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The Mathematical Intelligencer Flunks the Olympics

Abstract

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, 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). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals.

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Notes

  1. See further on circularity of Sergeyev’s definitions in footnote 5.

  2. The English word pathos is etymologically related to \(\pi \acute{\alpha }\theta o\varsigma \), passion.

  3. See our etymological comment in footnote 2.

  4. For each pair of complementary infinite subsets of \({\mathbb N}\), such a measure m “decides” in a coherent way which one is “negligible” (i.e., of measure 0) and which is “dominant” (measure 1).

  5. This point seems to have escaped Sergeyev, who claims it to be an advantage of the grossone system that the infinite numbers are found within \({\mathbb N}\), allegedly unlike nonstandard analysis; see Calude and Dinneen (2015, p. 95, note 3). Elsewhere Sergeyev claims that, on the contrary, ➀ is “the number of elements in \({\mathbb N}\)”, leading to a circularity already mentioned in footnote 1.

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Acknowledgments

We are grateful to Rob Ely for helpful suggestions. We thank the anonymous referee for Foundations of Science for helpful comments. M. Katz was partially funded by the Israel Science Foundation Grant No. 1517/12.

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Correspondence to Mikhail G. Katz.

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Gutman, A.E., Katz, M.G., Kudryk, T.S. et al. The Mathematical Intelligencer Flunks the Olympics. Found Sci 22, 539–555 (2017). https://doi.org/10.1007/s10699-016-9485-8

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  • DOI: https://doi.org/10.1007/s10699-016-9485-8

Keywords

  • Mathematical Intelligencer
  • Nonstandard Analysis
  • Transfer Principle
  • Conservative Extension
  • Algorithmic Problem