Infinity And The Mind

by Rucker, Rudy Von B.

• ISBN13: 9780691121277
• ISBN10: 0691121273
• Format: Paperback
• Copyright: 2004-11-01
• Publisher: Princeton Univ Pr

Summary

InInfinity and the Mind,Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Rucker acquaints us with Gouml;del's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations.Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gouml;del's incompleteness theorems. His personal encounters with Gouml;del the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism.

Table of Contents

Preface to the 2005 Edition	ix
Preface to the Paperback Edition	xvii
Preface	xix
Chapter One: Infinity	1	(52)
A Short History of Infinity	1	(8)
Physical Infinities	9	(26)
Temporal Infinities	10	(5)
Spatial Infinities	15	(9)
Infinities in the Small	24	(10)
Conclusion	34	(1)
Infinities in the Mindscape	35	(9)
The Absolute Infinite	44	(5)
Connections	49	(2)
Puzzles and Paradoxes	51	(2)
Chapter Two: All the Numbers	53	(40)
From Pythagoreanism to Cantorism	53	(11)
Transfinite Numbers	64	(14)
From Omega to Epsilon-Zero	65	(8)
The Alefs	73	(5)
Infinitesimals and Surreal Numbers	78	(9)
Higher Physical Infinities	87	(4)
Puzzles and Paradoxes	91	(2)
Chapter Three: The Unnameable	93	(64)
The Berry Paradox	93	(14)
Naming Numbers	95	(5)
Understanding Names	100	(7)
Random Reals	107	(36)
Constructing Reals	108	(12)
The Library of Babel	120	(6)
Richard's Paradox	126	(4)
Coding the World	130	(13)
What is Truth?	143	(9)
Conclusion	152	(3)
Puzzles and Paradoxes	155	(2)
Chapter Four: Robots and Souls	157	(32)
Gödel's Incompleteness Theorem	157	(7)
Conversations with Godel	164	(7)
Towards Robot Consciousness	171	(14)
Formal Systems and Machines	172	(3)
The Liar Paradox and the Non-Mechanizability of Mathematics	175	(5)
Artificial Intelligence via Evolutionary Processes	180	(3)
Robot Consciousness	183	(2)
Beyond Mechanism?	185	(2)
Puzzles and Paradoxes	187	(2)
Chapter Five: The One and the Many	189	(32)
The Classical One/Many Problem	189	(2)
What is a Set?	191	(5)
The Universe of Set Theory	196	(10)
Pure Sets and the Physical Universe	196	(6)
Proper Classes and Metaphysical Absolutes	202	(4)
Interface Enlightenment	206	(13)
One/Many in Logic and Set Theory	207	(2)
Mysticism and Rationality	209	(5)
Satori	214	(5)
Puzzles and Paradoxes	219	(2)
Excursion One: The Transfinite Cardinals	221	(46)
On and Alef-One	221	(5)
Cardinality	226	(12)
The Continuum	238	(15)
Large Cardinals	253	(14)
Excursion Two: Gödel's Incompleteness Theorems	267	(28)
Formal Systems	267	(13)
Self-Reference	280	(5)
Gödel's Proof	285	(7)
A Technical Note on Man-Machine Equivalence	292	(3)
Answers to the Puzzles and Paradoxes	295	(12)
Notes	307	(22)
Bibliography	329	(10)
Index	339