MathEd/Math 300 Review for Final – Davis & Hersh Chapter 1 1. What are some reasons that mathematics cannot be carefully defined? 2. What is Ulam’s dilemma, and what does it say about the nature of modern mathematics? Chapter 2 3. What beliefs about mathematics are apparent from the Ideal Mathematician? To what extent are these ideas apparent in the mathematical culture you are familiar with? Is the Ideal Mathematician consistent in his beliefs? 4. Mathematics is commonly thought of as a way of getting at truth. Discuss this in light of Davis and Hersh's discussion on the role of models and modeling in mathematics. Chapter 3 5. One commonly held belief about mathematics is that "God is a mathematician" or that the universe expresses itself mathematically. Contrast this idea with what Davis and Hersh call mathematics "by fiat." Give some examples of how mathematics might be imposed on the universe by us (instead of the other way around). 6. Explain the differences between mathematical Hardyism and mathematical Maoism, and how this relates to Davis and Hersh’s discussion of “utility” in mathematics. 7. What does a pure mathematician mean when he says that a piece of mathematics is useful? What does an applied mathematician mean by this? Chapter 4 8. Davis and Hersh claim that mathematical objects exist in the same way that unicorns exist. Explain what they mean by that. 9. Read the quote by George Berkeley on p. 161, together with the two paragraphs before and after. What does this have to say about what is “intuitively obvious” in mathematics? 10. In what sense can mathematics be said to be beautiful? Chapter 7 11. We have said that the working mathematician is a “Platonist on weekdays and a formalist on Sundays.” Describe what this means. 12. Describe what mathematical Platonism is and outline the major arguments for and against it. 13. How does what we know about “Pascal’s Triangle” support a Platonic view of mathematics? 14. How would a mathematical humanist like Reuben Hersh respond to the above argument about Pascal’s Triangle? 15. What is the Euclid Myth and why is it significant? To what extent do people believe the Euclid Myth today? 16. To what extent do people believe the Euclid Myth today? 17. Explain what is meant by the "crisis in foundations" in mathematics. Include the major figures and schools of thought. How was the crisis resolved? 18. How did logicism attempt to resolve the crisis in foundations? Who was involved? Why was it not successful? 19. How did intuitionism attempt to resolve the crisis in foundations? Who was involved? Why was it not successful? 20. How did formalism attempt to resolve the crisis in foundations? Who was involved? Why was it not successful? 21. Formalists believe that they cannot assert that a theorem is true, even if it is proven. What do they assert instead of truth? 22. What was the purpose of Lakatos's Proof and Refutations? 23. Describe how each of the following would view the notion of proof in mathematics: a. The Ideal Mathematician b. A Formalist c. A Fallibilist Place the letter of the mathematician or philosopher in the list below into the category that best describes his approach to mathematics. Logicism Formalism Constructivism / Intuitionism Fallibilism Humanism A. David Hilbert E. L. E. J. Brouwer B. Alfred North Whitehead F. Ludwig Wittgenstein C. Imre Lakatos G. Philip J. Davis D. Bertrand Russell H. Reuben Hersh.