We often times render the world around and within, with examples and renditions, based on simplifications or perhaps internalized ideals. Such as for example the circle, although it is only a specialized case of an ellipse, or of an equiangular spiral.
However manifestations are usually imperfect, even in principle. For example, homogeneity versus possible heterogeneity of Big Bang nucleosynthesis, anisotropy 10^-5 of cosmic background radiation, and clustering for early universe vs uniform distribution of galactic/cluster mass. Hence the homogeneity and isotropy of cosmological principle would seem to be just an idealization.Thus also would it seem that our world is just the result of such imperfections, such as matter / anti-matter asymmetry?
Also one has Newton’s emphasis on uniformity, such as for flow of time, and for initial base line of uniform change in general, such as inertia concept, and as implied by usage of suggestive terms, fluent/fluxion. Also one has the idealization of Minkowski manifold, rather than always curvature and mass, with Minkowski manifold just an idealized tangent surface.
Locally one has absence of perfect order; rather always an entropic description, such as unattainably of absolute zero Kelvin. Metaphorically one might express it as if similar to no ‘Dirac like’ one value function; rather always energy being partitioned more diversely i.e. ‘energy, the handmaiden of entropy’.
Conversely, also no idealized complete disorder i.e. not just an entropic description; rather always some information descriptive, such as emphasized in Ramsey’s theorem. That is, randomness, both in physical and mathematical sense (?), would seem to be just an idealization.
Also one has Lyle’s idealized uniformitarianism of geological change, and Darwin’s continuation of such simplification and assumption of uniformity (slower implied uniform rate of change) to biological evolution.
And in mathematics, one has clustering of primes, rather than a uniform distribution. Likewise for the rationals, since the primes are a specific case of rationals. Hence the distribution of Cantor dust would seem to be clustered i.e. not uniform.
Perhaps other mathematical examples, such as π?
J. Wallis: π/2 = 2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 …
Thus an irrational number can be constructed from infinite multiplicative series of rationals i.e. integer ratios. Thus can one have it both ways with π; that is both an irrational and also a rational rendering?
However although assuming it is a divergent series of rational terms, still the summation would seem statistically to reside in the greater ‘sea’ i.e. greater cardinality, of irrationals. This could be visualized by the complement of the clustered Cantor dust of discarded rationals; that is a greater ‘sea’ of irrationals.
Also does irrational rendition of π’s involvement with special case of the circle seem so unusual, especially since the circle is just a special case of an ellipse, wherein π is not descriptive?
Would this all suggest local imperfections i.e. non-idealization, being more suitable for our rendering, modeling, and even in a platonic sense, more typical for nature and even for mathematics? For example, would our world seem just an imperfect rendition i.e. for example, the result of such matter / anti-matter asymmetry? Is then the modified global scale of Spiral Rotation Model (with no parameters, nor degrees of freedom) a counter example, and complement, to such manifested imperfections?
see zankaon site
Is this discussion then relevant to Plato’s emphasis on inapparent idealized forms behind imperfect manifestations? However herein variation from such idealization, such as differing from forms, is emphasized. In overall sense, then are there similarities between not only philosophical and physical models, but also inclusive of ethical models? see zankaon site.
Also if rationals are clustered, and matchable to coarser to finer scale, then would such pattern also be consistent with manifested imperfections, such as early universe clustering, possible BB heterogeneous nucleosynthesis, CMB anisotropy, lack of randomness or perfect local order (such as zero Kelvin); and also would such clustering of rationals be consistent with mathematical imperfections, such as lack of randomness description (?) i.e. any proof or conjecture?
So is imperfection part of the scheme of things, both physical and even for mathematics?
Might there be other examples in nature, or in physical modeling, or even for mathematics per se, that also might be considered as overall intrinsic common characteristics?
Are manifolds difficult to deal with in mathematics, and also in physical models? That is, even though one can run up and down a binary tree i.e. merging and bifurcating, with manifolds as elements, still are such mathematical objects recalcitrant to manipulation?
For example, such as for merging 3-brane manifolds? Also budding (bifurcating) manifolds, such as for eternal chaotic inflation, would seem problematic, mathematically. Also creation of manifold i.e. continuity, from null set would seem problematic, even if not so for a number set? Also creation, and destruction, of manifolds, such as for ‘beginning’, or any ending, of 3-manifold ‘universe’, would seem difficult. Also disruption of 3-manifold, such as so-called tear (Big Rip model) or singularity construct, would seem problematic in regards to such continuity disturbance. And also alleged resistance to deformation of our 3-manifold i.e. so-called law of inertia of manifold, could be considered as if manifolds want to be left unperturbed.
So are manifolds another example, whether in nature, or for physical models, or even for mathematical world per se, of a common intrinsic characteristic, residing in intersection of such 2 world views (physical and mathematical), such as represented in a Venn diagram below? Likewise for alleged clustered rational number set and thus for primes, as discussed above?
Intersection of 2 world views – mathematical and physical
Therefore do such two examples denote built-in general common characteristics; thus serving as consistency arguments, and explanatory of general manifestations, such as imperfection and truculent manifolds? TMM
zankaon website for further elaboration.