Philosophy of Number, part 3

On Tuesday evening I was at a graduate institute party hosted by the college, and found myself talking with the GI director. Summer preceptorial choices sound really good; I’m pretty excited for them to come out officially, which should be this week or next. Anyway, I mentioned that I couldn’t figure out what Klein was trying to get at, and he asked me to try reading a later essay called The World of Physics and the “Natural” World. I still don’t quite understand what he’s saying, and should read it again, but I was encouraged nonetheless. It turns out the man could write in a way that’s fairly enjoyable and comprehensible. It’s like reading O K Bouwsma essays — he’s well informed and orderly, but obviously addressing himself to people who have some background in philosophy at least, and preferably the whole canon of Western liberal arts. And every now and again he breaks out into German, Latin, and Greek. After describing two methods often used by scientific historains, the one showing the continuity and the other the discontinuity between Greek and modern science, Klein goes on:

… There can be no doubt that the science of the seventeenth century represents a direct continuation of ancient science. On the other hand, neither can we deny their differences, differences not only in maturity, but above all, in their basic initiatives, in their whole disposition (habitus). The difficulty is precisely to avoid interpreting their differences and their affinity one-sidedly in terms of the new science. The new science itself did exactly that, in order to prove that its own procedure was the only correct one. The contemporary tendency to substitute admiration or tolerance of ancient cosmology for condemnations contributes little to our understanding of that cosmology. The issues at stake cannot be divorced from the specific conceptual framework within which they are interpreted. Conversely, these issues cannot even be seen within a conceptual framework unsuited to them: at best, they can only be imperfectly described. The best example comes from modern physics itself: the discussion of modern physical theories is ensnared in great difficulties when physicists and non-physicists alike try to ignore the mathematical apparatus of physics and present the results of research in a “commonsense” manner!

We need to approach ancient science on a basis appropriate to it, a basis provided by that science itself. Only on this basis can we measure the transformation ancient science underwent in the seventeenth century — a transformation unique and unparalleled in the history of man! Our modern “scientific consciousness” first arose as a result of this transformation. This modern 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, p 9-10. Saint John’s College Press 1985.)

Then, in concluding the essay, he says:

The world of mathematical physics built upon this presupposition, the world of natural processes according to law, determines the concept of nature in the new science generally, “Nature” means for it a system of laws, means — to speak with Kant — “the conformity to law of appearances in space and time.” All the concepts in this formula (as I have tried to show for “space” and “law”) can only be understood by contrast with the corresponding concepts of ancient science, Above all, the concept of conformity to law signifies a modification of the ancient concept of ταζις: ταζις is now understood as lex, that is, as order over time. The ascent from prima intentio to Secunda intentio is initiated here by the insertion of the time-dimension.

How, then, does the new science, on the basis of its intentionality, interpret ancient cosmology? How does it interpret the “natural” world of the ancients, the world of ταζις? It interprets it as the qualitative world in contrast to the “true” world, in contrast to the quantitative world. It understands the “naturalness” of this qualitative world in terms of the “naturalness” of the “true,” “lawful” world. Eddington, in the introduction to his recent book, speaks in a characteristic way of these two worlds: “there are duplicates of every object about me — two tables, two chairs, two pens.” The one table, the commonplace table, has extension, color, it does not fall apart under me. I can use it for writing. The other table is a “scientific” table. “It consists,” Eddington says, “mostly of emptiness. Sparsely scattered in that emptiness are numerous electrical charges rushing about with great speed.” (Klein, Lectures and Essays, The World of Physics and the “Natural” World, p 34.)

ταζις means order, and is pronounced taxis, like taxonomy. επιστημη means scientific knowledge in the Greek sense, and is said episteme. Prima intentio is a first intention: an idea referring to an object. Secunda intentio is a second intention: an idea referring to another idea or symbol, in turn referring to an object or the possibility of an object. Klein argues that Greek mathematics was always meant prima intentio; numbers are always seen as concrete, specific, numbers of something, though in theoretical arithmetic and theoretical logistic the of is “pure monads” which exist as noetic forms (in the Platonic understanding) or as neutral abstractions (in the Aristotelian understanding). Modern mathematics, especially algebra, exists primarily as secunda intentio — an equation like (ax+b)n=y expresses only possible determinacy, in contrast with the arithmetic of Diophantus, which is completely determinate, only we don’t happen to know some of the terms (in his case, ς) yet. In other words, ς is abbreviation for a number that we have not yet solved for, whereas a is a symbol for any number we might assign that place in the equation. There is no Greek equivalent to a variable. Λ is the diophantine abbreviation for “lacking,” but it may not mean precisely what we mean by – (minus), because Λ can only be applied to a determinate number, expressing something about the composition of that number, while – is primarily an operation preformed upon the number. I suppose negative numbers are different in this respect, but Greek mathematics does not seem to have negative numbers, so Λ would not mean that either.

Mr. D___ says that this is “very important,” though most people don’t have an “opportunity” to study it. Mr. Klein likewise thinks it’s very important, and puts in expressions like “a transformation unique and unparalleled in the history of man!” to show that. But he also calls it “a shift we can nowadays scarcely grasp,” which seems perhaps true. I certainly haven’t grasped it.

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