Shape
Geometry is the study of shapes. Here are some
evident truths about shapes in general that Euclid identified a few thousand
years ago (I took these from Wikipedia:
Euclidean Geometry):
(1)
A
straight
line exists from any point to any point.
(2)
For
any finite
straight line, there is a longer one.
(3)
There
is a circle for
every center and distance.
(4)
All
right angles are equal to one another.
(5)
The parallel postulate: If a straight line falling on
two straight lines make the interior angles on the same side less than two
right angles, the two straight lines, if infinite, meet on that side on which
are the angles less than the two right angles.
Each
of these seems true. Axiom (5) is controversial, however. It cannot be derived
from the other four, and mathematicians have been interested in studying
implications of denying or replacing (5). Still, I’ve yet to come across to a
good reason to think that (5) is false. And since (5) strikes me as prima facie
plausible, I’m inclined to accept it.
Oh,
I am aware that this is not a popular view. Some complain that you can’t “construct” the parallel postulate. But I don’t see
what construction has to do with truth. I’m not even sure what construction is
supposed to mean exactly. Do I
construct a line when I press the head of pen down along a piece of paper? Not
if lines are properties. What I do is
move material bits in such a way that they jointly exemplify a shape property, where a line is a part
of a shape. So, I don’t create lines, strictly speaking. Rather, I cause things
to exemplify lines.
An
important question is, what propositions do these
axioms express exactly? To answer that, we need a clear grasp of the primitive
terms. Some would say that the axioms are not expressing propositions about
some abstract realm but are rather just stipulated definitions of the terms, ‘point’ and ‘line’. I don’t understand
this view. Definitions are contingent
truths about words. For example, the proposition that ‘bachelor’ means
‘unmarried man’ is a contingent truth; our ancestors could have stipulated
‘bachelor’ to mean ‘purple giraffe’ instead. But the axioms above are clearly
not like that: they express propositions about shapes and lines, not merely the
words, ‘shape’ and ‘line’. The same propositions could be expressed using
different words, and indeed they are expressed using different words in other
languages. So, I do think the axioms express propositions about point and
lines. The question is, “What are points and lines?”
My
present hypothesis is this: A point is
the property of being extended to degree 0. A line is the property of being
extended to a degree greater than 0. Axiom 1 can, then, be translated as
follows:
(1*) Between any two things that are extended to
degree 0, there is a thing (a spatial relation)
that is extended to a degree greater than 0.
The
other four postulates can be translated accordingly. Axioms for N-dimensional
shapes, where N is greater than three could also be given. If someone knows a
good source on such axioms, let me know.