Theorems and Postulates

Date: 12/02/2006 at 04:05:45
From: Alona
Subject: difference between a theorem and a postulate

If SSS, SAS, and AAS are theorems, why do other books still use them
as postulates?  And can you show me the PROOFS that were used for
these theorems? :)

Sorry, I know you've answered questions about the topic many times but
as I was reading the answers I realized that you were trying to say
that SSS, SAS, and AAS are really theorems because they were proved
(theorems need proofs).  But, why do books and even teachers still
teach students like us SSS, SAS, and AAS "POSTULATES".  Do the words
"theorem" or "postulate" really matter?  I am a high school student
studying geometry right now and your answers to my questions would be 
a great help for me.

Thank you very much.

Date: 12/02/2006 at 23:29:48
From: Doctor Peterson
Subject: Re: difference between a theorem and a postulate

Hi, Alona.

These facts CAN be postulates, but they don't have to be.  It's a
matter of how an author chooses to present geometry to his audience.

Different geometry texts choose different starting points.  The best
way to do geometry is to start with as few assumptions as possible,
and prove everything from those.  Many texts "cheat" a bit by using as
postulates anything they don't want to bother proving (probably
because the proofs are difficult and wouldn't really help their
students understand the subject).  Others use a good, small set of
postulates, but state some theorems without proof, explaining that the
proof is beyond the level of the text.  I prefer the latter approach,
but I can understand the "cheating".

It is possible to take ONE of these congruence facts as a postulate
and prove the others from it (so they become theorems). It is also
possible to define congruence in such a way that all three can be
proved from some more basic postulate about congruence.

You can take them however your own text presents them; but be aware
that they are all really equivalent facts, and which you take as
postulates doesn't affect how you use them, which is what really
matters.  In other words, in answer to your question as to whether
"theorem" or "postulate" really matters: it matters in presenting a
specific systematic treatment of geometry, but not in USING the facts
you learn, which are true one way or another regardless.

See this page for more discussion:

  Theorem or Postulate?
    http://mathforum.org/library/drmath/view/62548.html 

See also:

  The Role of Postulates
    http://mathforum.org/library/drmath/view/62560.html 

If you have any further questions, feel free to write back.


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

Date: 12/10/2006 at 23:07:42
From: Alona
Subject: Proving theorems from theorems

Dear Dr. Math, thank you very much for answering my past question 
about "theorems" and "postulates".  I now know that one of the SSS, 
SAS and ASA theorems can be considered as a postulate.  It just 
depends on the starting point of the discussion.  But they are really 
theorems (I hope I understood it the way you want me to understand 
it).

My question is, is it really possible to prove theorems from theorems?
What I mean is, is it possible to call all the three congruency
theorems "theorems" and still prove each one using each theorem?

In our geometry class, it is possible to prove theorems from previous
theorems.  Then why do we need to assume one of the congruency
theorems as "postulate" when we could really prove it using "theorems".

Thank you very much for your previous reply.  It answered 70% of my 
questions.  These questions are the remaining 30%.  :)

Date: 12/11/2006 at 12:49:58
From: Doctor Peterson
Subject: Re: Proving theorems from theorems

Hi, Alona.

Certainly you can prove a theorem from a theorem; you do it all the 
time, I would think.  You can use any known fact, whether theorem or 
postulate, as the basis for a proof.

What you CAN'T do is prove A from B, and B from C, and C from A! 
Such circular reasoning is not allowed, because you have to start 
with something that is known to be true.  So if you call ALL THREE of 
these "theorems", then at least one of them has to be proved on the 
basis of something else (such as a definition of congruency that is 
more powerful than what elementary texts usually use).

That's why the best approach is to take one of them as a postulate, 
and then prove the others as theorems.  It doesn't matter which one 
you start with, but you have to start with one without assuming 
another is already true.

If you do merely prove each from another of them, then what you have 
done is to show that they are all EQUIVALENT--that is, IF one is true, 
then they all are.  But then either they are all true, OR they are all 
false.  You don't know which!

I think the pages I referred you to answer this question, by 
explaining the role of postulates as starting points. You may want 
to reread them with this new perspective in mind.


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

Date: 12/20/2006 at 01:11:37
From: Alona
Subject: Thank you (Proving theorems from theorems)

Thank you very very very much!!  I totally understood the topic now! :)