Приложение 9

Согласно закону Бенфорда, вероятность P, что цифра D появится на первом месте, составляет (логарифм по основанию 10)

P= log (1 + 1/D).

Следовательно, для D = 1

P= log (1 + 1) = log 2 = 0,30.

Для D = 2

P= log (1 + 1/2) = log 1,5 = 0,176,

И так далее. Для D = 9,

P= log (1 + 1/9) = log (10/9) = 0,046.

Согласно обобщенной формулировке закона вероятность того, что первые три цифры будут, к примеру, 1, 5 и 8, равна

P= log (1 + 1/158) = 0,0027.

Приложение 10

Доказательство Евклида, что существует бесконечное множество простых чисел, основано на методе reductio ad absurdum. Сначала Евклид предполагает, что верно противоположное: простых чисел существует лишь ограниченное множество. Однако, если это правда, одно из них должно быть самым большим простым числом. Обозначим самое большое простое число как P. Затем Евклид выводит новое простое число по следующему алгоритму: он перемножает все простые числа, начиная с 2 и до (включая) Р, и прибавляет к произведению единицу. Получается новое число

2 × 3 × 5 × 7 × 11 × … × P+ 1.

Согласно первоначальному предположению, это должно быть не простое, а составное число, поскольку оно, очевидно, больше Р, а мы решили, что Р – самое большое простое число. Следовательно, это число должно делиться по крайней мере на одно из существующих простых чисел. Однако из его конструкции следует, что если мы разделим его на любое простое число вплоть до (и включая) Р, получится остаток 1. А следовательно, если бы это число и в самом деле составное, оно должно делиться на какое-то простое число больше Р. Однако это предположение противоречит первоначальному утверждению, что Р – самое большое простое число, и мы, таким образом, доказали, что простых чисел бесконечно много.

Рекомендуемая литература

Только пустые, ограниченные люди не судят по внешности. Подлинная тайна жизни заключена в зримом, а не в сокровенном…

О. Уайлд (1854–1900) (Пер. М. Абкина)

Большинство книг и статей из этого списка – популярные, а не специальные. Те немногие, которые можно отнести к специальной литературе, отобраны за какие-то особые качества. Кроме того, я отобрал несколько веб-сайтов, где можно найти интересный материал.

1. Прелюдия к числу

Ackermann, F. “The Golden Section”, Mathematical Monthly, 2 (1895): 260–264.

Dunlap, R. A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997.

Fowler, D. H. “A Generalization of the Golden Section”, Fibonacci Quarterly, 20 (1982): 146–158.

Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions. Chicago: University of Chicago Press, 1987.

Ghyka, M. The Geometry of Art and Life. New York: Dover Publications, 1977.

Grattan-Guinness, I. The Norton History of the Mathematical Sciences. New York: W. W. Norton & Company, 1997.

Herz-Fischler, R. A Mathematical History of the Golden Number. Mineola, NY: Dover Publications, 1998.

Hoffer, W. “A Magic Ratio Occurs Throughout Art and Nature”, Smithsonian (December 1975): 110–120.

Hoggatt, V. E., Jr. “Number Theory: The Fibonacci Sequence”, in Yearbook of Science and the Future. Chicago: Encyclopaedia Britannica, 1977, 178–191.

Huntley, H. E. The Divine Proportion. New York: Dover Publications, 1970.

Knott, R. http://www.mcs.surrey.ac.uk/Personal/ R. Knott/Fibonacci/fib.html.

Knott, R. http://www.mcs.surrey.ac.uk/Personal/ R. Knott/Fibonacci/fibnet2.html.

Markowski, G. “Misconceptions about the Golden Ratio”, College Mathematics Journal, 23 (1992): 2–19.

Ohm, M. Die reine Elementar-Mathematik. Berlin: Jonas Veilags-Buchhandlung, 1835.

Runion, G. E. The Golden Section. Palo Alto: Dale Seymour Publications, 1990.

2. Гаммы и пентаграммы

http://search.britannica.com/search?query=fibonacci.

Barrow, J. D. Pi in the Sky. Boston: Little, Brown and Company, 1992.

Beckmann, P. A History of π. Boulder, CO: Golem Press, 1977.

Boulger, W. “Pythagoras Meets Fibonacci”, Mathematics Teacher, 82 (1989): 277–282.

Boyer, C. B. A History of Mathematics. New York: John Wiley & Sons, 1991.

Burkert, W. Lore and Science in Ancient Pythagoreanism. Cambridge, MA: Harvard University Press, 1972.

Conway, J. H., and Guy, R. K. The Book of Numbers. New York: Copernicus, 1996.

Dantzig, T. Number: The Language of Science. New York: The Free Press, 1954.

de la Füye, A. Le Pentagramme Pythagoricien, Sa Diffusion, Son Emploi dans le Syllaboire Cuneiform. Paris: Geuthner, 1934.

Guthrie, K. S. The Pythagorean Sourcebook and Library. Grand Rapids, MI: Phanes Press, 1988.

lfrah, G. The Universal History of Numbers. New York: John Wiley & Sons, 2000.

Maor, E. e: The Story of a Number. Princeton, NJ: Princeton University Press, 1994.

Paulos, J. A. Innumeracy. New York: Vintage Books, 1988.

Pickover, C. A. Wonders of Numbers. Oxford: Oxford University Press, 2001.

Schimmel, A. The Mystery of Numbers. Oxford: Oxford University Press, 1994.

Schmandt-Besserat, D. “The Earliest Precursor of Writing”, Scientific American (June 1978): 38–47.

Schmandt-Besserat, D. “Reckoning Before Writing”, Archaeology, 32–33 (1979): 22–31.

Singh, S. Fermat’s Enigma. New York: Anchor Books, 1997.

Stanley, T. Pythagoras. Los Angeles: The Philosophical Research Society, 1970.

Strohmeier, J., and Westbrook, P. Divine Harmony. Berkeley, CA: Berkeley Hills Books, 1999.

Turnbull, H. W. The Great Mathematicians. New York: Barnes & Noble, 1993.

von Fritz, K. “The Discovery of Incommensurability of Hipposus of Metapontum”.

Annals of Mathematics, 46 (1945): 242–264.

Wells, D. Curious and Interesting Numbers. London: Penguin Books, 1986.

Wells, D. Curious and Interesting Mathematics. London: Penguin Books, 1997.

3. В пирамиде, к звездам обращенной

Beard, R. S. “The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh”, Fibonacci Quarterly, 6 (1968): 85–87.

Burton, D. M. The History of Mathematics: An Introduction. Boston: Allyn and Bacon, 1985.

Doczi, O. The Power of Limits. Boston: Shambhala, 1981.

Fischler, R. “Théories Mathématiques de la Grande Pyramide”, Crux Mathematicorum, 4 (1978): 122–129.

Fischler, R. “What Did Herodotus Really Say? or How to Build (a Theory of) the Great Pyramid”, Environment and Planning B, 6 (1979): 89–93.

Gardner, M. Fads and Fallacies in the Name of Science. New York: Dover Publications, 1957.

Gazalé, M. J. Gnomon. Princeton, NJ: Princeton University Press, 1999.

Gillings, R. J. Mathematics in the Time of the Pharaohs. New York: Dover Publications, 1972.

Goff, B. Symbols of Prehistoric Mesopotamia. New Haven, CT: Yale University Press, 1963.

Hedian, H. “The Golden Section and the Artist”, Fibonacci Quarterly, 14 (1976): 406–418.

Lawlor, R. Sacred Geometry. London: Thames and Hudson, 1982.

Mendelssohn, K. The Riddle of the Pyramids. New York: Praeger Publishers, 1974.

Petrie, W. The Pyramids and Temples of Gizeh. London: Field and Tuer, 1883.

Piazzi Smyth, C. The Great Pyramid. New York: Gramercy Books, 1978.

Schneider, M. S. A Beginner’s Guide to Constructing the Universe. New York: Harper Perennial, 1995.

Spence, K. “Ancient Egyptian Chronology and the Astronomical Orientation of the Pyramids”, Nature, 408 (2000): 320–324.

Stewart, I. “Counting the Pyramid Builders”, Scientific American (September 1998): 98–100.

Verheyen, H. F. “The Icosahedral Design of the Great Pyramid”, in Fivefold Symmetry.

Singapore: World Scientific, 1992, 333–360.

Wier, S. K. “Insights from Geometry and Physics into the Construction of Egyptian

Old Kingdom Pyramids”, Cambridge Archaeological Journal, 6 (1996): 150–163.

4. Второе сокровище

Borissavlievitch, M. The Golden Number and the Scientific Aesthetics of Architecture. London: Alec Tiranti, 1958.

Bruckman, P. S. “Constantly Mean”, «Fibonacci Quarterly», 15 (1977): 236.

Coxeter, H. S. M. Introduction to Geometry. New York: John Wiley & Sons, 1963.

Cromwell, P. R. Polyhedra. Cambridge: Cambridge University Press, 1997.

Dixon, K. Mathographics. New York: Dover Publications, 1987.

Ghyka, M. L’Esthetique des proportions dans la nature et dans les arts. Paris: Gallimard, 1927.

Heath, T. A History of Greek Mathematics. New York: Dover Publications, 1981.

Heath, T. The Thirteen Books of Euclid’s Elements. New York: Dover Publications, 1956.

Jowett, B. The Dialogues of Plato. Oxford: Oxford University Press, 1953.

Kraut, R. The Cambridge Companion to Plato. Cambridge: Cambridge University Press, 1992.

Lasserre, F. The Birth of Mathematics in the Age of Plato. London: Hutchinson, 1964.

Pappas, T. The Joy of Mathematics. San Carlos, CA: Wide World Publishing, 1989.

Trachtenberg, M., and Hyman, I. Architecture: From Prehistory to Post Modernism/The

Western Tradition. New York: Harry N. Abrams, 1986.

Zeising, A. Der goldener Schnitt. Halle: Druck von E. Blochmann & Son in Dresden, 1884.

5. Сын доброй матери-природы

http://cedar.evansville.edu/~ck6/index.html.

Adler, I., Barabe, D., and Jean, R. V. “A History of the Study of Phyllotaxis”, Annals of Botany, 80 (1997): 231–244.

Basin, S. L. “The Fibonacci Sequence as It Appears in Nature”, Fibonacci Quarterly, 1 (1963): 53–64.

Brousseau, Brother A. An Introduction to Fibonacci Discovery. Aurora, SD: The Fibonacci Association, 1965.

Bruckman, P. S. “Constantly Mean”, Fibonacci Quarterly, 15 (1977): 236.

Coxeter, H. S. M. “The Golden Section, Phyllotaxis, and Wythoff’s Game”, Scripta Mathematica, 19 (1953): 135–143.

Coxeter, H. S. M. Introduction to Geometry. New York: John Wiley & Sons, 1963.

Cook, T. A. The Curves of Life. New York: Dover Publications, 1979.

Devlin, K. Mathematics. New York: Columbia University Press, 1999.

Douady, S., and Couder, Y. “Phyllotaxis as a Physical Self-Organized Process”, Physical

Review Letters, 68 (1992): 2098–2101.

Dunlap, R. A. The Golden Ratio and Fibonacci Numbers. Singapore: World Scientific, 1997.

Fibonacci, L. P. The Book of Squares. Orlando, FL: Academic Press, 1987.

“The Fibonacci Numbers”, Time, April 4, 1969, 49–50.

Gardner, M. Mathematical Circus. New York: Alfred A. Knopf, 1979.

Gardner, M. “The Multiple Fascination of the Fibonacci Sequence”, Scientific American (March 1969): 116–120.

Garland, T. H. Fascinating Fibonaccis. White Plains, NY: Dale Seymour Publications, 1987.

Gies, J., and Gies, F. Leonard of Pisa and the New Mathematics of the Middle Ages. New

York: Thomas Y. Crowell Company, 1969.

Hoggatt, V. E. Jr. “Number Theory: The Fibonacci Sequence”, Chicago: Encyclopaedia

Britannica, Yearbook of Science and the Future, 1977, 178–191.

Hoggatt, V. E. Jr., and Bicknell-Johnson, M. “Reflections Across Two and Three Glass Plates”, Fibonacci Quarterly, 17 (1979): 118–142.

Horadam, A. F. “Eight Hundred Years Young”, The Australian Mathematics Teacher, 31 (1975): 123–134.

Jean, R. V. Mathematical Approach to Pattern and Form in Plant Growth. New York: John Wiley & Sons, 1984.

O’Connor, J. J. and Robertson, E. F. www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Fibonacci.html.

Pickover, C. A. Keys to Infinity. New York: John Wiley & Sons, 1995.

Rivier, N., Occelli, R., Pantaloni, J., and Lissowdki, A. “Structure of Binard Convection”.

Cells, Phyllotaxis and Crystallography in Cylindrical Symmetry”, Journal Physique, 45 (1984): 49–63.

Singh, P. “The So-Called Fibonacci Numbers in Ancient and Medieval India”, Historia Mathematica, 12 (1985): 229–244.

Smith, D. E. History of Mathematics. New York: Dover Publications, 1958.

Stewart, I. “Fibonacci Forgeries”, Scientific American (May 1995): 102–105.

Stewart, I. Life’s Other Secret. New York: John Wiley & Sons, 1998.

Thompson, D. W. On Growth and Form. New York: Dover Publications, 1992.

Vajda, S. Fibonacci & Lucas Numbers, and the Golden Section. Chichester: Ellis Horwood Limited, 1989.

Vorob’ev, N. N. Fibonacci Numbers. New York: Blaisdell, 1961.

6. Божественная пропорция

Arasse, D. Leonardo Da Vinci. New York: Konecky & Konecky, 1998.

Beer, A. and Beer, P., eds., Kepler: Four Hundred Years. Vistas in Astronomy, vol. 18, New York: Pergamon Press, 1975.

Calvesi, M. Piero Della Francesca. New York: Rizzoli, 1998.

Caspar, M. Kepler. New York: Dover Publications, 1993.

Cromwell, P. R. Polyhedra. Cambridge: Cambridge University Press, 1997.

Gingerich, O. “Kepler, Galilei, and the Harmony of the World”, in Music and Science in the Age of Galileo, ed. V. Coeltho, 45–63. Dordrecht: Kluwer, 1992.

Gingerich, O. “Kepler, Johannes”, in Dictionary of Scientific Biography, ed. Charles Coulston Gillespie, vol. 7, 289–312. New York: Scribners, 1973.

Ginzburg, C. The Enigma of Piero. London: Verso, 2000.

James, J. The Music of the Spheres. New York: Copernicus, 1993.

Jardine, N. The Birth of History and Philosophy of Science: Kepler’s “A Defense of Tycho against Ursus” with Essays on Its Provenance and Significance. Cambridge: Cambridge University Press, 1984.

Kepler, J. The Harmony of the World. Philadelphia: American Philosophical Society, 1997.

Kepler, J. Mysterium Cosmographicum. New York: Abaris Books, 1981.

Leonardo Da Vinci. NY: Artabras/Reynal and Company, 1938.

MacKinnon, N. “The Portrait of Fra Luca Pacioli”, Mathematical Gazette, 77 (1993): 130–219.

Martens, R. Kepler’s Philosophy and the New Astronomy. Princeton, NJ: Princeton University Press, 2000.

O’Connor, J. J., and Robertson, E. F. www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Pacioli.html/Durer.html.

Pacioli, L. Divine Proportion. Paris: Librairie du Compagnonnage, 1988.

Pauli, W. “The Influence of Archetypal Ideas on the Scientific Theories of Kepler”, in The Interpretation of Nature and the Psyche, 147–240. New York: Parthenon, 1955.

Stephenson, B. The Music of the Spheres: Kepler’s Harmonic Astronomy. Princeton, NJ: Princeton University Press, 1994.

Strieder, P. Albrecht Dürer. New York: Abaris Books, 1982.

Taylor, R. E. No Royal Road. Chapel Hill: University of North Carolina Press, 1942.

Voelkel, J. R. Johannes Kepler. Oxford: Oxford University Press, 1999.

Westman, R. A. “The Astronomer’s Role in the Sixteenth Century: A Preliminary Survey”, History of Science, 18 (1980): 105–147.

7. Равноправие поэтов и живописцев

Altschuler, E. L. Bachanalia. Boston: Little, Brown and Company, 1994.

d’Arcais, F. F. Giotto. New York: Abbeville Press Publishers, 1995.

Bellosi, L. Cimabue. New York: Abbeville Press Publishers, 1998.

Bergamini, D. Mathematics. New York: Time Incorporated, 1963.

Bois, Y.-A., Joosten, J., Rudenstine, A. Z., and Janssen, H. Piet Mondrian. Boston: Little, Brown and Company, 1995.

Boring, E. G. A History of Experimental Psychology. New York: Appleton-Century-Crofts, 1957.

Bouleau, C. The Painter’s Secret Geometry. New York: Harcourt, Brace & World, 1963.

Curchin, L., and Fischler, R. “Hero of Alexandria’s Numerical Treatment of Division in Extreme and Mean Ratio and Its Implications”, Phoenix, 35 (1981): 129–133.

Curtis, W. J. R. Le Corbusier: Ideas and Forms. Oxford: Phaidon, 1986.

Duckworth, G. E. Structural Patterns and Proportions in Vergil’s Aeneid. Ann Arbor: University of Michigan Press, 1962.

Emmer, M. The Visual Mind. Cambridge, MA: MIT Press, 1993.

Fancher, R. E. Pioneers of Psychology. New York: W. W. Norton & Company, 1990.

Fechner, G. T. Vorschule der Aesthetik. Leipzig: Breitkopf & Härtel, 1876.

Fischler, R. “How to Find the ‘Golden Number’ Without Really Trying”, Fibonacci Quarterly, 19 (1981): 406–410.

Fischler, R. “On the Application of the Golden Ratio in the Visual Arts”, Leonardo, 14 (1981): 31–32.

Fischler, R. “The Early Relationship of Le Corbusier to the Golden Number”, Environment and Planning B, 6 (1979): 95–103.

Godkewitsch, M. “The Golden Section: An Artifact of Stimulus Range and Measure of Preference”, American Journal of Psychology, 87 (1974): 269–277.

Hambidge, J. The Elements of Dynamic Symmetry. New York: Dover Publications, 1967.

Herz-Fischler, R. “An Examination of Claims Concerning Seurat and the Golden Number”, Gazette des Beaux-Arts, 125 (1983): 109–112.

Herz-Fischler, R. “Le Corbusier’s ‘regulating lines’ for the villa at Garches (1927) and other early works”, Journal of the Society of Architectural Historians, 43 (1984): 53–59.

Herz-Fischler, R. “Le Nombre d’or en France de 1896 а 1927”, Revue de l’Art, 118 (1997): 9–16.

Hockney, D. Secret Knowledge. New York: Viking Studio, 2001.

Howat, R. Debussy in Proportion. Cambridge: Cambridge University Press, 1983.

Kepes, G. Module, Proportion, Symmetry, Rhythm. New York: George Braziller, 1966.

Larson, P. “The Golden Section in the Earliest Notated Western Music”, Fibonacci Quarterly, 16 (1978): 513–515.

Le Courbusier. Modulor I and II. Cambridge, MA: Harvard University Press, 1980.

Lendvai, E. Béla Bartók: An Analysis of His Music. London: Kahn & Averill, 1971.

Lowman, E. A. “Some Striking Proportions in the Music of Bela Bartók”, Fibonacci Quarterly, 9 (1971): 527–537.

Marevna. Life with the Painters of La Ruche. New York: Macmillan Publishing Co., 1974.

McManus, I. C. “The Aesthetics of Simple Figures”, British Journal of Psychology, 71 (1980): 505–524.

Nims, J. F. Western Wind. New York: McGraw-Hill, 1992.

Nuland, S. B. Leonardo da Vinci. New York: Viking, 2000.

Osborne, H. ed. The Oxford Companion to Art. Oxford: Oxford University Press, 1970.

Putz, J. F. “The Golden Section and the Piano Sonatas of Mozart”, Mathematics Magazine, 68 (1995): 275–282.

Sadie, S. The New Grove Dictionary of Music and Musicians. New York: Grove, 2001.

Schiffman, H. R., and Bobka, D. J. “Preference in Linear Partitioning: The Golden Section Reexamined”, Perception & Psychophysics, 24 (1978): 102–103.

Schillinger, F. Joseph Schillinger. New York: Da Capo Press, 1976.

Schillinger, J. The Mathematical Basis of the Arts. New York: Philosophical Library, 1948.

Schwarz, L. “The Art Historian’s Computer”, Scientific American (April 1995): 106–111.

Somfai, L. Béla Bartók: Compositions, Concepts, and Autograph Sources. Berkeley: University of California Press, 1996.

Svensson, L. T. “Note on the Golden Section”, Scandinavian Journal of Psychology, 18 (1977): 79–80.

Tatlow, R., and Griffiths, P. “Numbers and Music”, in The New Grove Dictionary of Music and Musicians, 18 (2001): 231–236.

Watson, R. I. The Great Psychologists. Philadelphia: J. B. Lippincott Company, 1978.

White, M. Leonardo the First Scientist. London: Little, Brown and Company, 2000.

Woodworth, R. S., and Schlosberg, H. Experimental Psychology. New York: Holt, Rinehart and Winston, 1965.

Zusne, L. Visual Perception of Form. New York: Academic Press, 1970.

8. Звездное небо над нами и плиточный пол у нас под ногами

Cohen, J., and Stewart, I. The Collapse of Chaos. Discovering Simplicity in a Complex World. New York: Penguin Books, 1995.

Fischer, R. Financial Applications and Strategies for Traders. New York: John Wiley & Sons, 1993.

Gardner, M. Penrose Tiles to Trapdoor Ciphers. New York: W. H. Freeman and Company, 1989.

Gleick, J. Chaos. New York: Penguin Books, 1987.

Lesmoir-Gordon, N., Rood, W., and Edney, R. Introducing Fractal Geometry. Cambridge: Icon Books, 2000.

Mandelbrot, B. B. Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1988.

Mandelbrot, B. B. “A Multifractal Walk Down Wall Street”, Scientific American (February 1999): 70–73.

Matthews, R. “The Power of One”, New Scientist, July 10, 1999, 27–30.

Peitgen, H.-O., Jürgens, H., and Saupe, D. Chaos and Fractals. New York: Springer-Verlag, 1992.

Peterson, I. “Fibonacci at Random”, Science News, 155 (1999): 376–377.

Peterson, I. The Mathematical Tourist. New York: W. H. Freeman and Company, 1988.

Peterson, I. “A Quasicrystal Construction Kit”, Science News, 155 (1999): 60–61.

Prechter, R. R. Jr., and Frost, A. J. Elliot Wave Principle. Gainesville, GA: New Classics Library, 1978.

Schroeder, M. Fractals, Chaos, Power Laws. New York: W. H. Freeman and Company, 1991.

Steinhardt, P. J., Jeong, H. C., Saitoh, K., Tanaka, M., Abe, E., and Tsai, A. P. “Experimental Verification of the Quasi-Unit-Cell Model of Quasicrystal Structure”, Nature, 396 (1998): 55–57.

Stewart, I. Does God Play Dice? London: Penguin Books, 1997.

Walser, H. The Golden Section. Washington, DC: The Mathematical Association of America, 2001.

Может быть, Бог – математик?

Baierlein, R. Newton to Einstein: The Trail of Light. Cambridge: Cambridge University Press, 1992.

Barrow, J. D. Impossibility. Oxford: Oxford University Press, 1998.

Chandrasekhar, S. Truth and Beauty. Chicago: University of Chicago Press, 1987.

Chown, M. “Principia Mathematica III”, New Scientist, August 25, 2001, 44–47.

Davis, P. J., and Hersh, R. The Mathematical Experience. Boston: Houghton Mifflin Company, 1998.

Dehaene, S. The Number Sense. Oxford: Oxford University Press, 1997.

Deutsch, D. The Fabric of Reality. New York: Penguin Books, 1997.

Hersh, R. What Is Mathematics, Really? New York: Oxford University Press, 1997.

Hill, T. P. “The First Digit Phenomenon”, American Scientist, 86 (1998): 358–363.

Kleene, S. C. “Foundations of Mathematics”, Chicago: Encyclopaedia Britannica (1971), 1097–1103.

Lakatos, I. Mathematics, Science and Epistemology. Cambridge: Cambridge University Press, 1978.

Lakoff, G., and Nüсez, R. Where Mathematics Comes From. New York: Basic Books, 2000.

Livio, M. The Accelerating Universe. New York: John Wiley & Sons, 2000.

Maor, E. To Infinity and Beyond: A Cultural History of the Infinite. Princeton, NJ: Princeton University Press, 1987.

Matthews, R. “The Power of One”, New Scientist, July 10, 1999, 26–30.

Penrose, R. The Emperor’s New Mind. Oxford: Oxford University Press, 1989.

Penrose, R. Shadows of the Mind. Oxford: Oxford University Press, 1994.

Pickover, C. A. The Loom of God. Cambridge, MA: Perseus Books, 1997.

Popper, K. R., and Eccles, J. C. The Self and Its Brain. New York: Springer International, 1977.

Raimi, R. “The Peculiar Distribution of the First Digit”, Scientific American (December 1969): 109–119.

Raskin, J. http://www.jefraskin.com/forjef2/jefweb-compiled/unpublished/effectiveness_mathematics/

Robinson, A. “From a Formalist’s Point of View”, Dialectica, 23 (1969): 45–49.

Russell, B. A History of Western Philosophy. New York: Simon and Schuster, 1945.

Russell, B. Human Knowledge, Its Scope and Its Limits. New York: Simon and Schuster, 1948.

Weisstein, E. http://matworld.wolfram.com/BenfordsLaw.html.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2002.