I ask students, “What is mathematics?” The answer is frequently, “Numbers.” No! Instead, mathematics is logic. Let us examine geometry for example.

 

Geometry is based upon postulates. A postulate is often incorrectly defined as

“Something taken to be true without proof” (American Heritage Talking Dictionary. Copyright © 1997 The Learning Company, Inc. ).

 

A postulate is also defined as “A proposition regarded as self-evidently true without proof.” This is from Mathworld, http://mathworld.wolfram.com/Axiom.html.

 

The word “axiom” is a slightly archaic synonym for postulate.

 

The correct definition is this. A postulate is an arbitrary statement, consistent with the other postulates of the system.

 

Arbitrary means you can say whatever you please.

 

A mathematical system is a collection of arbitrary (made-up) statements that do not contradict each other.

 

Of course, the statements in different mathematical systems do not have to agree.

 

Mathematics is a human activity. All that we ask from a mathematical system is consistency.

 

SEE THE BOOK FOR MORE DETAILS.