The Ethics of Geometry: A Genealogy of Modernity

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cited on p. 23 (PDF p. 31) of Jeff Kalb's Music and Measurement: An the Eidetic Principles of Harmony and Motion (St. Cecilia's Feast Day, 2016):

A ratio for Euclid … is a relation between magnitudes; it is not a magnitude or a quantity in its own right. (Hence, it is most emphatically not a “rational number”…) Therefore, operations to which magnitudes are “naturally” subject (such as addition of line-segments, multiplication of numbers and its geometrical counterpart…) would appear to be alien intruders once transplanted to the domain of ratios (or, indeed, the domain of proportions, as will happen in algebra when equations are added to, or multiplied by, one another, and so on.) Nonetheless, on the most plausible reading of “compound ratio” in Euclid, we are being asked to allow some such alien operation to be applied to a ratio or, more precisely, to a pair of ratios.

Euclid's definition of ratio in Element. (l. 5.) defins. as a "similitudo duarum proportionum " is mentioned in the context of "whether mind-independent relations themselves, precisely as such, are able to found other relations." Scotus thought so (cf. Deely's Four Ages of Understanding pp. 376-7, esp. where he summarizes, on the first full quoted ¶ on p. 377, Scotus's view: "unica requirit [Scotus] solum distinctionem inter res, quæ sunt extrema, non inter rationes fundandi.D. Thomas utrumque requirit "; cf. the Summa article "Whether there is equality in God?"); St. Thomas did not. cf. John of St. Thomas's Tractatus de Signis pp. 102 line 23ff.

Euclid:

3. A ratio (Λόγος) is a sort of relation in respect of size between two magnitudes of the same kind.
6. Let magnitudes which have the same ratio be called proportional (ἁνάλογον).

The Ethics of Geometry is a study of the relationship between philosophy and mathematics. Essential differences in the ethos of mathematics, for example, the customary ways of undertaking and understanding mathematical procedures and their objects, provide insight into the fundamental issues in the quarrel of moderns with ancients. Two signal features of the modern ethos are the priority of problem-solving over theorem-proving, and the claim that constructability by human minds or instruments establishes the existence of relevant entities. These figures are combined in the emblematic statement of Salomon Maimon, "In mathematical construction we are, as it were, gods." Construction is the mark of modernity. The disciplines of classical philology, literary interpretation and the history of philosophy and of mathematics are woven together in this volume.

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