Students need to be reminded about the basis of mathematics.

 

A mathematical system is a collection of axioms and postulates, which are arbitrary self-consistent statements.

 

Axioms and postulates are the same. Theorems are logical consequences of the postulates. Corollaries are theorems that are very easy to prove.

 

Truth in mathematics means logical consistency. Contrast this with truth in science, which means experimental verification or observation. Reminding them of this will help them stay on their toes, as they have to be self-critical, and cannot rely upon authority.

 

The teacher can encourage this behavior by asking, “Do you think this is correct?” If they reply that they are not sure, then tell them that they have to be sure one way or the other, and be able to defend it.

 

This is very different from politics. Each side gives arguments, responds to the other side’s comments, but does not have to be completely consistent. If we find the slightest inconsistency, if we can prove two theorems that say different things, the mathematical system is not valid.