×

Borel on the heap. (English) Zbl 1419.62009

Summary: In 1907 Borel published a remarkable essay on the paradox of the Heap (“Un paradoxe économique: le sophisme du tas de blé et les veérités statistiques”, see [Zbl 1419.62008] for a translation), in which Borel proposes what is likely the first statistical account of vagueness ever written, and where he discusses the practical implications of the sorites paradox, including in economics. Borel’s paper was integrated in his book Le Hasard, published 1914, but has gone mostly unnoticed since its publication. One of the originalities of Borel’s essay is that it puts forward a model of vagueness as imprecision, making particular use of the Gaussian law of measurement errors to model categorization. The aim of our paper is to give a presentation of the historical context of Borel’s essay, to spell out the mathematical details of his model, and to provide a critical assessment of his theory. Three aspects of Borel’s account are particularly discussed: the first concerns the comparison between Borel’s statistical account and posterior degree-theoretic accounts of vagueness. The second concerns the anti-epistemicist flavor of Borel’s approach, whereby the idea of statistical fluctuation is used to undermine the notion of sharp boundary for vague predicates. The third concerns the problematic link between Borel’s model of vagueness as imprecision and the notion of semantic indeterminacy. An English translation of Borel’s original essay is appended to this paper (Erkenntnis, this issue).

MSC:

62-03 History of statistics
01A60 History of mathematics in the 20th century
01A70 Biographies, obituaries, personalia, bibliographies
62A01 Foundations and philosophical topics in statistics
91B99 Mathematical economics

Biographic References:

Borel, Émile

Citations:

Zbl 1419.62008
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baldwin, J. M., & Peirce, C. S. (1902). “Sophism” and “sorites”. In J. M. Baldwin (Eds.), Dictionary of philosophy and psychology. (Vol. 2, pp. 556-557). New York: Macmillan.
[2] Bastiaanse, H. (2011). The rationality of round interpretation. In R. Nouwen, et al. (Eds.), Vagueness in communication. LNAI (Vol. 6517, pp. 37-50). · Zbl 1308.03012
[3] Bergson, H. (1907). L’évolution créatrice. Les Presses universitaires de France. (Creative Evolution. Translated in English by Arthur Mitchell, Ph.D. New York: Henry Holt and Company 1911). · JFM 39.0090.03
[4] Benovsky, J. (2011). Vagueness: A statistical epistemicist approach. Teorema, 30(3), 97-111.
[5] Black, M. (1937). Vagueness: An exercise in logical analysis. Philosophy of Science, 4, 427-455. · doi:10.1086/286476
[6] Borel, É. (1906). Sur les principes de la théorie cinétique des gaz. Annales scientifiques de l’Ecole normale supérieure , t., 23, 9-32. · JFM 37.0944.01
[7] Borel, É. (1907). Sur un paradoxe économique: Le Sophisme du tas de blé et les vérités statistiques. Revue du Mois 4, 688-699. (repr. in Oeuvres de Emile Borel t. IV, pp. 2197-2208).
[8] Borel, É. (1907a). La logique et l’intuition en mathématiques. Revue de Métaphysique et de Morale, 15(3), 273-83. · JFM 38.0081.02
[9] Borel, É. (1907b). L’évolution de l’intelligence géométrique. Revue de Métaphysique et de Morale, 15, 747-54. · JFM 38.0081.03
[10] Borel, É. (1908a). Réponse à M. Bergson. Revue de Métaphysique et de Morale, 16, 244-45. · JFM 39.0090.03
[11] Borel, É. (1908b). Le calcul des probabilités et la méthode des majorités. Année psychologique, 14, 125-151. · doi:10.3406/psy.1907.3740
[12] Borel, É. (1909a). La théorie des ensembles et les progrès récents de la théorie des fonctions. Revue générale des sciences, 20, 315-324. · JFM 40.0060.02
[13] Borel, É. (1909b). Éléments de la théorie des probabilités. Armand Colin. · JFM 40.0296.10
[14] Borel, É. (1909c). Le continu physique et le continu mathématique. Scientia, t. 6, 21-25. (Reprinted as note III in Borel (1922), 228:42).
[15] Borel, É (1913). La mécanique statistique et l’irréversibilité. Journal de physique, t. 3, 189-196. · JFM 44.0788.01
[16] Borel, É. (1914). Le Hasard, Félix Alcan, Nouvelle édition 1938.
[17] Borel, É. (1915). Mécanique statistique. Translation in French and voluminous additions to P. and T. Ehrenfest’s milestone review paper, Encyclopédie des sciences mathématiques pures et appliquées, Tome IV, Mécanique. · JFM 27.0185.01
[18] Borel, É. (1922). L’Espace et le temps, Félix Alcan, 6ème édition, 1939. English edition translated by A. S. Rappoport, & J. Dougall, Space and Time, Blackie and Son Limited, London and Glasgow, 1946. · JFM 40.0060.02
[19] Borel, É (1924). A propos d’un traité de probabilités. Revue philosophique, 98, 321-336.
[20] Borel, É. (1946). Les paradoxes de l’infini. Nrf, Gallimard, 7ème édition. · Zbl 0063.00521
[21] Borel, É. (1950). Probabilité et certitude. Presses Universitaires de France, Collection “Que Sais-je?”. (English translation by D. Scott, Probability and Certainty, Walker and Company: New York, 1963).
[22] Borel, É., & Deltheil R. (1923). Probabilités, erreurs. Armand Colin (5ème édition 1940). · JFM 49.0383.05
[23] Borel, É., Deltheil R., & Huron, R. (1954). Probabilités, erreurs. Armand Colin, 9ème édition. · Zbl 0056.38303
[24] Dawkins, R. (2004). The Ancestor’s tale. Phoenix paperback.
[25] Dubois, D., & Prade, H. (2001). Possibility theory, probability theory and multiple-valued logics: A clarification. Annals of Mathematics and Artificial Intelligence, 32, 35-66. · Zbl 1314.68309 · doi:10.1023/A:1016740830286
[26] Edgington, D. (1997). Vagueness by degrees. In R. Keefe, & P. Smith (Eds.), Vagueness: A reader, (pp. 294-316).
[27] Égré, P., & Bonnay, D. (2010). Vagueness, uncertainty and degrees of clarity. Synthese, 174(1), 47-78. · Zbl 1198.03009 · doi:10.1007/s11229-009-9684-8
[28] Égré, P. (2009). Soritical series and fisher series. In A. Hieke & H. Leitgeb (Eds.), Reduction: Between the mind and the brain. Frankfurt: Ontos-Verlag. · JFM 37.0944.01
[29] Égré, P. (2011a). Review of “<Emphasis Type=”Italic“>Vagueness and Degrees of Truth” by N. Smith. Australasian Journal of Philosophy, 89(1), 177-180.
[30] Égré, P. (2011b). Perceptual ambiguity and the Sorites. In R. Nouwen, et al. (Eds.), Vagueness in communication. LNAI (Vol. 6517, pp. 64-90). New York: Springer. · Zbl 1311.03012
[31] Fara, D. (2000). Shifting sands: An interest-relative theory of vagueness. Philosophical Topics, 28, 45-81. (Originally published under the name Delia Graff). · Zbl 0311.02011
[32] Faulkner, N. (2010). Wittgenstein’s philosophical grammar: A neglected discussion of vagueness. Philosophical Investigations, 33(2), 159-183. · doi:10.1111/j.1467-9205.2009.01381.x
[33] Fechner, G. (1860). Elements of Psychophysics (tran: Adler H. E., Howes, D. H., & E. G. Boring). Henry Hold editions in Psychology. 1966 (volume 1).
[34] Fine, K. (1975). Vagueness, truth and logic. Synthese, 30(3), 265-300. · Zbl 0311.02011 · doi:10.1007/BF00485047
[35] Frazee, J., & Beaver, D. (2010). Vagueness is rational under uncertainty. In M. Aloni, H. Bastiaanse, T. de Jager, & K. Schulz (Eds.), Logic, language and meaning: 17th Amsterdam Colloquium, Amsterdam, The Netherlands, December 16-18, 2009, Revised Selected Papers Lecture notes in artificial intelligence, (Vol. 6042, pp. 153-162). New York: Springer.
[36] Frege, G. (1879). Begriffsschrift, Halle a. S.: Louis Nebert, (English trans Bauer-Mengelberg S., & van Heijenoort, J). From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Cambridge, MA: Harvard University Press, 1967.
[37] Fischer, H. (2010). A History of the Central Limit Theorem. From classical to modern probability theory. New York: Springer.
[38] Franke, M., Jäger, G., & van Rooij, R. (2010). Vagueness, signaling & bounded rationality. In Proceedings of LENLS.
[39] Fults, S. (2011). Vagueness and scales. In P. Égré & N. Klinedinst (Eds.), Vagueness and language use, (pp. 25-50). Great Britain: Palgrave Macmillan. · Zbl 0184.00903
[40] Galton, F. (1892). Hereditary genius (1st edn). Macmillan: New York.
[41] Gomperz, T. (1905). Greek thinkers: A history of ancient philosophy, (vol. 2), (trans Berry, G. G.). New York: Charles Scribner’s Sons.
[42] Goguen, J. (1969). The logic of inexact concepts. Synthese, 19, 325-373. · Zbl 0184.00903 · doi:10.1007/BF00485654
[43] Goodman, N. (1951). The structure of appearance. New York: Springer.
[44] Gispert, H. (2012). Emile Borel, les multiples formes d’un engagement “politique”. In Mélanges pour Christian Gilain, (pp. 665-676). Presses universitaires de Nancy.
[45] Graham, L., & Kantor, J-M. (2009). Naming infinity. Belknap: Harvard.
[46] Guiraldenq, P. (1999) Émile Borel (1871-1956). L’espace et le temps d’une vie sur deux siècles. (published by the author).
[47] Hajek, P. (1998). Metamathematics of fuzzy logic. New York: Springer. · Zbl 0937.03030 · doi:10.1007/978-94-011-5300-3
[48] Hampton, J. (2007). Typicality, graded membership, and vagueness. Cognitive Science, 31(3), 355-384. · doi:10.1080/15326900701326402
[49] Hardin, C. L. (1988). Phenomenal colors and the sorites. Noûs, 22(2), 213-23. · doi:10.2307/2215860
[50] Hobbs J. (2000). Half orders of magnitude. In L. Obrst, & I. Mani (Eds.), Proceeding of the workshop on semantic approximation, granularity, and vagueness, a workshop of the seventh international conference on principles of knowledge representation and reasoning (KR’2000), Breckenridge, Colorado, April 11, 2000, pp. 28-38.
[51] Horgan, T. (1994). Robust vagueness and the forced-march sorites paradox. Philosophical Perspectives, 8, 159-188. · doi:10.2307/2214169
[52] Kamp, H.; Keenan, E. L. (ed.), Two theories about adjectives (1975), Cambridge · Zbl 0347.02038
[53] Kamp, H., & Partee, B. (1995). Prototype theory and compositionality. Cognition, 57, 129-191. · doi:10.1016/0010-0277(94)00659-9
[54] Keefe, R. (2000). Theories of Vagueness. Cambridge University Press.
[55] Krifka, M. (2007). Approximate interpretation of number words: A case for strategic communication. Cognitive Foundations of Interpretation, 111-126. · Zbl 0319.02016
[56] Lakoff, G. (1973). Hedges: A study in meaning criteria and the logic of fuzzy concepts. Journal of Philosophical Logic, 2(4), 458-508. · Zbl 0272.02047 · doi:10.1007/BF00262952
[57] Lassiter, D. (2011). Vagueness as probabilistic linguistic knowledge. In R. Nouwen et al. (Eds.), Vagueness in communication. LNAI (Vol. 6517, pp. 127-150). New York: Springer. · Zbl 1311.03019
[58] Le Cam, L. (1986). The Central Limit Theorem around 1935. Statistical Science, 1(1), 78-91. · Zbl 0603.60001 · doi:10.1214/ss/1177013818
[59] Lehrer, K., & Wagner, C. (1985). Intransitive indifference: The semi-order problem. Synthese, 65, 249-266. · doi:10.1007/BF00869302
[60] Lévy, P. (1925). Calcul des probabilités. Paris: Gauthier Villars et Compagnie. · JFM 51.0380.02
[61] Lewis, D. (1970). General semantics. Synthese, 22, 18-67. · Zbl 0214.00406 · doi:10.1007/BF00413598
[62] Lipman, B. (2009). Why is our language vague? Department of economics, Boston University. Manuscript available on the author’s webpage.
[63] Luce, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24(2), 178-191. · Zbl 0071.14006 · doi:10.2307/1905751
[64] Luce, R. D. (1959). Individual choice behavior. Wiley. (Reissue Dover Publications 2005). · Zbl 0093.31708
[65] Łukasiewicz, J. (1913). Logical foundations of probability theory. Repr. In Borkowski (Ed.), Jan Łukasiewicz - Selected Works, pp. 16-63, North Holland.
[66] Łukasiewicz, J. (1920). On Three-valued Logic. Ruch filozoficzny, 5, 170-171.
[67] Łukasiewicz, J. (1930). Philosophical remarks on many-valued systems of propositional logic. Repr. In Borkowski (Ed.), Jan Łukasiewicz - Selected Works, (pp. 153-178). North Holland.
[68] Machina, K. (1976). Truth, belief and vagueness. Journal of Philosophical Logic, 5, 47-78, repr. in R. Keefe and P. Smith eds., Vagueness: a Reader, MIT Press 1997. · Zbl 0336.02009
[69] Marbo, C. (1967). À travers deux siècles, souvenirs et rencontres (1883-1967). Paris: Grasset.
[70] McGee, V., & McLaughlin, B. (1995). Distinctions without a difference. In T. Horgan (Ed.), Vagueness, Spindel conference 1994, (vol. 33). Supplement of the Southern Journal of Philosophy, pp. 203-251.
[71] McNicol, D. (1972). A primer of signal detection theory. Lawrence Erlbaum Associates. Reedition 2005. · JFM 37.0944.01
[72] MacFarlane, J.; Dietz, R. (ed.); Moruzzi, S. (ed.), Fuzzy epistemicism, 438-463 (2010), Oxford · doi:10.1093/acprof:oso/9780199570386.003.0026
[73] Moline, J. (1969). Aristotle, Eubulides and the sorites. Mind, 78(311), 393-407. · doi:10.1093/mind/LXXVIII.311.393
[74] Parikh, R. (1994). Vagueness and utility: The semantics of common nouns. Linguistics and Philosophy, 17, 521-535. · doi:10.1007/BF00985317
[75] Pearson, K. (1896). Mathematical contributions to the theory of evolution. Philosophical transactions of the Royal society of London. Series A, Mathematical and physical sciences, 187, 253-318. · JFM 27.0185.01 · doi:10.1098/rsta.1896.0007
[76] Peirce, C. S. (1902). Vague. In J. M. Baldwin (Ed.), Dictionary of philosophy and psychology, (Vol. 2, p. 748). New York: Macmillan.
[77] Peirce, C. S., & Jastrow, J. (1885). On small differences in sensation. Memoirs of the National Academy of Sciences, 3, 73-83.
[78] Peirce, C. S. (1931-1935). In C. Hartshorne, & P. Weiss (Eds.), The collected papers of Charles Sanders Peirce, (vol. I-VI). Cambridge, MA: Harvard University Press.
[79] Poincaré, H. (1893). Le continu mathématique. Revue de métaphysique et de morale, 1, 26-34. Reissued as “La grandeur mathématique et l’expérience” in chap. 2 of La science et l’hypothèse (1902).
[80] Poincaré, H. (1902-1908). The foundations of science. New York: Science Press. · JFM 34.0068.01
[81] Quételet, A. (1849). Letters addressed to H.R.H. the Grand Duke of Saxe Coburg and Gotha on the theory of probabilities, as applied to the moral and political sciences. English edition translated by O. G Downes. London: Charles and Edwin Layton.
[82] Raffman D. (2011). Vagueness and observationality. In G. Ronzitti (Ed.), Vagueness: A guide (pp. 107-121). New York: Springer.
[83] Renouvier, C. (1874/1875). Traité de logique générale et de logique formelle. Seconde édition. Librairie Armand Colin (réimpression de 1912).
[84] Russell, B. (1923). Vagueness. Australasian Journal of Philosophy and Psychology, 1, 84-92. · doi:10.1080/00048402308540623
[85] Russell, B. (1940). An inquiry into meaning and truth. London: Allen Unwin.
[86] Savage, L. J. (1954). Foundations of Statistics, (2nd edn). Dover 1972. · Zbl 0055.12604
[87] Schiffer, S. (2003). The things we mean. Oxford: Clarendon Press. · doi:10.1093/0199257760.001.0001
[88] Simons, P.; Dietz, R. (ed.); Moruzzi, S. (ed.), Supernumeration: Vagueness and numbers, 482-490 (2010), Oxford · doi:10.1093/acprof:oso/9780199570386.003.0028
[89] Solt, S. (2011). Notes on the comparison class. In R. Nouwen et al. (Eds.), Vagueness in communication (pp. 189-206). LNAI 6517, Springer.
[90] Sorensen, R. (1988). Blindspots. Oxford: Clarendon Press.
[91] Sorensen, R. (2001). Vagueness and contradiction. Oxford: Oxford University Press. · Zbl 1136.00006
[92] Smith, N. J. (2008). Vagueness and degrees of truth. Oxford: Oxford University Press. · doi:10.1093/acprof:oso/9780199233007.001.0001
[93] Stevens, S. S. (1946). On the theory of scales of measurement. Science, 103(2684), 677-680. · Zbl 1226.91050
[94] Stigler, S. M. (1986). The history of statistics. The measurement of uncertainty before 1900. Belknap: Harvard. · Zbl 0656.62005
[95] Titchener, E. B. (1905). Experimental psychology. A manual of laboratory practice. vol. 2. Quantitative experiments. Part I. Student’s Manuel. The Macmillan company: New York.
[96] van Deemter, K. (2010). Not exactly: In praise of vagueness. Oxford: Oxford University Press.
[97] Rooij, R.; Ronzitti, G. (ed.), Vagueness in linguistics, 123-170 (2011), New York
[98] Voorhoeve, A., & Binmore, K. (2006). Transitivity, the sorites paradox, and similarity-based decision-making. Erkenntnis, 64(1), 101-114. · Zbl 1098.03524 · doi:10.1007/s10670-005-2373-1
[99] Williamson, T. (1992). Vagueness and ignorance. Proceedings of the Aristotelian Society, Supplementary Volumes, 66, 145-162.
[100] Williamson, T. (1994). Vagueness. London: Routledge.
[101] Wright, C.; Evans, G. (ed.); McDowell, J. (ed.), Language-mastery and the sorites paradox (1976), Oxford
[102] Zadeh, L. A. (1975). Fuzzy logic and approximate reasoning. Synthese, 30(3), 407-428. · Zbl 0319.02016 · doi:10.1007/BF00485052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.