Thinking About Proofs

Date: 09/24/97 at 15:29:27
From: Erika
Subject: PROOFS

How do you know what statement to write next when you're doing a 
proof?  And what are the reasons that you use?

ex. given:  m<4 + m<6 = 180                
    prove:  m<5 = m<6                           K
                                               / \ 
                                              /   \ 
                                             /     \
                                            /       \ 
                                     ____4_/5______6_\
                                           J          L

Date: 09/24/97 at 18:03:11
From: Doctor Pete
Subject: Re: PROOFS

Hi,

Probably the most difficult part of proving something is where to 
start. It takes a lot of practice, and trial and error. Since you are 
more interested in the thought process than the solution, let me tell 
you what went through my mind as I solved your example.

   "Well, I don't need to know anything about the lengths of the 
    triangle, because all I've been given are angles. I remember that 
    the sum of the angles of a triangle is always 180 degrees, but 
    that doesn't help because I don't know what angle 5 is nor do I 
    know what angle JKL is. Well, I know that angle 4 + angle 6 = 180. 
    Ah! But I also know that angle 4 + 5 = 180, because together they 
    form the straight line which contains JL. So if angle 4 + angle 6 
    = 180 = angle 4 + angle 5, then angle 4 + angle 6 = angle 4 + 
    angle 5, and once we subtract angle 4 from both sides of this 
    equation, we get our result, angle 6 = angle 5."

Notice that I tried to rule out some possible ways of proving the 
theorem, based on what I was given (and importantly, what I was not 
given). I then relied on previous theorems or things I already knew 
that were relevant. I applied what I was given in the problem to this 
previous knowledge, and I was able to determine that some of my ideas 
didn't work. Luckily, I recognized that I could apply one of my ideas 
to the solution, and at that point I was able to see that this was the 
right direction to take. The rest was a matter of finishing the 
details.

While this is not always how proofs are made, it gives you an idea of 
the process, which is more or less "educated trial-and-error."  You 
guess, you eliminate possibilities, narrow it down, and push the line 
of reasoning further. Some proofs, like this one, only required one 
"breakthrough" idea, namely, the fact that supplementary angles add to 
180 degrees. However, some proofs require applying many ideas, all in 
the right order. That's why they're difficult, because at each point 
in the proof there are many directions to take. Knowing which is the 
right one is something that takes practice and patience.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/