Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics

Johannes Lenhard

Philosophia Scientiae (2005)

  • Volume: 9, Issue: 2, page 97-107
  • ISSN: 1281-2463

Abstract

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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.

How to cite

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Lenhard, Johannes. "Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics." Philosophia Scientiae 9.2 (2005): 97-107. <http://eudml.org/doc/103763>.

@article{Lenhard2005,
abstract = {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.},
author = {Lenhard, Johannes},
journal = {Philosophia Scientiae},
language = {eng},
number = {2},
pages = {97-107},
publisher = {Éditions Kimé},
title = {Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics},
url = {http://eudml.org/doc/103763},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Lenhard, Johannes
TI - Axiomatics without foundations. On the model-theoretical viewpoint in modern axiomatics
JO - Philosophia Scientiae
PY - 2005
PB - Éditions Kimé
VL - 9
IS - 2
SP - 97
EP - 107
AB - 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.
LA - eng
UR - http://eudml.org/doc/103763
ER -

References

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  8. [8] Hilbert, David1918.— Axiomatisches Denken, Mathematische Annalen, 78, 405—415. JFM46.0062.03
  9. [9] Hilbert, David1919/20.— Natur und mathematisches Erkennen, Basel/Boston/Berlin: Birkhäuser, 1999. MR1316394
  10. [10] Hintikka, Jaakko1997.— Lingua Universalis vs. Calculus Ratiocinator: An Ultimate Presupposition of Twentieth-Century Philosophy, Dordrecht: Kluwer. MR1797984
  11. [11] Lenhard, Johannes and Michael Otte 2002.— Analyse und Synthese - Von Leibniz und Kant zum Axiomatischen Denken, Philoso-phia naturalis, 39 (2), 259–292. MR1954945
  12. [12] Majer, Ulrich2001.— 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. MR2042739
  13. [13] Pasch, Moritz1882.— Vorlesungen über neuere Geometrie, Berlin: Springer. 
  14. [14] Peckhaus, Volker1999.— 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. [15] Schlick, Moritz1918.— Allgemeine Erkenntnislehre, Frankfurt am Main: Suhrkamp, 1979. 

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