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The Growth of Mathematical Knowledge pp 153–176Cite as

The Growth of Mathematical Knowledge: An Open World View

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Part of the book series: Synthese Library ((SYLI,volume 289))

Abstract

In his book The Value of Science, Poincaré criticizes a certain view of the growth of mathematical knowledge:

The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past (Poincaré 1958, 14).

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References

  • Aleksandrov, A.D. (1963). “A General View of Mathematics” in (Aleksandrov et al. 1963).

    Google Scholar 

  • Aleksandrov, A. D. et al. (Eds.). (1963). Mathematics: Its Content, Method and Meaning. Cambridge: MIT Press.

    Google Scholar 

  • Aristotle. (1964). Analytica Priora et Posteriora, W.D. Ross (Ed.). Oxford: Oxford University Press.

    Google Scholar 

  • Barker-Plummer, D. (1992). “Gazing: an Approach to the Problem of Definition and Lemma Use.” Journal ofAutomated Reasoning. Vol. 8: 311–44.

    Article  Google Scholar 

  • Barnes, J. (1975). “Aristotle’s Theory of Demonstration.” in (Barnes, Schofield and Sorabji 1975, 65–87).

    Google Scholar 

  • Barnes, J., Schofield, M. and Sorabji, R. (Eds.). (1975). Articles on Aristotle. 1: Science, London: Duckworth.

    Google Scholar 

  • Bourbaki, N. (1949). “Foundations of Mathematics for the Working Mathematicians.” The Journal of Symbolic Logic. Vol. 14: 1–8.

    Article  Google Scholar 

  • Cambiano, G. (1967). “Il metodo ipotetico e le origini della sistemazione euclidea della geometria.” Rivista di filosofia. Vol. 58: 115–49.

    Google Scholar 

  • Cellucci, C. (1985). “I1 ruolo delle definizioni esplicite in matematica.” in (Mangione 1985, 419–34).

    Google Scholar 

  • Cellucci, C. (1996). Le ragioni della logica. Milan: il Saggiatore.

    Google Scholar 

  • Cialdea, Mayer M. and Pirri, F. (1993). “First Order Abduction via Tableau and Sequent Calculi.” Bulletin of the IGPL. Vol. 1: 99–117.

    Article  Google Scholar 

  • Corsi, G. and Sambin, G. (Eds.). (1991). Nuovi problemi della logica e della filosofia della scienza. Vol. II. Bologna: CLUEB.

    Google Scholar 

  • Davis, M. (Ed.). The Undecidable. Hewlett: Raven Press Eudemus. “Geometrikè istoría (fr. 140).” in (Wehrli–59, 57–66).

    Google Scholar 

  • Farmer, W. M., Guttman, J. D. and Thayer, F. J. (1992). “Little Theories.” in (Kapur 1992, 567–81).

    Google Scholar 

  • Feferman, S. (1979). “What Does Logic Have to Tell us about Mathematical Proofs?” The Mathematical Intelligencer. Vol. 2, No. 1: 20–4.

    Article  Google Scholar 

  • Frege, G. (1959). The Foundations ofArithmetic. Oxford: Blackwell Publishers.

    Google Scholar 

  • Frege, G. (1967). Begriffschrift. in (van Heijenoort 1967, 5–82).

    Google Scholar 

  • Frege, G. (1979). Posthumous Writings. Oxford: Blackwell Publishers.

    Google Scholar 

  • Frege, G. (1980). Philosophical and Mathematical Correspondence. Oxford: Blackwell Publishers.

    Google Scholar 

  • Frege, G. (1984). Collected Papers on Mathematics, Logic, and Philosophy. B. McGuinness. (Ed.). Oxford: Blackwell Publishers.

    Google Scholar 

  • Gödel, K. (1986). Collected Works. S. Feferman et al. (Eds.). Vol.. I. Oxford: Oxford University Press.

    Google Scholar 

  • Gödel, K. (1990). Collected Works. S. Feferman et al. (Eds.). Vol. II. Oxford: Oxford University Press.

    Google Scholar 

  • Gombrich, E. H. (1959). Art and Illusion. Washington: Trustees of the National Gallery of Art.

    Google Scholar 

  • Hadamard, J. (1949). The Psychology of Invention in the Mathematical Field. Princeton: Princeton University Press.

    Google Scholar 

  • Hilbert, D. (1900). “Über den Zahlbegriff.” Jahresbericht der Deutschen Mathematiker-Vereinigung. Vol. 8: 180–4.

    Google Scholar 

  • Hilbert, D. (1918). Prinzipien der Mathematik und Logik. Unpublished lecture notes. Göttingen: Mathematisches Institut.

    Google Scholar 

  • Hilbert, D. (1929). “Probleme der Grundlegung der Mathematik.” Mathematische Annalen. Vol. 102: 1–9.

    Article  Google Scholar 

  • Hilbert, D. (1931). “Die Grundlegung der elementaren Zahlenlehre.” Mathematische Annalen. Vol. 104: 485–94.

    Article  Google Scholar 

  • Hilbert, D. (1962). Grundlagen der Geometrie. Stuttgart: Teubner.

    Google Scholar 

  • Hilbert, D. (1967). “The Foundations of Mathematics.” in (van Heijenoort 1967, 464–79).

    Google Scholar 

  • Hilbert, D. (1970). Gesammelte Abhandlungen. Vol. 3. Berlin: Springer-Verlag.

    Google Scholar 

  • Hilbert, D. and Bernays, P. (1970). Grundlagen der Mathematik II. 2nd ed. Berlin: Springer-Verlag.

    Book  Google Scholar 

  • Kant, I. (1976). Critique of Pure Reason. London: Macmillan.

    Google Scholar 

  • Kant, I. (1992). Lectures on Logic. J. M. Young (Ed.). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Kapur, D. (Ed.). (1992). Automated Deduction CADE-11, LNCS 607. Berlin: Springer-Verlag.

    Google Scholar 

  • Kitcher, P. (1981). “Mathematical Rigor — Who Needs it?” Noûs. Vol. 15: 469–493.

    Article  Google Scholar 

  • Kleene, S. C. (1967). Mathematical Logic. New York: Wiley.

    Google Scholar 

  • Longo, G. (1991). “Notes on the Foundations of Mathematics and of Computer Science.” in (Corsi and Sambin 1991, 117–127).

    Google Scholar 

  • MacLane, S. (1986). Mathematics: Form and Function. Berlin: Springer-Verlag. Mangione, C. (Ed.). (1985). Scienza e filosofia. Milan: Garzanti.

    Book  Google Scholar 

  • Natorp, P. (1923). Die logischen Grundlagen der exakten Wissenschaften. Berlin: Teubner. Poincaré, H. (1952). Science and Hypothesis. New York: Dover.

    Google Scholar 

  • Poincaré, H. (1958). The Value of Science. New York: Dover.

    Google Scholar 

  • Polya, G. (1945). How to Solve It. Princeton: Princeton University Press.

    Google Scholar 

  • Post, E. (1965). “Absolutely Unsolvable Problems and Relatively Undecidable Propositions. Account of an Anticipation.” in (Davis 1965, 340–433).

    Google Scholar 

  • Proclus (1992). A Commentary on the First Book of Euclid “s Elements. G. R. Morrow (Ed.). Princeton: Princeton University Press.

    Google Scholar 

  • Quine, W. V. O. (1951). Mathematical Logic. New York: Harper and Row.

    Google Scholar 

  • Russell, B. (1903). Principles of Mathematics. London: Allen and Unwin.

    Google Scholar 

  • Stickel, M. E. (1994). “Upside-Down Meta-Interpretation of the Model Elimination Theorem-Proving Procedure for Deduction and Abduction.” Journal ofAutomated Reasoning. Vol. 13: 189–210. Tarski, A. (1946). Introduction to Logic and to the Methodology of Deductive Sciences. New York: Oxford University Press.

    Article  Google Scholar 

  • Todhunter, I. (1876). William Whewell: An Account of His Writings. Vol. II. London: Macmillan.

    Google Scholar 

  • Turing, A. (1965). “Systems of Logic Based on Ordinals.” in (Davis 1965, 155–222).

    Google Scholar 

  • van Heijenoort, J. (Ed.). (1967). From Frege to Gödel. Cambridge: Harvard University Press. van Heijenoort, J. (1985), Selected Essays. Naples: Bibliopolis.

    Google Scholar 

  • Wehrli, F. (Ed.). (59). Die Schule des Aristoteles, vol. VIII. Basel: B. Schwabe & Co.

    Google Scholar 

  • Wittgenstein, L. (1978). Bemerkungen über die Grundlagen der Mathematik. 2nd ed. Oxford: Blackwell Publishers.

    Google Scholar 

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Authors
  1. Carlo Cellucci
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Editors and Affiliations

  1. The Pennsylvania State University, USA

    Emily Grosholz

  2. Leibniz Archives, Hannover, Germany

    Herbert Breger

  3. University of Hannover, Germany

    Herbert Breger

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Cellucci, C. (2000). The Growth of Mathematical Knowledge: An Open World View. In: Grosholz, E., Breger, H. (eds) The Growth of Mathematical Knowledge. Synthese Library, vol 289. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9558-2_12

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  • DOI: https://doi.org/10.1007/978-94-015-9558-2_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5391-6

  • Online ISBN: 978-94-015-9558-2

  • eBook Packages: Springer Book Archive

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