April 12, 2017 Wednesday
Bedtime Story
Fifth Postulate in an Elliptical Geometry
Let me try to explain the discomfiture that the nineteenth century
geometers were feeling.
To empathize with their feelings, you will need to see how the
fifth postulate sounds in the elliptic geometry.
The fifth postulate in an elliptic geometry goes something as
follows:
Through a given point outside a line, no line parallel to it can
be drawn.
Now with such an out of the world axiom, how can one ever be sure
of the consistency of theorems that would be the fall out of it?
Just because so far it has been consistent, how can one ever be
sure that in future it will not give rise to theorems that would be self
contradictory?
Such concerns were genuine and valid given that the theorems that
would arise or were arising would literally unworldly.
So to solve this problem, someone devised the idea of a model or
an interpretation.
For example, the Euclidean geometry is applicable to our space
that we live in.
So our space is the model and as long as the postulates of
Euclidean geometry do not go against the nature of the space, its consistency
is assured.
Though let me tell you, this isn’t a sound way to demonstrate the
consistency of a formal system.
Anyway, this system was tried for the plane elliptic geometry.
This was based on Euclidean geometry wherein the plane of elliptic
geometry was the surface of Euclidean sphere.
The point of Euclidean geometry would become a pair of anti nodal
points on this elliptical surface or sphere.
A straight line of this new geometry would be the great circle of
this Euclidean sphere.
I think you get the idea.
Then it can easily be shown by drawing on the sphere that through
a point on the sphere, no great circle can be drawn parallel to a given great
circle.
Other theorems can perhaps also be shown similarly.
Apparently such a model-based proof sounds pretty neat.
Yet, it has its weakness.
That weakness is that this new system of proof rests on the old
system being full proof.
So in essence, what it claims is that elliptic geometry is
consistent if Euclidean geometry is consistent.
Then the question arises, are the axioms of Euclidean geometry
themselves consistent?
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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