We are all familiar with the parable of the Blind Men and
the Elephant: although it
does suggest something about the limits of human perception (and, by parabolic
implication, cognition), it is essentially a testimony to the complexity of
elephants.**
Had the blind men been, rather, fondling a billiard-ball, they would have agreed on
its characteristics. From whatever
angle the blindman came, he would conclude: Smooth!
Round!***
[** Likewise characteristic is the excellence of elephants.
For proof and examples, click here.]
[*** If wise, each blindman would testify only to the local properties of the surface, such as
the Gaussian curvature.
Additionally, even a sighted
man, given free run all over the object, must needs refrain from asseverating,
that what he felt and saw is all there is
to the geometry -- the perceptual two-sphere might rather have been a cross-section of an unperceived
hypersphere, of which it forms but a negligeable part.]
Less obviously, such blindmen would concur as to the nature
of the Stone–Čech compactification of
any given set. For, from whatever angle
this grand object were approached, the verdict would be identical: Maximal! Universal!
~
Contrast the subject of Algebraic Topology.
I just finished reading one elementary introduction to this
subject, and leafing through two others entitled Algebraic Topology which, though scarcely elementary, claim
the reassuring subtitle of An Introduction or A First Course.
The first labors long in the messy,
mucking-about-with-triangles world of simplexes, as a lead-in to simplicial
homology.
The second spurns these “rigid gadgets”, and hews to the
purer path of singular homology.
The third abjures homology
altogether.
[Appendix]
Weiteres zur Elefanten-Mechanik:
It is well known that slowly-moving
things obey classical mechanics.
-- Robert Lindsay & Henry
Margenau, Foundations of Physics (1936), p. 269
Corollary:
Elephants obey classical mechanics.
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