Abstract
The term “mathematics” is seldom defined by historians of the subject: for instance, Boyer’s History of Mathematics explicitly avoids the task, saying merely that “much of the subject [⋯] is an outgrowth of thought that originally centered on the concepts of number, magnitude and form.”1 Were we to take this as the basis for a definition, mathematics would not only go back to paleolithic times—and indeed long before, since one can talk about the “mathematical abilities” of various animals, and research has been done on the issue—but would encompass even the Neapolitan smorfia, a series of rules for extracting from dreams information supposed to be helpful in predicting winning lottery numbers. This too, one must admit, deals with questions centered on the concept of number.
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References
[Boyer], p. 1.
[Neugebauer: ESA], p. 151.
These statements by Eudemus (whose work has perished) are reported in Proclus, In primum Euclidis Elementorum librum commentarii, 157:10–11; 299:1–3, ed. Friedlein = [FV], I, 79:8–9+13–15, Thales A20.
Proclus, In primum Euclidis Elementorum librum commentarii, 352:14–18, ed. Friedlein = [FV], 79:15–19, Thales A20.
The main source for Zeno’s paradoxes is Aristotle, Physica, VI, ix, 239b–240a. This passage and all other relevant sources are reported in [FV], I, 247–258. The paradoxes are discussed in [Heath: HGM], vol. I, pp. 271–283.
Reconstructions of this episode are based primarily on two sources. The older one is a passage of Aristotle (Analytica priora, I, xxiii,41a:26–27), which says that the diagonal is not commensurable [with the side] because, if it were, odd and even numbers would coincide. The second source is a (probably spurious) passage in Euclid–s Elements (X, 408–411, ed. Heiberg, vol. III), containing a complete proof of the incommensurability, consistent with Aristotle’s brief remark.
This can reasonably be deduced from several elements: the fact, reported by many sources, that the Pythagoreans based geometry on the integers; the Pythagorean theories of “figurative numbers” (for which see [Heath: HGM], vol. I, pp. 76–84, and [Knorr: EEE], Chapter 5); and above all Aristotle’s assertion that the Pythagoreans attributed a magnitude to the units that made up material bodies (Aristotle, Metaphysica, XIII, vi, 1080b:16–21 + 1083b:8–18 = [FV], I, 453:39–454:9, Pythagoreans B9, B10). Sextus Empiricus seems to still be thinking in Pythagorean mode when he says that it is impossible to bisect a segment formed by an odd number of points: Adversus physicos I (= Adv. dogmaticos III = Adv. mathematicos IX), §283; Adversus geometras (= Adv. math. III), §§110–111.
Philolaus, as quoted by Stobaeus (Eclogae, I, xxi §7c, 188:9–12 = [FV], I, 408:7–10, Philolaus B5); Aristotle, Metaphysica, I,v, 986a:18+23–24. Evenness and oddness were still at the basis of arithmetic for Plato (Gorgias, 451a-b). For a discussion of the Pythagorean ideas about even and odd numbers, see [Knorr: EEE], pp. 134–14
Plato, Meno, 84e–85b.
Plato, Timaeus, 81b–c.
Plato, Republic, VI, 509c–511a.
Aristotle, Analytica posteriora, I, ii,71b:26–28.
The distinction between problems and theorems is discussed at length by Proclus (In primum Euclidis elementorum librum commentarii, 77–81, ed. Friedlein), and appears twice in Pappus (Collectio, III,30:3–24; VII, 650:16–20). Euclid does not differentiate between the two types of propositions in these terms, but the distinction is clear from the formula that closes the demonstration, which is either “as was to be shown” or “as was to be done”.
Euclid, Elements, I, proposition 47.
Euclid, Elements, I, proposition 46.
The problems that can be solved in this way are those that we would express in terms of algebraic equations of the first or second degree. For example, the determination of the fourth proportional of three given segments (Euclid, Elements, V, proposition 12) is equivalent to the calculation of a ratio, once one segment is chosen as the unit of measurement. The determination of the proportional mean of two given segments (Euclid, Elements, V, proposition 13) amounts to taking a square root. Obviously the algebraic formulation is not necessary for applications: every problem solvable by taking a square root can be solved equivalently by finding a proportional mean.
[Boyer], p. 173 (1st ed.), p. 157 (2nd ed.).
A précis of Apollonius’ system can be found in Pappus, Collectio, II, 6–28.
Euclid, Elements, I, proposition 46.
Plato, Republic, VI, 510c.
Aristotle, Analytica posteriora, I, x,76a:40.
Euclid, Elements, V, definition 4.
A more general version of the postulate, applying not only to surfaces but also to lines and solids, appears in Archimedes, De sphaera et cylindro, 11:16–20 (ed. Mugler, vol. I).
[Rosenfeld], p. 11.
[Boyer], p. 237 (1st ed.), p. 215 (2nd ed.).
[Geymonat], vol. I, p. 354.
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Russo, L. (2004). Hellenistic Mathematics. In: The Forgotten Revolution. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18904-3_3
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