# Math 235

**Instructor:** Dr. Clayton Shonkwiler

**Time:** Tuesday, Thursday 1:00–2:00

**Location:** Engineering E105

**Office:** Weber 216

**Office Hours:** Monday 11:00–12:00 AM, Wednesday 2:00–3:00 PM

**Email Address:** clay@shonkwiler.org

**Text:** *Mathematical Reasoning: Writing and Proof*, by Ted Sundstrom

**Syllabus**

## What’s this all about?

What, exactly, do mathematicians do? Of course there are many different answers to this question, but very common thing that mathematicians do is try to discover new things. These things might be statements about numbers, but they could just as well be statements about topological spaces or commutative rings or dynamical systems or any number of other things.

In general, discovering new statements (or lemmas, propositions, theorems, etc.) is not very much like solving calculus problems: there’s usually no textbook telling you which statements to prove and, most frighteningly, *nobody even knows which statements are true* (yet). Consider a statement like the following:

The real part of every non-trivial zero of the Riemann zeta function is \(\frac{1}{2}\).

Nobody knows whether this is true or not, but since the 19th century various people (some serious, some complete cranks) have claimed that it is true, and some day somebody will probably definitively resolve the question of whether it is true or false. So the question is: what would such a resolution look like? In other words, if you want to convince the world that (say) the statement is true, how would you go about it?

In short: give a proof. In mathematics, a proof is a rigorous justification of a statement. “Rigorous” and “justification” are, in the end, just words that convey a certain meaning and aren’t really formally justified, but there are conventions in the mathematical community about what constitutes a valid proof that (eventually, usually) lead to a consensus about which statements are true and which are false.

The goal of this class is to start learning those conventions and to develop your skills in giving convincing mathematical arguments (a.k.a., proofs). In other words, the goal is to learn how to write (and, more generally, communicate) mathematics.