ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
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| Let us say no more then about this objection, but enlarge our thesis, thus:
| If there be in the universe a 'q' other than the 'q' in the feeling,
| the latter may have acquaintance with an entity ejective to itself;
| an acquaintance moreover, which, as mere acquaintance, it would be
| hard to imagine susceptible either of improvement or increase,
| being in its way complete; and which would oblige us (so long
| as we refuse not to call acquaintance knowledge) to say not
| only that the feeling is cognitive, but that all qualities
| of feeling, 'so long as there is anything outside of them
| which they resemble', are feelings 'of' qualities of
| existence, and perceptions of outward fact.
|
| James, "Func of Cog", pages 15-16.
|
| William James, "The Function Of Cognition",
| Read before the Aristotelian Society, 1 Dec 1884.
| First published in 'Mind', 10 (1885). Reprinted in
|'The Meaning Of Truth, A Sequel To "Pragmatism"',
| Longmans, Green, & Company, London, UK, 1909.
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| We use transversality as a sufficient condition under which the fiber product
| of two morphisms exists. We recall that in any category, the 'fiber product'
| of two morphisms f : X -> Z and g : Y -> Z over Z consists of an object P
| and two morphisms
|
| g_1 : P -> X and g_2 : P -> Y
|
| such that f o g_1 = g o g_2, and satisfying the universal mapping property:
|
| Given an object S and two morphisms
|
| u_1 : S -> X and u_2 : S -> Y
|
| such that f o u_1 = g o u_2, there exists a unique morphism u : S -> P
| making the following diagram commutative:
|
| S
| o
| /|\
| / | \
| / | \
| u_1 / u \ u_2
| / | \
| / | \
| v v v
| X o<------P------>o Y
| \ g_1 g_2 /
| \ /
| \ /
| f \ / g
| \ /
| \ /
| v v
| o
| Z
|
| The triple (P, g_1, g_2) is uniquely determined,
| up to a unique isomorphism (in the obvious sense),
| and P is also denoted by X x_Z Y.
|
| Lang, DARM, page 29.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
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| o---------------------------------------o o-------------------o
| | X | | E_i |
| | | | |
| | | | o |
| | | | / \ |
| | o | | / \ |
| | / \ | | / \ |
| | / \ | | / \ |
| | / \ q_i | | / q_i U_i \ |
| | / o---------------------->| o o o |
| | / \ | | \ / \ / |
| | / \ | | \ / \ / |
| | / U_i \ | | o Eij o |
| | / \ | | \ / |
| | / \ | | \ / |
| | o o o | | o |
| | \ / \ / | | |
| | \ / \ / | | |
| | \ / \ / | o---------|---------o
| | \ / \ / | |
| | o Uij o | q_j o q_i^-1
| | / \ / \ | |
| | / \ / \ | o---------v---------o
| | / \ / \ | | E_j |
| | / \ / \ | | |
| | o o o | | o |
| | \ / | | / \ |
| | \ / | | / \ |
| | \ U_j / | | o Eji o |
| | \ / | | / \ / \ |
| | \ / | | / \ / \ |
| | \ o---------------------->| o o o |
| | \ / q_j | | \ q_j U_j / |
| | \ / | | \ / |
| | \ / | | \ / |
| | o | | \ / |
| | | | \ / |
| | | | o |
| | | | |
| | | | |
| o---------------------------------------o o-------------------o
|
| Figure 1. Manifold Of Ejective Impressions
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References And Incidental Nuances (RAIN)
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html
http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm
http://www.philosophy.ru/library/kant/01/cr_pure_reason.html
http://ez2www.com/go.php3?site=book&go=0387943382
http://hallmathematics.com/mathematics/1433.shtml
http://hallmathematics.com/mathematics/630.shtml
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