Goedel's Way: Exploits into an undecidable world

Gregory Chaitin, Francisco A Doria e Newton C.A. da Costa

CRC Press | 2011 | 162 páginas | pdf | 1,1 Mb

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Kurt Gödel (1906-1978) was an Austrian-American mathematician, who is best known for his incompleteness theorems. He was the greatest mathematical logician of the 20th century, with his contributions extending to Einstein’s general relativity, as he proved that Einstein’s theory admits time machines. 
The Gödel incompleteness theorem - one cannot prove nor disprove all true mathematical sentences in the usual formal mathematical systems- is frequently presented in textbooks as something that happens in the rarefied realm of mathematical logic, and that has nothing to do with the real world. Practice shows the contrary though; one can demonstrate the validity of the phenomenon in various areas, ranging from chaos theory and physics to economics and even ecology. In this lively treatise, based on Chaitin’s groundbreaking work and on the da Costa-Doria results in physics, ecology, economics and computer science, the authors show that the Gödel incompleteness phenomenon can directly bear on the practice of science and perhaps on our everyday life.
This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no technicalities. Besides theory, the historical report and personal stories about the main character and on this book’s writing process, make it appealing leisure reading for those interested in mathematics, logic, physics, philosophy and computer science.
See also: http://www.youtube.com/watch?v=REy9noY5Sg8

Contents
1. Gödel, Turing 
2. Complexity, randomness 
3. A list of problems 
4. The halting function and its avatars 
5. Entropy, P vs. NP
6. Forays into uncharted landscapes.

The Child’s Conception of Geometry

 

Jean Piaget, Barbel Inhelder e Alina Szeminska

International Library of Psychology

Routledge |1999 - 2ª edição | 420 páginas | rar - pdf |7,9 Mb

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The Child’s Conception of Geometry examines the development of geometric concepts in young children.

This volume from Piaget’s laboratory in Geneva deals primarily with the development of notions of measurement and geometrical concepts like coordinates, angles, and areas. It is a companion piece to The Child’s Conception of Space.

CONTENTS
Preface page vii
PART ONE - INTRODUCTION
I Change of Position 3
II Spontaneous Measurement 27
PART TWO - CONSERVATION AND MEASUREMENT OF LENGTH
III Reconstructing Relations of Distance 69
IV Change of Position and the Conservation of Length 90
V The Conservation and Measurement of Length 104
VI Subdividing a Straight Line 128
PART THREE - RECTANGULAR COORDINATES, ANGLES AND CURVES
VII Locating a Point in Two or Three Dimensional Space 153
VIII Angular Measurement 173
IX Two Problems of Geometrical Loci: the Straight Line and the Circle 209
X Representation of Circles, Mechanical and Composite Curves 226
PART FOUR - AREAS AND SOLIDS
XI The Conservation and Measurement of an Area and Subtracting Smaller Congruent Areas from Larger Congruent Areas 
XII Subdivision of Areas and the Concept of Fractions 302
XIII Doubling an Area or a Volume 336
XIV The Conservation and Measurement of Volume 354
PART FIVE CONCLUSIONS
XV The Construction of Euclidean Space: Three Levels 389
Index 409

Mathematics of the Transcendental: Onto-logy and being-there

 

Alain Badiou, A.J. Bartlett e Alex Ling

Bloomsbury Academic | 2014 | 291 páginas | rar - pdf | 2,74 Mb

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In Mathematics of the Transcendental, Alain Badiou painstakingly works through the pertinent aspects of Category Theory, demonstrating their internal logic and veracity, their derivation and distinction from Set Theory, and the 'thinking of being'. In doing so he sets out the basic onto-logical requirements of his greater and transcendental logics as articulated in his magnum opus, Logics of Worlds. This important book combines both his elaboration of the disjunctive synthesis between ontology and onto-logy (the discourses of being as such and being-appearing) from the perspective of Category Theory and the categorial basis of his philosophical conception of 'being there'.
Hitherto unpublished in either French or English, Mathematics of the Transcendental provides Badiou's readers with a much-needed complete elaboration of his understanding and use of Category Theory. The book is an essential aid to understanding the mathematical and logical basis of his theory of appearing as elaborated in Logics of Worlds and other works and is essential reading for his many followers.

TABLE OF CONTENTS
Translators’ Introduction: The Categorial Imperative 1
PART ONE TOPOS, OR LOGICS OF ONTO-LOGY: AN INTRODUCTION FOR PHILOSOPHERS 11
1 General Aim 13
2 Preliminary Definitions 17
3 The Size of a Category 21
4 Limit and Universality 27
5 Some Fundamental Concepts 29
6 Duality 37
7 Isomorphism 41
8 Exponentiation 45
9 Universe, 1: Closed Cartesian Categories 51
10 Structures of Immanence, 1: Philosophical Considerations 55
11 Structures of Immanence, 2: Sub-Object 59
12 Structures of Immanence, 3: Elements of an Object 63
13 ‘Elementary’ Clarification of Exponentiation 67
14 Central Object (or Sub-Object Classifier) 71
15 The True, the False, Negation and More 77
16 The Central Object as Linguistic Power 85
17 Universe, 2: The Concept of Topos 89
18 Ontology of the Void and Difference 95
19 Mono., Epi., Equ., and Other Arrows 99
20 Topoi as Logical Places 113
21 Internal Algebra of 1 123
22 Ontology of the Void and Excluded Middle 141
23 A Minimal Classical Model 147
24 A Minimal Non-Classical Model 151
PART TWO BEING THERE: MATHEMATICS OF THE TRANSCENDENTAL 163
Introduction 165
A. Transcendental Structures 171
B. Transcendental Connections 183
B.1. Connections between the transcendental and set-theoretic ontology: Boolean algebras 183
B.2. Connections between the transcendental and logic in its ordinary sense (propositional logic and first order predicate logic) 195
B.3. Connection between the transcendental and the general theory of localizations: Topology 202
C. Theory of Appearing and Objectivity 217
D. Transcendental Projections: Theory of Localization 235
E. Theory of Relations: Situation as Universe 249
Appendix: On Three Different Concepts of Identity Between Two
Multiples or Two Beings 265
Translator’s Endnotes 269
Index 277

150 Puzzles in Crypt-Arithmetic

Maxey Brooke

Dover | 1963 | 77 páginas | pdf | 1,1 Mb

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Harmony of the World - 75 Years of Mathematics Magazine

Gerald L. Alexanderson e Peter Ross 

 Mathematical Association of America | 2007 | 302 páginas | rar - pdf |3,8 Mb

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Who would expect to find in the pages of Mathematics Magazine the first full treatment of one of the more important and oft-cited twentieth century theorems in analysis, the Stone-Weierstrass Theorem in an article by Marshall Stone himself? Where else would one look for proofs of trigonometric identities using commutative ring theory? Or one of the earliest and best expository articles on the then new Jones knot polynomials, an article that won the prestigious Chauvenet Prize? Or an amusing article purporting to show that the value of has been time dependent over the years? These and much more are in this collection of the best from Mathematics Magazine. Readers are inundated with new material in the many mathematical journals. Gems from past issues of Mathematics Magazine or the Monthly or the College Mathematics Journal are read with pleasure when they appear but get pushed into the background when the next issues arrive. So from time to time it is rewarding to go back and see just what marvelous material has been published over many years, articles now to some extent forgotten. There is history of mathematics (algebraic numbers, inequalities, probability and the Lebesgue integral, quaternions, Pólya s enumeration theorem, and group theory) and stories of mathematicians (Hypatia, Gauss, E. T. Bell, Hamilton, and Euler). The list of authors is star-studded: E. T. Bell, Otto Neugebauer, D. H. Lehmer, Morris Kline, Einar Hille, Richard Bellman, Judith Grabiner, Paul Erdos, B. L. van der Waerden, Paul R. Halmos, Doris Schattschneider, J. J. Burckhardt, Branko Grunbaum, and many more. Eight of the articles included have received the Carl B. Allendoerfer or Lester R. Ford Awards.

Contents

The name of each article is followed by a notation indicating the field of mathematics from which it comes: (Al) algebra, (AM) applied mathematics, (An) analysis, (CG) combinatorics and graph theory, (G) geometry, (H) history, (L) logic, (M) miscellaneous, (NT) number theory, (PS) probability/statistics, and (T) topology.

Introduction .  .vii

A Brief History of Mathematics Magazine .  . xi

Part I: The First Fifteen Years .. .1

Perfect Numbers, Zena Garrett (NT) .. .3

Rejected Papers of Three Famous Mathematicians, Arnold Emch (H) .  5

Review of Men of Mathematics, G. Waldo Dunnington (H) .. 9

Oslo under the Integral Sign, G. Waldo Dunnington (H) . 11

Vigeland’s Monument to Abel in Oslo, G. Waldo Dunnington (H) . .19

The History of Mathematics, Otto Neugebauer (H) . .. .23

Numerical Notations and Their Influence on Mathematics, D. H. Lehmer (NT) .  .29

Part II: The 1940s  .33

The Generalized Weierstrass Approximation Theorem, Marshall H. Stone (An) . . .35

Hypatia of Alexandria, A. W. Richeson (H) .. . 45

Gauss and the Early Development of Algebraic Numbers, E. T. Bell (Al)  . . 51

Part III: The 1950s . . 69

The Harmony of the World, Morris Kline (M) .. 71

What Mathematics Has Meant to Me, E. T. Bell (M) .. .79

Mathematics and Mathematicians from Abel to Zermelo, Einar Hille (H) .  81

Inequalities, Richard Bellman (An) . . 95

A Number System with an Irrational Base, George Bergman (NT) . 99

Part IV: The 1960s  .107

Generalizations of Theorems about Triangles, Carl B. Allendoerfer (G) . .109

A Radical Suggestion, Roy J. Dowling (NT) . .115

Topology and Analysis, R. C. Buck (An, T). 117

The Sequence fsin ng, C. Stanley Ogilvy (An) . .. . 121

Probability Theory and the Lebesgue Integral, Truman Botts (PS) . . . 123

On Round Pegs in Square Holes and Square Pegs in Round Holes, David Singmaster (G) . . 129

t : 1832–1879, Underwood Dudley (M) . . . . 133

Part V: The 1970s .  . 135

Trigonometric Identities, Andy R. Magid (Al) . .  .137

A Property of 70, Paul Erdos (NT) . . . .139

Hamilton’s Discovery of Quaternions, B. L. van der Waerden (Al, H) .  . 143

Geometric Extremum Problems, G. D. Chakerian and L. H. Lange (G . .151

Polya’s Enumeration Theorem by Example, Alan Tucker (CG) . 161

Logic from A to G, Paul R. Halmos (L) . . . . 169

Tiling the Plane with Congruent Pentagons, Doris Schattschneider (G) .. . 175

Unstable Polyhedral Structures, Michael Goldberg (G) . . 191

Part VI: The 1980s .  .197

Leonhard Euler, 1707–1783, J. J. Burckhardt (H) .  . .199

Love Affairs and Differential Equations, Steven H. Strogatz (An)  . 211

The Evolution of Group Theory, Israel Kleiner (Al) . . . 213

Design of an Oscillating Sprinkler, Bart Braden (AM) . . . 229

The Centrality of Mathematics in the History of Western Thought, Judith V. Grabiner (M) . ..237

Geometry Strikes Again, Branko Gr¨unbaum (G) .. 247

Why Your Classes Are Larger than “Average”, David Hemenway (PS) . 255

The New Polynomial Invariants of Knots and Links, W. B. R. Lickorish and Kenneth C. Millett (CG, T)  . 257

Briefly Noted .. 273

The Problem Section . . . 279

Index . . . .281

About the Editors . . . 287

Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics

(Synthese Library) 

Ulianov Montano 

Springer | 2014 | 224 páginas | rar - pdf | 1,1 Mb

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This book develops a naturalistic aesthetic theory that accounts for aesthetic phenomena in mathematics in the same terms as it accounts for more traditional aesthetic phenomena. Building upon a view advanced by James McAllister, the assertion is that beauty in science does not confine itself to anecdotes or personal idiosyncrasies, but rather that it had played a role in shaping the development of science. Mathematicians often evaluate certain pieces of mathematics using words like beautiful, elegant, or even ugly. Such evaluations are prevalent, however, rigorous investigation of them, of mathematical beauty, is much less common. The volume integrates the basic elements of aesthetics, as it has been developed over the last 200 years, with recent findings in neuropsychology as well as a good knowledge of mathematics.

The volume begins with a discussion of the reasons to interpret mathematical beauty in a literal or non-literal fashion, which also serves to survey historical and contemporary approaches to mathematical beauty. The author concludes that literal approaches are much more coherent and fruitful, however, much is yet to be done. In this respect two chapters are devoted to the revision and improvement of McAllister’s theory of the role of beauty in science. These antecedents are used as a foundation to formulate a naturalistic aesthetic theory. The central idea of the theory is that aesthetic phenomena should be seen as constituting a complex dynamical system which the author calls the aesthetic as processtheory.

The theory comprises explications of three central topics: aesthetic experience (in mathematics), aesthetic value and aesthetic judgment. The theory is applied in the final part of the volume and is used to account for the three most salient and often used aesthetic terms often used in mathematics: beautiful, elegant and ugly. This application of the theory serves to illustrate the theory in action, but also to further discuss and develop some details and to showcase the theory’s explanatory capabilities.

Contents
Introduction.
Part 1. Antecedents.
Chapter 1. On Non-literal Approaches.
Chapter 2. Beautiful, Literally.
Chapter 3. Ugly, Literally.
Chapter 4. Problems of the Aesthetic Induction.
Chapter 5. Naturalizing the Aesthetic Induction.
Part 2. An Aesthetics of Mathematics.
Chapter 6. Introduction to a Naturalistic Aesthetic Theory.
Chapter 7. Aesthetic Experience.
Chapter 8. Aesthetic Value.
Chapter 9. Aesthetic Judgement I: Concept.
Chapter 10. Aesthetic Judgement II: Functions.
Chapter 11. Mathematical Aesthetic Judgements.
Part 3. Applications.
Chapter 12. Case Analysis I: Beauty.
Chapter 13. Case Analysis II: Elegance.
Chapter 14. Case Analysis III: Ugliness, Revisited.
Chapter 15. Issues of Mathematical Beauty, Revisited.

A History of Astronomy

Walter William Bryant

London Methuen 1907

online: archive.org

Routledge | 2013 | 434 páginas | rar - pdf | 5,52 Mb

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A History of Astronomy, first published in 1907, offers a comprehensive introduction to the steady development of the science since its inception in the ancient world up to the momentous progress of the nineteenth century. It includes biographical material relating to the most famous names in the study of astronomy – Copernicus, Galileo, Newton, Herschel – and their contributions, clear and accessible discussions of key discoveries, as well as detailing the incremental steps in technology with which many of the turning points in astronomy were intimately bound up.

CONTENTS
CHAP. PAGE
I. EARLY NOTIONS
II. THE EASTERN NATIONS OF ANTIQUITY 8
III. THE GREEKS 14
IV. THE ARABS 25
V. THE REVIVAL-COPERNICUS-TYCHO BRAHE 28
VI. KEPLER-GALl LEO 39
'VII. NEWTON 47
VIII. NEWTON'S SUCCESSORS: LAPLACE 53
IX. FLAMSTEED-HALLEy-BRADLEy-HERSCHEL 63
X. THE EARLY NINETEENTH CENTURy-NEPTUNE 73
XI. HERSCHEL-BESSEL-STRUVE • 83
XII. COMETS • 96
XIII. THE SUN-EcLIPSES-PARALLAX 103
XIV. GENERAL ASTRONOMY AND CELESTIAL MECHANICS 118
XV. OBSERVATORIES AND INSTRUMENTS • 132
XVI. ADJUSTMENT OF OBSERVATIONS. PERSONAL ERRORS 141
XVII. THE SUN 146
XVIII. SOLAR SPECTROSCOPY 159
XIX. SOLAR ECLIPSES-SPECTROSCOPY 169
XX. THE MOON 183
XXI. THE EARTH 192
XXII. THE INTERIOR PLANETS 201
XXIII. MARS 209
XXIV. MINOR PLANETS 219
XXV. THE MAJOR PLANETS 226
XXVI. THE SOLAR SYSTEM • 24I
XXVII. COMETS, METEORS, ZODIACAL LIGHT 247
XXVIII. THE STARS-CATALOGUES-PROPER MOTION-PARALLAX-MAGNITUDE 27I
XXIX. DOUBLE STARS 292
XXX. VARIABLE STARS 303
XXXI. CLUSTERS-NEBULIE-MILKY WAY. 318
XXXII. STELLAR SPECTROSCOPY 327
XXXIII. CONCLUSION • 340