Fundamentals of Mathematics represents a new kind of mathematical publication. While excellent technical treatises have been written about specialized fields, they provide little help for the nonspecialist; and other books, some of them semipopular in nature, give an overview of mathematics while omitting some necessary details. Fundamentals of Mathematics strikes a unique balance, presenting an irreproachable treatment of specialized fields and at the same time providing a very clear view of their interrelations, a feature of great value to students, instructors, and those who use mathematics in applied and scientific endeavors. Moreover, as noted in a review of the German edition in Mathematical Reviews, the work is "designed to acquaint [the student] with modern viewpoints and developments. The articles are well illustrated and supplied with references to the literature, both current and 'classical.'"
The outstanding pedagogical quality of this work was made possible only by the unique method by which it was written. There are, in general, two authors for each chapter: one a university researcher, the other a teacher of long experience in the German educational system. (In a few cases, more than two authors have collaborated.) And the whole book has been coordinated in repeated conferences, involving altogether about 150 authors and coordinators.
Volume I opens with a section on mathematical foundations. It covers such topics as axiomatization, the concept of an algorithm, proofs, the theory of sets, the theory of relations, Boolean algebra, and antinomies. The closing section, on the real number system and algebra, takes up natural numbers, groups, linear algebra, polynomials, rings and ideals, the theory of numbers, algebraic extensions of a fields, complex numbers and quaternions, lattices, the theory of structure, and Zorn's lemma.
Volume II begins with eight chapters on the foundations of geometry, followed by eight others on its analytic treatment. The latter include discussions of affine and Euclidean geometry, algebraic geometry, the Erlanger Program and higher geometry, group theory approaches, differential geometry, convex figures, and aspects of topology.
Volume III, on analysis, covers convergence, functions, integral and measure, fundamental concepts of probability theory, alternating differential forms, complex numbers and variables, points at infinity, ordinary and partial differential equations, difference equations and definite integrals, functional analysis, real functions, and analytic number theory. An important concluding chapter examines "The Changing Structure of Modern Mathematics."
Author(s): H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle
Edition: 1
Publisher: MIT Press
Year: 1974
Language: English
Pages: 561
City: Cambridge, Massachusetts
Tags: Foundations of Mathematics, Algebra, Real number System
CONTENTS
TRANSLATOR’S FOREWORD IX
FROM THE PREFACE (TO THE 1958 EDITION),
HEINRICH BEHNKE AND KUNO FLADT X
PART A 1
FOUNDATIONS OF MATHEMATICS H. HERMES AND W. MARKWALD
1. CONCEPTIONS OF THE NATURE OF MATHEMATICS 3
2. LOGICAL ANALYSIS OF PROPOSITIONS 9
3. THE CONCEPT OF A CONSEQUENCE 20
4. AXIOMATIZATION - 26
5. THE CONCEPT OF AN ALGORITHM 32
6. PROOFS 41
7. THEORY OF SETS 50
8. THEORY OF RELATIONS 61
9. BOOLEAN ALGEBRA 66
10. AXIOMATIZATION OF THE NATURAL NUMBERS 71
11. ANTINOMIES 80
BIBLIOGRAPHY 86
PART B 89
ARITHMETIC AND ALGEBRA INTRODUCTION, W. GROBNER , 91
CHAPTER 1
CONSTRUCTION OF THE SYSTEM OF REAL NUMBERS, G. PICKERT AND L.GORKE 93
1. THE NATURAL NUMBERS 93
2. THE INTEGERS 105
3. THE RATIONAL NUMBERS 121
4. THE REAL NUMBERS 129
APPENDIX: ORDINAL NUMBERS, D. KUREPA AND A. AYMANNS 153
CHAPTER 2
GROUPS, W. GASCHIITZ AND H. NOACK 166
1. AXIOMS AND EXAMPLES 167
2. IMMEDIATE CONSEQUENCES OF THE AXIOMS FOR A GROUP 178
3. METHODS OF INVESTIGATING THE STRUCTURE OF GROUPS 182
4. ISOMORPHISMS 188
5. CYCLIC GROUPS 191
6. NORMAL SUBGROUPS AND FACTOR GROUPS 194
7. THE COMMUTATOR GROUP 197
8. DIRECT PRODUCTS 198
9. ABELIAN GROUPS 199
10. THE HOMOMORPHISM THEOREM 212
11. THE ISOMORPHISM THEOREM 214
12. COMPOSITION SERIES, JORDAN-HOLDER THEOREM 215
13. NORMALIZER, CENTRALIZER, CENTER 217
14. P-GROUPS 219
15. PERMUTATION GROUPS 220
16. SOME REMARKS ON MORE GENERAL INFINITE GROUPS 230
CHAPTER 3
LINEAR ALGEBRA, H. GERICKE AND H. WÉISCHE 233
1. THE CONCEPT OF A VECTOR SPACE 235
2. LINEAR TRANSFORMATIONS OF VECTOR SPACES 246
3. PRODUCTS OF VECTORS 266
CHAPTER 4
POLYNOMIALS, G. PICKERT AND W. RTICKERT 291
1. ENTIRE RATIONAL FUNCTIONS ' 291
2. POLYNOMIALS 296
3. THE USE OF INDETERMINATES AS A METHOD OF PROOF 312
CHAPTER 5
RINGS AND IDEALS, W. GROBNER AND P. LESKY 316
1. RINGS, INTEGRAL DOMAINS, FIELDS I 316
2. DIVISIBILITY IN INTEGRAL DOMAINS 327
3. IDEALS IN COMMUTATIVE RINGS, PRINCIPAL IDEAL RINGS, RESIDUE
CLASS RINGS 338
4. DIVISIBILITY IN POLYNOMIAL RINGS ELIMINATION 346
CHAPTER 6
THEORY OF NUMBERS, H.-H. OSTMANN AND H. LIERMANN 355
1. INTRODUCTION 355
2. DIVISIBILITY THEORY 355
3. CONTINUED FRACTIONS 372
4. CONGRUENCES 380
5. SOME NUMBER-THEORETIC FUNCTIONS; THE MÖBIUS INVERSION
FORMULA 388
6. THE CHINESE REMAINDER THEOREM; DIRECT DECOMPOSITION OF
C/(M) 391
7. DIOPHANTINE EQUATIONS; ALGEBRAIC CONGRUENCES 395
8. ALGEBRAIC NUMBERS 401
9. ADDITIVE NUMBER THEORY 405
CHAPTER 7
ALGEBRAIC EXTENSIONS OF A FIELD, 0. HAUPT AND P. SENGENHORST 409
1. THE SPLITTING FIELD OF A POLYNOMIAL 410
2. FINITE EXTENSIONS 418
3. NORMAL EXTENSIONS 420
4. SEPARABLE EXTENSIONS 422
5. ROOTS OF UNITY 425
6. ISOMORPHIC MAPPINGS OF SEPARABLE FINITE EXTENSIONS 431
7. NORMAL FIELDS AND THE AUTOMORPHISM GROUP (GALOIS GROUP) 433
8. FINITE FIELDS 438
9. IRREDUCIBILITY OF THE CYCLOTOMIC POLYNOMIAL AND STRUCTURE OF
THE GALOIS GROUP OF THE CYCLOTOMIC FIELD OVER THE FIELD OF RATIONAL NUMBERS 448
10. SOLVABILITY BY RADICALS. EQUATIONS OF THE THIRD AND FOURTH DEGREE 452
CHAPTER 8
COMPLEX NUMBERS AND QUATERNIONS, G. PICKERT AND H.-G.STEINER . 456
1. THE COMPLEX NUMBERS 456
2. ALGEBRAIC CLOSEDNESS OF THE FIELD OF COMPLEX NUMBERS 462
3. QUATERNIONS 467
CHAPTER 9
LATTICES, H. GERICKE AND H. MARTENS 483
1. PROPERTIES OF THE POWER SET 485
2. EXAMPLES 490
3. LATTICES OF FINITE LENGTH 495
4. DISTRIBUTIVE LATTICES 497
5. MODULAR LATTICES 501
6. PROJECTIVE GEOMETRY 505
CHAPTER 10
SOME BASIC CONCEPTS FOR A THEORY OF STRUCTURE, H. GERICKE AND H. MARTENS 508
1. CONFIGURATIONS 509
2. STRUCTURE 515
CHAPTER 11
ZORN’S LEMMA AND THE HIGH CHAIN PRINCIPLE, H. WOLFI" AND H. NOACK 522
1. ORDERED SETS 522
2. ZORN’S LEMMA 524
3. EXAMPLES OF THE APPLICATION OF ZORN’S LEMMA 525
4. PROOF OF ZORN’S LEMMA FROM THE AXIOM OF CHOICE 529
5. QUESTIONS CONCERNING THE FOUNDATIONS OF MATHEMATICS 534
BIBLIOGRAPHY 536
INDEX 537
