Reasoning About Quantity
About Mathematics - Book Sources and Quotations
Allen Klinger, © 4/9/2014

Mathematics is interesting ... and sometimes difficult. But it isn't what we think it is. The mathematician Sonya Kovalevsky (Sofia Kovalevskaya) wrote "many who have never had the occasion to discover more about mathematics, confuse it with arithmetic and consider it a dry and arid science. In reality, however, it is a science which demands the greatest imagination." [See Osen or Kovalevskaya Quotations** and Becky Wilson article.] Indeed, imagination and observations led to "the arithmetic that was developed in Sumer as early as c. 3200 B.C. ... (one that was) ... so matched (to) the celestial order as to amount ... to a revelation." [See Campbell, p. 128]

Many aspects of mathematics usually encountered in school are only recent items. [Yandell, p. 26] states 'the = sign instead of the word "equals" came from an Englishman, Robert Recorde, in the 1550s. A German named Michael Stifel supplied the + sign for addition, the - sign for subtraction, and the √ sign for square root."

Another woman mathematician, Antonia Bluher wrote that although "Mathematics is conceived as an austere subject" ... "In fact, there is a whole category of results of the type well known to experts for quite some time which cannot be attributed to any particular individual, but somehow got proven in conversation and passed on by word of mouth." [Bluher Quotation] This link enables jumping within this file to the Cultural basis of mathematics.

"Beauty and insight" (Hoffman, p. 44) play important roles in mathematical creation. Sometimes both come from an amateur (see Bell on Fermat). But the changes introduced since the 1940's due to digital computer technology have blurred the line between mathematical and other traditions. Indeed deep questions akin to the replacement of words by signs like =, + and -, are the subject of [Iverson, 1980]. Going back from technology to mathematics, simple statements can be beautiful, as in the following paper title:

Erdos, Paul, and John L. Selfridge, "The Product of Consecutive Integers is Never a Power," Illinois Journal of Mathematics, 19, 2, June 1975.

A Nobel-prize winning physicist put it this way:

To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. -Richard Feynman (1918-1988)

Even mathematics' simplest aspects involve symbols, abbreviation and conventions. These may form an awkward barrier to understanding. Even more important, the conventions aren't always the best way to indicate the idea involved. [A Western convention and Asian (Vietnamese) equivalent tallying one through five (Compare Methods) shows the latter as quickly recognizable.] But these were invented because they are needed. In the book Ancient Puzzles, listed in two categories below, Dominic Olivastro says Solving a problem ... without the essential tools, is ... like thinking without words. But the ancients indeed lacked words. Like Olivastro, Marcel Danesi, believes in The Puzzle Instinct. [Campbell, pp. 115-130] describes some ancients' achievements under section titles "Mythic Time" and "The Mythic Flood" evoking problems, tools, and order found in the sky, by many cultures. Achievements such as the Maya's understanding of astronomical events are simply astounding.

Astronomy is a source of traditional mathematics. Many practical or popular issues also contribute to the field. Problems of quantity may deal with area, quantity and weight, as from land, cloth, food, and games.

Many readable books support exploring Mathematics: see list below or go to additional reading or reference. Among them these are worth special note: Enzensberger The Number Devil: A Mathematical Adventure (addressing children); Devlin Mathematics: The Science of Patterns (a comprehensive overview); and Dunham Journey Through Genius: The Great Theorems of Mathematics (for accessibility and general historic material). Hoffman and Kanigel each have written readable biographies. Their books also convey mathematical ideas. There are extensive resources on the worldwide web, including a partially-implemented problem solver Quickmath and an encyclopedia of mathematics, mathworld.

There are many Mathematical puzzle books. Some consider them delightful. (That delight often comes after reviewing the solution section.) Chernyak/Rose' The Chicken from Minsk is both interesting and filled with difficult things to think about. The file True leads to some facts about the two transcendental quantities π and e, images that suggest how many lines and dots ... or just a few ... create a scene or object, and curves with names that show the interaction between mathematical analysis and visual display.

General Book References*
Bronowski, J., The Ascent of Man, Boston MA: Little Brown, 1973.

Enzensberger, Hans Magnus, The Number Devil: A Mathematical Adventure (Translated by Michael Henry Heim, Illustrated by Rotraut Susanne Berner) NY: Metropolitan Books, Henry Holt and Company, 1998.

Devlin, Keith J., Mathematics: The Science of Patterns, NY: W. H. Freeman and Co., 1994.

[An article in the same vein as Devlin's book is: Steen, Lynn Arthur, "The Science of Patterns," Science, pp. 611-616, 29 April 1988, The American Association for the Advancement of Science.]

Dunham, William, Journey Through Genius: The Great Theorems of Mathematics, NY: John Wiley & Sons, Inc., 1990.

Hoffman, Paul, The Man Who Loved Only Numbers - The Story of Paul Erdos and the Search for Mathematical Truth, NY: Hyperion, 1998.

Kasner, Edward, and James R. Newman, Mathematics and the Imagination, NY: Simon and Schuster, 1940; ISBN 1-55615-104-7 Redmond: WA, Tempus Books of Microsoft Press, 1989.

Singh, Simon, Fermat's Enigma, ISBN 0-385-49362-2, NY: Random House Anchor Books, 1997, 1998.

Kanigel, Robert, The Man Who Knew Infinity - A Life of the Genius Ramanujan, NY: Simon and Schuster, first published Charles Scribner's Sons, 1991.

Kline, Morris, Mathematics for the Nonmathematician, New York: Dover, 1985.

van der Waerden, B. L., Geometry and Algebra in Ancient Civilizations, New York: Springer-Verlag, 1983.

Stevenson, Harold W. and Stigler, James, W., The Learning Gap - Why Our Schools Are Failing and What We Can Learn from Japanese and Chinese Education.

Beckmann, Petr, A History of Pi, Boulder, Colorado: Golem Press, 1982.

Dantzig, Tobias, Number The Language of Science, New York, The Macmillan Company, 1930, 1954.

Davis, Philip J., The Thread, A Mathematical Yarn, Second Edition, NY: Harcourt Brace Javanovich, Publishers, 1983, 1989.

Bell, Eric Temple, The Last Problem, Washington, D.C.: Mathematical Association of America, 1990.

Nahin, Paul J., An Imaginary Tale - The Story of √ -1, Princeton NJ: Princeton Univ. Press, 1998.

Conway, John H. and Guy, Richard K., The Book of Numbers, NY: Springer-Verlag, 1996.

Nelsen, Roger B., Proofs Without Words, The Mathematical Association of America, 1993; and Proofs Without Words II: More Exercises in Visual Thinking, ibid., 2000.

Adams, James L., Conceptual Blockbusting, San Francisco: W. H. Freeman, 1974.

King, Jerry, The Art of Mathematics, NY: Fawcett Columbine, 1992.

Sobel, Dava, Longitude -The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time, London, U.K.: Fourth Estate, 1995.

Dunham, William, The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems, and Personalities, NY: John Wiley & Sons, Inc., 1994.

Rothstein, Edward, Emblems of Mind: The Inner Life of Music and Mathematics, NY: Random House (Times Books), 1995.

Andrews, W. S., Magic Squares and Cubes, Dover Publications, Inc., NY, 1960, replica of Open Court Publishing Company, Second Edition, 1917.

Golomb, S., Polyominoes [with more than 190 diagrams by Warren Lushbaugh, NY: Charles Scribner's Sons, 1965]; Second Edition, Princeton NJ: Princeton Univ. Press, 1994.

Osen, Lynn M., Women in Mathematics, ISBN 0-262-15014-X, Cambridge MA: The MIT Press, 1974.

Kline, Morris, Mathematics and the Search for Knowledge, ISBN 0-19-503533-X, NY: Oxford University Press, 1985.

Barrow, John D., Pi in the Sky - Counting, Thinking, and Being, NY: Oxford Univ. Press, 1992.

Wallace, David Foster, Everything and More - A Compact History of Infinity, NY: W.W. Norton & Co., 2003. (Wordy but addresses a general reader.)

Spencer, Adam, Book of Numbers, NY: Four Walls Eight Windows, 2004.

Yandell, Benjamin H., The Honors Class - Hilbert's Problems and Their Solvers, Natick MA: A K Peters, 2002.

Humez, Alexander; Humez, Nicholas; Maguire, Joseph, Zero to Lazy Eight - The Romance of Numbers , NY: Simon and Schuster, 1993.
*Use these books to follow your interests. Try to relate their key ideas to your own experlences. Read to gain vocabulary and familiarity.

More Technical References

Iverson, Kenneth E., "Notation as a tool of thought," Communications of the ACM, Vol. 23, Issue 8, Aug. 1980, pp. 444-465, NY: ACM Press.

Stark, Harold M., An Introduction to Number Theory, Cambridge, Mass.: MIT Press, 1978.

Stewart, Ian, Flatterland Cambridge MA: Perseus Publishing, 2001.

Maor, Eli, e: The Story of a Number, Princeton NJ: Princeton Univ. Press, 1994.

Steen, Lynn Arthur, On the Shoulders of Giants: New Approaches to Numeracy, Washington D.C.: National Academies Press, 1990; also see the Bibliography.

Devlin, Keith J. The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, NY: Basic Books, 2002.
Historic and Cultural Aspects

In ancient Egypt the culture took an approach to dividing quantities into equal-sized portions that we would consider to be very unusual. [See some details in Mathematical Exposition or Shorter Version ... or just view the Image. The book by Gillings below is a detailed treatise on Egyptian number and calculation. The second chapter of Olivastro Ancient Puzzles covers Egyptian arithmetic and fractions.] In South America the shorter-lived Inca Empire fit blocks together of varied sizes and levied taxes over vast distances with no written numbers [but used knots and notions central to computing that are detailed in the book below by the two Ascher's]. Study of both these variations on mathematical ideas, and those from other areas [see the books listed below by Joseph, and by Davis] gives insight into history from a different perspective. The cultural achievement of humanity clearly rests upon many sources within the domain we normally call Mathematics. Interconnections with Culture, Art, and Architecture, as well as Science are the subject of several books below. (For the connection to art see material about Fibonnaci and the Parthenon, Gaudi and the Sacred Family Cathedral in Barcelona, and compositional systems in painting as described in Quick, Michael, et. al., The Paintings of George Bellows, ISBN 0-8109-3119-2, 1992, p. 21. These words appear there: "... rebatement, in which the shorter side of the rectangle is rotated upon the longer ... creating a square area ...")

Cultural and Historical Book References

Temple, Robert, The Genius of China, New York: Simon and Schuster, 1986.

Neugebauer, O., The Exact Sciences in Antiquity, New York: Dover, 1957, 1969.

Gillings, Richard J., Mathematics in the Time of the Pharaohs, Cambridge, MA: The MIT Press, 1972; NY: Dover Publications, Inc., 1982, ISBN 0-486-24315-X.

Ascher, Marcia and Ascher, Robert, Mathematics of the Incas: Code of the Quipu, Dover, 1997, ISBN 0486295540.

Joseph, George Gheverghese, The Crest of the Peacock - Non-European Roots of Mathematics, London UK: I.B. Tauris & Co. Ltd Publishers, 1991.

Ascher, Marcia, Ethnomathematics, A Multicultural View of Mathematical Ideas, Pacific Grove, CA: Brooks/Cole Publishing Company, Wadsworth, Inc., Belmont, CA, 1991, ISBN 0-534-14880-8.

Selin, Helaine, ed., Mathematics Across Cultures - The History of Non-Western Mathematics, Dordrecht, The Netherlands: Kluwer Academic Publishers, 2000, ISBN 0-7923-6481-3.

Bronowski, J., The Ascent of Man, Boston MA: Little Brown, 1973.

Ascher, Marcia, Mathematics Elsewhere, Princeton NJ: Princeton University Press, 2002.

Al-Daffa, Ali Abdullah, The Muslim Contribution to Mathematics, Prometheus Books, 1977.

Olivastro, Dominic, Ancient Puzzles: Classic Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries, NY: Bantam Books, 1993.

Campbell, Joseph, The Masks of God: Oriental Mythology, NY: Viking Press, 1962.

Danesi, Marcel, The Puzzle Instinct: The Meaning of Puzzles in Human Life, Bloomington and Indianapolis, IN: Indiana University Press, 2002.

Ifrah, Georges, tr. by Bair, Lowell, From One to Zero, A Universal History of Numbers, NY: Viking Press, 1987.
Encryption and Language Book References

Paul, Doris A., The Navajo Code Talkers, Pittsburgh PA: Dorrance, 1973.

Chadwick, John, The Decipherment of Linear B, Cambridge, UK: Cambridge University Press, 1987.

Tuchman, Barbara, The Zimmerman Telegram, NY: Ballantine, 1994.

Kahn, David, The Codebreakers, NY: Scribner, 1996.

Singh, Simon, The Code Book: The Evolution of Secrecy from Mary Queen of Scots to Quantum Cryptography, NY: Doubleday, 1999. [Includes 15 web site references.]

Lakoff, George and Nunez, Rafael, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, 2000.

Books on How Mathematics is Seen and Learned
Puzzle Book References

Chernyak, Yuri B., and Robert M. Rose, The Chicken from Minsk, NY: HarperCollins BasicBooks, 1995. [Preface - ... "If you sign up for this course," said another student, "do not make any other plans. Do not think you will have time to socialize, date, take showers or sleep. If you do sleep, you will have nightmares about the problems-and they will haunt you in the shower too!" ...]

Gardner, Martin (Selected and Edited), Mathematical Puzzles of Sam Loyd, NY: Dover Publications, 1959. [Includes (18, 71) duplicated cube and dissection problems. A number cubed that is square - true of any number which is itself square, e.g., 4 or 9.]

Konhauser, Joseph D. E., Velleman, Dan, Wagon, Stan, Which Way Did The Bicycle Go? ... And Other Intriguing Mathematical Mysteries, Mathematical Association of America, 1996, ISBN 0-88385-325-6.

Gardner, Martin, The Unexpected Hanging and Other Mathematical Diversions, Chicago Illinois: The University of Chicago Press, 1969, 1991, ISBN 0-226-28256-2.

Gardner, Martin, The Numerology of Dr. Matrix, NY: Simon and Schuster, 1967.

Olivastro, Dominic, Ancient Puzzles: Classic Brainteasers and Other Timeless Mathematical Games of the Last 10 Centuries, NY: Bantam Books, 1993.

Danesi, Marcel, The Puzzle Instinct: The Meaning of Puzzles in Human Life, Bloomington and Indianapolis, IN: Indiana University Press, 2002.

Enzensberger, Hans Magnus (illustrations by Berner, R.S.; translated by Heim, M. H.), The Number Devil: A Mathematical Adventure, paperback, May 2000.
Visually Oriented

Livio, Mario, The Equation That Couldn't Be Solved - How Mathematical Genius Discovered the Language of Symmetry, NY: Simon & Schuster, 2005.

Nelsen, Roger B., Proofs Without Words, The Mathematical Association of America, 1993; and Proofs Without Words II: More Exercises in Visual Thinking, ibid., 2000.

Adams, James L., Conceptual Blockbusting, San Francisco: W. H. Freeman, 1974.

Steinhaus, H., Mathematical Snapshots, New York: G.E. Stechert & Co., 1938.
References, sources, and ideas about mathematics follow from these pointers:
Geometric Solids Mathematics Book Lists By Educational Level. Mathematics Fiction List*# Words and Ideas Computing Tools Big Integers Size Numbers Many


Brain Teasers* Seeds - Widely-Played Play - Five Flex - Resistors Poem Pictures - Math Signs


*Newman, Philip





Statistics Evolution of Number


**from School of Mathematics and Statistics, University of St Andrews, Scotland UK
***by Dr. Scott W. Williams, Professor, Mathematics Department, The State University of New York at Buffalo
*#by Alex Kasman, Professor, Department of Mathematics, College of Charleston, Charleston, SC 29424-0001, kasmana at

2/8/16 Version
©2010 Allen Klinger