Student Suggested Questions for the Final Exam

1. Discuss the development of the concept of infinity throughout history, from the times of the Kabbalists and Greeks until the birth of the "continuum hypothesis" and its controversies.

2. Discuss the similarities and differences between the approaches that Borges uses in his short stories that relate to infinity.

3. What was the significant shift in thought Cantor made that allows him to begin modern set theory? Why was he so certain he could prove the continuum hypothesis? What do we now know about the continuum hypothesis that Cantor did not consider?

4. How can an understanding of mathematical infinity contribute to the discourses concerning the existence of God? How does Brendan Kneale use John Locke in his essay "God and Mathematical Infinity" to define his point?

5. Discuss the distinction between potential and actual infinity: what the distinction is, who made it, and a few examples of people working with actual infinity.

6. Discuss Cantor's work on infinity, including the discoveries he made and how the world received these discoveries.

7. Explain how both Max Bill and M. C. Escher attempted to represent infinity with physical objects, Compare their styles and the aspect of infinity they chose to represent.

8. In what three ways does M. C. Escher work relate to infinity? How did he refine his work to better portray the infinite? Was he successful?

9. From Jumpers by Tom Stoppard: What are George's views of God's existence and power? How do his views differ from the other characters in the play?

10. Zeno's paradox: based on the many readings we have had this semester, what are the different examples of Zeno's paradox? Which is the best example (i.e., which one does the best job in exemplifying the properties of the paradox)? Why?

11. Based on the material this semester, do you feel some mediums of expression (i.e. writing, art, etc) are more effective in portraying the infinite? Give some examples.

12. The link between God and the Infinite appears over and over in the readings throughout the semester. Choose one of the readings and analyze the specific connections made by the author between God and Infinity. Do you agree with the author's ideas?

13. We have seen many artistic depictions of qualities of the Infinite. Choose one author or painter whose work struck you in some way. What was interesting about this person's work? How did it further your understanding of concepts relating to infinity?

14. Briefly outline the progression of our picture of the universe. How have different scientists and philosophers justified their concept of an infinite universe?

15. Discuss the meaning of potential infinity in terms of mathematics, the Kabbalah, and art. How does potential infinity differ from actual infinity in each of these cases?

16. Both in context of our writing at the beginning of the term, and in the splay Jumpers by Tom Stoppard, discuss the relevance of Zeno's paradox to infinity.

17. How did the play Jumpers pertain to infinity? What were some of George's main points in the speech he was writing?

18. How does Zeno's paradox pertain to infinity? Was Zeno correct? If so, why? If not, what is one possible explanation?

19. Describe the role of the Axiom of Choice in the mathematics of infinity. Why is it controversial? Was it ever proven true or false? Why was it originally considered so important to the Continuum Hypothesis and was the original assumption correct?

20. What is the meaning of infinity that Tom Stoppard refers to it in his play Jumpers? How does it contribute to the entire interpretation of the play?

21. During the semester we have met many different forms of infinity: How does the notion of Ein Sof develop over the centuries and what forms does it acquire?

22. In the story "The Indefatigable Frog," Philip Dick makes reference to Zeno's paradox of the frog and the well. Explain the paradox and show why it really is a paradox. How is the paradox resolved? Does Philip Dick provide resolution for the paradox?

23. In his book To Infinity and Beyond: A Cultural History of the Infinite, Eli Maor asks questions about the universe. One of these questions is "Is the universe infinite or finite?" From your understanding, what do you presume the universe to be and why?

24. The real numbers are "dense" on the number line. The countable set of rational numbers and uncountable set of irrational numbers make up the set of real numbers. What does the work "dense" mean? What does it mean to say that a set is countable or uncountable? How did Cantor show that the rational numbers are countable?

25. If you were to go back in history and talk to one of the mathematicians and philosophers that are discussed in either The Mystery of the Aleph or To Infinity and Beyond: A Cultural History of the Infinite, whom would you choose? Why is he appealing to you in relation to our course's themes?

26. Borges refers to the "secret" as the metaphor of "truth." Describe the paradox Borges presents us with in his short stories. How does Borges demonstrate the irony of the philosophical history of reality?

27. It has been suggested that Cantor's manic depression was due to a bipolar disorder. Cantor viewed himself as the "lone protector of truth under attack by sinister forces intent on quashing it." What drove and intensified Cantor's mental illness?

28. In an article published in 1959, Escher writes that "periodic surface division" is not "infinity" but "certainly a fragment/part of the reptilian universe." What does Escher mean by ‘reptilian universe'? What is the problem Escher is referring to in his statement? What solution does he come up with? What other approaches does he take, other than "periodic surface division" to describe infinity? Compare and contrast the different ways in which other artists have approached infinity.