• Trick or Truth Essay Contest (2015)

  • The Physical Limitations on Mathematical Abstraction, the Representational Effect of Mathematics on Physical Explanation, and the Resulting Expansion of Computability by Steven P Sax

Constantinos,

We seem to have similar views, might I suggest that the conundrum of what distinguishes a measurement as mass from an equivalent value of energy can be found in a proposition that; for a discrete quantity of energy to exhibit inertia, some (small) portion of the quantity must exist as the greatest density at constant density in a direct universal proportion to the whole quantity. That would be the Relative Requisite Inertial Density:

I,=Ec^2=mc^4.

It has worked pretty well for me. Try it if you like. Cheers, jrc

Constantinos,

No. Picture a small volume at center of a free rest mass at a constant density throughout. That density would be the greatest density, as proportional to the whole mass/energy quantity. Outside that core volume the density of the rest of the energy quantity would drop off in accord with the inverse square law along any radii. The lower density bound of gravitational integrity would be theoretical.

What I'd meant to say is that the postulate can provide a basis for solution of the conundrum inherent to the mass energy equivalence, which exists because it is an equivalence and does not say where energy becomes mass or vice-versa. It provides an answer to what it is about inertia that is identical for any mass independent of its state of motion, and is thus a general definition of inertia. It also provides a means to determine a finite quantity in the core volume where both General Relativity and Maxwell's equations prescribe no limit to upper bound of intensity-density and consequently mathematically result in a singularity.

I also postulate that density varies in direct inverse relation to velocity which is at variance with Lorentz. But I argue that Lorentz is two dimensional and given greater degrees of freedom limits out at light velocity as a proportion of linear contraction, lateral expansion and diminution of density which would still result in the infinite electric bill to maintain a mass at light velocity. The sinusoidal wave of EMR is evidence of a sequential acceleration and deceleration of an electric charge dependent on rate of change in its motion, and the 'c' proportion of difference of intensity of a static electric charge and its accompanying magnetic field is physically rationalized as the electric density at rest moment reduces to magnetic density at peak periodic moment of the wave event.

The two propositions argue that while light velocity is the limit to acceleration of any mass, a mass small enough to prescribe an inertial density which is less than inelastic would more readily conform to linear contraction and lateral expansion when subjected to an impulse of accelerant energy and be capable of being propelled to momentary light velocity. Perhaps similar to a solitonic wave as Dr. Kadin theorizes, the accelerant charge would be recovered in the deceleration phase as the mass portion of the Planck Quantum Action ( which I amuse myself by calling a "planckton') seeks inertial stability at periodic rest moment. Modeling the Transition Zone from a spherical electron is on a back burner.

That's about it. The nut's shell. ;-) jrc

Dear Steve,

Your essay is to me a precise and comprehensive treatment of the questions raised by the contest topic.

Just one question: couldn't what you called the self referential state be BETTER interpreted to be simply in any system of observables the observer proper? Even more so when you also allow ultimately that the self referential trait may actually explain self awareness (consciousness).

You say also: "The limit of computability thus marks the ultimate interface of mathematics and physics."

This assumption in one wrap is the whole thesis of my essay: Observer as the Mathematician's "constant" and the physicist's "quantum".

But am yet to see just one professional who has actually read through it. I can understand that I being neither a mathematician nor a physicist the attitude is that not much worth can come from my end, especially when people have got their time to optimize.

But let me say, Mr Sax, you seem to see the kind of spark I myself see (forgive my grandstanding). Yet could you read and comment frankly on my line of argument. I think your far-going insight will be rewarded.

Bests,

Chidi Idika

I am glad that you enjoyed my essay. The Einscheidungsproblem of Hilbert turned out to have this strange impact on mathematics that Hilbert never imagined at the time. On the other hand I have read that Goedel discussed with Einstien on how he was fairly unhappy that his result seemed not to have practical impact on mathematics. However, in some ways that may now be the case. The formulation of mathematical physics might involve recognition of these matters.

Your recognition that a quantum system in a superposition of two states in a qubit has undecidable nature is interesting. I think a quantum system in a superposition of states could reflect a Goedelian undecidable situation in some problem involving einselection, or maybe even deeper with problems with quantum error correction codes (QECC) in black holes. It discuss hypercomputing in my essay, and this could involve some aspect of how QECC in black holes and the erasure of quantum bits that accumulate. This may be an undecidable problem, and hypercomputing might indicate something that is concealed from observability.

I will try to look up Gentzen's proof of consistency for Peano axioms. I thought I had scored your essay earlier, but I had not, so I just now scored it.

Cheers LC

JRC,

Thanks for all that. I cannot comment on this. For me this is yet another pretty picture claiming Truth of "what is". And so in essence antithetical to my view. Claiming Truth of "what is" is Metaphysics.

I have been arguing we need to purge Physics of Metaphysics. And I am proposing Physics can and should be founded on Mathematical Identities (Truisms) that describe the interactions of measurements. We need not make any physical assumptions, like the physical existence of energy quanta.

Planck's Law for blackbody radiation, for example, was derived using such physical assumption. I have shown Planck's Law is in fact a Mathematical Identity and can be derived without assuming energy quanta. And that explains why the experimental blackbody spectrum fits so identically with the theoretical curve using Planck's Law.

Constantinos

Dear Steven,

You decided to illustrate the maths/phys correspondance with a few well choosen examples.

I like the case of a qubit where you talk about self-reference.

Classically you have the CF (coin flip) gate (that randomizes the inputs) and is self-referential (that is idempotent) in the sense that CF^2=CF.

For qubits QCF= H.Z where H is the Hadamard gate and Z the Pauli (phase) gate (here H or QCF create the superposition of the input qubits). And it is easy to calculate QCF^(1/2)= X (the Pauli shift gate that you may well call NOT because it switches the input qubits). A good account is in "The square root of not by Bryan Hayes"

http://bit-player.org/wp-content/extras/bph-publications/AmSci-1995-07-Hayes-quantum.pdf

To conclude you write "Self referential operations, first seen to matematically limit a physical computer, indeed can be the underpinning of qubit manipulation, and the physical foundation of quantum computing".

I just polished your argument but I agree that passing from the classical bit to the qubit allows to reconsider the Halting Problem in a fresh (quantum) way and this has consequences on Goedel's theorem.

You had an essay on time entanglement (here the CNOT gate is relevant) so that you know that the (claimed) incompleteness of quantum theory relates to the EPR paradox. My view is that it can be further clarified by the use of advanced mathematics.

Best,

Michel

    Constantinos,

    Briefly, as I do not want to be impolite and clutter Steve's page. I quite agree that what I propose as a general definition of inertia is metaphysical and rests on an assumption of energy being a materialization of spacetime. I can conceive of no way to actually prove that, and perhaps because I am not heavily invested in the market forces of physics it does not trouble me in the least that any others would disagree with my naïve model-making. I treat it as a toy myself.

    Founding physics on mathematical identities gets a little abstract for me, and I often wonder if mathematicians realize how phenomenal their memory capacity must be. That's where I run into trouble, recalling into application the rules of operations arising from definitions, even when tallying up my monthly costs of living on a limited budget.

    Speaking of identity, several years ago I was reading a compendium of math history and gather that it has been only in the recent past that in Conventions it has been decided that the exponential rate unit can only be used as the base or as the radicand of a power. Given that many physical interactions operate exponentially, I wonder if in certain applications 'e' could be used as the power of a constant (such as light velocity) or the index when a compounding of light velocity might be found in superimposed gravitational fields such as by the gravitational collapse and aggregation of gaseous nebulae. The exponential rate unit has long been used as the base of natural logarithms so the analytical proofs would be heavily weighted, but is there any axiomatic objection to 'e' being the index of a constant as radicand? Or is it simply assumed that in some instance 'e' might wind up being its own root? Perhaps that could happen if the radicand were a variable.

    Thanks for the dialogue, Constantinos, jrc

    JRC,

    Don't misunderstand my intellectual attitude behind my last comment to you. It would be the same as I have also argued against more established models of "what is" by respected physicists. Mathematical or not.

    What makes my proposed mathematical formulation of Physics different are the following:

    1) My mathematical formulation is NOT a model of "what is". There are no Universal Laws of Physics, nor any physical assumptions made.

    2) The Mathematical Identities I speak of as providing a foundation for Physics describe the interactions of measurements.

    What connects this mathematical formulation (not model) to "what is " ( which we cannot know in essence) are our 'measurements' of "what is " (which we can only know).

    Constantinos

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    • Post #33

    Constantinos,

    Thanks for the say-so, I hadn't taken it otherwise anyway. I found in some comments of yours quite a while ago that your perspective was operational which I think is quite valid. I confess to liking a crutch of stuff I can think of as being substantive.

    How do you see the operations of interaction of measurements, for example; gravitation. In the Newtonian regime it is treated as an instantaneous action across measured distance, but even using Newton's formulation in a relativistic regime the action should propagate at light velocity. If we dispense with physical Law, how does gravitation operate? I find it rather interesting that the Gravitational Constant is derived from measurement and thus empirical, but has no known causality. Yet it is used in both Newtonian mechanics and GR, both of which are causal theories.

    There is a similarity to QM in your approach, in that it is the measurement of 'whatever it is', which is of prime import. jrc

    Constantinos and John,

    Thank you both for the interesting dialogue, and I'm glad you both were able to "be on the same page" (and I'm honored to offer the page :) ) and work off each other's ideas! Also Constantinos I plan to read your essay in depth. It's a very interesting approach to view Planck's Law purely from the mathematical side, and I'm eager to see how you apply the identities in light of your discussion above. In the meantime, John I noticed you mentioned it was difficult to access one of my references (on the Rubidium atom). Here's another reference which gives a thorough yet straightforward explanation of the phenomena, and may relate to some of the other points you were discussing:

    http://www.bgu.ac.il/atomchip/Theses/Amir_Waxman_MSc_2007.pdf

    Hope this helps!

    It's quite unifying to see the correspondence between undecidability and superposition, especially in view of complementary observables:

    The basis change and corresponding superposition represents a new undecidability resulting from the measurement of a complementary observable. Furthermore, the new basis is represented by the states of the original observable. Meanwhile mathematically the undecidability is an expression of Gödel's incompleteness.

    A succinct way to state all this is to consider Gödel's incompleteness and Heisenberg's uncertainty as being equivalent. It then makes sense that the self-referential operation is the underlying motivator from both perspectives - mathematically, and as physically manifested in the nature of the measurement. Quantum computation provides the arena in which this equivalence may be played out, and would be applicable to some of the pursuits you discussed.

    Thank you Steve, for your forbearance and patience with this discussion. Like John, I too have been little concerned we may have been intruding in your cyberspace.

    So John, if you like, we can continue this under my essay, "The 'man-made' Universe", where I have posted my latest reply to you.

    Best wishes,

    Constantinos

    Steve,

    Thank you for your interest in my results. May I suggest my Chapter, "The Thermodynamics in Planck's Law", where you can find a most comprehensive exposition of my Planck' Law mathematical derivation and many other results that emerge from this.

    I look forward to your comments and further discussion on these.

    Best,

    Constantinos

    Steven,

    Thank-you for the reference on Rubidium research! Something// is undecided in the intermediate period between half-pulses, like a suspended animation. I very much liked the clarity of your presentation in comparison of geometries by which anything is measured. Best wishes and I hope your essay attracts fruitful interests. jrc

    What is really undecidable is how a quantum system emerges in either |+> or |-> state in a measurement. We could consider the case of the two state atom in a cavity with a photon. The dipole interaction is P*A = g(s_+a + s_-a^†), in the rotating wave approximation, which contributes as an interaction term in the Lagrangian. This result in and oscillation between |+. and |-> that is periodic as cos(gt). The system is not so much undecided as it is inthe oscillating superposition. What is decided by physics is which state the atom is in when one turns on a detector in the cavity. This is decided by physics, or at least our phenomenological experience of nature, but it is not decided by quantum mechanics. It is in this sense that the proposition on whether the atom is in the excited or unexcited state, |+> or |->, when an observation is made is undecided.

    LC

    Dear Steven Sax,

    I very much enjoyed your essay. While ordinarily I tend to think of coordinates as simply a labeling convention, I liked your discussion of inertia and the fact that "Any mathematical representation still depends on physical assumptions, and changing the mathematical representation changes the physical explanation we use." And, per gravity, changing the physical assumptions changes the mathematical representation of space-time.

    This perspective certainly applies to Bell's theorem. When one changes the physical assumption from "precession in a constant field" to "scattering in a non-constant field" the representation changes from Pauli's provisional binary map to a continuum-based scattering spectrum, with consequent changes in correlation.

    Your treatment of computation is excellent, beginning with "every finitely realizable physical system can be perfectly simulated by a ... computing machine..." My Automatic Theory of Physics explores this point and [page 10 in my essay] I show how the automaton's 'next-state-address' corresponds to the physics 'potential' by linking the canonical automaton to a typical Feynman QFT kernel.

    Your explanation that undecidability of self-referential statements can be traced to endless loops that destroy causality is excellent. And, although I tend to resist certain interpretations, I found your discussion of half pulses, NOT gates, and your insights for future research especially fascinating. Thank you for your excellent essay.

    My best regards,

    Edwin Eugene Klingman

    This is the domain that might be where physics is undecidable, or where there is no decidable logic and where nature does this anyway. This contact point between the quantum and classical worlds could be a form of self-referential truth.

    LC

    (Response to March 25th post) That's right, just looking at the very nature of measuring a single observable is very fundamental to undecidability. My comparison mentioned above is focused on complementary observables. In that analysis the phenomenological experience of nature involved in measuring one observable would have as its underpinning some self-referential operation of the other observable - this is manifested physically in the specific measuring action (such as direction of the magnetic field, the way the detector interacts, etc.). A bit subtle is the linking between the two observables - the SR operation relates to one observable in the act of measuring the other.

    Regarding a single observable, consider that every observable has a complementary conceptual observable out there. So the |+> and |-> oscillating superposition in some way perhaps corresponds to another observable with an already decided value. If a measurement specific to this 'another observable' were made, it would not affect the already oscillating superposition of |+> and |->. Then, physically doing what is necessary to measure the|+> and |-> system (i.e. to determine if in the excited or unexcited state) would effectively be a SR operation on that 'another observable.' The undecidability of |+> and |-> is now swapped for undecidability in this 'another' domain. This is the same reasoning applied to pairs of observables already known to be complementary, like Sx and Sz. This again shows not everything can be decided together.

    Now, one might say this was still a physical undecidability; if so then by our new measurement we physically transferred the undecidability so to speak. But what did we really do? We changed the nature of our measuring environment, that is we sought to explain the system differently by conceptually distinguishing a different observable. Remember that changing a mathematical representation requires a change in physical explanation. "Change in physical explanation" is more vague than "change a mathematical representation" so more is needed to state this conversely: By distinguishing out a different observable, we're changing how we physically explain a system. If we seek to explain by distinguishing an observable that can't be described in a particular mathematical basis, then we must represent by a different basis. But what if we just try to change the representation to begin with? This can be done symbolically. If we have a measurement in Sx (let's say it's UP) and now we wish to represent that using a basis of (X up, X down), well we can't - that's mathematically undecidable as you brought out above. For example, (X up + X down)/в€љ2 does not give you a decided measurement of Sx. It was an attempt to represent it with respect to itself, and is thus self-referential. This mathematical self-referential operation is undecidable, but can be used to describe a decided value of a new concept Sz whose value would be UP. When physically manifested this changes the paradigm of explanation - it destroys any measured value of Sx and instead produces a new decided observable Sz.

    That not everything can be decided together squashes Hilbert's attempts as we noted above. (Maybe the Halting Program can be reformulated with the basis B and states T(n) as the inputs). That not everything can be explained together reflects complementarity. That each of these leads to each other shows the equivalence of mathematical incompleteness to physical uncertainty.

    (Response to March 26th posting) Thanks Lawrence this is a very interesting article. Yes perhaps the exact nature of where the self-referential operation manifests physically could be considered at this contact point. I definitely want to explore that more. I tend to not use the term "classical world" per se but I see how the article uses it, and that contact point is really where the environment provides just enough decoherence to explain it through emergent properties, like the point of einselection as you described earlier. In this case it appears the environment is associated with the attenuator and the number of bi photon pairs. It's very fascinating, and I want to check out the specific physical mechanism and see how self-referencing may be interpreted. Thank you :)