Essay Abstract

Despite how fundamental the concepts of undecidability, uncomputability, and unpredictability may be to our understanding, these concepts are, at best, not well defined and, at worst, part of an interpretation that reinforces long held misapprehensions of reality.

Author Bio

Professional programmer/systems analyst, self-taught metamathematician

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Dear Jason,

thank you for a well elaborated and very interesting piece of paper that tries to tackle the questions that have been posed by the essay contest's current theme. I very much enjoyed reading your lines of reasoning and also your good explanations of what Gödel's theorems say, according to Gödel himself. Good work!

Let me annotate a crucial detail: at page 7 you come to the conclusion that a formal system that is subject to Gödel's theorem must - logically - considered to be a liar.

I think this conclusion critically hinges on a special understanding of a negation within deductive formal systems. For example in physics, if I assume the fundamental workings of the physical world to be *not* deterministic, in the framework of thinking about the world in terms of *formalizability* this assumption forces me to adopt a mathematically probabilistic view of the universe. It is hard to escape this either-or game - and consequently the theorem ¬ ¬φ = φ seems to be universally true.

But wait a minute, is this theorem really universally true - and your conclusion at page 7 therefore should then be somewhat mandatory?

I think the answer is no, since if something isn't fully deterministic, is mustn't therefore be necessarily probabilistic - it well could be

1) either a mixture of both

2) or it could be non of both

Here case 2) is interesting. What could it be beyond being deterministic *or* probabilistic? In my essay I annotated that for a computer program to be conscious and to understand the halting problem, that computer program had also to realize that there is something beyond its in-build step-by-step deductions. From the point of view of formalizability of all-there-is being the one-and-only fundamental description of reality, this "something that is beyond step-by-step deductions" is really hard to find. What could it be?

Well, the reason that computer programs exist is due to a design process of human beings - and if a computer program would be conscious and could transcend its pre-determined paths, it would have to realize that its origins are beyond what its lines of coding say about the world (or about some particular part of the word).

In the same sense the LEM is not universally valid - not because logics isn't valid, but because logics and Gödel's results point to a reality that isn't fully formalizable. Trying to understand Gödel's results as a guarantee for the universality and fundamentality of (mathematical) classical logics is misleading in the same manner in which a computer program might conclude that its origins must lie somewhere within its source code - instead beyond the latter in the realm of a deliberate creation. So the truth that has been discovered by Gödel's results is a truth about the limits of any formal system to ever grasp the *true* origins of itself: its a negation, but a passive one that does not negate that there exist these origins. It only says that it cannot determine it from within itself. Consequently there has to exist a profound truth that cannot ever be discovered with the help of some sufficiently robust formal system and I take this as a hint that my mental world isn't exclusively only a formal system.

So, in my humble opinion Gödel's results point to the limits of what formal systems can truthfully say about the whole of reality: those systems say something profoundly about the formalizability of reality, namely that the latter cannot be fully formalized in principle. If true, then astonishingly this truth can be "inferred" by conscious beings - but I am well aware that "inferring" does not mean "proof". On the other hand, to avoid your conclusion at page 7 this "inferring" seems to be mandatory to me, at least if one wants the world and logics to be universally consistent. So under the premise that the world is indeed universally consistent I would consider the fundamental impossibility of formal systems to encapsulate all truths of reality. Indeed I would consider Gödel's results as an impossibility proof for all formal systems to ever discover more of ultimate reality than merely its formalizable parts.

Thanks again for a thought-provoking essay!

Best wishes in the contest,

Stefan

    Well said Stefan Weckbach,

    I feel Godels theporem is not applicable in Cosmology. I created and tested Dynamic Universe Model for the last forty years, I never encountered some situation as said by Godel.

    You may please have a look at my essay A properly deciding, Computing and Predicting new theory's Philosophy

    Best Regards

    =snp

    Stefan,

    Thank you for your detailed review/response. And I agree with most of your analysis. Where I do not agree I do not disagree (pun intended).

    I especially agree with your identification of the problem (or at least part of it) as "negation within deductive formal systems." In fact, this is one of the (many) things I had to cull from this short essay.

    However, I would have a different focus. Consider a text entry field for the input of some kind of inventory. In this case I would have three distinct options; 1) enter a non-zero inventory, 2) enter a zero inventory, or 3) enter nothing (i.e., do not enter anything into the field). 2 and 3, although quite similar, are quite different. 2 could be interpreted as "no-thing" and 3 as "nothing". I will not belabor this perspective.

    The perspective you presented, "1) either a mixture of both; 2) or it could be non of both" sounds something like the Buddhist catuskori that Graham Priest talks about (see, for instance, Pointing at the Moon - Buddhism, Logic, Analytic Philosophy), which (simply) says there are four options: True, False, Both True and False, Neither True or False.

    Of course, no perspective could be as bad as what could be called the "probabilistic gradient" proposed by some (i.e., that there is True and False and then also shades of gray in between).

    As interesting as this topic may be, I think the question of to what extent our scientific (formal) systems reflect reality or if they are simply employed (like a Pragmatist or neo-Pragmatist would) to predict reality is a more fundamental question. In that case, it doesn't matter if negation is unrealistic, only if it works. Subsequently, a more interesting question is if the two aspects mentioned, reflecting reality and predicting reality are complementary, in the sense of Bohr's principle of complementarity, which would mean that they are (intrinsically) incompatible perspectives.

    What would it mean (i.e., what would it say about reality) if our scientific (formal) systems could be realistic or effective, but not both?

    In any case, thanks again for your consideration and the discussion.

    Jason

    Hi Jason,

    thanks for your reply.

    Although I am not into Buddhism, the latter surely has produced some profound intellectual heritage for the world.

    In my last two essays I argue that to overcome the "catuskori", one could label all attempts to reflect reality as being generically incomplete.

    Predicting reality in my view critically hinges on whether or not reality is completely predictable. If there are areas where it isn't, it may be useful trying to figure out what the reasons could be. Maybe that could also shed some light on how to more meaningful reflect about reality (maybe, but not necessarily).

    "What would it mean (i.e., what would it say about reality) if our scientific (formal) systems could be realistic or effective, but not both? "

    Very deep question and interesting in its own rights. Although there surely would be a multitude of interpretations about that question (just like for QM), spontaneously I think of the world as a kind of simulation (abstraction) - it works effectively, but it is hard to realize that it is rather an unrealistic environment - compared to the reality outside of that simulation.

    I you like and are interested, I would be happy if you would read and comment on my own essay, I would be interested in your opinion.

    Best wishes,

    Stefan

    Note that there are two typos on page three. In the discussion of line 10 the two instances of the formula φ(x) are missing the negation symbol (the negation symbol is not missing from line 10). In other words, the two instances of φ(x) should be ¬φ(x) in the interpretation of line 10.

    I am surprised that you do not mention quantum logic as an argument against the law of the excluded middle. The idea is that you can explain the double-slit experiment by saying that the particle goes thru one slit or the other, but if we deny the LEM, then we cannot say that the particle definitely goes thru one of the slits. That is, the disjunction (particle passes thru slit A OR particle passes thru slit B) is true, but particles that just go thru one slit do not show the diffraction pattern.

      Your question has two answers (pun not intended). First, I did actually hint in that direction by mentioning Bohr's principle of complementarity. Although complementarity violates the LNC, if the LEM and LNC are actually equivalent, then complementarity would also violate the LEM.

      Second, and I think more importantly, to claim that the wave nature of light (i.e., the double-slit experiment that you mentioned) violates the LEM would rely on interpretation and thus it would be a weak argument that is not very clear and thus would require much more examination/discussion, which is not available in this relatively short essay. This is because an established interpretation of quantum theory does not exist.

      In other words, in the same way that it is said that quantum theory seems to refute determinism but it is not generally accepted as a conclusive refutation of determinism, it would equally be the case that quantum theory does not constitute a conclusive refutation the LEM. For example, consider a holistic interpretation of quantum theory that assumes an underlying holism and thus also that particles and waves are just convenient fictions. The LEM would not necessarily be violated in that interpretation.

      Hence, I chose to focus primarily on Gödel's proof because his proof provides a compelling and clear potential refutation of the LEM whereas the argument from quantum logic relies on interpretation and thus only provides a weak argument that is not very clear.

      However, some interpretations, such as the one you implied, would provide a refutation of LEM so, behind Gödel's proof, it could certainly be counted as further evidence worth considering.

      Dear Jason,

      While I am not familiar with the Godel incompleteness theorem, I learned a lot of things from your essay.

      I like the following point.

      the real underlying question is whether the fundamental goal should be to

      construct formal systems that accurately reflect reality or to construct formal systems that can accurately predict reality.

      On quantum mechanics, what do you think about the "verification" of quantum mechanics in this context? This verification context in computation is shortly written in my essay. I think that you are interested in this point.

      Best wishes,

      Yutaka

        very nice essay, love the mathematical aspect of your essay.you earned my votes. For theories are manufactured by humans.Can perceptual limits to Cognitive abilities be a source of incompleteness.Kindly read here-https://fqxi.org/community/forum/topic/3525.will appreciate your feedback. Thank you, All the best in the contest.

          As I stated in my comments on your paper, yes, I do think that "verification" of quantum mechanics could be of utmost importance. This idea is certainly relevant to the ideas that my essay was exploring, especially within the context of the line that you quoted. Would a truly random number be invented (generated ex nihilo) or discovered?

          Thank you for your kind words.

          You wrote: "Can perceptual limits to Cognitive abilities be a source of incompleteness."

          Or also, could cognitive limits to Perceptual abilities be a source of incompleteness? The plot thickens.

          Dear Jason,

          Thanks a lot for answering my comment.

          > This idea is certainly relevant to the ideas that my essay was exploring, especially within the context of the line that you quoted.

          Does it mean that "verification" and the Godel incompleteness theorem are relevant?

          > Would a truly random number be invented (generated ex nihilo) or discovered?

          Let me see. Our trials keep going now.

          Dear Jason,

          This was a very stimulating and well-argued essay.

          If I would have any complaint, it would be that the notation appeared to me quite obscure at times, especially in the part on Godel's proof.

          I would have some questions:

          From LEM and LNC we can construct reductio ad absurdum arguments in logic and mathematics. Do you have any thoughts on the value of this type of argument in physics?

          Could you please elaborate on what you mean by "meaning" in the introduction?

          Many thanks and good luck for the contest.

          Best wishes,

          Fabien and Matthew

            Dear Jason,

            I believe that the 3 Un's , as creations of skuld entities, are certainly intertwined with object reality, as we are part of that reality ourselves.

            Even if the Universe is not infinite, it will always be beyond our reckoning, and thus we are forced to make assumptions, which inevitably lead to paradoxes. Paradoxes are good if they force us to re-evaluate our assumptions. Thus over a long period of time, using the scientific method, we will progress to a better understanding with less paradoxes, especially if we drop LEM when formulating our theories. Whether the Universe is deterministic or not will not really matter, as long as the future is open.

            Best Regards

            Lockie Cresswell

              You wrote: "From LEM and LNC we can construct reductio ad absurdum arguments in logic and mathematics. Do you have any thoughts on the value of this type of argument in physics?"

              Yes, as a matter of fact, your question identifies what I would consider one of the principal ideas of my essay. The only way a proof by contradiction (which is based on the principle of reductio ad absurdum) could work in physics would be if reality is not contradictory. This is one of the fundamental ideas underlying (the famous psychologist) Carl Gustav Jung, which is that it doesn't appear that human beings are very adept at separating ontology (what is) from epistemology (what we can know). Are propositions "part of reality?" Or are they wholly mental and thus "merely part of our mind?" In other words, is reality logical or is logic simply a feature of the mind that thinks about reality? Obviously, this requires much more discussion, which is why I merely hinted at it it my essay. I actually emphasized this aspect of Ian Durham's essay in my comments on his essay.

              You wrote: "Could you please elaborate on what you mean by "meaning" in the introduction?"

              Within the present context what I meant to imply was simply that it is "more philosophical" and less "technical/formal." Or, furthermore, within the context of your preceding question, you could interpret my statement to imply that the concept of predictability is inextricably intertwined with the mind that predicts, whereas it is not obvious, at least not directly, that the concepts of undecidability and uncomputability are so entangled. I would say that my comments on Flavio Del Santo's essay almost perfectly illustrates the esoteric philosophical issues that I was trying to avoid.

              Thank you for your nice comments and thoughtful questions.

              Hi Jason!

              Thank you so much for your essay! It was a very nice read. Just for fun, what are your thought about a reality in which there are no observers? Do you think such a thing is possible? And second, do you think the 3 Uns remain the same for any possible observer? (Basically, I'm wondering just how objective reality is with respect to every possible observer, and how this could change our intuition about the three un's.)

              Cheers!

              Alyssa

                I agree with your conclusion that HUP is a fact of reality. But I believe its interpretation has been overstretched, way beyond what Heisenberg originally intended. As I have mentioned in other posts I do not believe uncertainty in momentum can be translated to meaning enormous energies for preon particles acting at much smaller scales than protons or neutrons. Objective reality is what it is. It is only undecidable, uncomputable and unpredictable when we apply our approximations to it, and I believe that is all we can ever do. However the current coarse-graining of the 3 un's will improve over time thanks to the scientific method, better engineering and better philosophy.

                LL&P

                Lockie Cresswell

                Dear Jason,

                I greatly appreciated your work and discussion. I am very glad that you are not thinking in abstract patterns.

                While the discussion lasted, I wrote an article: "Practical guidance on calculating resonant frequencies at four levels of diagnosis and inactivation of COVID-19 coronavirus", due to the high relevance of this topic. The work is based on the practical solution of problems in quantum mechanics, presented in the essay FQXi 2019-2020 "Universal quantum laws of the universe to solve the problems of unsolvability, computability and unpredictability".

                I hope that my modest results of work will provide you with information for thought.

                Warm Regards, `

                Vladimir

                You wrote: "I believe that the 3 Un's , as creations of skuld entities, are certainly intertwined with object reality, as we are part of that reality ourselves."

                Ah ha! I had to read your essay to understand what "skuld" is! I think your perspective is interesting and it is one that I certainly do not disagree with. However, I would not say that we need to "drop LEM" because I think reality beckons a more "sophisticated characterization." Personally, I would like to begin with the Hindu Sﾄ［khya philosophy or possibly the Daoist Yin/Yang.

                You wrote: "I agree with your conclusion that HUP is a fact of reality."

                Oh no. I did not say that it was. I simply said that was the "generally accepted interpretation." However, I do not disagree with that interpretation. Although I do not explicitly support it either. I think the Copenhagen interpretation is better than all the other interpretations that have been considered and thus I also think the quantum particles are nothing more than convenient fictions. But I do not think the Copenhagen interpretation is "the final word" either.

                You wrote: "Objective reality is what it is."

                Obviously.

                You wrote: "It is only undecidable, uncomputable and unpredictable when we apply our approximations to it, and I believe that is all we can ever do."

                I disagree with framing the question within the context of the "subjective/objective dichotomy" because I think that dichotomy is absolutely wrong. I do not think the 3 un's are merely epistemic, but neither do I think they are somehow an aspect of "objective reality". They are part of what is, as all things are. And the phrase "all things" does not accurately account for all that is because "everything that is" are not all "things." This where (the English) language fails us. But that may be going a bit too far, don't you think?

                You wrote: "Whether the Universe is deterministic or not will not really matter, as long as the future is open."

                The future has always been open and will, in all likelihood, always remain open regardless of what we think or believe or what "scientific theories" we conjure. I would only add that, even though it may be questioned whether contradictions are permitted by reality or not, I do not think reality permits discontinuities. The future is always constrained by the present. Or, in other words, what will be always "proceeds from" (i.e., is an extension of; is built on) what is. There are no gaps. Now I have certainly went too far.