DREAMING AND DOING

IN THE HISTORY OF MATHEMATICS

National Council of Teachers of Mathematics - 2005 Annual Meeting

Anaheim, California

Session 652, Friday, April 8, 1:00 - 2:30 pm

Hilton Anaheim, Room Avila-B, on the 4th floor.

**
http://pages.towson.edu/shirley/impossible.html
**

**
Lawrence H. Shirley,
Professor of
Mathematics and
Associate DeanCollege of Graduate Studies and Research,
Towson University,
Towson,
Maryland 21252
mailto:LShirley@towson.edu
;
phone 410-704-3500;
personal webpage: http://pages.towson.edu/shirley
**

**
Abstract
**
Using examples from the history of mathematics, we will observe the surprises, paradoxes, confusions, abstractions,
and insights that result when mathematicians believe in

I. Believing in the

II. Ancient mathematical power

III. Greek impossibilities

a. Zeno's paradoxes Can Achilles win?

b. Irrational numbers: Keep it a secret!

c. Construction problems square circles? trisect?

IV. Challenges and new ideas

a. negative and imaginary numbers numbers in new directions?

b. Ptolemy and Copernicus Where is Mars going?

c. more problems with infinitesimals Angels, yes! infinitesimals, no!

V. Revolution

a. non-Euclidean geometry Was Euclid wrong?

b. infinite sets beyond counting

c. math is incomplete "This statement cannot be proven"

VI. "Pathological" results

a. topology (see below) weird properties!

b. four (and more) dimensions Flatland and much more!

c. fractals infinite repetition!

d. giant proofs who can check them?

VII. ..but in the end: it's good!

a. solutions we can win some!

b. challenges ...but not all (at least, not yet)

c. creativity ...but we keep trying!

__Instructions
for topological objects:__

__Mbius Ring:__ Take a
strip of paper and bend it around into a ring, but give one end a 180-degree
twist before fastening the two ends. This has only one face (side) and one
edge. Cutting along the ring--either down the middle or about one-fourth
of the way in from the edge--yields interesting results.

__Klein
Bottle:__ Take a bag that home-delivery newspapers come in (or any long thin
plastic bag) and cut off the closed end to yield a cylinder open on both
ends. Also cut a small hole about one-third of the way along from one
end. Take the other end, squeeze it into a neck, stuff it through the
small hole, and flare it out to join the other end from the inside. The
resulting bottle has no inside or outside. It is really a four-dimensional
creature and in four dimensions the neck does not really pass through the side
(there is no small hole). If you could perceive four dimensions, you could
also make a Klein Bottle by fastening two Mbius rings together completely
along their edges--you can try this task in three dimensions, but you will get
caught up in the twists! (see the link below for more on Klein Bottles)

__
Links
__

-- Alice *
Through the Looking Glass* (Chapter 5 "Wool
and Water") and Lewis Carroll
(or Charles Dodgson, or
with more math emphasis)

--Biographies
of historical mathematicians (nearly
1700 names!) and other topics

----including Pythagoras,
Zeno,
Ptolemy,
Al-Khwarizmi,
Cardano,
Copernicus,
Descartes,
Newton,
Leibniz,
Berkeley,
Euler,
Bolyai,
Lobachevsky,
Hamilton,
Cantor,
Russell,
Godel,
Mandelbrot,
Wiles,
and hundreds more!

--a great reference source on many aspects of higher mathematics:
MathWorld ("Eric
Weisstein's World of Mathematics")

--geometric construction problems (including details of circling squaring, cube
duplication, and angle trisection)

--non-Euclidean
geometry

--topology: four-color theorem;
Klein
bottle details and pictures and Klein bottles for sale

--four-dimensional hypercube
(tesseract) in 3-D (note: If you have
red-blue 3-D glasses, you can use them. Otherwise, press the"stereo" button twice to get a double image. Then cross your eyes to
produce a third image between the two. Watch that one. The page also has
some instructions and other "cool" images below the tesseract. Enjoy!)

--Banach-Tarski Paradox--reassembling
pieces of a ball to make a larger one (it's OK not to believe this one!)

--fractals, and
fractint,
which is software for playing with fractal images.

--the "enormous"
theorem of classification of finite simple groups

--the Riemann zeta function
(Has it now been proven?
__Not__ confirmed)

--Here is a page of
Hilbert's 23 problems, presented in 1900 as a "homework assignment" for pure
mathematicians for the 20th century. Click on its "table of
contents to see the actual problems.

--In 2000, the Clay Mathematics Institute
offered $1
million prizes for the solution of each of the currently unsolved "Millennium Problems"

--For hot news from mathematics,
MathWorld Headline News (also,
from this link you can also jump to the rich *MathWorld* files of
mathematics content)

--Believe in Unicorns

more interesting mathematics links at:
http://www.towson.edu/~shirley/ShirleyBody.html#math-educ

__Print Sources and Suggestions for Further Reading
(with Amazon--or other--links, if available)__

Abbott, Edwin; and Stewart, Ian
(2001) * The Annotated Flatland: A Romance of Many Dimensions*, Perseus Publishing.

Bell, E.T. (1937)
*Men of Mathematics*, Simon & Schuster.

Boyer, Carl (1989)
*A History of Mathematics*, Princeton University Press.

Burton, David M. (1991)
*The History of Mathematics: An Introduction*, Wm. C. Brown, Publishers.

Calinger, Ronald (editor) (1995)
*Classics of Mathematics*, Prentice Hall.

Carroll, Lewis (1988) * The Annotated Alice: Alice's Adventures in Wonderland and Through the Looking Glass*

(intro,annotation by Martin Gardner), Penguin.

Casti, John (1995)
*Five Golden Rules: Great Theories of 20th Century Mathematics and Why They Matter*, Wiley.

Casti, John (2001) *Mathematical Mountaintops: The Five Most Famous Theorems of All
Time,* Oxford.

Davis, Philip and Hersh, Reuben (1981)
*The Mathematical Experience*, Houghton Mifflin.

Devlin, Keith (2002)
*The Millennium Problems*, Basic Books.

Dunham, William (1990)
*Journey through Genius: The Great Theorems of Mathematics*, Wiley.

Eves, Howard (1990)
*An Introduction to the History of Mathematics*, Saunders College Publishing.

Grinstein, Louise S. and Campbell, Paul J. (1987)
*Women of Mathematics*, Greenwood Press.

Kasner, Edward, and Newman, James (2001,originally 1940) * Mathematics and the Imagination*, Dover

Katz, Victor (1998)
*A History of Mathematics: An Introduction,* Harper-Collins.

Kline, Morris (1953)
*Mathematics in Western Culture*, Oxford University Press.

Kline, Morris (1962)
*Mathematics: A Cultural Approach*, Addison-Wesley.

Mazur, Barry (2003)
*Imagining Numbers,* Farrar, Straus and Giroux.

Nahin, Paul J.(1998)
*An Imaginary Tale: The Story of -1*
Princeton
University Press.

**National Council of Teachers of Mathematics (1969,1989)
Historical Topics for the Mathematics Classroom (31^{st} Yearbook), NCTM.**

Osen, Lynn M. (1974)

Seife, Charles (2000)

Shirley, Lawrence (2000) "Twentieth Century Mathematics: A Brief Review of the Century"

Smith, D.E. (1929,1959)

Smith, Sanderson M. (1996)

Stewart, Ian (1996)

Struik, D.J. (1967,1995)

Wills, Herbert, III (1985)

------------------------

Maryland Council of Teachers of Mathematics Annual Conference--October 18, 2002

National Council of Teachers of Mathematics Eastern Regional Conference, Baltimore--October 14, 2004

*This page was last updated on **April
13, 2005; links
checked on April 4, 2005*