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### Are Different Proofs of a Theorem Really the Same?

```Date: 07/05/2006 at 20:42:28
From: Eric
Subject: Are all mathematical proofs basically the same

When I was in Junior High School I remember being surprised to find
out that there are many different proofs of the Pythagorean Theorem.
I remember wondering if all of those proofs are in some way really the
same?

A few nights ago it occurred to me that there is a better way to ask
this question.  If you have a mathematical system with several axioms
(call them A, B, C, D, E and F), is it possible to have two proofs of
a theorem in this system where one proof uses only axioms A, B and C
and the other proof uses only axioms D, E and F?  In other words is
it possible for two proofs to use no common axioms?

```

```
Date: 07/05/2006 at 22:43:58
From: Doctor Tom
Subject: Re: Are all mathematical proofs basically the same

Hi Eric -

Very nice question!  The study of this sort of thing is called
"metamathematics", or "foundations of mathematics", and is incredibly
deep and interesting.

Let's take a simple example.  Suppose we have a system that consists
of objects and an operator * on those objects, and contains the
following two axioms (and perhaps others):

A) There exists an identity element e in the system such that if a
is any object in the system, e*a = a*e = a.

B) The system contains at least two objects.

Here's my "theorem":

There is at least one object in the system.

Clearly this can be proven from A alone or from B alone, and clearly
axioms A and B are independent.

Now this seems trivial, but it satisfies your conditions, and it's
possible to construct more interesting axiom systems and theorems that

So the answer is no--not all proofs are essentially equivalent.  A
much more common situation is that a theorem can be proved starting
from different subsets (possibly overlapping) of the axioms, but if
the axioms are independent, meaning that none of them is a consequence
of the fact that some subset of others forces it to be true, then the
two proofs are obviously not equivalent.

- Doctor Tom, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Logic
High School Logic

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