- When I speak of ‘the infinite’ in this paper the infinitely large is always meant
- 1960 . Metaphysics 1021 – 1021 . Ann Arbor trans. Richard Hope a 5
- 1965 . Critique of pure reason B 17 – B 17 . New York trans. Norman Kemp Smith
- Loemker , L. E. , ed. 1969 . Philosophic papers and letters , 2nd ed. 267 – 267 . Dordrecht trans. and ed Henceforth cited as ‘PPL’.
- PPL , 130 – 130 .
- PPL , 168 – 168 .
- 1954 . Averroes' Tahafut Al-Tahafut London trans. with an introduction and notes by S. van den Bergh: 2 vols. vol. 1, 14.
- 1930 . Physics 204 – 204 . London trans. R. P. Hardie and R. K. Gaye a 25
- Scholz , H. 1928 . Warum haben die Griechen die Irrationalzahlen nicht aufgebaut? . Kantstudien , 33 : 35 – 72 . (p. 55)
- Wolfson , Compare H.A. 1929 . Crescas' critique of Aristotle 137 – 137 . Cambridge, Mass.
- Averroes' Tahafut Al-Tahafut 14 – 14 . London
- Why the proposition that one infinite cannot be greater than another should have been taken as evident is something hard to account for. Consider, for example, the following statement by Locke: ‘if a man had a positive idea of infinite, either duration or space, he could add two infinites together; nay, make one infinite infinitely bigger than another—absurdities too gross to be confuted’ An essay concerning human understanding London 1689 book II, ch. XVII, sect. 20). Perhaps some of those who thought it evident that an infinite cannot be greater than another may have done so because they thought of an infinite as an ‘unlimited’ quantity, and because they thought further that once a quantity is unlimited, it is as big as it can be. Such ideas seem to underlie the argument given by Lucretius against the infinite complexity of matter: If there are no atoms, and consequently an infinite number of parts in even very small bodies, ‘what difference will there be between the whole universe and the very least of things? None at all. For, however endlessly infinite the universe may be, yet the smallest things will equally consist of an infinite number of parts’ (The nature of the universe (trans. R. T. Latham: 1951, Baltimore), 45)
- Pines , S. 1967–1968 . “ Thābit B. Qurra's conception of number and theory of the mathematical infinite ” . In Actes du XIe congrès international d'histoire des sciences Vol. 3 , 160 – 166 . Wroclaw 5 vols Compare the following remark by Descartes, in a letter to Mersenne: A propos of infinity, you asked me a question in your letter of 14 March … You said that if there were an infinite line it would have an infinite number of feet and fathoms, and consequently that the infinite number of feet would be six times as great as the number of fathoms. I agree entirely. ‘Then this latter number is not infinite’. In deny the consequence. ‘But one infinity cannot be greater than another’. Why not? Where is the absurdity? (Descartes philosophical letters (trans. and ed. A. Kenny: 1970, Oxford), 12; Descartes's italics.)
- Wolfson , Compare H.A. 1929 . Crescas' critique of Aristotle 189 – 189 . Cambridge, Mass.
- Murdoch , J.E. “ The “equality” of infinites in the middle ages ” . In Actes Vol. 3 , 171 – 174 . Wroclaw
- Altabrizi's argument is translated in Rabinovitch N.L. Rabbi Hasdai Crescas (1340–1410) on numerical infinites Isis 1970 61 224 230 (pp. 225–226). Although this article contains some valuable quotations, the commentary, in my opinion, often misinterprets them.
- Translated in Rabinovitch Rabbi Hasdai Crescas (1340–1410) on numerical infinites Isis 1970 61 224 230
- 1914 . Two new sciences 31 – 31 . New York trans. Henry Crew and Alfonso de Salvio repr. 1951
- 1914 . Two new sciences 33 – 33 . New York trans. Henry Crew and Alfonso de Salvio repr. 1951
- Fraenkel , A. 1953 . Abstract set theory 40 – 40 . Amsterdam
- Gerhardt , C.J. , ed. 1875–1890 . Die philosophischen Schriften von G. W. Leibniz Vol. 1 , 338 – 338 . Berlin 7 vols English trans. in B. Russell, A critical exposition of the philosophy of Leibniz (2nd ed. 1937, London), 244.
- Rescher , N. 1955 . ‘Leibniz’ conception of quantity, number and infinity . The philosophical review , 64 : 108 – 114 . (p. 112)
- All references in this paragraph are from PPL 159 159
- Schriften Vol. 6 , 629 – 629 . London trans. in Russell (footnote 21), 244
- PPL , 514 – 514 .
- Schriften Vol. 2 , 315 – 315 . London trans. in Russell (footnote 21), 244. Compare Russell, The principles of mathematics (1903, Cambridge), sect. 140.
- Gerhardt , C.I. , ed. 1849-1855 . Leibnizens mathematische Schriften 28 – 28 . Berlin 7 vols vol. 1, 85. English trans. in G. Martin, Leibniz: logic and metaphysics (trans. K. J. Northcott and P. G. Lucas: 1964, Manchester),
- Bolzano , B. 1973 . Theory of science Edited by: Berg , J. Dordrecht trans. B. Terrell sect. 87. This book is an edited translation of parts of Bolzano's Wissenschaftslehre (4 vols., 1937, Sulzbach).
- Two manuscripts pretaining to the Grössenlehre ‘Einleitung zur Grössenlehre’ and ‘Erste Begriffe der allgemeinen Grössenlehre’ Bad Cannstatt Stuttgart 1975 have been published as volume 7 of series 2, part A of Bolzano's Nachlass in Bernard Bolzano Gesamtausgabe edited by E. Winter, J. Berg, F. Kambartel, J. Louzil and B. van Rootselaar. I will refer to this volume as ‘Grössenlehre’.
- Two manuscripts pretaining to the Grössenlehre ‘Einleitung zur Grössenlehre’ and ‘Erste Begriffe der allgemeinen Grössenlehre’ Bad Cannstatt Stuttgart 1975 169 169 have been published as volume 7 of series 2, part A of Bolzano's Nachlass in Bernard Bolzano Gesamtausgabe edited by E. Winter, J. Berg, F. Kambartel, J. Louzil and B. van Rootselaar. I will refer to this volume as ‘Grössenlehre’.
- Theory of science Dordrecht sect. 87; Grössenlehre, 183f (Theorem 143). Compare also Grössenlehre; 219, 225 (para. 4) and 238 (para. 33).
- Grössenlehre , 219 – 219 .
- Bolzano maintained that infinite sets are always in one-one correspondence with some proper parts of themselves, and that this is a characteristic peculiar to them, in sect. 20 of his Paradoxien des Unendlichen Prihonsky F. Leipzig 1851 The latest German edition was edited by B. van Rootselaar and published in 1975 at Hamburg; the English edition is Paradoxes of the infinite (trans. and ed. D. Steele: 1950, London). This work is cited hereafter as ‘Paradoxes’.
- Paradoxes , sect. 21. Here one may doubt the propriety of the use of the word ‘inequality’. What is involved here is not the negation of an equality relation (for none has even been specified). Bolzano does not give us an equality relation which is demonstrably incompatible with the greater than relation which he likes to think of as an ‘inequality’. I will show that he does fall into an error in reasoning concerning this point. Moreover, Bolzano was unjustified in holding the proposition that equipollence always implies (numerical) equality to be a ‘mistake’ (Paradoxes, sect. 42). He does not seem to have realized that it is all a matter of definition, and what is demonstrable from definitions.
- Paradoxes , sect. 49
- Cantor , G. 1932 . Gesammelte Abhandlungen mathematischen und philosophischen Inhalts Edited by: Zermelo , E. 119 – 119 . Berlin repr. 1962, Hildesheim
- Bolzano , Compare . Grössenlehre , 184 – 184 . Bolzano assumed more than the comparability proposition, for he assumed mistakenly (not taking account of the reflexivity of infinite sets) that for any two sets M and N, either M and N are equipollent or one is equipollent to a part of the other, and (what is false) that these are exclusive alternatives. The comparability theorem was first established (using the axiom of choice) in 1904 by Ernst Zermelo.
- Gesammelte Abhandlungen 117 – 117 . Berlin compare p. 143
- Gesammelte Abhandlungen 278 – 281 .
- Russell , B. 1901 . Recent work on the principles of mathematics . The international monthly , 4 : 83 – 101 . (p. 95). (This paper was reprinted as ch. 5 of Russell's Mysticism and logic (1917, repr. 1963, London).)
- Russell , B. 1919 . Introduction to mathematical philosophy 136 – 136 . London
- For English translations of these papers, together with editorial comment, see van Heijenoort J. From Frege to Gödel … Cambridge, Mass. 1967 199 215 and 393–413
Annals of Science
Volume 34, 1977 - Issue 2
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