- 6. The Significance of Some Basic Mathematical Conceptions for Physics
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C H A P T E R 6 THE S I G N I F I C A N C E OF SOME BASIC M A T H E M A T I C A L C O N C E P T I O N S FOR P H Y S I C S * THE PRESENT chapter supplements and continues the preceding one, and we will again make observations on the relations between mathematics and physics on the level of abstractions and conceptualizations; in the preceding chapter we emphasized mechanics, and at present we will be more concerned with physics in general. The Greeks created rational philosophy and a general conception of rational philosophy, and they created a general conception of mathematics and a general conception of physics. Their mathematics is justly renowned; the Greeks imparted to it from the beginning a certain metaphysical accent which has become a distinguishing mark of mathematics in general. They developed an astronomy and geometrical optics which were quite mathematical , and the Greeks themselves wondered whether, and in what sense, these could be distinguished from mathematics proper. Yet if one undertakes to assess in retrospection and pragmatically the outcome of Greek mathematics and physics, and if one confronts the mathematics and physics of Greek antiquity with modern mathematics and physics, then one may nevertheless find that Greek physics as a whole, and in its inward structure, never developed into a "theoretical" system, that the system of Greek mathe- * Originally in his, Vol. 54 (1963), pp. 179-205. Revised. I209] M A T H E M A T I C A L C O N C E P T I O N S JN P H Y S I C S matics had severe limitations and shortcomings, and that these inadequacies made Greek mathematics unsuited to promote the rise of theoretical physics as we have it or to mold scientific thinking in our sense. A mathematics that was much more suited to do this began only after the Middle Ages, and its first impact on physics and other science was to promote the rise of a Rational Mechanics, which held the stage of science during the 17th and 18th centuries and without which the general mathematization of physics and then, progressively, of ever larger areas of so-called exact science and also of other (even social) science would not have come about. In what follows, some features of this mathematics, and their involvement with physics will be presented. I. THE SIGNIFICANCE OF MULTIPLICATION 1. GREEK MAGNITUDES From whatever reasons, Greek mathematics did not create a conception of real numbers. Instead, it was satisfied with, and became arrested on the precarious notion of magni tude ( = μέγεθος), which the Greeks never defined, in or out of mathematics, but which was their substitute for real number. For instance, lengths were one kind of magnitude, areas were another kind of magnitude, weights were still another kind of magnitude, etc. The Greeks could envisage the ratio Pi : P2 for any two values of the same magnitude, and the ratio L1 : L^ for any two values of any other magnitude, and they could envisage the proportion P1 IP2 = L1 Ii2 (1) between them; they could verify, by a famous criterion which was a summit achievement of Greek mathematics, whether the proportion does or does not hold for the four specific values given. They also had a calculus of ratios and proportions with which they could operate competently and skilfully. But the calculus was restrictive and stultify- [210] MATHEMATICAL CONCEPTIONS IN PHYSICS ing, it did not have a dynamism of evolution, and even Archimedes could not break out of its constrictions. The fact is that the "classical" Greek mathematics of Euclid, Archimedes, and Apollonius could never form the conceptual product PL for two magnitudes P and L in general. In the very special case in which P and L were both lengths, a product did "exist" for them, and it was an area; similarly if one of the factors was a length and the other an area, a product did exist and it was a volume. But these exceptions were made unconsciously and unreflectively; nobody "philosophized" about them or in any way suggested to extend the product to other magnitudes. And, as we have already stated in Chapter 5, in consequence of this limitation, the formation of a momentum in mechanics was retarded by 2,000 years. Before becoming too impatient with the Greeks for not having created a product ab of...

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