Document Zbl 1429.00007

Abstraction, axiomatization and rigor: Pasch and Hilbert. (English) Zbl 1429.00007

Hellman, Geoffrey (ed.) et al., Hilary Putnam on logic and mathematics. Cham: Springer. Outst. Contrib. Log. 9, 161-178 (2018).

Summary: In the late nineteenth century, Pasch made a well known statement concerning the conditions of attaining rigor in geometrical proof. The criterion he offered called not only for the elimination of appeals to geometrical figures, but of appeals to meanings of geometrical terms more generally. Not long after Pasch, Hilbert (and others) proposed an alternative standard of rigor. My aim in this paper is to clarify the relationship between Pasch’s and Hilbert’s standards of rigor. There are, I believe, fundamental differences between them.
For the entire collection see [Zbl 1411.03004].

MSC:

00A30	Philosophy of mathematics
03A05	Philosophical and critical aspects of logic and foundations

Keywords:

rigor; proof; Pasch; Hilbert; Lambert; Freudenthal; premisory surreption; abstraction from meaning; semantic abstraction; abstraction condition; axiomatic method; axiomatizaton; formalization

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[1]	Berkeley, G. (1734). The analyst; or, a discourse addressed to an infidel mathematician. Wherein it is examined whether the object, principles, and inferences of the modern analysis are more distinctly conceived, or more evidently deduced, than religious mysteries and points of faith, Printed for J. Jonson, London.
[2]	Bour, P.-E., Rebuschi, M., & Rollet, L. (Eds.). (2010). Construction. London: College Publications.
[3]	Brouwer, L. E. J. (1923). Über die Bedeutung des Satzes vom ausgescholessenen Dritten in der Mathematik, insbesondere in der Funktiontheorie. Journal für die reine und angewandte Mathematik, 154, 1-7. English translation in van Heijenoort (1967), pp. 334-345. Page references are to this translation.
[4]	Brouwer, L. E. J. (1928). Intuitionistische Betrachtungen über den Formalismus. Koninklijke Akademie van wetenschappen de Amsterdam, Proceedings of the Section of Sciences, 31, 324-329. English translation with introduction in van Heijenoort (1967), pp. 490-492. Page references are to this translation. · JFM 54.0053.01
[5]	Cellucci, C., Grosholz, E., & Ippoliti, E. (Eds.). (2011). Logic and knowledge. Newcastle upon Tyne: Cambridge Scholars Publishing.
[6]	Detlefsen, M. (2005). Formalism. In Shapiro (2005), pp. 236-317. · doi:10.1093/0195148770.003.0008
[7]	Detlefsen, M. (2010). Rigor, Re-proof and Bolzano’s Critical Program. In Bour et al. (2010), pp. 171-184. · Zbl 1225.03003
[8]	Detlefsen, M. (2011). Dedekind against intuition: Rigor, scope and the motives of his logicism. In Cellucci et al. (2011), pp. 273-288.
[9]	Engel, F., & Stäckel, P. (1895). Die Theorie der Parallelinien von Euklid bis auf Gauss. Leipzig: Teubner. · JFM 26.0057.06
[10]	Frege, G. (1906). Über die Grundlagen der Geometrie, Jahresbericht der Deutschen Mathematiker-Vereinigung, 15, 293-309, 377-403, 423-430. English translation in Frege (1971). · JFM 37.0485.01
[11]	Frege, G. (1971). Gottlob Frege: On the foundations of geometry and formal theories of arithmetic, trans. and ed., with an introduction, by Eike-Henner W. Kluge. New Haven: Yale University Press. · Zbl 0236.02003
[12]	Freudenthal, H. (1962). The main trends in the foundations of geometry in the 19th century. In Suppes et al. (1962), pp. 613-621. · Zbl 0136.00205
[13]	Hilbert, D. (1899). Grundlagen der Geometrie, Published in Festschrift zur Feier der Enthüllung des Gauss-Weber Denkmals in Göttingen. Leipzig: Teubner. · JFM 30.0424.01
[14]	Hilbert, D. (1900). Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung, 8, 180-194. · JFM 31.0165.02
[15]	Hilbert, D. (1922). Neubegründung der Mathematik. Erste Mitteilung. Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, 1, 157-77. Page references are to the reprinting in Hilbert (1935), Vol. III. · JFM 48.1188.01 · doi:10.1007/BF02940589
[16]	Hilbert, D. (1926). Über das Unendliche. Mathematische Annalen, 95, 161-190. · JFM 51.0044.02 · doi:10.1007/BF01206605
[17]	Hilbert, D. (1928). Die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6, 65-85. Reprinted in Hamburger Einzelschriften, 5, pp. 1-21. Leipzig: Teubner. Page references are to the latter. · JFM 54.0056.03
[18]	Hilbert, D. (1930). Naturerkennen und Logik. Die Naturwissenschaften, 18, 959-963. Printed in Hilbert (1935), Vol. III. Page references are to this printing. · JFM 56.0050.01
[19]	Hilbert, D., & Bernays, P. (1934). Grundlagen der Mathematik (Vol. I). Berlin: Springer. Page references are to Hilbert and Bernays (1968), the second edition of this text.
[20]	Hilbert, D. (1932-1935). Gesammelte Abhandlungen, three volumes. Bronx: Chelsea Pub. Co. · JFM 58.0048.03 · doi:10.1007/978-3-642-50831-8
[21]	Hilbert, D., & Bernays, P. (1968). Grundlagen der Mathematik (Vol. I, 2nd ed.). Berlin: Springer. · Zbl 0191.28402
[22]	Huntington, E. V. (1911). The fundamental propositions of algebra. Long Island University, Brooklyn: Galois Institute Press. · Zbl 0067.00505
[23]	Lambert, J. H. (1786). Theory der Parallelinien I. Leipziger Magazin für reine und angewandte Mathematik, 1(2), 137-164. Reprinted in Engel and Stäckel (1995), pp. 152-176.
[24]	Lambert, J. H. (1786). Theory der Parallelinien II. Leipziger Magazin für reine und angewandte Mathematik, 1(3), 325-358. Reprinted in Engel and Stäckel (1995), pp. 176-207.
[25]	Pasch, M. (1882). Vorlesungen über neuere Geometrie. Leipzig: Teubner. · JFM 14.0498.01
[26]	Pasch, M. (1918). Die Forderung der Entscheidbarkeit. Jahresbericht der Deutschen Mathematiker-Vereinigung, 27, 228-232. · JFM 46.0069.01
[27]	Pollard, S. (2010). ‘As if’ reasoning in Vaihinger and Pasch. Erkenntnis, 73, 83-95. · Zbl 1200.01023 · doi:10.1007/s10670-009-9205-7
[28]	Poncelet, J.-V. (1822). Traité des propriétés projectives de figures. Paris: Bachelier.
[29]	Putnam, H. (1984). Proof and experience. Proceedings of the American Philosophical Society, 128, 31-34.
[30]	Russell, B. (1902). The teaching of Euclid. The Mathematical Gazette, 2(33), 165-167. · JFM 33.0542.15 · doi:10.2307/3604768
[31]	Shapiro, S. (Ed.). (2005). Oxford handbook of the philosophy of mathematics and logic. Oxford: Oxford University Press. · Zbl 1081.03001
[32]	Smith, C., & Bryant, S. (1901). Euclid’s elements of geometry: Books I-IV, VI and XI. London: Macmillan. · JFM 32.0495.02
[33]	Suppes, P., Nagel, E., & Tarski, A. (Eds.) (1962). Logic, methodology, and philosophy of science: Proceedings of the 1960 International Congress. Stanford: Stanford U Press. · Zbl 0128.24101
[34]	Todhunter, I. (1869). The elements of Euclid for the use of schools and colleges. London: Macmillan.
[35]	van Heijenoort, J. (1967). From Frege to Gödel: A source book in mathematical logic, 1879-1931. Cambridge: Harvard University Press. · Zbl 0183.00601
[36]	Weyl, H. (1928). Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6, 86-88. Reprinted in Hamburger Einzelschriften, 5, pp. 22-24, Teubner, Leipzig, 1928. English translation in van Heijenoort (1967). · JFM 54.0056.03
[37]	Weyl, H. (1944). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612-654. · Zbl 0060.01521 · doi:10.1090/S0002-9904-1944-08178-0
[38]	Whitehead, A. N. (1906). The axioms of projective geometry. Cambridge: University Press. · JFM 37.0559.01
[39]	Young, J.

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