Paradoxes are abounding in humanity's ability to perform transformations in analyzable "space." This usage of such a universally accepted assumption that framework metric space and actual space are one and the same is the root of why such impossible riddles subsist.
For example, just take the all too clever formulations for the inconsistencies inherent in the spatial matrix transforms: the point at infinity AND the extended large infinity. Now to be sure, there are ways to account for this drawback while staying within the same assumption framework by creating "special functional domains." This in essence is the nature of Dirac sea matter from another imagined realm and as well as the renormalization criteria of quantum mechanics formulae. The whole particle physics adoption of transform spaces like SU2 to reign in this mismatch in the two disparate spaces.
Overall, the foregoing metamathematical gedankenexperiment is in fact a proof of the "irreality" that conformal geometry is just that "conformal," yet not quite real. This was Riemann's hypotheses that lie at the basis of such merely metrical space was the whole point of opening up visual perception of curvature to a higher standpoint. That of non-Euclidian and integrated hyper dimensional curved space. Infinities can be created that are not magical algebraic calculus add-ons like the development of renormalization when confronted with the very same paradoxical absurdities.
It was only this reworking of the metrical field mathematical geometry that led to the fluid mechanics of extra dimensional spacetime. Felix Klein explicitly said that the formulation of relativity via a special four dimensional spacetime was only a part of Riemann's overall system. And Klein was irrefutably correct as much as Fechner's extension of the meaning of a particular sort of communicative consciousness to both the vegetative and even possible inert realms. In the latter case, may we not say that the astronomical domain exhibits simpler form of quasi directedness? Recall that Riemann cites Fechner's schema of rhythmic variation across the spectrum of living spaces of reckoning of time.
As we look ahead to redefining time in the context of humanity's mission of conquering the extreme reality's necessary alterable conditions of living in outer space... take, for instance, the twin problem parameter scope, (i.e. "space") of engineering the technological creation of artificial gravity and shielding our frail biology from cosmic radiation.
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