Ebook: Selecta Mathematica: Volume 1

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.

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Content:
Front Matter....Pages i-x
Introduction....Pages 1-5
Karl Menger and Vienna’s Golden Autumn....Pages 7-21
Commentary on Menger’s Work on Dimension Theory....Pages 23-32
Front Matter....Pages 33-33
Zur Dimensions- und Kurventheorie....Pages 35-56
Das Hauptproblem ?ber die dimensionelle Struktur der R?ume....Pages 57-63
Dimension und Zusammenhangsstufe....Pages 65-80
“Allgemeine R?ume und Cartesische R?ume”. (Erste Mitteilung)....Pages 81-87
“Allgemeine R?ume und Cartesische R?ume”. Zweite Mitteilung: „Ueber umfassendste n~dimensionale Mengen“....Pages 89-92
?ber die Dimension von Punktmengen III....Pages 93-118
Axiomatische Einf?hrung des Dimensionsbegriffes....Pages 119-127
Remarques sur la th?orie axiomatique de la dimension....Pages 129-134
?ber die Hinweise auf Brouwer in Urysohns M?moire....Pages 135-139
Commentary on Menger’s Work on Curve Theory and Topology....Pages 141-152
Front Matter....Pages 153-153
Einige ?berdeckungss?tze der Punktmengenlehre....Pages 155-178
Grundz?ge einer Theorie der Kurven....Pages 179-208
Grundz?ge einer Theorie der Kurven....Pages 209-213
On the Origin of the n-Arc Theorem....Pages 215-217
Commentary on Menger’s “Untersuchungen ?ber allgemeine Metrik”....Pages 219-228
Front Matter....Pages 229-233
Untersuchungen ?ber allgemeine Metrik....Pages 235-235
Front Matter....Pages 237-325
Bemerkungen zur zweiten Untersuchung ?ber allgemeine Metrik....Pages 235-235
Untersuchungen ?ber allgemeine Metrik. Vierte Untersuchung. Zur Metrik der Kurven.....Pages 327-331
Commentary on Menger’s Work on the Calculus of Variation and Metric Geometry....Pages 333-368
Front Matter....Pages 369-376
Sur un th?or?me g?n?ral du calcul de? variations....Pages 377-377
Calcul des variations dans les espaces distanci?s g?n?raux....Pages 379-381
Courbes minimisantes non rectifiables et champs g?n?raux de courbes admissibles dans le calcul des variations....Pages 383-385
Metric Methods in Calculus of Variations....Pages 387-389
A Theory of Length and its Applications to the Calculus of Variations....Pages 391-397
Commentary on Menger’s Work on the Algebra of Geometry....Pages 399-403
Front Matter....Pages 405-416
Bemerkungen zu Grundlagenfragen. IV....Pages 417-417
New Foundations of Projective and Affine Geometry....Pages 419-435
A Note on a Previous Paper “New Foundations of Projective and Affine Geometry”....Pages 437-463
A New Foundation Of Non- Euclidean, Affine, Real Projective and Euclidean Geometry....Pages 465-465
Mathematical Notes....Pages 467-471
The Projective Space....Pages 473-474
Frammenti piani autoduali e relative sostituzioni....Pages 475-488
The New Foundation of Hyperbolic Geometry....Pages 489-493
Commentary on Menger’s Expository Papers on Geometry....Pages 495-506
Front Matter....Pages 507-514
Some Applications of Point-Set Methods....Pages 515-515
Generalized Vector Spaces. I....Pages 517-538
Front Matter....Pages 539-549
The Theory of Relativity and Geometry....Pages 515-515
The Formative Years of Abraham Wald and his Work in Geometry....Pages 551-566
Mathematical Implications of Mach’s Ideas: Positivistic Geometry, the Clarification of Functional Connections....Pages 567-573
Back Matter....Pages 575-593
....Pages 595-611