Paradoxes and Sophisms in Calculus

Sergiy Klymchuk e Susan G. Staples

The Mathematical Association of America | 2013 | páginas | rar - pdf |965 kb

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In the study of mathematics, surprising and counterintuitive examples can offer a fascinating insight into the development of the subject, and inspire a learner's passion for discovery. With a carefully chosen selection of so-called paradoxes and sophisms, this book offers a delightful supplementary resource to enhance the study of single variable calculus. By paradox, the authors mean an unexpected statement that looks invalid, but is in fact true. The word sophism describes intentionally invalid reasoning that looks formally correct, but, in fact, contains a subtle mistake or flaw. This collection of over fifty paradoxes and sophisms showcases the subtleties of calculus and leads students to contemplate the underlying concepts. Sophisms and paradoxes from the areas of functions, limits, derivatives, integrals, sequences and series are explored, with full explanations provided for each example. The book is an ideal resource for those studying or teaching calculus at high school and university level.

Contents
Introduction;
Part I. Paradoxes:
1. Functions and limits;
2. Derivatives and integrals;
Part II. Sophisms:
3. Functions and limits;
4. Derivatives and integrals;
Part III. Solutions to Paradoxes:
5. Functions and limits;
6. Derivatives and integrals;
Part IV. Solutions to Sophisms:
7. Functions and limits;
8. Derivatives and integrals;
References.

Interdisciplinarity, Creativity, and Learning: Mathematics with Literature, Paradoxes, History, Technology, and Modeling

(Montana mathematics enthusiast,  7)

Bharath Sriraman, Viktor Freiman e Nicole Lirette-Pitre 

Information Age Publishing | 2009 | 261 páginas | rar - pdf | 2,7 Mb

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A Volume in The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education Series Editor Bharath Sriraman, The University of Montana Interdisciplinarity is increasingly viewed as a necessary ingredient in the training of future oriented 21st century disciplines that rely on both analytic and synthetic abilities across disciplines. Nearly every curricular document or vision statement of schools and universities include a call for promoting creativity in students. Yet the construct of creativity and giftedness across disciplines remains elusive in the sense that the prototypical examples of such work come from eminent scientists, artists and mathematicians, and little if any work has been conducted with non-eminent individuals. This monograph is an attempt to fill this gap by putting forth the view that interdisciplinarity and creativity are related constructs, and that the cultivation of domain general creativity is possible. Mathematics has historically been anchored to numerous disciplines like theology, natural philosophy, culture and art, allowing for a flexibility of thought that is difficult to cultivate in other disciplines. In this monograph, the numerous chapters from Australia, U.S.A., Canada, Cyprus, Denmark and Japan provide a compelling illustration of the intricate connection of mathematics with literature, paradoxes, history, technology and modeling, thus serving as a conduit for interdisciplinarity, creativity and learning to occur.

ContentsSection I Interdisciplinarity in Mathematics and Literature
1 The Interdisciplinary Nature of Inductive Processes in Forming Generalizations.... 3
Bharath Sriraman and Harry Adrian
2 The Existential Void in Learning: Juxtaposing Mathematics and Literature.... 13
Bharath Sriraman and Harry Adrian
3 Mathematics and Literature: Synonyms, Antonyms or the Perfect Amalgam?...... 31
Bharath Sriraman
4 Mathematics and Literature (The Sequel): Imagination as a Pathway to Advanced Mathematical Ideas and Philosophy..... 41
Bharath Sriraman
Section II Mathematics and Paradoxes 
5 1 or 0?: Cantorian Conundrums in the Contemporary Classroom.... 55
Bharath Sriraman and Libby Knott
6 Understanding Mathematics through Resolution of Paradoxes...... 61
Margo Kondratieva
7 Mathematical Paradoxes as Pathways into Beliefs and Polymathy.... 75
Bharath Sriraman
Section II I Geometry and History
8 Voronoi Diagrams.... 97
Michael Mumm
9 An In-Depth Investigation of the Divine Ratio.... 109
Birch Fett
10 Cyclide Manipulation........ 133
Akihiro Matsuura
Section IV Interdisciplinarity and Modeling
11 Modeling Interdisciplinary Activities Involving Mathematics and Philosophy... 147
Steffen M. Iversen
12 Integrating Engineering Education within the Elementary and Middle School Mathematics Curriculum.... 165
Lyn D. English and Nicholas G. Mousoulides
13 Mathematical Modelling in the Early School Years........... 177
Lyn D. English and James J. Watters
Section V Technology and the NET Generation
14 Connected Giftedness: Mathematical Problem Solving by Means of a Web Technology: Case of the CASMI Project...... 205
Viktor Freiman and Nicole Lirette-Pitre
15 Teaching and Learning for the Net Generation: A Robotic-Based Learning Approach....217
Samuel Blanchard
16 Does Technology Help Building More Creative Mathematical Environments?... 233
Dominic Manuel

Paradoxes in Mathematics

(Dover Books on Mathematics) 

Stanley J. Farlow

Dover Publications | 2014 | 192 páginas | rar - epub | 6,48 Mb


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There's more than one way to define a paradox, and this intriguing book offers examples of every kind. Stanley J. Farlow, a prominent educator and author, presents a captivating mix of mathematical paradoxes: the kind with surprising, nonintuitive outcomes; the variety that rely on mathematical sleight-of-hand to impress the unwary observer; and the baffling type with a solution that passes all understanding.
Students and puzzle enthusiasts will find plenty of thought-provoking enjoyment mixed with a bit of painless mathematical instruction among these twenty-eight conundrums. Some of them involve counting, some deal with infinity, and others draw on principles of geometry and arithmetic. None requires an extensive background in higher mathematics. Challenges include The Curve That Shook the World, a variation on the famous Monty Hall Problem, Space Travel in a Wineglass, Through Cantor's Looking Glass, and other fun-to-ponder paradoxes.

Table of Contents
Introduction
Paper Folding: Do Not Try This at Home
Rope Around the Earth
Oh No! The Monty Hall Problem!
The Birthday Paradox: You’re NOT Going to Believe This
Mathematical Masonry
Aaaaaaaaaaaaaagh: The Curry Paradox
The Not-So-Uncommon Anomaly of Mr. Simpson
Some Paradoxes Can (Literally) Kill You
The Curve That Shook the World
Mary’s Paint Can Conundrum
The China Paradox
The Two-Envelope Paradox
St. Petersburg Paradox
Through Cantor’s Looking Glass
Aristotle’s Wheel Paradox
Did I Ever Tell You About My Uncle Charlie?
Parrondo’s Paradox: A Paradox for Losers
Zeno’s Paradox
Will the Real Numbers Please Stand Up?
The Four Bears Paradox
Uncle George’s Will
Milos and the Tainted Wine
Space-Filling Curve
Wheels: More Than Just Round and Round
Space Travel in a Wineglass
The Mysterious Transcendental Numbers
Gabriel’s Horn Paradox
Russell’s Paradox: Why We Need Rules

Diamond: A Paradox Logic

Series on Knots and Everything (Book 23)

N. S. Hellerstein


World Scientific Publishing Company; 2nd Revised edition | 2010 |  297 páginas | pdf | 1,8 Mb

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This book is about 'diamond', a logic of paradox. In diamond, a statement can be true yet false; an 'imaginary' state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown modulator"; and Metamathematical Dilemma, which relates diamond to Godelian meta-mathematics and dilemma games.

Contents
Introduction xi
Part One: Paradox Logic 1
1. Paradox 3
A. The Liar 3
B. The Anti-Diagonal 7
C. Russell’s Paradox 8
D. Santa and the Grinch 10
E. Antistrephon 17
F. Parity of Infinity 18
G. The Heap 19
H. Finitude 21
I. Game Paradoxes 24
J. Cantor’s Paradox 26
K. Paradox of the Boundary 27
2. Diamond 29
A. The Buzzer 29
B. Diamond Values 31
C. Harmonic Functions 32
D. Gluts and Gaps 35
E. Diamond Circuits 38
F. Brownian Forms 40
G. Boundary Logic 46
3. Diamond Algebra 49
A. Bracket Algebra 49
B. Laws 58
C. Normal Forms 64
D. Completeness 69
4. Self-Reference 71
A. Re-Entrance and Fixedpoints 71
B. Phase Order 74
C. The Outer Fixedpoints 79
5. Fixedpoint Lattices 85
A. Relative Lattices 85
B. Seeds and Spirals 88
C. Shared Fixedpoints 91
D. Examples 93
6. Limit Logic 109
A. Limits 109
B. Limit Fixedpoints 114
C. The Halting Theorem 116
7. Paradox Resolved 119
A. The Liar and the Anti-Diagonal 119
B. Russell’s Paradox 120
C. Santa and the Grinch 122
D. Antistrephon 126
E. Infinity, Finitude and the Heap 128
F. Game Paradoxes 130
8. The Continuum 131
A. Cantor’s Paradox 131
B. Dedekind Splices 132
C. Null Quotients 134
D. Cantor’s Number 136
E. The Line within the Diamond 139
F. Zeno’s Theorem 144
G. Fuzzy Chaos 145
9. Clique Theory 151
A. Cliques 151
B. Clique Equality 156
C. Clique Axioms 160
D. Graph Cliques 164
E. Clique Circuits 166
Part Two: The Second Paradox 169
10. Orthogonal Logics 171
A. Analytic Functions 171
B. Function Types 173
C. Dihedral Conjugation 177
D. Star Logic 180
E. Harmonic Projection 182
F. Diamond Types? 183
11. Interferometry 185
A. Quadrature 185
B. Diffraction 193
C. Buzzers and Toggles 200
D. Analytic Diffraction 204
E. Diffracting “Two Ducks in a Box” 206
12. How to Count to Two 209
A. Brownian and Kauffman Modulators 209
B. Diffracting the Modulators 212
C. Rotors, Pumps and Tapes 218
D. The Ganglion 222
Part Three: Metamathematical Dilemma 225
13. Metamathematics 227
A. Gödelian Quanta 227
B. Meta-Logic 231
C. Dialectic 236
D. Dialectical Dilemma 238
14. Dilemma 241
A. Milo’s Trick 241
B. Prisoner’s Dilemma 245
C. Dilemma Games 249
D. Dilemma Diamond 253
E. Banker’s Dilemma 257
F. The Unexpected Departure 260
Notes 265
Bibliography 287
Index 291

Vicious Circles and Infinity, A Panoply of Paradoxes

Patrick Hughes, George Brecht

Jonathan Cape | 1976 | 112 páginas | Djvu | 1,63 Mb

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The anthology has no particular ax to grind, apart from that of having no particular ax to grind. 

We hope you enjoy the paradoxes, verbal and visual, as much as we do. We have tried to let the paradoxes speak 

for themselves. The ratio of comment to material is great in the field of paradox; the paradoxes are few and brief, the commentaries many and lengthy. As is often the case, it is easier to talk about a thing than it is to do it. 

One of the bonds between the authors of this book is the sense of humor; perhaps the paradoxes should be seen in this light.

Logic & mathematical paradoxes

Sindy Dunbar

White Word Publications | 2012 | 99 páginas | PDF | 629 kb

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Table of Contents

Chapter 1 - Accuracy Paradox & Apportionment Paradox

Chapter 2 - All Horses are the Same Color & Infinite Regress

Chapter 3 - Drinker Paradox & Lottery Paradox

Chapter 4 - Paradoxes of Material Implication

Chapter 5 - Raven Paradox

Chapter 6 - Unexpected Hanging Paradox

Chapter 7 - Banach–Tarski Paradox

Chapter 8 - Coastline Paradox & Paradoxical Set

Chapter 9 - Gabriel's Horn & Missing Square Puzzle

Chapter 10 - Smale's Paradox & Hausdorff Paradox

Chapter 11 - Borel–Kolmogorov Paradox & Berkson's Paradox

Chapter 12 - Boy or Girl Paradox & Burali-Forti Paradox

Chapter 13 - Elevator Paradox

Chapter 14 - Gödel's Incompleteness Theorems

Chapter 15 - Gambler's Fallacy

Impossible?: Surprising Solutions to Counterintuitive Conundrums

Julian Havil

Princeton University Press | 2008 | 264 páginas | PDF | 1,04 MB

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In Nonplussed!, popular-math writer Julian Havil delighted readers with a mind-boggling array of implausible yet true mathematical paradoxes. Now Havil is back with Impossible?, another marvelous medley of the utterly confusing, profound, and unbelievable--and all of it mathematically irrefutable. Whenever Forty-second Street in New York is temporarily closed, traffic doesn't gridlock but flows more smoothly--why is that? Or consider that cities that build new roads can experience dramatic increases in traffic congestion--how is this possible? What does the game show Let's Make A Deal reveal about the unexpected hazards of decision-making? What can the game of cricket teach us about the surprising behavior of the law of averages? These are some of the counterintuitive mathematical occurrences that readers encounter inImpossible? Havil ventures further than ever into territory where intuition can lead one astray. He gathers entertaining problems from probability and statistics along with an eclectic variety of conundrums and puzzlers from other areas of mathematics, including classics of abstract math like the Banach-Tarski paradox. These problems range in difficulty from easy to highly challenging, yet they can be tackled by anyone with a background in calculus. And the fascinating history and personalities associated with many of the problems are included with their mathematical proofs. Impossible? will delight anyone who wants to have their reason thoroughly confounded in the most astonishing and unpredictable ways. Julian Havil is a retired former master at Winchester College, England, where he taught mathematics for thirty-three years. In addition to Nonplussed!, he is the author of Gamma: Exploring Euler's Constant (both Princeton). Table of Contents: Acknowledgments xi Introduction 1 Chapter 1: It's Common Knowledge 3 Chapter 2: Simpson's Paradox 11 Chapter 3: The Impossible Problem 21 Chapter 4: Braess's Paradox 31 Chapter 5: The Power of Complex Numbers 39 Chapter 6: Bucking the Odds 50 Chapter 7: Cantor's Paradise 68 Chapter 8: Gamow-Stern Elevators 82 Chapter 9: The Toss of a Coin 88 Chapter 10: Wild-Card Poker 103 Chapter 11: Two Series 113 Chapter 12: Two Card Tricks 131 Chapter 13: The Spin of a Needle 146 Chapter 14: The Best Choice 165 Chapter 15: The Power of Powers 176 Chapter 16: Benford's Law 190 Chapter 17: Goodstein Sequences 201 Chapter 18: The Banach-Tarski Paradox 210 The Motifs 217 Appendix 221 Index 233

The Pea and the Sun: A Mathematical Paradox

L Wapner

AK Peters | 2005 | 232 páginas |

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Take an apple and cut it into five pieces. Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski s magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

Mathematical Fallacies and Paradoxes

Bryan Bunch

Dover Publications | 1997 | 224 páginas | djvu

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Descrição: Stimulating, thought-provoking analysis of a number of the most interesting intellectual inconsistencies in mathematics, physics and language. Delightful elucidations of methods for misunderstanding the real world of experiment (Aristotle’s Circle paradox), being led astray by algebra (De Morgan’s paradox) and other mind-benders. Some high school algebra and geometry is assumed; any other math needed is developed in text.

Riddles in Mathematics

Eugene P. Northrop

D Van Nostrand Company, Inc. | 1944 | 262 páginas

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Descrição: Presents over two hundred brain-teasers drawn from arithmetic, algebra, geometry, trigonometry and calculus.

Diamond: A Paradox Logic

(Series on Knots and Everything, Vol 14)
N. S. Hellerstein

World Scientific Publishing Company | 1997 | 257 páginas |
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Descrição: This book is about "diamond", a logic of paradox. In diamond, a statement can be true yet false; an "imaginary" state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. Diamond is a new way to solve the dilemmas of higher mathematics. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book consists of two sections: Elementary; which covers the classic paradoxes of mathematical logic and shows how they can be resolved in this new system; and Advanced, which relates diamond to Boolean logic, three-valued logic, Gödelian meta-mathematics and dilemma games.