Discussion Questions for Theme 6
based on pages 233-240, 405-453 from the text
Please respond in complete sentences, providing sufficient detail to
fully answer the questions posted. Submit both question and answer inyour individual and group responses:
1. Determine if the following are true or false, and indicate the page in the text on which you found the discussion that led to your conclusion:
a. The list of primes on page 235 of the text exhibits certain patterns.
b. There are many unsolved problems and unanswered questions about numbers.
c. There are simple questions about the number pi that mathematicians never expect to be able to answer.
d. The law of the excluded middle says that every statement is true or false.
e. Every real number is either positive or negative.
2. Define a prime number. 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. How is the use of a computer to solve the four color conjecture different from its use in applied Mathematics?
7. Name six different interpretations for the word “intuition” listed in your text. Why are these six interpretations of intuition considered “vague” by the authors of the text?
8. Look up the word “intuition” in a standard dictionary, and give all the definitions listed. Compare these definitions with the interpretations of the word “intuition” given in your text? Do any coincide? If so, which? If not, would you characterize the dictionary
definitions as “vague”? Why or why not?
9. In the view of the authors of your text, what kind of role should the study of mathematical intuition play in a philosophy of mathematics?
10. Give the specific definition of mathematics described in your text which recognizes the role of intuition.
11. Explain the role that intuition plays in the Platonist philosophy of mathematics.
12. What do the authors of your text see as the double meaning of the world formalism? That is, how does its meaning in Mathematics differ from its meaning in philosophy?
13. Explain the difference between teaching mathematical concepts formally and teaching them intuitively.
14. How does intuitive reasoning differ from formal reasoning?
15. Give examples of zero, one, two, three, and four dimensional objects.
16.Howis a four-dimensional object described in the context of Albert Einstein’s relativity theory?
17.Howdoes 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.
18. What is a hypercube and whatspecific methodscan mathematicians use to describe it?
19. Create an interesting question for this reading and answer it, indicating on which page you found the discussion to support your answer.
20. If you were preparing a final examination for this course and you wanted to test your students’ overall knowledge of the text, what question would you pose and why? How would you answer your own question? For this final discussion question of the semester, report on each of the individual group members’ answers. Did any of your questions coincide?