July 07, 2018 Saturday
Bedtime Story
Why Riemann's 1854 Lecture Excited Gauss
The reception that the inaugural lecture of
Bernhard Riemann at Göttingen received is a study in contrast.
After most dissertations or lectures of
this sort that were presented to Gauss, he would often tell the speakers or
defenders that he himself had thought about the problem and had worked on it.
But to Riemann it was nothing of this sort;
Gauss not only was left astonished but excited as well after comprehending the
depth of Riemann’s ideas.
In his official report of the thesis, Gauss
praised Riemann lavishly mainly on the point of originality of mathematical
ideas.
This lecture was the launching pad of what
in future would go on to become Riemannian geometry where he delivered a very
broad and abstract generalization of the differential geometry of surfaces in R3 though it went largely
ignored for three decades in both the world of mathematics and mathematical
physics.
Differential geometry of surfaces is one of
those branches of mathematics that lies outside the domain of visual
imagination of most human apes though these days there are very good (and
rather colorful) computer simulations of Riemann manifolds that provides some
kind of visual scaffold for an interested ape to cling on to.
But for most parts it needs the language of
mathematics to make any sense out of it or to develop it further.
One of the reasons why Gauss immediately grasped
the formidable ideas of Riemann was the fact that he himself had selected the
topic of Riemann’s habilitation lecture or thesis which in German is called
Habilitationsschrift.
Yet the bigger and chief reason why Gauss
so fascinated with Riemann’s lecture was that Riemann was expounding the very
ideas that had struck Gauss himself but he never had actually published any of
it.
Now how can I make a claim as profound as
this when there is no evidence in print to justify it?
Well, in 1832 a Hungarian mathematical genius
János Bolyai published a treatise on a complete system on non-Euclidean
geometry which served as a mere appendix to a textbook of mathematic published
by his father Farkas Bolyai (who actually dissuaded his son not to venture into
any geometry that contradicted that of Euclid).
Farkas was a very good friend of Gauss and
he showed his work and the appendix to the book.
On reading the appendix Gauss wrote to his
friend, “I regard this young geometer Bolyai as a genius of the first order.
To praise it (the work) would amount to
praising myself.
For the entire content of the work…coincides
almost exactly with my own meditations which have occupied my mind for the past
thirty or thirty-five years.”
This remark clearly shows that Gauss was
also thinking of a new kind of geometry that went against the basic tenants of
Euclid, or at least the fifth postulate.
But that fact that Gauss was excited after
the lecture makes me believe that Riemann had ventured into space (quite
literally) that even Gauss had never contemplated.
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.
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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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