Readings and other materials:
All from Davis, Hersh and Marchisotto, The Mathematical Experience, Study Edition. Boston:Birkhauser, 1995.
1) "Platonism, Formalism, Constructivism" pps 356-358.
2) "The Philosophical Plight of the Working Mathematician" pps 359-360.
3) "Foundations, Lost and Found" pps 368-376.
4) "The Formalist Philosophy of Mathematics" pps 377-382.
Introduction:
Our text (459 & 461) defines these three as follows:
PLATONISM - As used in this book, Platonism is the position that the whole of mathematics exists externally, independently of man, and the job of the mathematician is to discover these mathematical truths.
FORMALISM - The position that mathematics consists merely of formal symbols or expressions which are manipulated or combined according to preassigned rules or agreements. Formalism makes no inquiry as to the meaning of the expressions.
CONSTRUCTIVISM - (also intuitionism) The doctrine which asserts that only those mathematical objects have real existence and are meaningful which can be 'constructed' from certain primitive objects in a finitistic way. Associated with L.E.J. Brouwer and his followers.
Possible Discussion Directions and Threads:
1. Discuss which of the three or combination of the three most closely describes your current mathematical philosophy. Explain.
2. Select a mathematical entity such as pi, the set of integers, the Fibonacci sequence or a Moebius band and compare and contrast what each of the three philosophies would say about it.
3. For each of these three discuss what constitutes a "proof".
4. Constructionist emphasize the algorithmic nature of mathematics. Make connections between this and computer science.
5. According to Maxime Bocher (1904) "Until a system of axioms is established, mathematics can not begin its work!" while according to Morris Kline (circa 1980) "When a mathematical subject is ready for axiomatization, it is ready for burial and the axioms are its obituary!" Which of these two statements best reflects your view of mathematics. Why?
Schedule:
Introduction on web and distributed by Tuesday, September 12, 2:50 PM
Individual initial posts by Sunday, September 17, midnight.
Follow up posts/discussion through Wednesday, September 27, midnight.
Wrap-up posted here and to mailing list Friday, September 29.