Foundations of Mathematics

Math 281 Dave’s Syllabus Spring 2017

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:

How do you establish mathematical certainty?

Important Facts:

Professor:

Dave Kung

175 Schaefer Hall, x4433

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

dtkung@smcm.edu

TA:

Emily Stark (enstark@smcm.edu)

Textbook:

Book of Proof (free to download, cheap at the bookstore)

Office Hours:

Mon	2:00-3:00pm	and by appointment.
Wed	1:00-2:00pm
Fri	11:00am-noon

Where to go for help: To learn the key concepts of FOM, we’ll use a variety of classroom activities, homework, and writing assignments (both online and on paper). We’ll do lots of work on the whiteboard tables in 161 – please avoid scratching the tables with three-ring binders or sharp jewelry. 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, Emily Stark. 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 frequently, but not between 11pm and 7am).

Assignments: There will be three different types of assignments: the problem of the week, your blog and other homework from the book, and written proofs.

Every Tuesday I will post a Problem of the Week (PoW) on the math wing. Please stop by and read the problem. Solutions are due one week later. (The first one is up and will be due Tuesday, Jan. 24^th) PoW solutions are graded largely on the quality of your attempt – and your lowest grade will be dropped before averaging the rest.

For the blog, I recommend using Wordpress.com (which allows for some math symbols). See the separate “Guide to Writing a FOM Blog” for more details. In addition to your blog, problems from the book will be occasionally assigned. Some will be done on your blog, others on paper.

Written Proofs will be assigned about once a week. They will be graded on a credit/no credit basis, but you may revise and resubmit them as many times as you’d like – with two catches:

1. You must turn something in on the due-date of each proof.

2. What you turn in must be at least your second draft. Demonstrate this by stapling them together (first drafts in back).

We encourage you to work with others to develop your proofs but the writing must be entirely your own.

Grading:

Assessment Date Percent

Midterm	Thursday, March 9th	15
Blog & Homework	all semester	15
Written Proofs	all semester	30
POWs	all semester	10
Class Participation	all semester	10
Take-home Final	Due Sat., May 6^th, 2pm	20
Total		100

The mid-term will be in class – though you may stay late if you need. The final will be a take-home exam that must be done without consulting other people or other books.

By the end of the semester, you will understand the amazing role proofs play in mathematics, as well as how and why to write them.

Learning Outcomes: At the completion of this course, you’ll be able to…

· implement the rules of mathematical logic as demonstrated by incorporating logic into valid mathematical proofs.

· use the principles of set theory as demonstrated by computing the results of set operations and proving properties of sets and set operations.

· judge the validity of a mathematical argument as demonstrated by analyzing attempts at mathematical proofs and assessing whether or not these attempts constitute actual proofs.

· solve novel problems in mathematics as demonstrated by successfully completing problems and proofs that involve unfamiliar mathematical definitions.

· construct clear and rigorous mathematical arguments as demonstrated by writing proofs that are readable and mathematically correct.

Fine Print

Rules for academic misconduct are contained in your student handbook. Anyone violating those rules will be dealt with quickly and effectively, to preserve the academic integrity of your fellow students and SMCM.

If you have a documented learning disability, please see me in the first week of classes to discuss your accommodations.