Abstract
It is a commonplace in the analytic geometry of physical space and time that an extended straight line segment, having positive length, is treated as “consisting of” unextended points, each of which has zero length. Analogously, time intervals of positive duration are resolved into instants, each of which has zero duration.
A detailed clarification and correction of this chapter is given in § 10 of the Appendix and on pp. 352–533 of ch. 16
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Notes
S. Luria: “Die Infinitesimaltheorie der antiken Atomisten,” Quellen und Studien zur Geschichte der Mathematik, Astronomie, und Physik (Berlin, 1933), Abteilung B: Studien, II, p. 106.
Aristotle: On Generation and Corruption, Book I, Chap, ii, 316a15–317a17; A. Edel: Aristotle’s Theory of the Infinite (New York: Columbia University Press; 1934), pp. 48–49, 76–78; T. L. Heath: Mathematics in Aristotle (Oxford: Oxford University Press; 1949), pp. 90, 117.
C. B. Boyer: The Concepts of the Calculus (New York: Hafner Publishing Company; 1949), p. 140.
B. Russell: The Philosophy of Leibniz (London: G. Allen & Unwin, Ltd.; 1937 ), p. 114.
P. du Bois-Reymond: Die Allgemeine Funktionentheorie (Tübingen: Lauppische Buchhandlung; 1882), Vol. I, p. 66.
G. Cantor: Gesammelte Abhandlungen, ed. E. Zermelo (Berlin: Julius Springer; 1932), pp. 275, 374.
P. W. Bridgman: “Some Implications of Recent Points of View in Physics,” Revue Internationale de Philosophie, Vol. III, No. 10 (1949), p. 490.
H. Hasse and H. Scholz: Die Grundlagenkrisis der griechischen Mathematik ( Charlottenburg: Pan-Verlag; 1928 ), p. 11.
K. Menger: Dimensionstheorie (Leipzig: B. G. Teubner; 1928), p. 244.
A. Grünbaum: “A Consistent Conception of the Extended Linear Continuum as an Aggregate of Unextended Elements,” Philosophy of Science, Vol. XIX (1952), pp. 290–95
E. V. Huntington: The Continuum and Other Types of Serial Order (2nd ed.; Cambridge: Harvard University Press; 1942), pp. 10, 44.
S. Lefschetz: Introduction to Topology ( Princeton: Princeton University Press; 1949 ), p. 28
H. Cramér: Mathematical Methods of Statistics (Princeton: Princeton University Press; 1946), pp. 11, 19.
R. Courant and H. Robbins: What is Mathematics? (New York: Oxford University Press; 1941), p. 249.
P. R. Halmos: Measure Theory (New York: D. Van Nostrand Company, Inc.; 1950.)
C. Hartshorne and P. Weiss (eds.): The Collected Papers of Charles Sanders Peirce (Cambridge: Harvard University Press; 1935), Vol. VI, para. 125.
P. Tannery: “Le Concept Scientifique du Continu: Zenon d’illee et Georg Cantor,” Revue Philosophique, Vol. XX, No. 2 (1885), pp. 391–92.
Cf. also G. H. Hardy: A Course of Pure Mathematics ( 9th ed.; New York: The Macmillan Company; 1945 ), pp. 145–47.
See E. W. Hobson: The Theory of Functions of a Real Variable (2nd ed.; Cambridge: Cambridge University Press; 1921) Vol. I, pp. 56–57.
B. Russell: Our Knowledge of the External World (London: George Allen & Unwin, Ltd.; 1926), pp. 138, 139–40; my italics.
B. Russell: The Principles of Mathematics (Cambridge: Cambridge University Press; 1903), p. 444; my italics. A similar criticism applies to Dedekind. He maintains that if we postulated a discontinuous space consisting of the algebraic points alone, then “the discontinuity of this space would not be noticed in Euclid’s science, would not be felt at all” (R. Dedekind: Essays on the Theory of Number [Chicago: Open Court Publishing Company; 19011, pp. 37-38). Since the set of algebraic points is still denumerable (A. Fraenkel: Einleitung in die Mengenlehre [New York: Dover Publications, Inc.; 1946], p. 40), the length (measure) of a segment consisting entirely of such points is paradoxically both zero and positive. I therefore cannot follow F. Waismann, who comments approvingly on Dedekind’s statement by saying: “As to physical space one has become accustomed to conceding the justification of this concept” (F. Waismann: Introduction to Mathematical Thinking [New York: F. Ungar Publishing Company; 1951 ], p. 212 ).
For the historical details, see K. von Fritz: “The Discovery of Incommensurability by Hippasus of Metapontum,” Annals of Mathematics, Vol. XLVI (1945).
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© 1973 D. Reidel Publishing Company, Dordrecht, Holland
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Grünbaum, A. (1973). The Resolution of Zeno’s Metrical Paradox of Extension for the Mathematical Continua of Space and Time. In: Philosophical Problems of Space and Time. Boston Studies in the Philosophy of Science, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2622-2_6
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