Selecta Mathematica : Volume 2

Karl Menger, one of the founders of dimension theory, belongs to the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced some of his colleagues. The Selecta Mathematica contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, as well as writings on economics, sociology, logic, philopsophy and mathematical results. The two volumes are a monument to the diversity and originality of Menger's ideas. Front Matter....Pages i-x Front Matter....Pages 1-1 Commentary on Menger and Intuitionism....Pages 3-8 Bemerkungen zu Grundlagenfragen....Pages 9-22 Front Matter....Pages 23-23 Commentary on Karl Menger’s Contributions to Analysis....Pages 25-34 On Cauchy’s Integral Theorem in the Real Plane....Pages 35-39 On Green’s Formula....Pages 41-45 The Behavior of a Complex Function at Infinity....Pages 47-48 A Characterization of Weierstrass Analytic Functions....Pages 49-50 Analytische Funktionen....Pages 51-54 Une Caractérisation Des Fonctions Analytiques....Pages 55-56 Weierstrass Analytic Functions....Pages 57-74 Front Matter....Pages 75-75 Commentary on Menger’s Path Length Papers....Pages 77-80 On Shortest Polygonal Approximations to a Curve....Pages 81-86 Définition intrinsèque de la notion de chemin....Pages 87-89 Stieltjes Integrals Considered as Lengths....Pages 91-93 What paths have length?....Pages 95-104 A Topological Characterization of the Length of Paths....Pages 105-108 Front Matter....Pages 109-109 Commentary on the Algebra of Analysis and Algebra of Functions....Pages 111-126 The Influence of Menger’s Ideas on the Work of Nöbauer and His School....Pages 127-134 Algebra of Analysis....Pages 135-180 Tri-Operational Algebra....Pages 181-188 Front Matter....Pages 109-109 General Algebra of Analysis....Pages 189-203 An Axiomatic Theory of Functions and Fluents....Pages 205-224 The algebra of functions: past, present, future....Pages 225-246 On Substitutive Algebra and its Syntax....Pages 247-270 Superassociative Systems and Logical Functors....Pages 271-288 Two theorems on the generation of systems of functions....Pages 289-300 Front Matter....Pages 301-301 Commentary on Didactics, Variables, Fluents....Pages 303-316 A Mengerian Tour Along Caratheodory’s Royal Road....Pages 317-323 Calculus — A Modern Approach....Pages 325-349 The Mathematics of Elementary Thermodynamics....Pages 351-365 Random Variables from the Point of View of a General Theory of Variables....Pages 367-381 What are x and y ?....Pages 383-392 Rates of change and derivatives....Pages 393-406 Front Matter....Pages 407-407 Commentary on Probabilistic Geometry....Pages 409-432 Statistical Metrics....Pages 433-435 Probabilistic Theories of Relations....Pages 437-439 Probabilistic Geometry....Pages 441-444 Ensembles flous et fonctions aléatoires....Pages 445-447 Front Matter....Pages 449-449 Commentary on Menger’s Work on Algebra....Pages 451-461 Beiträge zur Gruppentheorie. I.....Pages 463-485 Front Matter....Pages 449-449 Une théorie axiomatique générale des déterminants....Pages 487-489 Front Matter....Pages 491-491 Commentary on Menger’s Work on Sociology....Pages 493-500 An Exact Theory of Social Groups and Relations....Pages 501-509 On Social Groups and Relations....Pages 511-523 Front Matter....Pages 525-525 Commentary on Menger’s ‘Austrian Marginalism and Mathematical Economics’....Pages 527-530 Austrian Marginalism and Mathematical Economics....Pages 531-553 Front Matter....Pages 555-555 Editor’s Comments....Pages 557-557 Otto Schreier....Pages 559-564 Hans Hahn....Pages 565-568 Memories of Moritz Schlick....Pages 569-589 Introduction to the Sixth American Edition, 1960....Pages 591-607 Introduction....Pages 609-618 You Will Like Geometry....Pages 619-654 Back Matter....Pages 655-678 Karl Menger, one of the founders of dimension theory, belongs to the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and after his emigration to the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced some of his colleagues. The "Selecta Mathematica' contain Menger's major mathematical papers, based on a personal selection by himself out of his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, as well as writings on economics, sociology, logic, philosophy and mathematical results. The two volumes are a monument to the diversity and originality of Menger's ideas Karl Menger, one of the founders of dimension theory, is among the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced many of his colleagues. These two volumes contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, and also include material on economics, sociology, logic and philosophy. The Selecta Mathematica is a monument to the diversity and originality of Menger's ideas.