Lecture 1: Introduction and Axioms

What is Mathematics at its Core?

Mathematics can be understood in several different ways:

• Manipulation of numbers and variables: This is the computational aspect that most people first encounter in school.
• Extending logic to its limits (and sometimes beyond...): Mathematics pushes reasoning to its absolute boundaries, sometimes revealing paradoxes and surprising results.
• Abstract representation of the real world: Mathematical models help us understand and predict natural phenomena.
• Search for / discovery (invention?) of Universal Absolute Truths: This is perhaps the most profound aspect—mathematics seems to uncover truths that are independent of human culture, time, or even physical reality.

YBC 7289: The Ancient Clay Tablet

The YBC 7289 Cuneiform Tablet

This Babylonian clay tablet from ~1800-1600 BCE shows a square with its diagonal

The Mystery: Imagine you find this ball of clay in the ground, dated to approximately 2000 BCE. What is it, and what does it mean?

The Babylonians and Mesopotamians used a base-60 number system. (They're the reason why today there are 60 minutes in an hour, and 60 seconds in a minute!) The numbers on this tablet, when converted to our decimal system, give us:

$$1 + \frac{24}{60} + \frac{51}{3600} + \frac{10}{60^3} \approx 1.414212$$

This approximation of \(\sqrt{2} \approx 1.414213\) is accurate to within one part in a million—far more precision than would ever be needed for any engineering or economic application of that era.

This extraordinary precision distinguishes this object as pure mathematics rather than engineering. It represents knowledge pursued for its own sake, with no practical application.

Why √2? The diagram on the tablet shows the diagonal of a unit square. If we use the side of the square as our unit of measurement, then by what we now call the Pythagorean Theorem (Pythagoras lived around 500 BCE, over 1000 years after this tablet was made...), the diagonal has length \(\sqrt{1^2 + 1^2} = \sqrt{2}\).

This tablet now lives in New Haven, CT at Yale University. "YBC" stands for the Yale Babylonian Collection.

Plimpton 322: Pythagorean Triples

The Plimpton 322 Tablet

Another ancient object from ~2000 BCE, containing large integer solutions to the Pythagorean equation x² + y² = z²

This remarkable tablet (which now lives in NYC at Columbia University) contains sophisticated integer solutions to the Pythagorean equation \(x^2 + y^2 = z^2\). The Babylonians had discovered what we now call Pythagorean triples (again, a millennium before Pythagoras), including some with very large numbers that would have required considerable mathematical sophistication to calculate.

What these ancient objects are "missing" to be Mathematics (as we understand it today) is: PROOF.

The Babylonians had discovered mathematical relationships and could calculate with impressive accuracy, but they worked primarily through observation and computation. They found patterns and verified them through examples, but they did not yet have the concept of proving that these relationships must always hold.

Why did people eventually come to feel that proof was a necessary component of Understanding?

Thales's Theorem: The First Proof

The interactive above shows a circle centered at A, with diameter BD, and a random point C on the circle.

The Question: How does the angle ∠BCD depend on the position of C? What happens as C moves around the circle?

Observation: Every time we measure the angle, we find it equals 90°! But why does that happen?

If we were ancient Babylonians, simply stating this observed fact might have been sufficient. But for true mathematical understanding, this feels inadequate. We need to know why this is always true.

To the rescue comes Thales of Miletus (c. 624-546 BCE), with the earliest (currently known) proof:

The Need for Axioms

This absurd "proof" illustrates a crucial point: our entire mathematical edifice could collapse if we unknowingly built it upon false foundations. The whole house of cards comes tumbling down if any link in our infinite chain of reasoning contains a false statement.

The Solution: We must start with something that we consciously and conspicuously declare to be true, without giving proof. These foundational statements are called axioms or postulates.

Every mathematical discovery now requires what we might call a "sacred triplet": (Axioms on which the theory is being built, Fact to be established, Proof).

Euclid's Elements

What document represents the ancient epitome of this axiomatic ideal? The answer is Euclid's Elements, written around 300 BCE.

Euclid's Elements is perhaps the most influential mathematics textbook ever written. For over 2000 years, it served as the primary textbook for teaching geometry, and it established the template for how mathematics should be organized: starting with clear definitions and axioms, then building up complex theorems through logical deduction. Until perhaps recently, it was the second most published book of all time, after... The Bible.

You can explore the complete text at David Joyce's edition of Euclid's Elements. Let's dive in!

Euclid's Definitions

Euclid begins by defining the basic objects of geometry; let's look at some of them. Notice how these definitions attempt to capture our intuitive understanding of geometric concepts:

In other words, a point has no size—no length, width, or thickness. It represents pure position.

Note: By "line," Euclid means what we would call a "curve"—any one-dimensional path with length but no width.

This poetic definition tries to capture the idea that a straight line has the same direction at every point.

Why doesn't Euclid just say: a right angle is one measuring 90°? To do that, he would have to explain what it means to measure an angle! To avoid this, he has a clever idea: if two straight lines cross so that adjacent angles are equal, then that's what it will mean for the lines to be perpendicular, meeting at a right angle.

That is, a circle is the set of all points at equal distance from a center point.

Parallel lines are simply ones which never intersect, no matter how far they are extended.

Euclid's Postulates (Axioms)

After establishing definitions, Euclid states his postulates—the fundamental assumptions upon which all geometric reasoning will be based:

This just says that you can use a straight-edge to connect two points with a line segment.

In other words, if you've already drawn a line segment on the page, you can make it a little longer in either direction; this, too, would be accomplished with a straightedge.

This is the "compass postulate" — Euclid is stating that you can draw a circle centered at any point with any radius.

Huh? Why is this postulate needed at all? Isn't it obvious that 90° = 90°? Look back at Definition 10: it is a local statement, about two lines intersecting perpendicularly. When two other lines are also perpendicular, maybe their right angle measures some other amount? This postulate ensures that the concept of a right angle is a global, universal quantity, the same whereever it may be measured.

Wait, what?! This last one is a "Postulate"? Really? An obvious fact that requires no further justification?? Read again what it's saying: if two lines are crossed by a transversal, so that the sum of interior angles on one side add up to less than two right angles,

then if you extend those lines on that side, they will eventually intersect (and hence not be parallel). It's the most controversial of Euclid's postulates and thousands of years later led to the development of non-Euclidean geometries... (More on this later.)