I gave it a read.  Very interesting.  I learned some notational symbols.  If I were to rewrite the article myself though, it would be much longer because I would try to set up more basic ideas that say what can be done in a proof.  What is math and what is obvious and what is an axiom.  How certain axioms can be used for further theorms.
There seems to be some "common sense" thrown into mathematical proofs and it really bothers me.
If you are going to do a real proof and call it a proof, then start with an environment that talks about
how you manipulate ideas first.

I only skimmed over it, but here's what I think.

Our ability to recognize even and odd numbers is our instictive ability to recognize divisibility by 2.  Even though it is instinctive and we don't actually think it, it's what's going on.  So we should use the mathimatical definition for divisible by 2.  This eliminates the first few pages of dicussion.

But maybe that's just me.

Patrick, the first two pages focus solely on the definition of even and odd.  This is what I was referring to.

Patrick wrote:

I guess the "common sense" you're talking about is that we take even/odd numbers as a fact without thinking it over first?

Well what I'm getting at is first of all maybe I've missed the courses in the subject matter I am referring to, but is there some subject matter that provides an environment to manipulate ideas in a proof.  Sort of, an analogy would be a computer program, you have to stick to its syntax, but ofcourse this environment would be more human probably and allow for different types of logic and inferences, and they could be categorized.  So perhaps 20 or 30 or 50 different types of thinking could be outlined with examples for starters to try to figure out if we can even agree on what thinking is, and what can be taken as simply given or common sense, and what cannot.

So if there is any written material on this subject, I would love to read it.  Does anyone know??
It would be like proof-environment or some other name.  And perhaps when leaps of genius appear in someones proof, the environment may have to change to accommodate a new type of thinking that has evolved if it doesn't fit into the existing framework.

I'm not entirely what you mean John, but I think you're talking about the following.  There are different ways to prove things:

We wish to prove that if p is true, then q is true.

Direct: Assume that p is true, and follow mathimatical steps till you arrive at q.

Contrapositive: Assume that q is not true, and follow mathimatical steps till you arrive at not p.  Since if q is false, then p is false, it must be that when p is true, q is true.

Contradiction: Assume p and not q.  Arrive at a statement you have already proved to be false.  Then it must be that if p is true, q is true.

Induction: This is a bit different.  Assume that a_0 is true.  Then show that if a_n is true (for any n), then it must be that a_n+1 is true.  Since a_0 is true, it must be that a_1 is true, and so it must be that a_2 is true, and so it must be that...  and so on.  As it seems, induction can only be used in a very limited number of proofs.

Is this what you mean?

Ricky wrote:

bla bla bla
Is this what you mean?

Yes!  That is what I am talking about!  And also something called "proof theory", which is way over my head I just started reading about.

Great analogy, George.  I really like it.