Spring 2013 Professor: R. Sutcliffe (Room NSC 17)
Math 400-1 (Directed Studies in Mathematics), partially in lieu of MATH 480: Foundations of Mathematics (3 Sem. Hrs.)
Weekly Homework Assignments (Due always on the first day of each week. Class is W at 1600-1715. 1. Jan 9; due Jan 14 Read Kline, the introduction and chapter 1.
a. Summarize the views of Plato, Aristotle, and Pythagoras on the existence, reality, and truth of mathematical entities. Comment specifically on the differences between the views of Plato and Aristotle.
b. Comment on the difference between Mathematical and Biblical views of truth
2. Jan 16; due Jan 21 Read chapter 2.
a. How did Descartes view the relationship between mathematics and the physical world, and how did he think one acquired mathematical truths?
b. How did Galileo rate scientific and mathematical knowledge as opposed to Scripture?
3. Jan 23; due Jan 28 Read Chapter 3
a. What was Newton's principal motivation for his mathematical and scientific work and what was his belief regarding the nature of scientific theories?
b. How did Liebniz explain the acquisition of mathematical truth, and how did he account for the concord between mathematics and the physical world?
4. Jan 30; due Feb 4 Read Chapter 4
a. Why did the success of Mathematics harm religious beliefs and why in turn did this further undermine confidence in Mathematics as truth?
b. What is unusual about the quaternions and how did this affect the structure of mathematics?
5. Feb 6; Due Feb 12 Read Chapter 5
a. What problem did the discovery of the irrationals pose to the Greeks? The negative and complex numbers to later mathematicians?
b. Why did many 17th century mathematicians object to the use of Algebra?
6. Feb 13; Due Feb 25; Read Chapter 6
a. What difficulties arose in the justification of differential calculus? of integral calculus?
b. Why were infinite series introduced into calculus; what questions or problems were introduced because of this; and how did the use of them lead to strange or contradictory results?
7. Feb 27; Due Mar 4; Read Chapter 7
a. Why did many 19th century mathematicians object to the use negative and complex numbers?
b. Why did mathematicians struggle so much to put a logical foundation under irrational, negative and complex numbers, algebra, and calculus?
8. Mar 6; Due Mar 11; Read Chapter 8
a. How did mathematicians propose to build a proper logical foundation for mathematics, why did they start with rigorizing calculus, and what new problems resulted by this axiomatization of mathematics?
b. What was the main defect of Aristotelian logic?
c. Comment on the development of symbolic logic.
9. Mar 13; Due Mar 21 Read Chapter 9
a. State three paradoxes worked on by Russell and Poincare, and how they attempted to resolve them. What were the shortcomings of this attempt?
b. Explain the main tenant of the logistic school and how Russell and Whitehead avoided paradoxes. State their initial motivation and final conclusion about logicism. What were the shortcomings of this attempt?
c. What are some of the contributions and criticisms of intuitionism?
10. Mar 20; Due Mar 25 Read Chapter 10-11
a. Outline the main criticisms of the logistic approach.
b. How did the intuitionists hope to establish the truth of Mathematics?
c. what are some of the criticisms of intuitionism?
d. What were the main theses of Hilbert's formalism and what were the two main objections to it?
e. How did the set theorists hope to provide a foundation for math; how did they avoid paradoxes, and what were some of the objections to their approach?
11. Mar 27; Due Apr 2 Read Chapter 12-13
a. What were the two devastating assertions of Godel's 1931 incompleteness theorem?
b. What does Church's Threorem of 1936 say about the decision problem?
c. What is the main thesis of Robinson's non-standard analysis?
d. Comment on the dichotomy between pure and applied math and on the reasons why there was a movement from the former to the latter in the last century.
e. Comment on Kline's attack on Pure Math.
12. Apr 3; Due Apr 8 Read Chapter 14
a. Why is the notion of proof the basic issue in the mathematicians' dilemma?
b. How are logic and intuition related?
13. Apr 10; Due Apr 15 Read Chapter 15
a. Compare and contrast the two views that (i) mathematics is discovered (ii) mathematics is invented by mathematicians. Do this from a distinctively Christian point of view.
b. Discuss the idea that the soundness of a piece of mathematics can be judged primarily by its applicability to the physical world. As you do, critique the Kantian and Poincare explanations of why mathematics is applicable.
c. Kline claims that mathematics is man's supreme intellectual development and the most original creation of the human spirit. He sees God as an ideal, a product of human civilization. Critique these views.
These last 15 minutes each; ten to talk, five for questions. (Two presenters each of three times.)
Sign up first class for these
Presentations in weeks 2 (Chell & Lee), 6, (Trick & Chell) and 10 (Lee & Trick)
NOTE: please stay on topic. Your time is short.
1. Mathematics and proofs for the existence of God. Jiwon week 2 Jan 16
2. Infinity in mathematics and Theology.
3. Truth in mathematics and theology.
4. The epistemology of Mathematics vs that in science.
5. The interaction between mind and matter: physics (and/or mathematics) and consciousness.
6. Cosmology and eschatology.
7. The implications of Godel's theorems beyond mathematics.
8. Time and eternity. Matt Week 2 Jan 16
9. God as a mathematician. Jiwon week 10 March 20
10. Mathematics as language. Matt week 6 Feb 13
11. Intuition and mathematics.
12. What does it mean to "prove" a theorem?
13. A Christian philosophy of mathematics.
14. Quantum mechanics, free will, and divine action. Ian week 6 Feb 13
15. The absolute and the relative in Theology and Science
16. Computing science and mathematics. Ian week 10 March 20
17. The nature and status of scientific theories.
Weeks 4 (Lee vs Trick) 8 (Chell vs Lee), and 12 (Trick vs Chell) First name is positive; second name is negative)
(sign up second class for these after talking things over with your opponent.
1. Science is a better source of truth than is religion.
2. Big Bang cosmology is consistent with Christianity.
3. Mathematics is created by God.
4. The goal of strong AI (machines that can think as well as or better than humans) is reachable. Trick vs Chell Week 12 Apr 3
5. Mathematics is more an art than a science. Lee vs Trick Week 4 Jan 30
6. Mathematics, if not certain, is at least precise.