Essay Abstract

Development of Theory Of Everything (TOE) has a fundamental issue of baggage which is understood as a set of notions and assumptions underlying a theory. TOE should at best be baggage-free but this is problematic with theories based on algorithmic representations. We show how uncomputability may provide a new conceptual framework for the TOE without baggage. Special uncomputable sequences are introduced and it is shown how a looped self-referential system of symbols from sequences and sequences from symbols is formed. Properties of this system are analyzed emphasizing symmetry requiring operation of infinite permutation groups. It is shown how mathematical structures emerge and how it is possible to tackle nothingness and its relation to uncomputability. Our development points that TOE is a special mathematical structure arising from the uncomputable substrate intrinsically tied with nothingness and dissolving symmetry of this structure results in physics. This indicates that uncomputability forms ultimate foundation of physics enabling to answer deepest questions about the Universe.

Author Bio

I am retired Professor of Information Technology in the area of Signal Processing and Networked Multimedia Systems. My career was in on what became the information backbone of the current life but around 2008 I noticed that essential stuff in this area was ready and got interested in the informational approach to physics. I decided to study fundamental aspects of this problem and when I retired a couple of years ago it became the main focus of my brain activity.

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Dear Irek,

I found your essay to be most thought provoking. I enjoyed reading it, and I would like to make a few comments.

Early on you state "Maybe at the deepest level there are no algorithms, infinity is common, real numbers are basic? The Uncomputability signpost looks then not so surely pointing into a blind alley but maybe to something foundational. At least it fits perfectly into the Wheeler line of thinking which makes its exciting to exploration." If you consider a material Universe made entirely of particles (matter and aether) then what you say may be true, although I suspect 'infinity' is may not be needed (or true).

You state: "If that is not strange enough we are now able to tell something about nothing. Nothingness would typically be thought of like an empty set in the category of sets. This sounds like an algorithmic description but nothing should not be algorithmic because algorithms describe 'something'. Nothingness thus should be somehow non-algorithmic." Zero is a number that has an important role in many algorithmic structures, and I think you can apply this to your 'nothingness' argument as well. I would then ask why should nothingness be non-algorithmic?

"Where is the TOE lurking? Or, you can ask, where is the physics here? If anything, we could only see fleeting appearance of some basic mathematical structures on top of uncomputable sequences. Where is then the usual stuff of physics, fields, forces, waves, particles, matter and the Universe? The answer is that we only covered the foundational TOE level from nothingness to simple mathematical structures which are indispensable for physics. We also see that there are colossal permutation groups in action which makes possible emerging of mathematical structures with much higher complexity." It is a good question that you pose. We need very special structures to create the stuff of physics. Our best approach, whether top-down or bottom-up, is to use some or all of the physical framework that we have already created through centuries of scientific experimentation. We cannot do that by sifting randomly though endless mathematical permutations, although Wolfram shows us the path to this methodology. I truly believe all mathematical structures are created by intelligent entities, and that there is no Platonic realm, as envisaged by many.

In your conclusion you argue "We indicate here that mathematics exists, mathematical structures emerge virtually out of nothingness tied with uncomputability. Due to the richness of possibilities in our framework physics appears then as a very special mathematical structure which provides strong support to the notion that physics can be completely reduced to mathematics and all mathematical structures exist." I support some of this statement, but I do not agree that all mathematical structures exist. I am not even sure that a computational TOE is a mathematical structure.

I cover some of this ground in my essay "Wandering towards a 'Theory of Everything' and how I was stopped from achieving my goal by Nature", where I look at my own physical TOE from a computability point of view.

It may be that the mathematical structures you seek are the proverbial "Emperor's new clothes'.

Best wishes on your well constructed essay.

Lockie Cresswell

    Dear Lockie,

    Thank you for reading my essay. Now I also read your essay and I am impressed by the breadth of your thinking, e.g. I was totally not aware about the Norse mythology.

    You certainly grasped a lot from my essay and I see your points of disagreement. It was difficult to state precisely my ideas in short essay and the questions you rise I see as partly resulting from the lack of in-depth elaboration by myself, work is waiting to be done on this. For example you state "I suspect 'infinity' is may not be needed (or true)". I am not saying that infinity is needed or true in the Universe. What I am saying is that the substrate from which the Universe emerge is uncomputable/inifinite. One could think that the substrate is 'pre-Universe'. This goes towards solving apparent contradictions that infinities should not be present in physics/Universe but on the other hand theory landscape is full of them. Now about nothingness and zero. I am glad you mention zero in this context. The innocent-looking zero has in fact a lot of baggage behind it, number theory, algebra and so one, you can write e.g. 0=5-5. This context is thus algorithmic, and what is needed is the concept of nothingness which is free of such baggage which points into nonalgorithmic. This is why a I came to the concept of nothingness rooted in the special kind of uncomputability described in the essay. Besides as I mention nothingness is of this special type of object which is tricky, if you say nothingness exists it could mean object with some properties and baggage but nothingness has no properties and baggage.

    The belief that mathematical structures are created by intelligent entities is widespread, but in my view the opposite belief is gaining track and I am showing arguments for it: mathematical structures *emerge* from uncomputability. This is not Platonism in which the structures are living in some ideal world of ideas.

    In my development mathematical structures exists intermittently due to the action of symmetry groups, only some of them are 'stable' and one of those is physics.

    The crucial, absolutely fundamental issue about the TOE is the theory baggage. Let's say you claim the foundation of TOE are cellular automata, Hilbert spaces, algebraic topology, etc. Then there is a question, where they are coming from. This is the same with any other claim that TOE is this or that. In the end it must be then so that the TOE is non-algorithmic because whatever algorithm, equation, one shows it carries a baggage and the question is why. TOE with no baggage has to be hence self-referential which of course is extremely tricky. I am furnishing idea how this could be possible with uncomputability. Relaying on uncomputability has an advantage that it is beyond finite and algorithmic baggage.

    Thank you,

    Irek

    Dear Professor Irek Defee,

    You have presented a wonderful essay with deep optimism that a TOE is possible mathematically. May be it is possible, but with sophisticated mathematics and with complex numbers with imaginary axes.... we dont know how to understand the final or intermediate results and to interpret them. Leading to "Going nowhere situation".

    Probably we can go for a simple mathematics, and see how the mathematics is explaining the physical situations. Then Uncomputability problem will not be there.

    Hope you can spend some time with my essay

    "A properly deciding, Computing and Predicting new theory's Philosophy" where I discussed these issues of mathematics in Physics and in Pure Mathematics

    Best Regards

    =snp

    Dear Irek,

    I answered your message in my essay forum. Here is only a copy:

    "thanks for your interest and your words about my essay. I also read your essay with great interest. I understand now much better what you mean that uncomputability is central for you approach.

    Ineed to say more about your essay but give with some days."

    All the best

    Torsten

    your focus is offering some explanations about something interesting, my interest is more in how you do that. i saw you commented on my essay.

    4 days later

    Dear Irek,

    interesting essay, a phyisical process can be also seen as a sequence of measurement values. It is a extreme view but also true. Now the correlations inside of the sequence give you the physics back. Also symmetries are encoded in this sequence. It is not easy to recover all these properties from the sequence. Even symmetries are not directly viewable. Every symmetry group must be represented by a permutation subgroup. The most interesting point is the introduction of the real numbers. It is clear that one needs infinite sequences to get real numbers. Interestingly, here is an ambiguity. You used the standard completion of the rational numbers to the real numbers. But there is a second one (and only a second one). This completion leads to the p-adic numbers and your discussion of the tree of symbols seems to imply that this completion is the natural one. Therefore I recommend to investigate the p-adic numbers

    I also investigate models containing discrete information but which are given by continuous manifolds (wild embedding like Alexanders horned sphere, a fractal space). Maybe here we disagree: I think real numbers are needed to represent the model. The informational content is discrete. I discuss it in a previous essay.

    I hope these points are the start of our discussion.

    Best wishes

    Torsten

      Dear Torsten,

      Thank you for your remarks. They are very valuable since my essay introduces notions which are outside of standard thinking and thus they have to be explained with utmost clarity and examples to be understood the way I intended, otherwise it be seen as a junk. In the essay explanations were compressed to minimum and thus clarity is not at the highest level. This obviously implies that I must prepare extended exposition of these ideas and I am looking into it.

      P-adic numbers are covered recently by Tim Palmer in an interesting way and he also has an essay here. However my questions are more fundamental: If p-adic then where they are emerging from? In particular there will be issue, why this particular p?

      Uncomputable sequences appear in my essay due to critical problem of theory baggage and background which is critical for TOE. TOE is very tricky in this respect, if somebody claims having developed TOE and starts with e.g. 'assume Minkowski space' there is immediate question why this and where it is coming from? Even the usage of real numbers can not be taken for granted in the TOE. In this sense QM is at a significant distance from TOE since it assumes huge amount of background stuff.

      So I am starting from the level below real numbers, everything should be emerging due to symmetries.

      I will be happy to discuss the issues further, if you wish you can contact me directly to my email which is in the essay headline.

      Br,

      Irek

      5 days later

      Dear Irek

      I just read your essay which is well thought and written. As you know from my essay, I agree that the physical theories must be written in mathematical terms, however, I also support the view that we should develop a physical understanding to have a complete view of reality. In your last part, you mention that "all mathematical structures exist". In my experience, there are many mathematical structures that have not found any applications in any science, so I would say that only some mathematical structures exist. Your proposal is interesting although I feel that I need to study some aspects of group theory and symmetries. You may wish to read Sabine Hossenfelder, she also denies the existence of infinity and real numbers.

      Good luck in the contest!

      Israel

      5 days later

      Dear Irek,

      I have given your essay a high rating as it was very worthwhile. It is disappointing to see you have been trolled with 1's. I hope others will rate you up higher, because your essay desrves better.

      Regards

      Lockie

      Irek - A stunning tour de force, thank you. You've been able to articulate in mathematical structures what I was searching for In my 2015 essay The Hole at the Center of Creation (https://fqxi.org/community/forum/topic/2381 ) - the infinite void that lies at the frontiers of our our understanding. Yes, nothingness hides in infinity and vice-versa, and the TOE resides in the structures of perfect symmetry beyond the axiomatic realm. Which means it resides nowhere. I love it!

      I hope you get a chance to read my essay about the foundational importance of autonoetic consciousness.

      George Gantz: The Door That Has No Key: https://fqxi.org/community/forum/topic/3494

      This came to mind as well: (Rubaiyat v XXIX)

      Into this Universe, and Why not knowing

      Nor Whence, like Water, willy-nilly flowing;

      And out of it, as Wind along the Waste,

      I know not Whither, willy-nilly blowing.

      Dear Irek,

      Excellent essay. Good approach, nicely expressed. Yes, I DO suspect we can close in on a TOE, if not with current methods. The one thing I'd like to pick up on is your suggestion; "maybe at the deepest level there are no algorithms".

      I consider that an important comment, but do you see it as a statement of ontological methodology? or as a final limit once we HAVE a widely consistent ontological system? I ran into that in my essay, from proposed better foundations, but wondered if higher order precision really mattered!

      I hope you'll study mine and respond. But in the meantime it seems yours may have been hit with multiple 1's like mine, my score reflects it true high value.

      Very best.

      Peter

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