Very interesting analytical essays in the spirit of deep Cartesian doubt. I note your particular important ideas, conclusions and fundamental questions for the "fundamental science" that I liked:
"If we use Wigner's limitation of the study of nature to that of physics, then we immediately remove a huge swath of nature and knowledge - that of animate (or living) nature. In response to Mr. Wigner's original statement (the title of his article) we must expand our view of natural science to more than physics and include animate nature. We must also consider how reasonable (or unreasonable) mathematics is in representing all of nature - not simply physics. Let us first expand our view of 'natural science' beyond Wigner's admittedly limited view. "
"The distinction between animate and inanimate nature still remains a mystery, as the question of what is lost when a living being stops living is still unknown."
"There is another aspect of the animate world we should consider - the importance of uniqueness to animate objects."
"Should the goal of science be simply about developing 'invariant' laws or are we attempting to describe nature? Can invariant laws, even theoretically, completely describe nature? This is an assumption, even presumption, of the scientific method and certainly of some 'Theory of Everything'. However, consider a hypothesis (more properly a 'conjecture') that a universal aspect of nature both animate and inanimate, is the uniqueness of objects, beings, and events. It would seem an obvious conjecture, since it is a common experience for us. However, this conjecture would mean that no two objects have precisely the same characteristics and the more accurate (precise) we care to be the more we will see these differences."
"If science is about finding a set of universal laws, which are invariant across all objects and could (theoretically) predict all motion and change, then this seems uniquely at odds with a universal property that all objects are unique. More to the point, to achieve a full description of nature means to be able to describe individual situations, in all their individuality and uniqueness. A methodology, like the scientific method, that removes individual characteristics would seem at odds with this goal. As well, the unique character of animate objects might undermine a scientific methodology that removes the uniqueness of the objects it studies.
The above discussion should bring into question whether the methodology and tools being applied to inanimate nature are adequate for completely describing nature. Further, do the (and how do the) methodology and underlying tools impact the study of all of nature? If the goal of science is to describe nature, including the unique aspects of nature, then are (mathematical) tools, which remove the unique aspects of nature, short-circuiting this goal? If this is the case, then mathematics (and science using this methodology) cannot represent nature in its entirety."
"So there is an historic precedent we might need to consider: We are using a concept of number and representational system for numbers that is limiting our models - in both mathematics and science. What evidence is there for this situation, especially given how amazingly (unreasonably?) accurate our current theories appear to be? The place to start might be with our assumption that Real numbers, especially as represented today (via decimals, logarithms, and various bases) are the end-all of linear numbers (comprising a line or continuum). What if we have limited our conception of number, and of a geometric continuum, to our conception of a 'Real' number - because representations like decimals for Real numbers are all we know? This would be similar to the situation of the Pythagoreans, where ratios (Rational numbers) were all they knew and comprised a continuum that 'irrational' numbers did not have a seat at. What if 'more' numbers than Real numbers can exist on a continuum?"
"We need to question the adequacy of the underlying mathematics we apply to nature. We generally presume that our current representation of complex numbers (and quaternions) is adequate to the task of modeling nature. Do we have evidence of this? We are aware that we do not have fully formed values for complex or quaternion numbers. Our current representation always involves unknowns (i.e.. 'i', 'k', 'j'). So how can a theory built on these unknowns be 'real'? This situation is like pre-decimal times, when a Real number (at least an 'irrational' one) was not considered a real number. Only rational numbers (represented as integer ratios) were considered 'numbers'. Some people saw that solutions to certain algebraic problems could not be solved using ratios, but they could not properly represent these irrational (and 'transcendental') values - so the numbers were somehow phantom numbers. Only after decimals came into common use was representing them as values possible. We still do not have an adequate representation of complex or quaternion values. Since such numbers pervade many physical models, how can we expect these models to be adequate representations of nature?"
Mathematics overcome their crises through the dialectic breakthroughs, connecting the unconnected, thereby overcoming the regular limit on the path of knowledge. Modern Mathematics and Physics- sciences without ontological justification (basification), as well as all "the fundamental knowledge". How to choose a path? Deeper limit ontology and dialectic in the spirit of Plato - Cusa - Hegel: "coincidence of opposites." We have now a lot of logic. To grasp the Universum as a whole (to connect Cartesian "res cogitans" and "res extensa") need the dialectical logic. Requires the deepest synthesis of all the accumulated knowledge, including Tradition. The "Occam's razor" should be extremely sharp. Necessary unification "matter" at all levels in the spirit of Plato's, ontological justification (basification) and extension "space" based on absolute (unconditional, extreme) forms of existence of matter: "continnum" + "discretuum" + "dis-continuum". We need to understand the ontology and dialectic "three in one" in Nature.
E.Husserl gave good tips in "Origin of Geometry": "Only to the extent, to which in case of idealization, the general content of spatio-temporal sphere is apodictically taken into account, which is invariant in all imaginable variations, ideal formation may arise , that will be clear in any future for all generations and in such form will be transferable by the tradition and reproducible in identical intersubjective sense."
I believe that only the deepest ontological turn of the fundamental science will provide an opportunity to get out of the "crisis of understanding", "crisis of interpretation and representation" to the new heuristics.
I invite you to see my analysis of the philosophical foundations of mathematics and physics, the method of ontological constructing of the primordial generating structure, "La Structure mère" as the ontological framework, carcass and foundation of knowledge, the core of which - the ontological (structural, cosmic) memory and information - polyvalent phenomenon of the ontological (structural) memory of Universum as a whole. I believe that the scientific picture of the world should be the same rich senses of the "LifeWorld» (E.Husserl), as a picture of the world lyricists , poets and philosophers.