Dear Vladimir,
thanks for comments.
I've read your essay; it was not simple reading, and I did not catch very much.
An attempt to substantiate all of mathematics seems to me like telling for all Odessa. Mathematics is toov diverse and must meet different demands of the human practice (incorrect inverse problems, computer science, etc). Physics (fundamental) is another deal and should engage some special, dedicated math, especially beautiful and worthy math, and be the only possible choice in a sense.
There is a natural need in parceling the world, the being, into parts, smaller and smaller pieces, down to a point. Following this way we arrive to the concepts of continuum and field theory. Earlier, in natural philosophy, there were an empty space (absolute, and the absolute time) and material points; according to Newton, the body weight was determined by the number of material points in the body.
At present, the Lorentz group is a sort of engineering science, and it is not good idea to consider the time separately (from the space-time).
Absolute parallelism in fact is not a very good name; the better name is just the frame field theory. The name AP is due to the fact that the field equations can be written using a covariant differentiation with asymmetric connection which is "compatible" with the frame (i.e. the frame can be brought through this differentiation, changing Greek indices to Latin ones, and vice-versa). The curvature tensor for such a differentiation (with asymm. connection) is obviously zero.
However there are the metric and the usual covariant differentiation with symmetrical connection (Levi-Civita), for which the curvature tensor (Riemann tensor) is not zero. In my opinion, it is better to use this usual covariant differentiation, because it leads to the energy-momentum tensor, Riemanian geodesic lines, etc.
By the way, any linear combination of two connections is a connection too, so the range of covariant differentiations in AP is really great; but the symmetry of connection is the most important feature, that simplifies matters!
In general, the AP theory is about the fact that the general covariance (the group of coordinate diffeomorphisms) can coexist with the Lorentz group (however, generally, without inertial coordinates).
One can try to say something about AP in philosophical terms (philosophy is supposed to help in understanding the meaning of some math that tries to describe the objective reality, the being).
Well, the Hegel's statement about the identity of the being and not-being can be stated in a bit more concrete form: locally, or rather, at a point, in zero jets, the being is identical to the not-being.
That is, staying in a point, without differentiating, at one point, it is impossible to distinguish the being from the not-being.
In the AP, sure, the not-being means the trivial solution, the Minkowski space-time, where there are absolutely no differences between the points, nothing able to catch the eye. And the being is a solution of general position (of some perfect field equations) where all points are regular (there are no emergent singular points where the rank of the frame matrix drops per unit).
In vacuum General relativity (no fields other than the metric) it would seem that there is a stronger similarity of points: even in the first jets all points are indistinguishable from each other and from a point of the trivial empty space-time.
However, in solutions of the GRT (of general position) singular points do appear, points where the covariant metric is singular (of co-rang 1). And, of course, these singular points are sharply different from regular points (and points of the not-being).
Best regards