References
- See Sheynin O.B. On the prehistory of the theory of probability Archive for history of exact sciences 1974 12 97 141 (p. 135): ‘An argument about randomness and necessity, possibly the first one in recent times is due to Kant’.
- See Newton Isaac Opticks Dover Publ. New York 1952 based on 1730 edition), query 31 (p. 402): ‘… some inconsiderable irregularities … which may have arisen from the mutual actions of comets and planets upon one another, and which will be apt to increase, till this system wants a reformation’.
- Kant , Immanuel . 1905 . Der einzig mögliche Beweisgrund zu einer Demonstration des Dasein Gottes (Gesammelte Schriften) Vol. 2 , 111 – 111 . Berlin Die Erfahrung stimmt auch mit dieser Abhängigkeit sogar der freiesten Handlungen von einer grossen natürlichen Regel überein. Denn so zufällig wie auch immer die Entschliessung zum Heirathen sein mag, so findet man doch in eben demselben Lande, dass das Verhältniss der Ehen zu der Zahl der Lebenden ziemlich beständig sei, wenn man grosse Zahlen nimmt, und dass z.E. unter 110 Menschen beiderlei Geschlechter sich ein Ehepaar findent. Jedermann weiss, wie viel die Freiheit der Menschen zu Verlängerung oder Verkürzung des Lebens beitrage. Gleichwohl müssen selbst diese freie Handlungen einer grossen Ordnung unterworfen sein, weil im Durchschnitte, wenn man grosse Mengen nimmt, die Zahl der Sterbenden gegen die Lebenden sehr genau immer in eben demselben Verhältniss steht …. die Gesetze der Freiheit keine solche Umgebundenheit in Ansehung der Regeln einer allgemeinen Naturordnung mit sich führen, das nicht eben derselbe Grund, der in der übrigen Natur schon in der Wesen der Dinge selbst eine unausbleibliche Beziehung auf Vollkommenheit und Wohlgereimtheit bestetigt, auch in dem natürlichen Laufe des freien Verhaltens wenigstens eine grössere Lenkung auf ein Wohlgefallen des höchsten Wesens ohne vielfältige Wunder verursachen sollte. Therefore miracles are rarely necessary, and it is strange that they should occur.
- For details see Rabinovitch N.L. Probability and statistical inference in ancient and medieval Jewish literature University of Toronto Press Toronto 1973
- Avot , 5 4 – 4 . The numerals are inserted
- See Leviticus 23 10 11
- See Leviticus 23 17 17
- Exodus , 25 30 – 30 .
- Pesahim , 1 5 – 5 .
- Exodus , 12 15 – 20 .
- Leviticus , 7 12 – 14 .
- Our printed texts omit the name of the teacher. However, citations in early medieval sources as well as a parallel passage in the Babylonian Talmud (Pesahim, 13b) all confirm that Rabbi Hanina is meant. See also Lieberman Saul HaYerushalmi Ki Phshuto Jerusalem 1934 381 381 Rabbi Hanina came in mature age from Babylon to Israel where he became a disciple of Rabbi, and after the death of Rabbi's successor, Hanina took his place as head of the academy at Sepphoris.
- Sanhedrin , 1 4 – 4 . A similar prodecure is described there in connection with the redemption of the first-born sons (Numbers, 3 : 44–49). That text is discussed in detail in my (footnote 4), 29.
- See, for example Guttman Julius Philosophies of Judaism Anchor Books Garden City, N.Y. 1966 45ff 45ff and the literature there cited. See also S. Sambursky, ‘On the possible and the probable in ancient Greece’, Osiris, 12 (1956), 35–48.
- Mishnah Avot , 3 18 – 18 . Most ancient and modern commentators understand Rabbi Akiva's dictum as an attempt to hold fast to both horns of the dilemma: ‘All is foreseen’=God's foreknowledge co-exists with ‘freedom is granted’=man's freedom of will. Recently Professor Ephraim Urbach (The sages—their concepts and beliefs [in Hebrew] (Jerusalem, 1969), 229ff.) has questioned this interpretation. He proposes to translate the passage as follows: ‘Everything is observed and freedom is granted …’. The point is then that God observes all that man does and holds him accountable for it. The issue turns on the precise meaning of a single Hebrew word, and Professor Urbach claims that although its usage in the sense of foreknowledge is well attested in the Talmud, this was a later development, whereas in Rabbi Akiva's time the meaning was quite general. The argument appears far from convincing. In any case, even if Rabbi Akiva is understood differently, this does not affect our conclusion.
- Exodus , 3 14 – 14 .
- See, for example Kasher M.M. Torah Shelemah New York 1954 8 150 150 n. 176
- Ezekiel , 20 33 – 33 .
- 1878 . Midrash Shemot Rabbah Vol. 2 , 22 – 22 . Vilna 3 : 6. This Midrash contains mainly exegetical and homiletical material on selected verses in Exodus. The first fourteen chapters of Shemot Rabbah form a separate unit which is thought to have been assembled in the seventh century. Some date the final editorial compilation of this Midrash as a whole as late as the eleventh century, although the material is of course much older.
- Compare Exodus 10 1 2 ‘Then the Lord said to Moses, “Go in to Pharoah … that I may show these signs of mine among them, and that you may tell in the hearing of your son and of your son's son … what signs I have done among them; that you may know that I am the Lord”’.
- Maimonides , Moses . 1972 . Iggerot , 98 – 100 . Jerusalem : Joseph Kafih . (Arabic original, edited and translated into Hebrew by) I have summarised and paraphrased most of the discussion with the occasional quotations of salient points so marked.
- Exodus , 7 10 – 10 .
- Exodus , 9 22 – 22 . 20
- Compare I Kings 13 5 5 ‘The altar also was torn down, and the ashes poured out from the altar, according to the sign which the man of God had given by the word of the Lord’. This is but one of Maimonides's illustrations. In the following classifications too, I have not cited all of the examples in Maimonides's text.
- Exodus , 9 6 – 6 .
- Leviticus , 26 3 – 45 .
- For instances of probability computations and statistical reasoning by Maimonides in legal and other contexts see Rabinovitch N.L. On probability theory in Rabbinic literature Tarbiz 1972 42 79 89 [in Hebrew] ‘The concept of “possibility” in Maimonides’ [in Hebrew], Tarbiz, 44 (1974), 159–171; ‘A fifteenth-century law of large numbers’, Isis, 65 (1974), 229–238; and ‘Early antecedents of error theory’, Archive for history of exact sciences, 13 (1974), 348–358.
- Maimonides's greatest work is his Code entitled Mishneh Torah which is a complete compendium of rabbinic law as it had developed up to his time. Parts of the Mishneh Torah were translated into Latin in the seventeenth century by John Selden. Recently the Yale Judaica Series has included a number of volumes of the Mishneh Torah rendered into English. I used the standard rabbinic edition of Mishneh Torah published at Vilna in 1900.
- Mishneh Torah: Issurei Biah , 22 18 – 18 .
- Ecclesiastes , 7 14 – 14 .
- Hagigah , 15a
- Thus Rabbi Hai Gaon (939–1038) reputedly wrote sharply: ‘This is not true …’ (see Otzar ha-Geonim Lewin B.M. Jerusalem 1931 4 Tractate Hagigah, 62
- Maimonides , Moses . Mishneh Torah: Teshuvah , 6 5 – 5 .
- Maimonides , Moses . 1963 . The guide for the perplexed , 534 – 535 . Chicago : University of Chicago Press . translated by S. Pines part 3, ch. 34
- Maimonides , Moses . 1963 . The guide for the perplexed , 560 – 560 . Chicago : University of Chicago Press . translated by S. Pines ch. 41
- Aquinas , Thomas . 1966 . Summa theologiae Vol. 28 , 139 – 141 . London I, ii, 96, 6: English translation by Thomas Gilby
- Aquinas , Thomas . 1966 . Summa theologiae Vol. 28 , 119 – 119 . London I, ii, 96, 6: English translation by Thomas Gilby 96, 1; Gilby ‘The Jurist’ is the usual medieval designation of the compendium of Roman law entitled the Digest which was published by authority of the Emperor Justinian in 533.
- For example, one of the most radical of fourteenth-century probabilists was Nicholas of Autrecourt. The modern editor of his work, J. Reginald O'Donnell, has pointed out dependences on Maimonides The philosophy of Nicholas of Autrecourt and his appraisal of Aristotle Medieval studies 1942 4 97 125 (p. 108 n. 3)). See also J. R. Weinberg, Nicholas of Autrecourt (Princeton, 1948), 84 ff. Some further details are given in my (footnote 4), 166ff.
- Aramah , Isaac . 1848 . Aqedat Yitzhaq Edited by: Pollack , H.J. Vol. 5 , Pressburg vol. 1, ch. 28' folio 221a. Isaac ben Moses Aramah (1420–1494), rabbi, philosopher and preacher, lived in Spain until the expulsion of the Jews in 1492, when he fled to Naples where he lived until his death. His major work is Aqedat Yitzhaq, which is written in the form of philosophical homilies on the Pentateuch. It consists of 105 chapters (called ‘Gates’ or ‘Portals’), and in it Aramah develops his views on the major philosophical problems of his time. For other aspects of Aramah's contribution to statistical thought see my paper ‘A fifteenth-century law’ (footnote 31).
- Aramah , Isaac . 1848 . Aqedat Yitzhaq Edited by: Pollack , H.J. Vol. 5 , Pressburg vol. 1, ch. 28' folio 221a. Isaac ben Moses Aramah (1420–1494), rabbi, philosopher and preacher, lived in Spain until the expulsion of the Jews in 1492, when he fled to Naples where he lived until his death. His major work is Aqedat Yitzhaq, which is written in the form of philosophical homilies on the Pentateuch. It consists of 105 chapters (called ‘Gates’ or ‘Portals’), and in it Aramah develops his views on the major philosophical problems of his time. For other aspects of Aramah's contribution to statistical thought see my paper ‘A fifteenth-century law’ (footnote 31).
- Keynes , J.M. 1961 . A treatise on probability , 3rd ed. 41 – 41 . Oxford
- Bava Kama , 4 1 – 1 . For a fuller discussion of the context in which this statement appears see my (footnote 4), 94ff.
- Maimonides . 1963 . The guide for the perplexed , Chicago : University of Chicago Press . translated by S. Pines part 3, ch. 15, pp. 460–461. See also N. L. Rabinovitch, ‘A fifteenth-century law’ (footnote 31)
- 1805 . Responsa Lemberg no. 372. See also my (footnote 4), 73
- Oresme , Nicole . 1966 . De proportionibus proportionum , 385 – 385 . Madison, Wis. : University of Wisconsin Press . edited and translated into English by Edward Grant It is clear from his applications to the commensurability of celestial motions that Oresme identified equi-possible with equi-probable; see my (footnote 4).
- 1861 . “ Praeclare distinguit passim Maimonides inter intellectionem et imaginationem, docetque non hanc, sed illam de possibilitate judicare ” . In A. Foucher de Careil, Leibniz, la philosophiejuive et la cabale Paris quoted from his Leibnitii observationes ad Rabbi Mosis Maimonidis librum qui inscribitur doctor perplexorum (reprint by Gregg Int. Publ., 1969), 44. He had noted the same point earlier in a marginal note on Part 3, ch. 15: ‘Non fiunt possibilia quae imaginari licet sed quae intelligere’ (ibid., 37. Leibniz devoted considerable thought to the question whether probabilities could be determined empirically rather than a priori by counting all the possible alternatives. In an exchange of letters with Jacob Bernoulli, Leibniz refused to be convinced. However, nine years after Bernoulli's death, he wrote (22 March 1714) to Bourguet: ‘On estime encore les vraisemblances a posteriori, par l'experience, et on y doit avoir recours au défaut des raisons a priori; par exemple, il est egalement vraisemblable que l'enfant qui doit naistre soit garçon ou fille, parce que le nombre des garçons et des filles se trouve à peu près egal dans ce Monde’ (B. L. van der Waerden (ed.), Die Werke von Jakob Bernoulli, vol. 3 (Basel, 1975), 513).
- Abrabanel , Don Isaac . “ Aqedat Yitzhaq ” . In Perush ha-Torah Vol. 1 , Warsaw ch. 28, folio 213b 5622 A.M.
- The modern literature on the subject is too vast and ramified to be detailed here. Yet the following passages, redolent as they are of Aramah, almost beg to be cited. The first quotation is from an essay by Maxwell dated 11 February 1873 which constitutes chapter XIV. Maxwell on determinism and freedom of will p. 434-444 of Campbell Lewis Garnett William The life of James Clerk Maxwell MacMillan and Co. London 1882 ‘… the effects of widespread causes, though very different in each individual, will produce an average result on the whole nation’ (p. 439). Our second quotation is from William James; from ‘The dilemma of determinism’, in Essays on faith and morals (sel. R. B. Perry: New York, Longmans, Green & Co., 1949): The belief in free-will is not in the least incompatible with the belief in Providence … Suppose him to say, I will lead things to a certain end, but I will not now decide on all the steps thereto … The creator's plan of the universe would thus be left blank as to many of its actual details … But … its final upshot would be rigorously determined once for all.