July 06, 2018 Friday
Bedtime Story
'Application to Space' - Part 3
Tonight once again we continue with
Riemann’s paper or rather the lecture of the year 1854 made in Göttingen.
“Questions about measure-relations of space
in the infinitely small are not therefore superfluous questions.
If we suppose that bodies exist
independently of position, the curvature is everywhere constant, and it then
results from astronomical measurements that it cannot be different from zero;
or at any rate its reciprocal must be an area in comparison with which the
range of our telescopes may be neglected.
But if this independence of bodies from
position does not exist, we cannot draw conclusions from metric relations of
the great, to those of the infinitely small; in that case the curvature at each
point may have an arbitrary value in three directions, provided that the total
curvature of every measurable portion of space does not differ sensibly from
zero.
Still more complicated relations may exist
if we no longer suppose the linear element expressible as the square root of a
quadratic differential.
Now it seems that the empirical notions on
which the metrical determinations of space are founded, the notion of a solid
body and of a ray of light, cease to be valid for the infinitely small.
We are therefore quite at liberty to suppose
that the metric relations of space in the infinitely small do not conform to
the hypotheses of geometry; and we ought in fact to suppose it, if we can
thereby obtain a simpler explanation of phenomena.
The question of the validity of the
hypotheses of geometry in the infinitely small is bound up with the question of
the ground of the metric relations of space.
In this last question, which we may still
regard as belonging to the doctrine of space, is found the application of the
remark made above; that in a discrete manifoldness, the ground of its metric
relations is given in the notion of it, while in a continuous manifoldness,
this ground must come from outside.
Either therefore the reality which
underlies space must form a discrete manifoldness, or we must seek the ground
of its metric relations outside it, in binding forces which act upon it.
The answer to these questions can only be
got by starting from the conception of phenomena which has hitherto been
justified by experience, and which Newton assumed as a foundation, and by
making in this conception the successive changes required by facts which it
cannot explain.
Researches starting from general notion,
like the investigation we have just made, can only be useful in preventing this
work from being hampered by too narrow views, and progress in knowledge of the
interdependence of things from being checked by traditional prejudices.
This leads us into the domain of another
science, of physics, into which the object of this work does not allow us to go
today.”
What an inaugural lecture for a young
mathematician to deliver!
Tomorrow we shall do a moderate analysis of
the paper and the kind of reception that it got.
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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