Dear Helmut,

I also wouldn't say that math is the ultimate guide for a fundamental theory of reality, but only that it is the language we should use. And I do agree that math has its own blind spots and most of them are precisely on the foundations of mathematics. A also believe math must be improved for being used in such a fundamental theory.

About Gödel's incompleteness theorem, I think the point is about completeness. I do not want to make a theory that is complete in this sense, but consistent, useful and insightful. And this theorem may also affect any other formal language, every language has some limitations.

A theory is complete when every formula in its language either is provable or its negation is provable. In other words "if it is consistent, and none of its proper extensions is consistent." But, a fundamental theory of reality must have proper extensions because every theory that describes an aspect of reality is an extension of it.

Self-referential system can be very powerful, but they are also very dangerous. Well, you may find a consistent one, but most of the paradoxes of logic and mathematics were found in self-referential systems. So we need extra care when dealing with these systems. But I believe Gödel's incompleteness theorem represents no danger for metaphysics because of what I mentioned before. I think you don't need to worry about this theorem. Make your theory and if someone says it is impossible then simply show you have already done it!

P.S. I'm taking a look on your paper. I'll rate it as an author; please rate mine.

Dear Frederico,

I am not quite sure whether we agree with respect to the role that mathematics is playing in physics.

To discuss the case of metaphysics: In my view of a modern metaphysics all formulas without any exception have to fail, otherwise the metaphysical concept of TRANSCENDENCE does not have any physical meaning. This demand implies that we have to extend our theories to infinite values, f.e. to an infinite velocity, but just these values are usually regarded as a sign that the underlying concept is physically meaningless.

What do you think about this? Are there infinite values involved in a truly fundamental theory of the universe?

Kind Regards

Helmut

    Dear Helmut

    My point is that math is a precise language, and any theory, when it gets enough precise and clear, it can be formulated in a mathematical language. Therefore, even a metaphysical theory can at some moment be, in principle, formulated in mathematical language. Can you already talk about transcendence using math? That's the question. However, not everything that can be described by a mathematical equation is a physical observable. In the theory I propose in "On the Nature of Reality", everything is described mathematically but there are many elements that are not physical observables. But can you explain what you think of transcendence?

    Well, I see no problem of infinities on physics. The problem is only with math which cannot hand infinities very well. Once our math could handle them properly and elegantly they will begin to appear on our physical theories.

    Best Regards

    Frederico

    Dear Hải.Caohoàng,

    Thanks for your encouraging comments. Definitely, to find a clearly defined basis for foundations physics is the greatest of all intellectual challenges; and we should work together to accomplish it.

    However I have to admit that I don't have much knowledge about the standard model and Higg's bosons. I have yet no opinion about these things and so I cannot answer your questions. I hope we could find something else to discuss. I'm reading and I will rate you essay as an author. Please rate mine.

    I believe no theory is simply correct or incorrect. Its correctness depends on where it is being applied, that is, if its assumptions holds in a specific domain, then the theory is correct on that domain, if the assumptions does not holds on that domain, then it is incorrect in that domain.

    Best Regards!

    Frederico

    Dear Frederico,

    transcendence is, of course, an epistemological term indicating that the most fundamental branch of reality (i.e. the ONE) cannot be described and observed in any way. Transcendence is the common feature of all religious systems.

    But to fill this purely epistemological term with a physical resp. metaphysical content you have to choose an ontological one. The most appropriate term in this regard is the term of INVISIBILITY, because something to which this property is ascribed, cannot be described or observed in any way. It is epistemologically a property of transcendent nature.

    Though the ONE itself being invisible cannot be described or observed -- the visible resp. observable Universe, of cource, can be described and observed.

    Now at this point I asked myself: How must the visible Universe be organized if it shall base on something invisible?

    Answer: It must have unavoidably a sort of radical non-dual conception.

    Argument: As the property of visibility (of the universe) depends on the existence of specific differences, something, that shall be invisible, must have solved all differences into ONENESS. Otherwise it would be visible.

    And just this general philosophical idea of ONENESS that is as old as Platon can physically be precised. I am calling this precised idea the --- radical non-dual conception of the universe ---. This metaphysical conception implies physical consequences that are provable from an empirical point of view.

    It implies f.e. the prediction, that at the outermost edge at the universe there must be a coincidence, otherwise the invisiblility of the ONE would not be secured.

    Such an empircial coincidence can actually be found at the edge of our universe. In modern physics it is known as Mach0. You can google this; choose the key word - Mach Principle -; there have been listed 10 versions of this principle. Mach0 is the first one: It is indeed the only one, that is explicitly expressed as a purely observational fact.

    Surprisingly this observational fact represents an ANOMALY within modern physics that cannot yet be explained convincingly. To my opinion it is - to make use of an Einsteinian metaphor - a Signature of GOD inside our universe.

    Kind Regards

    Helmut

    4 days later

    Dear Frederico

    your essay and associated paper are thought provoking and deep. It will take time to assimilate it. My main comment for the present refers to this statement of yours:

    "I have means to say that the main wrong assumption of physics is not

    a physical assumption, but a millenary logical assumption: the principle of excluded middle .. This principle says that a proposition is either true or false, in other words, either the proposition or its negation is true" I think that you might be saying that the truth or falsity of a proposition may depend on its context. That is very close to the concept of contextual effects that I discuss in my essay.

    George Ellis

      Dear George,

      Thanks for your comments and sorry for the late reply. Actually, what I mean by the violation of the principle of excluded middle can be understood when we think of a system in a superposed state. For example, when a cat is not in a superposed state, we can say that it is dead when the state is [math]|D\rangle [/math] or it is alive when the state is [math]|A\rangle [/math]. But then, when it is in a superposition [math]|D\rangle |A\rangle [/math], it is not dead but it is also not alive! It is in a superposition of dead and alive! But according to that principle, if it is not dead, then it is alive. Therefore, superposition means precisely violation of the principle of excluded middle! In the classical world there is no superposition, therefore the principle holds in a particular case. I hope it could make the idea clearer.

      Best Regards,

      Frederico

      Dear Frederico,

      i read your essay and you did a good job in examining the suspicious premises that are widly held high. Especially what you write about Commensurability is of interest for me, but i have to think about it some more time and also read your arxiv-paper. Nonetheless you struck me with your statement

      "There, reality becomes a closed and completely precise notion. Therefore, I have means to say that the main wrong assumption of physics is not a physical assumption, but a millenary logical assumption: the principle of excluded middle. It is the source of most of the paradoxes and misunderstandings about quantum theory and is precisely the assumption that gives rise to the classical world."

      For this statement alone you deserved - in my humble opinion - a positive score, nonetheless that my own usage of some exceptions to the principle of the excluded middle are more than epistemological in nature. My own interpretation is that there are ontological states that aren't anymore driven by this principle and i tried to explain this in my own essay. Especially i explain why the "collapse of the wave function" occurs, namely because at the point this does "happen", the causal structure of the system has consistently changed and the mathematical description of the former - unmeasured dynamics - isn't valid anymore. Indeed - as i believe - the schroedinger equation is just an illusion that mimics causality. It is no more than a mathematical tool, misinterpreted as an ontological dynamic process! I call that whole process of quantum mechanical mimics "physical retrodiction". If you like, i would be happy you could have the time to visit my essay and leave a comment.

      Its very easy to read, informative as well as entertaining (i guess) and you don't have to dive into some complex mathematics.

      Best wishes,

      Stefan

      Dear Stefan,

      Thanks for your encouraging comments. My arXic paper might be very interesting because it shows how the ideas proposed in the essay can be implemented. Well the principle of excluded middle is a hidden assumption that is part of the language we use. It is almost impossible to understand how it can be violated using only ordinary language, specially English. Try to say something like the cat is not (not dead)...

      I've explained this idea on a reply to George Ellis some topics before. Take a look and then ask me any questions. It'll be a pleasure to make this idea clearer. I've started reading your essay; I'll rate it as an author, please rate mine!

      The principle of excluded middle is deeply connected to the wave function collapse. The point is that, after the collapse, the principle of excluded middle becomes valid for the set of projectors you used for measuring! Think about it, as I said, most of the paradoxes are related to this principle, and the classical world is precisely where this principle holds.

      Best regards!

      Frederico

        Dear Frederico,

        thanks for replying to my comment.

        I read George Ellis' comment and yours and i agree with both comments.

        George states that classical information depends on the context. He uses the term "proposition" to indicate the premises we have built in to come to a certain conclusion about truth/false values. From the reference frame of a classical observer the opposite of the principle of the excluded middle (for convenience i will call it "non-ex") is the opposite of the principle itself (for convenience i will call it "ex"). So, "ex" and "non-ex" are opposites, and due to the logics and due to "ex", - both the governing-laws of the classical world ("ex" and its illusion of time, causal order and forces) and "non-ex" for us are contextual. Contextual in the sense that both principles are relative, they refer to each other and can be differenciated from each other only by referring to the othe principle. I think this is a hint that beyond the classical world there must be indeed a non-classical world - only due to logical considerations. QM is an indirect proof of this assumption, especially the feature of superposition.

        In my essay i assume the "cat" to be neither dead nor alive, rather than being dead and alive at the same time. I think this is a difference to common thinking about superposed states. I also avoid the conclusion of many worlds which could be constructed (by assuming an extended pure state for every mixed state by assuming the mixed state) is just an illusion due to our kind of thinking in a classical reference frame that assumes causality to be more fundamental than consistency. I think the latter is more fundamental and we should accept incompleteness of information in the classical world. Extrapolating the Schroedingers' wave function to be universally valid only leads to many-worlds; they may be complete in a certain sense and consistent, but the measurement problem for me seems to be not solved (the problem why the mathematical description of Schroedinger does "collapse" at the "moment" of measurement).

        At the weekend i will read your arxiv paper with great interest. I already gave your essay here a positive vote.

        After reading your arxiv-paper, i will give you another feedback on that and maybe you can profit from my points of view (i would hope so).

        Best wishes,

        Stefan

        Frederico,

        I think it's a rather bold conjecture that "The quest for giving a precise meaning to our fundamental concepts cannot be accomplished using natural language" ... without giving an example of a mathematical statement that cannot in principle be translated to natural language.

        So I'm in the uncomfortable position of agreeing with your conclusions while disagreeing with the way that you got there. It is surprising to me that you venture into meta-mathematics in your commentary without mentioning Chaitin's leading-edge research (particularly since he is a Brazilian Professor). For example, Chaitin's Omega gives a clear example of an algorithmic compression of a number that is uncompressible (the halting probability of a Turing machine). This example would seem to stand your conjecture on its head: an example of a meta-mathematical result whose natural language translation is straightforward: The number is algorithmically compressible IFF the halting probability of a universal Turing machine asymptotically approaches zero, regardless of the machine language by which such probability is calculated. With this extreme example, I am doubtful that your conjecture is true.

        Chaitin has extended his meta-mathematical program to life itself, with the recent publication of *Proving Darwin: Making Biology Mathematical.*

        Nevertheless, as a separate subject, I have to agree that what you call a closed theory (and which I would characterize as a closed logical judgment of a mathematically complete theory) is sine qua non -- not only to a final theory -- but to any scientific theory. Closed logical judgements, such as those in relativity, correspond to physical results that make the theory mathematically complete. I like your discussion of quantum theory interpretation, because it clearly exposes why anti-systematic analyses cannot impose a logically objective meaning. Quantum theory is mathematically incomplete. (I don't understand how you reconcile Wittgenstein's anti-systematic philosophy to your philosophy of science, though I would be interested to know.)

        Anyway, thanks for a great read and all best wishes in the contest. If you would like to see an information-theoretic take on the Schrodinger's cat problem, and which explicitly uses the excluded middle, please visit my essay ("The Perfect First Question.")

        Tom

        Minor corrections to my latest post:

        instead

        "by assuming an extended pure state for every mixed state by assuming the mixed state"

        please read

        "by assuming an extended pure state for every mixed state"

        i wrote

        "Extrapolating the Schroedingers' wave function to be universally valid only leads to many-worlds; they may be complete in a certain sense and consistent, but the measurement problem for me seems to be not solved (the problem why the mathematical description of Schroedinger does "collapse" at the "moment" of measurement)."

        i would add

        It's not only the moment of a "collapse" that is problematic (in many worlds the "collapse" does not exist and therefore is not a problem), but also the fact that for every binary decision i make, another "me" realizes the other alternative. The question here is if this alternative "me" existed before *i* made my binary decision or is it somewhat "cloned" by the Schroedinger equation? I assume that the two branches arising out of *my* decision aren't symmetrical in the sense that the other branch (the alternative "me") is forced to decide between *my* binary decision and gains exactly the "opposite" of *my* decision. Here the question arises why there should be multiple quantum experimentalists within a Schroedinger equation that branches those experimentalists in a non-deterministic manner. Not enough, where is the borderline between a classical decision and a quantum decision? A quantum decision surely seems to be random, but the Schroedinger equation says it is deterministic. So even *my* own decisions (for example my decision to write this post) should seem to be as random as single quantum events seem. An interpretation of QM as strictly deterministic does not alter the mystery why *i* got entangled such that i had to write down this comment. It further does not elucidate that human behaviour in most cases is consistent and makes some sense. Although this could be due to past entanglements, but nothing in the whole QM theory and the Schroedinger equations neccessitates that human behaviour should make sense and shouldn't be non-sensical when averaged over the whole human population.

        Dear Frederico,

        i now read your arXiv-paper. Although i must confess that i did not check every part of your statements to be reasonable or not, my main interest was to look at what you have to say in your plain text.

        You wrote

        "The importance of the distinction between pure and mixed states, and the fact that only pure states can be associated with a vector (ket) is dismissed. The distinction show totally different mathematical properties and meaning of each of them. We say that the pure state [ψ] is something objective, actually that it represents reality itself. And the mixed states ρ represents something

        epistemic, that they are the state of knowledge of some observers. This two elements are not, however, completely independent, there is a condition for a density operator to be considered a possible state of knowledge. However, the state of knowledge is not totally defined by reality, and that's why we say that it is subjective. We can say that this theory is a theory of knowledge and truth, and this the explanation of why there are elements of this two natures: epistemic and ontic."

        As i understand this, it is not possible to unambigiously assign some reality to the "time" between two measurements?

        I think this would be difficult, because in the case of Schroedingers Cat (if one leaves decoherence processes out of the discussion) you write

        "Note that we have propositions that are neither true, neither false; and that is something new, because according to classical logic, a proposition is either true or false."

        In the time in between the measurement of the Cat's state, there must be some "time" elapsing, i guess. In my own interpretation, with the help of some ideas of decoherence, i tried to interpret what's the meaning of the "time" between those two measurements.

        It is true that classical logic leads to either true or false values, but if we consider the opposite of the classical logics to return neither true or false values, we cut out a huge part of reality, if we take this new logics to be a passive negation of the classical logic. We can interpret that as the subjective lack of knowledge (the epistemologic part), but this interpretation in my opinion is not sufficient to understand why QM is the way it is, this interpretation is only sufficient to illustrate why QM *itself* cannot give the answer why it is how it is. That's an important difference.

        My own interpretation assumes a meta-physical (meta-physical in the sense that classical logic is only a special case of a logic that isn't anymore bound to causal thinking - but nonetheless to reasons - and to physical time) level of reality where the *reasons* for the non-classical logic reside. In my opinion this level has no classical timeframe, but only conistent interdependencies. Therefore no assumption are needed in this realm and all answers to questions are rendered senseless because questions in a timless, interconnected realm don't make anymore sense.

        Maybe we are not so far apart from each others point of views, except that i speculate about what's beyond the classical logic and epistemological knowledge, and you seem to not do this.

        Thanks again for an exciting essay and an interesting arxiv-paper!

        Best wishes,

        Stefan

        Dear Thomas

        Thanks for your comments. Well, the problem is that we need to translate math to natural language and natural language to math, we need both, and what I call an interpretation is what allows it. Try to read the equation F=ma. In the context of classical mechanics you would say the total force equals to the product of the mass and the acceleration. But out of context, or better, without any interpretation, you cannot say it.

        The problem of natural language is that it is not as precise as math, and there are too undefined concepts or open concepts. For example, define reality? Define proposition? Define truth? These concepts are very hard if not impossible to define in natural language. I cannot prove you it is impossible to define these concepts in natural language, but if you look to a dictionary you will see that some definitions are at most circular and you cannot remove the circularity.

        Really, I was not aware of Chaitin's work. I liked the way you characterized a closed theory, although we need to take with the meaning of mathematically complete, because if it is understood in the sense Godel then we have a problem with his incompleteness theorem. I've read his book but I'm not a specialist on Wittgenstein philosophy, and so I cannot really answer how my ideas are in relation to his philosophy. If you have something more specific...

        There is no problem on agreeing with my conclusions without agreeing with the initial discussion simply because it is only a motivation for studying the notion of closed theory. And I agree this notion applies to all scientific theories.

        I'll visit you essay. Thanks for visiting mine, and pleas rate me if you haven't! I wish you all the best in this contest, and that we could keep our talks beyond this scope.

        Best Regards

        Frederico

          Dear Stefan

          In my paper I do not give much emphasis to the measurement problem, not because it is not important, but because I do not propose any solution for it; at least for now. So I don't have answer to your questions about it. Try to think this way: what can we say before any measurement is performed? That is what I try to answer. In the future I'll try to explain the process but for now I can't.

          I do not negate classical logic, but I say that it holds only in a particular case, so I'm not cutting any part of reality. The point is that, in my approach you can describe truth and falsity (objective) and also epistemological knowledge (subjective)! You can even talk about necessity and possibility! It is much richer than classical propositional logic. And well, lets talk about the simpler things. That is a challenge, and so I'm challenging you. My theory is before the notion of time. The notion of time and dynamics will be introduced in the future as an extension to it. I don't know how it will be. After we introduce time, then we can talk about causality and measurement process, but each one is a new step, and I hope i won't be alone building them. My theory introduces a new level of precision in physics, it is a closed theory. So you must pay attention to its form and precision, that is where it differs from anything before. It is the first closed theory!

          Take a look on chapter 4. it has a lot of math, but is where you can get the essence of my work. Every STATEMENT section contains a theorem and a reading of it in natural language. DEFINITION defines a new concept. Concepts like truth and falsity are defined mathematically and the basis for logic are established.

          I added new elements to quantum theory, and then you have to see how the new theory is for you to understand it. It is more than simply a different interpretation.

          My next paper will be on the nature of logic. I'll show all the logic that arises from this previous one.

          "except that i speculate about what's beyond the classical logic and epistemological knowledge, and you seem to not do this." That is something should not remain true, because I'm trying to go beyond them both, so help me understanding what is missing.

          Thanks for reading and discussing my paper. That is really important for me, and is good to know there are others interested in my work. Take the time you, but please, take a look on the mathematical part, that is the essence of my work; the rest are discussions and motivations, the real content is on this part.

          Best wishes

          Frederico

          Frederico Pfrimer wrote:

          "I have means to say that the main wrong assumption of physics is not

          a physical assumption, but a millenary logical assumption: the principle of excluded middle1."

          Niels Bohr talking about it

          "There are trivial truths and the great truths. The opposite of a trivial truth is plainly false. The opposite of a great truth is also true."

          Read more at http://www.brainyquote.com/quotes/authors/n/niels_bohr.html#u2LwwTk8pFUyOzeb.99

          Few peoples understand it unfortunately.

            Dear Frederico,

            o.k., now i understand your approach better. I will take a closer look at the mathematical part and if i can say anything helpfull, i will post it here. Unfortunately i am not a mathematician, so i first must research the meaning of some expressions like idempotent and a few others.

            Thanks again for explaining your work.

            Best wishes,

            Stefan

            Hi Frederico,

            I enjoyed reading your interesting essay. I agree that the non-Boolean logic inherent to quantum mechanics is precisely what gives rise to all the weirdness that conflicts with our classical intuitions. Personally, however, I'm not sure it's enough to say that quantum logic is non-Boolean and leave it at that - that's merely a description, not an explanation. I think it remains important to ask, in Wheeler's famous phrase, why the quantum? Why non-Boolean logic?

            I take a speculative stab at that question in my essay, suggesting that non-Boolean quantum logic expresses a radical frame-dependence in the nature of reality, one most strongly argued for by holography and Lenny Susskind's notion of "horizon complementarity".

            Great work.

            Regards,

            Amanda