Discussion Abstract: To Be or Not To Be
In relation to the current situation in fundamental science, the estimates of the role and state of rigorous, mathematical description of reality apparently evolve towards two diverging groups.
One of them, the "mainstream mathematical physics", defends the existing, traditional development of mathematical description of reality by always simplified but internally technically powerful modelling, with its "unreasonable" successes and not less mysterious failures, "unsolvable" problems and accumulating "dark matters". One considers in this mainstream paradigm that one can continue to solve a large enough part of fundamental and practical problems with if not "unreasonable", but at least sufficiently high efficiency and precision, while the remaining "mysteries" (e. g. of time, quantum mechanics, or dark matter), "unsolvable" problems (real interaction) and "non-computable" phenomena (e. g. from the humanities) can be accepted in the form of "inexplicable postulates" or basically empirical and only mechanistically quantifiable knowledge.
In the second attitude, one considers that the truly consistent, not only rigorous, but also causally complete and unified description of reality is still possible, certainly beyond traditional limited "models" but within a qualitatively extended mathematical framework. Actually this is a "strong version of science" as objectively reliable form of knowledge, where one accepts the challenge of ultimately complete science covering eventually all knowledge and providing the totally consistent picture of reality.
The present essay describes a working version of the second approach, in the form of Universal Science of Complexity confirmed by various applications, from elementary particles to all high-level systems (biology, society, consciousness), now within the causally complete and totally unified description, with dynamically emerging, physically real space, time, intrinsic properties and laws. It is obtained as explicit extension of usual mathematics framework, in the form of dynamically multivalued unreduced solution to arbitrary real interaction problem, while the traditional theory modelling corresponds to the dynamically single-valued, effectively zero-dimensional (point-like) projection of the unreduced dynamics of any real system or process.
That explicit projection relation between the traditional and new mathematics provides also a transparent explanation of the "unreasonable efficiency" of the former strangely intermingled with its unsolvable problems, persisting mysteries and simply rigorously indescribable phenomena. Indeed, any example of geometrical projection of a three-dimensional object to lower-dimension spaces shows immediately that the limited image of low-dimensional projection can vary essentially in its correspondence to the unlimited three-dimensional prototype, depending on the direction of projection "view" with respect to essential structural features of the object (consider a pencil projection varying from a realistic "rod" to the ambiguous "thick point"). In the same way, the ultimately limited point-like dynamical projection of real system behaviour within the traditional model approach can provide either "surprisingly" realistic or strangely "mysterious" image of the multivalued real system feature depending on the more or less successfully guessed "projection kind/direction" (becoming much less obvious for more complex systems).
The transition from the traditional, artificially simplified description to the proposed intrinsically complete mathematical framework corresponds thus to the transition from inevitably separated and often "strangely looking" usual projections to the full-dimensional and therefore causally complete, dynamically unified image of unreduced reality (see the essay for details and references). That (mathematically specified) vision shows that the "extreme" program of the ultimately complete and unified science can be quite realistic and natural (as opposed to the accumulating pessimism of the traditional science framework).