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THE ARITHMETIC OF THE EVEN AND THE ODD

Published online by Cambridge University Press:  18 January 2016

VICTOR PAMBUCCIAN
VICTOR PAMBUCCIAN*
Affiliation:
School of Mathematical and Natural Sciences, Arizona State University
*
*SCHOOL OF MATHEMATICAL AND NATURAL SCIENCES ARIZONA STATE UNIVERSITY - WEST CAMPUS P. O. BOX 37100, PHOENIX, AZ 85069-7100 E-mail: pamb@asu.edu
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Abstract

We present several formal theories for the arithmetic of the even and the odd, show that the irrationality of $\sqrt 2$ can be proved in one of them, that the proof must involve contradiction, and prove that the irrationality of $\sqrt {17}$ cannot be proved inside any formal theory of the even and the odd.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 2 , June 2016 , pp. 359 - 369
Copyright
Copyright © Association for Symbolic Logic 2016 

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References

BIBLIOGRAPHY

Artmann, B. (1994). A proof for Theodorus’ theorem by drawing diagrams. Journal of Geometry, 49 3–35.Google Scholar
Becker, O. (1936). Die Lehre von Geraden und Ungeraden im neunten Buch der euklidischen Elemente. Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien, 3, 533–553.Google Scholar
Becker, O. (1954). Grundlagen der Mathematik in geschichtlicher Entwicklung. Freiburg/München: Verlag Karl Alber.Google Scholar
Becker, O. (1957). Das mathematische Denken der Antike. Göttingen: Vandenhoeck und Ruprecht.Google Scholar
Burnyeat, M. F. (1978). The philosophical sense of Theaetetus’ mathematics. Isis, 69, no. 249, 489–513.Google Scholar
Burnyeat, M. F. (2005). Archytas and optics. Science in Context, 18, no. 249, 35–53.Google Scholar
Casas, J. M., Omirov, B. A., & Rozikov, U. A. (2014). Solvability criteria for the equation x q = a in the field of p-adic numbers. Bulletin of the Malaysian Mathematical Sciences Society. Second Series, 37, no. 3, 853–863.Google Scholar
Caveing, M. (1996). The debate between H. G. Zeuthen and H. Vogt (1909-1915) on the historical source of the knowledge of irrational quantities. Centaurus, 38, 277–292.Google Scholar
Conrad, K. (2015). Hensel’s lemma. http://www.math.uconn.edu/∼kconrad/blurbs/gradnumthy/hensel.pdf.Google Scholar
Gardies, J.-L. (1998). Sur l’axiomatique de l’arithmétique euclidienne. Oriens Occidens, 2, 125–140.Google Scholar
Heller, S. (1956). Ein Beitrag zur Deutung der Theodoros-Stelle in Platons Dialog, “Theaetet”. Centaurus 5, 1–58.Google Scholar
Huffman, C. A. (1993). Philolaus of Croton: Pythagorean and Presocratic. Cambridge: Cambridge University Press.Google Scholar
Huffman, C. A. (2005). Archytas of Tarentum: Pythagorean, philosopher and mathematician king. Cambridge: Cambridge University Press.Google Scholar
Itard, J. (1961). Livres arithmétiques d’Euclide. Histoire de la Pensée, X. Paris: Hermann.Google Scholar
Jeřábek, E. (2012). Sequence encoding without induction. Mathematical Logic Quarterly, 58, 244–248.Google Scholar
Kahane, J.-P. (1984). Platon et mathématique. Bulletin de la Société Franco-Japonaise des Sciences Pures et Appliquées. 40, 16–30.Google Scholar
Kaye, R. (1991). Models of Peano Arithmetic. Oxford: Oxford University Press.Google Scholar
Knorr, W. (1975). The evolution of the Euclidean Elements: A study of the theory of incommensurable magnitudes and its significance for early Greek geometry. Synthese Historical Library, 15. Dordrecht: D. Reidel.Google Scholar
Knorr, W. (1979). Methodology, philology, and philosophy. With a reply by M. F. Burnyeat. Isis, 70, no. 254, 565–570.Google Scholar
Lučič, Z. (2015). Irrationality of the Square Root of 2: The Early Pythagorean Proof, Theodorus’s and Theaetetus’s Generalizations. The Mathematical Intelligencer, 37, no. 3, 26–32.Google Scholar
Malmendier, N. (1975). Eine Axiomatik zum 7. Buch der Elemente von Euklid. Mathematisch-physikalische Semesterberichte, 22, 240–254.Google Scholar
McCabe, R. L. (1976). Theodorus’ irrationality proofs. Mathematics Magazine, 49, 201–203.Google Scholar
Mueller, I. (1997). Greek arithmetic, geometry and harmonics: Thales to Plato. In: Taylor, C. C. W., editor. Routledge History of Philosophy, Vol. 1: From the Beginning to Plato, Chapter 8, pp. 271–322, London: Routledge.Google Scholar
Ofman, S. (2010). Une nouvelle démonstration de l’irrationalité de racine carrée de 2 d’après les Analytiques d’Aristote. In: Laks, A., Narcy, M. editors, Philosophie et mathématiques, Philosophie Antique, 10, 81–138. Villeneuve d’Ascq: Presses Universitaires du Septentrion.Google Scholar
Ofman, S. (2014). Comprendre les mathématiques pour comprendre Platon — Théétète (147d–148b). Lato Sensu, 1, 71–80.Google Scholar
Pambuccian, V. (2016). A problem in Pythagorean arithmetic. Notre Dame Journal of Formal Logic, to appear.Google Scholar
Reidemeister, K. (1949). Das exakte Denken der Griechen. Hamburg: Claassen & Goverts.Google Scholar
Salzmann, H., Grundhöfer, T., Hähl, H., & Löwen, R. (2007). The classical fields. Cambridge: Cambridge University Press.Google Scholar
van der Waerden, B. L. (1948). Die Arithmetik der Pythagoreer. I. Mathematische Annalen 120, 127–153.Google Scholar
van der Waerden, B. L. (1949). Die Arithmetik der Pythagoreer. II. Die Theorie des Irrationalen. Mathematische Annalen 120, 676–700.Google Scholar
Vandoulakis, I. M. (1998). Was Euclid’s approach to arithmetic axiomatic? Oriens Occidens 2, 141–181.Google Scholar
Waterhouse, W. C. (1978). Review of McCabe (1976). Mathmetical Reviews, 54, 4893.Google Scholar
Zeeman, E. C. (2008). What’s wrong with Euclid Book V. Bulletin of the London Mathematical Society, 40, 1–17.Google Scholar
Zeuthen, H. G. (1910). Sur la constitution des livres arithmétiques des Éléments d’Euclide et leur rapport à la question de I’irrationalité. Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger, 395–435.Google Scholar
Zhmud, L. (2012). Pythagoras and the Early Pythagoreans. Oxford: Oxford University Press.Google Scholar