Karl Menger

Untersuchungen über allgemeine Metrik

Von Metrik ist in verschiedenen Teilen der Geometrie die Rede, in der Elementargeometrie, in der Differentialgeometrie, in der Mastheorie, in der Punktmengenlehre. Zwischen den verschiedenen Gesichtspunkten, unter denen das Messen untersucht wird, einen Zusammenhang auf allgemeinster (mengentheoretischer) Grundlage herzustellen, ist eine der Aufgaben dieser Untersuchungen.

On the Origin of Money

Publisher Summary This chapter discusses the origin of money. Money is used as a token in trade to reassure traders in such a sequence that they are not making an egregiously bad deal. This leads to an alternate line of investigation, recognition that a theory of a medium of exchange is inter alia a theory of the liquidity or saleability of commodities. In primitive traffic, the economic man is awaking but very gradually to an understanding of the economic advantages to be gained by exploitation of existing opportunities of exchange. Under such conditions each man is intent to get by way of exchange just such goods as he directly needs, and to reject those of which he has no need at all, or with which he is already sufficiently provided. Thus, equipped he has the prospect of acquiring such goods as he finally wishes to obtain, not only with greater ease and security, but also by reason of the steadier and more prevailing demand for his own commodities, at prices corresponding to the general economic situation—at economic prices.

Reminiscences of the Vienna Circle and the Mathematical Colloquium

I The Historical Background.- II The Cultural Background.- III The Philosophical Atmosphere in Vienna.- IV Why the Circle invited me. The Theory of Curves and Dimension Theory.- V Vignettes of the Members of the Circle in 1927.- VI Reminiscences of the Wittgenstein Family.- VII Ludwig Wittgensteins Austrian Dictionary.- VIII Wittgensteins Tractatus and the Early Circle.- IX On the Communication of Metaphysical Ideas. Wittgensteins Ontology.- X Wittgenstein, Brower, and the Circle.- XI Discussions in the Circle 1927-30.- XII Poland and the Vienna Circle.- XIII The United States 1930-31.- XIV Discussions in the Circle 1931-34.- XV The Circle on Ethics.- XVI Moritz Schlicks Final Years.- Memories of Kurt Godel.- Index of Names.

Selected papers in logic and foundations, didactics, economics

I. Papers Introducing Logical Tolerance.- Logical Tolerance in the Vienna Circle.- 1 The New Logic (1933).- Appendix (1937).- 2 On Intuitionism (1930).- II. Opuscula logica.- 3 Meaningfulness and Structure (1930).- Appendix (1978).- 4 A New Point of View on the Logical Connectives (1978).- 5 An Intuitionistic-Formalistic Dictionary of Set Theory (1928).- 6 Ultrasets and the Paradoxes of Set Theory (1928).- 7 A Logic of the Doubtful. On Optative and Imperative Logic (1939).- III. Fundamental Concepts in Pure and Applied Mathematics.- 8 A Counterpart of Occams Razor (1960, 1961).- 9 A Theory of the Application of the Function Concept to Science (1970).- 10 Variables, Constants, Fluents (1961).- 11 Wittgenstein on Formulae and Variables (1978).- IV. Didactics of Mathematics.- 12 A New Approach to Teaching Intermediate Mathematics (1958).- 13 Why Johnny Hates Math (1956).- 14 On the Formulation of Certain Questions in Arithmetic (1956).- 15 On the Design of Grouping Problems and Related Intelligence Tests (1953).- 16 The Geometry Relevant to Modern Education (1971).- V. Philosophical Ramifications of some Geometric Ideas.- 17 On Definition, Especially of Dimension (1921-1923, 1928).- 18 Square Circles (The Taxicab Geometry) (1952, 1978).- 19 The Algebra of Geometry (1978).- 20 Geometry and Positivism. A Probabilistic Microgeometry (1970).- VI..- 21 My Memories of L. E. J. Brouwer (1978).- VII. Economics. Meta-Economics.- 22 The Role of Uncertainty in Economics (1934).- 23 Remarks on the Law of Diminishing Returns. A Study in Meta-Economics (1936).- VIII. Gullivers Interest in Mathematics.- 24 Gulliver in the Land without One, Two, Three (1959).- 25 Gullivers Return to the Land without One, Two, Three (1960).- 26 Gulliver in Applyland (1960).- Bibliography of Works by Karl Menger.- Index of Names.

The Role of Uncertainty in Economics

The following essay was essentially completed by 1923. In 1927, I presented the paper to the Viennese Economic Society; in 1934, it was published in the Viennese Zeitschrift fur Nationalkonomie under the title ‘Das Unsicherheitsmoment in der Wertlehre’, [1934, 8 and 8a].

New Foundations of Projective and Affine Geometry

Projective geometry is often called geometry of projection and section (Geometrie des Verbindens und Schneidens). In this paper foundations of projective geometry are given in terms of these two operations. We start from a single class of undefined entities, corresponding to the linear parts of a space, and two undefined operations denoted by + and ., corresponding to the join and the intersection, respectively, of these linear parts. Thus if A, B are two undefined entities, A + B corresponds to the least dimensional part of which both A and B are parts, while A • B corresponds to the highest dimensional part which is part both of A and of B. In this way we obtain a far-reaching analogy with abstract algebra where, in defining a field, one also starts with a Single class of undefined elements and two undefined operations, addition and multiplication. Moreover, we obtain an analogy with the algebra of logics, in particular with the calculus of classes.1 In fact, this paper presents what might be called an algebra of elementary geometry.

Untersuchungen über allgemeine Metrik. Vierte Untersuchung. Zur Metrik der Kurven.

Bei einer der klassischen Definitionen der Bogenlange geht man folgendermasen vor: Man setzt, wenn ein Bogen B zwischen den Punkten a und c gegeben ist, auf B einen Richtungssinn fest, etwa von a nach c. Man ordnet sodann, wenn E eine endliche, etwa n Punkte enthaltende Teilmenge von B ist, die Punkte von E in jene Reihenfolge, in welcher sie bei der Durchlaufung von B in der festgesetzten Richtung angetroffen werden, und numeriert sie in dieser Reihenfolge mit b1, b2,…,bn. Man bildet hierauf, wenn bibi+1 den Abstand der Punkte bi und bi+1 bezeichnet, die Zahl l(E, B)

The algebra of functions: past, present, future

= \,\sum\limits_{i = 1}^{n - 1} {{b_i}} {b_{i + 1}}