July 05, 2018 Thursday
Bedtime Story
'Application to Space' - Part 2
We shall continue tonight with Riemann’s
‘Application to Space’ part of his paper that he had presented to Gauss in 1854
as the first of his lectures or maybe his very first lecture.
Mind you, this paper was never published by
him.
Truth is that this paper which he presented
as a lecture at Göttingen to Gauss among others was only published two years
after his premature death by Dedekind (Riemann passed away in 1866 from
tuberculosis at the age of 39).
Dedekind and Riemann had worked together
for two years at the University of Berlin just about the time Riemann presented
this paper.
“This consideration becomes important in
the extensions of these empirical determinations beyond the limits of
observation to the infinitely great and infinitely small; since the latter may
clearly become more inaccurate beyond the limits of observation, but not the
former.
In the extension of space-construction to
the infinitely great, we must distinguish between unboundedness and infinite
extent, the former belongs to the extent relations, the latter to the
measure-relations.
That space is an unbounded three-fold
manifoldness, is an assumption which is developed by every conception of the
outer world; according to which every instant the region of real perception is
completed and the possible positions of a sought object are constructed, and
which by these applications is for ever confirming itself.
The unboundedness of space possesses in
this way a greater empirical certainty than any external experience.
But its infinite extent by no means follows
from this; on the other hand if we assume independence of bodies from position,
and therefore ascribe to space constant curvature, it must necessarily be
finite provided this curvature has ever so small a positive value.
If we prolong all the geodesics starting in
a given surface-element, we should obtain an unbounded surface of constant
curvature, i.e., a surface which is a flat manifoldness of three dimensions
would take the form of a sphere, and consequently be finite.
§ 3. The questions about the infinitely
great are for the interpretation of nature useless questions.
But this is not the case with the questions
about the infinitely small.
It is upon the exactness with which we
follow phenomena into the infinitely small that our knowledge of their causal
relations essentially depends.
The progress of recent centuries in the
knowledge of mechanics depends almost entirely on the exactness of the
construction which has become possible through the invention of the
infinitesimal calculus, and through the simple principles discovered by
Archimedes, Galileo and Newton, and used by modern physics.
But in the natural sciences which are still
in want of simple principles for such constructions, we seek to discover the
causal relations by following the phenomena into great minuteness, so far as
the microscope permits.”
We shall continue with Riemann’s paper 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