Theorem or Postulate?

Date: 11/03/2002 at 17:36:59
From: Maureen Hamilton
Subject: Geometry textbook inconsistencies

I am homeschooling my daughter in math. I want to teach her proofs, 
which have just been removed from the high school curriculum. My 
problem is that there seems to be a great discrepancy between 
definitions, postulates, and theorems, from textbook to textbook. What 
is a postulate in one book is a theorem in the next, and vice versa.  

Is there a textbook that faithfully follows Euclid's Elements?  I have 
the translation, but the language is quite formal and of course there 
aren't any problems to work out.  Your help would be most appreciated 
in this matter.

Date: 11/04/2002 at 08:59:07
From: Doctor Peterson
Subject: Re: Geometry textbook inconsistencies

Hi, Maureen.

It's important to be aware that there is no one "correct"
axiomatization of geometry; a number of different schemes have been 
developed by reputable mathematicians, not to mention by textbook 
authors who are trying to keep things simple for students. Certainly 
the variety of postulates used in texts makes it hard for me as a 
Math Doctor to answer questions about proofs, since I have to ask 
what facts are available to the student; but in a sense that helps in 
our mission, since our goal is to help in understanding, not to give 
specific answers. By making the student look for whatever postulate 
or theorem in their book corresponds to a fact I mention, I help them 
learn how to put ideas together.

That's relevant to you, because your goal is not to teach a particular 
"correct" set of postulates, but to teach reasoning (using whatever 
postulates and theorems are available) and geometric facts (all of 
which are agreed upon, even if we don't agree on which to call 
postulates and which to call theorems, or on what the theorems are 
ultimately based on). The educational results don't really depend 
on the details of the system used.

It is definitely not true that Euclid is the one right way to do 
geometry. Despite the centuries when the Elements were treated as the 
Bible of geometry, there are many flaws in his treatment, and many 
new ideas have been introduced since then. For example, Euclid does 
not use the concept of "congruence" as we understand it today. So 
although going through Euclid can be a very enlightening experience, 
it need not (and probably should not) be a student's first exposure 
to axiomatic geometry.

So I would recommend not looking for the text closest to Euclid, but 
for the text that best presents the concepts and demonstrates how 
proofs work.

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

Date: 11/11/2002 at 10:06:51
From: Maureen Hamilton
Subject: Geometry textbook inconsistencies

Thank you for your response. However, I have many more questions. In 
all the textbooks that I am perusing I notice that there are three
triangle postulates - SSS, ASA, and SAS. Shouldn't these be theorems?
Are they postulates because the proof is beyond the scope of the high
school student, or are they postulates because they cannot be proved?

Date: 11/11/2002 at 12:08:32
From: Doctor Peterson
Subject: Re: Geometry textbook inconsistencies

Hi, Maureen.

As I said before, there are different axiomatizations of geometry. 
Some of them prove SSS, SAS, and ASA as theorems, as Euclid does; to 
do so properly they need a better definition of congruence than 
Euclid's concept of "superposition," with a clear set of axioms 
governing the motions that are allowed when one shape is put on top 
of another. Hilbert's famous axiomatic system takes SAS as a 
postulate, and derives SSS and ASA from it as theorems. One textbook 
I have makes ASA and SAS postulates, but proves SSS. Many texts seem 
to make all three theorems, probably, as you suggest, in order to 
avoid confusing students with a rigorous proof when they are not ready 
for such details. (I would prefer to present a good set of postulates 
and just say honestly that we are skipping the proof of some of these 
theorems because they are too difficult to go into yet.) In each case, 
whatever is chosen as a postulate is such because it can't be proved 
_from the other postulates that have been chosen for that 
presentation_. Some of these systems are clearly deficient, but it can 
be hard to say which is best.

Here are some discussions of Euclid's, Hilbert's, and other systems 
that I have found:

   Euclid's Postulates - Canadian Society for Biomechanics
   http://www.health.uottawa.ca/biomech/csb/laws/euclid.htm 

   Lecture Notes: The Axioms of Euclid and Hilbert, and the Parallel 
   Postulate (PDF) - Jeff Connor
   http://www.math.ohiou.edu/~connor/geometry/chap3/betweenlct.pdf 

   Axiomatic Systems for Geometry - George Francis (PDF)
   http://www.math.uiuc.edu/~gfrancis/M302/handouts/postulates.pdf 

   Teaching Geometry According to Euclid - Robin Hartshorne (PDF)
   http://www.ams.org/notices/200004/fea-hartshorne.pdf 

   Euclid's Elements, Book I, Proposition 4 - David Joyce
   http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI4.html 

   Congruence Of Triangles - Jim Loy
   http://www.jimloy.com/geometry/congruen.htm 

It is true that many of the first theorems students are asked to 
prove are so obvious that they seem not worth proving; but that is 
only because they are deliberately easy, and are little more than a 
concatenation of postulates and definitions. That may well leave some 
students wondering why we need proofs, when they are so trivial. They 
need to be introduced early to a surprising proof, even if they can't 
follow it all yet, just so they can see the value of proofs. And they 
should also be shown a false proof, so they can see the need for care 
in each step, and will not think that the "obvious" is always true.

The goal of an axiomatic system is to reduce the number of 
assumptions we make to a minimum (as far as possible) so that all our 
reasoning is based on readily accepted 'facts'. So postulates should 
be as 'obvious' as possible; yet the fact that something seems obvious 
is not enough to make it a postulate, since it may be provable from 
existing postulates, so that it would be redundant. On the other hand, 
it is not required that we prove a set of postulates is minimal in 
order to use them. As the last link I gave above mentions, SSS, SAS, 
and ASA are all equivalent postulates, so that only one of them need 
be postulated; but it is common to make them all postulates, and that 
is not illegal, just unnecessary. The problem is that this produces a 
bloated set of postulates and gives a false sense that geometry has to 
make unnatural assumptions.

You will note that although Euclid's fifth postulate is such because 
it could not be proved from his other postulates, there have been 
many alternative ways to phrase it in order to make it seem less 
arbitrary. Any of those versions is a valid postulate. Math is at 
root a somewhat arbitrary endeavor; there are many ways to choose 
starting points for the same field. Yet ultimately it makes no 
difference; who cares whether SAS is a postulate or a theorem, when 
you need to use it in a later proof? You just write SAS and know that 
it is true, one way or the other. So although the variations in 
postulates will make a difference in the details of a student's work, 
and might cause some confusion if he moves to a different text in the 
middle of a course, none of the important things is affected: all the 
same facts are true, and the importance of proof is still being 
demonstrated.

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

Date: 03/19/2003 at 14:21:43
From: Rahul 
Subject: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES?

Dear Dr. Math,

I see in my textbook that SAS is a postulate and others (ASA, SSS...) 
are theorems. I know that using SAS you can prove the others; 
therefore they are theorems but SAS is a postulate.

But in some books SAS is named as a theorem and proved using 
superposition.

I read on the Internet that Euclid first proved these using 
superposition, but he was not happy with his proof as he tried to 
avoid superposition in other situations. And now superposition is not 
legal as it contains complicated assumptions.

In the Dr. Math archives I find some doctors saying that SAS, SSS, 
ASA, etc. are postulates and cannot be proved, but some of the math 
doctors still refer to them as theorems. Which are they?

Thanking you,
Rahul Radhakrishnan.

Date: 03/19/2003 at 17:03:06
From: Doctor Peterson
Subject: Re: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES?

Hi, Rahul.

One man's theorem can be another's postulate. This is because we can 
choose different starting points for our development of mathematics, 
and often we find that two different facts can be proved from one 
another, so that it makes no difference which we take to be more 
fundamental. In our own answers, since we don't know the particular 
approach being taken by a student's text, we often use the word 
"theorem" to refer to any fact that is known to mathematicians, 
whether proven or taken as an axiom. In the best systems, no more 
than one of the congruence "theorems" needs to be a postulate; but it 
doesn't really matter which one that is! In some system, any of them 
can be a theorem.

You will be especially interested in the last of the links I gave 
above:

   Congruence Of Triangles - Jim Loy
   http://www.jimloy.com/geometry/congruen.htm 

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

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

Date: 03/23/2003 at 08:02:53
From: Rahul 
Subject: Re: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES? 

Dear Dr. Peterson,

Thank you very much for your prompt and lengthy reply. I understand 
that all three congruence theorems (or postulates or whatever we call 
them) can be considered postulates though it is unnecessary. By 
considering one as a postulate we can prove others. So far I am clear.
   
But if I say all three are theorems, am I right? I mean, is there any 
way of proving them without considering one a postulate (except 
superposition)?
   
Second, is superposition valid now? Is it legal?

Thanking you,
Rahul Radhakrishnan

Date: 03/23/2003 at 23:15:52
From: Doctor Peterson
Subject: Re: Are SSS, SAS and ASA Congruences THEOREMS or POSTULATES? 

Hi, Rahul.

In order to turn all three rules into theorems, in the sense of being 
proved from postulates, you need something equivalent to superposition 
on which to base them. I would not say this is "illegal" in general; 
it simply is not used in the most popular current presentations of 
Euclidean geometry. As I said before, there is no one "correct" system 
of postulates, but many alternatives that can be chosen; and with a 
careful definition of the motions allowed when you move one figure 
onto another, as in transformational geometry, I think there is no 
reason not to turn superposition into a clearly defined concept on 
which congruence can be based.

Here are some references to the validity of superposition in 
transformational geometry:

   Summary of Axioms of Congruence - Draga Vidakovic, 
     Georgia State University
   http://www.cs.gsu.edu/~matdnv/Math6371/ReadingNotes/ch3congr/congruence.html
  
   Euclid did not assume SAS (Axiom 6) as an axiom but tried to
   prove it as a theorem. His proof involved taking one triangle and
   theoretically placing it on top of another. Because he had never
   stated an axiom that allows figures to be moved without changing
   their size and shape (a concept called superposition), the proof
   was invalid. Later on, transformational geometries would be
   created that would allow this proof to be valid. However, based
   on the given axioms, it is not considered a valid proof. 

   Geometric Congruence - Eric Weisstein, World of Mathematics
   http://mathworld.wolfram.com/GeometricCongruence.html 

   Two geometric figures are said to exhibit geometric congruence
   (or "be geometrically congruent") iff one can be transformed
   into the other by an isometry (Coxeter and Greitzer 1967, p. 80).

This idea of transformation by an isometry (movement that preserves 
distances) is essentially "superposition".

   Analytic Foundations of Geometry - Robert S. Wilson
   http://www.sonoma.edu/users/w/wilsonst/Papers/Geometry/default.html

The above is an example of a development of geometry from this 
perspective; look in section 6 for the definition of congruence, and 
a proof of SAS, SSS, and ASA based on carefully defined superposition.

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

Date: 03/28/2003 at 06:47:43
From: Rahul 
Subject: Thank you (Are SSS, SAS and ASA Congruences THEOREMS or
POSTULATES?)

Dear Dr. Peterson,

Thank you very very much for your prompt replies to my questions. It 
has helped me so much. The Math Forum is a very useful site. I thank 
Drexel University for creating such a marvelous site.

Rahul Radhakrishnan