August 16, 2018 Thursday
Bedtime Story
Felix Klein Associates Group Theory to Geometry
From the abstract algebra of polynomials of
the higher orders we have found a link to the group theory.
Now we have a more grueling task ahead.
The coverage of my wider bedtime story line
makes it mandatory to understand how the group theory is related to mathematics
of symmetry.
One possible means of linking the group
theory with the mathematics of symmetry is through the path of geometry; that
is, by finding the connection between the abstract algebra of polynomials to
geometry.
The one person who made a significant
contribution for this to take place was Felix Klein through his Erlangen
program.
What exactly did Klein see in his mind that
was completely new?
Well, you must understand that Euclid and
his geometry ruled the world of mathematics completely till eighteenth century;
Euclidean geometry was, in the minds of mathematicians, the perfect and
flawless system of axiomatic mathematics that accurately described the known nature
and physical data gleaned so far from the experiments.
The perfect geometries of two dimensions
and the three dimensions (solid geometry) had impeccably established themselves
and their axioms were labeled as “self-evident truths”.
Even to cast a bare minimal doubt or
aspersions to their truthfulness amounted to a sacrilege in the eyes of high
priests of mathematical establishment.
But during the first half of the nineteenth
century several mathematicians did begin to question the solidity of these two
geometries.
The greatest danger to the establishment
was from the mathematical physicists because the developing physics required
dimensions of more than three dimensions and the fifth or the parallel
postulate was coming under close scrutiny.
The realization that the fifth postulate was
an assumption with no firm backing resulted in the development of non-Euclidean
geometry.
When Klein was in Göttingen before joining
Erlangen, he published two papers in 1871 titled ‘On the So-Called
Non-Euclidian Geometry’ where he had indirectly shown or rather proved that
non-Euclidean geometry was consistent if and only if Euclidean geometry was.
From the title you can make it that
non-Euclidean geometry was still a new and controversial subject and Felix
Klein was being careful in not to upset the establishment.
Felix Klein to a large extent ended that
controversy raising non-Euclidean to the same pedestal as Euclidean in a most
diplomatic manner, not stating that the Euclidean geometry was based on an
axiom that was disputable, but rather the new type of strange geometry is
consistent if and only if the traditional geometry is.
He actually went further than that.
We shall continue to explore how Felix
Klein linked geometry to group theory in the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
Advertisements
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:
No comments:
Post a Comment