The view of mathematics contained in the last quotation did not arise all at once. A mathematics that looks at patterns rather than individual properties of individual mathematical objects was what Descartes’ sought in mathesis universalis, ‘universal mathematics’, which he associated with the then new subject of algebra. This ‘general science’, he said, existed “…to explain that element as a whole which gives rise to problems about order and measurement, restricted as these are to no special subject matter” (Descartes, 1970 , p.13).[i] What Descartes was suggesting, in other words, was that when one writes an expression like x2-y2=k one may look at it as a purely symbolic expression, an abstract pattern, to be manipulated and studied; one should not have to tie it to square figures whose sides have lengths x and y. Descartes said his algebraic approach was only a rediscovery of a mathematics secretly practiced by the Greeks; in fact mathematics itself was being reborn in a new form For this reason Felix Browder rightly points out that “From the 17th century on…a broader vision of mathematics arose in the minds of such intellectual innovators as Leibniz and Descartes, a vision of mathematics as the total science of intellectual order, as the science of pattern and structure” (Browder, 1975 , p.14)
We do not always appreciate how far the symbolic character of modern mathematics, which began to take shape in Descartes’ time, distinguishes modern mathematics from, for example, Greek mathematics. Greek mathematicians typically began with specific mathematical objects, such as a circle or a section of a cone, and then proved that those objects possess certain properties. They did not begin with some property and then find an object possessing it or a set of objects that could be related by it. For Greek mathematics was a non-algebraic mathematics (Klein, 1968; Grattan-Guinness, 1996; Fried & Unguru, 2001 ), and to begin with a property abstracted from any particular object is precisely what symbolic algebra allows us to do supremely well, indeed, what it is made for. Such abstracted properties are what we are looking for when we are looking for patterns. And this is what Hardy had in mind, surely, when he said the mathematician’s patterns “are made with ideas.” The symbolic nature of modern mathematics, then, is what allows mathematics to be a science of patterns, and it is now, indeed, a science of patterns; but because mathematics was not always symbolic we ought to take care and say that mathematics is the science of patterns because it has grown to be so. With that, let us turn to Euclid and Steiner. First, Euclid.
[i] Just a few lines before the quotation above, Descartes also says that “…all those matters only were referred to Mathematics in which order and measurement are investigated, and it makes no difference whether it be in numbers, figures, stars, sounds or any other object that the question of measurement arises.” In many ways, that sentiment encapsulates the modern view. Thus it is no accident that one hears strong echoes of it in this statement by Whitehead: “The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours” (Whitehead, 1964 , p.9).