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A Consistent Conception of The Extended Linear Continuum as an Aggregate of Unextended Elements

Published online by Cambridge University Press:  14 March 2022

Adolf Grünbaum
Adolf Grünbaum*
Affiliation:
Lehigh University
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Extract

It is a commonplace in the analytic geometry of physical space-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.

Type
Research Article
Information
Philosophy of Science , Volume 19 , Issue 4 , October 1952 , pp. 288 - 306
Copyright
Copyright © 1952, The Williams & Wilkins Company

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Footnotes

∗

This paper is based on part of a dissertation presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Yale University, and written during the tenure of a pre-doctoral fellowship from the American Council of Learned Societies for 1948–1950.

References

1. Aristotle, On Generation and Corruption, Book I, Chap. II, 316a15–317a17.Google Scholar
2. Bolzano, B., Paradoxes of the Infinite (ed. D. A. Steele), London, 1950 and New Haven, 1951.Google Scholar
3. Bourbaki, N. Éléments de Mathématique, Première Partie, Livre III: Topologie Générale, Actualités series No. 858, Paris, 1940.Google Scholar
4. Boyer, C. B., The Concepts of the Calculus, New York, 1949.Google Scholar
5. Bridgman, P. W., “Some Implications of Recent Points of View in Physics,” Revue Intern. de Phil. vol. 3, No. 10, 1949.Google Scholar
6. Burington, R. S., and Torrance, C. C., Higher Mathematics, New York, 1939.Google Scholar
7. Cajori, F., “The Purpose of Zeno's Arguments on Motion,” Isis S, 1920–1.CrossRefGoogle Scholar
8. Cantor, G., Gesammelte Abhandlungen (ed. Zermelo, E.), Berlin, 1932.CrossRefGoogle Scholar
9. Courant, R., and Robbins, H., What is Mathematics?, New York, 1941.Google Scholar
10. Cramér, H., Mathematical Methods of Statistics, Princeton, 1946.CrossRefGoogle Scholar
11. Dedekind, R., Essays on the Theory of Number, Chicago and London, 1924.Google Scholar
12. Du Bois-Reymond, P., Die Allgemeine Funktionentheorie, Erster Teil, Tübingen, 1882.Google Scholar
13. Edel, A., Aristotle's Theory of the Infinite, New York, 1934.Google Scholar
14. Fano, G., “Die Gruppentheorie als geometrisches Einteilungsprinzip,” Enzyk. d. math. Wissensch. III A.B. 4b.Google Scholar
15. Federer, H., “Dimension and Measure,” Trans. Am. Math. Soc. 62, 1947.CrossRefGoogle Scholar
16. Fraenkel, A., Einleitung in die Mengenlehre, New York, 1946.Google Scholar
17. Freund, H., Ontologische Untersuchungen zum Cantor'schen Mengenbegriff (diss.) Quackenbrück, 1933.Google Scholar
18. Graves, L. M., The Theory of Functions of Real Variables, New York, 1946.Google Scholar
19. Grünbaum, A., “Relativity and the Atomicity of Becoming,” The Review of Meta-physics vol. 4, No. 2, 1950.Google Scholar
20. Grünbaum, A., “Some Recent Writings in the Philosophy of Mathematics,” The Review of Metaphysics vol. 5, No. 2, 1951.Google Scholar
21. Halmos, P. R., Measure Theory, New York, 1950.CrossRefGoogle Scholar
22. Hardy, G. H., A Course of Pure Mathematics (9th ed.), New York, 1945.Google Scholar
23. Hartshorne, C., and Weiss, P. (eds.), The Collected Papers of Charles Sanders Peirce, Cambridge, 1935.Google Scholar
24. Hasse, H., and Scholz, H., Die Grundlagenkrisis der griechischen Mathematik, Charlottenburg, 1928.CrossRefGoogle Scholar
25. Heath, T. L., A History of Greek Mathematics, I, Oxford, 1921.Google Scholar
26. Heath, T. L., Mathematics in Aristotle, Oxford, 1949.Google Scholar
27. Hjelmslev, J., “Die natürliche Geometrie,” Abhandl. aus dem math. Sem. d. Hamb. Univ. 2, 1923.Google Scholar
28. Hobson, E. W., The Theory of Functions of a Real Variable and the Theory of Fourier's Series, (second ed.), I, Cambridge, 1921.Google Scholar
29. Huntington, E. V., The Continuum and Other Types of Serial Order, (second ed.), Cambridge, 1942.Google Scholar
30. Hurewicz, W., and Wallman, H., Dimension Theory, Princeton, 1941.CrossRefGoogle Scholar
31. Kant, I., Werke (ed. Cassirer, E.), Berlin, 1912.Google Scholar
32. Kasner, E., and Newman, J., Mathematics and the Imagination, New York, 1941.Google Scholar
33. Klein, F., Elementary Mathematics from an Advanced Standpoint, II, New York, 1945.Google Scholar
34. Klein, F., “Vergleichende Betrachtungen über neuere geometrische Forschungen,” Math. Annalen 43, 1893.CrossRefGoogle Scholar
35. Lee, H. D. P., Zeno of Elea, Cambridge, 1936.Google Scholar
36. Lefschetz, S., Introduction to Topology, Princeton, 1949.CrossRefGoogle Scholar
37. Lefschetz, S., “The Structure of Mathematics,” American Scientist 38, 1950.Google Scholar
38. Luria, S., “Die Infinitesimaltheorie der antiken Atomisten,” Quellen und Studien zur Gesch. d. Math. Astr. u. Phys., Abt. B: Studien, II. Berlin, 1933.Google Scholar
39. Menger, K., “Bericht über die Dimensionstheorie,” Jahresbericht der Deutsch. Math.- Ver. 35–36, 1926–7.Google Scholar
40. Menger, K., Dimensionstheorie, Leipzig, 1928.10.1007/978-3-663-16056-4CrossRefGoogle Scholar
41. Menger, K., “What is Dimension?,” Am. Math. Monthly 50, 1943.CrossRefGoogle Scholar
42. Poincaré, H., “Pourquoi l'espace a trois dimensions,” Rev. de Mét. et de Mor. 20, 1912.Google Scholar
43. Russell, B., An Essay on the Foundations of Geometry, Cambridge, 1897.Google Scholar
44. Russell, B., Our Knowledge of the External World, London, 1914.Google Scholar
45. Russell, B., The Philosophy of Leibniz, London, 1937.Google Scholar
46. Russell, B., The Principles of Mathematics, New York, 1903.Google Scholar
47. Szpilrajn, E., “La dimension et la mesure,” Fund. Math. 28, 1937.CrossRefGoogle Scholar
48. Tannery, P., “Le Concept Scientifique du Continu: Zénon d'Élée et Georg Cantor,” Rev. Phil., 20 No. 2, 1885.Google Scholar
49. The Thirteen Books of Euclid's Elements, (tr. T. L. Heath), Cambridge, 1926.Google Scholar
50. van der Waerden, B. L., “Zenon und die Grundlagenkrise der griechischen Mathematik,” Math. Ann. 117, 1940.Google Scholar
51. von Fritz, K., “The Discovery of Incommensurability by Hippasus of Metapontum,” Annals of Mathematics 46, 1945.Google Scholar
52. von Laue, M., Die Relativitätstheorie, II, Braunschweig, 1923.Google Scholar
53. Waismann, F., Introduction to Mathematical Thinking, New York, 1951.Google Scholar
54. Weyl, H., Philosophy of Mathematics and Natural Science, Princeton, 1949.CrossRefGoogle Scholar