17 results back to index

**
Massive: The Missing Particle That Sparked the Greatest Hunt in Science
** by
Ian Sample

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Albert Einstein, Arthur Eddington, cuban missile crisis, dark matter, Donald Trump, double helix, Ernest Rutherford, Gary Taubes, Isaac Newton, John Conway, John von Neumann, Menlo Park, Murray Gell-Mann, Richard Feynman, Richard Feynman, Ronald Reagan, Stephen Hawking, uranium enrichment, Yogi Berra

Like anyone who has a calling, particle physicists go where the jobs are. As high-energy facilities rise and fall on different continents, scientists migrate to wherever they have the greatest chance of finding something new in nature. With modern computer networks, some make the move a virtual one and analyze collision data from the comfort of their university offices. Others up sticks to follow the action. John Conway is a case in point. An experimentalist at the University of California at Davis, he spent years at CERN with the Aleph team, the group that went on to see tantalizing hints of the Higgs particle. Long before the excitement broke out, Conway returned to the United States to help revamp the CDF detector at Fermilab. A while later, in December 2006, he was back at CERN again, this time to arrange the delivery of some exquisite electronics designed to track particles inside the Compact Muon Solenoid, or CMS, detector for the still-in-progress Large Hadron Collider.

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Thinking it over some time later, Conway recalled the feeling he had that Saturday morning at CERN when, all alone, he had seen the bump for the first time. It is a feeling that drives many people to do science. “You have this hope that someday you’ll see something that is genuinely new, that no one else in the world has ever seen,” he said. “You want to make a discovery.” At Fermilab, there are two detectors that physicists use to hunt for the Higgs particle. John Conway’s team searched for evidence of the elusive boson amid collisions recorded by the CDF detector. Other groups use the DZero detector. One of the spokesmen for the DZero collaboration is Dmitri Denisov, a Russian-born scientist who was educated in Moscow by some of the country’s most respected physicists. Denisov was at Fermilab when the top quark was discovered in 1995. He is not a glass-half-empty man.

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I would also like to say a special thanks to my colleagues Alok Jha, David Adam, Karen McVeigh, and James Kingsland for covering for me. Countless scientists and engineers gave up some of their precious time to talk with me while I was researching the book, and I’m profoundly grateful to all of them. The end result was vastly improved thanks to those who checked my clumsy drafts, including Steven Weinberg, John Ellis, Michael Fisher, Lyn Evans, John Conway, Gerry Guralnik, and Dick Hagen. Peter Higgs provided comprehensive and invaluable comments on key chapters and put me right on many occasions. His help is a debt I cannot repay. Thanks to Freeman Dyson for digging back through his memories to tell me about Peter Higgs’s visit to the Institute for Advanced Study in 1966 and for his reflections on Robert Oppenheimer. For explaining their contributions to the theory of the origin of mass, thanks to the six men who came up with the idea: François Englert, Robert Brout, Peter Higgs, Gerry Guralnik, Dick Hagen, and Tom Kibble.

**
A Beautiful Mind
** by
Sylvia Nasar

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Al Roth, Albert Einstein, Andrew Wiles, Brownian motion, cognitive dissonance, Columbine, experimental economics, fear of failure, Henri Poincaré, invisible hand, Isaac Newton, John Conway, John Nash: game theory, John von Neumann, Kenneth Rogoff, linear programming, lone genius, market design, medical residency, Nash equilibrium, Norbert Wiener, Paul Erdős, prisoner's dilemma, RAND corporation, Ronald Coase, second-price auction, Silicon Valley, Simon Singh, spectrum auction, The Wealth of Nations by Adam Smith, Thorstein Veblen, upwardly mobile

Harold Kuhn, personal communication, 8.97. 12. Armand Borel, professor of mathematics, Institute for Advanced Study, interview, 3.1.96. 13. F. Browder, interview. 14. Ibid. 15. Joseph Kohn, interview, 7.19.95. Phrasing the question precisely, Ambrose would have used the adverb “isometrically” — meaning “to preserve distances” — after “embedding.” 16. Shlomo Sternberg, professor of mathematics, Harvard University, interview, 3.5.96. 17. Mikhail Gromov, interview. 12.16.97. 18. John Forbes Nash, Jr., Lcs Prix Nobel 1994, op. cit. 19. Gromov, interview. 20. John Conway, professor of mathematics, Princeton University, interview, 10.94. 21. Jürgen Moser, e-mail, 12.24.97. 22. Richard Palais, professor of mathematics, Brandeis University, interview, 11.6.95. 23. Moser, interview. 24. Donald J.

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There has been some tendency in recent decades to move from harmony to chaos. Nash says chaos is just around the corner.19 John Conway, the Princeton mathematician who discovered surreal numbers and invented the game of Life, called Nash’s result “one of the most important pieces of mathematical analysis in this century.”20 It was also, one must add, a deliberate jab at then-fashionable approaches to Riemannian manifolds, just as Nash’s approach to the theory of games was a direct challenge to von Neumann’s. Ambrose, for example, was himself involved in a highly abstract and conceptual description of such manifolds at the time. As Jürgen Moser, a young German mathematician who came to know Nash well in the mid-1950s, put it, “Nash didn’t like that style of mathematics at all. He was out to show that this, to his mind, exotic approach was completely unnecessary since any such manifold was simply a submanifold of a high dimensional Euclidean space.”21 Nash’s more important achievement may have been the powerful technique he invented to obtain his result.

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Interviews with Marvin Minsky, professor of science, MIT, 2.13.96; John Tukey, 9.30.97; David Gale, 9.20.96; Melvin Hausner, 1.26.96 and 2.20.96; and John Conway, professor of mathematics, Princeton University, 10.94; John Isbell, e-mails, 1.25.96, 1.26.97, 1.27.97. 3. Isbell, e-mails. 4. Letter from John Nash to Martin Shubik, undated (1950 or 1951); Hausner, interviews and e-mails. 5. William Poundstone, Prisoner’s Dilemma, op. cit.; John Williams, The Compleat Strategist (New York: McGraw Hill, 1954). 6. Poundstone, op. cit. 7. Solomon Leader, interview, 6.9.95. 8. Martha Nash Legg, interview, 8.1.95. 9. Isbell, e-mails. 10. Hartley Rogers, interview, 1.26.96. 11. Ibid. 12. Ibid. 13. Nash may have had the idea while he was at Carnegie. This, in any case, is Hans Weinberger’s recollection, interview, 10.28.95. 14. Martin Gardner, Mathematical Puzzles and Diversions (New York: Simon & Schuster, 1959), pp. 65–70. 15.

**
Zero: The Biography of a Dangerous Idea
** by
Charles Seife

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Albert Einstein, Albert Michelson, Arthur Eddington, Cepheid variable, cosmological constant, dark matter, Edmond Halley, Georg Cantor, Isaac Newton, John Conway, place-making, probability theory / Blaise Pascal / Pierre de Fermat, retrograde motion, Richard Feynman, Richard Feynman, Solar eclipse in 1919, Stephen Hawking

All the children cheer. December 31, 1999, is the evening when the great odometer in the sky clicks ahead. The Zeroth Number Waclaw Sierpinski, the great Polish mathematician…was worried that he’d lost one piece of his luggage. “No, dear!” said his wife. “All six pieces are here.” “That can’t be true,” said Sierpinski, “I’ve counted them several times: zero, one, two, three, four, five.” —JOHN CONWAY AND RICHARD GUY, THE BOOK OF NUMBERS It may seem bizarre to suggest that Dionysius and Bede made a mistake when they forgot to include zero in their calendar. After all, children count “one, two, three,” not “zero, one, two.” Except for the Mayans, nobody else had a year zero or started a month with day zero. It seems unnatural. On the other hand, when you count backward, it is second nature.

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Worst of all, if you wantonly divide by zero, you can destroy the entire foundation of logic and mathematics. Dividing by zero once—just one time—allows you to prove, mathematically, anything at all in the universe. You can prove that 1 + 1 = 42, and from there you can prove that J. Edgar Hoover was a space alien, that William Shakespeare came from Uzbekistan, or even that the sky is polka-dotted. (See appendix A for a proof that Winston Churchill was a carrot.) Multiplying by zero collapses the number line. But dividing by zero destroys the entire framework of mathematics. There is a lot of power in this simple number. It was to become the most important tool in mathematics. But thanks to the odd mathematical and philosophical properties of zero, it would clash with the fundamental philosophy of the West.

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The sound was usually dissonant and sometimes even worse. Often the tone wobbled like a drunkard up and down the scale. To Pythagoras, playing music was a mathematical act. Like squares and triangles, lines were number-shapes, so dividing a string into two parts was the same as taking a ratio of two numbers. The harmony of the monochord was the harmony of mathematics—and the harmony of the universe. Pythagoras concluded that ratios govern not only music but also all other types of beauty. To the Pythagoreans, ratios and proportions controlled musical beauty, physical beauty, and mathematical beauty. Understanding nature was as simple as understanding the mathematics of proportions. Figure 7: The mystical monochord This philosophy—the interchangeability of music, math, and nature—led to the earliest Pythagorean model of the planets.

**
Turing's Vision: The Birth of Computer Science
** by
Chris Bernhardt

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Ada Lovelace, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, British Empire, cellular automata, Claude Shannon: information theory, complexity theory, Conway's Game of Life, discrete time, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Henri Poincaré, Internet Archive, Jacquard loom, Jacquard loom, John Conway, John von Neumann, Joseph-Marie Jacquard, Norbert Wiener, Paul Erdős, Turing complete, Turing machine, Turing test, Von Neumann architecture

Cellular automata We only looked briefly looked at cellular automata, but they have a long and interesting history. They were first studied by Ulam and von Neumann as the first computers were built. Nils Barricelli was at Princeton during the 1950s and used the computer to simulate the interaction of cells. George Dyson’s Turing’s Cathedral gives a good historical description of this work John Conway, in 1970, defined Life involving two-dimensional cellular automata. These were popularized by Martin Gardner in Scientific American. William Poundstone’s The Recursive Universe is a good book on the history of these automata and how complexity can arise from simple rules. (This book was first published in 1985, but was been republished by Dover Press in 2013.) Stephen Wolfram’s A New Kind of Science is an encyclopedia of one-dimensional cellular automata with extensive notes.

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It was enough to get him elected as a Fellow: a position that provided money, board and lodging for three years with the only requirement that he would concentrate on mathematical research. Now he had to prove himself. He had to do something original. What better way than to tackle a problem of the world’s leading mathematician and prove him wrong? This is exactly what Turing set out to do. He would tackle Hilbert’s Entscheidungsproblem. Before we describe what Turing did, it is helpful to understand why Hilbert stated his problem. This requires introducing some of developments in mathematics that occurred during the second half of the nineteenth century and first part of the twentieth century. In particular, we will look at the rise of mathematical logic, the attempts to find a firm axiomatic foundation for mathematics, and the role of algorithms. Mathematical Certainty Mathematics is often seen as the epitome of certainty.

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Other mathematicians, notably Hilbert, supported Cantor and felt that what he was doing was not only correct, but important for the future of mathematics. In the first chapter, we discussed the foundations of mathematics. There was the formalist approach of Hilbert and the logicist approach of Russell and Whitehead. The logicists wanted to show that all of mathematics could be derived from logic. The formalists wanted a formal system in which you could construct arguments about the consistency and completeness of the axioms. Both approaches assumed that there was nothing fundamentally wrong with mathematics. The paradoxes that had cropped up in Cantor’s work could be eliminated with a more careful definition of a set. However, another group of mathematicians took a completely different view. They regarded the work of Cantor as being seriously wrong, so wrong that the foundations of mathematics should be rewritten specifically to exclude the types of arguments that Cantor was using.

**
Hackers: Heroes of the Computer Revolution - 25th Anniversary Edition
** by
Steven Levy

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air freight, Apple II, Bill Gates: Altair 8800, Buckminster Fuller, Byte Shop, computer age, computer vision, corporate governance, El Camino Real, game design, Hacker Ethic, hacker house, Haight Ashbury, John Conway, Mark Zuckerberg, Menlo Park, non-fiction novel, Paul Graham, popular electronics, RAND corporation, reversible computing, Richard Stallman, Silicon Valley, software patent, speech recognition, Steve Jobs, Steve Wozniak, Steven Levy, Stewart Brand, Ted Nelson, Whole Earth Catalog, Y Combinator

Would the Golden Age, now drawing to its close, really have meant anything? • • • • • • • • It was in 1970 that Bill Gosper began hacking LIFE. It was yet another system that was a world in itself, a world where behavior was “exceedingly rich, but not so rich as to be incomprehensible.” It would obsess Bill Gosper for years. LIFE was a game, a computer simulation developed by John Conway, a distinguished British mathematician. It was first described by Martin Gardner, in his "Mathematical Games" column in the October 1970 issue of Scientific American. The game consists of markers on a checkerboard-like field, each marker representing a “cell.” The pattern of cells changes with each move in the game (called a “generation”), depending on a few simple rules—cells die, are born, or survive to the next generation according to how many neighboring cells are in the vicinity.

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It was a great moment for Solomon, seeing the computer he had helped bring to the world making a color television set run beautiful patterns. Then they tried another program: LIFE. The game-that-is-more-than-a-game, created by mathematician John Conway. The game that MIT wizard Bill Gosper had hacked so intently, to the point where he saw it as potentially generating life itself. The Altair version ran much more slowly than the PDP-6 program, of course, and with none of those elegantly hacked utilities, but it followed the same rules. And it did it while sitting on the kitchen table. Garland put in a few patterns, and Les Solomon, not fully knowing the rules of the game and certainly not aware of the deep philosophical and mathematical implications, watched the little blue, red, or green stars (that was the way the Dazzler made the cells look) eat the other little stars, or make more stars.

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Sometimes, after a number of generations, patterns would alternate, flashing between one and the other: these were called oscillators, traffic lights, or pulsars. What Gosper and the hackers were seeking was called a glider gun. A glider was a pattern which would move across the screen, periodically reverting to the same pointed shape. If you ever created a LIFE pattern, which actually spewed out gliders as it changed shape, you’d have a glider gun, and LIFE’s inventor, John Conway, offered fifty dollars to the first person who was able to create one. The hackers would spend all night sitting at the PDP-6’s high-quality “340” display (a special, high-speed monitor made by DEC), trying different patterns to see what they’d yield. They would log each “discovery” they made in this artificial universe in a large black sketchbook, which Gosper dubbed the LIFE scrapbook. They would stare at the screen as, generation by generation, the pattern would shift.

**
What Technology Wants
** by
Kevin Kelly

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Albert Einstein, Alfred Russel Wallace, Buckminster Fuller, c2.com, carbon-based life, Cass Sunstein, charter city, Clayton Christensen, cloud computing, computer vision, Danny Hillis, dematerialisation, demographic transition, double entry bookkeeping, en.wikipedia.org, Exxon Valdez, George Gilder, gravity well, hive mind, Howard Rheingold, interchangeable parts, invention of air conditioning, invention of writing, Isaac Newton, Jaron Lanier, John Conway, John von Neumann, Kevin Kelly, knowledge economy, Lao Tzu, life extension, Louis Daguerre, Marshall McLuhan, megacity, meta analysis, meta-analysis, new economy, out of africa, performance metric, personalized medicine, phenotype, Picturephone, planetary scale, RAND corporation, random walk, Ray Kurzweil, recommendation engine, refrigerator car, Richard Florida, Silicon Valley, silicon-based life, Skype, speech recognition, Stephen Hawking, Steve Jobs, Stewart Brand, Ted Kaczynski, the built environment, the scientific method, Thomas Malthus, Vernor Vinge, Whole Earth Catalog, Y2K

This utter allegiance to a path predetermined by its previous state is the foundation of the “laws of physics.” Yet a particle’s spontaneous dissolution into subparticles and energy rays is not predictable, nor predetermined by laws of physics. We tend to call this decay into cosmic rays a “random” event. Mathematician John Conway proposed a proof arguing that neither the mathematics of randomness nor the logic of determinism can properly explain the sudden (why right now?) decay or shift of spin direction in cosmic particles. The only mathematical or logical option left is free will. The particle simply chooses in a way that is indistinguishable from the tiniest quantum bit of free will. Theoretical biologist Stuart Kauffman argues that this “free will” is a result of the mysterious quantum nature of the universe, by which quantum particles can be two places at once, or be both wave and particle at once.

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But the weird and telling thing about this experiment, which has been done many times, is that the wave/particle only chooses its form (either a wave or a particle) after it has already passed through the slit and is measured on the other side. According to Kauffman, the particle’s shift from undecided state (called quantum decoherence) to the decided state (quantum coherence) is a type of volition and thus the source of free will in our own brains, since these quantum effects happen in all matter. As John Conway writes,Some readers may object to our use of the term “free will” to describe the indeterminism of particle responses. Our provocative ascription of free will to elementary particles is deliberate, since our theorem asserts that if experimenters have a certain freedom, then particles have exactly the same kind of freedom. Indeed, it is natural to suppose that this latter freedom is the ultimate explanation of our own.

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Hendrik Lorentz, a theoretical physicist who studied light waves, introduced a mathematical structure of space-time in July 1905, the same year as Einstein. In 1904 the French mathematician Henri Poincare pointed out that observers in different frames will have clocks that will “mark what one may call the local time” and that “as demanded by the relativity principle the observer cannot know whether he is at rest or in absolute motion.” And the 1911 winner of the Nobel Prize in physics, Wilhelm Wien, proposed to the Swedish committee that Lorentz and Einstein be jointly awarded a Nobel Prize in 1912 for their work on special relativity. He told the committee, “While Lorentz must be considered as the first to have found the mathematical content of the relativity principle, Einstein succeeded in reducing it to a simple principle.

**
Haskell
** by
Graham Hutton

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Eratosthenes, John Conway, Simon Singh, type inference

Using this approach, the function births can be rewritten as follows: births b = [p | p ← rmdups (concat (map neighbs b)), isEmpty b p, liveneighbs b p == 3] The auxiliary function rmdups removes duplicates from a list, and is used above to ensure that each potential new cell is only considered once: rmdups :: Eq a ⇒ [a ] → [a ] rmdups [ ] = [] rmdups (x : xs) = x : rmdups (ﬁlter ( = x ) xs) The next generation of a board can now be produced simply by appending the list of survivors and the list of new births: nextgen :: Board → Board nextgen b = survivors b ++ births b 9.9 EXERCISES Finally, we deﬁne a function life that implements the game of life itself, by clearing the screen, showing the living cells in the current board, waiting for a moment, and then continuing with the next generation: life :: Board → IO () life b = do cls showcells b wait 5000 life (nextgen b) The function wait is used to slow down the game to a reasonable speed, and can be implemented by performing a given number of dummy actions: wait :: Int → IO () wait n = seqn [return () | ← [1 . . n ]] For fun, you may like to try out the life function with the glider example, and experiment with some patterns of your own. 9.8 Chapter remarks The use of the IO type to perform other forms of side effects, including reading and writing from ﬁles, and handling exceptional events, is discussed in the Haskell Report (25). A formal meaning for input/output and other forms of side effects is given in (24). A variety of libraries for performing graphical interaction are available from the Haskell home page, www .haskell .org . The game of life was invented by John Conway, and popularised by Martin Gardner in the October 1970 edition of Scientiﬁc American. 9.9 Exercises 1. Deﬁne an action readLine :: IO String that behaves in the same way as getLine , except that it also permits the delete key to be used to remove characters. Hint: the delete character is ’\DEL’, and the control string for moving the cursor back one character is "\ESC[1D". 2. Modify the calculator program to indicate the approximate position of an error rather than just sounding a beep, by using the fact that the parser returns the unconsumed part of the input string. 3.

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For example, attempting to divide by zero or select the ﬁrst element of an empty list will produce an error: > 1 ‘div ‘ 0 Error > head [ ] Error In practice, when an error occurs the Hugs system also produces a message that provides some information about the likely cause. For reference, appendix A presents some of the most commonly used deﬁnitions from the standard prelude, and appendix B shows how special Haskell symbols, such as ↑ and ++, are typed using a normal keyboard. 2.3 Function application In mathematics, the application of a function to its arguments is usually denoted by enclosing the arguments in parentheses, while the multiplication of two values is often denoted silently, by writing the two values next to one another. For example, in mathematics the expression f (a, b) + c d means apply the function f to two arguments a and b , and add the result to the product of c and d . Reﬂecting its primary status in the language, function application in Haskell is denoted silently using spacing, while the multiplication 2.4 HASKELL SCRIPTS of two values is denoted explicitly using the operator ∗.

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Reﬂecting its primary status in the language, function application in Haskell is denoted silently using spacing, while the multiplication 2.4 HASKELL SCRIPTS of two values is denoted explicitly using the operator ∗. For example, the expression above would be written in Haskell as follows: f a b+c∗d Moreover, function application has higher priority than all other operators. For example, f a + b means (f a ) + b . The following table gives a few further examples to illustrate the differences between the notation for function application in mathematics and in Haskell: Mathematics Haskell f (x ) f (x , y ) f (g (x )) f (x , g (y )) f (x )g (y ) f f f f f x x y (g x ) x (g y ) x ∗g y Note that parentheses are still required in the Haskell expression f (g x ) above, because f g x on its own would be interpreted as the application of the function f to two arguments g and x , whereas the intention is that f is applied to one argument, namely the result of applying the function g to an argument x .

**
Programming in Haskell
** by
Graham Hutton

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Eratosthenes, John Conway, Simon Singh, type inference

Using this approach, the function births can be rewritten as follows: births b = [p | p ← rmdups (concat (map neighbs b)), isEmpty b p, liveneighbs b p == 3] The auxiliary function rmdups removes duplicates from a list, and is used above to ensure that each potential new cell is only considered once: rmdups :: Eq a ⇒ [a ] → [a ] rmdups [ ] = [] rmdups (x : xs) = x : rmdups (ﬁlter ( = x ) xs) The next generation of a board can now be produced simply by appending the list of survivors and the list of new births: nextgen :: Board → Board nextgen b = survivors b ++ births b 9.9 EXERCISES Finally, we deﬁne a function life that implements the game of life itself, by clearing the screen, showing the living cells in the current board, waiting for a moment, and then continuing with the next generation: life :: Board → IO () life b = do cls showcells b wait 5000 life (nextgen b) The function wait is used to slow down the game to a reasonable speed, and can be implemented by performing a given number of dummy actions: wait :: Int → IO () wait n = seqn [return () | ← [1 . . n ]] For fun, you may like to try out the life function with the glider example, and experiment with some patterns of your own. 9.8 Chapter remarks The use of the IO type to perform other forms of side effects, including reading and writing from ﬁles, and handling exceptional events, is discussed in the Haskell Report (25). A formal meaning for input/output and other forms of side effects is given in (24). A variety of libraries for performing graphical interaction are available from the Haskell home page, www .haskell .org . The game of life was invented by John Conway, and popularised by Martin Gardner in the October 1970 edition of Scientiﬁc American. 9.9 Exercises 1. Deﬁne an action readLine :: IO String that behaves in the same way as getLine , except that it also permits the delete key to be used to remove characters. Hint: the delete character is ’\DEL’, and the control string for moving the cursor back one character is "\ESC[1D". 2. Modify the calculator program to indicate the approximate position of an error rather than just sounding a beep, by using the fact that the parser returns the unconsumed part of the input string. 3.

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For example, attempting to divide by zero or select the ﬁrst element of an empty list will produce an error: > 1 ‘div ‘ 0 Error > head [ ] Error In practice, when an error occurs the Hugs system also produces a message that provides some information about the likely cause. For reference, appendix A presents some of the most commonly used deﬁnitions from the standard prelude, and appendix B shows how special Haskell symbols, such as ↑ and ++, are typed using a normal keyboard. 2.3 Function application In mathematics, the application of a function to its arguments is usually denoted by enclosing the arguments in parentheses, while the multiplication of two values is often denoted silently, by writing the two values next to one another. For example, in mathematics the expression f (a, b) + c d means apply the function f to two arguments a and b , and add the result to the product of c and d . Reﬂecting its primary status in the language, function application in Haskell is denoted silently using spacing, while the multiplication 2.4 HASKELL SCRIPTS of two values is denoted explicitly using the operator ∗.

…

Reﬂecting its primary status in the language, function application in Haskell is denoted silently using spacing, while the multiplication 2.4 HASKELL SCRIPTS of two values is denoted explicitly using the operator ∗. For example, the expression above would be written in Haskell as follows: f a b+c∗d Moreover, function application has higher priority than all other operators. For example, f a + b means (f a ) + b . The following table gives a few further examples to illustrate the differences between the notation for function application in mathematics and in Haskell: Mathematics Haskell f (x ) f (x , y ) f (g (x )) f (x , g (y )) f (x )g (y ) f f f f f x x y (g x ) x (g y ) x ∗g y Note that parentheses are still required in the Haskell expression f (g x ) above, because f g x on its own would be interpreted as the application of the function f to two arguments g and x , whereas the intention is that f is applied to one argument, namely the result of applying the function g to an argument x .

**
You Are Not a Gadget
** by
Jaron Lanier

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1960s counterculture, accounting loophole / creative accounting, additive manufacturing, Albert Einstein, call centre, cloud computing, crowdsourcing, death of newspapers, digital Maoism, Douglas Hofstadter, Extropian, follow your passion, hive mind, Internet Archive, Jaron Lanier, jimmy wales, John Conway, John von Neumann, Kevin Kelly, Long Term Capital Management, Network effects, new economy, packet switching, PageRank, pattern recognition, Ponzi scheme, Ray Kurzweil, Richard Stallman, Silicon Valley, Silicon Valley startup, slashdot, social graph, stem cell, Steve Jobs, Stewart Brand, Ted Nelson, telemarketer, telepresence, The Wisdom of Crowds, trickle-down economics, Turing test, Vernor Vinge, Whole Earth Catalog

If you search online for math and ignore the first results, which are often the Wikipedia entry and its echoes, you start to come across weird individual efforts and even some old ThinkQuest pages. They were often last updated around the time Wikipedia arrived. Wikipedia took the wind out of the trend.* The quest to bring math into the culture continues, but mostly not online. A huge recent step was the publication of a book on paper by John Conway, Heidi Burgiel, and Chaim Goodman-Strauss called The Symmetries of Things. This is a tour de force that fuses introductory material with cutting-edge ideas by using a brash new visual style. It is disappointing to me that pioneering work continues primarily on paper, having become muted online. The same could be said about a great many topics other than math. If you’re interested in the history of a rare musical instrument, for instance, you can delve into the internet archive and find personal sites devoted to it, though they probably were last updated around the time Wikipedia came into being.

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Wikipedia has already been elevated into what might be a permanent niche. It might become stuck as a fixture, like MIDI or the Google ad exchange services. That makes it important to be aware of what you might be missing. Even in a case in which there is an objective truth that is already known, such as a mathematical proof, Wikipedia distracts the potential for learning how to bring it into the conversation in new ways. Individual voice—the opposite of wikiness—might not matter to mathematical truth, but it is the core of mathematical communication. * See Norm Cohen, “The Latest on Virginia Tech, from Wikipedia,” New York Times, April 23, 2007. In 2009, Twitter became the focus of similar stories because of its use by protestors of Iran’s disputed presidential election. † See Jamin Brophy-Warren, “Oh, That John Locke,” Wall Street Journal, June 16, 2007

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There’s an aversion to talking about it much, because we don’t want our founding father to seem like a tabloid celebrity, and we don’t want his memory trivialized by the sensational aspects of his death. The legacy of Turing the mathematician rises above any possible sensationalism. His contributions were supremely elegant and foundational. He gifted us with wild leaps of invention, including much of the mathematical underpinnings of digital computation. The highest award in computer science, our Nobel Prize, is named in his honor. Turing the cultural figure must be acknowledged, however. The first thing to understand is that he was one of the great heroes of World War II. He was the first “cracker,” a person who uses computers to defeat an enemy’s security measures. He applied one of the first computers to break a Nazi secret code, called Enigma, which Nazi mathematicians had believed was unbreakable.

**
Free as in Freedom
** by
Sam Williams

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Asperger Syndrome, cognitive dissonance, Debian, East Village, Hacker Ethic, informal economy, Isaac Newton, John Conway, Maui Hawaii, Murray Gell-Mann, profit motive, Richard Feynman, Richard Stallman, Silicon Valley, slashdot, software patent, Steven Levy, Ted Nelson, urban renewal, VA Linux, Y2K

There was Gerald Sussman, original author of the robotic block-stacking program HACKER. And there was Bill Gosper, the in-house math whiz already in the midst of an 18-month hacking bender triggered by the philosophical implications of the computer game LIFE.See Steven Levy, Hackers (Penguin USA [paperback], 1984): 144. Levy devotes about five pages to describing Gosper's fascination with LIFE, a math-based software game first created by British mathematician John Conway. I heartily recommend this book as a supplement, perhaps even a prerequisite, to this one. Members of the tight-knit group called themselves " hackers." Over time, they extended the "hacker" description to Stallman as well. In the process of doing so, they inculcated Stallman in the ethical traditions of the "hacker ethic ." To be a hacker meant more than just writing programs, Stallman learned.

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Like most members of the Science Honors Program, Stallman breezed through the qualifying exam for Math 55, the legendary "boot camp" class for freshman mathematics "concentrators" at Harvard. Within the class, members of the Science Honors Program formed a durable unit. "We were the math mafia," says Chess with a laugh. "Harvard was nothing, at least compared with the SHP." To earn the right to boast, however, Stallman, Chess, and the other SHP alumni had to get through Math 55. Promising four years worth of math in two semesters, the course favored only the truly devout. "It was an amazing class," says David Harbater, a former "math mafia" member and now a professor of mathematics at the University of Pennsylvania. "It's probably safe to say there has never been a class for beginning college students that was that intense and that advanced.

…

As a kid who'd always taken pride in being the smartest mathematician the room, it was like catching a glimpse of his own mortality. Years later, as Chess slowly came to accept the professional rank of a good-but-not-great mathematician, he had Stallman's sophomore-year proof to look back on as a taunting early indicator. "That's the thing about mathematics," says Chess. "You don't have to be a first-rank mathematician to recognize first-rate mathematical talent. I could tell I was up there, but I could also tell I wasn't at the first rank. If Richard had chosen to be a mathematician, he would have been a firstrank mathematician." For Stallman, success in the classroom was balanced by the same lack of success in the social arena. Even as other members of the math mafia gathered to take on the Math 55 problem sets, Stallman preferred to work alone.

**
I Am a Strange Loop
** by
Douglas R. Hofstadter

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Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Brownian motion, double helix, Douglas Hofstadter, Georg Cantor, Gödel, Escher, Bach, Isaac Newton, James Watt: steam engine, John Conway, John von Neumann, mandelbrot fractal, pattern recognition, Paul Erdős, place-making, probability theory / Blaise Pascal / Pierre de Fermat, publish or perish, random walk, Ronald Reagan, self-driving car, Silicon Valley, telepresence, Turing machine

Page 91 radicals, such as Évariste Galois… The great Galois was indeed a young radical, which led to his absurdly tragic death in a duel on his twenty-first birthday, but the phrase “solution by radicals” really refers to the taking of nth roots, called “radicals”. For a shallow, a medium, and a deep dip into Galois’ immortal, radical insights into hidden mathematical structures, see [Livio], [Bewersdorff ], and [Stewart], respectively. Page 95 there is a special type of abstract structure or pattern… “Real Patterns” in [Dennett 1998] argues powerfully for the reality of abstract patterns, based on John Conway’s cellular automaton known as the “Game of Life”. The Game of Life itself is presented ideally in [Gardner], and its relevance to biological life is spelled out in [Poundstone]. Page 102 I am sorry to say, now hackneyed… I have long loved Escher’s art, but as time has passed, I have found myself drawn ever more to his early non-paradoxical landscapes, in which I see hints everywhere of his sense of the magic residing in ordinary scenes.

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kits, electronic KJ, Himalayan peak, unscalability of Klagsbrun, Francine Klee, Paul Klüdgerot, the Klüdgerotic condition knees: awareness level of; as candidates for consciousness; reflex behavior of knobs of Twinwirld knowing, elusive nature of knurking and glebbing; not physical processes but subjective sensations; reliably evoked independently of brain’s wiring koans Kolak, Daniel Krall, Diana Kriegel, Uriah Külot, Gerd, drama critic, review of Prince Hyppia: Math Dramatica by L lambs as edible beings landmark integers language: acquisition of; as unperceived code; without self-reference “language”, vagueness of the term lap loop; photo of large-souled vs. small-souled beings Latin leaf piles: as endowed with Leafpilishness; intrinsic nature of; as macroscopic entities Leafpilishness, Capitalized Essence of leather, purchase of leatherette dashboard Leban, Roy and Bruce leg that is asleep Leibniz, Gottfried Wilhelm von Le Lionnais, François Leonardo di Pisa (Fibonacci) letters of the alphabet, as meaningless level-confusion, prevalence of, in discussions of brain/mind level-crossing feedback loops level-shifts, perceptual levels of description: causality at different; oscillation between Lexington (in “Pig”) liar paradox liberty and imprisonment as flipped sensations life: defined; as illusion; in Universe Z Life, John Conway’s Game of “light on inside”; suddenly extinguished linguistic sloppiness in reference to robots Linus (“Peanuts”) lions: compassion of; conscience of; possible vegetarianism of liphosophy lists: abstract patterns having great reality for us; abstractions in brain having causal powers; accidental attachments of Leafpilishness dollops; actions launched by self; brain structures, in descending order; Carol’s losses; causes and effects; composers whose style the author borrowed from; concepts in canine minds; concepts involved in “grocery store checkout stand”; concepts involved in “soap digest rack”; conscious entities, according to panpsychists; copycat actions by the author; determining factors of identity; emotionladen verbs; entities without selves; epiphenomena at human size; episodes in one’s memory; famous achievements influencing the author; high-level causal agents; high-level phenomena in brain; high-level phenomena in mind; ideas beyond Ollie’s ken; importable mannerisms of other people; items of dubious reality in newspaper; items in hog’s environment; leaf-pile enigmaslist of principal lists in I Am a Strange Loop; low-level phenomena in brain; macroscopic reliabilities; macroscopic unpredictables; magnanimous souls; memories from Carol’s youth; mentalistic verbs; mundane concepts beginning with “s”; mythical symbols; names morphing from “Derek Parfit” to “Napoleon Bonaparte”; objects of study in literary criticism; obstacles that crop up at random in life; Parfit book’s chapter titles; people with diverse influences on the author; phrases denying interpenetration of souls; physical phenomena that lack consciousness; physical structures lacking hereness; potential personal attributes; potential symbols in mosquito brain; problems with Consciousness as a Capitalized Essence; prototypically true sentences; qualia; questions triggered by Gödel’s theorem; rarely thought-of things; realest things of all; recipients of dollops of Consciousness; scenic events perceived by no one; self-referential sentences; shadowy abstract patterns in brain; simultaneous experiences in one brain; small-souled beings; stuff without inner light; synonyms for “consciousness”; synonyms for “eagerness”; things I wasn’t but could imagine being; things of unclear reality beginning with “g”; traits of countries; unlikely substrates for “I”ness; video-feedback epiphenomena; video-feedback knobs; what makes the world go round; words with ill-defined syllable-counts; words for linguistic phenomena literary criticism, objects of study in Little Tyke, allegedly vegetarian lion living inside someone else; see also survival; visitation Löb, Martin Hugo locking-in: of epiphenomena on TV screen: of “I”; of perceptions; of self lockstep synchrony of Gödel numbers and PM formulas logic of simmballs’ dance Logical Syntax of Language, The (Carnap) logicians’ use of blurry concepts long sentence loophole in set theory, Russell’s love: for children leading to soul-entanglement; halo of concepts with which we understand love; as cause for marriage; inseparability from “I” concept; poorly understood so far in terms of quantum electrodynamics; profound influence on us of those whom we “lower” animals, see hierarchy lower-level events, see substrate lower-level meaning of Gödel’s formula; ignoring of low-level view of brains low notes gliding into rumbles low-resolution copies, see fidelity Lucas, see Natalie Lucy (“Peanuts”) M Machine Q vs.

…

You might object, “But those aren’t mathematical notions! Berry’s idea was to use mathematical definitions of integers.” All right, but then show me a sharp cutoff line between mathematics and the rest of the world. Berry’s definition uses the vague notion of “syllable counting”, for instance. How many syllables are there in “finally” or “family” or “rhythm” or “lyre” or “hour” or “owl”? But no matter; suppose we had established a rigorous and objective way of counting syllables. Still, what would count as a “mathematical concept”? Is the discipline of mathematics really that sharply defined? For instance, what is the precise definition of the notion “magic square”? Different authors define this notion differently. Do we have to take a poll of the mathematical community? And if so, who then counts as a member of that blurry community?

**
Fermat’s Last Theorem
** by
Simon Singh

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Albert Einstein, Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Arthur Eddington, Augustin-Louis Cauchy, Fellow of the Royal Society, Georg Cantor, Henri Poincaré, Isaac Newton, John Conway, John von Neumann, kremlinology, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, Simon Singh, Wolfskehl Prize

The hours I spent quizzing and chatting with them were enormously enjoyable and I appreciate their patience and enthusiam while explaining so many beautiful mathematical concepts to me. In particular I would like to thank John Coates, John Conway, Nick Katz, Barry Mazur, Ken Ribet, Peter Sarnak, Goro Shimura and Richard Taylor. I have tried to illustrate this book with as many portraits as possible to give the reader a better sense of the characters involved in the story of Fermat’s Last Theorem. Various libraries and archives have gone out of their way to help me, and in particular I would like to thank Susan Oakes of the London Mathematical Society, Sandra Cumming of the Royal Society and Ian Stewart of Warwick University. I am also grateful to Jacquelyn Savani of Princeton University, Duncan McAngus, Jeremy Gray, Paul Balister and the Isaac Newton Institute for their help in finding research material.

…

First, it developed the idea of proof. A proven mathematical result has a deeper truth than any other truth because it is the result of step-by-step logic. Although the philosopher Thales had already invented some primitive geometrical proofs, Pythagoras took the idea much further and was able to prove far more ingenious mathematical statements. The second consequence of Pythagoras’ theorem is that it ties the abstract mathematical method to something tangible. Pythagoras showed that the truth of mathematics could be applied to the scientific world and provide it with a logical foundation. Mathematics gives science a rigorous beginning and upon this infallible foundation scientists add inaccurate measurements and imperfect observations. An Infinity of Triples The Pythagorean Brotherhood invigorated mathematics with its zealous search for truth via proof.

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When you have told me what you mean by ‘entity’, we will resume the argument. Russell’s work shook the fundations of mathematics and threw the study of mathematical logic into a state of chaos. The logicians were aware that a paradox lurking in the foundations of mathematics could sooner or later rear its illogical head and cause profound problems. Along with Hilbert and the other logicians, Russell set about trying to remedy the situation and restore sanity to mathematics. This inconsistency was a direct consequence of working with the axioms of mathematics, which until this point had been assumed to be self-evident and sufficient to define the rest of mathematics. One approach was to create an additional axiom which forbade any class from being a member of itself. This would prevent Russell’s paradox by making redundant the question of whether or not to enter the catalogue of catalogues which do not list themselves in itself.

**
Protocol: how control exists after decentralization
** by
Alexander R. Galloway

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Ada Lovelace, airport security, Berlin Wall, bioinformatics, Bretton Woods, computer age, Craig Reynolds: boids flock, discovery of DNA, double helix, Douglas Engelbart, easy for humans, difficult for computers, Fall of the Berlin Wall, Grace Hopper, Hacker Ethic, informal economy, John Conway, Kevin Kelly, late capitalism, linear programming, Marshall McLuhan, means of production, Menlo Park, mutually assured destruction, Norbert Wiener, packet switching, phenotype, post-industrial society, profit motive, QWERTY keyboard, RAND corporation, Ray Kurzweil, RFC: Request For Comment, Richard Stallman, semantic web, SETI@home, stem cell, Steve Crocker, Steven Levy, Stewart Brand, Ted Nelson, telerobotics, the market place, theory of mind, urban planning, Vannevar Bush, Whole Earth Review, working poor

He writes: “I proposed to create a very large, complex and inter-connected region of cyberspace that will be inoculated with digital organisms which will be allowed to evolve freely through natural selection”94—the goal of which is to model the 92. For other examples of artiﬁcial life computer systems, see Craig Reynolds’s “boids” and the ﬂocking algorithm that governs their behavior, Larry Yaeger’s “Polyworld,” Myron Krüger’s “Critter,” John Conway’s “Game of Life,” and others. 93. Tom Ray, “What Tierra Is,” available online at http://www.hip.atr.co.jp/~ray/tierra/ whatis.html. 94. Tom Ray, “Beyond Tierra: Towards the Digital Wildlife Reserve,” available online at http://www1.univap.br/~pedrob/PAPERS/FSP_96/APRIL_07/tom_ray/node5.html. Power 109 spontaneous emergence of biodiversity, a condition believed by many scientists to be the true state of distribution of genetic information in a Nature that is unencumbered by human intervention.

…

However their allchannel network is not identical to a distributed network, as their senatorial example betrays: “an all-channel council or directorate” (p. 8). Truly distributed networks cannot, in fact, support all-channel communication (a combinatorial utopia), but instead propagate through outages and uptimes alike, through miles of dark ﬁber (Lovink) and data oases, through hyperskilled capital and unskilled laity. Thus distribution is similar to but not synonymous with allchannel, the latter being a mathematical fantasy of the former. Chapter 1 32 bureaucracies and vertical hierarchies toward a broad network of autonomous social actors. As Branden Hookway writes: “The shift is occurring across the spectrum of information technologies as we move from models of the global application of intelligence, with their universality and frictionless dispersal, to one of local applications, where intelligence is site-speciﬁc and ﬂuid.”5 Computer scientists reference this historical shift when they describe the change from linear programming to object-oriented programming, the latter a less centralized and more modular way of writing code.

…

Fragmentation allows the message to be ﬂexible enough to ﬁt through a wide range of networks with different thresholds for packet size. Whenever a packet is created via fragmentation, certain precautions must be taken to make sure that it will be reassembled correctly at its destination. To this end, a header is attached to each packet. The header contains certain pieces of vital information such as its source address and destination address. A mathematical algorithm or “checksum” is also computed and amended to the header. If the destination computer determines that the information in the header is corrupted in any way (e.g., if the checksum does not correctly correlate), it is obligated to delete the packet and request that a fresh one be sent. At this point, let me pause to summarize the distinct protocological characteristics of the TCP/IP suite: • TCP/IP facilitates peer-to-peer communication, meaning that Internet hosts can communicate directly with each other without their communication being buffered by an intermediary hub. • TCP/IP is a distributed technology, meaning that its structure resembles a meshwork or rhizome. • TCP/IP is a universal language, which if spoken by two computers allows for internetworking between those computers. • The TCP/IP suite is robust and ﬂexible, not rigid or tough. • The TCP/IP suite is open to a broad, theoretically unlimited variety of computers in many different locations.

**
Advances in Artificial General Intelligence: Concepts, Architectures and Algorithms: Proceedings of the Agi Workshop 2006
** by
Ben Goertzel,
Pei Wang

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AI winter, artificial general intelligence, bioinformatics, brain emulation, combinatorial explosion, complexity theory, computer vision, conceptual framework, correlation coefficient, epigenetics, friendly AI, information retrieval, Isaac Newton, John Conway, Loebner Prize, Menlo Park, natural language processing, Occam's razor, p-value, pattern recognition, performance metric, Ray Kurzweil, Rodney Brooks, semantic web, statistical model, strong AI, theory of mind, traveling salesman, Turing machine, Turing test, Von Neumann architecture, Y2K

We should be devising carefully controlled experiments to ask about the behavior of different kinds of systems, rather than exploring a few plausible systems chosen by instinct, or augmenting the same kind of instinctually-chosen systems with mathematics as a way to make them seem less arbitrary and more rigorous. Both of those old approaches involve assumptions about the relationship between the high-level functionality of AI systems and their low-level mechanisms which, from the point if view of the Complex Systems Problem, are untenable. References [1] Waldrop, M. M. (1992) “Complexity: The emerging science at the edge of order and chaos.” Simon & Schuster, New York, NY. [2] Holland, J. H. (1998) “Emergence.” Helix Books, Reading, MA. [3 ]Horgan, J. (1995) “From complexity to perplexity.” Scientific American 272(6): 104-109. [4] Wolfram, S. (2002) “A New Kind of Science.” Wolfram Media: Champaign, IL. 737-750. [5] Gardner, M. (1970) “Mathematical Games: The fantastic combinations of John Conway's new solitaire game ‘life’.”

…

Interestingly, as the connectionist movement matured, it started to restrict itself to the study of networks of neurally inspired units with mathematically tractable properties. This shift in emphasis was probably caused by models such as the Boltzmann machine [11] and backpropagation learning [10], in which the network was designed in such a way that mathematical analysis was capable of describing the global behavior. But if the Complex Systems Problem is valid, this reliance on mathematical tractability would be a mistake, because it restricts the scope of the field to a very small part of the space of possible systems. There is simply no reason why the systems that show intelligent behavior must necessarily have global behaviors that are mathematically tractable (and therefore computationally reducible). Rather than confine ourselves to systems that happen to have provable global properties, we should take a broad, empirical look at the properties of large numbers of systems, without regard to their tractability. 170 4.2.

…

That the integers have a compact structure is evident from the fact that all of their properties are determined by 5 axioms-- but beyond this you know algorithms that you can use to rapidly solve many problems involving them (for example, to determine if a 50 digit number is even). My working hypothesis is that our mathematical abilities arise from Occam's razor. Roughly speaking, we have these abilities because there is a real a priori structure underlying mathematics, and evolution discovered modules that exploit it, for example modules that know how to exploit the structure of Euclidean 2 and 3 space. By evolving such modules we were able to solve problems important to evolution such as navigating around the jungle, but such modules perforce generalize to higher problems. Mathematical reasoning is one example of an ability that has arisen this way, but of course the collection of modules we use to understand the world extends far beyond, as seen for example from the explanation of metaphors above.

**
Machines of Loving Grace: The Quest for Common Ground Between Humans and Robots
** by
John Markoff

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A Declaration of the Independence of Cyberspace, AI winter, airport security, Apple II, artificial general intelligence, augmented reality, autonomous vehicles, Baxter: Rethink Robotics, Bill Duvall, bioinformatics, Brewster Kahle, Burning Man, call centre, cellular automata, Chris Urmson, Claude Shannon: information theory, Clayton Christensen, clean water, cloud computing, collective bargaining, computer age, computer vision, crowdsourcing, Danny Hillis, DARPA: Urban Challenge, data acquisition, Dean Kamen, deskilling, don't be evil, Douglas Engelbart, Douglas Hofstadter, Dynabook, Edward Snowden, Elon Musk, Erik Brynjolfsson, factory automation, From Mathematics to the Technologies of Life and Death, future of work, Galaxy Zoo, Google Glasses, Google X / Alphabet X, Grace Hopper, Gödel, Escher, Bach, Hacker Ethic, haute couture, hive mind, hypertext link, indoor plumbing, industrial robot, information retrieval, Internet Archive, Internet of things, invention of the wheel, Jacques de Vaucanson, Jaron Lanier, Jeff Bezos, job automation, John Conway, John Maynard Keynes: Economic Possibilities for our Grandchildren, John Maynard Keynes: technological unemployment, John von Neumann, Kevin Kelly, knowledge worker, Kodak vs Instagram, labor-force participation, loose coupling, Mark Zuckerberg, Marshall McLuhan, medical residency, Menlo Park, Mother of all demos, natural language processing, new economy, Norbert Wiener, PageRank, pattern recognition, pre–internet, RAND corporation, Ray Kurzweil, Richard Stallman, Robert Gordon, Rodney Brooks, Sand Hill Road, Second Machine Age, self-driving car, semantic web, shareholder value, side project, Silicon Valley, Silicon Valley startup, Singularitarianism, skunkworks, Skype, social software, speech recognition, stealth mode startup, Stephen Hawking, Steve Ballmer, Steve Jobs, Steve Wozniak, Steven Levy, Stewart Brand, strong AI, superintelligent machines, technological singularity, Ted Nelson, telemarketer, telepresence, telepresence robot, Tenerife airport disaster, The Coming Technological Singularity, the medium is the message, Thorstein Veblen, Turing test, Vannevar Bush, Vernor Vinge, Watson beat the top human players on Jeopardy!, Whole Earth Catalog, William Shockley: the traitorous eight

Shannon, known as the father of “information theory,” had created a simple chess-playing machine in 1950, and there was early interest in biological-growth simulating programs known as “automata,” of which John Conway’s 1970 Game of Life would become the most famous. Minsky was largely distracted by his impending wedding, but McCarthy made the most of his time at Bell Labs, working with Shannon on a collection of mathematical papers that was named at Shannon’s insistence Automata Studies.11 Using the word “automata” was a source of frustration for McCarthy because it shifted the focus of the submitted papers away from the more concrete artificial intelligence ideas and toward more esoteric mathematics. Four years later he settled the issue when he launched the new field that now, six decades later, is transforming the world. He backed the term “artificial intelligence” as a means of “nail[ing] the idea to the mast”12 and focusing the Dartmouth summer project.

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Artificial intelligence as a field of study was originally rooted in a 1956 Dartmouth College summer workshop where John McCarthy was a young mathematics professor. McCarthy had been born in 1927 in Boston of an Irish Catholic father and Lithuanian Jewish mother, both active members of the U.S. Communist Party. His parents were intensely intellectual and his mother committed to the idea that her children could pursue any interests they chose. At twelve McCarthy encountered Eric Temple Bell’s Men of Mathematics, a book that helped determine the career of many of the best and brightest of the era including scientists Freeman Dyson and Stanislaw Ulam. McCarthy was viewed as a high school math prodigy and only applied to Caltech, where Temple Bell was a professor, something he later decided had been an act of “arrogance.” On his application he described his plans in a single sentence: “I intend to be a professor of mathematics.” Bell’s book had given him a realistic view of what that path would entail.

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In contrast, it noted, the human brain was composed of ten billion responsive cells and a hundred million connections with the eyes. The earliest work on artificial neural networks dates back to the 1940s, and in 1949 that research had caught the eye of Marvin Minsky, then a young Harvard mathematics student, who would go on to build early electronic learning networks, one as an undergraduate and a second one, named the Stochastic Neural Analog Reinforcement Calculator, or SNARC, as a graduate student at Princeton. He would later write his doctoral thesis on neural networks. These mathematical constructs are networks of nodes or “neurons” that are interconnected by numerical values that serve as “weights” or “vectors.” They can be trained by being exposed to a series of patterns such as images or sounds to later recognize similar patterns.

**
Practical OCaml
** by
Joshua B. Smith

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cellular automata, Debian, domain-specific language, general-purpose programming language, Grace Hopper, hiring and firing, John Conway, Paul Graham, slashdot, text mining, Turing complete, type inference, web application, Y2K

The third is the logical and of the sample and the random (Figure 27-4). Figure 27-2. Random BMP 620Xch27final.qxd 9/22/06 1:22 AM Page 389 CHAPTER 27 ■ PROCESSING BINARY FILES Figure 27-3. xor BMP Figure 27-4. and BMP 389 620Xch27final.qxd 390 9/22/06 1:22 AM Page 390 CHAPTER 27 ■ PROCESSING BINARY FILES Conway’s Game of Life In 1970, a British mathematician named John Conway created the field of cellular automata when he published the first article on the subject. Conway’s “game” isn’t so much a game played by people as it is a mathematical experiment. The game is an example of emergent behavior because there are only four simple rules that generate an amazing amount of complexity. Conway’s game is also Turing Complete, which means that (given the right initial conditions) the game is as powerful as any “real” computer. The game itself is represented (in its original version) by a matrix of cells.

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OCaml supports regular expressions, and strings are a native type. Let’s not forget research and analysis applications. Many companies write their applications in another language and then write verification and analysis code in OCaml (or another meta-language [ML] dialect). Functional programming in general is designed to make computer programs more like mathematical processes (for example, complex numbers and arbitrary precision-number modules are in the standard library). The precedence features mimic normal mathematical precedence. Also, real numbers and floats are treated differently. Who Uses OCaml? This is often the second question people ask about OCaml. The answer is this: a lot of people. From hedge fund users to graduate students, the list of people using OCaml to solve problems grows every day. Airbus and Microsoft are two of the many companies that use OCaml to help avoid problems in programs written in languages other than OCaml.

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He completed an undergraduate degree in English and proceeded to use those skills in tech support. Joshua became a Unix administrator and programmer in the financial industry. After completing his MBA, he moved to the suburbs of Washington DC, where he now works and lives with his wife and son. xxi 620Xfmfinal.qxd 9/22/06 4:21 PM Page xxii 620Xfmfinal.qxd 9/22/06 4:21 PM Page xxiii About the Technical Reviewer ■RICHARD JONES studied mathematics and computer science at Imperial College, London, before working at a number of companies involved in everything from crystallography to high-speed networks to online communities. He is currently employed by Merjis, studying web site usability and search engine advertising, and training developers in the finer points of the Google AdWords API. Richard’s significant contributions to OCaml include mod_caml (bindings for Apache), perl4caml (using Perl code within OCaml), PG’OCaml (typesafe bindings for PostgreSQL), and the Merjis AdWords Toolkit.

**
The Beginning of Infinity: Explanations That Transform the World
** by
David Deutsch

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agricultural Revolution, Albert Michelson, anthropic principle, artificial general intelligence, Bonfire of the Vanities, conceptual framework, cosmological principle, dark matter, David Attenborough, discovery of DNA, Douglas Hofstadter, Eratosthenes, Ernest Rutherford, first-past-the-post, Georg Cantor, Gödel, Escher, Bach, illegal immigration, invention of movable type, Isaac Newton, Islamic Golden Age, Jacquard loom, Jacquard loom, John Conway, John von Neumann, Joseph-Marie Jacquard, Loebner Prize, Louis Pasteur, pattern recognition, Richard Feynman, Richard Feynman, Search for Extraterrestrial Intelligence, Stephen Hawking, supervolcano, technological singularity, The Coming Technological Singularity, the scientific method, Thomas Malthus, Thorstein Veblen, Turing test, Vernor Vinge, Whole Earth Review, William of Occam

One expression of this within mathematics is the principle, first made explicit by the mathematician Georg Cantor in the nineteenth century, that abstract entities may be defined in any desired way out of other entities, so long as the definitions are unambiguous and consistent. Cantor founded the modern mathematical study of infinity. His principle was defended and further generalized in the twentieth century by the mathematician John Conway, who whimsically but appropriately named it the mathematicians’ liberation movement. As those defences suggest, Cantor’s discoveries encountered vitriolic opposition among his contemporaries, including most mathematicians of the day and also many scientists, philosophers – and theologians. Religious objections, ironically, were in effect based on the Principle of Mediocrity. They characterized attempts to understand and work with infinity as an encroachment on the prerogatives of God. In the mid twentieth century, long after the study of infinity had become a routine part of mathematics and had found countless applications there, the philosopher Ludwig Wittgenstein still contemptuously denounced it as ‘meaningless’.

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Sometimes politicians have been so perplexed by the sheer perverseness of apportionment paradoxes that they have been reduced to denouncing mathematics itself. Representative Roger Q. Mills of Texas complained in 1882, ‘I thought…that mathematics was a divine science. I thought that mathematics was the only science that spoke to inspiration and was infallible in its utterances [but] here is a new system of mathematics that demonstrates the truth to be false.’ In 1901 Representative John E. Littlefield, whose own seat in Maine was under threat from the Alabama paradox, said, ‘God help the State of Maine when mathematics reach for her and undertake to strike her down.’ As a matter of fact, there is no such thing as mathematical ‘inspiration’ (mathematical knowledge coming from an infallible source, traditionally God): as I explained in Chapter 8, our knowledge of mathematics is not infallible. But if Representative Mills meant that mathematicians are, or somehow ought to be, society’s best judges of fairness, then he was simply mistaken.* The National Academy of Sciences panel that reported to Congress in 1948 included the mathematician and physicist John von Neumann.

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Consequently, the reliability of our knowledge of mathematics remains for ever subsidiary to that of our knowledge of physical reality. Every mathematical proof depends absolutely for its validity on our being right about the rules that govern the behaviour of some physical objects, like computers, or ink and paper, or brains. So, contrary to what Hilbert thought, and contrary to what most mathematicians since antiquity have believed and believe to this day, proof theory can never be made into a branch of mathematics. Proof theory is a science: specifically, it is computer science. The whole motivation for seeking a perfectly secure foundation for mathematics was mistaken. It was a form of justificationism. Mathematics is characterized by its use of proofs in the same way that science is characterized by its use of experimental testing; in neither case is that the object of the exercise. The object of mathematics is to understand – to explain – abstract entities.