This article describes one of the most interesting classroom discussions I ever had in a geometry class. The discussion was based on a homework assignment regarding geometric postulates (also known as axioms).

Many students have a difficult time understanding the difference between axioms and theorems. How is an axiom different from a theorem?

The simple answer is that a theorem is something we "prove", while an axiom is something we accept without proof. In other words, axioms are things we are so utterly convinced of that we don't feel the need to prove them.

After we had a brief discussion on this idea, I gave my students a rather unusual homework assignment. I told them to make a list of several things which they accept as "axiomatic". I explained that these did not necessarily have to be related to mathematics, and of course, they all liked that.

The next day I asked each student to share one or more of their axioms. Their answers were very interesting, and included the following:
 

• The sun will rise tomorrow


• My parents love me


• Even when I can't see them because of the clouds, the sun, moon and stars are still there


• I'm going to get accepted to college
• There is a God


• There is no God


 

The last two items on the list were especially interesting, because on the question of the existence of God, we had students with diametrically opposed "axioms". I asked the students how their differing axioms about God affected the way they lived and acted. This produced some interesting contrasts:

• I do/don't go to church


• I do/don't pray


• This physical existence is all we've got/Life is eternal

It was clear that there were fundamental differences in the belief systems and actions of these students, based on their fundamental difference in axioms.

This provided me the opportunity to discuss Euclidean and Non-Euclidean geometry; Euclid had a set of axioms--things he accepted as true without proving them--and others, such as Gauss and Lobachevsky, had very different axioms, which resulted in different theorems, or proofs.

This exercise is valuable not just from a mathematical standpoint, but also from a general life experience standpoint; it forces students to examine the nature of their own beliefs about the universe, and to come to grips with the fact that their every action and behavior is based on a set of fundamental beliefs that they have accepted without proof. This revelation may shake some of them, but sometimes a little shaking is good.

I concluded the discussion with a quotation from the Bible which clearly defines the nature of an axiom. It is found in Hebrews 11:1.

Now faith is the assurance of things hoped for, the conviction of things not seen.