Foundations of Mathematics

Math 281 Dave’s Syllabus Fall 2010

There’s some irony to the name of this course. You’ve probably taken math classes for 13 straight years and now you get to the Foundations!?! What’s all of your math knowledge built on anyway, sand? Nothing?

Actually your path through mathematics mirrors the historical development of those same ideas. Limits and derivatives were being used for 170 years before good definitions were developed. Various cultures talked about a concept of infinity for centuries before Georg Cantor provided the foundations for the mathematical study of infinity. (He proved a stunning fact that we will hopefully get to in this course – not only are there different sizes of infinity, but there are actually an infinite number of sizes of infinity!)

In this semester of FOM, we’ll work our way through the following topics, all of which will be vital in future math courses (and, actually, in life):

What it means to prove something & why we prove things.
Set theory - sets, power sets, combinations of sets.
Logical statements - the real language underlying mathematics.
Quantifiers – mathematically precise uses of “for all” and “there exists”.
Induction – proving an infinite number of cases – all at once!
Problem solving skills – systematic ways to get unstuck when you’re stuck on a problem.

The thread that connects all of these topics, and the main point of this course, is to answer this question:

What does it take to establish mathematical certainty?

Important Facts:

Professor:

Dave Kung

175 Schaefer Hall, x4433

(or 240-895-4433 from off-campus)

dtkung@smcm.edu

TA:

Christiana Sabett

cmsabett@smcm.edu

Office Hours:

Mon	1:10-2:10	and by appointment.
Tue	3:00-4:00
Thurs	1:20-2:20

Text: Foundations of Higher Mathematics: Exploration and Proof, by Dan Fendel and Diane Resek.

Where to go for help: To learn the key concepts of FOM, we’ll use a variety of classroom activities, homework, and writing assignments. Also, you’ll be expected to spend a significant amount of time reading the textbook. When you get stuck, you’ll have three main resources to draw on. The first and most important is your fellow classmates. This course will be hard – at times very hard. It will go much smoother for all of us if you start getting to know your classmates and start studying with them outside of class early in the semester. The second is your able teaching assistant, Lydia Garcia, fresh back from a math-packed semester in Budapest. Your third resource is me - contact info and office hours appear above. I will also be around at other times - feel free to drop by and say hi. If you can't find me, email or call and we'll schedule an appointment that works for both of us. If an emergency comes up and you are forced to miss class, you should drop me an email (I check it very frequently).

Assignments: There will be four different types of assignments: the problem of the week, your journal, written proofs, and problem solutions.

Every Monday I will post a Problem of the Week on the math wing. Please stop by and read the problem. Solutions are due one week later. (Twice during the semester this will happen on Wednesday, since Monday classes are cancelled on Labor Day and Fall Break. PoW solutions are graded largely on the quality of your attempt – and your lowest grade will be dropped before averaging the rest.

For the journal, you may choose any type of notebook/binder/daily diary. See the separate “Guide to Writing a FOM Journal” for more details.

Written Proofs will be assigned about once a week and collected in class. You will be graded on how complete and understandable your proofs are. For your first two proofs, you will be encouraged to revise and resubmit them. This will give you some time to adjust to our expectations. We encourage you to work with others to develop your proofs but the writing must be entirely your own.

Grading:

Assessment Date Percent

Midterm	Wednesday, October 27^th	15
Journal	all semester	15
Written Proofs	all semester	15
Problem Solutions	all semester	10
Class Participation	all semester	10
Take-home Final	Due Dec. 13^th, 10am	20
Final Project	Due Dec. 6^th , in class	15
Total		100

The mid-term will be in class – though you may start as early as 8am if you’d like. Anyone who has an 8am class will be given an opportunity to have a similar amount of time. The final will be a take-home exam which must be done without consulting other people or other books. The final project is your chance to be creative. Past projects include short films, skits, board games, mathematical sculptures, short stories, musicals, and a South Park takeoff called Math Park.