Essay Instructions: Question:
Support each of the following statements, using at least two specific mathematical examples:
a. A change in context often alters an accepted result in mathematics.
b. Results are sometimes accepted in mathematics before proof can be demonstrated.
c. New branches of mathematics are often the conse?quence of investigating familiar facts.
d. A discovery in one field can precipitate major crises in the mathematical world and beyond.
Write a five paragraph essay that illustrates certain factors which contribute to the growth of Mathematics and what modern developments have resulted largely due to these factors. For example, to illustrate how Mathematics grows, you might tell your reader about the how the discovery or invention of geometries other than Euclidean (e.g., fractal geometry, hyperbolic geometry, etc.) came about as a result of examining familiar facts (e.g., Euclid?s parallel postulate, the characteristics of natural objects like mountains and clouds, etc). You might show your reader how it often takes centuries to find proofs for mathematical statements that are believed to be true (e.g. Four Color theorem, Fermat?s theorem, Goldbach?s conjecture, etc.), but that in the process new Mathematics is discovered or invented. You could discuss how mathematicians are able to explore the fourth dimension (even though it is not observable) by generalizing the patterns of our three-dimensional world. Give your essay an exciting title to motivate your reader.
The American poet John Godfrey Saxe (1816-1887) based the following poem on a fable which was told in India many years ago.
It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind
The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
God bless me! but the Elephant
Is very like a wall!
The Second, feeling of the tusk,
Cried, Ho! what have we here
So very round and smooth and sharp?
To me ?tis mighty clear
This wonder of an Elephant
Is very like a spear!
The Third approached the animal,
And happening to take
The squirming trunk within his hands,
Thus boldly up and spake:
I see, quoth he, the Elephant
Is very like a snake!
The Fourth reached out an eager hand,
And felt about the knee.
What most this wondrous beast is like
Is mighty plain, quoth he;
?Tis clear enough the Elephant
Is very like a tree!
The Fifth, who chanced to touch the ear,
Said: E?en the blindest man
Can tell what this resembles most;
Deny the fact who can
This marvel of an Elephant
Is very like a fan!
The Sixth no sooner had begun
About the beast to grope,
Than, seizing on the swinging tail
That fell within his scope,
I see, quoth he, the Elephant
Is very like a rope!
And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
Moral:
So oft in theologic wars,
The disputants, I ween,
Rail on in utter ignorance
Of what each other mean,
And prate about an Elephant
Not one of them has seen!
What does this poem have to do with the topic for this assignment? Each blind man accurately describes what he has derived from his investigation of the elephant. Although the correct description of the animal eludes the group, each has accurately reported on a piece of the picture. It is much the same with 1) trying to describe Mathematics and how it grows, and 2) capturing the challenge of investigating the Mathematics of objects we cannot see and teaching students for whom the picture of Mathematics is obscured by its abstractness. Many factors are responsible for the growth of Mathematics. But if we try to enumerate them, it is likely that in the end we will have a string of fragments rather than an overall picture. Furthermore, in a certain sense, the exploration of unseen objects such as the fourth dimension put mathematicians in the role of blind men or women. And students often feel like Mathematics leave them in the dark.
NEED:
WORKS CITED LIST.
CITATIONS.
TWO OUTSIDE REFERENCES
Some hint to help:
2. Name three questions that mathematicians have not been able to answer concerning prime numbers.
3. Explain how the prime number theorem is an example in number theory of order out of chaos.
4. What is the law of trichotomy for the real number system?
5. In what sense is computer software a kind of Mathematics?
6. What is the double meaning of the world formalism, i.e., how does its meaning in Mathematics differ from its meaning in philosophy?
7. How is the use of a computer to solve the four color conjecture different from its use in applied Mathematics?
8. Give examples of zero, one, two, three, and four dimensional objects.
9. How is a four dimensional object described in the context of Albert Einstein?s relativity theory?
10. How does a mathematician describe the idea of four dimensions, without appealing to relativity and space-time. To answer this question you can describe how to take a zero-dimensional figure to a four dimensional figure.
SUGGESTED RESOURCES:
Appel, K. and W. Haken. The Solution of the Four-Color Map Problem. Scientific. American, Vol. 237, 108-121, 1977.
Appel, K. and W. Haken. The Four Color Proof Suffices. Mathematical Intelligencer, Vol. 8, 10-20 and 58, 1986.
Cipra, Barry A. Fermat's theorem proved?. Science 239.n4846 (March 18, 1988): 1373(1). InfoTrac OneFile. See the series of article by Cipra on this topic. They give a revealing look at the process of mathematical growth.
Cipra, Barry A. Doubts over Fermat proof. (Pierre Fermat's Last Theorem). Science 239.n4847 (March 25, 1988): 1481(1). InfoTrac OneFile.
Cipra, Barry A. Fermat's Last Theorem remains unproved. (includes related article on mathematical proofs). Science 240.n4857 (June 3, 1988): 1275(2)
Cipra, Barry. Math attendees find there's life after Fermat proof. (joint Jan 1994 meetings of American Mathematical Society and the Mathematical Association of America)(includes related information on Fermat proof). Science 263.n5147 (Feb 4, 1994): 614(2). InfoTrac OneFile.
Cipra, Barry. Fermat proves points to next challenges.(number theorist Andrew Wiles)(includes related article on right-triangle problem in need of proof). Science 271.n5256 (March 22, 1996): 1668(2). InfoTrac OneFile
Cipra, Barry. Fermat proves points to next challenges.(number theorist Andrew Wiles)(includes related article on right-triangle problem in need of proof). Science 271.n5256 (March 22, 1996): 1668(2). InfoTrac OneFile
Cipra, Barry. Third time proves charm for prime-gap theorem.(NUMBER THEORY). Science 308.5726 (May 27, 2005): 1238(1). InfoTrac OneFile
Devlin, Keith. Fermat's Last Theorem: the Story of a Riddle that Confounded the World's Greatest Minds for 358 Years. Nature 387.n6636 (June 26, 1997): 868(1). InfoTrac OneFile
Downing, Ben. Fermat's Last Theorem.(poem). Parnassus: Poetry in Review 20.n1-2 (Spring-Fall 1995): 320(2). InfoTrac OneFile
Thomas Banchoff; Davide P. Cervone
Leonardo, Vol. 25, No. 3/4, Visual Mathematics: Special Double Issue. (1992), pp. 273-280. Illustrating Beyond the Third Dimension
URL: http://links.jstor.org/sici?sici=0024-094X%281992%2925%3A3%2F4%3C273%3AI%22TTD%3E2.0.CO%3B2-L
Bishop, Randal J. The Use of Realistic Imagery to Represent the Relationships
in a Four-Dimensional Coordinate System. Humanistic Mathematics Journal 17, 1998, pages 5-9. Available on-line.
There are faxes for this order.
