Can I lay out, in a somewhat orderly fashion, what it is Klein is trying to say about ancient and modern mathematics?
1) What is the being of number? Do numbers have being, as the Platonists believe; are they neutral abstractions, as the Aristotelians say; or does it not matter, as high school textbooks would lead one to surmise? Is 1 a number or a monad? Is √2 a number or an operation? Is π a number or a proportion? Does it matter if something is a number; are numerical units something with ontological interest? They were to the ancient Greeks, especially the platonists and the pythagoreans. The latter were shocked and horrified to discover incommensurable magnitudes, because that meant that the world could not be measured in its entirety by number. We apparently defer the problem by saying that √2 is a number, it just happens to be irrational, which is awkward. But it can be operated on algebraically with accurate results, so whether it’s a number or a number-like-thing or simply the proportion between the sides of a square and the diagonal, it’s a legitimate number-like-thing.
The chief distinction Klein seems to be making is that to the Greeks the numerability of a number depended upon its position as a thing which counted other things; ultimately as a thing that counted either indivisible pure monads or divisible neutral abstractions from the act of counting. The operations would therefore be determined by the number; this is especially the case with division, where the possibility of many division problems is determined by whether one sees numbers as being essentially indivisible monads (in which case they cannot be divided in such a way as to produce fractions) or as neutral abstractions (in which case fractions are allowable but irrationals are not). In modern mathematics, on the other hand, it is not the number itself, but the operations that determine the validity of a number or procedure. If it can be operated upon in the ways outlined by algebraic rules then it’s legitimate, even if one ends up having to deal in approximations of irrationals and oddly divided fractions.
2) There’s a tension between method and object. Because for the Greeks numbers always count specific things, are always numbers of something, there’s a tension in Greek mathematics between generality of method, as in Diophantus’ arithmetic, and specificity of object. There is no Greek equivalent to an indeterminate a, much less to a variable x. Thus, they excelled far more in geometric proofs, where the numerical relationship of the sides is not an object of consideration, than in arithmetic. Euclid, in his book on numbers, does not actually use any numbers. Diophantus uses numerical examples rather than general formulas, though his equations can be so converted by people with a symbolic understanding of numerical equations. The 16th century mathematician, Viete, noticed this and supposed that Diophantus must have calculated “by species” and then cleverly converted his calculations into numerical examples, because it is much more difficult to find the answers to his problems otherwise. Klein, however, argues that calculation “by species” was not only unknown but impossible in a conceptual framework based upon determinate monads. In Viete and thereafter, however, it is possible to resolve the tension between a general method and a determinate object by applying the general method to similarly general symbolic objects.
3) While the above mentioned tension is most apparent in arithmetic, it is present in geometry as well. To illustrate his point in The World of Physics and the “Natural” World (p. 12-20), Klein discusses the difference between geometric forms in Euclid and (especially) Appolonius, and in Descartes. IN the former, there is certainly a generality of method: the proofs are meant to be true not only of this particular triangle or cone, but of every possible triangle or cone that meets the requirements of the proof. However, the figure before us is precisely this figure, and not just any figure. Book 5 of Euclid, on proportions, might be an exception. But for Descartes the drawing is not “this figure here;” his ellipse is not “this ellipse,” but “any ellipse” — a general object to match the generality of what he’s seeking to prove. He quotes Descartes on this on p. 18, but I’m not sure it’s worth reproducing here.
4) There’s a large shift from the Greek understanding of the natural order as ταζις, based upon proportion and the modern understanding of law, based upon motion.