Changes

===2.5.  Differential Geometry===

===2.5.  Differential Geometry===

One of the difficulties I've had finding guidance toward the proper form of a differential calculus for logic has been the variety of ways that the classical subjects of real analysis and differential geometry have been generalized.  As a first cut, two broad philosophies may be discerned, epitomized by their treatment of the differential df of a function f : X → R.  Everyone begins with the idea that df ought to be a locally linear approximation dfu(v) or df(u,v) to the difference function Dfu(v) = Df(u,v) = f(u+v) – f(u).  In this conception it is understood that "local" means in the vicinity of the point u and that "linear" is meant with respect to the variable v.

One of the difficulties I've had finding guidance toward the proper form of a differential calculus for logic has been the variety of ways that the classical subjects of real analysis and differential geometry have been generalized.  As a first cut, two broad philosophies may be discerned, epitomized by their treatment of the differential d''f'' of a function f : ''X'' → '''R'''.  Everyone begins with the idea that d''f'' ought to be a locally linear approximation d''f''_''u''(''v'') or d''f''(''u'', ''v'') to the difference function D''f''_''u''(''v'') = D''f''(''u'', ''v'') = ''f''(''u'' + ''v'') – ''f''(''u'').  In this conception it is understood that "local" means in the vicinity of the point ''u'' and that "linear" is meant with respect to the variable ''v''.

====2.5.1.  Local Stress and Linear Trend====

====2.5.1.  Local Stress and Linear Trend====

But one school of thought stresses the local aspect, to the extent of seeking constructions that can be meaningful on global scales in spite of coordinate systems that make sense solely on local scales, being allowed to vary from point to point, e.g. (Arnold, 1989).  The other trend of thinking accents the linear feature, looking at linear maps in the light of their character as representations or homomorphisms (Loomis & Sternberg, 1968).  Extenuations of this line of thinking go to the point of casting linear functions under the headings of the vastly more general morphisms and abstract arrows of category theory (Manes & Arbib, 1986), (MacLane, 1971).

But one school of thought stresses the local aspect, to the extent of seeking constructions that can be meaningful on global scales in spite of coordinate systems that make sense solely on local scales, being allowed to vary from point to point, for example, (Arnold, 1989).  The other trend of thinking accents the linear feature, looking at linear maps in the light of their character as representations or homomorphisms (Loomis & Sternberg, 1968).  Extenuations of this line of thinking go to the point of casting linear functions under the headings of the vastly more general morphisms and abstract arrows of category theory (Manes & Arbib, 1986), (MacLane, 1971).

=====2.5.1.1.  Analytic View=====

=====2.5.1.1.  Analytic View=====