MATH 320 – History of mathematics
Semester: Spring 2015 Instructor: Lucian M. Ionescu
Contact Info: STV 312J, 438-7167; ; www.ilstu.edu/~lmiones/mat320new.htm
Office Hours: TBA and by appointment at other times.
Text: An Episodic History of Mathematics, by Steven G. Krantz, 2006.
Content: In this course, students will study the development of mathematics from the growth of algebraic reasoning of early Mesopotamia and the geometry of ancient Greece, through the mathematics of the Renaissance, all the way to modern times. We will learn how mathematics was discovered and developed, by whom and why. Learning about the life of mathematicians will provide examples of what impacts one’s interest to mathematics and later on, his/her career as mathematician; especially our role as teachers. Mathematics is at the same time entertainment (for some of us), an art and a science; the interactions with other sciences shaped the problem solving process into mathematical modeling. This journey through mathematics will allow us to touch upon various topics, sometimes quite advanced, without having to master it; rather marvel at the ingenuity of human mind, and be advised when choosing our learning path. The emphasis will be on notable mathematicians who set the tone in various branches of mathematics, for decades and sometimes for many generations.
Prerequisites: C or better in MAT 147.
Course Format: You are responsible for reading assignments, and this is not a trivial task. The reading assignment is to be completed before the next scheduled meeting of the class. During the class, we will discuss the main people in the forefront together with the corresponding mathematics. Written outlines of the corresponding reading assignment are due the next class and will be used as a base for discussions.
Homework: Solutions for the exercises stated in class, consisting in details and/or proofs supplementing the class explanations, are expected the next class period, but will be collected on a weekly basis.
Projects: There will be one essay to be concurrently developed during the course, with several submissions and revisions. The final submission will be as an e-mail attachment MS-word format or txt/rtf/pdf. In the essay “Mathematics: art or science?” discuss the mathematics discovered from the point of view of its roots and mathematician’s motivation; was it discovered out of pure curiosity (“math as an art”) or prompted by a practical need (“math as a science”)? Even if discovered out of curiosity, explain its later role within mathematics itself or how it was used by another science. Other “math events” arose from the need to model reality as investigated by another science, so it is predominantly “math as a science”. There is no right/wrong approach here; we will discuss your project as we go and you will develop it towards a final version to be submitted the last day of class. Feel free to take a personal attitude and spend more time on your favorite kind of math and problems.
Graduate Credit: For graduate credit, research the connection between the material presented and the current curriculum (elementary/middle/high school levels and mathematics areas). Plan ahead; e.g. “I will tell this nice story in this / that class” or how math education could benefit from incorporating the historical roots and motivation in the current curriculum etc.
Exam: There will be one Final Exam; it will be based on the material covered, class discussions and problems solved as part of the homework (More info on the study guide).
Evaluation: Grades will be assigned based on the following points:
Class activities (Attendance and participation): 100
Homework and Outlines 200
Essay (2 submissions x 50 points each, and final submission 100) 200
Final exam 100
Total 600 points.
The grading scale is based on: A [90-100%], B [80-90%), C [70-80%), D [60,70%), F [0,60%).
N.B.: Students who believe they may need accommodations in this class/program are encouraged to contact the Disability Access Center as soon as possible to better ensure that such accommodations are implemented in a timely fashion.
“Warning: Plagiarism and cheating are serious offenses. Penalties can range from a minimum of a zero grade on the invalid instrument to expulsion from the University”
Suggestions for Learning (Mathematics)
Make an outline: For each assigned section of the text, make an outline containing a description of the major concepts and procedures. As you are making this outline, make a list of questions you want to ask your study partner or instructor.
Do the assignments: Work on your project each day. The best strategy is to write the outlines in a file and use it to develop your Project on a daily basis. Develop the list of questions to ask your study partner or instructor.
Topics assigned
Week / Sections / Topics / Ex. / Week / Sections / Topics / Ex.1)
1/12 / Ch.1 / Introduction
Ancient Greeks: Pythagoras (alg., NT & geometry) / 9)
3/9 / Spring Break
2)
1/19 / Ch.2 & Ch.3 / (Monday: no class)
Euclid, Archimedes, Zeno & Quantum ST
Hypanthia & conics / 10)
3/16 / Ch.13, / Prime numbers
Euclid’s trick & fusion; cryptography & Pratt trees
3)
1/26 / Ch.4 / Arabs and developments of algebra; congruence arithmetic / 11)
3/23 / Ch.15 / Riemann and Differential Geometry
Complex Analysis
4)
2/2 / Ch.5 / Cardano, Abel, Galois and Solvability of Polynomial equations / 12)
3/30 / Ch.17 / Number systems
Infinities: small & big (Rp vs. Qp)
5)
2/9 / Ch.6 / Descartes & analytic geometry
Pythagorean triples & Algebraic-Geometry / 13)
4/6 / Ch. 18 / Henry Poincare & topology
Poincare-Hopf
Gauss-Bonnet
6)
2/16 / Ch.7, Ch. 8 / Fermat: NT & Calc
Leibnitz and Newton, F. Th. of Calculus / 14)
4/10 / Ch. 20 / Emmy Noether: Modern Algebra & Betti numbers
7)
2/23 / Ch.9
& Ch.10 / Complex Numbers & F.Th. of Algebra
Gauss: The Prince of Mathematics / 15)
4/17 / Ch. 22 / Alan Turing & Cryptography
Elliptic Curves
8)
3/2 / Ch.10
Ch.12 / Gaussian integers & Gaussian primes (UF?)
Cauchy & Analysis
Real vs p-adic numbers / 16)
4/23 / Supplements and Review