers led to the development of formal proofs in mathematics, epitomised by Euclid's Elements, because it was felt that every statement required a rigorous justification, nothing could be taken as obvious.

Much later, Platonism suffered an increasing series of attacks. One of these was the invention (discovery?) of complex numbers. They were originally considered as purely imaginary fictions or ideal elements, which were very useful for the solution of cubic equations. Nevertheless complex numbers were increasingly used in a wide variety of contexts, perhaps the most concrete demonstration of their value is in the description of wave-forms in quantum physics by complex functions. Gradually the view emerged that complex numbers were not fictions but had a real existence of their own. The extension of this view was that any mathematical construction, provided it was consistent, existed in the Platonist realm.

Later, mathematicians like Lobachevsky, Bolyai, Gauss, Riemann and others showed that Euclidean geometry was not the one and true geometry, but one among many alternative geometries. In an interesting example of a genuine experiment in mathematics, Gauss questioned the assumption that our universe had a Euclidean geometry by taking three triangulation points and summing the angles of the triangle. In fact, given the experimental accuracy that was available, he found the sum to be , consistent with a Euclidean geometry. It was much later, with Einstein's general relativity, that it was shown that the geometry of the universe is not Euclidean but curved.

But the greatest crisis and the eventual overthrow of Platonism occurred in the mathematical revolution² which happened between 1870 and 1935.