Why Does Math Need Proofs?

Date: 03/24/2000 at 18:22:41
From: Lesa
Subject: Proofs

Why does math need to have proofs? I don't understand the importance 
of them. Please explain.

Thank you for your time,
Lesa Schultz

Date: 03/25/2000 at 22:55:45
From: Doctor Peterson
Subject: Re: Proofs

Hi, Lesa.

We need proofs in math, first, because we want to be sure that what we 
do is right. There are enough sources of error in our calculations, 
from imprecise measurement to misunderstanding of the formulas we 
should use, that it's important to make sure that our thinking doesn't 
add more error. Proof just means checking our reasoning.

In math, unlike science or any other field, we CAN prove that what we 
do is absolutely right. That's because math is not dependent on 
partially known physical laws or unpredictable human behavior, but 
simply on reason. In math, unlike the real world, we set the rules, so 
we can know everything we need to know in order to be certain what 
will happen. For example, we can define what we mean by addition, and 
then prove that if we add b + a we will always get the same as a + b. 
Since we can do it, we should take advantage of the possibility. Truth 
is rare enough to value highly.

But the reason we really HAVE to prove things is that we can be easily 
fooled. Some things that seem perfectly reasonable turn out to be 
wrong. In fact, even if something is true whenever we try it, that 
isn't enough to be sure that it always will be. The reason we can be 
sure that a + b = b + a, for example, is not that we've always seen it 
work that way, but that we can understand what is happening when we 
add, and know that this rule is a natural result of the way addition 
works. Often I see students trying to find a formula for some relation 
(say, the number of diagonals in a polygon) by making a table and 
looking for a pattern. Sometimes they find a formula that works for 
the numbers they have; but if you add a line to the table, the formula 
will no longer work. What they have to do is go back to the way the 
table is made and see how a pattern will develop naturally. That's a 
proof, and when you've done that, you KNOW it's right. You don't need 
to guess and risk being fooled by a false pattern.

The Greeks were, as far as I know, the first to develop this love of 
certainty. They saw that math was not just a tool they could use, but 
a way to build a world of absolute truth, building one fact on another 
so that they knew they were right. But they weren't perfect. They 
originally built large parts of their geometrical thinking on the 
assumption that any two lines could be compared by finding some unit 
small enough that both lengths were whole-number multiples of that 
unit; that is, all lines were assumed to be "commensurable." But it 
was discovered that the diagonal of a square was incommensurable with 
the side of the square - that is, the square root of two was 
irrational. That shook them, and forced them to rethink their proofs, 
since a lot of what they knew was based on a false assumption. They 
were able to rebuild their geometry (Euclid's geometry incorporated 
this rethinking) and as far as I know, nothing turned out to be wrong; 
but the incident reinforced mathematicians' awareness of the 
importance of really proving everything.

On the other hand, we can also be fooled in the other direction: there 
are some things that are hard to believe without seeing a proof. For 
example, I think it's hard to believe that the Pythagorean theorem 
should always be true. I need a proof to convince me that I can always 
use it and it will always work.

A different kind of proof can be useful in saving effort: the 
existence proof. Sometimes it can take a lot of work to solve a 
problem; a mathematician may first be able to prove whether a solution 
exists, without having to do all the work of finding it. That can 
either save us from bothering to try it, or allow us to work in 
confidence, knowing there is an answer.

Not only mathematicians, but you yourself can benefit from learning to 
do proofs. The skills you develop in learning to prove mathematical 
statements are useful in many other areas of life. You learn logic, 
which lets you recognize when a supposed "proof" (whether in math or 
life) is flawed and shouldn't be believed. See our FAQ section on 
False Proofs: http://mathforum.org/dr.math/faq/faq.false.proof.html   . 
You learn how to reason carefully and find links between facts. I 
myself am a computer programmer, and though I don't prove theorems all 
the time, I'm often checking whether a program will do what I expect, 
and using those logical skills. Other people, from lawyers to 
consumers, need to use logic in all sorts of ways.

You can find some other answers to your question by looking in our FAQ 
under 

   Why study math?
   http://mathforum.org/dr.math/faq/faq.why.math.html   

and 

   Proofs
   http://mathforum.org/dr.math/faq/faq.proof.html    

The latter includes this link, which is worth looking at:

   Proofs in Mathematics, Bogomolny
   http://www.cut-the-knot.org/proofs/   

If I haven't fully answered your question, feel free to write back.

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