Euclidean and Non-Euclidean Geometry
Bob Matthews
Department of Mathematics and Computer
Science
January, 2000
Administrivia
- Meeting times: 9:00 - 9:50 MTThF Thompson 125
- Final Exam: 4:00 PM Monday, May 8 required
- Instructor:
- Bob Matthews (email matthews@ups.edu)
- Thompson 502
- Extension 3561
- Office hours (tentative):
- 11:00 - 11:50 MTThF
- Or by appointment.
- If you catch me free at any time, please feel free to drop
in. Messages sent via email are welcome, and can be used to ask
a question or to set up an appointment.
- Textbook
- Required: Greenberg, Marvin J.: Euclidean and
Non-Euclidean Geometries W.H. Freeman and Company, 1993. We
will work our way through the first eight chapters of the
textbook.
- Other readings as assigned. In addition to readings from
the textbook (which will be our primary reference), we will,
from time to time, leave the textbook to explore ideas and
questions generated by the text using other sources.
Weekly reading and lecture schedule
Exam Reviews
Assignments
Evaluation (Please note changes discussed 3/10)
- Four hour exams + a comprehensive final: (The final exam will
have the weight of two hour exams). If we do manage four hour
exams, I may drop the lowest hour exam score.
- Written exercises will be given the weight of one hour exam. I
will make assignments from the textbook on a routine basis, and
will select problems from each assignment to grade. You will also
be asked to present your work on the board (though this will not
generally be graded, except for participation).
- Term Paper: The term paper will
have the weight of one hour exam.
Some comments on the course:
A course in Euclidean and non-Euclidean geometries serves several
purposes in the undergraduate mathematics curriculum. For prospective
teachers, it is a course required by most states for teacher
certification. For many, it is the first course that involves
rigorous proof. For students interested in the philosophy and history
of mathematics, it provides an important example of how mathematics
works, how one does mathematics, how mathematics has developed over
time (together with false starts and wonderful surprises), and gives
insight into what are commonly called 'foundational issues' (What is
the nature of proof? What is the nature of mathematical truth? What,
if anything, does this all mean? What is the geometry of the space
around us?).
We will approach all of these issues in the course of this term.
We will study geometry by doing it. From the practical point of view,
this means that we will spend time learning how to prove things and
how to present results both orally and in writing. Along the way we
will talk about and work with the process of discovery, the
uncovering of assumptions, the rigorous presentation of results, and
the logical and philosophical foundations of mathematics (and some of
the issues surrounding those foundations).
It will be hard work, but it should also be a great deal of
fun.
Some Important Dates:
Please check the Master
Calendar for important dates in the term (last day to add/drop,
etc.).
Hour Exams will be held on the following dates:
- Exam 1: Friday, Feb. 4
- Exam 2: Tuesday, Feb. 29 (Note: this has been
changed from Friday, Feb. 25)
- Exam 3: Monday, March 27 (Note: this has been changed
from Friday, March 24)
- Exam 4: Monday, April 24 (Note: this has been changed
from Friday, April 14)
The final exam for this class will be at 4:00 PM Monday, May 8. It
will be a comprehensive, two hour in-class examination.
Syllabus
HONORS 213
QUANTIFICATION: Foundations of
Geometry
I. Introduction
A. Catalog Description
This course will present a rigorous
treatment of the foundations of Euclidean geometry and an
introduction to non-Euclidean geometry. The discovery of
non-Euclidean geometries shattered the traditional conception
of geometry as the true description of physical space. The
discovery led to a revolution in geometry as scientifically
profound as that of the Copernian revolution in astronomy.
Students will learn the history and foundations of geometry by
actually proving theorems based upon Hilbert's axioms for
geometry. Emphasis will be placed on logic, the axiomatic
method, and mathematical models.
B. Objectives
This course is designed specifically for
the Honors program. The emphasis will be on constructing
logical arguments, logic, the axiomatic method, and modeling.
This will give the students a flavor of modern methods in
mathematics. Students will see the birth of these ideas by
examining the history and the foundations of
geometry.
II. Required Topics
1. Set theory
2. Logic: truth tables, negation,
quantifiers, proofs.
3. Hilbert's Axioms: incidence, betweenness,
congruence, continuity, parallelism
4. Models
5. Neutral Geometry: geometry without
parallel axiom, exterior,angle theorems, angle sum of a
triangle
6. History of the Parallel
Postulate
7. Discovery of Non-Euclidean
Geometry
III. Bibliography
Greenberg Euclidean and Non-Euclidean
Geometries
Hilbert Foundations of
Geometry
Moise Geometry
Poincaré Science and
Method
Reid Hilbert
Lakatos Proofs and Refutations, The Logic
of Mathematical Discovery
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