Paradoxes: Guiding Forces in Mathematical Exploration

1800·7.67 MB·english

by  Hamza Alsamraee

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Paradoxes
Guiding Forces in
Mathematical Exploration
2
Paradoxes
Guiding Forces in
Mathematical Exploration
Hamza E. Alsamraee
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This page is intentionally left blank
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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
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