×

The Mathematical Intelligencer flunks the Olympics. (English) Zbl 1392.03061

Summary: The Mathematical Intelligencer recently published a note by Y. D. Sergeyev [Math. Intell. 37, No. 2, 4–8 (2015; Zbl 1329.90074)] that challenges both mathematics and intelligence. We examine Sergeyev’s claims [loc. cit.] concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson [Nederl. Akad. Wet., Proc., Ser. A 64, 432–440 (1961; Zbl 0102.00708)], and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). L. H. Kauffman, who published an article on a grossone [Appl. Math. Comput. 255, 25–35 (2015; Zbl 1338.68083)], places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals.

MathOverflow Questions:

What is... A Grossone?

MSC:

03H05 Nonstandard models in mathematics
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
90B50 Management decision making, including multiple objectives
90B90 Case-oriented studies in operations research

Software:

MathOverflow
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Avigad, J. (2005). Weak theories of nonstandard arithmetic and analysis. Reverse mathematics, Lecture notes in logic 21. La Jolla, CA: Association for Symbolic Logic.
[2] Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Schaps, D., Sherry, D., & Shnider, S. (2013). Is mathematical history written by the victors? Notices of the American Mathematical Society, 60(7), 886-904. http://www.ams.org/notices/201307/rnoti-p886 and arXiv:1306.5973 · Zbl 1334.01010
[3] Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., Nowik, T., Sherry, D., & Shnider, S. (2014). Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow. Notices of the American Mathematical Society, 61(8), 848-864. http://www.ams.org/notices/201408/rnoti-p848 and arxiv:1407.0233. · Zbl 1338.26001
[4] Benci, V; Nasso, M, Numerosities of labelled sets: A new way of counting, Advances in Mathematics, 173, 50-67, (2003) · Zbl 1028.03042 · doi:10.1016/S0001-8708(02)00012-9
[5] Borovik, A; Katz, M, Who gave you the Cauchy-Weierstrass tale? the dual history of rigorous calculus, Foundations of Science, 17, 245-276, (2012) · Zbl 1279.01017 · doi:10.1007/s10699-011-9235-x
[6] Bradley, R., & Sandifer, C. (2009). Cauchy’s Cours d’analyse. An annotated translation. Sources and studies in the history of mathematics and physical sciences. New York: Springer. · Zbl 1189.26001
[7] Calude, C., & Dinneen, M. (Eds). (2015). Unconventional Computation and Natural Computation. In 14th International Conference, UCNC 2015, Auckland, New Zealand August 30-September 3, 2015, Proceedings, Springer. · Zbl 1318.68012
[8] Dauben, J; Guicciardini, N; Lewis, A; Parshall, K; Rice, A, Ivor grattan-guinness (June 23, 1941-December 12, 2014), Historia Mathematica, 42, 385-406, (2015) · Zbl 1327.01037 · doi:10.1016/j.hm.2015.09.002
[9] Day, P. (2006). Review of “Sergeyev, Yaroslav D. ‘Mathematical foundations of the infinity computer’. Annales Universitatis Mariae Curie-Sklodowska. Sectio AI. Informatica, \(4\), 20-33”. http://www.ams.org/mathscinet-getitem?mr=2325643.
[10] Goedel, K, The consistency of the axiom of choice and of the generalized continuum-hypothesis, Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences), 24, 556-557, (1938) · Zbl 0020.29701 · doi:10.1073/pnas.24.12.556
[11] Gutman, A; Kutateladze, S, On the theory of the grossone [(Russian) sibirskii matematicheskii zhurnal 49(5), 1054-1063; translation in], Siberian Mathematical Journal, 49, 835-841, (2008) · Zbl 1224.03045 · doi:10.1007/s11202-008-0082-0
[12] Hamkins, J. D. (2015). http://mathoverflow.net/questions/226277/what-is-a-grossone.
[13] Henson, CW; Kaufmann, M; Keisler, HJ, The strength of nonstandard methods in arithmetic, Journal of Symbolic Logic, 49, 1039-1058, (1984) · Zbl 0587.03048 · doi:10.2307/2274260
[14] Henson, CW; Keisler, HJ, On the strength of nonstandard analysis, Journal of Symbolic Logic, 51, 377-386, (1986) · Zbl 0624.03051 · doi:10.1017/S0022481200031248
[15] Hewitt, E, Rings of real-valued continuous functions. I, Transactions of the American Mathematical Society, 64, 45-99, (1948) · Zbl 0032.28603 · doi:10.1090/S0002-9947-1948-0026239-9
[16] Iudin, D; Sergeyev, Y; Hayakawa, M, Interpretation of percolation in terms of infinity computations, Applied Mathematics and Computation, 218, 8099-8111, (2012) · Zbl 1252.82059 · doi:10.1016/j.amc.2011.11.044
[17] Kanovei, V., Katz, K., Katz, M., & Schaps, M. (2015). Proofs and retributions, or: Why Sarah can’t take limits. Foundations of Science, 20(1), 1-25. doi:10.1007/s10699-013-9340-0 and http://www.ams.org/mathscinet-getitem?mr=3312498. · Zbl 1368.00021
[18] Kanovei, V., Katz, K., Katz, M., & Sherry, D. (2015). Euler’s lute and Edwards’ oud. The Mathematical Intelligencer, 37(4), 48-51. doi:10.1007/s00283-015-9565-6 and arxiv:1506.02586. · Zbl 1342.01017
[19] Kanovei, V; Katz, M; Mormann, T, Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics, Foundations of Science, 18, 259-296, (2013) · Zbl 1392.03012 · doi:10.1007/s10699-012-9316-5
[20] Katz, K., & Katz, M. (2011). Meaning in classical mathematics: Is it at odds with Intuitionism? Intellectica56(2), 223-302. arxiv:1110.5456.
[21] Katz, K., & Katz, M. (2012). A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Foundations of Science, 17(1), 51-89. doi:10.1007/s10699-011-9223-1 and arxiv:1104.0375. · Zbl 1283.03006
[22] Katz, M., & Kutateladze, S. (2015). Edward Nelson (1932-2014). The Review of Symbolic Logic, \(8\)(3), 607-610. doi:10.1017/S1755020315000015 and arxiv:1506.01570. · Zbl 1323.01037
[23] Katz, M., & Leichtnam, E. (2013). Commuting and noncommuting infinitesimals. American Mathematical Monthly, 120(7), 631-641. doi:10.4169/amer.math.monthly.120.07.631 and arxiv:1304.0583. · Zbl 1280.26047
[24] Katz, M., & Sherry, D. (2012). Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), 1550-1558. http://www.ams.org/notices/201211/rtx121101550p and arxiv:1211.7188. · Zbl 1284.03064
[25] Katz, M., & Sherry, D. (2013). Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis, 78(3), 571-625. doi:10.1007/s10670-012-9370-y and arxiv:1205.0174. · Zbl 1303.01012
[26] Kauffman, L., & Lins, S. (1994). Temperley-Lieb recoupling theory and invariants of 3-manifolds. Annals of mathematics studies (Vol. 134). Princeton, NJ: Princeton University Press. · Zbl 0821.57003
[27] Kauffman, L, Infinite computations and the generic finite, Applied Mathematics and Computation, 255, 25-35, (2015) · Zbl 1338.68083 · doi:10.1016/j.amc.2014.06.054
[28] Kauffman, L. (2015b). MathOverflow answer. http://mathoverflow.net/questions/226277/what-is-a-grossone.
[29] Keisler, H. J. (1986). Elementary calculus: An infinitesimal approach (2nd ed.). Prindle, Weber & Schimidt, Boston. http://www.math.wisc.edu/ keisler/calc.html. · Zbl 0655.26002
[30] Kreinovich, V. (2003). Review of “Sergeyev, Yaroslav D. Arithmetic of infinity. Edizioni Orizzonti Meridionali, Cosenza, 2003”. http://www.ams.org/mathscinet-getitem?mr=2050876. · Zbl 1076.03048
[31] Kreisel, G. (1969). Axiomatizations of nonstandard analysis that are conservative extensions of formal systems for classical standard analysis. Applications of model theory to algebra, analysis, and probability (International Symposium, Pasadena, CA, 1967) (pp. 93-106) Holt, Rinehart and Winston, New York. · Zbl 0188.32203
[32] Kutateladze, S. (2011). Letter to the Editor. On the Grossone and the infinity computer. Newsletter of the European Mathematical Society, 79, 60. https://www.ems-ph.org/journals/newsletter/pdf/2011-03-79.
[33] Lolli, G, Metamathematical investigations on the theory of grossone, Applied Mathematics and Computation, 255, 3-14, (2015) · Zbl 1338.03118 · doi:10.1016/j.amc.2014.03.140
[34] Łoś, J. (1955). Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. Mathematical interpretation of formal systems (pp. 98-113). Amsterdam: North-Holland Publishing Co. · Zbl 0068.24401
[35] Nelson, E, Internal set theory: A new approach to nonstandard analysis, Bulletin of the American Mathematical Society, 83, 1165-1198, (1977) · Zbl 0373.02040 · doi:10.1090/S0002-9904-1977-14398-X
[36] Robinson, A. (1961). Non-standard analysis. Nederl Akad Wetensch Proceedings Series A 64 = Indagationes Mathematicae, 23, 432-440 [reprinted in Selected Works, see item (Robinson 1979), pp. 3-11] · Zbl 0102.00708
[37] Lightstone, A., & Robinson, A. (1975). Nonarchimedean fields and asymptotic expansions. North-Holland Mathematical Library (Vol. 13). Amsterdam-Oxford: North-Holland Publishing Co., New York: American Elsevier Publishing Co., Inc. · Zbl 0303.26013
[38] Robinson, A. (1979). Selected papers of Abraham Robinson. Vol. II. Nonstandard analysis and philosophy. Edited and with introductions by W. A. J. Luxemburg and S. Körner. Yale University Press, New Haven, CT. · Zbl 1338.03118
[39] Sergeyev, Y. (2003). Arithmetic of infinity. Cosenza: Edizioni Orizzonti Meridionali. · Zbl 1076.03048
[40] Sergeyev, Y, Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers, Chaos, Solitons and Fractals, 33, 50-75, (2007) · doi:10.1016/j.chaos.2006.11.001
[41] Sergeyev, Y, Solving ordinary differential equations on the infinity computer by working with infinitesimals numerically, Applied Mathematics and Computation, 219, 10668-10681, (2013) · Zbl 1303.65061 · doi:10.1016/j.amc.2013.04.019
[42] Sergeyev, Y, The olympic medals ranks, lexicographic ordering, and numerical infinities, The Mathematical Intelligencer, 37, 4-8, (2015) · Zbl 1329.90074 · doi:10.1007/s00283-014-9511-z
[43] Sergeyev, Y. (2015b). Keynote address, Las Vegas. http://www.world-academy-of-science.org/worldcomp15/ws/keynotes/keynote_sergeyev. · Zbl 1338.68083
[44] Sergeyev, Y, Letter to the editor, The Mathematical Intelligencer, 37, 2-3, (2015) · doi:10.1007/s00283-015-9600-7
[45] Sergeyev, Y, The exact (up to infinitesimals) infinite perimeter of the Koch Snowflake and its finite area, Communications in Nonlinear Science and Numerical Simulation, 31, 21-29, (2016) · Zbl 1467.28011 · doi:10.1016/j.cnsns.2015.07.004
[46] Shamseddine, K, Analysis on the Levi-Civita field and computational applications, Applied Mathematics and Computation, 255, 44-57, (2015) · Zbl 1356.46062 · doi:10.1016/j.amc.2014.04.108
[47] Skolem, T. (1933). Über die Unmöglichkeit einer vollständigen Charakterisierung der Zahlenreihe mittels eines endlichen Axiomensystems. Norsk Mat. Forenings Skr., II. Series No. 1/12, pp. 73-82. · Zbl 0007.19305
[48] Skolem, T, Über die nicht-charakterisierbarkeit der zahlenreihe mittels endlich oder abzählbar unendlich vieler aussagen mit ausschliesslich zahlenvariablen, Fundamenta Mathematicae, 23, 150-161, (1934) · JFM 60.0025.02 · doi:10.4064/fm-23-1-150-161
[49] Skolem, T. (1955). Peano’s axioms and models of arithmetic. Mathematical interpretation of formal systems (pp. 1-14). Amsterdam: North-Holland Publishing. · Zbl 0068.24603
[50] Tall, D, The calculus of Leibniz-an alternative modern approach, Mathematical Intelligencer, 2, 54-55, (1979) · Zbl 0499.26009 · doi:10.1007/BF03024388
[51] Tao, T. (2014). Hilbert’s fifth problem and related topics. Graduate studies in mathematics (Vol. 153). Providence, RI: American Mathematical Society. · Zbl 1298.22001 · doi:10.1090/gsm/153
[52] Tarski, A, Une contribution à la théorie de la mesure, Fundamenta Mathematicae, 15, 42-50, (1930) · JFM 56.0089.05 · doi:10.4064/fm-15-1-42-50
[53] Vakil, N. (2012). Interpreting Sergeyev’s numerical methodology within a hyperreal number system. http://vixra.org/abs/1209.0070. · Zbl 0587.03048
[54] Zlatoš, P. (2009). Review of (Gutman & Kutateladze 2008). http://www.ams.org/mathscinet-getitem?mr=2469053. · Zbl 1028.03042
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.