Dear Sir,
We thoroughly enjoyed your essay.
When you say: "The emergence principle is a general property of the universe", it needs some clarification. What is emerging? Is it the laws of Nature or their revelation to us? In the present context, the obvious answer is the second. But the first cannot be ignored. We find a set of rules that remain invariant through space and time. The same with objects (matter) and forces (energy). But then, we are also finding hints about their unification. They must have emerged from some common source. Our goal is to find that common source by moving from diversity to unity - not emergence, but convergence, which you have also agreed.
You have rightly pointed the way out: applying a deterministic perspective to replace the probability arguments. Even probability is always within a band. It is never unlimited. Within the band, everything is deterministic. Thus, probability implies our inability to understand local dynamics beyond a scale. The reason is uncertainty inherent in Nature, which arise not chaotically, but from step by step changes due to other influencing factors in the environment. Like the butterfly effect, we are not capable enough to know these steps or measure it - bringing in uncertainty in our knowledge. But when mathematics enters the field, problems multiply by division. Position is not absolute, but is a fixed coordinate with reference to some origin. Similarly, momentum not absolute, but is a mobile coordinate with reference to some origin. If both these are assigned to the same object, if it has a fixed position, its momentum is zero. If it has momentum, it does have a fixed position - position is zero. If we multiply both, the result will not be h, but zero for all values of position and momentum. Yet, we are running after this wrong notion.
You have rightly pointed out that: "Generic atomism model's failure is attributed to lack of knowledge or the predictions required are too complex". The basic difference between micro and macro physics is the difference between the individual and the composite (apart from scale factor). When we deal with QM or QED or QFT, we deal with the energetic particles, such as quarks, leptons, etc. individually. The macro world is a collective of the same particles in large numbers, where the energy gets confined due to the regulated combination of opposites. This brings in new characteristics. The properties of water is not the same as that of hydrogen or oxygen or a combination of both. This non-linear behavior creates the disparities. Hence they are not different, but different manifestation of the same thing. We have to understand the non-linear dynamics - not treat both separately. Thus, you have rightly described it as a fractal system. Only we have to identify the similarities.
We have presented a paper here to physically explain 10 dimensions, where we have briefly discussed about mathematics. It is essentially the same as your view except your treatment of geometry as extended object. Geometry essentially deals with fields with volumes as an extension. While objects associated with numbers are discrete - quantized or digitized, fields are analog. While digital can form a part of analog (like our use of space, time, coordinates or a bucket of water from the ocean), the reverse is not always true. While our measure of space, time, etc. can be extended to the absolute and analog space and time, the bucket of water cannot be extended to imply ocean. Treating both as one has misled mathematicians and physicists alike. We can chart a length on a graph in two dimensions. But length of what? The object itself has three dimensions. Thus, the graph shows one dimension less than the object. Hence it is not a true representation. When we plot several points on a graph and assign numbers to them, we are extending those objects like we blow up a balloon. We are marking different lengths - equal or different in magnitude and direction. At the universal level, if the line is the locus of a point, then it has only two parts: closed or open ended. Closed represents all digitized structures. Open represents the analog nature of motion. Both are not the same like space/time and a bucket of water. You also seem to agree with this concept in later pages. Regarding mathematical transformation, we have discussed at length in our paper.
Regards,
basudeba