Unproven Fundamentals of Geometry

Date: 05/18/99 at 00:21:55
From: Han
Subject: The Unproven Fundamentals of Geometry: Postulates, Points, 
Lines, Etc.

Hi, 

I was inspired by some of the answers in your archives to further 
investigate why the fundamentals of geometry are necessarily unproven/ 
undefined. It seems that in every human system of thought discoveries 
and inventions must be built upon faith. Less vaguely, in geometry, 
the most basic unit - the point - cannot be defined. What are some 
other important postulates or axioms that geometry cannot exist 
without, but cannot prove, either?

Date: 05/18/99 at 17:49:01
From: Doctor Rick
Subject: Re: The Unproven Fundamentals of Geometry: Postulates, 
Points, Lines, Etc.

Hi, Han, I like thought-provoking questions like this.

I agree with you about the necessity of faith as it relates to our
knowledge of and interaction with the real world. In math, though, I 
see things a little differently.

Math in itself is not intrinsically connected to the real world. It is
possible, and perfectly okay, to develop a mathematical system that 
doesn't relate to anything in the real world. It is, as you say, 
necessary to have "undefined terms" describing entities in the system, 
and "postulates" (unproven facts relating those entities). But these 
are not so much matters of faith as "rules of the game." They are 
rules that we must adhere to if we are going to prove theorems within 
the particular mathematical system. 

We aren't allowed to introduce additional assumptions (undefined terms 
or postulates) or alter them without explicitly stating the new 
assumptions. When we do so, we are no longer working in the same 
mathematical system. It may be a perfectly valid system, but it isn't 
the same one once its rules have been changed, even the slightest bit.

Many mathematical systems - probably all until the last two centuries 
or so - were motivated by attempts to describe and explain things in 
the real world. At this point, math overlaps with science, and faith 
becomes relevant. Do the undefined terms and postulates of our system 
correspond to elements of the real world and their interactions? We 
can't know. In all likelihood, they don't correspond exactly, but they 
may make a good approximation.

For instance, a "point" in geometry can be thought of as something 
with no length, width, or breadth. Everything in the real world has 
some length, width, and breadth; we can only approximate a point by 
making a dot with the sharpest pencil we can get. (Physicists now 
think that electrons may actually be points, but electrons obey the 
laws of quantum physics, which is rather more complicated than 
ordinary geometry.)

Still, somehow, geometry is very useful in describing the real world, 
even though strictly speaking, it describes things that don't exist in 
the real world.

I said that you can change the rules and come up with a new system. 
Euclid had 5 postulates in his system of geometry. You can see them 
here, along with his undefined terms (he called them "definitions", 
but not all of them are) and "common notions" (actually postulates 
that are more fundamental than geometry):

  Euclid's Elements, Book I (David Joyce)
  http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html   

The fifth postulate was a lot more complicated than the others. It 
wasn't very pretty, but it seemed to be needed in order to prove some 
basic facts about real-world geometry - for instance, that the angles 
in a triangle add up to 180 degrees.

Over the years, people tried to prove the fifth postulate, thinking 
that something so complex must somehow follow from the simpler 
postulates. They failed. In the nineteenth century, mathematicians 
tried a different tack: try changing the postulate, and see what 
happens. They found that they ended up with several varieties of 
"non-Euclidean" geometry that were completely self-consistent, but 
different from Euclid's geometry. Changing the "rules" made a new but 
perfectly good game.

So what do you think happened next? Einstein came along and discovered 
that these non-Euclidean geometries were just the thing to describe 
the real-world interactions of objects with mass - that is, to 
describe gravity. This is a case where the mathematical system was 
invented with no consideration of the real world (and therefore no 
faith element), but it turned out that this system does appear to 
describe the real world.

The experiments to show that Einstein's theory of general relativity 
do describe the real world better than any other mathematical system 
are very tricky; it is still possible that another system would do 
better. We can be absolutely sure that the results of general 
relativity theory follow from its assumptions; the only question is 
whether or not those assumptions match the way the real world is.

Let me put it another way. There are two kinds of truth; I'll call 
them mathematical truth and real-world truth. Mathematical truth means 
that a statement is consistent with the assumptions of a particular 
mathematical system. In a sense, people created that system, and they 
can tell absolutely whether the statement is true within that system.

(However, Kurt Goedel threw a monkeywrench in the works earlier in 
this century. He proved that any sufficiently rich mathematical system 
must include statements that are true but that cannot be proved within 
that system. We can't tell what these statements might be. This is
mind-boggling!)

Real-world truth is of a different order: it means that a statement is
consistent with the particular system that is the real world. There is 
only one real world, and no human created it; no one knows exactly 
what the rules are. Scientists try to make rules that seem to describe 
the real world, but they can't possibly know whether these rules 
really describe everything in the universe.

So yes, faith is necessary, because we did not create the real world, 
so we can't know absolutely what the rules of this system are ... 
unless someone from outside this system - the creator of the system - 
lets us know the rules.

I know I went far beyond answering your question. I hope my references 
to geometry as an example give you an idea of how undefined terms and
postulates work. Thanks for writing.

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