Why Proofs? Definitions and Axioms

Date: 09/16/2001 at 18:49:30
From: Allie
Subject: Geometry, proofs

Why is a proof needed? Why do you think proofs are important in the 
development of a mathematical system such as geometry?

Thank you,
Allie

Date: 09/18/2001 at 17:30:49
From: Doctor Achilles
Subject: Re: Geometry, proofs

Hi Allie,

Thanks for writing to Dr. Math.

Proofs are the foundation of all mathematical systems.  Geometry is 
perhaps the paradigm (or clearest) case of how to build a mathematical 
system on proofs.

To do this, you start with definitions and axioms. (I should note that 
the example definitions and axioms I give below are by no means 
official, they are just samples that I am using to illustrate what a 
definition and an axiom are.)

Definitions are things like "A triangle is a set of three line 
segments joined at the tips" and the like. Definitions are just a way 
to create vocabulary. They don't actually say anything meaningful, 
they just give a word meaning. Basically, it's very cumbersome and 
difficult to write "a set of three line segments joined at the tips" 
over and over again every time you want to refer to such an object.  
So instead you make up a word "triangle" for such an object and save 
yourself ink.

Axioms are things that you claim are true about the world. For 
example, "Given a line and a point not on the line, there is one and 
only one parallel line that crosses that point." This is not simply a 
meaning of a term, it is a substantive sentence about the world: it's 
a claim that something is true. Axioms are usually chosen to be 
obvious things that people won't dispute.

Once you have a set of definitions and axioms you see what "follows" 
from it; i.e. you assume that your axioms are true, and then use logic 
to see what else MUST be true. Basically what mathematics does is show 
that if we accept certain _basic_ facts, then we must also accept 
certain other _derived_ facts. So if you accept the axioms of 
geometry, then you have to accept all the theorems (for example that 
the area of a triangle is one half its base times its height).

Proofs are not just for historic significance. Math grows through new 
proofs. Mathematicians are constantly using proofs to find out new 
things. In a way, math is about pushing the limits of what we MUST 
accept if we choose to accept a few things that seem obvious at first.  
Without proofs there really is no math.

I hope this helps. If you have other questions about this or other 
math 
topics, please write back.

- Doctor Achilles, The Math Forum
  http://mathforum.org/dr.math/