Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-19T02:11:32.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Who were the American postulate theorists?

Published online by Cambridge University Press:  12 March 2014

Michael Scanlan
Michael Scanlan*
Affiliation:
Philosophy Department, Oregon State University, Corvallis, Oregon 97331
Get access Rights & Permissions [Opens in a new window]

Abstract

Articles by two American mathematicians, E. V. Huntington and Oswald Veblen, are discussed as examples of a movement in foundational research in the period 1900–1930 called American postulate theory. This movement also included E. H. Moore, R. L. Moore, C. H. Langford, H. M. Sheffer, C. J. Keyser, and others. The articles discussed exemplify American postulate theorists' standards for axiomatizations of mathematical theories, and their investigations of such axiomatizations with respect to metatheoretic properties such as independence, completeness, and consistency.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 981 - 1002
Copyright
Copyright © Association for Symbolic Logic 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

I appreciate comments and corrections on a draft of this paper from William Aspray, John Corcoran, David McCarty, Gregory Moore, Woosuk Park, and George Weaver. I also appreciate a Travel to Collections grant from the National Endowment for the Humanities, which enabled me to consult the Veblen papers in the Library of Congress.

References

REFERENCES

Aspray, W. [1988] The emergence of Princeton as a world center for mathematical research, 1896–1939, History and philosophy of modern mathematics (Aspray, W. and Kitcher, P., editors), Minnesota Studies in the Philosophy of Science, Vol. XI, University of Minnesota Press, Minneapolis, Minnesota, pp. 346–366.Google Scholar
Aspray, W. [1991] Oswald Veblen and the origins of mathematical logic at Princeton, Perspectives on the history of mathematical logic (Drucker, Thomas, editor), Birkhäuser, Boston, Massachusetts, pp. 54–70.Google Scholar
Baldus, R. [1928] Zur Axiomatik der Geometrie. I: Über Hilberts Vollständigkeitsaxiom, Mathematische Annalen, vol. 100, pp. 321–333.CrossRefGoogle Scholar
Bernays, P. [1967] Hilbert, David, The encyclopedia of philosophy. Vol. 3, Macmillan, New York, pp. 499–500.Google Scholar
Bliss, G. A. and Dickson, L. E. [1936] Biographical memoir of Eliakim Hastings Moore, Biographical Memoirs of the National Academy of Sciences, vol. 17, pp. 83–102.Google Scholar
Calinger, R. S. [1972] Moore, Eliakim Hastings, Dictionary of Scientific Biography, vol. 9, pp. 501–503.Google Scholar
Carnap, R. and Bachmann, F. [1936] Über Extremalaxiome, Erkenntnis, vol. 6, pp. 166–188; English translation (by H. G. Bohnert), On extremal axioms, History and Philosophy of Logic , vol. 2 (1981), pp. 67–85.CrossRefGoogle Scholar
Church, A. [1927] Alternatives to Zermelo's assumption, Transactions of the American Mathematical Society, vol. 29, pp. 178–208.Google Scholar
Corcoran, J. [1980a] Categoricity, History and Philosophy of Logic, vol. 1, pp. 187–207.CrossRefGoogle Scholar
Corcoran, J. [1980b] On definitional equivalence and related topics, History and Philosophy of Logic, vol. 1, pp. 231–234.Google Scholar
Corcoran, J. [1981] From categoricity to completeness, History and Philosophy of Logic, vol. 3, pp. 113–119.Google Scholar
Corcoran, J. [1983] Editor's Introduction to the second edition of Tarski [1956].Google Scholar
Corcoran, J. [1986] Undefinability tests and the Erlanger Programm: historical footnotes, Abstracts, International Congress of Mathematicians, Berkeley, California, p. 339.Google Scholar
Dreben, B. and van Heijenoort, J. [1986] Introductory remarks to Gödel's dissertation, in Gödel, K., Collected works, Vol. 1, Oxford University Press, New York, pp. 44–59.Google Scholar
Freudenthal, H. [1957] Zur Geschichte der Grundlagen der Geometrie, Nieuw Archief voor Wiskunde, ser. 3, vol. 5, pp. 105–142.Google Scholar
Hilbert, D. [1899] Grundlagen der Geometrie, Festschrift zur Feier der Enthüllung des Gauss-Weber-Denkmals in Göttingen, Teubner, Leipzig (plus many later editions as a separate book); English translation (by E. J. Townsend), The foundations of geometry , Open Court, LaSalle, Illinois, 1902. Page citations are to the latter.Google Scholar
Freudenthal, H. [1990a] Mathematische Probleme, Vortrag, gehalten auf dem internationalen Mathematiker-Kongress zur Paris 1900, Archiv der Mathematik und Physik, ser. 3, vol. 1 (1901), pp. 44–63, 213–237; English translation (by M. W. Newson), Mathematical problems, Bulletin of the American Mathematical Society , vol. 8 (1902), pp. 437–479.Google Scholar
Freudenthal, H. [1900b] Les principes fondamentaux de la géométrie (translated by L. Laugel), Gauthier-Villars, Paris.Google Scholar
Hilbert, D. and Ackermann, W. [1928] Grundzüge der theoretischen Logik, Springer, Berlin.Google Scholar
Holder, E. [1972] Hölder, Otto Ludwig, Dictionary of Scientific Biography, vol. 6, pp. 472–474.Google Scholar
Hölder, O. [1901] Die Axiome der Quantität und die Lehre vom Mass, Berichte über die Verhandlungen der Königlich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Classe, vol. 53, pp. 1–64.Google Scholar
Huntington, E. V. [1902] A complete set of postulates for the theory of absolute continuous magnitude, Transactions of the American Mathematical Society, vol. 3, pp. 264–279.CrossRefGoogle Scholar
Huntington, E. V. [1905a] A set of postulates for real algebra comprising postulates for a one-dimensional continuum and for the theory of groups, Transactions of the American Mathematical Society, vol. 6, pp. 17–41.CrossRefGoogle Scholar
Huntington, E. V. [1905b] A set of postulates for ordinary complex algebra, Transactions of the American Mathematical Society, vol. 6, pp. 209–229.CrossRefGoogle Scholar
Huntington, E. V. [1913] A set of postulates for abstract geometry, expressed in terms of the simple relation of inclusion, Mathematische Annalen, vol. 73, pp. 522–559.CrossRefGoogle Scholar
Kleene, S. C. [1987] Kurt Gödel, Biographical Memoirs of the National Academy of Sciences, vol. 56, pp. 135–178.Google Scholar
Lane, S. Mac [1974] Veblen, Oswald, Dictionary of Scientific Biography, vol. 13, p. 599–600.Google Scholar
Moore, E. H. [1902] On the projective axioms of geometry, Transactions of the American Mathematical Society, vol. 3, pp. 147–158.CrossRefGoogle Scholar
Moore, G. [1988] The emergence of first-order logic, History and philosophy of modern mathematics (Aspray, W. and Kitcher, P., editors), Minnesota Studies in the Philosophy of Science, Vol. XI, University of Minnesota Press, Minneapolis, Minnesota, pp. 95–135.Google Scholar
Padoa, A. [1900a] Essai d'une théorie algébrique des nombres entiers, précédé d'une introduction logique à une théorie déductive quelconque. Premier congrès international de philosophie, 1900, Vol. 3 Armand Colin, Paris, 1901, pp. 309–365. Cited from partial translation in van Heijenoort [1967], pp. 118–123.Google Scholar
Padoa, A. [1900b] Un nouveau système irréductible de postulats pour l'algèbre, Compte rendu du deuxième congrès international des mathématiciens, Gauthier-Villars, Paris, 1902, pp. 249–256.Google Scholar
Padoa, A. [1903] Le probleme n. 2 de M. David Hilbert, L'Enseignement Mathématique, vol. 5, pp. 85–91.Google Scholar
Parshall, K. H. [1984] Eliakim Hastings Moore and the founding of a mathematical community in America, 1892–1902, Annals of Science, vol. 41, pp. 313–333.CrossRefGoogle Scholar
Peano, G. [1900a] Les définitions mathématiques, Premier congrès international de philosophie, 1900, Vol. 3, Armand Colin, Paris, 1901, pp. 279–288.Google Scholar
Peano, G. [1900b] Remark on Hilbert's talk, reported in Compte rendu du deuxième congrès international des mathématiciens, Gauthier-Villars, Paris, 1902, p. 21.Google Scholar
Quine, W. V. [1934] A method of generating part of arithmetic without use of intuitive logic, Bulletin of the American Mathematical Society, vol. 40, pp. 753–761; reprinted in his Selected logic papers , Random House, New York, 1966, pp. 52–60. Citation is to the reprint pagination.CrossRefGoogle Scholar
Quine, W. V. [1936] Truth by convention, Philosophical essays for A. N. Whitehead, Longmans, New York, pp. 90–124.Google Scholar
Quine, W. V. [1961] Reply to Professor Marcus, Synthèse, vol. 13, pp. 323–330.CrossRefGoogle Scholar
Quine, W. V. [1985] The time of my life, MIT Press, Cambridge, Massachusetts.Google Scholar
Quine, W. V. [1986] Autobiography of W. V. Quine, The philosophy of W. V. Quine (Hahn, L. E. and Schilpp, P. A., editors), Open Court, LaSalle, Illinois, pp. 3–46.Google Scholar
Scanlan, M. [1983] Veblen and the definability of congruence, this Journal, vol. 48, pp. 906–907. (abstract)Google Scholar
Skolem, T. [1928] Über die mathematische Logik, Norsk Matematisk Tidsskrift, vol. 10, pp. 125–142; English translation in van Heijenoort [1967], pp. 508–524.Google Scholar
Sommer, J. [1900] Hilbert's foundations of geometry, Bulletin of the American Mathematical Society, vol. 6, pp. 287–299.CrossRefGoogle Scholar
Tarski, A. [1924] Sur les principes de l'arithmétique des nombres ordinaux (transfinis), Annales de la Société Polonaise de Mathématique, vol. 3, pp. 148–149; reprinted in Tarski [1986], Vol. 4, pp. 533–534.Google Scholar
Tarski, A. [1934] Some methodological investigations on the definability of concepts, Przeglad Filozoficzny, vol. 37, pp. 438–460 (Polish); German translation, Erkenntnis , vol. 5, pp. 80–100; revised English translation, Tarski [1956], pp. 296–319.Google Scholar
Tarski, A. [1935] Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1, pp. 261–405; English translation, Tarski [1956], pp. 152–278.Google Scholar
Tarski, A. [1936] Über den Begriff der logischen Folgerung, Actes du Congrès International de Philosophie Scientifique. Vol. 7, Actualités Scientifiques et Industrielles, vol. 394, Hermann, Paris, 1936, pp. 1–11. English translation, Tarski [1956], pp. 409–420.Google Scholar
Tarski, A. [1956] Logic, semantics, metamathematics: papers from 1923 to 1938, Clarendon Press, Oxford; second revised edition, with editor's introduction and an analytic index (J. Corcoran, editor), Hackett Publishing Company, Indianapolis, Indiana, 1983.Google Scholar
Tarski, A. [1986] Collected papers (Givant, S. R. and McKenzie, R. N., editors), Vols. 1-4, Birkhäuser, Basel.Google Scholar
Tarski, A. and Lindenbaum, A. [1926] Sur l'indépendance des notions primitives dans les systèmes mathématiques, Annales de la Société Polonaise de Mathématique, vol. 5, pp. 111–113. (abstract)Google Scholar
van Heuenoort, J. (editor) [1967] From Frege to Gödel: a source book in mathematical logic, Harvard University Press, Cambridge, Massachusetts.Google Scholar
Veblen, O. [1903] Review of Hilbert [1899], The Monist, vol. 13, pp. 303–309.CrossRefGoogle Scholar
Veblen, O. [1904] A system of axioms for geometry, Transactions of the American Mathematical Society, vol. 5, pp. 343–384.CrossRefGoogle Scholar
Veblen, O. [1905] Euclid's parallel postulate, Open Court, vol. 19, pp. 752–755.Google Scholar
Veblen, O. [1906] The foundations of geometry, Popular Science Monthly, vol. 68 (01 1906), pp. 21–28.Google Scholar
Veblen, O. [1925] Remarks on the foundations of geometry, Bulletin of the American Mathematical Society, vol. 31, pp. 121–141.CrossRefGoogle Scholar
Veblen, O. and Whitehead, J. H. C. [1932] Foundations of differential geometry, Cambridge University Press, Cambridge.Google Scholar
Veblen, O. and Young, J. W. [1910] Projective geometry. Vol. 1, Ginn, Boston, Massachusetts.Google Scholar
Veblen, O. and Young, J. W. [1918] Projective geometry. Vol. 2, Ginn, Boston, Massachusetts.Google Scholar
Wiener, N. [1953] Ex-prodigy, Simon and Schuster, New York; 2nd ed., MIT Press, Cambridge, Massachusetts, 1964. Page citation is to the latter.Google Scholar
Young, J. W. [1911] Lectures on fundamental concepts of algebra and geometry, Macmillan, New York.Google Scholar