Trinity Western University Faculty of Natural and Applied Sciences
Fall 2011 Professor: R. Sutcliffe (Room NSC 17)
MATH 480: Foundations of Mathematics (3 Sem. Hrs.)
A study of basic issues in the history and philosophy of mathematics and physics. Topics include logic, infinity, Godel’s theorems, time, space, determinism, the nature of mathematical and scientific truth, the ontological status of theoretical entities, implications of relativity, quantum mechanics, and modern cosmology. Particular attention is paid to philosophical/theological presuppositions, implications, and applications in the mathematical sciences. NB: This course meets the University’s upper level IDIS core requirement. In combination with NATS 490, it meets the Faculty of Natural and Applied Sciences’ Interdisciplinary Studies core requirement.
NOTE: For the 2011 offering only, the course Math 480 will meet only once per week, Students will also attend and be responsible for all material covered in NATS 487, including the IDIS lectures and the major paper, and the grade in that course will be combined with the one for this portion to produce an overall grade for Math 480. NATS 487 by itself is not sufficient to replace the Mathematics requirements of Math 480. Debate and presentation topics will be posted online.
Advanced standing in mathematics (including Math 150 and at least one 300-level course) or special consent from the instructor.
To initiate the student into a study of issues in the history and philosophy of mathematics; to encourage the development of an individual philosophy of mathematics.
Morris Kline Mathematics: The Loss of Certainty. Oxford: Oxford University Press, 1980.
Online Materials: are in http://www.csc.twu.ca/sutcliffe/M480/index.html
Course Outline: The following topics, among others, will be discussed:
1. Symbolic logic and proofs
2. Set Theory and Infinity: cardinality, continuum hypothesis
3. Logicism, formalism, intuitionism
5. The ontological status of mathematical entities
1. Sept 12; due Sept 19 Read Kline, the introduction and chapter 1.
Summarize the views of Plato, Aristotle, and Pythagoras on such topics as the existence, reality, and truth of mathematical entities.
2. Sept 19; due Sept 26 Read chapter 2.
How did Descartes view the relationship between mathematics and the physical world, and how did he think one acquired mathematical truths?
3. Sept 26; due Oct 3 Read Chapter 3
What was Newton's principal motivation for his mathematical and scientific work and what was his belief regarding the nature of scientific theories?
4. Oct 3; due Oct 17 Read Chapter 4-5 (no class on Oct 10, so we need to read two chapters.)
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?
c. What problem did the discovery of the irrationals pose to the Greeks? The negative and complex numbers to later mathematicians?
5. Oct 17; Due Oct 24 Read Chapter 6
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?
6. Oct 24; Due Oct 31; Read Chapter 7
Why did mathematicians struggle so much to put a logical foundation under irrational, negative and complex numbers, algebra, and calculus?
7. Oct 31; Due Nov 7; Read Chapter 8
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?
8. Nov 7; Due Nov 14; Read Chapters 9 and 10
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.
c. What are some of the contributions and criticisms of intuitionism?
9. Nov 14; Due Nov 21 Read Chapter 11-12 (We have covered the main issues of chapter 12 in class, however and will not have questions on it for homework, just on the exam.)
How did the set theoretists hope to provide a foundation for math; how did they avoid paradoxes, and what were some of the objections to their approach?
10. Nov 21; Due Nov 28 Read Chapters 13-14
a. 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.
b. 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.
11. Nov 28; Due Dec 5
a. 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.
b. 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.
Presentations: These last 15 minutes each; ten to talk, five for questions. (Two presenters the first few times, three later.)
Sept 19 (2), Oct 3 (2), Oct 17 (2), Nov 7 (2), Nov 14 (3), Nov 28 (3)
NOTE: please stay on topic. Your time is short.
1. Mathematics and proofs for the existence of God. Alex Oct 3
2. Infinity in mathematics and Theology.
3. Truth in mathematics and theology. Caitlyn Sept 19
4. The epistemology of Mathematics vs that in science.
5. The interaction between mind and matter: physics (and/or mathematics) and consciousness. Alex Nov 7
6. Cosmology and eschatology. Alan Oct 17
7. The implications of Godel's theorems beyond mathematics. Justin Nov 7
8. Time and eternity. Travis Sept 19
9. God as a mathematician. Michelle Oct 3
10. Mathematics as language.
11. Intuition and mathematics.
12. What does it mean to "prove" a theorem?Edward Oct 17
13. A Christian philosophy of mathematics.
14. Quantum mechanics, free will, and divine action. Alan Nov 14
15. The absolute and the relative in Theology and Science
16. Computing science and mathematics.
17. The nature and status of scientific theories.
Debates: Two vs two the first few; three vs three if needed on the last.
Sept 26, Oct 24, Nov 21
1. Science is a better source of truth than is religion. Justin Nov 28
2. Big Bang cosmology is consistent with Christianity. Michelle Nob 28
3. Mathematics is created by God. Caitlyn Nov 28
4. The goal of strong AI (machines that can think as well as or better than humans) is reachable.
5. Mathematics is more an art than a science.
6. Mathematics, if not certain, is at least precise.
7. Without mathematics, science would never have come about. Travis Nov 14
8. Mathematics describes real world entities.
9. Infinity has a real existence. Travis & Caitlyn vs Michelle & Justin Nov 21
10. The physical world has an inherent randomness, such that not even God can know the future with certainty.