The quest for true geometry

There are many problems and paradoxes with standard geometry and topology (and differential equations) based on the classical notions of differentiability and smoothness which fall short of the conceptual perfection of analyticity or its extensions. Indeed the classical view of the real line is problematic.  The work of Lawvere in synthetic differential geometry is of outstanding importance. And yet from a higher philosophical and scientific point of view, what is the true geometry ? Perhaps it lies within the inner mind of the classical painter, sculptor and writer.

But let us understand better the qualitative topological revolution of Poincaré.  Qualitative and topological knowledge only is meaningful in the context of a strictly quantitative situation (a given equation, a given problem, a given theory). On the other hand quantitative analysis is meaningless without being able to draw some qualitative conclusions, all intelligible descriptions of the results of such an analysis must involve or implicitly posit qualitative notions.

What being is in its essence determines its qualitative characteristics and correlates them with certain definite quantitative changes by the fundamental law of its own nature. The magnitude does not determine the quality, nor does the quality determine the magnitude, but the roots, both of the quantitative and qualitative elements in being, lie deeply concealed in the fundamental essence.  - J.G. Hibben, Hegel's Logic, An essay in interpretation.

We should investigate the fundamental mathematical, scientific and philosophical problem of the strange separation between the concepts of smooth (infinitely differentiable) maps and analytic maps (real and complex). This is closely connected to locality and determinism. 

Analytic functions are mysterious objects. We are told that analytic functions form a sheaf. For disjoint open sets if we take constant functions taking different values on each of them, then the (trivial) 'gluing' exists but if we take the complement of the domain of holomorphy of this  analytic function on the union, then it must topologically separate the two open sets ! This is all trivial, but is shows how connectivity or simple connectivity are bound up with the nature of analytic functions.