Abstract
This article is not so much intended as a recapitulation of all the results achieved by the Vienna Mathematical Colloquium1 during its six years of existence-such a recapitulation would in fact require a much larger space-but intends instead to expound the direction of the ideas that were pursued in this colloquium. Moreover, we limit ourselves to shedding light on the tendencies of the colloquium in the direction of geometry, although it certainly did not deal exclusively with geometrical studies. And in truth, apart from the fact that the geometric researches cultivated in this colloquium often required investigations into other fields of mathematics, other studies, above all in logic2, but also regarding new applications of the exact sciences to problems of a sociological character, were carried out.3
I thank Dr. L. GEYMONAT for having helped me in writing the present article in Italian.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Notes
See: Ergebnisse eines mathematischen Kolloquiums. Unter Mitwirkung von K. GÖDEL und G. NÖBELING herausgegeben von K. MENGER, volumes 1–6 (Deuticke, Vienna). In the sequel we cite them as “Ergebnisse”.
Apart from a short lecture of K.GÖDEL (Ergebnisse 3, p.12) on his fundamental results concerning the completeness and the consistency of axiomatic systems [that is: in every theory that contains the arithmetic of PEANO one finds propositions that cannot be proved by means of this theory itself, and in particular it is impossible to prove the consistency of such a theory without stepping outside of the axiomatic system on which it is based], there are investigations of classical mathematical logic, of many-valued logic (GöDEL, TARSKI, PARRY) and of intuitionistic logic, among them a proof of the surprising fact that the so-called intuitionistic arithmetic is only apparently more restrictive than classical arithmetic, whereas in an appropriate interpretation it contains the latter (GöDEL, Ergebnisse 4, p.34).
For instance on the existence and uniqueness of the equations of production in mathematical economics (See SCHLESINGER-WALD, Ergebnisse 6, p. 10).
See MENGER: Neuere Methoden und Probleme der Geometrie, Verhandl. d. Intern. Math. Kongr., Zürich 1932, vol. I, p. 310.
See MENGER: Dimensionstheorie, Leipzig 1928, where one finds the complete references up to 1928, including in particular the works of URYSOHN. See also NÖBELING: Die neuesten Ergebnisse der Dimensionstheorie, Jahresbericht d. deutsch. Math. Ver., 41, 1931, p. 1.
See MENGER: Kurventheorie, Teubner, Leipzig, 1932.
For the literature see NÖBELING, Die neuesten Ergebnisse der Dimensionstheorie, Jahresbericht d. deutsch. Math. Ver., 41, 1931, p. 1 loc.cit., footnote 5.
For a short proof see FLORES, Ergebnisse 5, p. 17 and vol.6, p. 4.
See MENGER, Mathem. Annalen, 100, p.76, and for an abstract Ergebnisse 1, p. 20.
For a simple new proof of this theorem, see ARONSZAJN, Ergebnisse vol. 6, p. 45.
See MENGER, Math. Annalen 100, p. 113 and, for an abstract, Ergebnisse 1, p. 20.
See WALD, Ergebnisse 5, pp. 32–42.
See MENGER, Ergebnisse 2, p. 34; WALD-FLEXER, Ergebnisse 5, p. 32 and, for an even more farreaching generalization, TAUSSKY, Ergebnisse 6, p. 20.
This definition and those of a real metric space and a positive metric space given on the following pages should not be confused with the concept of a metric space in the sense of FRÉCHET (See p. 2).
loc.cit. footnote 11.
See MENGER, Mathem. Annalen, 100, pp. 120–128.
See MENGER: Zur Theorie der Bogenkrümmung, Mathem. Annalen, 103, p. 480; Jahresber. d. deutsch. Math. Ver., 50, pp. 211–213, 218.
See WALD, Ergebnisse 6, p. 29.
See WALD, Ergebnisse volume 7, that will appear in autumn 1935. This article will also contain a simplification of the definition of curvature stated above.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Wien
About this chapter
Cite this chapter
Menger, K. (1998). On the direction of ideas and the principal tendencies of the Vienna Mathematical Colloquium. In: Dierker, E., Sigmund, K. (eds) Karl Menger. Springer, Vienna. https://doi.org/10.1007/978-3-7091-6470-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-7091-6470-9_6
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-7330-5
Online ISBN: 978-3-7091-6470-9
eBook Packages: Springer Book Archive