Introduction

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Welcome!

This is Math 239, Introduction to Mathematical Proof.

I’m Doug Baldwin.

Pre-Assessment

A reference against which the math department can assess what this course has taught you about proofs by the end of the semester.

Preview

Math has lots of surprising results, most of them somehow tied to logic and proof. So this course prepares you to understand those surprises, and in many cases shows them to you.

Some of my favorite examples, as questions for short discussion:

• True or false: every pink leprechaun in this room is dancing on the front table.
  • This is “vacuously” true, i.e., true because there are no pink leprechauns in this room (although we might want to be more careful about exactly what a pink leprechaun is, e.g., if pink leprechauns can be invisible maybe there are some in the room). Consider that the number of pink leprechauns is 0, and the number dancing on the table is 0, i.e., the same as the number of all pink leprechauns in the room. Vacuous truth is something we’ll see lots more of.
• Which are there more of, whole (i.e., natural) numbers, fractions (rational numbers), or real numbers?
  • There are infinitely many of all 3 kinds of number. But it turns out there are different magnitudes of infinity, and in particular there are the same number of naturals and rationals, but more reals. This is another subject we will talk a lot about at the end of the semester.
• Do irrational numbers really exist, or are they just an excuse for lazy people not to look at enough decimal places to see a repeating pattern? How do you know?
  • Irrationals do exist, and we’ll see proofs later in this course. Even though most people intuitively believed in irrational numbers, really defending that belief needs a proof.
• Is it in principle possible to find a proof for every true mathematical theorem?
  • Consider this theorem -- Theorem 1: Theorem 1 has no proof.
  • If Theorem 1 is provable, then its proof is a proof of a falsehood, and so proofs don’t always establish truth (the next point). On the other hand, if Theorem 1 isn’t provable, then it’s an example of a true statement that can’t be proven.
  • This is the essence of work done by Gödel around 1930, which shook up mathematics in ways similar to how such people as Einstein and Heisenberg were shaking up physics at roughly the same time.
  • We won’t take this topic any further, but take the mathematical logic course (Math 301) to see more along these lines.
• Is every theorem that has a mathematical proof true?

Next

How this course will work and the underlying learning philosophy.

Read the syllabus.

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