John Conway Concepts

Prime Obsession:: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics

by John Derbyshire · 14 Apr 2003

of Sciences or its affiliated institutions. Library of Congress Cataloging-in-Publication Data Derbyshire, John. Prime obsession : Bernhard Riemann and the greatest unsolved problem in mathematics / John Derbyshire. p. cm. Includes index. ISBN 0-309-08549-7 1. Numbers, Prime. 2. Series. 3. Riemann, Bernhard, 1826-1866. I. Title. QA246

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don’t understand the Hypothesis after finishing my book, you can be pretty sure you will never understand it. ***** Various professional mathematicians and historians of mathematics were generous with their help when I approached them. I am profoundly grateful to the following for their time, freely given, for their advice,

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a sort of salon, full of brilliant people saying brilliant things to each other. Euler was a very brilliant man indeed, but unfortunately only in mathematics. His opinions on matters of philosophy, literature, religion, and worldly affairs, while well-informed and sensible, were commonplace and uninspired. Further, Frederick was a

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mathematician alive, after Gauss. Dirichlet married Rebecca Mendelssohn, one of the sisters of the composer Felix Mendelssohn, thereby forming one of the many Mendelssohn-mathematics connections.33 We have some sketches of Dirichlet and his teaching style during his Berlin years from Thomas Hirst, an English mathematician and diarist who

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anywhere, a sensational achievement. Its reading was, declares Hans Freudenthal in the Dictionary of Scientific Biography, “one of the highlights in the history of mathematics.” The ideas contained in this paper were so advanced that it was decades before they became fully 128 PRIME OBSESSION accepted, and 60 years before

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who attended their lectures (technically by the university, which forwarded the students’ fees to the lecturers). 132 PRIME OBSESSION There were few students of mathematics at Göttingen at this time— Riemann’s first lecture drew eight—and lectures were frequently canceled because nobody enrolled for them. Riemann and Dedekind seem

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and imaginary numbers. A lot of people are alarmed by this topic. They believe imaginary numbers are scary, or fantastic, or impossible—have leaked into mathematics from science fiction somehow. This is all nonsense. Complex numbers (of which imaginary numbers are a special case) came into math from very practical

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tactfully informing Hilbert of the situation was delegated to his assistant, Richard Courant. Knowing the pleasure Hilbert took in strolls in the countryside while talking mathematics, Courant invited him for a walk. Courant managed matters so that the pair walked through some thorny bushes, at which point Courant informed Hilbert

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went on to speak of the importance of difficult problems in concentrating the attention of mathematicians, inspiring new developments and new symbols, and in pushing mathematics to higher and higher levels of generalization. He ended with a list of 23 particular problems “from the discussion of which an advancement of

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, is “Riemannian.”) HILBERT’S EIGHTH PROBLEM 195 What has been happening these past few decades, very roughly, is this. For most of its development, mathematics has been firmly rooted in number. Most of nineteenth-century math was concerned with numbers: whole numbers, rational numbers, real numbers, complex numbers. In the

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) becomes untrue or irrelevant—though it might become unfashionable, or be subsumed as a particular case of some more general theory. (And note that in mathematics, “more general” does not necessarily mean “more difficult.” There is a theorem in projective geometry, Desargues’ Theorem, which is easier to prove in three

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a result of a remarkable encounter I shall write about in due course, a physical thread emerged, linking the Hypothesis to 198 PRIME OBSESSION the mathematics of particle physics. While all this was going on, analytic number theorists were still working steadily away, continuing the tradition begun by Riemann himself,

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Landau, 1877−1938. These three men, half a generation after Hilbert, were all pioneers in the early assaults on the Riemann Hypothesis. II. British mathematics in the nineteenth century had been oddly asymmetrical in its development and achievements. Great advances were made by British mathematicians in the least abstract areas

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spent the next few years publishing papers on analysis. One fruit of Hardy’s early analytical obsession was an undergraduate textbook, A Course of Pure Mathematics, first published in 1908 and never subsequently out of print. I learned analysis from this book, as did most twentieth-century British undergraduates. We

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August 1914, when that war began. University professors were civil servants and so came under the scope of the decree. Of the five professors teaching mathematics at CLIMBING THE CRITICAL LINE 255 Göttingen, three—Edmund Landau, Richard Courant, and Felix Bernstein—were Jewish. A fourth, Hermann Weyl (who had succeeded

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VIII. Finally, I cannot resist mentioning the most indirect approach of all, the one through non-deductive logic. This is not, properly speaking, a mathematical topic. Mathematics demands rigorous logical proof before a result can be accepted. Most of the world is not like this, however. In our daily lives we work

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the natural sciences. This line of thought has most recently been taken up by Australian mathematician James Franklin. His 1987 paper “Non-deductive Logic in Mathematics,” in The British Journal for the Philosophy of Science, included a section headed “Evidence for the Riemann Hypothesis and other Conjectures.” Franklin approaches the

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the different approaches I have mentioned: algebraic, analytic, computational, and physical. AIM was established in 1994 by Gerald Alexanderson, a senior figure in American mathematics (and author of a very good book about George Pólya), and John Fry, a California businessman. Fry comes from a family of entrepreneurs. His parents

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in Chapter 9 of The Book of Numbers, by John Conway and Richard Guy. PRIME OBSESSION 370 Though I have not described it properly in this book, the very observant reader will glimpse the Euler-Mascheroni number in Chapter 5. 20. In the mathematics department of my English university, all undergraduates were expected

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In the modern municipality of Verbania. 140. Weender Chaussee has since been renamed Bertheaustrasse. Appendix THE RIEMANN HYPOTHESIS IN SONG T om Apostol, Professor of Mathematics Emeritus at Caltech, wrote the following tribute to the Riemann Hypothesis (RH) in 1955 and performed it at the Caltech Number Theory conference held in

Advances in Artificial General Intelligence: Concepts, Architectures and Algorithms: Proceedings of the Agi Workshop 2006

by Ben Goertzel and Pei Wang · 1 Jan 2007 · 303pp · 67,891 words

fulfilled. In May of 1997, Deep Blue defeated grandmaster and world chess champion Garry Kasparov. Later that year, the sixty-year-old Robbins conjecture in mathematics was proved by a general-purpose, automatic theorem-prover [2]. Narrow AI had come of age. S. Franklin / A Foundational Architecture for Artificial General

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should be such that powerful E. Baum / A Working Hypothesis for General Intelligence 61 superstructures will emerge in new domains such as reasoning about higher mathematics. Finding new meaningful modules is a hard computational problem, requiring substantial search. This suggests a mechanism by which human mental abilities differ so broadly from

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(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’.” Scientific American 223(4): 120-123. [6] Guy, R. K. (1985) “John Horton Conway,” in Albers and G

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effective, context-sensitive inference control heuristics is why the general ability of current automated theorem provers is considerably weaker than that of a mediocre university mathematics major [42]. B. Goertzel and S.V. Bugaj / Stages of Cognitive Development 179 3.Novamente and Probabilistic Logic Networks Novamente’s knowledge representation consists

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the “cat” ConceptNode to the “dog” ConceptNode gets a higher probability weight than the one joining the “cat” ConceptNode to the “washing machine” ConceptNode). The mathematics of 180 B. Goertzel and S.V. Bugaj / Stages of Cognitive Development transformations involving these probabilistic weights becomes quite involved-particularly when one introduces SchemaNodes

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called Probabilistic Logic Networks (PLN) which exists specifically to carry out reasoning on these relationships, and will be described in a forthcoming publication [8]. The mathematics of PLN contains many subtleties, and there are relations to prior approaches to uncertain inference including NARS [40] and Walley’s theory of interval probabilities

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of indirect learning about inference itself) are fully developed. Formal stage tasks are centered entirely around abstraction and higher-order inference tasks such as: x Mathematics and other formalizations. x Scientific experimentation and other rigorous observational testing of abstract formalizations. x Social and philosophical modeling, and other advanced applications of empathy

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mechanisms for uncertain inference is central to the development of Artificial General Intelligence systems. While probability theory provides a principled foundation for uncertain inference, the mathematics of probability theory has not yet been developed to the point where it is possible to handle every aspect of the uncertain inference process in

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this narrowing occurs. An interval of this sort is what Walley calls an “imprecise probability.” Walley’s approach comes along with a host of elegant mathematics including a Generalized Bayes’ Theorem. However it is not the only approach to interval probabilities. For instance, one alternative is Weichselberger’s ([5]) axiomatic

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foundations of the “indefinite probability” notion. In the course of developing the indefinite probabilities approach, we found that the thorniest aspects lay not in the mathematics M. Iklé et al. / Indefinite Probabilities for General Intelligence 199 or software implementation, but rather in the conceptual interpretation of the truth values and

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matters here would take us too far afield. Walley’s approach to representing uncertainty is based explicitly on a Bayesian, subjectivist interpretation; though whether his mathematics has an alternate frequentist interpretation is something he has not explored, to our knowledge. Similarly, our approach here is to take a subjectivist perspective on

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end, we consider the various interpretations of probability to be in the main complementary rather than contradictory, providing different perspectives on the same very useful mathematics. Moving on, then: To adopt a pragmatic frequency-based interpretation of the second-order plausibility in the definition of indefinite plausibility, we interpret “I

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pp. 3–57, (1996). [6] Stanford Encyclopedia of Philosophy, “Interpretations of Probability,” 2003, http://plato.stanford.edu/ entries/probability-interpret/. [7] Frank Ramsey, Foundations of Mathematics, 1931. [8] Bruno de Finetti, Theory of Probability, (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974–5. [9] R. T

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. Cox, “Probability, Frequency, and Reasonable Expectation,” Am. Jour. Phys., 14, 1–13, (1946). [10] Michael Hardy, Advances in Applied Mathematics, August 2002, pages 243–292. [11] Pei Wang, Rigid Flexibility: The Logic of Intelligence, Springer, 2006. [12] Pei Wang, Non-Axiomatic Reasoning System: Exploring the

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well. Looks’ paper discusses the MOSES probabilistic evolutionary learning system created for incorporation into Novamente; and the Ikle’ et al. paper discusses some of the mathematics underlying Novamente’s Probabilistic Logic Networks (PLN), including a simple example of PLN inference applied to embodied learning in the AGISim simulation world. This paper

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agents conversation with humans reading in structured data from databases o databases of general knowledge constructed for other purposes o relational databases o the Mizar mathematics database o quantitative scientific, financial, etc. data o knowledge DB’s constructed specifically for AI’s and other software programs everyday knowledge oriented DB’

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may update the probabilities of conjunctions at various levels in the hierarchy using PLN probabilistic inference, which is ultimately just a different rearrangement of the mathematics used in Bayes nets (though PLN’s arrangement of probability theory has significantly more general applicability). It may be noted that this combination of attentional

Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else

by Jordan Ellenberg · 14 May 2021 · 665pp · 159,350 words

description of Newton (Isaac, not John): “A mind forever / Voyaging through strange seas of Thought, alone.” Hamilton was fascinated with all forms of scholastic knowledge—mathematics, ancient languages, poetry—from his earliest youth, but found his interest in math hyperactivated by a childhood encounter with Zerah Colburn, the “American Calculating Boy

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expansions of the sizes of all the piles, but you can learn all about it in Elwyn Berlekamp, John Conway, and Richard Guy’s astonishingly colorful, profound, and idea-rich book Winning Ways for Your Mathematical Plays, along with other games like Hackenbush, Snort, and Sprouts, and why every game is, in the

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; Word2vec is not a magic meaning machine. More often than not, its proposed “analogy” is no more than a synonym (female “boring” is “uninteresting,” female “mathematics” is “math,” female “amazing” is “incredible”) or a misspelling (female “vicious” is “viscious”) or just wrong: male “duchess” is “prince,” female “pig” is “piglet,” female

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, or so it seemed to the Dutch math team. But sometime in 1983 the Look-and-Say sequence made its way to John Conway, for whom making amusements into mathematics (and mathematics into amusement) was a way of life. Conway showed that the Look-and-Say sequence never contains a number greater than 3

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which they called the graphic notation. Mathematicians, too, were inspired by the new geometric questions the chemists had uncovered, and quickly transposed them into pure mathematics. How many different structures were there, and how should this wild geometric zoo be organized? The algebraist James Joseph Sylvester was one of the first

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. G. Herbst, “Establishing a Custom of Proving in American School Geometry: Evolution of the Two-Column Proof in the Early Twentieth Century,” Educational Studies in Mathematics 49, no. 3 (2002): 283–312. gradient of confidence: Ben Blum-Smith, “Uhm Sayin,” Research in Practice (blog), http:// researchinpractice.wordpress.com/2015/08/01

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. 1 (Dec. 1999): 25. “victorious and intact”: Poincaré, “The Present and the Future,” 38. “crap”: In German, “Mist.” Colin McLarty, “Emmy Noether’s first great mathematics and the culmination of first-phase logicism, formalism, and intuitionism,” Archive for History of Exact Sciences 65, no. 1 (2011): 113. “[S]he discovered methods

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and the Fundamental Theorem of Arithmetic,” Historia Mathematica 21, no. 2 (1994): 162–73. Nim is first attested: L. Rougetet, “A Prehistory of Nim,” College Mathematics Journal 45, no. 5 (2014): 358–63. those hardy microbial spores: W. Fajardo-Cavazos et al., “Bacillus Subtilis Spores on Artificial Meteorites Survive Hypervelocity Atmospheric

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material in this section is adapted from Jordan Ellenberg, “A Fellow of Infinite Jest,” Wall Street Journal, Aug. 14, 2015. “the murder weapon”: István Hargittai, “John Conway—Mathematician of Symmetry and Everything Else,” Mathematical Intelligencer 23, no. 2 (2001): 8–9. He was a compulsive: R. H. Guy, “John Horton Conway

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: J. Conway, “The Weird and Wonderful Chemistry of Audioactive Decay,” Eureka 46 (Jan. 1986). “the first we may compare”: Karl Fink, A Brief History of Mathematics: An Authorized Translation of Dr. Karl Fink’s Geschichte der Elementarmathematik, 2nd ed., trans. Wooster Woodruff Beman and David Eugene Smith (Chicago: Open Court Publishing

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. 4 (2006): 561–89. The Eggenberger-Pólya paper referred to is Florian Eggenberger and George Pólya, “Über die statistik verketteter vorgänge,” ZAMM—Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 3, no. 4 (1923): 279–89. slang for an unclassifiable: Jim Warren, “Feeling Flulike? It’s

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Application of the New Atomic Theory to the Graphical Representation of the Invariants and Covariants of Binary Quantics, with Three Appendices [Continued],” American Journal of Mathematics 1, no. 2 (1878): 109. “In poetry and algebra”: Sylvester, “On an Application.” the phrase “graphic notation”: Material on the origin of the term “graph

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with Rita Dove,” Agni 54 (2001), 175. “you also realize”: “A Chorus of Voices,” 175. “What I have just said”: Henri Poincaré, “The Future of Mathematics” (1908), trans. F. Maitland, appearing in Science and Method (Mineola, NY: Dover Publications, 2003), 32. Perelman himself: Luke Harding, “Grigory Perelman, the Maths Genius Who

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, 137, 138–39, 141, 155, 203, 419 cholera, 244 Chomsky, Noam, 262–63 chords, 191, 298 Christianity, 213 church doctrine, 87. See also religion and mathematics Church of Scientology, 325n cicadas, 149–50 ciphers, 132–33, 136–37. See also cryptography circles, 146–51, 188, 189–90, 196 Clay Foundation, 419

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rotation, 147–48, 148n and scronch geometry, 57 Conic Sections Rebellion, 319–20 connected components, 313 Connect Four, 127 Conrad of Hirsau, 105 consensus in mathematics, 420–21 conservation, 43, 54, 54n, 58, 62–63 consuegro, 190 consumption, 245 “Contagion of Probability, The” (Eggenberger), 299 contiguity, 387, 387n continuous change, 240

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), 415 Geometry No Friend to Infidelity (Jurin), 413 geometry of all geometries, 418 Geometry of Statistics (Pearson), 77 Geometry Revisited (Coxeter and Greitzer), 56 German mathematics and Analytical Society, 253 influence on the French, 38–39 and math pedagogy, 17 and Pearson’s background, 75 and Sylvester’s background, 319, 323

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. McLean, 352 pizza theorem, 308 Planck, Max, 78 Planck’s constant, 297, 297n plane geometry, 62, 411–13 Plato, 215–16, 235, 284 playfulness of mathematics, 121, 223 Plot, Robert, 258–59 Poe, Edgar Allan, 155, 236n poetry chemistry compared to, 318–19 and cryptography, 128–29, 129–30 and Dorfman

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–57, 114 Reform Act of 1832, 350–51 regularity, 283 reinfection rates, 221, 221n Reish Lakish (Shim’on ben Lakish), 159n relativity, 417n religion and mathematics, 85, 87, 276–78, 314, 322n, 413, 420–21 Rényi, Alfréd, 313, 313n, 338n Replacements, The (film), 310 Republican Party. See redistricting restriction system, 138

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-19, the proportion of deaths is thankfully small enough that it’s okay to run the model without this for starters. * Also a collaborator of John Conway’s, on the geometric problem of packing very high-dimensional oranges as tightly as possible into a very high-dimensional box. * A good reminder that

Symmetry and the Monster

by Ronan, Mark · 14 Sep 2006 · 212pp · 65,900 words

presenting intriguing pieces of the subject. This book also presents some interesting gems, but in the service of explaining one of the big quests of mathematics: the discovery and classification of all the basic building blocks for symmetry. Some mathematicians were sceptical of explaining it in a non-technical way, but

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I would like to thank them. In particular I owe thanks to those mathematicians who read all, or large parts, of the manuscript: Jon Alperin, John Conway, Bernd Fischer, Bill Kantor, and Richard Weiss. I also thank my son and daughter who were always positive about the outcome, and finally my editor

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Cambridge in December – he’d been visiting the Institute for Advanced Study in Princeton when he received McKay’s letter – he mentioned these coincidences to John Conway, who had found some of the new symmetry objects himself. Conway had masses of data on the Monster, and used it to produce other sequences

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will always be deeper levels to uncover and further surprises in store. Carl Friedrich Gauss, one of the greatest mathematicians of all time, has called mathematics ‘The queen of sciences’, and it is a subject that compels creativity, driving mathematicians forward on quests that are beyond the power of any individual

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equations of degree 3 reached the ears of Girolamo Cardano (1501–76), who had published work on medicine, astrology, astronomy, and philosophy, as well as mathematics. He asked Tartaglia for the formula but Tartaglia refused. There the matter rested for four years, but Cardano would not be denied. He cajoled Tartaglia

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it appears increasingly likely that this resolution is impossible and contradictory.’* That same year, Paolo Ruffini (1765–1822), a professor of clinical medicine and applied mathematics at Modena, inspired by Lagrange’s work, published a ‘proof’ of the fact that there could be no recipe for solving equations of degree 5

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approximated using a remarkable sequence discovered by the medieval mathematician Leonardo of Pisa, also know as Leonardo Fibonacci. He wrote the first original book on mathematics published in Europe, in about 1200, well before the Renaissance. Leonardo had been brought up in North Africa, where he had learned the Arabic

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tradition of mathematics. He then visited Egypt, Syria, Greece, Sicily, and Provence before settling in Pisa. His book Liber abaci (book of the abacus) introduced Hindu-Arabic

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this. An important result, one that will be used elsewhere, has to be proved to everyone’s satisfaction. Theorems are essential, and this is how mathematics makes progress. It is rather different from theoretical physics in this sense. As the famous physicist Richard Feynman said, ‘The whole purpose in physics is

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Two weeks later he presented results generalizing those of Kummer. Cauchy was an astonishingly productive mathematician and wrote research papers at a terrific pace. French mathematics had, and still has, a regular bulletin called the Comptes Rendus, in which research notes are published very swiftly. In a period of less than

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believe, the results are correct, he has performed an outstanding service. Generally speaking, now the theory of transformation groups ... will reign over vast areas of mathematics.’ It is really unfortunate that Killing, working as he did at a school for the training of future clerics, had no research students. He had

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details were renamed and new details were added, and the result is now known as the Killing–Cartan classification. Abstraction is a vital part of mathematics. It is essential for simplifying and merging difficult technical ideas, so that new progress can be made. Cartan’s attitude to abstraction is well illustrated

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by a talk he gave much later in life (1940 in Belgrade): More than any other science, mathematics develops through a sequence of consecutive abstractions. A desire to avoid mistakes forces mathematicians to find and isolate the essence of problems and the entities

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in importance, and in 1974 the French mathematician Jean Dieudonné wrote, ‘Lie theory is in the process of becoming the most important field in modern mathematics. It had gradually become apparent that the most unexpected theories, from arithmetic to quantum physics, all circle around this field, as around a giant axis

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will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.* Burnside continued turning out excellent results, and in 1904 he published an important theorem about

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and early 1970s, but even the shortest of these cannot compare in elegance and comprehensibility with the original.’* Elegance and clarity are markers of excellent mathematics, and Burnside had used a sophisticated new technique called ‘character theory’ that we shall hear more of later. Other proofs of his theorem were

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was Bourbaki, and in the mid-twentieth century his namesake worked assiduously with his collaborators to create a series of books called The Elements of Mathematics. These books developed an abstract and logical approach to various branches of the subject. The attitude behind them is illustrated by an early Bourbaki paper

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and Armand Borel, a later contributor to the Bourbaki project, describes the early years in the following terms: In the early thirties the situation of mathematics in France at the university and research levels ... was highly unsatisfactory. World War I had essentially wiped out one generation.... Little information was available about

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because so many of its young mathematicians were sent to the front lines and died there. A wartime directory of the most prestigious establishment for mathematics in France shows that about two-thirds of the student population was killed in the war. In Germany, by contrast, young mathematicians were often

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negative assessment of his commitment after a compulsory National Socialist course for lecturers during August 1937. He was reinstated and served as a professor of mathematics in Hamburg until his retirement in 1979. He remained rather honest and naive, never quite realizing how shocked people could be by his flirtation with

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problem in radio broadcasting. 12 The Leech Lattice Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos. E. T. Bell, Mathematics, Queen and Servant of Science In the early days of radio broadcasting, reception was often disturbed by background noise and distortion. You could be sitting

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, And by his frequent lack of socks.* The remarkable Conway will appear again before the end of this book. 13 Fischer’s Monsters In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy. G. H. Hardy, A Mathematician’s Apology In any creative activity

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can sometimes be very useful to go back to fundamentals. The Italian Renaissance, for example, harked back to the ideals of classical art and architecture. Mathematics is like this too, and some wonderful advances are made by going back to basic questions. Bernd Fischer did what many excellent mathematicians have done

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extraordinary group of students and visitors together.’ Some visitors came for extended periods, others just to give a talk. ‘Almost everyone in this area of mathematics came to Frankfurt. Tits was a frequent visitor, Thompson came, Janko came; there were very few exceptions.’ Before the war, in the spring of

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England, and he wrote lecture notes analysing all cases. These notes were freely available, and when I needed a copy some ten years later the mathematics department at Warwick happily obliged. But of course the notes were not available in university libraries, and where another mathematician might have elaborated these notes

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groups, and working out character tables. Fischer made lengthy visits to Birmingham, and he and Livingstone got on well because they had similar tendencies in mathematics. They both loved intense work on technical details, with an adequate supply of cigarettes and coffee, and both were rather diffident about writing up their

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s special prime numbers yield what I will call mini-j-functions*. When Thompson returned to Cambridge at the beginning of 1979 he explained to John Conway that by adding character degrees in the first column of the Monster’s character table he could get the first six coefficients of the j

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using different columns of the character table, were combinations of Monster characters. Thompson wrote a couple of fairly short papers on his recent work, and John Conway and Simon Norton wrote a more expansive paper with the title ‘Monstrous Moonshine’. It dealt with all columns in the Monster’s character table, and

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They had appeared a couple of years earlier as ‘vertex algebras’, but most mathematicians had never heard of them. Moreover, vertex operators originated not in mathematics, but in physics. They come from string theory, and describe the interaction of strings, which are models for elementary particles. This suggested a connection between

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C. Tompson (eds), The Collected Papers of William Burnside, two volumes, Oxford University Press, 2004. Chapter 8: After the War 90 N. Bourbaki, Foundations of mathematics for the working mathematician, Journal of Symbolic Logic, 14 (1949), 1–8. 91 A. Borel, Twenty-five years with Nicolas Bourbaki (1949–1973), Notices of

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happened to work in the same area of mathematics. 150 Quotations from Conway appear in Thomas Thompson, From Error-correcting Codes through Sphere Packings to Simple Groups, Carus Mathematical Monograph 21, Mathematical Association of America, 1983. 156 John Conway, On Numbers and Games, Academic Press, 1976; Elwyn Berlekamp, John Conway, and Richard Guy, Winning Ways for Your

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work of Richard Ewen Borcherds, Documenta Mathematica, extra volume, 1 (1998), 99–108. L. Gårding and L. Hörmander, Why is there no Nobel Prize in mathematics?, Mathematical Intelligencer, 7 (1985), 73–4. 227 This special feature also yields a fact, first noticed by Euler, that the formula x2 − x + 41 gives prime

Practical OCaml

by Joshua B. Smith · 30 Sep 2006

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

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’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

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to learn more about types and their impact on programming. Unfortunately, type theory and category theory are not very approachable without a fairly serious formal mathematics background. You can read a few books on the subject that might be useful to nonmathematicians, however. Then there is OCaml itself. We have scratched

The Grand Design

by Stephen Hawking and Leonard Mlodinow · 14 Jun 2010 · 124pp · 40,697 words

that most laws of nature exist as part of a larger, interconnected system of laws. In modern science laws of nature are usually phrased in mathematics. They can be either exact or approximate, but they must have been observed to hold without exception—if not universally, then at least under a

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mathematics and physical picture are different from that of the original formulation of quantum physics, but the predictions are the same. In the double-slit experiment

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B, you add the phases, or arrows, associated with every path connecting A and B. There are an infinite number of paths, which makes the mathematics a bit complicated, but it works. Some of the paths are pictured below. Feynman’s theory gives an especially clear picture of how a Newtonian

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Davy. Faraday would stay on for the remaining forty-five years of his life and, after Davy’s death, succeed him. Faraday had trouble with mathematics and never learned much of it, so it was a struggle for him to conceive a theoretical picture of the odd electromagnetic phenomena he observed

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requirement that one include all the histories by which an interaction can occur—for example, all the ways the force particles can be exchanged—the mathematics becomes complicated. Fortunately, along with inventing the notion of alternative histories—the way of thinking about quantum theories described in the last chapter—Feynman also

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number of ways to curl up the tiny dimensions? In M-theory those extra space dimensions cannot be curled up in just any way. The mathematics of the theory restricts the manner in which the dimensions of the internal space can be curled. The exact shape of the internal space determines

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equations. In 1922, Russian physicist and mathematician Alexander Friedmann investigated what would happen in a model universe based upon two assumptions that greatly simplified the mathematics: that the universe looks identical in every direction, and that it looks that way from every observation point. We know that Friedmann’s first assumption

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ideas of space and time do not apply to the very early universe. That is beyond our experience, but not beyond our imagination, or our mathematics. If in the early universe all four dimensions behave like space, what happens to the beginning of time? The realization that time can behave like

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can help us think about issues of reality and creation is the Game of Life, invented in 1970 by a young mathematician at Cambridge named John Conway. The word “game” in the Game of Life is a misleading term. There are no winners and losers; in fact, there are no players. The

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which we could not have written this book. Moreover, they always knew where to find the best pubs. STEPHEN HAWKING was the Lucasian Professor of Mathematics at the University of Cambridge for thirty years, and has been the recipient of numerous awards and honors including, most recently, the Presidential Medal of

The End of Theory: Financial Crises, the Failure of Economics, and the Sweep of Human Interaction

by Richard Bookstaber · 1 May 2017 · 293pp · 88,490 words

who experienced the effects of these crises firsthand, and was prepared to bring new tools to the job. Jevons was the first modern economist, introducing mathematics into the analysis and initiating what became known as the marginalist revolution—a huge leap forward that reshaped our thinking about the values of investment

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analysis and with a focus on the human consequences of the dominance of capital, fomented revolution that would engulf the world. The other, based on mathematics, emulated the mechanics of the natural sciences while ignoring the human aspect completely, forming the foundation for today’s standard economic model, that of

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elements of human nature and the limits that these imply: computational irreducibility means that the complexity of our interactions cannot be unraveled with the deductive mathematics that forms the base—even the raison d’être—for the dominant model in current economics. As the novelist Milan Kundera has written, we

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not left merely watching the phenomenon and taking notes. The primary tool for executing these shortcuts, the tool of the scientist-cum-cartographer, is mathematics, and mathematics deductively applies a general axiomatic structure, a structure that begins with the statement of laws. If we instead are left to traverse the map to

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on the design of the machines, conquering the universe—with exponential efficiency. The universal constructor caught the interest of John Conway, a British mathematician who would later hold the John von Neumann Chair of Mathematics at Princeton, and over “eighteen months of coffee times,” as he describes it, he began tinkering to simplify

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and philosopher, began an intense ten-year study that would culminate with the completion of a three-volume, 1,800-page tome of nearly impenetrable mathematics titled Principia Mathematica. This study, for which he enlisted his teacher, the mathematician Alfred North Whitehead, as a co-laborer, aimed to demonstrate that

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and Related Systems,” which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable. (This work, one of the keystones of mathematics and logic, was, incredibly enough, written as part of Gödel’s qualifying dissertation for entrance into the teaching profession.) Gödel proved that it would

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of self-referential systems. HEISENBERG’S UNCERTAINTY PRINCIPLE Just four years before Gödel defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his undecidability theorem, the German physicist Werner Heisenberg’s celebrated uncertainty principle delineated the limits of inquiry into the

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, we must work with the limits that address these essential aspects of our human condition. And, more important, we must refute the use of mathematics because the essential problems are computationally irreducible; refute the notion that we all can be represented by a proxy, which already should be refuted based

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of looking at how people interact and seeing where that drives the world employs an inductive process. It uses experience and observation; it might use mathematics, but, as with historical analysis, it might proceed through narrative. This is a pivotal difference between the methods of economics and the methods I

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, in addition to the roles I have already mentioned in developing game theory and conceptualizing replicating machines, and in addition to his foundational work in mathematics, physics, computer science, and economics, also was central in this effort. 2. This fits within an emerging interest among the socially minded in the

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Bouwman. 2009. “Bank Liquidation Creation.” Review of Financial Studies 22, no. 9: 3779–3837. doi: 10.1093/rfs/hhn104. Bloch, William Goldbloom. 2008. The Unimaginable Mathematics of Borges’ Library of Babel. Oxford: Oxford University Press. Boccaletti, Stefano, Ginestra Bianconi, Regino Criado, Charo I. Del Genio, Jesus Gómez-Gardeñes, Miguel Romance, Irene

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Zagaglia. 2011. “Measuring Market Liquidity: An Introductory Survey.” Quaderni DSE Working Paper no. 802. doi: 10.2139/ssrn.1976149. Gardner, Martin. 1970. “Mathematical Games: The Fantastic Combinations of John Conway’s New Solitaire Game ‘Life.’” Scientific American 223: 120–23. Gigerenzer, Gerd. 2008. Rationality for Mortals: How People Cope with Uncertainty. Evolution and

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Qualities, The, 178. See also Musil, Robert map versus territory, 25 Mark Humphrys. See Humphrys, Mark Marx, Karl, 4 Marxism, 58. See also Marx, Karl mathematics: and axioms optimization method, 74; and the deductive approach, 86–87; and determinism, 116 MBIA, 165 Mencken, H. L., 88 Menger, Carl, 194 Meriwether,

The Man From the Future: The Visionary Life of John Von Neumann

by Ananyo Bhattacharya · 6 Oct 2021 · 476pp · 121,460 words

protein crystallography from Imperial College London. To geeks and nerds everywhere, but especially for the three closest to me. ‘If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.’ John von Neumann INTRODUCTION: Who Was John von Neumann? ‘Von

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he joined the Manhattan Project, von Neumann was finishing, with the economist Oskar Morgenstern, a 640-page treatise on game theory – a field of mathematics devoted to understanding conflict and cooperation. That book would change economics, make game theory integral to fields as disparate as political science, psychology and evolutionary

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artificial intelligence and influenced the development of neuroscience. Von Neumann was a pure mathematician of extraordinary ability. He established, for example, a new branch of mathematics, now named after him, that was richly productive: half a century later, Vaughan Jones won the Fields Medal – often called the Nobel Prize of

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to the gimnázium, just different in scope and somewhat more practical then the “gentlemanly” gimnázium,’ according to one historian and ‘boasted extraordinary students in mathematics and the sciences’.15 Among them were Fejér, Leo Szilard, who first conceived of the nuclear chain reaction that powers reactors and bombs, and Dennis

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at stake. The seventeen-year-old von Neumann stepped in to put things right. 2 To Infinity and Beyond A teenager tackles a crisis in mathematics ‘Mathematics is the foundation of all exact knowledge of natural phenomena.’ David Hilbert, 1900 Von Neumann’s unique talents were spotted as soon as he started

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pursuit would help to birth a truly revolutionary machine: the modern computer. 3 The Quantum Evangelist How God plays dice ‘If only I knew more mathematics!’ Erwin Schrödinger, 1925 After his doctoral examination, von Neumann quickly secured a grant from the Rockefeller Foundation and headed to Hilbert’s Göttingen, the

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in the abridged version of the paper that was published by the prestigious journal Naturwissenschaften.62 Quite possibly she felt that rigorous philosophy, not more mathematics, was required to save determinism.63 Not until 1966, thirty years after Hermann had published her critique, were the limitations of the impossibility proof

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the early twentieth century. An unlikely turn of history would entangle the intellectual roots of the modern computer with Hilbert’s challenge to prove that mathematics was complete, consistent and decidable. Soon after Hilbert issued his challenge, the intellectually dynamic but psychologically frail Austrian mathematician Kurt Gödel would demonstrate that

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it is impossible to prove that mathematics is either complete or consistent. Five years after Gödel’s breakthrough, a twenty-three-year-old Turing would attack Hilbert’s ‘decision problem’ (Entscheidungsproblem)

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in a way completely unanticipated by any other logician, conjuring up an imaginary machine to show that mathematics is not decidable. The formalisms of these two logicians would help von Neumann crystallize the structure of the modern computer. The result of his musings

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of the type of Goldbach or Fermat) that, while contentually true, are unprovable in the formal system of classical mathematics.’ In other words, there are truths in mathematics that cannot be proven by mathematics. Mathematics is not complete. His off-hand references to Goldbach and Fermat were portentous. Golbach’s conjecture (that all

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retired as the Drummond Professor of Political Economy at Oxford University. As a result, after many years of indifference, Morgenstern developed a newfound respect for mathematics. ‘In Germany,’ Morgenstern wrote after Edgeworth’s death the following year, ‘every beginner believes … that he should create entirely “new” foundations and a completely

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new methodology. In England, however, the use of mathematics is more common than elsewhere, a further circumstance to frighten away dilettantes who are only half interested in economics.’ During his fellowship, Morgenstern developed an

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sought to explain booms and busts by careful analysis of economic statistics with the eventual goal of anticipating downturns. But what Morgenstern had learned about mathematics made him pessimistic. His dissertation was a sustained attack on economic forecasting, which he argued was impossible. When his screed was published in 1928,

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Neumann, he absolutely didn’t see this,’ Wolfram says, insisting that he was the first to truly understand that enormous complexity can spring from automata. ‘John Conway, same thing.’51 A New Kind of Science is a beautiful book. It may prove in time also to be an important one. The jury

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Martians of Science: Five Physicists Who Changed the Twentieth Century, Oxford University Press, Oxford. Heims, Steve J., 1982, John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, MIT Press, Cambridge, Mass. Hoddeson, Lillian, Henriksen, Paul W., Meade, Roger A. and Westfall, Catherine, 1993, Critical

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you congratulating me?’ a bewildered Jancsi had asked him. Because, Wigner said, Johnny had been very close. 7. See, for example, Harry Henderson, 2007, Mathematics: Powerful Patterns into Nature and Society, Chelsea House, New York, p. 30. 8. Despite decades of research, whether chess ability is correlated with general intelligence

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.highereducation. 42. Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory. 43. Steve J. Heims, 1982, John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, MIT Press, Cambridge, Mass. 44. Quoted in Dyson, Turing’s Cathedral. 45. Quoted in ibid. 46. See Leonard

What Algorithms Want: Imagination in the Age of Computing

by Ed Finn · 10 Mar 2017 · 285pp · 86,853 words

; each of them generates cultural power based on the inherent tension between reality and representation. The link between spoken language and abstract symbolic systems, particularly mathematics, has created new avenues for mystical connections between numbers, universal truths, and the fundamental structure of reality. Jewish kabbalah, Isaac Newton’s fascination with

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turn to the engineers and computer scientists who implement computational systems. Rooted in computer science, this version of the algorithm relies on the history of mathematics. An algorithm is a recipe, an instruction set, a sequence of tasks to achieve a particular calculation or result, like the steps needed to

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level, computationalism suggests that our knowledge of computation will answer many fundamental questions: computation becomes a universal solvent for problems in the physical sciences, theoretical mathematics, and culture alike. The quest for knowledge becomes a quest for computation, a hermeneutics of modeling. But of course models always compress or shorthand

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both struggling with the boundary problems of mathematics. In one framing, posed by mathematician David Hilbert, known as the Entscheidungsproblem, the question is whether it’s possible to predict when or if

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from quantum mechanics to the circuits inside the human brain. After World War II, a new field emerged to pursue that promise, struggling to align mathematics and materiality, seeking to map out direct correlations between computation and the physical and social sciences. In its heyday cybernetics, as the field was known

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to model these processes in simulation. Mathematician John Conway’s game of life, for example, seeks to model precisely this kind of spontaneous generation of information, or seemingly living or self-perpetuating patterns, from simple rule-sets. It, too, has been shown to be mathematically equivalent to a Turing machine, and indeed

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By occupying and defining that awkward middle ground, algorithms and their human collaborators enact new roles as culture machines that unite ideology and practice, pure mathematics and impure humanity, logic and desire. To discuss implementation is thus to join a conversation about materiality and the embodied subjects that enact, transmit,

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opportunity for direct, unmediated human contact, adds a little to the space between computation and human experience. Algorithms work as complex aggregates of abstraction, incantation, mathematics, and technical memory. They are material implementations of the cathedral of computation.87 When we interact with them, we are speaking to oracles, gods,

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other without the need for a trusted third party.”24 The straightforward paper describes a system for exchanging currency based purely on computing power and mathematics (which I describe in more detail below), with no dependence on a central bank, a formal issuing authority, or other “faith and credit” standards

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. Morozov, Evgeny. To Save Everything, Click Here: The Folly of Technological Solutionism. New York: PublicAffairs, 2013. Moschovakis, Yiannis N. What Is an Algorithm? In Mathematics Unlimited: 2001 and Beyond, edited by Björn Engquist and Wilfried Schmid, 919–936. Berlin: Springer-Verlag, 2001. Nakamoto, Satoshi. “Bitcoin: A Peer-to-Peer Electronic

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and, 39–40 LCARS (Star Trek) and, 67–68 machine learning and, 2, 15, 28, 42, 62, 66, 71, 85, 90, 112, 181–184 mathematics and, 49–50 Mechanical Turk and, 12, 135–145 media scholars and, 54 mining value and, 175–176 movie rentals and, 88–89 (see also

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and, 26–29 fungible space of, 35 how to think of, 36–41 imagination and, 46, 93, 192–193 implementation and, 47–49, 54 mathematics and, 40 metaphor and, 34 Netflix and, 93 pragmatist approach and, 25–26 process methodology and, 53 proofs and, 24 quest for universal knowledge and

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24, 40 as intellectual technology, 4 intelligent assistants and, 11, 57, 62, 64–65, 77 machine learning and, 2, 112 many registers of, 1–2 mathematics and, 2, 55 meaning and, 1 metaphor and, 183–184 (see also Metaphor) natural language processing (NLP) and, 62–63 of new media, 112,

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146 Wall Street and, 16, 66, 109, 151, 153, 171, 185 Marx, Karl, 165 Master Algorithm, The (Domingos), 183 Materiality, 26, 47–49, 53, 133 Mathematics abstract symbolism, 2, 55 algebra, 17 Babylonian, 17 Berlinski and, 9, 181 calculus, 24, 26, 30, 34, 44–45, 98, 148, 186 complexity, 28 computationalist

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culture machines and, 62–65, 68–69 Google and, 159–160 ideology and, 68 imagination and, 69, 73–74 of information, 8, 63, 69–71 mathematics and, 84 meaning and, 8, 21–22, 26, 39 money and, 156–159, 178–179 Netflix and, 92, 94, 96 Siri and, 62–65,

What Technology Wants

by Kevin Kelly · 14 Jul 2010 · 476pp · 132,042 words

observations into the very condensed container of E = mc2. Every scientific theory and formula—whether about climate, aerodynamics, ant behavior, cell division, mountain uplift, or mathematics—is in the end a compression of information. In this way, our libraries packed with peer-reviewed, cross-indexed, annotated, equation-riddled journal articles are

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, such as the arboreal splay of branches in a tree and coral or the swirling spiral of petals on a flower, are based on the mathematics of growth. They repeat because the math is eternal. All life on Earth is protein based, and the way those proteins fold and unfold inside

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rest of humanity, the inevitable happens on schedule. The technium’s trajectory is more fixed in certain realms than in others. Based on the data, “mathematics has more apparent inevitability than the physical sciences,” wrote Simonton, “and technological endeavors appear the most determined of all.” The realm of artistic inventions—those

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a retreat from the rest of us. Kaczynski buckled under the many rules and expectations society put up for him as an aspiring professor of mathematics. He said, “Rules and regulations are by nature oppressive. Even ‘good’ rules are reductions in freedom.” He was deeply frustrated at not being able to

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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

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) 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

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frequency of intersections times length. In other words, the ants discovered an approximate value for pi derived by intersecting diagonals, a technique now known in mathematics as Buffon’s Needle. Headroom in the potential ant house is measured by the ants with their bodies and then “multiplied” with the calculated area

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paint and canvas unleashed the talents of painters through the centuries. The technology of film created cinematic talents. The soft technologies of writing, lawmaking, and mathematics all expanded our potential to create and do good. Thus in the course of our lives as we invent things and create new works that