1 Paradoxes Guiding Forces in Mathematical Exploration 2 Paradoxes Guiding Forces in Mathematical Exploration Hamza E. Alsamraee 3 This page is intentionally left blank 4 Copyright © 2020 by Hamza E. Alsamraee. All rights reserved. No part of this book may be reproduced or distributed in any form, stored in any data base or retrieval system, or transmitted in any form by any means — electronic, mechanical, photocopy, recording, or otherwise — without prior written permission of the publisher, except as provided by United States of America copyright law. For permission requests, e-mail the publisher at [email protected]. 2020 First edition published September Typeset using LATEX Edited by Rick Keys 978 1 7357156 1 2 ISBN - - - - (paperback) 978 1 7357156 0 5 ISBN - - - - (ebook) 2020917274 LCCN 10 9 8 7 6 5 4 3 2 1 www.hamzaalsamraee.com 5 To the DailyMath community. CONTENTS 11 Author’s Note i introduction 1 why mathematical paradoxes matter 15 11 26 . A Shift in Mathematics Education? 12 29 . The Paradox as an Illusion 121 32 . . The Paradox as a Magic Trick 13 33 . New Mathematics 14 34 . A Guide on How to Use This Book 2 paradoxes in science 37 ii foundational paradoxes 3 zeno s paradoxes 51 ’ 31 56 . The Three Paradoxes 311 56 . . The Dichotomy Argument 312 57 . . The Arrow in Flight 313 57 . . Achilles and the Tortoise 32 58 . Some Resolutions 321 58 . . A Purely Mathematical Attempt 33 65 . Legacy 4 russell s paradox the necessity of rules 67 ’ : 41 70 . Naive Set Theory 42 71 . Russell and his Paradox 43 73 . The Tragic Story of a Genius 44 77 . Resolution 45 78 . Grelling’s Paradox 5 gödel s incompleteness theorems 81 ’ 51 81 . Math wars 511 82 . . Logicism 512 83 . . Formalism 7 8 contents 513 85 . . Intuitionism 52 85 . Gödel’s Surprise 521 87 . . Background 522 95 . . The Toppling of a Regime 53 99 . Hilbert’s Dream: Post-Gödel Era iii infinity 6 does 999 1 107 . ...= ? 61 108 . An Algebraic Argument 62 110 . The Construction of the Real Numbers 63 111 . Geometric Series 64 112 . Alternate Number Systems 641 112 . . Ultrafinitism 642 112 . . Infinitesimals and Hyperreal Numbers 7 hilbert s hotel paradox 115 ’ 71 117 . What Do Our Methods Tell Us? 72 117 . Rethinking the Problem 721 117 . . Finite Guests 722 118 . . Infinitely Many Guests 723 118 . . InfinitelyManyBusesWithInfiniteGuests 73 119 . The New Mathematics 8 are some infinities bigger than others 121 ? 81 123 . Some Simple Observations 82 125 . Diagonalization 83 126 . In Popular Culture 84 126 . Transfinite Numbers 841 127 . . The Continuum Hypothesis 842 127 . . Even Larger Infinities? 9 grandi s series 129 ’ 91 130 . History 911 130 . . Grandi 912 131 . . Leibniz 913 132 . . Euler 914 132 . . Other Approaches 92 133 . Resolution contents 9 93 133 . Legacy 931 134 . . Thomson’s Lamp 10 1+2+3+4+... 137 101 138 . Some Approaches 1011 139 . . Ramanujan’s Approaches 1012 140 . . Zeta Function Regularization 102 142 . Applications in Physics iv geometry that defies intuition 11 gabriel s horn 149 ’ 111 149 . History 112 150 . Derivation 113 153 . Insight 12 the banach tarski paradox 155 - 121 155 . A Useful Analogy 122 156 . A Little Background 1221 156 . . History 1222 158 . . How is This Theorem Unique? 1223 161 . . Hyper-webster 1224 162 . . Measure 123 163 . A Simplified Explanation 124 166 . Real World Examples? v mind bending statistics - 13 berkson s paradox 173 ’ 131 174 . Medical Data 132 175 . Are Attractive People Jerks? 133 180 . More Examples 1331 180 . . DoesHollywoodRuinFilmAdaptations? 14 simpson s paradox 183 ’ 141 184 . College Admission Scandal? 142 185 . Visualization is Key 15 a send off 189 -