www.EngineeringBooksPDF.com Science, SETI, and Mathematics www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com SCIENCE, SETI, AND MATHEMATICS Carl L DeVito berghahn NEW YORK • OXFORD www.berghahnbooks.com www.EngineeringBooksPDF.com Published in 2014 by Berghahn Books www.berghahnbooks.com © 2014 Carl L DeVito All rights reserved Except for the quotation of short passages for the purposes of criticism and review, no part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system now known or to be invented, without written permission of the publisher Library of Congress Cataloging-in-Publication Data DeVito, Carl L., author Science, SETI and mathematics / Carl L DeVito pages cm Includes bibliographical references and index ISBN 978-1-78238-069-6 (hardback : alk paper) — ISBN 978-1-78238-070-2 (institutional ebook) Science—Mathematics Extraterrestrial beings I Title Q175.32.M38D48 2013 999.01’51 dc23 2013015469 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Printed in the United States on acid-free paper ISBN: 978-1-78238-069-6 hardback ISBN: 978-1-78238-070-2 institutional ebook www.EngineeringBooksPDF.com Contents Preface viii Chapter Where Are We? Remark: Natural Numbers, Sets, and Subsets Chapter Naïve Questions Remark: Infinite Sets, Correspondences, Unions, and Intersections 15 Chapter Are We Special? 17 Remark: Systems of Enumeration, Powers of Ten, Positional Notation, and Casting Out Nines 23 Chapter Stories—Part One 26 Remark: Human Perception of Motion, and Mathematical Description of Physical Fields 32 Chapter Measuring Our Solar Neighborhood 36 Remark: Euclid’s Fifth Postulate, Non-Euclidean Geometries, and How Choice of Geometry Affects Physics 43 Chapter The Scotsman Remark: The Fundamental Wave Equation, Partial Differential Equations, Equations of Mathematical Physics, and the Function Concept 51 www.EngineeringBooksPDF.com 47 Chapter The Birth of SETI 54 Remark: Two Functions and Why They Are Special, the Power of Trigonometry, and Fourier Series 60 Chapter The Conference at Green Bank 64 Remark: The Drake Equation, Drake’s Postcard, and Prime Numbers 69 Chapter Stories—Part Two 73 Remark: Development of Calculus, Models for Time, Differential Calculus and the Science of Motion, and Derivatives and Partial Derivatives 84 Chapter 10 Talking to E.T 89 Remark: Continuity of Space, Area, Integral Calculus and the Founding of Carthage, Line Integrals and the CAT Scan 94 Chapter 11 Languages 99 Remark: Real Numbers as the Basis for Calculus, Complex Numbers and the Calculus of Complex Functions, Complex Integration, and Whether Mathematical Objects Are Real 109 Chapter 12 Paradoxes 113 Remark: Group Theory in Algebra and Geometry 115 Chapter 13 The Universal Science 119 Remark: Atomic Weights and the Avogadro Number 127 Chapter 14 The Special Theory of Relativity Remark: Space-Time, Higher Dimensional Spaces, and Hilbert Space 138 www.EngineeringBooksPDF.com 129 Contents • vii Chapter 15 The General Theory of Relativity 143 Remark: The Geometry of Minkowski’s 4-World, and Why Points Are Zero Dimensional 149 Chapter 16 The University of Colorado Study 152 Remark: Space as Multi-Dimensional, the Dimension of Sets, and General Topology and Functional Analysis 160 Chapter 17 Surprise! 163 Remark: Fibonacci Numbers and the Golden Ratio, Logarithms, Exponentials, and the Number e, Connections to the Complex Numbers 165 Chapter 18 Epilogue 169 Remark: Ramanujan 176 Appendix I Infinite Sets 177 Appendix II Mars 182 Appendix III The DeVito-Oehrle Language 185 Bibliography 198 Index 203 www.EngineeringBooksPDF.com Preface This book is intended for my colleagues in the humanistic and natural sciences who share my interest in the search for extraterrestrial intelligence (SETI) It is about the role mathematics might play in this endeavor Since I am writing for a wide audience, an audience of people with very diverse backgrounds, I have focused on ideas and avoided mathematical symbolism and technical jargon No prior knowledge of mathematics is assumed and, since this subject may be new to many of my readers, I also present the history of, and the science behind, this search My goal is to stimulate a discussion, among scientists interested in this area, of the ideas presented here Many contend that a great deal of our mathematics would be understandable, even familiar, to the members of any technologically sophisticated race—the only kind of society our current methods of searching will enable us to find I examine this contention in detail The astronomical environment of our planet, in particular our large moon, human evolutionary history, and our reliance on the sense of sight, have all influenced our mathematics The subject is very much a part of our humanity, somewhat like our music and art But mathematics has a way of becoming useful either as a model for some aspect of reality or in solving practical problems, and it can be more easily communicated to another, distant, society I have tried to show that, in doing so, we say quite a lot about ourselves The early workers in SETI were concerned with the technical problems of sending and receiving radio signals across inter-stellar distances Slowly, however, the deeper www.EngineeringBooksPDF.com Preface • ix questions inherent in this endeavor rose to prominence: questions about the possible nature of extraterrestrial intelligence, the nature of language, and the philosophical/ psychological motivation for this search In recent years these questions have attracted scholars from a remarkably wide variety of disciplines Several recent books1 contain articles written by philosophers, psychologists, anthropologists, archaeologists, artists, and religious scholars These scholars bring valuable insight into the many deep problems posed by SETI As we broaden the scope of our discussions, however, it is important to remember the realities of this endeavor Our method of searching, the radio telescope, restricts the kind of society we might contact to those capable of sending electro-magnetic signals over inter-stellar distances (yes, some search for optical signals, others for evidence of alien technology, but communication, if it occurs, will be by some form of electro-magnetic radiation) Thus the early insights of astronomers, physicists, and mathematicians are still relevant and provide a framework for ongoing research In this book I try to bring the early work to the attention of those new to the field Also, at the end of each chapter I have a section labeled “remark.” Here I present some aspect of mathematics that, I think, might illuminate the ongoing discussion Anyone who expresses an interest in SETI is, sooner or later, confronted by someone, sometimes a very belligerent someone, who claims the subject is inane and pointless As “proof” such people will relate stories of UFO (unidentified flying object) sightings that, they claim, show that aliens exist and visit us often This can be very disconcerting, especially if it happens when one is giving a public lecture But some familiarity with the major incidents shows such people and anyone else listening that you are neither ignorant of, nor afraid to face, these “facts”—just not impressed by them www.EngineeringBooksPDF.com 196 • Science, SETI, and Mathematics (405 < t) → (∀ p)( NH3 @ t deg @ p atm) = γ NH3) (t ≤ 405) → (∃ p)(NH3 @t deg @ p atm) = λ NH3) The first line says that when t is above 405 degrees, ammonia is a gas no matter what the pressure is The second line says that when t is 405 degrees or less there is a pressure at which ammonia will liquefy We now use the concept of critical pressure to communicate the value of atm (112 < p) → (NH3 @ 405 deg @ p atm) = λ NH3 (p < 112) → (NH3 @ 405 deg @ p atm) = γ NH3 The first line says that when the pressure is above 112 atm, ammonia at 405 degrees is a liquid The second line says that when the pressure is below 112 atm, ammonia at 405 degrees will be a gas From this information our unit of pressure can be found We now give some useful facts about water, the solvent that, as far as we know, is essential for life (1g, σ H2O @ 273 deg @ atm) + 79.7 cal Δ → (1g, λ H2O @ 273 deg @ atm) (1g, λ H2O @ 373 deg @ atm) + 539 cal Δ → (1g, γ H2O @ 373 deg @ atm) The first of these says that to change one gram of ice to one gram of water requires 79.7 calories, and the second says that to change on gram of water to one gram of steam requires 539 calories We have discussed two of the three variables necessary to describe the gaseous state The third one is, of course, volume It is not easy to give a general definition of volume However, so many common phenomena, besides the study of gases (e.g., the expansion of liquids when heated, density, etc.), involve this concept that a society with a technology will have come to terms with this notion So we take the position that the aliens understand volume www.EngineeringBooksPDF.com The DeVito-Oehrle Language • 197 and our task is to make clear that this familiar idea is what we are trying to communicate L = 1g, λ H2O, Vol(L @ 277 deg) = cm3, 1000 cm3 = liter Vol (4g, He @ 273 deg @ atm) = 22.4 liter Vol (4g, He @ 273 deg @ atm) = 11.2 liter Vol (4g, He @ 546 deg @ atm) = 44.8 liter, and so on We give the volume of grams of helium (He) at 273 degrees and under 1atm, then show what happens when we double the pressure or double the temperature; we are stating Boyle’s law and Charles’s law Finally we shall give the ideal gas equation for the case of He and the value of the proportionality constant If our units of pressure and temperature have been understood, the aliens can use the value of this constant to compute our unit of volume independently of our gram Once that is done the gram can be recalculated from our definition of cm3 This gives another way of finding our gram: L = 4g, He, press(L) × Vol(L) = R × Temp(L), R = 0.8027 and so on The final stage of the paper contains a technical discussion of the real numbers based on the work of Dedekind (Chapter 11) As we discussed in that chapter once we have communicated these numbers we can, in principle, communicate all of mathematical analysis www.EngineeringBooksPDF.com Bibliography Adler, Ronald, Maurice Bazin, and Menahem Schiffer 1965 Introduction to General Relativity New York: McGraw-Hill Book Company Ascher, Marcia 2002 Mathematics Elsewhere: An Exploration of Ideas Across Cultures Princeton, NJ: Princeton University Press Beckman, F S 1981 Mathematical Foundations of Programming Menlo Park, CA: Addison-Wesley Pub Co Bennett, Jeffrey 2001 On the Cosmic Horizon New York: Addison-Wesley Berliner, Don and Stanton T Friedman 2004 Crash at Corona New York: Paraview Special Editions Bilanuick, et al 1969 “Particles beyond the Light Barrier.” Physics Today: 22, 5: 43-51 Blum, Howard 1990 Out There New York: Simon & Schuster Inc Bruckner, Andrew M., and Jack Ceder 1975 On Improving Lebesgue Measure Normat Nordisk Matematisk Tidskrift 23: 59–68 Craig, Roy 1995 UFOs Denton, TX: University of North Texas Press Crownover, Richard 1995 Introduction to Fractals and Chaos Boston, MA: Jones and Bartlett Publishers Dantzig, Tobias 2000 Number: The Language of Science, 4th ed New York: Penguin Group Darling, David 2001 Life Everywhere: The Maverick Science of Astrobiology New York: Perseus Book Group Davies, Paul 1974 The Physics of Time Asymmetry Berkeley: University of California Press DeVito, Carl L 1978 Functional Analysis New York: Academic Press www.EngineeringBooksPDF.com Bibliography • 199 ——— 2007 Harmonic Analysis: A Gentle Introduction Sudbury, MA: Jones and Bartlett Publishers DeVito, Carl L., and Richard T Oehrle 1990 “A Language Based on the Fundamental Facts of Science.” Journal of the British Interplanetary Society 43: 561–68 Dickson, Leonard E 1957 Introduction to the Theory of Numbers New York: Dover Publications, Inc Dunham, William 1999 Euler: The Master of Us All Washington, DC: The Mathematical Association of America Ebbing, Darrell D 1987 General Chemistry, 2nd ed Boston: Houghton Mifflin Company Edwards, H M 1984 Galois Theory New York: Springer-Verlag Enderton, Herbert B 1977 Elements of Set Theory New York: Academic Press Folland, Gerald B 2010 “A Tale of Topology.” The American Mathematical Monthly 117: 663–72 Freudenthal, Hans 1960 Lincos: Design of a Language for Cosmic Intercourse Amsterdam: North-Holland Publishing Company ——— 1974 “Cosmic Language.” Current Trends in Linguistics 12: 1019–42 Friedman, M 1992 Kant and the Exact Sciences Cambridge, MA: Harvard University Press Grinspoon, David 2003 Lonely Planets New York: HarperCollins Publishers, Inc Harrison, Albert A 1997 After Contact: The Human Response to Extraterrestrial Life New York: Plenum Trade ——— 2007 Cosmic Visions in Science, Religion and Folklore New York: Berghahn Books Hart, M.H 1995 Extra-terrestrials—Where Are They? Cambridge, MA: Cambridge University Press Hawking, Steven W 1988 A Brief History of Time New York: Bantam Books ——— 1993 Black Holes and Baby Universes New York: Bantam Books Heard, Gerald 1951 Is Another World Watching? New York: Harper Herzing, Denise 2011 Dolphin Diaries: My 25 Years with Spotted Dolphins in the Bahamas New York: St Martin’s Press www.EngineeringBooksPDF.com 200 • Bibliography Hewitt, Edwin, and Karl Stromberg 1969 Real and Abstract Analysis New York: Springer-Verlag Hey, J W S 1983 The Radio Universe New York: Pergamon Press Hynek, J Allen 1966 “Are Flying Saucers Real?” Saturday Evening Post, December 17 ——— 1972 The UFO Experience: A Scientific Enquiry New York: Ballantine Books ——— 1977 The Hynek UFO Report New York: Barnes and Noble Books Jacobs, David Michael 1975 The UFO Controversy in America Bloomington: Indiana University Press Jaynes, Julian 2000 The Origin of Consciousness in the Breakdown of the Bicameral Mind Boston: Houghton Mifflin Co Jeffrey, R L 1956 Trigonometric Series Toronto: University of Toronto Press Jung, Carl 1959 Flying Saucers: A Modern Myth of Things Seen in the Skies Translated by R F C Hull Princeton, NJ: Princeton University Press Kaufman, Marc 2011 First Contact: Scientific Breakthroughs in the Hunt for Life Beyond Earth New York: Simon and Schuster Kaufmann, William J 1994 Universe, 4th ed New York: W H Freeman and Company Kitei, Lynne D 2010 The Phoenix Lights: A Skeptics Discovery That We Are Not Alone Charlottesville, VA: Hampton Road Publishing Company, Inc Kreyszig, Erwin 1999 Advanced Engineering Mathematics, 8th ed New York: John Wiley and Sons “Lack of Translators Hindering Terror Fight.” 2002 Arizona Daily Star, 10 November Lamphere, R L 2002 “Solution of the Direct Problem of Uniform Circular Motion in Non-Euclidean Geometry.” The American Mathematical Monthly 109: 650–55 Lemarchand, G A., and Jon Lomberg 2011 “Communication among Interstellar Intelligent Species.” In Communication with Extraterrestrial Intelligence, ed D A Vakoch, 371–96 New York: SUNY Press www.EngineeringBooksPDF.com Bibliography • 201 Maccone, Claudio 2012 Mathematical SETI New York: Springer Mackey, George W 1973 “Group Theory and its Significance for Mathematics and Physics.” Proceedings of the American Philosophical Society 117: 374–80 Marcel, Jesse Jr 2007 The Roswell Legacy Helena, MT: Big Sky Press McAndrews, James 1997 The Roswell Report: Case Closed Washington, DC: U.S Government Printing Office McLeary, John 2002 “Trigonometries.” The American Mathematical Monthly 109: 623–38 Michaud, Michael A G 2007 Contact with Alien Civilizations New York: Copernicus Books Parker, Barry 1991 Cosmic Time Travel: A Scientific Odyssey New York: Plenum Press Peaslee, D C 1955 Elements of Atomic Physics Englewood Cliffs, NJ: Prentice-Hall Peebles, Curtis 1994 Watch the Skies Washington, DC: Smithsonian Institution Press Poincare, Henri 1952 Science and Hypothesis New York: Dover Ramsay, Arlan, and Robert D Richtmyer 1995 Introduction to Hyperbolic Geometry New York: Springer-Verlag Randle, Kevin D., and Donald R.Schmitt 1994 The Truth About the UFO Crash at Roswell New York:Avon Books Reichenbach, Hans 1957 The Philosophy of Space and Time New York: Dover Publications, Inc Richardson, M 1950 Plane and Spherical Trigonometry New York: The Macmillan Company Sacks, Oliver 1998 The Man Who Mistook His Wife for a Hat New York: Simon and Schuster, Inc Sagan, Carl, and I S Shklovskii 1966 Intelligent Life in the Universe San Francisco: Holden-Day, Inc Saler, Benson, Charles A Ziegler, and Charles B Moore 1997 UFO Crash at Roswell Washington, DC: Smithsonian Institution Press Simmons, George F 1963 Introduction to Topology and Modern Analysis New York: McGraw-Hill Book Company, Inc www.EngineeringBooksPDF.com 202 • Bibliography Sobel, Dava 2005 The Planets New York: Penguin Group Steiger, Brad, and John White, eds “Other Worlds, Other Universes.” Science Digest (June 1973) Stewart, Ian 2007 The Story of Mathematics: From Babylonian Numerals to Chaos Theory United Kingdom: Quercus Books Story, Ronald D., with Richard J Greenwell 1981 UFOs and the Limits of Science New York: William Morrow & Company, Inc Stringfield, Leonard H 1997 Situation Red: The UFO Siege New York: Doubleday Sturrock, Peter A 1974 Evaluation of the Condon Report on the Colorado UFO Project Stanford, CA: Institute for Plasma Research Sullivan, Walter 1964 We Are Not Alone New York: McGrawHill Book Company Trudeau, Richard J 1987 The Non-Euclidean Revolution Boston: Birkhauser Vallee, Jacques 1965 Anatomy of a Phenomenon Chicago: Henry Regnery Company Verschuun, Gessit L 1974 The Invisible Universe: The Story of Radio Astronomy New York: Springer-Verlag Wapner, Leonard M 2005 The Pea and the Sun Natick, MA: A K Peters, Ltd Webb, Stephen 2002 Where Is Everybody? New York: Copernicus Books Whitrow, G J 1961 The Natural Philosophy of Time London: Thomas Nelson & Sons Limited Zill, Dennis G., and Michael R Cullen 2006 Advanced Engineering Mathematics, 3rd ed Boston: Jones and Bartlett Publishers www.EngineeringBooksPDF.com Index Abel, Niels Henrik, 116 absolute zero, 123, 128 Alamogordo, New Mexico, 79 Aleph (Hebrew letter), 103 algebra, 115, 173 algebraic topology, 174 Allen, Paul, 169 Alvin (submersible), 163 analytic geometry, 139 Andromeda galaxy, 65, 126 angular momentum, 66 Ansen, Fridt Jof, 23 anthropologists, 124–25 Apollo program, 170 Arabia, 40 Area, 95–97 Argand, Robert, 111 Armstrong, Mary Louise, 156 Arnold, Kenneth, 10, 11, 26, 73 Arroway, Elinor, 169 Astrolingustics, 112 astronomical unit, 40–42 Atchley, Dana, 68 Atom, 56, 121, 122, 127 atomic weight (mass), 122, 128 Attimo, 85 Augustin, plains of Saint, 76–78 Augustine, Saint, 144 Avogadro’s law, 128 Avogadro’s number, 23, 122, 127–28, 193 Banach, Stephan, 102, 172 barber paradox, 113 Barnett, Grady “Barney,” 75–77 Bell, Jocelyn, 148 Bell Labs, 50 Beltrami, Eugenio, 44 Berkeley, University of California at, 169 Berkner, Lloyd, 59, 64 Berry’s paradox, 114 Bessel, Friedrich Wilhelm, 111 Bessel functions, 102 Big Foot, 78 Big Splat, 18 black hole, 65, 147–49 Blanchard, Colonel William “Butch,” 73, 77 Boltzman, Ludwig, 131 Bolyai, Farkas, 44 Bombelli, Rafael, 111 Boyle’s law, 131, 197 Bradbury, Ray, Brazel, William “Mac,” 75–76, 79 Biv, Roy G (pneumonic), 50 Burgi, Jost, 166 Burroughs, Edgar Rice, Cahn, J P., 32 calorie, 94 calculus differential, 86–87, 94–95 integral, 95–96 of variations, 96 calendar Jewish, 21 Mayan, 21 www.EngineeringBooksPDF.com 204 • Index Camp Ripley, Minnesota, 30 Canada, 169, 182 canali, canals, Calvin, Melvin, 68–69 Cantor, Georg, 16, 103, 110, 160, 177 Capella, Martianus, 85, 147 Cardano, Gerolomo, 111 cardinal number, 19–20 Carthage, 97 CAT scan, 98 causality, violation of, 138 Cavett, Captain Sheridan, 77 Celsius, 123 cellular automata, 164 Charles’s law, 131, 197 Chaves County, 27, 77 Chern (laboratory), 136 Chinese remainder theorem, 21 Chronones, 147 Cibola, seven cites of, 72 Clausius, Rudolf, 131 Cocconi, Giuseppe, 55–57, 59–60, 68 co-domain, 53, 166 cold war, 11, 37, 67 Cook, Stewart W., 154 Columbus, Christopher, 48, 171 Coma cluster, 125–26, 135 compiler, 90 complex numbers, 111–12, 116 systems, 164 compound, 120 Compton, Arthur Holly, Condon, Edward U., 153–59 congruent numbers, 21, 24 conservative fields, 97 Contact (film), 68, 169 Corona, New Mexico, 73, 76 Coronado, Francisco Vasquez de, 72 Cosine (trigonometric function), 61–63 Cosmos (television program), 68 countable set, 178, 181 Craig, Roy, 81–82, 155–56 Creighton, Gordon, 74 cubic equation, 116, 118 curve (line integral), 97 cyclic group, 168 Danish Academy of Sciences, 111 dark energy, 127 dark matter, 119, 125–26 Dearborn Observatory, 153 Dedekind, Richard, 110, 197 Deimos (Martian moon), 183 de Milo, Venus, 48 denumerable, 178–80 derivative, 87 Descartes, Rene, 139, 174 DeVito, Carl, 98, 100, 107–8, 185–97 Dido, 96–97 differential calculus, 86 differential equations, 88, 161 differentiation, 87 dimension, 138–42, 160–62 distance Euclidean, 141 Minkowski (space-time), 146 dog (Laika), 37 dolphin, 68, 91 Dolphin, Order of the, 69–70 domain, 53, 166 Drake, Frank, 58–60, 65–71 DuBose, Colonel Thomas, 28 eccentricity (of an ellipse), 41 Eddington, Sir Arthur, 145 egg-head, 37, 47 www.EngineeringBooksPDF.com Index • 205 Egypt, 38 Einstein, Albert, 104, 129–30, 132–36, 138, 146, 149 electro-magnetic waves, 49–50, 52, 121 elements, 120–21, 125, 127–28 ellipse, 40–41 elves, 170 English, 89 Enterprise, U.S.S., 129 Epsilon Eridani, 60 equi-numerous, 8, 110 ether, luminiferous, 134 Euclid, 39, 43 Euclidean geometry, 39, 44–46, 63, 139, 141 Euclid’s 5th postulate, 43, 61 Euler, Leonard, 168, 173 exponential, 167–68 eye cephalad, 124 insect, 124 vertebrate, 124 Fermat, Pierre de, 106–7, 139 Fermi, Enrico, 9, 12–14 Fermilab, 136 Fermi, Laura, Fibonacci (Leonard of Pisa), 165–66 field gravitational, 34, 97 electric, 34, 48 magnetic, 34, 48 flying saucers, 10 Fontes, Doctor Olavo T., 80–81 Ford, President Gerald, 154 fortran, 90 Foster, Jodi, 169 Foster Ranch, 75, 78 four color problem, 173–74 Fourier, Joseph Louis, 62 fractal, 142 fraction, 33 frequency, 50–52 Freudenthal, Hans, 69, 99–100, 103, 105–7 Fuller, John G., 156 function (mapping, transformation), 51–53 function iteration, 164 functional analysis, 162, 175 fundamental theorem of arithmetic, 70–71 of calculus, 96 Gagarin, Yuri, 37 galaxy See also Andromeda galaxy, 65, 126 Galileo, Galilei, 8–9, 15, 84, 86, 132, 171 Galois group, 118 Gauss, Karl F., 148 Gebauer, Leo, 32 general topology, 160–62, 174 ghost stories, 78 Godmen Air Base, 29 Golden Fleece award, 171 golden mean (ratio, section), 165–66 Goldilocks region, 66, 67 Gould, Stephen, 14 graph theory, 173 Greeks, 38–39, 84, 165 Green Bank conference, 58–60, 65 Observatory, 58–60, 65 group (mathematical structure), 115, 117–18, 172–74 Hall, Asap, 183 Hardy, G H., 176 Harriot, Thomas, 111 www.EngineeringBooksPDF.com 206 • Index Harrison, Albert A., 100 Harvard University, 169 Hausdorff, Felix, 102, 172 Haut, Lt Walter, 27, 77 Hawking, Stephen, 149 Hertz, Heinrich, 49–50 Herzing, Denise L., 70 Hewish, Anthony, 148 Hewlett-Packard, 68 Hilbert, David, 104, 141–42, 162 hoof and mouth disease, 48 Huang, Su-Shu, 66, 68 hydrogen (spin-flip transition), 56–57 Hynek, J Allen, 143, 153–55, 159 hyperbolic geometry, 43 hypotenuse, 61, 151 India, 40 Indian-Islamic system of enumeration, 166 infinite process, 87, 172 infinite series, 167–68 infinite sums, 62, 142, 167 integral calculus See calculus, integral integral Lebesgue, 161 line, 97–98, 111–12 Riemann, 97 integration, 88,95 international geophysical year, 36 intersection, 16 inverse function, 167 Iran (Persia), 40 irrational numbers, 33–34, 110–11, 151 Islamic scholars, 40, 85 iteration (of a function), 164 Italy, 169 Jansky, Karl G., 50 Japan, 169 Jewish, Jewish calendar, 21 Joule, James P., 131 Jung, Carl, 11 Jupiter, 40 Kant, Immanuel, 44–45 Kaputnik, 37 Kelvin, Lord (William Thomson), 128, 194 Kemo Sabe, 169 Kennedy, President John, 37 Kentucky, Franklyn, 29 Kepler, Johannes, Konigsberg bridge problem, 173 Konopinski, Emil, 12 Korolev, Sergei Pavlovitch, 36–37 Kuiper belt, 41 Laika See dog language logistic, 103, 186 natural, 104–5, 186 languages, 99–109 La Place, Pierre Simon, 101 Lebesgue, Henri, 161 Lehmer, Derrick Henry, 72 Leonardo of Pisa (Fibonacci), 165–66 Levi-Civita, Tullio, 105 Levine, Norman E., 155–56 Lilly, John, 68, 70 Lincos, 99 linear algebra, 175 linear equation, 139–40, 142 lion (English speaking), 89 line integral See integral, line little green men (pulsars), 148, 171 Lobachevsky, Nicolai Ivonovitch, 44 Loch Ness, 78, 156 www.EngineeringBooksPDF.com Index • 207 logarithm common, 166–67 natural, 167 Lomberg, Jon, 65 Lone Ranger, 169 Lorentz equations, 134 Los Alamos, New Mexico, 12 Low, Robert, 154–56 Lowell, Percival, 3, 54, 182 Lucas, Edouard, 72 luminiferous ether See ether, luminiferous lunation, 20 Maccone, Claudio, 69 MACHOs, 126 Mackey, George, 117 Maimonides, Rabbi, 85 Maltais, Vern, 75–76 Mantell, Captain Thomas, 29–31 Marcel, Major Jesse, 28, 77 Marconi, Gugliemo, 49, 51 Mars, 3–4, 22, 40, 54, 182–84 Martian bees, 54 moon, 11, 183–84 matrix (rectangular array of numbers), 175 Maxwell, James Clerk, 47–49, 51–52, 131 Mayan calendar, 21 McMillan Observatory, 153 media, 170 Mendeleev, Dimitri Ivonovitch, 121, 127 Mersenne numbers, 72 Michaud, Michael A G., 100 Michelson-Morley experiment, 134 Michigan, 153–54 Milky Way galaxy, 2, 17, 23, 65–67 Minkowski, Hermann, 146, 149–50 Moon (of Earth), 17–18, 20–22, 170 Moore, Charles B., 79 Morrison, Phillip, 55–57, 59–60, 68 Napier, J., 166 natural language See language, natural navel, 166 Native American, 19, 48 Naval research laboratory, 36 Neptune, 40 network topology, 173 neutron star, 147–48 Newton, Isaac, 7, 44, 85, 126, 135, 175 Newton, Silas M., 31–32 Nobel prize, 9, 49, 68 non-Euclidean geometry, 139, 150 Northwestern University, 143, 153 n-space, 160 number cardinal, 19 complex, 111–12 natural, 5–6, 8–9, 15, 102, 110, 176–78, 191 ordinal, 20 rational, 33, 110, 180–81, 191 real, 109–10, 147, 177, 180–81, 197 transfinite, 103 numbers Fibonacci, 165 integers, 33, 191 irrational, 109–11, 151, 167, 180–81 prime, 70–72, 176 www.EngineeringBooksPDF.com 208 • Index Oehrle, Richard T., 98, 100, 107–8, 185–97 Ogilvie, Doctor Robert E., 81, 83 Ohio State University, 153 Olbers, Heinrich, 7, Oliver, Barney, 68 Ollongren, Alexander, 112 one-to-one correspondence, 8–9, 15, 112, 115, 160, 177–79 one-to-one function, 53, 166–67, 177–79 Oordt cloud, 41 Orbit, 40 Order of the Dolphin, 69 ordinal number, 19–20 Ozma, Project, 60 paradox Banach, Tarski, Hausdorff, 102, 172 Berry’s, 114 Russell’s, 113 paradoxes, 113–15 parallax, 42 parallel, 43–44 partial derivatives, 88 partial differential equations, 52, 88 Peano, Giuseppi, 103–5, 160–61, 186–87 Pearman, J P T., 64, 68 Pebbles, Curtis, 29 periodic table, 121, 127, 192 Persia (Iran), 40 Phobos (Martian moon), 183–84 Phoenix, Lights, 159 Planck length, 149 time, 149 Planck’s constant, 24 plate tectonics, 22 Pluto, 40–41 Poincare, Henri, 44 postulates, 43, 131, 133–35 power set, 114–15, 178–79, Presley, Elvis, 78 pressure, 123, 131, 195–97 prime number, 70–72 Project Blue Book, 30, 153 Grudge, 153 Mogul, 79 Ozma, 60 Sign, 153 projective geometry, 46 Proxima Centurii, 42, 138 Proxmire, Senator William, 171 pulsars, 148 Pythagoras,34, 39 Pythagorean Theorem, 151 Pythagoreans, 33 Q (rational numbers), 33–34, 110, 180–81, 191 quadratic equation, 115–16, 118 Quantum theory, 141, 174 Quasar, 157–58 quintic equation, 116 Rabbi Maimonides, 85 radians, 63 radio astronomy, 51 radio waves, 48–50, 137 Radon’s problem, 98 Ramanujan, Srinivasa, 176 Ramey, General Roger, 28 Rawin target balloon, 28 Reber, Grote, 51 redshift, 121, 125 Reichenbach, Hans, 145 Renaissance, 46, 116 Ricci-Curbastro, Gregorio, 105 Riemann, Bernard, 44 Riemann integral, 97–98 surface, 174 www.EngineeringBooksPDF.com Index • 209 right angle, 17, 61, 141 triangle, 61, 151 Ripley, Camp See Camp Ripley Roach, Franklyn E., 154 Roddenberry, Gene, 129 Roman taxation cycle, 21 roots of an equation, 118 of unity, 112, 168 Rosetta Stone, Roswell, New Mexico, 27–28, 73–74, 76, 78–79 Roush, J Edward, 157 Rubin, Vera, 126 Ruppelt, Captain Edward J., 29–30 Russell, Bertrand, 103–4, 109, 113, 178 Russia, 169 Sagan, Carl, 68, 171, 184 Sao Paulo, Brazil, 20 saucers See flying saucers Saunders, David R., 155–56 Schiaparelli, Giovanni, 3, 182–83 Scully, Frank, 31–32, 34, 73–74 Seamans, Robert C (Secretary of the Air Force), 153 series, 62–63 set, 6, 15–16, 33, 52–53, 61, 103, 113–15, 161, 166, 177–81, 189–93 SETI institute, 169 League, 169 Shaw, G B., 171 Sharpless, Bevan, 183 Shklovsky, Iosif S., 183–84 Shuch, Paul (SETI League), 170 similar triangles, 61 sine (trigonometric function), 61–63 sky hook balloon, 30 space-time, 138–39 spectrum, 50, 121 Sputnik, 36–38 Star Trek (television show), 129 Stewart, Ian, 164 Stringfield, Leonard H., 74 Struve, Otto, 65–66 Sturrock, Peter A., 157–59 subset, 6, 8, 15–16, 33, 114–15, 178–80 Sued, Ibrahim, 79–81 sums of angles (of a triangle), 43, 61 supernova type 1A, 126–27 swamp gas, 153–54 symmetry, 117–18 tachyons, 136 Tarski, Alfred, 102, 172 Tarter, Jill, 169 Tau Ceti, 60 Teller, Edward, 12 tensor calculus, 102, 105 Thales, 38–39 theory in law, 130 in mathematics, 130–31 in science, 131–32 time, 143–47 topology algebraic, 174 general, 161–62, 174 network, 173 triangle, 43 trigonometric functions, 62 series, 62, 142, 161 trigonometry, 39, 45, 61 Trobriand Islands, 20 U Tant (Secretary General of the U.N.), 143 Ubatuba, Brazil, 80, 82, 84 www.EngineeringBooksPDF.com 210 • Index UFO (unidentified flying object), 1, 26–32, 42–43, 54–55, 73–84, 152–59 union, 16 universe, University of California at Berkeley, 169 University of Colorado at Boulder, 154 University of Pennsylvania, 76 unknown radiation, 137 Vakuta (island), 20 vampires, 170 van de Hulst, Henk, 57 Vanguard, 36–37 Variety Magazine, 32 vector, 34 vector field, 34 Venus, 4, 5, 22, 30, 31 Venus de Milo See de Milo, Venus vibrating string, 52 visible spectrum, 50 Vivian, Senator Westan E., 154 Walker, Doctor Walter W., 81–82 warp drive, 129, 135 wave equation, 51 fundamental, 51 partial differential, 52 wave frequency of, 50–52 length, 51–52 Weierstrass, Karl, 109–10 Welles, Orson, 183 Wells, H G., 4, 183 werewolves, 170 Wessel, Caspar, 111 Weyl, Hermann, 145 Wheaton, Illinois, 51 Wheeler, John, 147–48 white dwarf, 147–48 white holes, 149 Whitehead, Alfred North, 103–4, 109 Wilcox, Sheriff George, 77 Wiles, Andrew, 107 Wilmot, Mr and Mrs., 74, 76 Wilson, Holis, 76 WIMPs, 126 witches,78 work, 97 worm hole, 76 Wright-Patterson Air Base, 153 Yeager, Chuck, 26 York, Herbert, 12 Zahlen, 33 Zeeman effect, 59 Zeno, paradoxes of, 84 Zweiky, Fritz, 125–26 www.EngineeringBooksPDF.com .. .Science, SETI, and Mathematics www.EngineeringBooksPDF.com www.EngineeringBooksPDF.com SCIENCE, SETI, AND MATHEMATICS Carl L DeVito berghahn NEW YORK... self-replicating probes to explore and colonize the galaxy in a very, by cosmic standards, short time (Webb 2002: 24) www.EngineeringBooksPDF.com 14 • Science, SETI, and Mathematics In response to this... powers The sym- www.EngineeringBooksPDF.com 24 • Science, SETI, and Mathematics bol 10–n means 1/10n So 10–2 is 1/102 or 0.01, and 10–3 is 1/103 or 0.001, and so on An important number in physics