The Role of Postulates

Date: 03/29/2003 at 16:53:02
From: Julia
Subject: Geometry based on non-proven postulates

I'm in Euclidean Geometry and the teacher said that theorems are 
proven; postulates are not. Why? Who decided what were postulates and 
what were theorems? I asked my teacher if postulates *could* be proven 
and simply weren't, and she said that they couldn't be proven.

This is my current question. Postulates come first, and then theorems 
are formed from those postulates (right?). So the entire geometry is 
based on postulates that weren't and can't be proven. That just 
doesn't seem right to me. Could you explain to me why it's okay that 
they're not proven? 

Thanks a lot. 

Date: 03/31/2003 at 09:48:12
From: Doctor Peterson
Subject: Re: Geometry based on non-proven postulates

Hi, Julia.

The basic answer to your question is that we have to start somewhere.

The essence of mathematics (in the sense the Greeks introduced to the 
world) is to take a small set of fundamental "facts," called 
postulates or axioms, and build up from them a full understanding of 
the objects you are dealing with (whether numbers, shapes, or 
something else entirely) using only logical reasoning such that if 
anyone accepts the postulates, then they must agree with you on the 
rest.

Now, these postulates may be (and were, for the Greeks) basic 
assumptions or observations about the way things really are; or they 
may just be suppositions you make for the sake of imagining something 
with no necessary connection with the real world. In the first case, 
we want to choose as postulates facts that are so "obvious" that no 
one would question them; in the second case, we are free to assume 
whatever we want. In both cases, we want a minimal set of postulates, 
so that we are assuming as little as possible, and can't prove one 
from another.

Euclid's problem was that one of the postulates (the fifth) didn't 
seem simple enough, so people over the centuries tried to prove it 
from the other postulates, rather than be forced to accept something 
that didn't seem immediately obvious. Eventually it was realized that 
there are in fact different kinds of geometry, some of which don't 
follow all of Euclid's postulates; and that you could replace his 
parallel postulate with a contradictory assumption and still have a 
workable system. In particular, spherical geometry - the way things 
work on a sphere, if you think of a "line" as a great circle - is a 
very real example of this, in which parallel lines just don't exist. 
Spherical geometry follows different rules, yet is just as valid as 
plane geometry.

So we have to take as our starting point some postulates that simply 
define the particular mathematical system we are studying. If we take 
a different set of postulates, we get a different system, which may be 
just as useful as the original - and therefore just as "true" - yet 
different in its conclusions. The postulates we choose are the 
connection between the abstract concepts about which we are making 
proofs, and the "real world" ideas that they model (if any). Without 
postulates, we would not have such a connection, and would be 
reasoning about nothing!

Here are a few Dr. Math archive discussions of the role of postulates 
in math:

   Unproven Fundamentals of Geometry
   http://mathforum.org/library/drmath/view/52290.html 

   Understanding Mathematics
   http://mathforum.org/library/drmath/view/52289.html 

   Properties and Postulates
   http://mathforum.org/library/drmath/view/52472.html 

   What is Mathematics?
   http://mathforum.org/library/drmath/view/52350.html 

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

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

Date: 03/31/2003 at 15:49:39
From: Julia
Subject: Thank you (Geometry based on non-proven postulates)

Thanks for your answer. "[...] or they may just be suppositions you 
make for the sake of imagining something with no necessary connection 
with the real world." I never really realized this - thanks. I 
appreciate it.