How Singular? How singular is the Singularity? Will it happen in an instant? Lets consider again the derivation of the word. In mathematics a singularity is a value that is beyond any limit - in essence, infinity. The Singularity as we have discussed it in this book, does not achieve infinite levels of computation, memory, as any other other measurable attribute. But it certainly achieves vast levels of all these qualities, including intelligence. With the reverse engineering of the human brain we will be able to apply the parallel, self-organizing, chaotic, algorithms of human intelligence to enormously powerful computational substrates. This intelligence will then be in a position to improve its own design, both hardware and software, in a rapidly accelerating iterative process.

"Following his blockbuster biography of Steve Jobs, The Innovators is Walter Isaacson's revealing story of the people who created the computer and the Internet. It is destined to be the standard history of the digital revolution and an indispensable guide to how innovation really happens. What were the talents that allowed certain inventors and entrepreneurs to turn their visionary ideas into disruptive realities? What led to their creative leaps? Why did some succeed and others fail? In his masterly saga, Isaacson begins with Ada Lovelace, Lord Byron's daughter, who pioneered computer programming in the 1840s. He explores the fascinating personalities that cr eated our current digital revolution, such as Vannevar Bush, Alan Turing, John von Neumann, J.C.R. Licklider, Doug Engelbart, Robert Noyce, Bill Gates, Steve Wozniak, Steve Jobs, Tim Berners-Lee, and Larry Page. This is the story of how their minds worked and what made them so inventive. It's also a narrative of how their ability to collaborate and master the art of teamwork made them even more creative. For an era that seeks to foster innovation, creativity, and teamwork, The Innovators shows how they happen"--

For decades, proponents of artificial intelligence have argued that computers will soon be doing everything that a human mind can do. Admittedly, computers now play chess at the grandmaster level, but do they understand the game as we do? Can a computer eventually do everything a human mind can do? In this absorbing and frequently contentious book, Roger Penrose--eminent physicist and winner, with Stephen Hawking, of the prestigious Wolf prize--puts forward his view that there are some facets of human thinking that can never be emulated by a machine. Penrose examines what physics and mathematics can tell us about how the mind works, what they can't, and what we need to know to understand the physical processes of consciousness. He is among a growing number of physicists who think Einstein wasn't being stubborn when he said his "little finger" told him that quantum mechanics is incomplete, and he concludes that laws even deeper than quantum mechanics are essential for the operation of a mind. To support this contention, Penrose takes the reader on a dazzling tour that covers such topics as complex numbers, Turing machines, complexity theory, quantum mechanics, formal systems, Godel undecidability, phase spaces, Hilbert spaces, black holes, white holes, Hawking radiation, entropy, quasicrystals, the structure of the brain, and scores of other subjects. The Emperor's New Mind will appeal to anyone with a serious interest in modern physics and its relation to philosophical issues, as well as to physicists, mathematicians, philosophers and those on either side of the AI debate.

"In 1950, mathematician-philosopher Norbert Wiener ended this classic book on the place of machines in society with a warning: "We shall never receive the right answers to our questions unless we ask the right questions.... The hour is very late, and the choice of good and evil knocks at our door." Wiener, the founder of the science of cybernetics--the study of the relationship between computers and the human nervous system--was widely mislabeled as an advocate for the automation of human life. As The Human Use for Human Beings reveals, his vision was much more complex and interesting, and is more relevant in today's world of AI than anyone could have anticipated. In his new introduction, Brian Christian aptly calls Wiener the "progenitor of contemporary AI-safety discourse." Wiener hoped that machines would release people from relentless and repetitive drudgery to achieve more creative pursuits, yet he anticipated the danger of dehumanizing and displacement. His pioneering views on the human-machine relationship as a "communicative process" are only more crucial now, as we carry in our pockets AI devices that we can literally speak to. His prescient warnings illuminate our contemporary relationships with language, art, and even social media. The Human Use of Human Beings examines the implications of cybernetics for education, law, language, science, technology, as Wiener anticipates the enormous impact--in effect, a third industrial revolution--that the computer has had on our lives."-- Provided by publisher

Lecture I The Early History of Fermat's Last Theorem.- 1 The Problem.- 2 Early Attempts.- 3 Kummer's Monumental Theorem.- 4 Regular Primes.- 5 Kummer's Work on Irregular Prime Exponents.- 6 Other Relevant Results.- 7 The Golden Medal and the Wolfskehl Prize.- Lecture II Recent Results.- 1 Stating the Results.- 2 Explanations.- Lecture III B.K. = Before Kummer.- 1 The Pythagorean Equation.- 2 The Biquadratic Equation.- 3 The Cubic Equation.- 4 The Quintic Equation.- 5 Fermat's Equation of Degree Seven.- Lecture IV The Naïve Approach.- 1 The Relations of Barlow and Abel.- 2 Sophie Germain.- 3 Congruences.- 4 Wendt's Theorem.- 5 Abel's Conjecture.- 6 Fermat's Equation with Even Exponent.- 7 Odds and Ends.- Lecture V Kummer's Monument.- 1 A Justification of Kummer's Method.- 2 Basic Facts about the Arithmetic of Cyclotomic Fields.- 3 Kummer's Main Theorem.- Lecture VI Regular Primes.- 1 The Class Number of Cyclotomic Fields.- 2 Bernoulli Numbers and Kummer's Regularity Criterion.- 3 Various Arithmetic Properties of Bernoulli Numbers.- 4 The Abundance of Irregular Primes.- 5 Computation of Irregular Primes.- Lecture VII Kummer Exits.- 1 The Periods of the Cyclotomic Equation.- 2 The Jacobi Cyclotomic Function.- 3 On the Generation of the Class Group of the Cyclotomic Field.- 4 Kummer's Congruences.- 5 Kummer's Theorem for a Class of Irregular Primes.- 6 Computations of the Class Number.- Lecture VIII After Kummer, a New Light.- 1 The Congruences of Mirimanoff.- 2 The Theorem of Krasner.- 3 The Theorems of Wieferich and Mirimanoff.- 4 Fermat's Theorem and the Mersenne Primes.- 5 Summation Criteria.- 6 Fermat Quotient Criteria.- Lecture IX The Power of Class Field Theory.- 1 The Power Residue Symbol.- 2 Kummer Extensions.- 3 The Main Theorems of Furtwängler.- 4 The Method of Singular Integers.- 5 Hasse.- 6 The p-Rank of the Class Group of the Cyclotomic Field.- 7 Criteria of p-Divisibility of the Class Number.- 8 Properly and Improperly Irregular Cyclotomic Fields.- Lecture X Fresh Efforts.- 1 Fermat's Last Theorem Is True for Every Prime Exponent Less Than 125000.- 2 Euler Numbers and Fermat's Theorem.- 3 The First Case Is True for Infinitely Many Pairwise Relatively Prime Exponents.- 4 Connections between Elliptic Curves and Fermat's Theorem.- 5 Iwasawa's Theory.- 6 The Fermat Function Field.- 7 Mordell's Conjecture.- 8 The Logicians.- Lecture XI Estimates.- 1 Elementary (and Not So Elementary) Estimates.- 2 Estimates Based on the Criteria Involving Fermat Quotients.- 3 Thue, Roth, Siegel and Baker.- 4 Applications of the New Methods.- Lecture XII Fermat's Congruence.- 1 Fermat's Theorem over Prime Fields.- 2 The Local Fermat's Theorem.- 3 The Problem Modulo a Prime-Power.- Lecture XIII Variations and Fugue on a Theme.- 1 Variation I (In the Tone of Polynomial Functions).- 2 Variation II (In the Tone of Entire Functions).- 3 Variation III (In the Theta Tone).- 4 Variation IV (In the Tone of Differential Equations).- 5 Variation V (Giocoso).- 6 Variation VI (In the Negative Tone).- 7 Variation VII (In the Ordinal Tone).- 8 Variation VIII (In a Nonassociative Tone).- 9 Variation IX (In the Matrix Tone).- 10 Fugue (In the Quadratic Tone).- Epilogue.- Index of Names.