Two conflicting interpretations of modern axiomatics will be considered. The logico-analytical interpretation goes back to Pasch, while the model-theoretical approach stems from Hilbert. This perspective takes up the distinction between logic as calculus ratiocinator versus lingua characterica that Heijenoort and Hintikka placed emphasis on. It is argued that the Heijenoort-Hintikka distinction can be carried over from logic to mathematical axiomatics. In particular, the model-theoretical viewpoint is deeply connected to a philosophy of mathematics that is not committed to a foundational perspective, but oriented more at applications and at mathematical practice.
@article{PHSC_2005__9_2_97_0, author = {Lenhard, Johannes}, title = {Axiomatics without foundations. {On} the model-theoretical viewpoint in modern axiomatics}, journal = {Philosophia Scientiae}, pages = {97--107}, publisher = {\'Editions Kim\'e}, volume = {9}, number = {2}, year = {2005}, language = {en}, url = {http://archive.numdam.org/item/PHSC_2005__9_2_97_0/} }
TY - JOUR AU - Lenhard, Johannes TI - Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics JO - Philosophia Scientiae PY - 2005 SP - 97 EP - 107 VL - 9 IS - 2 PB - Éditions Kimé UR - http://archive.numdam.org/item/PHSC_2005__9_2_97_0/ LA - en ID - PHSC_2005__9_2_97_0 ER -
Lenhard, Johannes. Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics. Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 97-107. http://archive.numdam.org/item/PHSC_2005__9_2_97_0/
[1] The Foundations of Mathematics, Amsterdam: North-Holland. | Zbl
1968.-[2] Towards a Philosophy of Real Mathematics, Cambridge: Cambridge University Press. | MR | Zbl
2003.-[3] From Kant to Hilbert: a Source Book in the Foundations of Mathematics, Oxford: Clare-don Press. | MR | Zbl
(Ed.) 1996.-[4] Logic in the Twenties: the Nature of the Quantifier, The Journal of Symbolic Logic, 44 (3), 351-368. | MR | Zbl
1979.-[5] Peirce between Logic and Mathematics, in (N. Houser, D. D. Roberts and J. van Evra, eds.) Studies in the Logic of Charles Sanders Peirce, Bloomington and Indianapolis: Indiana University Press, 23-42. | MR
1997.-[6] Logic as Calculus and Logic as Language, Synthese, 17, 324-330. | Zbl
1967.-[7] Grundlagen der Geometrie, Stuttgart: Teubner, 1999.
1899.-[8] Axiomatisches Denken, Mathematische Annalen, 78, 405-415. | JFM
1918.-[9] Natur und mathematisches Erkennen, Basel/Boston/Berlin: Birkhäuser, 1999. | MR
1919/20.-[10] Lingua Universalis vs. Calculus Ratiocinator: An Ultimate Presupposition of Twentieth-Century Philosophy, Dordrecht: Kluwer. | MR
1997.-[11] Analyse und Synthese - Von Leibniz und Kant zum Axiomatischen Denken, Philoso-phia naturalis, 39 (2), 259-292. | MR
and 2002.-[12] Hilbert's Axiomatic Method and the Foundations of Science: Historical Roots of Mathe-matical Physics in Göttingen (1900-1930), in (M. Rédei and M. Stöltzner, eds.) John von Neumann and the Foundations of Quantum Physics, Dordrecht: Kluwer, 11-33. | MR
2001.-[13] Vorlesungen über neuere Geometrie, Berlin: Springer.
1882.-[14] The Pragmatism of Hilbert's Programme, Lecture at GAP, Bielefeld. 1967 Mathematics Without Foundations, in (H. Putnam and P. Benacerraf, eds.), Philosophy of Mathematics: Selected Readings, Cambridge: Cambridge University Press, 1983, 295-311.
1999.-[15] Allgemeine Erkenntnislehre, Frankfurt am Main: Suhrkamp, 1979.
1918.-