Thank you Tom for your kind words, and I'm enjoying your essay very much as well (which I also plan on reading again in depth when I get back to town). Fascinating take on Bell, and your explanation of Popper is very enlightening. Looking forward to corresponding with you further. Steve
The Physical Limitations on Mathematical Abstraction, the Representational Effect of Mathematics on Physical Explanation, and the Resulting Expansion of Computability by Steven P Sax
Also congratulations on your being published by Springer in Conflict and Complexity!
Thanks George, and your discussion on Godel is very fascinating! A key factor regarding Euclid vs. Riemann as a mathematical representation of space, is the physical explanation one attempts for physical observation (including the assumed explanation of how those observations are measured, for example), when using such a representation. If the fifth postulate forces the sum of angles of a triangle to be 180, and then one measures them to be 200, (and your measuring device has previously given results consistent with all your other physical experience to the point you're confident in it) - then something has to change. When the physical explanation is compelling enough, it forces us to abandon the postulated assumptions of the mathematical representation. For example, the sum of the angles of a triangle on a sphere is 180°(1 + 4f), where f is the fraction of the sphere's surface that is enclosed by the triangle. That understanding based on new intuitive observation around a sphere shows is not compatible with a Euclidean representation, based on other assumptions we already maintain. All that being said, in MUH it may be possible to have such split physical universes based on the particular restrictions of the additional axioms. It's an interesting exercise to consider how that would play out.
I'm traveling now, but hope to address your other two points this weekend.
Thanks again, Steve
(reposted from my essay forum)
Thanks, Steve -- as I posted in your forum, we are in accord on many things, and the foundations of computability is, I think, the most important issue in frontier science.
Beyond the scope of the essay question, the growing fields of brain science and artificial intelligence depend strongly on resolving the issues of network robustness and integrity -- i.e., the amount of information that can be effectively used at each decision point such that positive feedback doesn't overpower the computing function.
It's a key point -- the number (1) in your concluding remarks, that twice applying the self-referential operation generates a true statement. It's the identical point I was making with the Popper example of pairwise correlations followed by a single result that may or may not be correlated with the pairwise value. Length restrictions kept me from exploring the basis of Popper's program -- which is Richard von Mises's theory of the independence of collectives -- Popper notes (p. 196) in *Realism and the Aim of Science*:
"von Mises's 'axiom' (which postulates the existence of a limit of the relative frequency of the occurrence of a property P in any probabilistic sequence of events or 'collective') may be written as a universal-existential-universal-existential-universal statement, of the following form: '*For every* probabilistic sequence, *there exists* a real number x between 0 and 1, called the limit of the relative frequency, such that *for every* given fraction y, however small, for which y > 0 holds, *there exists* a natural number n, such that *for every* natural number n (for which n > m holds) the relative frequency of m/n, of m occurrences of the property P up to the nth event of the sequence does not deviate from x by more than y, that is to say, - y =/< x - (m/n) =/< y."
In network terms, adding a time parameter, this implies that information lost to one decision node is not lost to the network hub at which it originated, such that continuously shifting hubs of decision activity are self-organized in the same context that you take to be self-referential.
As I think it is pertinent to the content of both of our essays, if you don't mind, I am going to repost this in toto in your forum.
Thanks again and all best,
Tom
Steven,
I don't know who in the FQXi community sandbagged the high rating your essay enjoyed, but there are several individuals whom make themselves conspicuous by a jealous regard of their own personal equations. That might serve to guard their position in the informal ranking, but also serves to warn others that if such an equation or applicable argument is indeed of independent merit and is then included and cited in another's work, they become burdened with defense of the source as well. Being employed in the federal bureaucracy (a thankless job) you are no stranger to the syndrome.
Being a member of the public myself, my rating is rather meaningless here, but I do thank you for professionally contributing to the public good. And I found your essay, informative, insightful and comprehensible. It is apparent that some original work is incorporated in the section on limits of computability, but presented in keeping with the Contest Topic and inclusive of real research that impacts the cornerstone of quantum mechanics and the role of mathematics. Overall, your essay addresses the Essay Contest Topic in the most appropriate manner of any that I have personally read. All politics aside; Excellent! jrc
Jose, the point I meant is that even the most reduced version of mathematical assumptions we attempt to make must yield predictions for measurements and interactions that are consistent with the explanations we give for our observations, if those explanations rely on those assumptions or are represented by a mathematical formalism based on those assumptions. Looking at it the other way around (and perhaps even more astounding), the explanation(s) we give to our observations and the predictions we make when physically interacting with the world, change based on the way we mathematically represent those observations and interactions. Physical reality is understood through the dynamic volleying between these two reckonings. Thanks for helping to flush that out with your question.
To appreciate the rigorous patience of this dynamism, consider how many centuries it took to reconsider Euclid's assumption about the flatness of space, or 'absolute direction.'
I liked your essay and am curious based on your 'physicalist' stand what your issue is with the quantum concept. In particular, what explanation would you alternatively give to explain the discrete energy representation? Or what else would you have changed from the assumptions and representations to account for the observed data of blackbody radiation?
Superposition, especially in view of the multiverse concept, offers an explanation based on the probabilistic representations given for measurement, while considering our other physical assumptions about measurement. What makes superposition compelling is the physical interpretation it offers to explain interference effects, which are very much the physical situation we appear to observe. (Consistent at least with the other assumptions and representations we afford to those observations).
Steve Sax
Thanks Lawrence, that's very insightful and ties in well with the end of my essay. When one measures a complementary operator, the basis changes for that measurement - and what was once undecided is now a decided measurement. For example you measure Spin x to be X up for an electron. So Spin x is decided but it's in the basis of Spin z, namely X up =(Z up + Z down)/в€љ2, and Spin z is undecided. Now you measure Spin z. You changed the basis to Spin x, and in doing so you're able to obtain a decided value for Spin z. Let's say it's Z up. But now with the changed basis, namely Z up =(X up + X down)/в€љ2, Spin x is undecided. This entails changing the paradigm of what's being focused on. Physically this amounts to changing the environment namely the measuring interactions (such as a magnetic field in a different direction, etc.) so yes, I see how this can be looked at from an environmental decoherence (and thus einselection) perspective. Thanks for filling in that gap, I like it.
The key point then is there is not one basis in which everything can be decided. This will change depending on the particular measurement paradigm and I related it to Gödel incompleteness. Modeling physically how a self referential operation may change the particular measurement paradigm opens many possibilities for quantum computing (pun intended ;) ). As I pointed out above, superposition in light of the multiverse concept offers an explanation for how the undecidability might be represented, and the physical interpretation it offers to explain physical observations like interference effects makes it very compelling.
Thanks again for an excellent comment, and I look forward to reading you essay.
Steve Sax
Dear Steven P Sax,
In your essay you wrote, "Photons, quantum mechanics, and the basis of modern technology arose by shifting the terrains of infinity itself. We see now that the two pillars of modern physics ‐ general relativity and quantum physics - owe their discovery in part to purely mathematical changes in representation." I claim that there are at least 3 empirical disproofs of the Lambda-CDM concordance cosmological model: the space roar, the photon underproduction crisis, and the empirical evidence for Milgrom's MOND. Google "Witten Milgrom" for more information. My guess is that string theory needs to explain the cosmological problems BUT with "mathematical changes in representation" -- using the monster group and the 6 pariah groups. One puzzle is the GZK paradox. Could the explanation for the GZK paradox be inverse Compton scattering from relativistic jets? Do you have any guesses for the explanation of the GZK paradox?
John R. Cox,
"I don't know who in the FQXi community" you are targeting but since your post here followed mine, let me quickly dispel any sly implication you may wish to make I had anything to do with your "sandbagging" accusations. The truth of the matter is I have yet to cast any vote for any essay!
Further, the ratings are meaningless to me. That is not the reason I participate in FQXi Contests. The reason is simple! I realize, however, it may be beyond your comprehension.
I seek open and honest discussions with others so interested on some ideas I have been thinking for years. My references in my posts are entirely for this reason.
Constantinos
This leads to the prospect there is no complete system for computing einselected bases. This could be an interesting thing to work on. Curiously this has some bearing on the nature of energy gaps. We usually think of quantum bases that are "natural" as ones that are diagonalized in an energy eigenbasis. This is largely because we measure quantum systems this way and the outcome to use Copenhagen language is collapsed in the way. However, quantum systems in a pure form have no intrinsic recognition of this basis over any other. A quantum system of an atom and photon that is Rabi oscillating between two energy states has no preferred basis.
This seems to have some bearing upon the problem of energy or mass gaps. How a nonabelian gauge field generates a mass-gap, which some associated Hilbert space of bounded states, is maybe not decidable.
I will try to elaborate later when I have a bit more time.
LC
Constantinos,
I really don't know who has trashed some high ratings, but when clicking to the page to select contest essays, which shows top ratings, it was noticeable that sudden drops occurred while comments clearly opposed to the metaphysical paradigm had been made. And like yourself, I don't that assume that I know anything for certain. And if I disagree I don't vote. Who am I to rate anyone down? Simply a matter of timing that my post in a separate thread followed yours in another, and anyone reading your essay would or should recognize you only come to learn.
Incidentally, 2010 was pre-computer for me and I hadn't seen your essay that year. For what its worth, I agree that the Planck Quantum is an action quantity, not an indivisible fundamental entity. Just when physics was getting interesting, the photo-electric effect was interpreted as a photon being an indivisible 'chunck' of energy, and the instantaneous quantum leap defied realism. The ad hoc inventions of quantum mechanics are in my view a failure of classical mechanics to rationalize the quantum, define electrical charge in terms which do not result in a singularity, and determine what a particle is. We can't fault success, so the results of QM can be used as clues to a realistic paradigm.
In his list of references, Prof. Sax gives #(10) a research paper at the University of Maryland to which he briefly refers concerning laser experiments with Rubidium. Other research there involves counting photons down to a mere (4) four. The Rubidium experiments he cites found that a half-pulse would not pump the single electron in the outer shell to a level of an excited state of emission, but a second half-pulse (presumably within the window of relaxation time) would! That suggests that the energy of the 1st half-pulse is stored in the region between the shells and the 2nd half-pulse supplies the rest needed for an electron mass quantity to condense into a volumetric particle, at least long enough to dissipate the full pulse energy as a shower of light. It is coming close to experimental capability to determine the quantum leap as time dependent. And the photon to be described as a bundle of volumetric wavetrains.
The quantum when viewed as a preferred quantity observed as the action through the duration of any single wavelength, would allow a component of that action energy to be determinant of spatial length of wave event. In reality it would be more like a Quantum Stroll. I see no reason not to take it leisurely. :-) jrc
John R.
Thank you for clarifying! Yes, I do come to learn! And to discuss and exchange ideas. The Rubidium experiments you mention is especially interesting. This fits well my explanation of the double-slit experiment. Central to this (and most all I write about in Physics) is the "accumulation of energy" (what in my writings is the quantity 'eta') and my claim that "before discrete manifestation there is continuous accumulation of energy".
Eric Reiter has done some experiments which also seem to confirm such "accumulation of energy". You may like to read his 2012 FQXi essay,
"A Challenge to Quantized Absorption by Experiment and Theory".
Constantinos
Constantinos,
Glad to have you reply, thank-you. I do like 'eta', and your statement that "before discrete manifestation there is continuous accumulation of energy" puts the proposition quite well. If I remember correctly (I'll have to re-search) Max Planck came to hold a similar view in his 'preloaded theory'.
Thanks for the link to Eric Reiter's essay, I recall seeing something of his experiments and a video. At the time I was a bit fuzzy as to whether his concept of a photon was 'chunck-like' or whether it could be a composite of serial wave-volume events, which is my personal view at present. I'll go back and read.
The Rubidium experiments Sax cited do seem potentially exciting, but the paper is archived on the University server requiring a student or researcher account as best I could tell, so all I could do was browse a bit. I'd like to hear your thoughts on the double slit phenomenon and any conceptualization of particulate matter as a unified field condensate.
I've learned quite a lot coming to this site, not without displaying ignorance all along. But I decided years ago that the only self-harm that comes from displaying ignorance is in pretending otherwise. And I've been a lot happier ever since. Cordially, jrc
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
John R.,
"greatest density at constant density" Did you mean "constant velocity"?
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,