1. This is a three part question that you should provide detailed answers to:
a. What is the principal function of symbols in Mathematics?
b. Why did mathematicians start to use symbols?
c. Name another field that uses symbols for the same or similar reasons. Describe how.
2. Describe the “trade-off” that mathematicians face when they use symbols.
3. Explain the difference between abstraction as idealization and abstraction as extraction. Give examples of each in Mathematics.
4. What do we mean by the process of formalismin Mathematics? Give an example to illustrate.
5. What do we mean by the process of generalization in Mathematics? Give a specific example and name one benefit of this process.
6. If a mathematical object is considered to “exist”, precisely what does its existence depend upon, in the eyes of mathematicians?
7. This is a multi-part question:
a. For mathematicians, what is the difference between a proof and a conjecture?
b. Look up “proof” and “conjecture” in a standard dictionary. Write down the definitions of each.
c. Go to the on-line mathematics dictionary MATHWORLD http://mathworld.wolfram.com/ . Write down the definitions of “proof” and conjecture”.
d. Determine (by specifically comparing definitions) if the definitions in MATHWORLD agree with those in the standard dictionary. If not, explain why not.
8. Christian Goldbach is introduced in your text in connection with a number theory conjecture.
a. Explain Goldbach’s conjecture and give one example different from your text to illustrate it.
b. Go to. http://www-groups.dcs.st-and.ac.uk/~history/BiogIndex.html to learn more about Goldbach, and write a short paragraph about him. What, if anything, did you find most interesting about him?
9. Who was Hobbes, and what does his comment (Davis et al, page 165) on Euclid’s proof allude to? In particular, what does his comment tell you about where proofs “start”?
10. Why does the “language of proof” have a formal and severely restricted quality?
11. Why do all mathematical arguments that involve the infinite have to be scrutinized?
12. Describe fully the paradox of Achilles. Be sure to give the context for the paradox.
13. Name four distinct benefits to the process of proof.
14. Based on your reading of The Stretched String (Davis et al. pages XXXXXXXXXXname a quality that describe what constitutes the “straightness” of a straight line, which Euclid neglected to mention.
15. Explain why a successful gambling system cannot be used in a game of predicting the outcomes of tossing a fair coin?
16. How are the ideas of order, chaos, and pattern related, both in general terms, and in mathematical terms?
17. A major theme of the section Pattern, Order, and Chose is the fact that patterns often "create mathematics" and mathematics often creates patterns. Give examples of this phenomena. You should have TWO examples: 1) how patterns create mathematics and 2) how mathematics creates patterns.
18. Determine if the following are true or false, and indicate on what page of the text you found discussion to support your conclusion:
a. A mathematical structure consists of a set of objects, a set of relationships between those objects, and a set of characteristics of those objects.
b. It has been known and accepted from the time of Zeno in antiquity that a segment of finite length can be divided into infinitely many parts.
c. The aesthetic appeal of the golden ratio that was enjoyed historically in art and architecture continues to be prevalent today.
d. The surprising appearance of the golden ratio in geometry and arithmetic that are discussed in the text can be explained by one underlying mathematical theory.
e. There are few formal descriptions of the aesthetic appeal of Mathematics.
f. There are those who believe that the dominant element in mathematical creativity is aesthetics rather than logic.
g. Mathematics has been used in recent years to measure beauty in art.
19. What attempts have been made to analyze aesthetic components of Mathematics? In your own experience, have you found some topic or aspect of Mathematics beautiful, surprising, visually or intellectually pleasurable? Would you be able to compare Mathematics to art or music in any of these considerations?
20. What is the ultimate goal of Mathematics with respect to order and chaos?