Question

Hi Professor Puzzler, I was wondering if you could explain what an axiom is?

Katie M, Grade 11
South Dakota

Answer

Hi Katie,

Well, I'm guessing that you're taking a Geometry class, because that's when many students first see that term.

An axiom is something that you believe or accept to be true without having ever proven it.  It is something that you consider to be "obviously true." You might not realize it, but your life is filled with axioms - things that have never been proven to you, but you believe anyway.  I once asked a group of high school students to name some things they couldn't prove, but they were confident of anyway.  Here were some of their answers:

• My parents love me.
• The sun will rise tomorrow.
• Gravity keeps me from falling out into space.
• There is a God.
• 1 + 1 = 2.

Why are axioms important? Because when you are proving something, there will always be fundamental building blocks of ideas that your proof is based on, and it's important to remember that the most fundamental building blocks are actually things you haven't proved at all.

In a way, an axiom is sort of like what the Bible says about faith: faith is the substance of things unseen.

Euclid had some axioms (or postulates), and one of his very important axioms was that if you have a line, and a point that's not on the line, there's only one line that goes through the point, and is parallel to the line.  That might seem obvious, but some other mathemeticians like Lobachevsky, Riemman, and Gauss had some different ideas, and their notions of geometry turned out to be very different from Euclid's.

The assumptions you start with can drastically change your results!

In modern usage, the "obviously true" concept is not how axioms are always viewed; in another sense, we could view an axiom as simply a "rule of the game" in mathematics.  In other words, mathematicians might say, "Let's start from assumption X, and see what sort of mathematics we develop!"  The assumption, or axiom they begin with may not be something that's "obviously true," but often valuable results come from such game playing.

Thanks for writing,
Professor Puzzler