Sunday, January 18, 2015

The Blind Men and the Billiard-Ball

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 a 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

Elephants obey classical mechanics.

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