1. Different versions of Euclid's parallel postulate. In what sense are they
equivalent?
All Euclid’s postulates are fairly “self-evident”, except the fifth, the so-called parallel
postulate, which says (in Heaths English translation): “That, 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 produced indefinitely, meet on that side on which
are the angles less than the two right angles.” We can easily understand that
mathematicians throughout the subsequent history found this rather less than
obvious. An accompanying figure does help to make the statement a little less
obscure:
If the sum of the two marked interior angles is less than two right angles, then the
two lines will eventually meet on the side of these angles, that is, to the right.
We are probably more familiar with the version we learned at school: “Through a
point outside a line we can draw one and only one parallel to the line.” Here, parallel
lines may be defined as lines lying in the same plane, which do not meet, no matter
how far they are “produced” on either “side”.
But there are other versions as well. Here I shall only briefly mention two of them:
1) Given any figure in space, we can draw a similar figure of different size. This
version could be named the postulate of spatial homogeneity of scale. And 2) The
sum of the angles in a triangle equals two right angles. This statement we are used
to prove as a simple theorem, using the Euclidean postulates, but if we assume this
as a postulate, we can prove the parallel postulate, in Euclid's original version or the
"school version".
If you choose this assignment, your task will be to pick some of the versions, and
examine just what is presupposed to make them equivalent, by giving the rigorous
equivalence proofs.
2. Relative consistency proofs for non-Euclidean geometries. (For simplicity, I
propose to delimit this assignment to elliptic and hyperbolic plane geometry.)
This assignment is of course related to the first one, as the non-Euclidean
geometries considered are exactly those (two-dimensional) geometries violating the
parallel postulate while retaining all the other Euclidean postulates (one of them
with a subtle twist to the meaning of the second and the third postulate). Now, using
the "school version" of the parallel postulate, it is clear that it can be violated in two
different ways. We can either postulate that there is no parallel to the given straight
line going through the given point outside it, or that there are more than one such
parallels. It is easy to prove that if there are more than one parallel, there are
infinitely many, so we don't get different geometries with 2, 3, 4 etc. parallels. The
"no parallel" geometry is called elliptic plane geometry, and the "many parallels"
one is called hyperbolic plane geometry.
The "no parallel" and the "many parallel" postulates were entertained by
mathematicians for more than two millenia with the sole purpose of producing a
contradiction. Such a contradiction would be an indirect proof of Euclid's parallel
postulate from his other postulates. As we know, they never succeeded. Eventually,
a few brave souls began suspecting that Euclid was right after all, in the sense that
he really needed his fifth postulate to make his geometry tick. If so, we will never
come up with a contradiction, no matter how ingeneously we search for it. But how
do we prove the impossibility of getting a contradiction from a set of postulates? We
do it by constructing a model satisfying the given set of postulates. The thought
behind this model approach is simply that the model is "real" in some sense, and
reality cannot be contradictory, so if the model satisfies all the postulates, there can
be no contradiction between those postulates. But as mathematicians, and scientists
in general, we are very reluctant to maintain any "metaphysical" assumptions about
what is and is not "real". And as for "reality" being necessarily non-contradictory,
what about the wave-particle duality in quantum physics? Does it constitute a
"contradiction" in "quantum reality" or not? Due to our unwillingness to become
entangled in such philosophical questions, we find ourselves unable to give any
"absolute" consistency proofs. But what we can do, and have done, is constructing
models of non-Euclidean geometries within Euclidean geometry. We take ordinary
Euclidean geometry, redefine the fundamental geometrical concepts, "point",
"straight line", "angle", "circle", "parallel", and we prove, within Euclidean geometry,
that these "new" points, straight lines, circles and parallels satisfy all the Euclidean
postulates except the parallel postulates, plus either the "no parallel" or the "many
parallels" postulate. Such a proof is called a "relative consistency proof" of
respectively elliptic and hyperbolic plane geometry. In effect, we prove that if the
good old Euclidean geometry is free from contradictions, then elliptic and
hyperbolic geometry are also free from contradictions. I think very few people in
their right mind really doubt that Euclidean geometry is free from contradictions,
whereas elliptic and hyperbolic geometry may seem a bit more dubious at first
glance, they do have some rather odd properties. Thus, even relative consistency
proofs give increased credibility to these "new" geometries.
If you choose this assignment, your task wil be to construct at least one Euclidean
model for each of the two non-Euclidean geometriess considered, and prove that
these models satisfy the relevant postulates.
3. Equivalence of different definitions of finite-dimensional vector spaces.
I assume most of you have encountered at least one definition of this concept. But,
as you probably have guessed by now, there are more. The difficulty here is not so
much the equivalence proofs as such, they are fairly easy once you have grasped the
subtle conceptual differences between the definitions.
The briefest definition of our concept is probably this: A finite-dimensional vector
space is a finite-dimensional free module over a field. But this is kind of "cheating",
as most of the "meat" of the concept is contained in the concept "module", which we
would have to know to understand the definition. A more frequently occurring
definition is to regard the n-dimensional vector space as simply the set of ordered ntuples of elements in a fixed field, which in most elementary texts is either R or C
(real or complex numbers). If u and v are such vectors and k is an element in the
field, we define u+v by componentwise addition, and kv by componentwise
multiplication. Alternatively, we could posit an abstract abelian group V and a
mapping from K x V to V satisfying a set of axioms. Here K is the field in question. Or
we could have an abstract abelian group and a set of unary operations on this group
respecting the group structure, define the sum of two operations by (a+b)(v) =
a(v)+b(v), and the product by composition, (ab)(v) = a(b(v)), and demand that the
set of operations with this sum and product satisfy the field axioms.
This assignment is really an exposure to what Saunders MacLane called the
"protean" character of mathematics, or the non-uniqueness of its fundamental
concepts. You may not gain much concrete knowledge from this assignment, but I
feel confident you will increase your mathematical maturity and get a wider
perspective.