• 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

Dear Steven P Sax,

From what you have stated, I interpret your 'physical reality' as follows: From observations to explanations to mathematical relations to correct predictions, such a sequence gives us some idea, and from the mathematical relations (that make correct predictions) back to observations and explanations, the reverse sequence helps to fine-tune that idea, and thus we can arrive at the physical reality. There are no pre-assumptions regarding reality.

That is exactly what I meant by the term 'mathematicalism'. As pointed out in the example in my essay (the case of A and B drifting apart), observation, explanation, mathematical relation and prediction perfectly agree, but still it is impossible to arrive at the underlying 'physical reality', because there are multiple options and we have to select one. Physicalism implies that there should be a pre-assumption that reality agrees with our commonsense ideas (about the 'basics').

I propose that light contains streams of fundamental particles moving along spiraling paths, and so it exhibits wave-nature also. The particles are further quantized into well separated physical units (quanta) having a fixed length. A 'quantum' is a thus real physical entity having internal structure. What I oppose is the dual-nature, especially, the 'instantaneity' of the dual nature; light is not an electromagnetic wave; it is particles having variable electromagnetic filed. The 'physical reality' implied by QM is incorrect, though the rest of it (its role as a mathematical tool) are correct.

Superposition is instantaneous; it is not 'body alternating between two positions', or 'body alternating between two forms' or 'two bodies alternately occupying the same position'; it is like the Schrodinger's cat being alive and dead at the same instant. It goes against physical reality. However I agree with your argument that in the 'present circumstances', 'proposing superposition' has some validity.

Let us do the 'one-slit' and 'two-slits' experiments using tennis balls instead of electrons, the sizes of the source, the slits, the gap between the slits, the thickness of the material in which the slits are made and also the distance between source and slit magnified proportionately. Now throwing the balls from all possible positions in the source, in all possible directions, we will get interference patterns in two-slits experiment, but not in a one-slit experiment. Here, there is no superposition, there is only close-stacking. Can you say the tennis ball has a dual-nature based on this?

George, thanks again for your stimulating questions. Regarding your second one, please see the discussion further down the thread initiated by Lawrence Crowell. Our discussion brought out the point that just like there is not one measurement paradigm that could be used to explain everything (every observable, interaction, etc.), so there is not one mathematical basis in which everything is decidable.

These two restrictions go hand in hand, and from the interfaced perspective of computation they in fact would be equivalent. Both can be related to Gödel incompleteness: a) the basis change and corresponding superposition representing a new undecidability resulting from a new self-referential operation, and b) a new environmental interaction setup physically corresponding to a self-referential action changing the measuring paradigm.

This, especially in view of your question, ties back to Hilbert's Entscheidungsproblem which asks for an algorithm that takes as input a statement of a mathematical formalization of axioms and determines whether the statement is universally valid in every structure satisfying the axioms. Thus it asks whether it's possible to have a fixed procedure to determine whether any specific mathematical statement can be proven within that system. This as we know was answered by Church and Turing in the negative - it can't be done and for example a Turing machine can't solve the self-referential nature of the Halting problem. Closely related is Hilbert's second problem, which asks for a proof that arithmetic is consistent (free of any internal contradictions). Gödel's second incompleteness theorem shows this can't be done. Interestingly, Gentzen showed a consistency proof for Peano arithmetic (a first order axiomatic formalism of arithmetic for natural numbers) that isn't a stronger form of first order arithmetic theory per se, but it is not finitistically within the same formalism either. Maybe this ties in to your speculation above.

John R.,

It is safe to say we are all ignorant of the Truth! And the more we claim we know it, the more ignorant we likely are! Best to be open and humble than to be arrogant and closed.

Planck, I believe, also believed in "accumulation of energy" (loading). Up until he gave it up in favor of the rising tide of "energy quanta" and Quantum Mechanics. He also is said to have said theories die out when the last "true believer" dies. (I am paraphrasing, of course). And that underscores the Metaphysics of Physics.

I have argued we cannot know the Truth of "what is" physical reality. And any attempt to claim such Truth inevitably leads to Metaphysics. And to consequent "religious wars" between various schools of thought. What I have always found telling of Physics and not of Math! Math only concerns "logical certainties" and not "what is".

I have proposed the way out of such predicament is to have Physics based on Mathematical Truisms that describe the interactions of measurements. I don't believe in Universal Physical Laws aside from this. But I don't believe in a Mathematical Universe Ontology. Since this brings us back to claiming the Truth of "what is". And to just another form of Metaphysics.

Concerning my plausible explanation of the double slit experiment, my basic argument is:

1) energy propagates continuously as a wave, but manifests discretely at the threshold of observability.

2) the "burst of energy" emitted at the source is not the same as that detected at the screen

3) before "manifestation" there is "accumulation" of energy.

As for "matter as a unified field condensate", I don't have a firm sense of how "mass" can be defined in my view that would result in the Law of Gravity. Though I do have some ideas about it. Too vague to share with others at this time.

Constantinos

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