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concept of a ‘real number’ in terms of length ratios. He also provided rules for the sum and product of such real numbers. There was a basic diVerence in viewpoint, however, between the Greek notion of a real number and the modern one, because the Greeks regarded the number system as basically ‘given’ to us, in terms of the notion of distance in physical space, so the problem was to try to ascertain how these ‘distance’ measures actually behaved. For ‘space’ may well have had the appearance of being itself a Platonic absolute even though actual physical objects existing in this space would inevitably fall short of the Platonic ideal. (However, we shall be seeing in §17.9 and §§19.6,8 how Einstein’s general theory of relativity has now changed this perspective on space and matter in a fundamental way.) A physical object such as a square drawn in the sand or a cube hewn from marble might have been regarded by the ancient Greeks as a reasonable or sometimes an excellent approximation to the Platonic geometrical ideal. Yet any such object would nevertheless provide a mere approximation. Lying behind such approximations to the Platonic forms—so it would have appeared—would be space itself: an entity of such abstract or notional existence that it could well have been regarded as a direct realization of a Platonic reality. The measure of distance in this ideal geometry would be something to ascertain; accordingly, it would be appropriate to try to extract this ideal notion of real number from a geometry of a Euclidean space that was assumed to be given. In eVect, this is what Eudoxos succeeded in doing. By the 19th and 20th centuries, however, the view had emerged that the mathematical notion of number should stand separately from the nature of physical space. Since mathematically consistent geometries other than that of Euclid had been shown to exist, this rendered it inappropriate to insist that the mathematical notion of ‘geometry’ should be necessarily extracted from the supposed nature of ‘actual’ physical space. Moreover, it could be very diYcult, if not impossible, to ascertain the detailed nature of this supposed underlying ‘Platonic physical geometry’ in terms of the behaviour of imperfect physical objects. In order to know the nature of the numbers according to which ‘geometrical distance’ is to be deWned, for example, it would be necessary to know what happens both at indeWnitely tiny and indeWnitely large distances. Even today, these questions are without clearcut resolution (and I shall be addressing them again in later chapters). Thus, it was far more appropriate to develop the nature of number in a way that does not directly refer to physical measures. Accordingly, Richard Dedekind and Georg Cantor developed their ideas of what real numbers ‘are’ by use of notions that do not directly refer to geometry. [3.4] Can you see how to formulate these? §3.2 CHAPTER 3