Ebook: 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.

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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.

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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.
Content:
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 Caract?risation 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
D?finition intrins?que 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 N?bauer 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 al?atoires....Pages 445-447
Front Matter....Pages 449-449
Commentary on Menger’s Work on Algebra....Pages 451-461
Beitr?ge zur Gruppentheorie. I.....Pages 463-485
Front Matter....Pages 449-449
Une th?orie axiomatique g?n?rale des d?terminants....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

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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.
Content:
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 Caract?risation 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
D?finition intrins?que 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 N?bauer 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 al?atoires....Pages 445-447
Front Matter....Pages 449-449
Commentary on Menger’s Work on Algebra....Pages 451-461
Beitr?ge zur Gruppentheorie. I.....Pages 463-485
Front Matter....Pages 449-449
Une th?orie axiomatique g?n?rale des d?terminants....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
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