Historic and Cultural Aspects of Reasoning About Quantity
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.] 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.
Mathematics is interesting ... and sometimes difficult. But it isn't what we
think it is. The mathematician Sonya Kovalesky 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." "Beauty and insight"
(Hoffman, p. 44) play important roles in mathematical creation. Sometimes both
come from an amateur (see Bell on Fermat). 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.
To explore Mathematics one could choose from many readable books. I
recommend those below, especially: 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 many Mathematical puzzle books. Some consider them delightful.
(That delight often comes after reviewing the solution section.)
Chernyak/Rose' The Chicken from Minskis both troubling and interesting.
General Book References
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, 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.
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.
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.
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.
Nelsen, Roger B., Proofs Without Words, The Mathematical Association of
America, 1993.
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.
Stark, Harold M., An Introduction to Number Theory,
Cambridge, Mass.: MIT Press, 1978.
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.
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.
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.
Cultural and Historical Book References
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.
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!" ...]
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.
Further references and sources follow from the pointers listed.