Fall 2009
"Mathematics is the part of science you could continue to do if you
woke up tomorrow and discovered the universe was gone."
Put on the web by Dave Rusin.
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Instructor |
Brian
Shelburne 329-E BDK Science |
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Class
Meetings |
Th
2:10 - 3:40 - Room 320 |
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Textbook |
Journey
Through Genius
by W. Dunham |
Course Objectives: This is a capstone course for mathematics majors. Its purpose is to let you think about and reflect on what mathematics is and to tie together your years of studying mathematics. Dunham's book, Journey Through Genius, covers the story of mathematics from the 5th century BC up to the 20th century AD by looking at some famous problems and theorems and the mathematicians who worked on them. The book is many things. It is a selective history of mathematics. It is a look at some of the more famous and colorful characters who were mathematicians. It presents rigorous but readable proofs of some interesting and famous theorems in mathematics.
Additional readings from papers by J. Grabiner that deals with the 19 century origins of the rigorous calculus and L. Becerra and R. Barnes covers the sweep of mathematics into the 21st century.
The course will also make use of a number of on-line resources; two in particular are The MacTutor History of Mathematics Archive at The University of St. Andrews in Scotland (http://www-history.mcs.st-andrews.ac.uk/history/index.html) which contains a wealth of biographical and historical essays on mathematicians and mathematics and Kevin Brown’s Mathpages.com (http://www.mathpages.com) which is source of articles on a number of unusual and interesting mathematical topics.
Dunham’s book is well written, fun to read. Augmented by the papers by Grabiner and Becerra & Barnes and your own explorations of on-line resources this course will give you a deeper appreciation for the unique endeavor we call mathematics.
Grading: The grade for the course will be based on the following criteria:
1. Class attendance, completing reading assignment before class, contributing to class discussions (15%)
2. Short in-class presentations on various mathematical topics (20%)
3. Write ups of math problems based on the readings and class discussions (30%)
4. A research paper on a mathematical
topic. Final copies (paper and electronic) are due Tuesday December 15, 2009.
Papers will be spiral bound and class members will receive copies. (30%)
5. Oral presentation on your paper given
during Final Exam Week (5%).
6. Taking the ETS Major Field Test in Mathematics (5%)
Class Attendance etc: It is expected that no student will deliberately miss class. Necessary for good class discussion is coming prepared; so it’s expected that the readings will be completed before each class.
Short In-Class Presentations: This class is a seminar so members are expected to research, prepare and present material to the rest of the class. Each week 2 or 3 volunteers will make oral presentations. Since there is a lot of interesting mathematics readily available on the internet, the websites listed should be used as starting points for your research. Good, interesting, and informative presentations require some work so don’t put them off to the last minute. Use of technology is encouraged. Feel free to use me as a resource person for ideas & suggestions.
Weekly Math Problem Write-Ups: This is a course about theorems and proving results. Based on the materials we cover each week I will assign one to three problems to be written up and handed in. Write up should be polished; neatness and legibility count! These are not pledged assignments. Collaboration on understanding the material is okay but write ups must be your own. Therefore don’t copy one another.
Mathematical Paper and Class Presentation: Since this is a writing intensive seminar, you will be expected to research some topic of interest in mathematics and present your findings to the rest of seminar. Paper should be expository (a research paper is not required) and should be approximately 12 – 15 pages in length. The paper will be graded on mathematical content and on context; for example presenting the solution of some mathematical problem or theorem within its the historical setting as a background much like what Dunham does in his book.
As part of this process, there are a number of milestones you must satisfy
Milestone #1 (Week 5): Deadline for Approval of Paper Topic
Note: This means running your paper topic by me before the deadline. The choice of a topic is up to you; the idea is to choose something that you find interesting.
Milestone #2 (Week 9): Outline of Mathematics Content Due (1 -2 pages)
Note: This does not have to be a set of math proofs completely worked out. I want to see an overview of the mathematics, just enough so that I’m confident that you have the basic math down.
Milestone #3 (Week 13): Outline of Context Due (1 – 2 pages)
Note: A good context will make the paper more interesting!
The final paper is due during exam week; both a paper copy and a electronic copy must be submitted.
For help in how to document papers check out Research and Documentation
On-Line by Diane Hacker. Check out the [Sciences] link on the lower right, [Finding Sources] and [Documenting
Sources]. Under Documenting Sources – CSE Style (Council of Science
Editors) follow the CSE Reference List
link and scroll down to see the reference examples.
During time allocated for the final everyone will give a 15 minute presentation on their paper topic.
ETS Major Field Test in Mathematics: "The Major Field Tests are objective, end-of-program tests .... Based on the Graduate Record Examination Subject Test they have been shortened to two hours each, made less difficult than the GRE tests, and revised to reflect undergraduate programs and to be appropriate for all seniors majoring in a field, not just those planning graduate study.
Score on these tests provide useful information for institutions seeking outcomes measures, for departments in evaluating their curriculum, and for faculty in measuring the progress of their students and considering curriculum changes." (from Major Field Tests Program Manual).
Performance on the ETS test will in no way affect your grade for the course or affect your graduation status unless you do not take the test. This is for our own internal departmental assessment. Of course we expect you to take this test seriously and do the best you can.
The test will be scheduled at a mutually agreeable time and date sometime in November. It will require a 3 hour block of time.
Syllabus - Senior Seminar Math 460 - Journey Through Genius
All reading
assignments must be completed before class
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Week: Calendar Dates |
Sections Covered – Topics |
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1 – Th Aug 27 |
Topic: Introduction to the Math Seminar |
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2 – Th Sept 3 |
Reading Assignment: Dunham Ch 1 (to be completed before class) Topic: Hippocrates Quadrature of the Lune |
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3 – Th Sept 10 |
Reading Assignment: Dunham Ch 2 Topic: Euclid's Proof of the Pythagorean Theorem |
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4 – Th Sept 17 |
Reading Assignment: Dunham Ch 3 Topic: Euclid and the Infinitude of Primes |
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5 – Th Sept 24 |
Reading Assignment: Dunham Ch 4 Topic: Archimedes' Determination of Circular Area Paper Milestone #1: Deadline for Approval of Paper Topics |
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6 – Th Oct 1 |
Reading Assignment: Dunham Ch 5 Topic: Heron's Formula for Triangular Area |
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7 – Th Oct 8 |
Reading Assignment: Dunham Ch 6 Topic: Cardano and the Solution of the Cubic |
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8 – Th Oct 15 |
Reading Assignment: Dunham Ch 7 Topic: A Gem from Isaac Newton |
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9 – Th Oct 22 |
Reading Assignment: Dunham Ch 8 Topic: The Bernoulli’s and the Harmonic Series ETS Major Field Test will be scheduled for sometime in
November |
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10 Th Oct 29 |
Reading Assignment: Dunham Ch 9 Topic: The Extraordinary Sums of Leonard Euler Paper Milestone #2: Math Content Outline Due |
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11Th - Nov 5 |
Reading Assignment: Dunham Ch 10 Topic: A Sample of Euler's Number Theory |
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12 Th – Nov 12 |
Reading Assignment: Dunham Ch 11 pp 245 – 251 Mathematics of the 19th Century Reading Assignment: Who Gave You the Calculus? Cauchy and the Origins of Rigorous Calculus; J. Grabiner |
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13 Th – Nov 19 |
Reading Assignment: rest of Dunham Ch 11 Topic: The Non-Denumerability of the Continuum Paper Milestone #3: Context Outline Due |
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14 Thanksgiving Break |
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15 Th – Dec 3 |
Reading Assignment: Dunham Ch 12 Topic: Cantor and the Transfinite Realm |
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16 Th – Dec 10 |
Reading Assignment: The Evolution of Mathematical Certainty; L. Becerra & R. Barnes |
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17 Tu – Dec 15 3:30 – 6:30 PM |
Oral Presentations on Paper Topics Final Copies of Papers Due |
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AI do not know what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinarily, whilst the great ocean of truth lay all undiscovered before me.@ - Isaac Newton |
Thales: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the “T” link to find the link to Thales
Pythagoras: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "P" link to find the link to Pythagoras
The Three Famous Problems of Classical Greek Mathematics: Go to http://www-history.mcs.st-andrews.ac.uk/Indexes/Greeks.html . The first three items under the column to the left are links to
- Squaring the circle
- Doubling the cube
- Trisecting an angle
Some solutions to these problems are covered in Three Famous Problems.pdf
What are constructable numbers? See On Constructable Numbers.pdf
Euclid's Elements On-Line. "The" Textbook on Geometry written by Euclid around 300 B.C.E. is brought forward into the 21th Century C.E
Eudoxus: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "E" link to find the link to Eudoxus
Euclid: Go to http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "E" link to find the link to Euclid.
Eves, Howard & Newsom, Carroll, An Introduction to the Foundations and Fundamental Concepts of Mathematics Revised Ed., Holt Rinehard & Winston, 1971; Appendix A.6 The Eudoxian Resolution of the First Crisis in the Foundation of Mathematics; pp 332 - 334
An Article on Non-Eucludean Geometry: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry.html
An outline of Euclid’s Elements
Prime numbers: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Prime_numbers.html
The Golden Ratio: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Golden_ratio.html to find a construction of the regular pentagon
Archimedes: Goto http://www-history.mcs.st-andrews.ac.uk/history/index.html , follow the Biographies Index link and click the "A" link to find the link to Archimedes
Kevin Brown's Archimedes on Spheres and Cylinders: http://mathpages.com/home/kmath343.htm
A History of Pi: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pi_through_the_ages.html
Archimedes’ result that the surface area of a sphere, Ssphere = 4πr2, is also equal to the lateral surface area of a cylinder that encloses it (i.e. a cylinder with radius r and height 2r), has an interesting application in cartography. If we project all points on the sphere radially outward from a north-south axis (not from the center), we obtain the Lambert cylindrical equal area projection of the sphere onto a plane which leaves all areas the same. See http://en.wikipedia.org/wiki/Lambert_cylindrical_equal-area_projection. Thus, unlike the “standard” Mercator projection map, Greenland and Africa are correctly proportioned with respect to their areas (although Greenland is smeared across the northern half of the projection) – TANSTAAFL!
At this point we consider the influence of Arab, Indian and Chinese cultures on mathematics
Arab Mathematics: Forgotten Brilliance: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html
An Overview of Chinese Mathematics: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Chinese_overview.html
An Overview of Indian Mathematics: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html
A History of Zero: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html
Tartaglia vs Cardano: Goto: http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Tartaglia_v_Cardan.html
Quadratic, Cubic and Quartic Equations: Goto: http://www-history.mcs.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations.html
Internet Resources for the History of Complex Numbers: Goto: http://math.fullerton.edu/mathews/c2003/HistoryComplexBib/Links/HistoryComplexBib_lnk_1.html
Gottfried Wilhelm von Leibniz: Goto http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Leibniz.html
A History of the Calculus: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/The_rise_of_calculus.html
Beyond Mere Convergence - A paper by James Sellers of Penn State discussing the convergence of other series by Jakob Bernoulli: Goto http://www.math.psu.edu/sellersj/p25.pdf
Fermat Numbers: Goto Wolfram's MathWorld http://mathworld.wolfram.com/FermatNumber.html to learn (?) about Fermat Numbers and Fermat Primes
Who was Sophie Germain? Goto http://www.pbs.org/wgbh/nova/proof/germain.html
Grabiner, Judith V. "Who Gave You The Epsilon? Cauchy and the Origins of Rigorous Calculus".The American Mathematical Monthly Vol 90, No. 3 (Mar.1983), pp. 185-194. Available on-line thru JSTOR. Goto http://www.jstor.org/search/ and Search using the title of the article.
The Function Concept: Goto http://www-history.mcs.st-andrews.ac.uk/HistTopics/Functions.html