As is evident from Riemann's great roadmap for scientific progress "The Hypotheses that are the Foundations of Geometry," Riemann, in spirit, followed Kepler's injunction against reliance on algebra for determining physical causality. Riemann, as well, avoided the error of Cartesianism that Leibniz warned against.
The which is encompassed in especially two main ideas from that work that operate together in the fashion of a musical dialogue in a great fugue. One that asserts "The questions about the infinitely great are for the interpretation of nature useless questions. (...)We are therefore quite at liberty to suppose that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena."
In other words, there is evidence of a breakdown in the infinitely large and infinitely small of normal geometrical relationships. Indeed, isn't this precisely what the quantum world challenges us with? Our normal assumptions based upon reasoning from our macroscopic world break down. For how can a thing be simultaneously a wave and particle as well as participating in superposition. This violates our geometrical assumptions based upon our visual manifold of what an identity of a "thing" is, after all.
At the other end of the spectrum in the infinitely large, we are confronting another daunting paradox. It seems that unknown forces are propelling galaxies to rotate out of synch with gravity. Additionally, the very fabric of spacetime itself is measured to be constantly expanding by a factor denoted as lambda.
The other principle that Riemann appended to his great Gedankenexperiment was that one must set aside all prejudices when embarking upon the realm of physics, "Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices. This leads us into the domain of another science, of physic, into which the object of this work does not allow us to go today."
Now it is the case that Einstein, in his breakthrough relativity theory, did precisely as Riemann had supplied in the body of his hypotheses by adopting a curved spacetime in a particular four dimensional manifold by challenging "traditional prejudices." However, Einstein encountered a roadblock in attempting to formulate a "unified field theory."
Now, it occurs to me that when speaking of forces qua geometrical relationships in the realm of the physically non-living, at the level of the infinitesimal, we have a breakdown of the principle of Cartesian "empty" metrical space. Otherwise, how can we justify the strong and weak nuclear forces as operating disparately in a continuous manifold. And at the same time that we can prove that the quantum realm is itself discontinuous?
Likewise, at the realm of the infinitely large the rotational forces holding together galaxies operate in a somehow different fashion than our conception of conventional gravity, or we must imagine the influence of unknown and hidden non luminous particles. But perhaps there is another type of strong force at this infinitely large realm analogous to that which holds the nucleus together? Likewise the weak force of radioactive decay might find a physical analogy somehow in the lambda expansion.
Now all of this foregoing returns me to Kepler, who approached the geometry of our solar system from a principle that there must be an overall unity of lawful harmony in the planetary elliptical orbits. It is from this standpoint that Leibniz developed his own Monadological approach to unification of what Vernadsky would later further refine as the realms of Lithosphere, Biosphere and Noosphere.
Because, echoing Riemann's warning above, when attempting to unify forces, one must depart the framework of the inert non living. This is so because the living lesser monads have a sort of primitive force, a connatus, that separates themselves hermetically from the non living. For while being constituted from the same atoms as the non living, the living force is radically non commutative, in an analogous manner that there are distinct transfinite numerical orderings as demonstrated by Georg Cantor. Likewise, within the realm of the Noosphere of creative human mentation there is a transfinite boundary separating all other lifeforms. We humans have the unique capacity of reason. However, this capacity is not some mystical substance. Rather it is our willful capability to overcome physical and biological limits of the lower realms of reality.
