How about I just try going to the heart of what I’m interested in saying in this essay, so that I’ll at least be able to say something, as opposed to what I’m otherwise inclined to do, which is to simply repeat Klein more or less word for word? Well, what is it that I’m looking for? There’s this thing Klein does in the introduction to the Greek math book and at some point in several essays: he makes this transition from the symbolism of mathematics as such — from algebra as a system of technical procedures for manipulating symbols, and then heads off toward physics. Once he gets to physics he says that this is essentially how we moderns see the world. That we are not only treating a second intention as though it were a first intention when we go off and do this separate thing called algebra, but that that we see the world that way (Klein, 1985, p 63). Well, what does he mean by that? There are all these conceptual distinctions bouncing around in Klein’s speech: the first and second intentions, symbols vs monads, the ontological status of numbers, taxis vs. law, rationalism as symbolic abstraction, and so on. So where’s the target?
I’m not in a position to say with any degree of confidence if Klein’s understanding of ancient mathematics is convincing. It may be, for instance, that when Diophantus represents the number he’s trying to find as an S and proceeds to operate with it inside the equation he’s doing something fundamentally different from what Vieta does in his Zetetics, and from what we do in our algebra. Certainly he refrains from operating with more than one unknown. It seems unlikely that he would solve those equations by eidos and then hide the fact to make his task appear more difficult, as Vieta supposed. Still, the precise nature of his reasoning and of the way in which he understands the significance of the designations of monad and the unknown is more or less opaque unless one assumes, as does Klein, that he must be assuming a Peripatetic understanding of number. The reasons given for that must seem fairly convincing but by no means necessary, and it would be a stretch but not completely unjustified to see the Arithmetic as a not yet fully formalized algebra. Not yet quite symbolic, but also not in opposition to becoming symbolic should a formal language develop. Which ends in “taking for granted what is in dispute” (Klein, 1968, p 130).
More interesting is the series of conclusions Klein begins to suggest in his first few Lectures and Essays. If modern algebraic formulations are essentially symbolic in the “second intention functioning as a first intention” sense, what does it mean when we then begin to base our physics — the fundamental way in which we understand the world — upon that rather dubious method of generalized abstraction? First, it becomes impossible to accurately represent the findings of mathematical physics accurately in non-algebraic terms (Klein, 1968, p 4; 1985 p 1-3). The reason for this must be the generalizing of the objects of our study. Physics comes to deal not so much with the being of any particular object, or even with the matter of which the object is made, but with the “laws” common to any possible matter (Klein, 1985, p 33). As algebra deals not with concrete monads, or even determinate “numbers” in whatever we might currently mean by that term (including strange things like irrationals), but rather with symbols representing the possible substitution of any number, so mathematical physics presumably deals not so much with particular objects exhibiting particular characteristics or motions, but rather with second intentioned symbolic abstractions representing the laws that govern any possible particular arrangement of matter, but which at the same time can be operated upon as though it were determinate.