Extended Mathematics of Unreduced Dynamic Complexity: The Exact Image of Unified Reality, from the Electron to Consciousness by Andrei Kirilyuk

Joe, you're citing item I of the Guidelines expressing general goals of all contests, but there's also item II expressing the particular theme and problems of interest for the current contest. I think one should not neglect the latter, at this particular place. However, at FQXi they also have the general Forum for discussion of all ideas, including the Ultimate Reality direction (with 12 topics). Please note that it's not my rules or subjective attitude and has nothing to do with the value of your ideas. I remain open to discussion of ideas relevant to the current contest and my essay. It seems logical that the discussion within any essay thread should be somehow related to the ideas of that essay (as well as the current contest agenda)...

Dear Michael, Thank you for your interest and comments. Based on your description here and your essay and paper, I feel that indeed our attitudes on the incompleteness of standard mathematical and physical framework have much in common. I actually propose a provable version of the necessary completion applicable at all levels of world dynamics, from elementary particles (causally complete quantum mechanics and relativity) to the highest-complexity systems like society, conscious brain dynamics or intelligent communication networks (see my arXiv papers and web page for references). It remains to hope that such extended visions and approaches will finally win and bring fundamental science (otherwise stagnating) to the state of new progress...

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).

Discussion Abstract: From Postulated Being to Dynamical Becoming

The growing loss of certainty of modern, apparently successful mathematics is largely due to its tacitly increasing departure from the fundamental principle of internal consistency closely related to the demand of realism understood as a close enough correspondence to the physically consistent natural structures this mathematical framework of science is supposed to efficiently describe.

In the same way as the elements of physical reality interact and give rise to ever more elaborated structures and phenomena, the "abstract" elements of mathematical framework progressively form respective abstract structures and laws, according to rigorous derivation rules ensuring the result consistency.

However, another, qualitatively more abstract approach to mathematical description of reality has appeared and quickly grown especially in the age of "new physics" starting from the beginning of the 20th century, where the "final", resulting mathematical structures and laws are somehow heuristically "guessed" and then simply postulated for further direct "confirmation by experiment", but without the real, consistent derivation and logical understanding of their origin. In particular, their growing number appears even to be quite inconsistent, i. e. incomplete, contradictory and "mysterious" (quantum mechanics, relativity, gravity, "dark matter", "hidden dimensions", "broken symmetries", etc.), but still "unreasonably efficient" and "confirmed by experiment". This tendency of "mathematical physics" is reduced to never-ending trial-and-error attempts of direct postulation of a new mathematical "language" or "manifold" with its following "adaptation" to a limited set of "experimental observations". No derivation and solution of a dynamic equation from first principles is involved any more in this now dominating approach (contrary to previous situation of few postulates and the majority of derived features), which strangely and increasingly resembles the ancient Ptolemaic kind of "exact science".

Such kind of mathematical description is the ultimate expression of the "modelling" logic from the previous Discussion Abstract, while the principle of internal consistency can also be formulated as self-referential, autonomous and "automatically correct" nature of the desired truly consistent and complete mathematical framework. The self-referential, "living" and "intelligent" framework of reality-based mathematics should naturally include the intrinsically creative, "becoming" character of physical reality and its "biting" time-change (as emphasized especially by Prigogine and his school and previously by Bergson), as opposed to only "being", mechanistically fixed and effectively timeless character of mainstream postulated structures.

Needless to say, the necessary qualitative transition from such postulated "Being" of dead abstract models to dynamically creative, intrinsically complete "Becoming" of consistently derived structures is accomplished with the help of unreduced interaction problem solution presented in this essay, due to its clearly specified extended mathematical features of dynamic multivaluedness, entanglement and probabilistic fractality (summarised by the universal dynamic complexity). Wouldn't it represent the best possible kind of natural (and otherwise increasingly missing) unification of the mathematical and physical realities and universes, without any limits on any high-level applications?

Discussion Abstract: Conceptual Mathematics and Causally Complete Science

If mathematics is not only a technical tool or language of knowledge, but is designed for a deeper role of consistent science basis, then it follows that irrespective of details it should contain special foundational directions, or conceptual mathematics, which should use the exact description methods specifically for essential extension and completion of scientific knowledge, rather than only technical description of new structures and phenomena at the same level of understanding.

While particular examples of such kind of fundamentally new features can eventually be found (e. g. non-Euclidean geometry or chaotic dynamics), this quality is rarely emphasized as the necessary intrinsic feature of mathematical framework, apart from applications. Moreover, the appearing foundational novelties of modern mathematics show quickly growing trial-and-error uncertainty (cf. Kline, 1980) compromising their real efficiency as a basis for essential science progress, in contrast to spectacular advances of technical tools and methods as such. Given the above starting assumption of a deeper role of mathematics and the ensuing universality of conceptual mathematics, one arrives at the necessity of essential upgrade of the latter confirmed by the resulting qualitative completion of scientific knowledge.

The extended mathematics of complexity presented in this essay provides a well-specified realisation of this absolutely general conclusion, thus not only solving particular problems, but also introducing the necessary, otherwise increasingly missing creative order into the entire huge system of empirical, technical-theoretic and conceptual-theoretic knowledge.

As a result of this proper extension of conceptual mathematics, one obtains a whole series of knowledge completion and unification features. Not only fundamental physics regains its otherwise increasingly broken unity, causality and consistency, but now it can also be correctly unified with the equally rigorous description of all higher levels of reality to form a single, universal and intrinsically comprehensible form of absolutely exact scientific knowledge (see the essay and its references for details). We also destroy thus the familiar annoying barrier between the technically rigorous description and transparent figurative understanding of reality merging now into the higher-level, causally complete and widely understandable kind of science.

In order to see the advantages of thus conceptually extended mathematics, one can compare this really emerging unification with the growing serious doubts of conventional science professionals about not only any possibility of unified knowledge (e. g. Gleiser 2010, 2010a, 2011, 2013, Wells, 2013, Noë, 2014), but even the very existence of truly universal and rigorous scientific laws (Kauffman, 2011, 2013).

The resulting ultimate scientific revolution puts therefore an end to usual, "antagonistic" scientific revolutions (Kuhn, 1980) and leads to permanently progressing kind of intrinsically complete knowledge, where the notorious internal, qualitative "incompleteness" of traditional science and mathematics is replaced by only external, "quantitative" limitations of currently accessible observation space (or the "infinity problem").

Discussion Abstract: From Complexity Crisis to the New Science Age

The accumulating "difficult", practically "unsolvable" and therefore eventually destructive problems of today's global civilisation, from economical crisis to conflicting interests and missing sustainability, have something deep in common: the "untreatable" dynamic complexity of unreduced multicomponent interaction processes or systems. The same kind of impasse occurs in (traditional) fundamental physics, from the persisting old and growing new "mysteries" (quantum mechanics, unification, dark matter/energy, time, ...) to the vital issues of sustainable energy sources and intelligent machines of all levels.

As shown in the present essay, usual mathematical framework of science can not be efficient in resolution of this omnipresent and dangerously growing complexity crisis, already because it does not propose the unreduced many-body problem solution, replacing it with various qualitatively limited point-like projections, or "models". We then reveal the origin and dynamical structure of the causally complete, dynamically multivalued problem solution and specify the new mathematical and physical qualities it discovers in real system origin and dynamics. It is due to those rigorously expressed and universally applicable new qualities that the obtained new mathematics of unreduced dynamic complexity can form the reliable basis for the complete solution of complexity crisis problems and further, now unlimited progress at the emerging superior levels of civilisation complexity.

It is important that the same new mathematics of unreduced complexity provides the causally complete and intrinsically unified description of fundamental physical systems, from elementary particles to the universe structure and evolution, now without postulated "mysteries", "dark" matters and other "hidden dimensions". Equally successful applications at all higher complexity levels confirm the proposed extended framework universality and put an end to the growing deep doubts of mainstream science about the power of scientific method as such (see also the end of the previous Discussion Abstract here).

In summary, the proposed new mathematics of unreduced complexity, unified by the universal symmetry of complexity of the dynamically probabilistic fractal, paves the way from the current deep complexity crisis to the new, basically unlimited civilisation progress driven by the extended, causally complete knowledge of the universal science of complexity.