Sorry, my post was supposed to be addressed to Lawrence.

Apologies

Mauro

Hi Lawrence,

I think your essay achieves these goals very well. Also it is nice how across history people have always remained interested in this area of study.

Cheers,

Antony

Undecidability is a universal property, such as it is known that not all propositions in Peano arithmetic are decidable. The Church-Turing thesis states that any computable function is run on a Turing machine that halts. Of course it is known that a universal Turing machine is not capable of determining whether all TMs are halting. Hence the λ calculus is incapable of verifying Hilbert's Entscheidungsproblem.

My paper works with certain correspondences between modal logic, Lob's theorem and the Godel theorem. I did not break this out in great detail to avoid turning the paper into a complicated discussion of symbolic logic. To address your question again; I think what you are asking is how can we determine what is undecidable. This is an area of research for people who delve into this subject. There are mathematics on proof theory and hierarchies of undecidability. I will confess that I am not an expert in this area of work, for this deviates seriously from physics.

My argument is then more physical than formal, where I consider a certain algebraic postulate about observables, that being associativity. The physical argument involves observables or quantum amplitudes in a region that contains a black hole.

Cheers LC

  • [deleted]

Anton,

I am sorry I can't respond in the length you write. The day is getting a bit on already and I need to attend to other things. There are a couple of points I can make.

If the universe had one charged particle something funny would happen. If we consider the spatial surface of the universe, say on the Hubble frame, as a 3-sphere the lines of electric force from this particle would wind around this sphere endlessly. Indeed these lines of force would densely fill the space. They would also interact with themselves or in effect the charged particle. The charged particle would be "driven" into a divergent condition to rapidly increasing energy. The system would in effect "explode." It is comparable to an undamped oscillator that is driven by a resonant frequency.

The recent developments with inflationary cosmology now involve bubble nucleations or O-regions that result from a vacuum transition to a small value. This is an aspect of the multiverse. There is some prospect that our O-region, bubble or sometimes called pocket universe interacted with another one in its early phase. This may have left an imprint on the CMB. The de Sitter spacetime where this inflationary expansion occurs may in turn be a vacuum-field configuration on a D-brane, where in this higher dimensional space of 10 or 11 dimensions there is a foliation of such branes. This then leaves the bulk 10/11 dimensional spacetme, which might in turn exist in a 26 dimensional spacetime and so forth. This sort of gets into the proverbial idea, "its turtles all the way down."

Cheers LC

Dear Crowell,

Now let us call the most natural automaton simply "heredity" (or indeed any cycle, Carnot or whatever). Then the idea of entropy or 2nd law of thermodynamics is that each stage of evolution within an automaton path does in fact depreciate possibility of return to the initial state; so there must be a halt or "fatalism"

My question is: doesn't your undecidableness amount to what we already know as uncertainty? Such that above a cut-off (Heisenberg Cut?) there is higher probability of return to initial state (what we know as determinism) but below the cut (being what we know as the quantum scale) there is no probability of return to initial state i.e. the system is not well-behaved.

Now I put it to you that once you assume that a well-behavedness (or conversely "uncertainty") is simply put an automaton then it qualifies as the ALGORITHM or computer proper. This computer/algorithm is what I call simply THE OBSERVER (perhaps Wheeler's anticipation).

It is a response to your position that: "...a quantum field is propagating on spacetime,but where spacetime is the quantum field. This heuristically appears self-referential, and the physical ansatz is this nonlocality is an undecidable proposition of the above modal theory of causality."

By quantum theory, this state is rather at once a Godel "incompleteness" (the uncertainty) and yet a Peano (or Planck) "constant" (i.e. the conservation law).

My "observer" is in other words then the SUPERPOSITION proper.See What a Wavefunction is

So Wheeler's proposition does work! Pls see my essay and prove me wrong on this particular approach.

    I hesitate to call the Heisenberg uncertainty principle some derivative of Godel's theorem. However, my ansatz that this incompleteness rests with the associativity of fields is really just another form of the quantum uncertainty. There are limited ways of knowing what propositions about a mathematical system are undecidable, so my assumption here is not a form derivation of any sort. So my idea here is more of a physical assumption than a formal proof or derivation.

    The role of an observer is in some ways similar to a self-reference in mathematics. Godel's trick is to let a mathematical system contain predicates that act on their own Godel numbers as the subject. An observer is ultimately made of quantum particles and the act of a measurement has the appearance of being a sort of physical form of self-reference. Of course my paper is not about the measurement problem or the role of observers, but ultimately the universe is as it is so that observers can exist. This is a sort of strong form of the anthropic principle.

    As for fatalism, it is of course the case that the universe will end up as a pure de Sitter vacuum in a sort of heat death. This will be reached in around 10^{110} years. It will probably quantum decay from there over a far longer period of time into a Minkowki vacuum. That might sound like fatalism, just as saying that we are all going to die sounds fatalist. However, in the mean time a lot of different things can happen. In the sense of Plato's final cause the outcome might be in a sense fated, but what happens before then is not.

    Cheers LC

    Lawrence,

    There are two kinds of scientists, the kind which can perfectly build further on the theories one learned at school, and the kind which tries to find alternative interpretation of observations. However invaluable an education is, its disadvantage is that, since you've learned many theories describing phenomena, it is very hard to dream up a different approach which perhaps might solve some of the many fundamental contradictions and enigmas of present physics. Though it is very understandable that man came up with the big bang idea, I'm afraid that it actually is even a worse 'theory' than creationism which at least honestly, boldly states that the universe has been created by some outside intervention. As far as I am aware of, the universe either has been created by some outside interference (which is the position of both creationism and big bang cosmology), a possibility I reject or it creates itself. If so, then fundamental particles must be as much the cause, the source of forces as their effect, the product of their interactions, meaning that a force cannot be either attractive or repulsive, always, which is the mistaken belief string theory is based upon. String theory, like big bang cosmology don't solve anything but are part of the problem.

    Cheers AB

    6 days later

    Dear Lawrence

    Your quote from comment to Platan essay:

    "What do you think of algebraic curves over [0, 1, ∞] and the Langlands program?"

    By coincidence i used long time ago similar trick with pseudoscalar mesons.See my essay. I signify as 1 mass of proton and then observed what position would take place other pseudoscalar mesons.That i got phenomenon 18 degrees.

    Do you see some explanation?

    Yuri

      The number 18 is important in Jewish mysticism. I am not sure that has any bearing on what you are saying though.

      Is your number something similar to a Cabibbo angle or Weinberg angle?

      I'll take a look at your paper later today.

      Cheers LC

      It is not common with a Cabibbo angle or Weinberg angle

      Just put the values of mass on y=tanx plot, then exploring angles.

      Dear Laurence,

      (I copy the reply to your post on my page)

      Your post is very stimulating. I need time to look at this possibility of relating black-hole physics and entanglement, and non-associativity. On the other hand, I don't consider that entanglement is a primary category in non-local/contextual questions. It may be that conformal arguments adapted to Grothendieck's approach may approach the subject you are talking about. I should say that I am not familiar enough with black-hole physics to have a motivated opinion I intend to read and understand this Maldacena-Susskind paper before discussing more with you on this topic. Meanwhile, may be you can have a look at recent papers by Frédéric Holweck and co-authors (we are now working together) about entanglement and algebraic geometry.

      Thanks and best wishes,

      Michel

        Dear Lawrence, apologies if this does not apply to you. I have read and

        rated your essay and about 50 others. If you have not read, or did not

        rate "link:fqxi.org/community/forum/topic/1756] my essay The

        Cloud of Unknowing[/link] please consider doing so. With best wishes.

        Vladimir

        • [deleted]

        The program of finding physics with [0, 1, ∞] can be found with the SL(2,C) group and the linear fractional transformation (LFT)

        f(z) = (az b)/(cz d),

        which has a correspondence with matrices of SL(2,C). The Mobius transformation or LFT is an automorphism group on the Argand plane, and this is equivalent to PSL(2,C). This projective linear group is then the automorphism group of C. If we let the constants a, b, c, d be points in C then the LFT

        f(z) = [(z - z_1)/(z - z_2)][z_3 - z_2)/(z_3 - z_1)]

        is for the identity f(z) = z a case where z_1 = 0, z_3 = 1, and z_2 = ∞. A matrix representation may be found by dividing through by z_i and taking the limit z_i --- > ∞.

        From this comparatively simple example we may move up to SL(2,H) and SL(2,O). In the case of SL(2,O) ~ SO(9,1), there is an embedding of SO(9) ~ B_4. This in turn is defined with the short exact sequence

        F_4: 1 --- > B_4 ---> F_{52/36} ---> OP^2 --- > 1

        where the strange symbol in the middle means that the 52 dimensions of F_4 - the 36 dimensions of B_4 ~ SO(9) defines the OP^2 projective Fano plane or OP^2 ~ F_4/B_4.

        The B_4 group is the SUSY group that Susskind employs with the holographic principle.

        The group F_4 is a centralizer in the E_8, which means it commutes with the automorphism of E_8, which is G_2. We then have a somewhat Rococo form of the same construction. A projective form of SL(2,O), PSL(2,O), defines matrices ~ aut(O) ~ G_2 which map three points to [0, 1, ∞] with the action of the 7 elements in the Moufang plane. I think I can find this matrix in the near future.

        Unfortunately I am moving shortly, so that is complicating plans to do much analysis. If I do this in the immediate future it will have to be in the next week.

        Cheers LC

        I read your essay sometime bzck. I have a list of these papers and which I have scored. I would probably have to reread or at least refresh myself about your paper. As I recall it is a bit of a metatheory.

        Cheers LC

        Lawrence

        Could you please explain where is your theory connected with Golden ratio?

        See part Symmetries... PSL(2,Z)etc

        http://en.wikipedia.org/wiki/Golden_ratio

        Yuri

          Lawrence,

          I loved your explanation of modal logic in causality. Very succinct! You brought back pleasant memories of doing symbolic logic in university.

          The notion that "causality is necessary incomplete" can also be appreciated from a quantum information point of view using a complex valued system. When an EPR state is prepared, all entropies are conditional on the observer, who is a subsystem in an "EPR-triplet". However, in making a measurement, the observer throws away her entanglement information so that the subsystem of the EPR pair is no longer conditional. (See my essay "A Complex Conjugate Bit and It".)

          A quantum correlated system thus becomes a classically correlated system. Using Venn diagrams, it becomes apparent that this process can be interpreted as a change in associativity.

          Best wishes,

          Richard

            Yuri,

            The polytope for the E8 grop, the Grosette polytope with 240 roots can be decomposed into the icosian of 120. The icosian or 120-cell has two quaternions with length (1/2)(1 +/- sqrt{5}) where the plus one has length 1.618..., which is the golden mean. In fact these quaternions define something called the golden field in a Galois ring. This is related to the Fibonacci sequence.

            Cheers LC

            Richard,

            Thanks for the kinds words. I agree that quantum measurements and even quantum teleportation involve the destruction of entanglement. Maybe better put the entanglement is transferred to a reservoir of states in an unpredictable manner. The entropy of the system is then indeed conditional, and a measurement loses this.

            I propose that the incompleteness has to do with associativity in QFT. My argument then involves the situation of fields near an event horizon. There is a profound difference between the classical case and the quantum case. I am not clear how this plays with standard quantum measurements. Zeh, as I recall, talks of a quantum horizon. Maybe there is some parallel situation there which makes associators play a role.

            I will try to read your essay soon. I am rather slowly getting around to as many as I can.

            Cheers LC