Choosing the Next Step in a Proof

Date: 11/03/2003 at 18:51:17
From: Stephanie
Subject: How do I know what theorem to use in a proof

When my geometry teacher is teaching proofs, he says to use this 
property or that property.  I don't understand where he gets them 
from.  

How do I know which theorem is the correct one to use next when I'm
trying to prove something?  Is there a list somewhere that I should be
using? 

Date: 11/03/2003 at 22:59:50
From: Doctor Ian
Subject: Re: How do i know what theorem to use in a proof

Hi Stephanie, 

That's sort of like asking: "How do I know what sentence to write next
in an essay?"

A list of "possible sentences" would be enormous.  So would a list of
theorems that you might use in a proof. 

But a list wouldn't really be all that helpful, because it would take
you hours to look through it!  What you really want to be doing is
learning to think about proofs in a certain way.  

For example, suppose I want to prove that the diagonals of a rectangle
have to be equal.  How might I go about that?  

Instead of thinking about theorems directly, I want to think about all
the ways that I might show two things to be equal.  In geometry, a
common way to do that is to show that they're corresponding parts of
congruent shapes.  For example, if I could show that each diagonal is
part of a triangle, and the two triangles are the same, that would
mean that the diagonals are the same.  

Does that make sense?  What I've done now is break one problem into
two smaller problems.  (One:  Find two congruent shapes, each
containing one of the diagonals.  Two: Show that the diagonals are
corresponding parts of the shapes.)  And this is basically how _all_
proofs work, although you don't get to see this when they show up in
books.  

Why not?  Because people _find_ proofs by stumbling blindly around,
following all kinds of false leads, until they find a path that leads
from the premises to the conclusion... or more often, from the
conclusion to the premises.  But they don't want to publish all their
mistakes!  So they tidy them up, and make it look like there's an
obvious way to see what the proof must be. 

Suppose I blindfold you and hide something in a room, then ask you to
find it without taking the blindfold off.  You'd be going all over the
place, groping around, picking up the wrong things, until you finally
found it.  Now, suppose we blindfold someone else, and put them where
you started, and ask you to give directions to the object.  Would you
lead them through the same path you took?  Or would you give them a
nice, simple set of instructions?  

It's the same thing with proofs.  So the short answer to your question
is:  No one knows what the correct theorem to use is, unless they're
already done stumbling around. 

So how do you get better at this?  The only thing that helps, really,
is practice.  It's sort of like getting better at any complicated
skill, like doing crossword puzzles, or hitting a baseball.  There's
only so far that you can get with instructions.  At a certain point,
you just have to sit down and do the thing over and over and over,
letting your brain build up the right associations.  

I hope this helps.  Write back if you'd like to talk more about this,
or anything else.   

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