MATH325: Lecture 1


Syllabus | Schedule | Grading Keys/Rubrics

When circumstances allow, I'll be typing up bits of my lecture notes and posting them online. These may or may not bear any resemblance to the actual lectures.

Motivation

Q: Why bother teaching geometry?

A: So that your students learn and practice linear, rational thought. These skills may also be learned in a computer programming or logical reasoning course.

Q: Why do I love geometry?

A: Because it is beautiful. The figures in Euclidean geometry (see pages 33 and 37) are aesthetically pleasing to me, and Euclid was the first western mathematician to try to use logic to create a mathematical system of absolute truths.

Q: What's the most important thing I can learn in this class?

A: How mathematics is invented/discovered.

Foundations of Mathematics

This is my version of the lecture on how modern mathematics is based on theorems which are proved from lemmas and definitions.

Suppose you're babysitting a 4 year old, and you give her a lollipop. She asks you what it's made of.

You: "Sugar."

Her: "What's sugar made of?"

You: "About half hydrogen; the rest is carbon and oxygen."

Her: "What's hydrogen made of?"

You: "Neutrons, protons, and electrons."

Her: "What are neutrons made of?"

You: "Uh, quarks maybe? I don't know -- I'm a babysitter, not a physicist!" (Unless you actually are a physicist, in which case you may make it all the way down to superstrings before giving up.)

Now imagine the same process with a math problem, asking "how do you know it's true?" rather than "what's it made of?" Eventually you get to questions like "How do you know two plus two is equal to four?" and you're stumped.

With young children and with mathematicians, the final answer is often "that's just the way it is." Mathematicians make a virtue of this necessity by collecting all the answers of this sort and identifying them as "axioms" or "definitions". Our goal is to have to say "that's just the way it is" as seldom as possible and to be able to explain everything else based on those very few axioms and definitions.

Axioms and definitions are the basic assumptions we base our arguments on.

So far, this is very much like what all scientists do when they try to describe the world. The difference between mathematicians and most other scientists is that at the very bottom, our work is based on intellectual theories rather than physical facts. Because of this, mathematicians are able to prove conclusions that must be absolutely true based on our axioms and definitions. (Of course, if we choose axioms like "2+2=5", our conclusions will not be very useful.)

The most important thing you can learn in this class is how mathematicians strive to start with some basic, simple truths (quadrilaterals are four sided polygons) and use them to prove big, complex conclusions (there are eight types of quadrilaterals: scalene, kite, dart, trapezoid, rhombus, parallelogram, rectangle, rhombus, square).

What Actually Happens

Mathematicians are only human; we make mistakes. Things that are "obvious" to us might not actually be true. For instance, if two edges of a polygon can cross each other, we have to add "bowtie" quadrilaterals to the list above.

We've spent over 2000 years examining our assumptions and correcting our mistakes. We've tried making subtle changes to our assumptions and starting with entirely different sets of assumptions. If different sets of assumptions lead to different conclusions we decide which conclusions we like best and adopt the assumptions that led to those conclusions. If it's too hard to reach any conclusions based on a set of assumptions, we add more assumptions until we can do what we need to.

The text book starts with Euclid's assumptions and adds more assumptions to make sure that the conclusions we reach match what we understand about figures drawn in the plane (or on paper or a white board). Because of this, we can start right off with some interesting (but confusing) theorems rather than spend a long time laying a foundation of simple (but still confusing) theorems.

Vocabulary and Notation

Look at the vocabulary listed on page 3 of the text. Review the difference between Supplementary angles (measures add to 180°) and Complementary angles (measures add to 90°).

Read the definition of a Degenerate Triangle -- this not a commonly used term and may be new to you. An example of a degenerate triangle appears below:

A-----B--------C
In this triangle, |AB| + |BC| = |AC|. In a non-degenerate triangle, the sum of two edge lengths is always greater than the length of the third edge. (For more information, read up on the triangle inequality.)

Some of the notation used in the book is described on page 4. The typed notes will mostly follow these conventions (it's hard to type a line over two letters!), but when writing solutions by hand we will use the abbreviations mentioned.

Unlike the textbook, we will denote the measure of the angle ∠ABC by m∠ABC. The example below shows an angle ∠ABC. The measure of angle ∠ABC is 90°.

A
|
|
|
B------C
Many of the exercises in the book are also theorems we might need to prove things later. Read them. A * on an exercise means it's difficult. A dagger symbol (†) next to an exercise means that it is especially important -- i.e. that it's almost certainly needed to prove things later.

Euclid's Elements

2300 years ago, Euclid of Alexadria attempted to organize theorems proved by mathematicians such as Pythagoras, Hippocrates, Theaetetus, and Eudoxus into one coherent body of work. An annotated translation of the result appears at http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI5.html.

Note that the Definitions given at the beginning of Book 1 are not very descriptive. The problem is that when you define something, you define it in terms of something else. So, in order to define something you need to already know the meaning of something else. But how was that "something else" defined? Realizing this, it's now customary to use undefined terms that everyone is expected to understand without definition. The text book quietly does this for terms like point, line and angle.

The Five Postulates listed in Book 1 are famous. Read them and compare them to the axioms on page 12 of the book. The first four are the same (and relatively simple). The fifth is different (and more complicated). We'll come back to that difference soon -- this subtle change to Euclid's assumptions turns out not to change any of the resulting conclusions.

Euclid's "Common Notions" are rules for measuring or comparing things (note that the online guide has found several common notions that Euclid used without listing). Axioms 6 through 8 on page 13 cover the notion of "equality" for the text -- two figures are equal if one can be superimposed on another.