Georg Cantor

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pages: 292 words: 88,319

The Infinite Book: A Short Guide to the Boundless, Timeless and Endless by John D. Barrow


Albert Einstein, Andrew Wiles, anthropic principle, Arthur Eddington, cosmological principle, dark matter, Edmond Halley, Fellow of the Royal Society, Georg Cantor, Henri Poincaré, Isaac Newton, mutually assured destruction, Olbers’ paradox, prisoner's dilemma, Ray Kurzweil, short selling, Stephen Hawking, Turing machine

The information conveyed by the words ‘the statement T is true’ is different from that contained in the statement T itself. Chapter five The Madness of Georg Cantor 1. (1888–1973) US naturalist and environmental activist 2. A. Christie, An Autobiography, HarperCollins, London, 1998. 3. Quoted in E. Schechter, Handbook of Analysis and its Foundations, Academic, New York, 1998. 4. J. Dauben, Georg Cantor, Princeton University Press, 1990, p. 1. 5. Photograph of Leopold Kronecker, c. 1885, copyright © akg-images. 6. D. Burton, History of Mathematics, 3rd edn, Wm. C. Brown, Dubuque, IA, 1995, p. 593. 7. Dauben, Georg Cantor, p. 134. 8. Ibid. 9. Ibid., p. 135. 10. Ibid., p. 136. 11. Ibid., p. 147. 12. Letter 15 Feb 1896 to Esser, H. Meschkowski, Arch.

Alas, within mathematics the story was quite different, as we shall see. chapter five The Madness of Georg Cantor ‘To be listened to is a nearly unique experience for most people. It is enormously stimulating. Man clamors for the freedom to express himself and for knowing that he counts.’ Robert C. Murphy 1 CANTOR AND SON ‘I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.’ Agatha Christie2 Cantor & Co. was a successful international wholesale business, and as a result young Georg Cantor was one of six children who grew up in comfortable circumstances, attending good private schools in Frankfurt.

The Norwegian Academy has recently created an international prize for mathematics, to rival the Swedish Academy’s long-standing Nobel Prizes in science, literature, economics and peace, called the Abel Prize in honour of Niels Abel (1802–29); it rhymes nicely with Nobel and has similar monetary value. The first winner, in 2003, was French mathematician J.P. Serre. 18. Photograph of Georg Cantor with his wife, Vally, c. 1880, reproduced from Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard University Press, 1979. 19. F. Hutcheson (1694–1746), Inquiry into the Original of Our Ideas of Beauty and Virtue, J. Darby, London, 1720, II, iii. 20. This proviso is to remove the ambiguity created by decimals that end with recurring 9’s. By eliminating decimal expansions that end in zeros, 27/100 is 0.26999 . . . not 0.27000. . . . 21.


pages: 158 words: 49,168

Infinite Ascent: A Short History of Mathematics by David Berlinski


Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Andrew Wiles, Benoit Mandelbrot, Douglas Hofstadter, Eratosthenes, four colour theorem, Georg Cantor, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, John von Neumann, Murray Gell-Mann, Stephen Hawking, Turing machine, William of Occam

The notation encompasses all of the elementary mathematical acts. But neither my own verbal gestures, nor any table of examples, succeed in providing what is really needed, and that is a moment of complete intellectual clarity, circumstances that may be read backward into the late seventeenth century, as Leibniz used one concept that he could not precisely define to explore other concepts that he could not precisely see. Two centuries were to pass before Georg Cantor was to discover the words that in 1684 Leibniz lacked; during all that time, mathematicians continued to use functions of the most remarkable variety, indifferent to their own inability to capture the concept completely. What follows is thus an exercise in anachronism. I am explaining what Leibniz thought as he would have wished to have thought it; in this way, the calculations that he imagined were easy to explain turn out against every expectation to be easy to explain.

The era of the stolid professor has commenced, almost every important mathematician in the years between 1850 and 1900 draping his well-upholstered bottom into a university chair and from those chairs controlling access to the learned journals and the ebb and flow of graduate students and disciples. Dissertations perish in their committees. If mathematics had since the death of Euler in 1785 lost something of its untamed rhapsodic aspect, what it gained was something even more considerable—intellectual mass, soberness, organization, discipline, self-confidence. All eyes on Berlin, of course. Let me interrupt myself to ask: Is a crack-up coming? And to answer: Of course it is. Georg Cantor was born in St. Petersburg in 1845 and spent the first eleven years of his life in the rich, warm, syrupy Russian milieu made accessible by his father’s success as a shrewd merchant. When, in 1856, his father moved his family to Wiesbaden in Germany, Cantor acquired a second language and so a second culture, but he retained throughout his life the notably dreamy disposition of a man inhabiting a larger imaginative space than his tidy German surroundings might have suggested.

His father had thought that his son might become an engineer, a profession that like medicine occupied the middle ground between the practicalities of Geschäft and the rapture of mathematics. The idea filled Cantor with dismay. He won his father’s permission to study mathematics, the exchange between the two men, far from being a tense domestic drama, apparently marked instead by the mutual tenderness of two romantic temperaments, both concerned to please the other. These details might suggest a young man embarking on a modest mathematical career. Nothing of the sort. Georg Cantor initiated a great upheaval in nineteenth-century thought, carrying out one of those revolutions that like certain earthquakes survive in the form of aftershocks long after the first tremor has subsided. The effort dominated his life and it drove him to madness, so that in his last years, when not resident in various lunatic asylums, he occupied himself in proving to his satisfaction that Shakespeare had not written his own plays.


pages: 210 words: 62,771

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


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

Turing did not want an undecidable problem that concerned properties of a hypothetical computing device. He wanted an undecidable problem that was immediately comprehensible to mathematicians. He found one using an ingenious argument of Georg Cantor. 8 Cantor’s Diagonalization Arguments “I see it, but I don’t believe it!” Georg Cantor “I don’t know what predominates in Cantor’s theory — philosophy or theology, but I am sure that there is no mathematics there.” Leopold Kronecker “No one shall drive us from the paradise which Cantor has created for us.” David Hilbert Georg Cantor 1845–1918 Georg Cantor moved with his family from St. Petersberg to Germany when he was eleven years old. After completing his dissertation at the University of Berlin in 1867 he took up a position at the University of Halle where he remained for his professional life.

Encodings and the Universal Machine A Method of Encoding Finite Automata Universal Machines Construction of Universal Machines Modern Computers Are Universal Machines Von Neumann Architecture Random Access Machines RAMs Can Be Emulated by Turing Machines Other Universal Machines What Happens When We Input 〈M〉 into M? 7. Undecidable Problems Proof by Contradiction Russell’s Barber Finite Automata That Do Not Accept Their Encodings Turing Machines That Do Not Accept Their Encodings Does a Turing Machine Diverge on Its Encoding? Is Undecidable The Acceptance, Halting, and Blank Tape Problems An Uncomputable Function Turing’s Approach 8. Cantor’s Diagonalization Arguments Georg Cantor 1845–1918 Cardinality Subsets of the Rationals That Have the Same Cardinality Hilbert’s Hotel Subtraction Is Not Well-Defined General Diagonal Argument The Cardinality of the Real Numbers The Diagonal Argument The Continuum Hypothesis The Cardinality of Computations Computable Numbers A Non-Computable Number There Is a Countable Number of Computable Numbers Computable Numbers Are Not Effectively Enumerable 9.

At times you have a proof, but feel that it is clumsy and that there must be a better one that needs to be found. This is what the Hungarian mathematician Paul Erdős was referring to when he talked about The Book. According to Erdős, there is a book in which God had written down all the shortest, most beautiful proofs. Erdős famously said “You don’t have to believe in God, but you should believe in The Book.” Turing’s proofs, along with those of Gödel and Georg Cantor on which they are based, are definitely in The Book. This book is for the reader who wants to understand these ideas. We start from the beginning and build carefully. The reader doesn’t have to know much mathematics — high school provides enough — but it does require some mathematical aptitude and also a little work. It needs to be read carefully, and some sections may need to be re-read. This is to be expected since Turing is not saying trivial things about computation, but saying deep and nonintuitive things.


pages: 315 words: 93,628

Is God a Mathematician? by Mario Livio


Albert Einstein, Antoine Gombaud: Chevalier de Méré, Brownian motion, cellular automata, correlation coefficient, correlation does not imply causation, cosmological constant, Dava Sobel, double helix, Edmond Halley, Eratosthenes, Georg Cantor, Gerolamo Cardano, Gödel, Escher, Bach, Henri Poincaré, Isaac Newton, John von Neumann, music of the spheres, probability theory / Blaise Pascal / Pierre de Fermat, The Design of Experiments, the scientific method, traveling salesman

First, the introduction of abstract geometric spaces and of the notion of infinity (in both geometry and the theory of sets) had blurred the meaning of “quantity” and of “measurement” beyond recognition. Second, the rapidly multiplying studies of mathematical abstractions helped to distance mathematics even further from physical reality, while breathing life and “existence” into the abstractions themselves. Georg Cantor (1845–1918), the creator of set theory, characterized the newly found spirit of freedom of mathematics by the following “declaration of independence”: “Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.”

Figure 49 The fact that one paradox could have such a devastating effect on an entire program aimed at creating the bedrock of mathematics may sound surprising at first, but as Harvard University logician W. V. O. Quine once noted: “More than once in history the discovery of paradox has been the occasion for major reconstruction at the foundation of thought.” Russell’s paradox provided for precisely such an occasion. Russell’s Paradox The person who essentially single-handedly founded the theory of sets was the German mathematician Georg Cantor. Sets, or classes, quickly proved to be so fundamental and so intertwined with logic that any attempt to build mathematics on the foundation of logic necessarily implied that one was building it on the axiomatic foundation of set theory. A class or a set is simply a collection of objects. The objects don’t have to be related in any way. You can speak of one class containing all of the following items: the soap operas that aired in 2003, Napoleon’s white horse, and the concept of true love.

Due to the perceived drawbacks of the axiom of choice, mathematicians started to wonder whether the axiom could either be proved using the other axioms or refuted by them. The history of Euclid’s fifth axiom was literally repeating itself. A partial answer was finally given in the late 1930s. Kurt Gödel (1906–78), one of the most influential logicians of all time, proved that the axiom of choice and another famous conjecture due to the founder of set theory, Georg Cantor, known as the continuum hypothesis, were both consistent with the other Zermelo-Fraenkel axioms. That is, neither of the two hypotheses could be refuted using the other standard set theory axioms. Additional proofs in 1963 by the American mathematician Paul Cohen (1934–2007, who sadly passed away during the time I was writing this book) established the complete independence of the axiom of choice and the continuum hypothesis.


pages: 233 words: 62,563

Zero: The Biography of a Dangerous Idea by Charles Seife


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

Do it again and again; the number will quickly zoom toward infinity or toward zero, except if you entered 1 or -1 to begin with. There is no escape.) The Infinite Zero My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? I have studied it…I have followed its roots, so to speak, to the first infallible cause of all created things. —GEORG CANTOR Infinity was no longer mystical; it became an ordinary number. It was a specimen impaled on a pin, ready for study, and mathematicians were quick to analyze it. But in the deepest infinity—nestled within the vast continuum of numbers—zero kept appearing. Most appalling of all, infinity itself can be a zero. Figure 41: Spiraling outward and inward on the plane… Figure 42:…are mirror images on the sphere.

It oscillates up and down faster and faster as it approaches the singularity, whipping from positive to negative and back again. In even the tiniest neighborhood around the singularity, the curve takes on almost every conceivable value over and over and over again. Yet as weird as these singularities behave, they were no longer mysterious to mathematicians, who were learning to dissect the infinite. The master anatomist of the infinite was Georg Cantor. Though he was born in Russia in 1845, Cantor spent most of his life in Germany. And it was in Germany—the land of Gauss and of Riemann—where infinity’s secrets were revealed. Unfortunately, Germany was also the land of Leopold Kronecker, the mathematician who would hound Cantor into a mental institution. Underneath Cantor’s conflict with Kronecker was a vision of the infinite, a vision that can be described with a simple puzzle.


pages: 204 words: 60,319

Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers by Amir D. Aczel


colonial rule, double entry bookkeeping, Georg Cantor, offshore financial centre, Y2K

“So you know about Cantor’s work?” I asked, surprised. “Yes, of course. I studied philosophy for many years, including the philosophy of mathematics.” What was so surprising—and something I had not realized before—was that what he told me based on the Jain text provided some proof of a real mathematical understanding of infinity so early, and so long before a great genius in Germany, the tormented mathematician Georg Cantor, was able to explain the same concepts. Cantor was a mathematician at the University of Halle in eastern Germany in the late 1800s, where he single-handedly developed the mathematics of infinity. He had been a student at the University of Berlin, one of the most important universities in Europe at that time, studying under a mathematics giant, Karl Weierstrass, who contributed hugely to our understanding of the real numbers: the numbers on the real number line, which include both rational numbers (integers or quotients of integers) and irrational numbers (numbers, such as pi, that cannot be expressed as quotients of integers).

, A Sourcebook on Indian Philosophy (Princeton, NJ: Princeton University Press, 1957), quoted in Graham Priest, “The Logic of the Catuskoti,” 25. Priest explains that “saint” is a poor translation and that what it means is someone who has reached enlightenment, a Buddha (or Tathagata). 4. Graham Priest, “The Logic of the Catuskoti,” 28. Chapter 17 1. For more on the story of Georg Cantor and the various levels of infinity, see Amir D. Aczel, The Mystery of the Aleph (New York: Washington Square Books, 2001). Chapter 22 1. For accurate radiocarbon dating of the Thera explosion see Amir D. Aczel, “Improved Radiocarbon Age Estimation Using the Bootstrap,” Radiocarbon 37, no. 3 (1995): 845–49. Chapter 24 1. I heard this story from another well-known mathematician and friend of Kakutani, Janos Aczel (no relation; it’s a common Hungarian last name).


pages: 212 words: 68,754

Thinking in Numbers by Daniel Tammet


Albert Einstein, Alfred Russel Wallace, Anton Chekhov, computer age, dematerialisation, Edmond Halley, four colour theorem, Georg Cantor, index card, Isaac Newton, Paul Erdős, Searching for Interstellar Communications

Most of these subsets, though, would have completely slipped his mind. Returning to Borges’s list of subsets of animals, several of the categories pose paradoxes. Take, for example, the subset (j): ‘innumerable ones’. How can any subset of something – even if it is imaginary, like Borges’s animals – be infinite in size? How can a part of any collection not be smaller than the whole? Borges’s taxonomy is clearly inspired by the work of Georg Cantor, a nineteenth-century German mathematician whose important discoveries in the study of infinity provide us with an answer to the paradox. Cantor showed, among other things, that parts of a collection (subsets) as great as the whole (set) really do exist. Counting, he observed, involves matching the members of one set to another. ‘Two sets A and B have the same number of members if and only if there is a perfect one-to-one correspondence between them.’

Ducks and geese need only follow their instinct for when to up sticks and migrate. I have read of oxen that carried their burden for precisely the same duration every day. No whip could persuade them to continue beyond it. We wear the tally of our years on our brow and cheeks. I doubt our body could ever lose its count. Like the ox, each knows intimately the moment when to stop. Higher than Heaven On 22 January 1886 Georg Cantor, who had discovered the existence of an infinite number of infinities, wrote a letter to Cardinal Johannes Franzen of the Vatican Council, defending his ideas against the possible charge of blasphemy. A devout believer, the mathematician considered himself a friend of the Church. God, he believed, had used his preoccupation with numbers to reveal a further aspect of His infinite nature. Fellow logicians had mostly sidestepped the young man’s thinking; hardly anyone yet took seriously the outstanding insights that would make his name.


pages: 262 words: 65,959

The Simpsons and Their Mathematical Secrets by Simon Singh


Albert Einstein, Andrew Wiles, Benoit Mandelbrot, cognitive dissonance, Erdős number, Georg Cantor, Grace Hopper, Isaac Newton, John Nash: game theory, mandelbrot fractal, Menlo Park, Norbert Wiener, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, Richard Feynman, Richard Feynman, Schrödinger's Cat, Simon Singh, Stephen Hawking, Wolfskehl Prize, women in the workforce

However, there was general agreement that Homer and Ned’s petty argument would have derailed the scriptwriting process, as it would have triggered a debate over the nature of infinity. So, is infinity plus one more than infinity? Is it a meaningful statement or just gobbledygook? Can it be proved? In their efforts to answer these questions, the mathematicians around the scripting table would doubtless have mentioned the name of Georg Cantor, who was born in St. Petersburg, Russia, in 1845. Cantor was the first mathematician to really grapple with the meaning of infinity. However, his explanations were always deeply technical, so it was left to the eminent German mathematician David Hilbert (1862–1943) to convey Cantor’s research. He had a knack for finding analogies that made Cantor’s ideas about infinity more palatable and digestible.

Slight variations of this argument can be repeated to show that there are lots of other numbers that are missing from the list of decimals. In other words, when we try to match up the two infinities, the list of decimals between 0 and 1 is doomed to be incomplete, presumably because the infinity of decimal numbers is greater than the infinity of counting numbers. This argument is a simplified version of Cantor’s diagonal argument, a watertight proof published in 1892 by Georg Cantor. Having confirmed that some infinites are bigger than others, Cantor was confident that the infinity that describes counting numbers was the smallest type of infinity, so he labeled it 0, with (aleph) being the first letter of the Hebrew alphabet. He suspected that the set of decimals between 0 and 1 illustrated the next and bigger type of infinity, so he labeled it 1 (aleph-one). Larger types of infinity, for they also exist, are logically named 2, 3, 4, ….


pages: 437 words: 132,041

Alex's Adventures in Numberland by Alex Bellos


Andrew Wiles, Antoine Gombaud: Chevalier de Méré, Black Swan, Black-Scholes formula, Claude Shannon: information theory, computer age, Daniel Kahneman / Amos Tversky, family office, forensic accounting, game design, Georg Cantor, Henri Poincaré, Isaac Newton, pattern recognition, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, random walk, Richard Feynman, Richard Feynman, SETI@home, Steve Jobs, The Bell Curve by Richard Herrnstein and Charles Murray, traveling salesman

‘Non-Euclidean’ geometry was a watershed for mathematics in that it described a theory of physical space that totally contradicted our experience of the world, and therefore was hard to imagine, but nevertheless contained no mathematical contradictions, and so was as mathematically valid as the Euclidean system that came before. Later that century an intellectual breakthrough of similar significance was made by Georg Cantor, who turned our intuitive understanding of the infinite on its head by proving that infinity comes in different sizes. Non-Euclidean geometry and Cantor’s set theory were gateways into two strange and wonderful worlds, and I’ll visit them both in the following pages. Arguably, together they marked the beginning of modern mathematics. Hyperbolic crochet. The Elements, to recap from much earlier, is easily the most influential maths textbook of all time, having set out the basics of Greek geometry.

Considered opinion is that the universe is either flat or spherical, although it is still possible that the universe might be hyperbolic. It is wonderfully ironic to think that a geometry originally thought to be nonsensical might actually reflect the way things really are. At around the same time that mathematicians were exploring the counter-intuitive realm of non-Euclidean space, one man was turning upside-down our understanding of another mathematical notion: infinity. Georg Cantor was a lecturer at Halle University in Germany, where he developed a trail-blazing theory of numbers in which infinity could have more than one size. Cantor’s ideas were so unorthodox that they initially provoked ridicule from many of his peers. Henri Poincaré, for example, described his work as ‘a malady, a perverse illness from which some day mathematics would be cured’, while Leopold Kronecker, Cantor’s former teacher and professor of maths at Berlin University, dismissed him as a ‘charlatan’ and a ‘corruptor of youth’.

Galileo’s conclusion was that, when it comes to infinityostmerical concepts ‘more than’, ‘equal to’ and ‘less than’ do not make sense. These terms may be understandable and coherent when discussing finite amounts, but not with infinite ones. It is meaningless to say there are more numbers than there are squares, or that there is an equal number of numbers and squares, since the totality of both numbers and squares is infinite. Georg Cantor devised a new way to think about infinity that made Galileo’s paradox redundant. Rather than thinking about individual numbers, Cantor considered collections of numbers, which he called ‘sets’. The cardinality of any set is the number of members in the collection. So {1, 2, 3} is a set with a cardinality of three and {17, 29, 5, 14} is a set with cardinality four. Cantor’s ‘set theory’ gets the pulse racing when considering sets with an infinite number of members.


pages: 364 words: 101,286

The Misbehavior of Markets by Benoit Mandelbrot


Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, buy low sell high, capital asset pricing model, carbon-based life, discounted cash flows, diversification, double helix, Edward Lorenz: Chaos theory, Elliott wave, equity premium, Eugene Fama: efficient market hypothesis, Fellow of the Royal Society, full employment, Georg Cantor, Henri Poincaré, implied volatility, index fund, informal economy, invisible hand, John von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, new economy, paper trading, passive investing, Paul Lévy, Plutocrats, plutocrats, price mechanism, quantitative trading / quantitative finance, Ralph Nelson Elliott, RAND corporation, random walk, risk tolerance, Robert Shiller, Robert Shiller, short selling, statistical arbitrage, statistical model, Steve Ballmer, stochastic volatility, transfer pricing, value at risk, volatility smile

At every scale you look, each element of the diagram is similar in shape to the element on the next scale higher up or lower down; “similar” means reduced in size with no deformation. Finance requires a different class of fractals called self-affine, meaning that the scaling happens faster horizontally than vertically. In more general fractals, the parts can get systematically twisted, rotated, or in other ways transformed. The Cantor dust. This is one of the oldest fractals, named after Georg Cantor, a Russian-German of the nineteenth century who radically changed the way mathematicians think about infinity, sets, and many other basic ideas previously taken for granted. The Cantor dust is typical of his paradoxes. It starts as a simple line: straight, continuous, and one-dimensional (here a thickened bar to make it possible to actually see it). Its generator is the same line with the middle third punched out.

price change departure from price swings following probability distributions as risk measured with three different Beta (β) analyzing investments with CAPM with definition of expected return in P/E effect with stock price with Bienaymé, Irénée Binomial time bending Black, Fischer background of eulogy of influence of modern finance influenced by options valued by Black Monday Black-Scholes formula calculation of origin of problem with results of risk evaluation with uses of volatility with Blindfolded archer’s score chance of Bond trading Paris exchange of Book-to-market Bouchaud, Jean-Philippe Bourbaki Boussinesq, Joseph Box-counting dimension fractals with Brahe, Tycho Bridge range in fractal geometry Bronchia fractals Brown, Robert Brownian motion charts with computer-simulated chart of dependence with Dow price movement compared to financial modeling with multifractal model with Nile river flooding with ordered appearance of pollen observed with price changes following risk values following Buffett, Warren E. Bull market Burton, Richard Francis Calculus Calvet, Laurent Canary Wharf Cantor, Georg Cantor dust fractals Capital Asset Pricing Model (CAPM) APT and discovery of finance with Merrill Lynch’s use of premise of results of study of time-scale with Capital budgeting Bachelier’s theories influencing Capital Fund Management CAPM. See Capital Asset Pricing Model Cartier-Bresson photograph Cartoon. See Fractal cartoon Cat brain activity Cauchy, Augustin-Louis distribution by exceptional chance seen by Center of gravity Cerf, Georges Certified Financial Adviser Chance blindfolded archer’s score with corporate finance based on determinism v.


pages: 573 words: 163,302

Year's Best SF 15 by David G. Hartwell; Kathryn Cramer


air freight, Black Swan, experimental subject, Georg Cantor, gravity well, job automation, Kuiper Belt, phenotype, semantic web

My theory stands as firm as a rock; every arrow directed against it will quickly return to the archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things. —Georg Cantor, German mathematician (1845–1918) In a finite world, Abdul Karim ponders infinity. He has met infinities of various kinds in mathematics. If mathematics is the language of Nature, then it follows that there are infinities in the physical world around us as well. They confound us because we are such limited things. Our lives, our science, our religions are all smaller than the cosmos. Is the cosmos infinite?

How much more palatable this is than the thought that the process stops somewhere, that at some point there is a pre-preon, for example, that is composed of nothing else but itself. How fractally sound, how beautiful if matter is a matter of infinitely nested boxes. There is a symmetry in it that pleases him. After all, there is infinity in the very large too. Our universe, ever expanding, apparently without limit. He turns to the work of Georg Cantor, who had the audacity to formalize the mathematical study of infinity. Abdul Karim painstakingly goes over the mathematics, drawing his finger under every line, every equation in the yellowing textbook, scribbling frantically with his pencil. Cantor is the one who discovered that certain infinite sets are more infinite than others—that there are tiers and strata of infinity. Look at the integers, 1, 2, 3, 4…Infinite, but of a lower order of infinity than the real numbers like 1.67, 2.93, etc.

What wonderful arrogance possessed them that they, puny things, could dream so large? He mentions this once to his friend, a Hindu called Gangadhar, who lives not far away. Gangadhar’s hands pause over the chessboard (their weekly game is in progress) and he intones a verse from the Vedas: From the Infinite, take the Infinite, and lo! Infinity remains… Abdul Karim is astounded. That his ancestors could anticipate Georg Cantor by four millennia! That fondness for science…that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic. —Al Khwarizmi, eighth century Arab mathematician Mathematics came to the boy almost as naturally as breathing.


pages: 541 words: 109,698

Mining the Social Web: Finding Needles in the Social Haystack by Matthew A. Russell


Climategate, cloud computing, crowdsourcing,, fault tolerance, Firefox, full text search, Georg Cantor, Google Earth, information retrieval, Mark Zuckerberg, natural language processing, NP-complete, profit motive, Saturday Night Live, semantic web, Silicon Valley, slashdot, social graph, social web, statistical model, Steve Jobs, supply-chain management, text mining, traveling salesman, Turing test, web application

A few of the most relevant ones for the upcoming work at hand include: smembers Returns all of the members of a set scard Returns the cardinality of a set (the number of members in the set) sinter Computes the intersection for a list of sets sdiff Computes the difference for a list of sets mget Returns a list of string values for a list of keys mset Stores a list of string values against a list of keys sadd Adds an item to a set (and creates the set if it doesn’t already exist) keys Returns a list of keys matching a regex-like pattern Skimming the pydoc for Python’s built-in set data type should convince you of the close mapping between it and the Redis APIs. We’re Gonna Analyze Like It’s 1874 Although the concepts involved in set theory are as old as time itself, it is Georg Cantor who is generally credited with inventing set theory. His paper, “On a Characteristic Property of All Real Algebraic Numbers,” written in 1874, formalized set theory as part of his work on answering questions related to the concept of infinity. For example: Are there more natural numbers (zero and the positive integers) than integers (positive and negative numbers)? Are there more rational numbers (numbers that can be expressed as fractions) than integers?

semi-standardized relational data, Motivation for Clustering sentence detection, Syntax and Semantics, Sentence Detection in Blogs with NLTK, Sentence Detection in Blogs with NLTK in blogs, using NLTK, Sentence Detection in Blogs with NLTK, Sentence Detection in Blogs with NLTK sentence tokenizer, Sentence Detection in Blogs with NLTK set operations, Elementary Set Operations, Elementary Set Operations, How Much Overlap Exists Between the Entities of #TeaParty and #JustinBieber Tweets? intersection of entities in #TeaParty and #JustinBieber tweets, How Much Overlap Exists Between the Entities of #TeaParty and #JustinBieber Tweets? Redis native functions for, Elementary Set Operations sample, for Twitter friends and followers, Elementary Set Operations set theory, invention by Georg Cantor, Elementary Set Operations SFDP (scalable force directed placement), Graphviz, Visualizing Community Structures in Twitter Search Results similarity metrics, What Entities Co-Occur Most Often with #JustinBieber and #TeaParty Tweets?, Common Similarity Metrics for Clustering, Finding Similar Documents, The Theory Behind Vector Space Models and Cosine Similarity, Clustering Posts with Cosine Similarity, Clustering Posts with Cosine Similarity, Clustering Posts with Cosine Similarity, Before You Go Off and Try to Build a Search Engine… common, for clustering, Common Similarity Metrics for Clustering cosine similarity, Finding Similar Documents, The Theory Behind Vector Space Models and Cosine Similarity, Clustering Posts with Cosine Similarity, Clustering Posts with Cosine Similarity, Before You Go Off and Try to Build a Search Engine… clustering posts using, Clustering Posts with Cosine Similarity, Clustering Posts with Cosine Similarity limitations of, Before You Go Off and Try to Build a Search Engine… theory behind vector space models and, The Theory Behind Vector Space Models and Cosine Similarity most frequent entities co-occurring with #JustinBieber and #TeaParty tweets, What Entities Co-Occur Most Often with #JustinBieber and #TeaParty Tweets?


Wireless by Stross, Charles


anthropic principle, back-to-the-land, Benoit Mandelbrot, Buckminster Fuller, Cepheid variable, cognitive dissonance, colonial exploitation, cosmic microwave background, epigenetics, finite state, Georg Cantor, gravity well, hive mind, jitney, Khyber Pass, Magellanic Cloud, mandelbrot fractal, peak oil, phenotype, Pluto: dwarf planet, security theater, sensible shoes, Turing machine

(True names have power, so the Laundry is big on call by reference, not call by value; I’m no more “Bob Howard” than the “Alan Turing” in room two is the father of computer science and applied computational demonology.) She continues. “The real Alan Turing would be nearly a hundred by now. All our long-term residents are named for famous mathematicians. We’ve got Alan Turing, Kurt Godel, Georg Cantor, and Benoit Mandelbrot. Turing’s the oldest, Benny is the most recent—he actually has a payroll number, sixteen.” I’m in five digits—I don’t know whether to laugh or cry. “Who’s the nameless one?” I ask. “That would be Georg Cantor,” she says slowly. “He’s probably in room four.” I bend over the indicated periscope, remove the brass cap, and peer into the alien world of the nameless K. Syndrome survivor. I see a whitewashed room, quite spacious, with a toilet area off to one side and a bedroom accessible through a doorless opening—much like the short-term ward.


pages: 311 words: 130,761

Framing Class: Media Representations of Wealth and Poverty in America by Diana Elizabeth Kendall


Bernie Madoff, blue-collar work, Bonfire of the Vanities, call centre, David Brooks, declining real wages, Donald Trump, employer provided health coverage, ending welfare as we know it, framing effect, Georg Cantor, Gordon Gekko, greed is good, haute couture, housing crisis, illegal immigration, income inequality, lump of labour, mortgage tax deduction, new economy, payday loans, Ponzi scheme, Ray Oldenburg, Richard Florida, Ronald Reagan, Saturday Night Live, telemarketer, The Great Good Place, Thorstein Veblen, trickle-down economics, union organizing, upwardly mobile, urban planning, working poor

Although based on people’s perceptions about their own communities, these statements also suggest that individuals living in the region share good middle-class moral values and a belief in the work ethic. National and regional news coverage about a community’s values are not unique. Like the report carried by CNN, articles in local newspapers such as the Detroit News extol the virtues of the middle class. Consider, for example, editorial writer George Cantor’s story titled “Middle-Class Livonia Turns into Wayne County Power”: Livonia is a seething hotbed of middle-class values. It has an almost invisible crime rate [and] neat residential streets, many of them looking as if they had been time-warped from 1956 Detroit. . . . But it is Livonia’s sheer lack of drama that is its charm. “The American dream writ large,” approvingly says an attorney friend of mine who specializes in municipal finance.

Brent Baker, “FNC’s Baier Corrects Washington Post’s Claim Obama ‘Rare’ Product of Middle Class,” NewsBusters, February 5, 2010, http://www.newsbusters .org/blogs/brent-baker/2010/02/05/fnc-s-baier-corrects-washington-post-s-claim -obama-rare-product-middle (accessed October 15, 2010). 40. Ehrenreich, Fear of Falling, 5. 41. Ehrenreich, Fear of Falling, 51. 42. Robin M. Williams Jr., American Society: A Sociological Interpretation, 3rd ed. (New York: Knopf, 1970). 43. Kim Fulcher Linkins, “Midwest Lures Family-Based IT,”, 1999, (accessed February 17, 2004). 44. George Cantor, “Middle-Class Livonia Turns into Wayne County Power,” Detroit News, January 26, 2002, d07-400393.htm (accessed February 18, 2004). 45. Jill Lawrence, “Values, Points of View Separate Towns—and Nation,” USA Today, February 18, 2002, A10. 46. Lawrence, “Values, Points of View Separate Towns.” 9781442202238.print.indb 255 2/10/11 10:47 AM 256 Notes to Pages 179–188 47.


pages: 229 words: 67,599

The Logician and the Engineer: How George Boole and Claude Shannon Created the Information Age by Paul J. Nahin


Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, Albert Einstein, Any sufficiently advanced technology is indistinguishable from magic, Claude Shannon: information theory, conceptual framework, Fellow of the Royal Society, finite state, four colour theorem, Georg Cantor, Grace Hopper, Isaac Newton, John von Neumann, knapsack problem, New Journalism, reversible computing, Richard Feynman, Richard Feynman, Schrödinger's Cat, Steve Jobs, Steve Wozniak, thinkpad, Turing machine, Turing test, V2 rocket

There is no minimum separation between the rationals.5 Nevertheless, despite their denseness, the rationals are still a countable infinity. In other words, the rationals and the integers are infinite sets of the same size, even though the set of the integers is included in the set of rationals. This astonishing result, totally at odds with intuition (what is often called “common sense’’), was discovered by the Russian-born German mathematician Georg Cantor (1845–1918) in 1874. Most mathematicians of his day thought Cantor was crazy, but it was they (not him) who were wrong (although, ironically, Cantor died in a mental institution). To show how the rationals are countably infinite, Cantor had two brilliant ideas. First, he showed how to systematically write down all the rationals so as not to overlook even one. He did that in the form of an infinitely large two-dimensional matrix array.


pages: 396 words: 112,748

Chaos by James Gleick


Benoit Mandelbrot, butterfly effect, cellular automata, Claude Shannon: information theory, discrete time, Edward Lorenz: Chaos theory, experimental subject, Georg Cantor, Henri Poincaré, Isaac Newton, iterative process, John von Neumann, Louis Pasteur, mandelbrot fractal, Murray Gell-Mann, Norbert Wiener, pattern recognition, Richard Feynman, Richard Feynman, Stephen Hawking, stochastic process, trade route

(Once, to Mandelbrot’s horror, a batch of data seemed to contradict his scheme—but it turned out that the engineers had failed to record the most extreme cases, on the assumption that they were irrelevant.) Engineers had no framework for understanding Mandelbrot’s description, but mathematicians did. In effect, Mandelbrot was duplicating an abstract construction known as the Cantor set, after the nineteenth-century mathematician Georg Cantor. To make a Cantor set, you start with the interval of numbers from zero to one, represented by a line segment. Then you remove the middle third. That leaves two segments, and you remove the middle third of each (from one-ninth to two-ninths and from seven-ninths to eight-ninths). That leaves four segments, and you remove the middle third of each—and so on to infinity. What remains? A strange “dust” of points, arranged in clusters, infinitely many yet infinitely sparse.


pages: 289 words: 85,315

Fermat’s Last Theorem by Simon Singh


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

But how can something which is undeniably smaller than an infinite quantity also be infinite? The German mathematician David Hilbert once said: ‘The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.’ To resolve the paradox of the infinite it is necessary to define what is meant by infinity. Georg Cantor, who worked alongside Hilbert, defined infinity as the size of the never-ending list of counting numbers (1, 2, 3, 4, …). Consequently anything which is comparable in size is equally infinite. By this definition the number of even counting numbers, which would intuitively appear to be smaller, is also infinite. It is easy to demonstrate that the quantity of counting numbers and the quantity of even numbers are comparable because we can pair off each counting number with a corresponding even number: If every member of the counting numbers list can be matched up with a member of the even numbers list then the two lists must be the same size.


pages: 1,402 words: 369,528

A History of Western Philosophy by Aaron Finkel


British Empire, Eratosthenes, Georg Cantor, invention of agriculture, Mahatma Gandhi, Plutocrats, plutocrats, the market place, William of Occam

There is one entry in the lower row for every one in the top row; therefore the number of terms in the two rows must be the same, although the lower row consists of only half the terms in the top row. Leibniz, who noticed this, thought it a contradiction, and concluded that, though there are infinite collections, there are no infinite numbers. Georg Cantor, on the contrary, boldly denied that it is a contradiction. He was right; it is only an oddity. Georg Cantor defined an “infinite” collection as one which has parts containing as many terms as the whole collection contains. On this basis he was able to build up a most interesting mathematical theory of infinite numbers, thereby taking into the realm of exact logic a whole region formerly given over to mysticism and confusion. The next man of importance was Frege, who published his first work in 1879, and his definition of “number” in 1884; but, in spite of the epoch-making nature of his discoveries, he remained wholly without recognition until I drew attention to him in 1903.

The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. “Continuity” had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.


The Haskell Road to Logic, Maths and Programming by Kees Doets, Jan van Eijck, Jan Eijck


Albert Einstein, Eratosthenes, Georg Cantor, P = NP

Their meaning may have been explained by definitions in terms of other notions. However, the process of defining notions must have a beginning somewhere. Thus, the need for notions that are primitive, i.e., undefined. For instance, we shall consider as undefined here the notions of set and natural number. Given the notion of a set, that of a function can be defined. However, in a context that is not set-theoretic, it could well be an undefined notion. Georg Cantor (1845-1915), the founding father of set theory, gave the following description. The Comprehension Principle. A set is a collection into a whole of definite, distinct objects of our intuition or of our thought. The objects are called the elements (members) of the set. Usually, the objects that are used as elements of a set are not sets themselves. To handle sets of this kind is unproblematic.


pages: 434 words: 135,226

The Music of the Primes by Marcus Du Sautoy


Ada Lovelace, Andrew Wiles, Arthur Eddington, Augustin-Louis Cauchy, computer age, Dava Sobel, Dmitri Mendeleev, Eratosthenes, Erdős number, four colour theorem, Georg Cantor, German hyperinflation, global village, Henri Poincaré, Isaac Newton, Jacquard loom, Jacquard loom, music of the spheres, New Journalism, Paul Erdős, Richard Feynman, Richard Feynman, Search for Extraterrestrial Intelligence, Simon Singh, Solar eclipse in 1919, Stephen Hawking, Turing machine, William of Occam, Wolfskehl Prize, Y2K

While he was out one day, running along the banks of the River Cam, Turing experienced the second flash of enlightenment that told him why none of these Turing machines could be made to distinguish between statements that had proofs and those that didn’t. As he paused for a breather, lying on his back in a meadow near Granchester, he saw that an idea which had been used successfully to answer a question about irrational numbers might be applicable to this question about the existence of a machine to test for provability. Turing’s idea was based on a startling discovery made in 1873 by Georg Cantor, a mathematician from Halle in Germany. He had found that there were different sorts of infinities. It may seem a strange proposition, but it is actually possible to compare two infinite sets and say that one is bigger than the other. When Cantor announced his discovery, in the 1870s, it was considered almost heretical or at best the ramblings of a madman. To compare two infinities, imagine a tribe that has a counting system that goes ‘one, two, three, lots’.


pages: 626 words: 181,434

I Am a Strange Loop by Douglas R. Hofstadter


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

The absolute nonexistence of a decision procedure for truth (or for provability) is discussed in [DeLong], [Boolos and Jeffrey], [Jeffrey], [Hennie], [Davis 1965], [Wolf], and [Hofstadter 1979]. Page 139 No formula can literally contain… [Nagel and Newman] presents this idea very clearly, as does [Smullyan 1961]. See also [Hofstadter 1982]. Page 139 an elegant linguistic analogy… See [Quine] for the original idea (which is actually a variation of Gödel’s idea (which is itself a variation of an idea of Jules Richard (which is a variation of an idea of Georg Cantor (which is a variation of an idea of Euclid (with help from Epimenides))))), and [Hofstadter 1979] for a variation on Quine’s theme. Page 147 “…and Related Systems (I)”… Gödel put a roman numeral at the end of the title of his article because he feared he had not spelled out sufficiently clearly some of his ideas, and expected he would have to produce a sequel. However, his paper quickly received high praise from John von Neumann and other respected figures, catapulting the unknown Gödel to a position of great fame in a short time, even though it took most of the mathematical community decades to absorb the meaning of his results.


pages: 903 words: 235,753

The Stack: On Software and Sovereignty by Benjamin H. Bratton


1960s counterculture, 3D printing, 4chan, Ada Lovelace, additive manufacturing, airport security, Alan Turing: On Computable Numbers, with an Application to the Entscheidungsproblem, algorithmic trading, Amazon Mechanical Turk, Amazon Web Services, augmented reality, autonomous vehicles, Berlin Wall, bioinformatics, bitcoin, blockchain, Buckminster Fuller, Burning Man, call centre, carbon footprint, carbon-based life, Cass Sunstein, Celebration, Florida, charter city, clean water, cloud computing, connected car, corporate governance, crowdsourcing, cryptocurrency, dark matter, David Graeber, deglobalization, dematerialisation, disintermediation, distributed generation, don't be evil, Douglas Engelbart, Edward Snowden, Elon Musk,, Eratosthenes, ethereum blockchain, facts on the ground, Flash crash, Frank Gehry, Frederick Winslow Taylor, future of work, Georg Cantor, gig economy, global supply chain, Google Earth, Google Glasses, Guggenheim Bilbao, High speed trading, Hyperloop, illegal immigration, industrial robot, information retrieval, intermodal, Internet of things, invisible hand, Jacob Appelbaum, Jaron Lanier, Jony Ive, Julian Assange, Khan Academy, linked data, Mark Zuckerberg, market fundamentalism, Marshall McLuhan, Masdar, McMansion, means of production, megacity, megastructure, Menlo Park, Minecraft, Monroe Doctrine, Network effects, new economy, offshore financial centre, oil shale / tar sands, packet switching, PageRank, pattern recognition, peak oil, performance metric, personalized medicine, Peter Thiel, phenotype, place-making, planetary scale, RAND corporation, recommendation engine, reserve currency, RFID, Sand Hill Road, self-driving car, semantic web, sharing economy, Silicon Valley, Silicon Valley ideology, Slavoj Žižek, smart cities, smart grid, smart meter, social graph, software studies, South China Sea, sovereign wealth fund, special economic zone, spectrum auction, Startup school, statistical arbitrage, Steve Jobs, Steven Levy, Stewart Brand, Stuxnet, Superbowl ad, supply-chain management, supply-chain management software, TaskRabbit, the built environment, The Chicago School, the scientific method, Torches of Freedom, transaction costs, Turing complete, Turing machine, Turing test, universal basic income, urban planning, Vernor Vinge, Washington Consensus, web application, WikiLeaks, working poor, Y Combinator

In twelfth century Majorca, Ramon Llull described logical machines, influencing Gottfried Leibniz, who developed a predictive calculus and a biliteral alphabet that, drawing on the I Ching, allowed for the formal reduction of any complex symbolic expression to a sequence of discrete binary states (zero and one, on and off). Later, the formalization of logic within the philosophy mathematics (from Pierre-Simon Laplace, to Gottlob Frege, Georg Cantor, David Hilbert, and so many others) helped to introduce, inform, and ultimately disprove a version of the Enlightenment as the expression of universal deterministic processes (of both thought and physics). In 1936, with his now-famous paper, “On Computable Numbers, with an Application to the Entscheidungsproblem,” a very young Alan Turing at once introduced the theoretical basis of modern computing and demonstrated the limits of what could and could not ever be calculated and computed by a universal technology.


pages: 551 words: 174,280

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


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

But if you believe that there are bounds on the domain in which reason is the proper arbiter of ideas, then you believe in unreason or the supernatural. Similarly, if you reject the infinite, you are stuck with the finite, and the finite is parochial. So there is no way of stopping there. The best explanation of anything eventually involves universality, and therefore infinity. The reach of explanations cannot be limited by fiat. 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.


pages: 1,079 words: 321,718

Surfaces and Essences by Douglas Hofstadter, Emmanuel Sander


affirmative action, Albert Einstein, Arthur Eddington, Benoit Mandelbrot, Brownian motion, Chance favours the prepared mind, cognitive dissonance, computer age, computer vision, dematerialisation, Donald Trump, Douglas Hofstadter, Ernest Rutherford, experimental subject, Flynn Effect, Georg Cantor, Gerolamo Cardano, Golden Gate Park, haute couture, haute cuisine, Henri Poincaré, Isaac Newton, l'esprit de l'escalier, Louis Pasteur, Mahatma Gandhi, mandelbrot fractal, Menlo Park, Norbert Wiener, place-making, Silicon Valley, statistical model, Steve Jobs, Steve Wozniak, theory of mind, upwardly mobile, urban sprawl

Pandora’s box is open and one jumps readily from four to five dimensions, then six, and so on, all the way to infinity. “What?! A space with an infinite number of dimensions? Balderdash!” Thus reacted many mathematicians at the end of the nineteenth century, yet such objections would just bring smiles to the lips of their counterparts today, for whom the idea seems self-evident. In fact, this is just the tip of the iceberg, for after the work of the German mathematician Georg Cantor, it became a commonplace that there is not just one infinity, but many infinities (of course, there are an infinite number of different infinities). Spaces with a countably infinite number of dimensions (this is the smallest version of infinity, and mathematicians would say that it is the cardinality of the set of natural numbers — that is, of the set of all whole numbers) are called “Hilbert spaces”, and for theoretical physicists, quantum mechanics “lives” in such a space; that is to say, according to modern physics, our universe is based on the mathematics of Hilbert spaces.


The Art of Computer Programming by Donald Ervin Knuth


Brownian motion, complexity theory, correlation coefficient, Eratosthenes, Georg Cantor, information retrieval, Isaac Newton, iterative process, John von Neumann, Louis Pasteur, mandelbrot fractal, Menlo Park, NP-complete, P = NP, Paul Erdős, probability theory / Blaise Pascal / Pierre de Fermat, RAND corporation, random walk, sorting algorithm, Turing machine, Y2K

It is possible to add and subtract mixed-radix numbers by using a straight- straightforward generalization of the usual addition and subtraction algorithms, provided of course that the same mixed-radix system is being used for both operands (see exercise 4.3.1-9). Similarly, we can easily multiply or divide a mixed-radix number by small integer constants, using simple extensions of the familiar pencil- and-paper methods. Mixed-radix systems were first discussed in full generality by Georg Cantor [Zeitschrift fiir Math, und Physik 14 A869), 121-128]. Exercises 26 and 29 give further information about them. Several questions concerning irrational radices have been investigated by W. Parry, Ada Math. Acad. Sci. Hung. 11 A960), 401-416. Besides the systems described in this section, several other ways to represent numbers are mentioned elsewhere in this series of books: the combinatorial num- number system (exercise 1.2.6-56); the Fibonacci number system (exercises 1.2.8-34, 5.4.2-10); the phi number system (exercise 1.2.8-35); modular representations (Section 4.3.2); Gray code (Section 7.2.1); and Roman numerals (Section 9.1).