[M]odern consciousness is to be understood not simply as a linear continuation of ancient επιστημη, but as the result of a fundamental conceptual shift which took place in the modern era, a shift we can nowadays scarcely grasp. –Klein, Lectures and Essays, The World of Physics and the “Natural” World, Saint John’s College. 1985. p 10
These explorations of Greek Mathematical Thought and the Origin of Algebra strike me as being irritatingly redundant and obtuse. I don’t especially want to write a redundant and obtuse preceptorial essay, but am not sure what else is possible. This is especially true since I’m not familiar with math stuff, but Klein is familiar, and his writing is still irritating, obtuse, and redundant. It would be terrific to not just write so as to fill up 15 pages and not fail the course. It would be really great to write and thereby to discover why the class is worth being in to begin with. But if Klein’s observations are correct, then it is true that his is a subject about which we have little familiarity and therefore requires a lot of time and effort to try to just understand what is meant, let alone why it’s interesting and important. Why might the conceptual shift in mathematics in the 17th century be interesting and important?
According to Klein, it is related to mathematical physics in particular in a way both immensely important and very difficult to demonstrate clearly. Philosophy, he says, “should never forget, even for an instant, that mathematical physics is the foundation of our mental and spiritual life, that we see the world and ourselves in this world at first quite ingeniously as mathematical physics has taught us to see it, that the direction, the very manner of our questioning is fixed in advance by mathematical physics, and that even a critical attitude towards mathematical physics does not free us from its dominion. The idea of science intrinsic to mathematical physics determines the basic fact of our contemporary life, namely, our ‘scientific consciousness'” (1985, p 3). In the introduction to Greek Mathematical Thought and the Origin of Algebra he says:
Before entering upon a discussion of the problems which mathematical physics faces today, we must therefore set ourselves the task of inquiring into the origin and conceptual structure of this formal language… the inquiry will never lose sight of the fundamental question, directly related as it is to the conceptual difficulties arising within mathematical physics today. (1968, p 4)
So, OK, “however far afield” he may go, what Klein is ultimately interested in is how mathematical physics came to be what it is. As it turns out, he goes very far afield, and spends most of his time trying to draw some manner of distinction between a mathematics of “pure, indivisible monads,” or of divisible rational “neutral abstractions” from the act of counting, and of variables where the actual determinacy of a number is replaced by the possibility of determinacy. That, apparently, is the gap between ancient and modern mathematics, and between mathematical expressions that can be conveyed in other ways, and expressions that can only be shown in algebraic symbolism: the variable. Also, there is a substitution of judging the validity of an operation by the sense of its’ outcome, to a supposition that if the equation was correctly constructed and the operations were all validly carried out then whatever the outcome is, that is the true answer.