Riemannian Surfaces and Biophysics: Branch Points to Nonlinear Functional Space

Recently, I came across some interesting research papers on medical imaging using conformal mapping techniques derived from Riemann. As I noted above in the caption to this blog Riemann's bold claim that the mind is itself a "compact, multiply connected thought mass with internal connections of the most intimate kind. It grows continuously as new thought masses enter it, and this is the means by which it continues to develop." 

Now the question that I shall pose here is how can one understand the difference between surfaces as mere mathematical relics and multiply connected functional phase spaces. This issue is akin to the use of the highly developed compositional tools of fine arts such as the form of the tragic or comedic drama. For instance, one may put on stage a five act play that outwardly resembles in its surface features, say Shakespeare's Othello, and yet its content be entirely vapid and without merit. 

So the issue that I want to bring to the fore is the question of what gives meaningful substance to Riemann's above hypothesis about the mind? To do this, we must inquire about what is meant by "thought masses."  Given that Riemann himself revolutionized non-Euclidian geometry, leading to Einstein's etablishment of a metric for multiply connected non-Euclidean spacetime, "thought masses" can only be taken as ideas of a very higher ordered class. That is, those ideas that propel humanity to a qualitatively new material and creative level of functioning. 

Now it is my contention that Riemann's critical geometric breakthroughs, indeed, had precisely this unique benefit for society. As Benjamin Franklin would have it, they performed the office across generations of "doing the good." In using the term "functional phase space" one must distinguish the domain that is referenced. Riemann has referenced the qualitatively highest domain in the quote I have cited from his philosophical fragments. It was Vladimir Vernadsky's systemization of such domains into three overarching and interacting "spheres," the noosphere or distinctly human mind, the organic or biosphere, and the inorganic or lithosphere.  

Thus, for instance, when applying a Riemannian mapping technique of surface features of the brain's sulcuses, it must be kept in mind that Riemann himself understood that it is the growth process of moving through a singular branch point from one functional manifold (surface) to another more developed transfinite domain that separates the truly human mind from the merely physical brain.