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CHARACTER AND OBJECT

Published online by Cambridge University Press:  20 June 2016

JEREMY AVIGAD and
REBECCA MORRIS
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Abstract

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In 1837, Dirichlet proved that there are infinitely many primes in any arithmetic progression in which the terms do not all share a common factor. Modern presentations of the proof are explicitly of higher-order, in that they involve quantifying over and summing over Dirichlet characters, which are certain types of functions. The notion of a character is only implicit in Dirichlet’s original proof, and the subsequent history shows a very gradual transition to the modern mode of presentation.

In this essay, we study the history of Dirichlet’s theorem with an eye towards understanding the methodological pressures that influenced some of the ontological shifts that occurred in nineteenth century mathematics. In particular, we use the history to understand some of the reasons that functions are treated as ordinary objects in contemporary mathematics, as well as some of the reasons one might want to resist such treatment.

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Type
Research Article
Information
The Review of Symbolic Logic , Volume 9 , Issue 3 , September 2016, pp. 480 - 510
Copyright
Copyright © Association for Symbolic Logic 2016 

References

BIBLIOGRAPHY

Avigad, J. Modularity in mathematics, in preparation.Google Scholar
Avigad, J., & Morris, R. (2014). The of “character” in Dirichlet’s theorem on primes in an arithmetic progression. Archive for History of Exact Sciences, 68(3), 265–326.Google Scholar
Avigad, J., & Morris, R. Character and object (expanded version), unpublished. http://arxiv.org/abs/1505.07238.Google Scholar
Curtis, C. W. (1999). Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. American Mathematical Society and London Mathematical Society, Providence, RI.Google Scholar
de la Vallée Poussin, C. J. (1895–1896). Démonstration simplifiée du théorèm de Dirichlet sur la progression arithmétique. Mémoires couronnés et autres mémoires publiés par L’Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, 53.Google Scholar
Dedekind, R. (1932). In Fricke, R., Noether, E., and Ore, Ö., editors. Gesammelte mathematische Werke, Vols. 1–3. Braunschweig: F. Vieweg & Sohn. Reprinted by Chelsea Publishing Co., New York, 1968.Google Scholar
Dirichlet, J. P. G. L. (1837a). Beweis eines Satzes über die arithmetische Progression. Bericht über die Verhandlungen der königlich Presussischen Akademie der Wissenschaften Berlin. Reprinted in Dirichlet (1889), pp. 309–312.Google Scholar
Dirichlet, J. P. G. L. (1837b). Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält. Abhandlungen der königlich Preussischen Akademie der Wissenschaften, 45–81. Reprinted in Dirichlet (1889), pp. 313–342. Translated by Stefan, Ralf as “There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime.”Google Scholar
Dirichlet, J. P. G. L. (1840). Über eine Eigenschaft der quadratischen Formen. Journal für die reine und angewandte Mathematik, 21, 98–100.Google Scholar
Dirichlet, J. P. G. L. (1841). Untersuchungen über die Theorie der complexen Zahlen. Journal für die reine und angewandte Mathematik, 22, 190–194.Google Scholar
Dirichlet, J. P. G. L. (1863). In Dedekind, R., editor. Vorlsesungen über Zahlentheorie . Braunschweig, Germany: Vieweg. Subsequent editions in 1871, 1879, 1894, with “supplements” by Dedekind, Richard. Translated by John Stillwell, with introductory notes, as Lectures on Number Theory, American Mathematical Society, Providence, RI, 1999.Google Scholar
Dirichlet, J. P. G. L. (1889). In Kronecker, L., editor. Werke. Berlin: Georg Reimer.Google Scholar
Edwards, H. M. (1984). Galois Theory. New York: Springer.Google Scholar
Euler, L. (1748). Introductio in analysin infinitorum, tomus primus. Lausannae. Publications E101 and E102 in the Euler Archive.Google Scholar
Frege, G. (1904). Was ist eine Funktion? In Meyer, S., editor, Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstage. Leipzig: J. A. Barth. Reprinted in Frege (2002) and translated as “What is a function?” In Geach, P. and Black, M., editors, Translations from the Philosophical Writings of Gottlob Frege. Oxford: Oxford University Press, 1980.Google Scholar
Frege, G. (2002). In Textor, M. editor, Funktion – Begriff – Bedeutung. Göttingen: Vandenhoeck and Ruprecht.Google Scholar
Gauss, C. F. (1801). Disquisitiones Arithmeticae. Leipzig: G. Fleischer. Reprinted in Gauss’ Werke, Königlichen Gesellschaft der Wissenschaften, Göttingen, 1863. Translated with a preface by Clarke, Arthur A., Yale University Press, New Haven, 1966, and republished by Springer, New York, 1986.Google Scholar
Gray, J. (1992). The nineteenth-century revolution in mathematical ontology. In Gillies, D., editor, Revolutions in Mathematics. Oxford: Oxford University Press, pp. 226–248.CrossRefGoogle Scholar
Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bulletin de la Société Mathématique de France, 24, 199–220.Google Scholar
Hawkins, T. (1971). The origins of the theory of group characters. Archive for History of Exact Sciences, 7, 142–170.Google Scholar
Kronecker, L. (1870). Auseinandersetzung einiger eigenschaften der klassenzahl idealer complexer zahlen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 881–882. Reproduced in Kronecker (1968), vol. I, pp. 271–282.Google Scholar
Kronecker, L. (1895–1930). In Hensel, K., editor, Werke, vol. 1–5. Leipzig: B. G. Teubner. Reprinted by Chelsea Publishing Co., New York, 1968.Google Scholar
Kronecker, L. (1901). In Hensel, Kurt, editor. Vorlesungen über Zahlentheorie. Leipzig: B. G. Teubner.Google Scholar
Kummer, E. E. (1846). Zur Theorie der complexen Zahlen. Koniglich Akademie der Wissenschaft Berlin, Monatsbericht, 87–97. Also in Journal für die reine und angewandte Mathematik, 35, 319–326, 1847, and in Kummer’s Collected Papers, edited by André Weil, Springer-Verlag, Berlin, 1975, vol. 1, 203–210.Google Scholar
Landau, E. (1909). Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1. Leipzig: B. G. Teubner.Google Scholar
Landau, E. (1927). Vorlesungen über Zahlentheorie. Leipzig: S. Hirzel.Google Scholar
Mach, E. (1960). The Science of Mechanics: A Critical and Historical Account of its Development. Translated by McCormack, T. J., La Salle, Illinois: The Open Court Publishing Co.Google Scholar
Mancosu, P., editor. (2008). The Philosophy of Mathematical Practice. Oxford: Oxford University Press.Google Scholar
Manders, K. The Euclidean diagram. In Mancosu (2008), pp 80–133.Google Scholar
Morris, R. (2011). Character and object. Master’s thesis, Carnegie Mellon University.Google Scholar
Quine, W. V. O. (1948). On what there is. The Review of Metaphysics, 2, 21–38. Reprinted in Quine, W. V. O. (1980) From a Logical Point of View. Cambridge: Harvard University Press.Google Scholar
Quine, W. V. O. (1969). Ontological Relativity, and Other Essays. New York: Columbia University Press.Google Scholar
Tignol, J.-P. (2001). Galois’ Theory of Algebraic Equations. New Jersey: World Scientific.Google Scholar
Urquhart, A. Mathematics and physics: strategies of assimilation. In Mancosu (2008), pp. 417–440.Google Scholar
Weber, H. (1882). Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist. Mathematische Annalen, 20, 301–329.Google Scholar
Wilson, M. (1994). Can we trust logical form? The Journal of Philosophy, 91, 519–544.Google Scholar
Wittgenstein, L. (1989). Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge, 1939. Chicago: University of Chicago Press.Google Scholar