SUO: Manifolds Of Sensuous Impressions (MOSI's)
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Stand! Unfold! Ontologists! --
Let me read you a story from one
of my favored books of manifolds:
| Serge Lang, 'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
In presenting this text I am obligated to change
many Greek characters into Latin letters, and so
by way of a slightly skewed form of compensation,
I will convert Roman numerals to Arabic decimals.
Notes from the translator [me] will be placed in
square brackets, to ease the transits to English.
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Chapt 2. Manifolds
Starting with open subsets of Banach spaces [think R^n for the moment],
one can glue them together with 'C^p'-isomorphisms [bijective mappings
that are continuously differentiable up to at least as far as order p].
The result is called a manifold. We begin by giving the formal definition.
We then make manifolds into a category, and discuss special types of morphisms.
We define the tangent space at each point, and apply the criteria following
the inverse function theorem to get a local splitting of a manifold when
the tangent space splits at a point.
We shall wait until the next chapter to give a manifold structure
to the union of all the tangent spaces.
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To be continued, and I dare say it, to be differentiated,
up to some order as yet to be predestinately determinate.
Note to critics who may happen to follow the style sheet
of the APA ("American Pedantical Association"). The "we"
that you see prevailing in this mannerism of mathematical
writing is not of necessity the "we" of plural authorship,
and of necessity not the "we" of birth through royal blood,
as it was discovered years ago that there is no royal robe
to mathematics, but it is the very democratic "we" of the
participatory demonstracy, and it begins to lose its title
to that with every citizen of this res publica who demurs
from their reponsibility and their right to follow along.
Jon Awbrey
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