Dear Ryu,
We have gone through your excellent analysis and logical approach. There are many similarities in our views, though we may be expressing it differently. We have published a book on number theory derived from natural principles and defined each term precisely.
You define 'digital' as the set composed of countable elements. Counting is done by segregating objects that are perceived as similars. Thus, we define number as a property of objects by which we differentiate between similars. Each digit is differentiated from others before we sum it up. Thus, each digit is one by itself. Depending upon the repetition of perception, we assign the digit sequence different names that are called numbers. You define analog as it's opposite. By opposite we assume opposite of clear perception. If the dimensions of an object are fully perceived - if it has finite dimensions, then it has a number. If the dimensions of an object are not perceived fully, then it cannot be said to have a number - it has infinite dimension. Thus, it is called infinity, which behaves like one - without similars (because if there is a similar object, then the dimensions of both would be perceptible). Thus, the mathematics involving one and infinities have similarities related to dimension only. For example, linear accumulation and reduction, i.e., addition and subtraction involving one is possible because of its finite dimension. Similar operations involving infinity are not possible because of its infinite dimension - thus the answer remains infinite. Non-linear accumulation and reduction, i.e., multiplication and division involving one does not change the number. The same is true for infinity except that since the dimension is not perceptible, the whole sequence becomes imperceptible. Infinity is different from a very big number, as the mathematics involving both are different. Thus, only space, time, direction (coordinates) are infinite in addition to cognition (perception). Hence only these are analog. All other objects are digital entities.
Cantor's views that infinite set has different sorts: integer set and real number set cannot be 1-1 corresponded to each other can have meaning only in the digital context if we remember that digital entities are composites of real numbers, which are related to perception. For example, if one object is decomposed to five different parts, each of the five parts has the number one associated with it. However, this one is different from the previous one. The continuum hypothesis flows from the above. The Axioms of Choice can be highly misleading. In the photoelectric effect, it was found that intensity of light has no effect on electron emission - only wave length is crucial. Let us apply AC and occam's razor, because the axiom doesn't affect the physical reality. Hence we first drop it and see how one can build mathematical structure for physics. We will realize that in the case of an electron held only weakly by its atom - as in the case of cesium - intensity of light does have effect on electron emission. Till we find the exceptional behavior out, any theory formulated by applying AC would be misleading.
Space is the interval between objects. Since it is not perceptible by itself, it is described through alternative symbolism by describing the objects contained in it and assigning their properties to space. For example, we define the position of objects by measuring the interval from another object or a point with reference to the arrangement of objects by comparing it with a unit interval and call it space measurement. Since there is no true vacuum, the "energy flow due to density variation" in space affects the position and movement of objects. We call this as the curvature of space. Since the density is different at different points and since space is analog, your statement that space will be supposed to be locally connected is proved. For more details, you can read our essay.
Having said the similarities, let us discuss the differences in our approach. Quantum theory has been developed based on observational findings from different experiments set up for different purposes. The theories are not based on the analysis of natural phenomena. Subsequently, natural phenomena was sought to be explained through it. It is well known that people learn in two ways: from experiences and from education. The first is more time consuming, painful and may be dangerous and can mislead. We agree with you that theoretical reasoning and experimental verifications are two cornerstones of physics. But this sequence should not be reversed. Nature is the greatest teacher. By ignoring it and choosing to learn from experience, we have landed in a scenario where there is not a single theory that can be said as quantum theory. It has various interpretations that are not consistent with one another. There is no unanimity as to what constitutes reality. The measurement problem is hinting at the inadequacy of many interpretations. The hypothetical particles like Higg's boson or graviton will never be found out. Yet, it provides a large number of people sustenance for leading a cozy life at public expenses by perpetuating the cult of incomprehensibility. Failures are never publicized. But possibility of success is highly exaggerated. One example is LHC experiment.
There are many questions and challenges to Big Bang, which are also well known. Now Big Bounce is taking over and we support this view, though we have a different explanation for this phenomenon (and a different mechanism). We treat gravity as a composite force that stabilizes - whether it is the planetary orbits or the atomic orbitals. In our essay we have discussed the uncertainty principle to show that it arises from the laws of Nature, but has been wrongly interpreted.
We warn you of one danger. Since we are going against "main stream physics" (which in other words means blind acceptance of majority opinion even on the face of evidence to the contrary), our comments are looked upon as "dirtying" the "clean" physics. But we are assured that TRUTH WILL ULTIMATELY PREVAIL.
Regards
basudeba
