Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics
Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 97-107.

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
DA  - 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  - 
%0 Journal Article
%A Lenhard, Johannes
%T Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics
%J Philosophia Scientiae
%D 2005
%P 97-107
%V 9
%N 2
%I Éditions Kimé
%G en
%F PHSC_2005__9_2_97_0
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] Beth, Evert Willem 1968.- The Foundations of Mathematics, Amsterdam: North-Holland. | Zbl

[2] Corfield, David 2003.- Towards a Philosophy of Real Mathematics, Cambridge: Cambridge University Press. | MR | Zbl

[3] Ewald, William (Ed.) 1996.- From Kant to Hilbert: a Source Book in the Foundations of Mathematics, Oxford: Clare-don Press. | MR | Zbl

[4] Goldfarb, Warren D. 1979.- Logic in the Twenties: the Nature of the Quantifier, The Journal of Symbolic Logic, 44 (3), 351-368. | MR | Zbl

[5] Grattan-Guinness, Ivor 1997.- 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

[6] Heijenoort, Jean Van 1967.- Logic as Calculus and Logic as Language, Synthese, 17, 324-330. | Zbl

[7] Hilbert, David 1899.- Grundlagen der Geometrie, Stuttgart: Teubner, 1999.

[8] Hilbert, David 1918.- Axiomatisches Denken, Mathematische Annalen, 78, 405-415. | JFM

[9] Hilbert, David 1919/20.- Natur und mathematisches Erkennen, Basel/Boston/Berlin: Birkhäuser, 1999. | MR

[10] Hintikka, Jaakko 1997.- Lingua Universalis vs. Calculus Ratiocinator: An Ultimate Presupposition of Twentieth-Century Philosophy, Dordrecht: Kluwer. | MR

[11] Lenhard, Johannes and Michael Otte 2002.- Analyse und Synthese - Von Leibniz und Kant zum Axiomatischen Denken, Philoso-phia naturalis, 39 (2), 259-292. | MR

[12] Majer, Ulrich 2001.- 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

[13] Pasch, Moritz 1882.- Vorlesungen über neuere Geometrie, Berlin: Springer.

[14] Peckhaus, Volker 1999.- 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.

[15] Schlick, Moritz 1918.- Allgemeine Erkenntnislehre, Frankfurt am Main: Suhrkamp, 1979.