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Linearity and Reflexivity in the Growth of Mathematical Knowledge

Published online by Cambridge University Press:  26 September 2008

Leo Corry
Leo Corry
Affiliation:
The Institute for the History and Philosophy of Science and IdeasTel Aviv University
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Abstract

Recent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of the body and images of mathematics.

Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those of any other exact science. In no other exact science can the standard methodological framework used within the discipline also be used to study the nature of the discipline itself.

Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics.

Type
Research Article
Information
Science in Context , Volume 3 , Issue 2 , Autumn 1989 , pp. 409 - 440
Copyright
Copyright © Cambridge University Press 1989

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References

Agassi, J. 1975. Science in Flux. Dordrecht: Reidel.CrossRefGoogle Scholar
Albers, D. J., and Alexanderson, G. L., eds. 1985. Mathematical People. Boston: Birkhauser.Google Scholar
Askey, R. 1988. “How Can Mathematicians and Mathematical Historians Help Each Other?” In Aspray and Kitcher 1988, 201–17.Google Scholar
Aspray, W., and Kitcher, P., eds. 1988. History and Philosophy of Modern Mathematics. Minnesota Studies in the Philosophy of Science, Vol.11. Minneapolis: University of Minnesota Press.Google Scholar
Bishop, E. 1975. “The Crisis in Contemporary Mathematics.” Historia Mathematica 2:507–17.CrossRefGoogle Scholar
Bourbaki, N. 1960. Eléments d'Histoire des Mathématiques. Paris: Hermann.Google Scholar
Bourbaki, N. 1968. Theory of Sets. Paris: Hermann.Google Scholar
Bunge, M. 1983. Treatise on Basic Philosophy. Vol.6: Understanding the World. Dordrecht: Reidel.Google Scholar
Bunge, M. 1985. Seudociencia e Ideología. Madrid: Alianza Universidad.Google Scholar
Crowe, M. 1975. “Ten ‘Laws’ Concerning Patterns of Change in the History of Mathematics.” Historia Mathematica 2:161–66.CrossRefGoogle Scholar
Crowe, M. 1976. “Ten ‘Laws’ Concerning Patterns of Change in the History of Mathematics.” Historia Mathematica 3:469–70.Google Scholar
Crowe, M. 1988. “Ten Misconceptions about Mathematics and Its History.” In Aspray and Kitcher 1988, 260–92.Google Scholar
Dauben, J. W. 1988. “Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics.” In Aspray and Kitcher 1988, 177–200.Google Scholar
Davis, P. J. 1972. “Fidelity in Mathematical Discourse.” American Mathematical Monthly 79:252–62. Reprinted in Tymoczko 1985.CrossRefGoogle Scholar
Davis, P. J. and Hersh, R.. 1986. Descartes' Dream: The World According to Mathematics. Harcourt Brace Jovanovich.Google Scholar
Detlefsen, M. and Luker, M., 1980. “The Four-Color Problem and Mathematical Proof.” Journal of Philosophy 77:803–24.CrossRefGoogle Scholar
Dieudonné, J. 1982. “The Work of Nicolas Bourbaki in the Last Thirty Years.” Notices AMS 29:618–23.Google Scholar
Elkana, Y. 1981. “A Programmatic Attempt at an Anthropology of Knowledge.” In Sciences and Cultures, edited by Mendelson, E. and Elkana, Y., 1–76. Sociology of the Sciences, Vol.5. Dordrecht: Reidel.CrossRefGoogle Scholar
Elkana, Y. 1982. “The Myth of Simplicity.” In Albert Einstein, Historical and Cultural Perspectives, edited by Holton, G. and Elkana, Y., 205–51. Princeton, N. J.: Princeton University Press.Google Scholar
Elkana, Y. 1986. “The Emergence of Second-Order Thinking in Classical Greece.” In Axial Age Civilizations, edited by Eisenstadt, S. N., 40–64. Albany: State University of New York Press.Google Scholar
Euclid, . 1956. The Thirteen Books of the Elements, 3vols. Translated and with introduction and commentary by 2nd. ed. London: Dover. 1956.Google Scholar
Feferman, S. 1978. “The Logic of Mathematical Discovery vs. the Logical Structure of Mathematics.” Philosophy of Science 2:309–27.Google Scholar
Feferman, S. 1984. “Foundational Ways.” Perspectives in Mathematics. Anniversary of Over-wolfach. Basel: Birkhäuser.Google Scholar
Feferman, S. 1985. “Working Foundations.” Synthese 62:229–54.CrossRefGoogle Scholar
Freudenthal, H. 1976. “What Is Algebra and What Has It Been in History?” Archive for the History of Exact Sciences 16:189–200.CrossRefGoogle Scholar
Grabiner, J. 1975. “The Mathematician, the Historian, and the History of Mathematics.” Historia Mathematica 2:439–47.CrossRefGoogle Scholar
Hacking, I. 1979. “Imre Lakatos' Philosophy of Science.” Review of Lakatos 1978, British Journal for the Philosophy of Science 30:381–410.CrossRefGoogle Scholar
Hallett, M. 1979. “Towards a Theory of Mathematical Research Programs.” British Journal for the Philosophy of Science 30:1–25, 135–59.CrossRefGoogle Scholar
Hersh, R. [1979] 1985. “Some Proposals for Reviving the Philososphy of Mathematics.” Advances in Mathematics 31:31–50. Reprinted in Tymozcko 1985, 9–28.CrossRefGoogle Scholar
Hilbert, D. [1899] 1930. Grundlagen der Geometrie, 2nd ed. Leipzig: Teubner.Google Scholar
Hilbert, D. 1971. Foundations of Geometry. Translated from the tenth German edition (1968) by Leo Unger; revised and enlarged by Paul Bernays. La Salle, Ill.: Open Court.Google Scholar
Holton, G. 1978. The Scientific Imagination: Case Studies. Cambridge: Cambridge University Press.Google Scholar
Kitcher, P. 1983. The Nature of Mathematical Knowledge. New York: Oxford University Press.Google Scholar
Kitcher, P. 1988.“Mathematical Naturalism.” In Aspray, and Kitcher, , 1988, 293–323.Google Scholar
Kolata, G. B. 1976. “Mathematical Proofs: The Genesis of Reasonable Doubt.” Science 192:989–90.CrossRefGoogle ScholarPubMed
Kreisel, G. 1971. “Observations on Popular Discussions on Foundations.” In Axiomatic Set Theory, edited by Scott, D., 189–98. Proc. Symp. in Pure Math. of the American Mathematical Society, New York.CrossRefGoogle Scholar
Lakatos, I. 1970. “Falsification and the Methodology of Scientific Research Programs.” In Lakatos and Musgrave 1970, 91–195.Google Scholar
Lakatos, I. 1976. Proofs and Refutations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Lakatos, I. 1978. Philosophical Papers, 2 vols. Vol. 1: The Methodology of Research Programs, vol. 2: Mathematics, Science and Epistemology. Cambridge: Cambridge University Press.Google Scholar
Lakatos, I. and Musgrave, A., eds. 1970. Criticism and the Growth of Knowledge. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Laudan, L. 1973. “Peirce and the Trivialization of the Self-Correcting Thesis.” In Foundations of Scientific Method: The Nineteenth Century, edited by Giere, R. N. and Westfall, R. S. Bloomington: Indiana University Press.Google Scholar
Lorenzo, J. de. 1977. La Matemática y el Problema de su Historia. Madrid: Tecnos.Google Scholar
MacLane, S. 1981. “Mathematical Models: A Sketch for the Philosophy of Mathematics.” American Mathematical Monthly 88:462–72.Google Scholar
MacLane, S. 1986. Review of Bourbaki 1960 (English edition, 1984). Mathematical Reviews 86h:01:005.Google Scholar
MacLane, S. 1987a. Mathematics: Form and Function. New York: Springer.Google Scholar
MacLane, S. 1987b. Address on the occasion of the 1986 Steele prizes award, Ninety-third Annual Meeting of the AMS, at San Antonio, Texas. Notices AMS 34:229–30.Google Scholar
May, K. O. 1975. “What Is Good History and Who Should Do It?” Historia Mathematica 2:449–55.CrossRefGoogle Scholar
Mehrtens, H. 1976. “T. S. Khun's Theories and Mathematics: A Discussion Paper on the ‘New Historiography of Mathematics’.” Historia Mathematica 3:297–320.CrossRefGoogle Scholar
Mehrtens, H. 1979. Die Entstehung der Verbandstheorie. Hildesheim: Gerstenberg.Google Scholar
Pasch, M. [1882] 1976. Vorlesungen über neuere Geometrie. Berlin: Springer.Google Scholar
Popper, K. 1963. Conjectures and Refutations. London: Routledge and Keagan Paul.Google Scholar
Putnam, H. 1967. “Mathematics without Foundations.” In Philosophical Papers, Vol.1:43–59. Cambridge: Cambridge University Press.Google Scholar
Rabin, M. 1976. “Probabilistic Algorithms.” In Algorithms and Complexity: New Directions and Recent Results, edited by Traub, J. F., 21–40. New York: Academic Press.Google Scholar
Russell, B. and Whitehead., A. N. 1910–1913. Principia Mathematica. Cambridge: Cambridge University Press.Google Scholar
Sheffer, H. M. 1913. “A Set of Five Independent Postulates for Boolean Algebras.” Transactions of the American Mathematical Society 14:481–88.CrossRefGoogle Scholar
Torreti, R. 1978. Philosophy of Geometry from Riemann to Poincaré. Dordrecht: Reidel. (Episteme, Vol. 7).CrossRefGoogle Scholar
Tymoczko, T. 1979. “The Four-Color Problem and Its Philosophical Significance.” Journal of Philosophy 76:57–83.CrossRefGoogle Scholar
Tymoczko, T. 1985. New Directions in the Philosophy of Mathematics. Boston: Birkhäuser.Google Scholar
Unguru, S. 1975. “On the Need to Rewrite the History of Greek Mathematics.” Archive for the History of Exact Sciences 15:67–114.CrossRefGoogle Scholar
Unguru, S. and Rowe, D.. 1981. “Does the Quadratic Equation Have Greek Roots? A Study of ‘Geometric Algebra,’ ‘Application of Areas,’ and Related Problems, I.” Libertas Mathematica 1:1–49.Google Scholar
Unguru, S. and Rowe., D. 1982. “Does the Quadratic Equation Have Greek Roots? A Study of ‘Geometric Algebra,’ ‘Application of Areas,’ and Related Problems, II” Libertas Mathematica 2:1–62.Google Scholar
van der Waerden, B. L. [1930] 1949–50. Moderne Algebra, 2 vols. Berlin: Springer. English translation of 2nd. ed.: Vol. 1 translated by Blum, Fred (1949), Vol. 2 translated by T. J. Benac (1950); includes revisions and additions by the author. New York: Ungar.CrossRefGoogle Scholar
van der Waerden, B. L. 1934–1936. “De logische Grondslagen der Euklidische Meetkunde.” Zeitschrift Christian Huygens 13–14: sec. 3.CrossRefGoogle Scholar
van der Waerden, B. L. 1976. “Defense of a “Shocking' Point of View.” Archive for the History of Exact Sciences 15:199–210.CrossRefGoogle Scholar
Weil, A. 1978. “Who Betrayed Euclid?” Archive for the History of Exact Sciences 19:91–94.CrossRefGoogle Scholar
Weil, A. 1980. “History of Mathematics: Why and How.” Proceedings of the International Congress of Mathematicians,1978, Vol.1, 227–36.Google Scholar
Wilder, R. L. 1952. Introduction to the Foundations of Mathematics. New York: Wiley.Google Scholar