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

Andrei Kirilyuk

Essay Abstract

The current crisis in exact description of fundamental and applied systems has the well-defined origin and rigorously substantiated resolution in the form of qualitatively extended, unified mathematical framework of unreduced dynamic complexity. It is based on the unreduced universal solution of arbitrary interaction problem revealing the new, extended qualities with respect to traditional mathematical constructions. We describe the origin of the problem, the proposed causally complete solution and its mathematical novelties confirmed by problem-solving applications in fundamental and applied sciences.

Author Bio

Andrei Kirilyuk is a theoretical physicist (since 1979) and Senior Researcher at the Institute of Metal Physics, Kyiv, Ukraine. He also worked in France (Paris, 1991-95). He is the author of the Universal Science of Complexity, the intrinsically unified, causally complete and problem-solving extension of conventional theories.

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John Hodge

Could the sequence to complexity be - technology ->new data -> mysteries (data inconsistent with models) ->ad hoc additions to the models (holding on by the fingernails to the status quo) ->`superstitions' (metaphysics) -> complexity.

Perhaps the solution is a revolution in the concept of the model to encompass the mysteries and the established models. That is, a radically `new' causality model is required. I had thought this is what the FQXi is seeking.

The Scalar Theory of Everything (STOE) model

started with a concept for one potential field to be the `space' of general relativity and the medium (plenum) for the quantum mechanical wave. It has addressed many mysteries and corresponds to the Big Bang model and to Quantum Mechanics

Andrei Kirilyuk

John, a priori there may be two general kinds of "revolutionary novelties" or "problem solutions", one centered on a "new entity" (until now unknown or neglected) and another one on a "new kind of dynamics/behaviour". My "unreduced complexity" and its "extended mathematics" is of the second kind, while your proposal is apparently closer to the first kind. While both kinds of novelties are not impossible in principle, I strongly favour the second kind (in any version) and would doubt the reality of the first one. This is because the addition of any new entity produces too many new features, including those we don't need, just due to the inevitable interaction of the new entity with the existing ones. Thus, any new "field" would correspond to new fundamental forces and even new dimensions (details can be found in my papers, but I hope it can be generally accepted here by evident "physical" considerations). That's why, by the way, I also reject the "standard" Higgs field idea. Those additional features it should produce are definitely not observed.

Or else you should deny the reality of the existing major entities (fields, particles, forces, etc.) and derive them and their effects from the new ones you propose. I could not catch it by a quick glance at your analysis and I doubt that it can be an efficient approach, since those standard major entities are variously confirmed by observations...

Christine Dantas

Dear Andrei,

From a first reading, I find that your essay offers a very intriguing presentation of ideas, summarizing a rather wide-ranging set of concrete applications. Your approach sounds interesting and certainly novel, and deserves further attention. This will need more time.

I find it very intriguing your term "unreduced" in the context of your dynamic complexity framework. Although I do not have a complete understanding of what you propose at this point, I intuitively have a picture, and I think your point is relevant if related to the fact that we cannot really understand nature, in its all complexity, by using just reducible frameworks (that is, idealizations that are carried all the way "down").

We need a kind of "non-factorizable" dynamics. It is not just a question of how accurate one can describe nature, but the need for not eliminating the essential "ingredients" from the very beginning, because those are what really matter, if we want to make progress beyond models. It is not just a question of "adequacy", but re-analising how we approach our dynamical universe out there in an equally self-dynamical framework. If this is a reasonable interpretation of some of what you propose, even if at a basic level, then it is certainly compatible with some of my points.

Best regards,

Christine

Andrei Kirilyuk

Thank you, Christine, for your kind estimate and keen understanding of my results. Yes, "unreduced" complexity is to make difference from imitative model "complexity" being just mechanistic "intricacy" of basically simple patterns (like entangled one-dimensional thread). The unreduced, "dynamically multivalued" version is indeed about non-factorizable and self-referential dynamics. And, following your hint, it's a kind of intrinsically creative and co-evolution framework, where interaction gives rise to complex-dynamical patterns that give rise to higher-level interactions, etc. That's why it emerges scientifically "as a whole", in the form of intrinsically unified multi-level structure and evolution.

Edwin Klingman

Dear Andrei Kirilyuk,

Your observation and analysis "shows a strange 'inverted-progress effect', where the number of difficult, 'unsolvable' problems, or 'mysteries', remarkably grows..." and you note that "usual mathematics does not propose the consistent, unreduced solution to the arbitrary real interaction problem."

I know you are discussing a far more general problem when you urge consideration of unreduced reality (i.e., reality not reduced to 1-D or oversimplified models) but in some cases even a 1-D treatment represents progress over a 0-D current treatment. For example, John Bell analyzes the scattering of a particle in an inhomogeneous field by assuming that the field is constant (thus zeroing out the gradient) and assumes that the resultant scattering continuum distribution is reduced to a binary result. A less reduced (i.e., simplified) local model of the interaction of the particle with the unreduced field produces exactly the quantum mechanical correlations that are impossible with Bell's oversimplified model.

Thus, while your generalized approach focuses on the mathematics involved in the unreduced solution to the arbitrary real interactions problem, it is also the case that 'de-simplifying" the current approach in specific cases to merely a "less reduced" physically realistic model may pay dividends. I hope you will read my essay with this perspective, and provide feedback to me.

Best regards,

Edwin Eugene Klingman

Andrei Kirilyuk

Dear Edwin, thank you for your interest. I have looked through your essay and I think we have similar attitudes against excessive simplification of physical reality within usual mathematical models. You can find more details in my comment to your paper and the details of my solution to quantum-mechanical problems in respective references in my essay (see also the web outline of my Universal Science of Complexity).

John Hodge

Thanks for your response.

The STOE proposal suggests 2 entities (`space' and QM wave medium) are only one entity. This is a simplification or replacement of the 2 with the one.

The idea of `correspondence' is that the successes of the Big Bang and QM are incorporated in the new model. Thus, I don't deny but affirm the successes. Additionally, the model solves some mysteries that have lead to complexity.

I presume the addition of components such as dark matter, dark energy, etc. is increased complexity. These are the `new entities' you suggest. Therefore, do you deny them as I do?

Andrei Kirilyuk

Yes, John, all conventional "dark" entities are but ghosts produced by lacking completeness of usual description (within my approach, I show why exactly, see e. g. [1,2]).

However, pretending for an extension of conventional picture, one should start from derivation of all usual fundamental entities and laws (hopefully without usual mechanistic "postulates" and contradictions), rather than only solution of some conventional problems. So what about electromagnetic, gravitational and other fundamental fields and intrinsic properties or elementary particle nature in your approach? Do you really derive this complete basic set only from your starting entities (even the latter should be somehow justified by real-world properties)? Because if you just add your new entity (at least one) to those usual basic ones, then there will be a new problem - see my previous comment.

Andrei Kirilyuk

Thank you, Joe, for your interest and the discussion. However, as you seem to propose ideas different from both my essay content and the contest topics, I am not sure that it can be the right place here to discuss them in detail. I am certainly not against the discussion of any kind of science-related vision, and as the Earth is round and small I hope there will be another opportunity for it.

Joe Fisher

Dear Dr. Kirilyuk,

The FQXi.org Guidelines state: The goals of the Foundational Questions Institute's Essay Contest (the "Contest") are to: •Encourage and support rigorous, innovative, and influential thinking connected with foundational questions;

My ideas concern reality. Yours concern abstraction. Please refute me or accept me.

Joe Fisher

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)...

Michael Goodband

Dear Andrei Kirilyuk

I am very impressed by your approach to address the issue that underlies the stagnation of results in fundamental science: unreducible states really exist in the world, and they exist in systems across the whole spectrum of science. There-exists a whole class of self-referential systems for which I have proven any physics theory modelling the system components directly will be subject to Gödel's incompleteness theorem (see my paper). In my 2012 FQXi essay I discussed this as being the underlying physics behind quantum theory. Quantum field theory says that a particle is a self-referential dynamic state of interactions. If you attempt to model the components of the dynamic particle interaction state in a theory, it would be subject to Gödel's incompleteness. The conclusion is that a particle is an unreducible state. However, since Gödel's result depends on natural-number based maths it can be bypassed by switching to a real-number basis in a physics theory. If you do this for the particle case, what you find is mathematically the same as quantum theory. Conclusion: the fields of quantum theory are a way to describe an unreducible state. I discuss further in my 2015 FQXi essay how quantum theory is inevitably what you get from having to bypass Gödel's incompleteness to get a usable physics theory.

The underlying proof of when Gödel's theorem applies in science is in terms of network dynamics, and this same dynamics can occur elsewhere in science. But the quantum theory trick won't work in these cases. Another approach is needed, and I think you have it. In Agent Physics I look for alternative approaches to modelling unreducible dynamics states that self-reference and independently arrived at the same conclusion about approach as you:

"In this way, the universal science of complexity presented here clearly specifies the origin and avoids various manifestations of the intrinsic incompleteness of usual mathematical framework of science, including the famous Gödel's incompleteness."

Best regards

Michael Goodband

Andrei Kirilyuk

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...

Joe Fisher

Dear Andrei,

You do not think that the most important question of the Foundational Questions Institute does not concern how the real Universe operates.

I state real light is inert and there is no physical space. You just keep repeating other folk's abstract mathematics and physics.

Joe Fisher

Andrei Kirilyuk

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).

Andrei Kirilyuk

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?

Andrei Kirilyuk

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").

Joe Fisher

Dear Andrie,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

Andrei Kirilyuk

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.