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

Wow! All the way down the line -- from special relativity and metric spaces, to the foundations of Planck's constant, to the liar's paradox, 3-valued logic and computability -- I think we agree in full. This is an essay I am certain to read several times, and the first vote I'll cast.

If you don't mind revealing your email address, I have something else I'd like to share privately. thomasray1209@comcast.net

All best,

Tom

Steven - A brilliant essay, thank you. Remarkably, it seems designed to answer some questions Marc Seguin and I were just asking about in the essay that appears just prior to yours in the competition. That is so curious...

I was speculating, in response to Mark's essay, on the fact that you can respond to a Godellian truth - the ones that cannot be proved - by adding it as an axiom and getting to a higher level mathematical theory. Or you could add it's negation as an axiom, which would then split the Mathematical universe in two, just as the physical universe gets split in two when a superposition collapses. So at the fifth postulate in geometry, mathematics splits into Euclidean or Reimmanian space.

But Godel's problem is tied to the self-referential nature of mathematics, and as you point out, that seems to resolve if you do the self-referencing twice... Beautiful!

In your conclusion #2, you note that physical reality can not be entirely reduced as "there will never be one measurement paradigm that can be used to explain everything else."

Would you agree as well that mathematics can never be reduced as there is never one formulation that will ever explain everything - no universal set of axioms (finite or infinite) that can produce all the answers. And,of course, if you open the door to the MUH with the ultimate "no axioms" (infinite degrees of freedom) you also get no answers.

Your conclusion #4 introduces consciousness, causality, non locality and self-awareness. That's a lot to cover in one paragraph and I'm not sure I follow your leaps. But would you agree that these concepts will share in the indeterminacy you describe for math and physics?

That would suggest - no TOE?

I would be delighted if you have a chance to read my essay "The Hole at the Center of Creation."

Many thanks! - George Gantz

Dear Steven P Sax,

Yours is the last essay posted in this contest. I was eager to know what it contained. You say, "But even those postulates rely on some type of intuition, which ultimately must come to terms with our knowledge of physical reality if they are to represent it". Everyone will agree with this, though each will have his own version of 'physical reality'. What is your version of the 'physical reality'?

"When he presented his theory, most scientists (including Planck!) didn't consider this quantum concept to be realistic but believed it to be just a mathematical trick." This should have been the stand even now, a 'physicalist' stand, as I would like to call it. Please go through my essay: A physicalist interpretation of the relation between Physics and Mathematics.

"But if you shine the laser for only half the time - a half pulse - the electron goes into a superposition of both the ground and excited states". A superposition? The actual physical situation has been explained by you clearly: "What this means physically is if you attempted to measure the energy of the electron you would have a 50% chance of measuring the energy at the ground state, and a 50% chance of measuring it at the excited state". Explaining the '50% probability' as superposition is illogical, especially when you argue that this 'superposition' is a 'physical situation' that quantum computer offers. In my opinion, superposition goes against physical reality.

I am not sure about how you treat undecidability. The two state system with eigenstates |0> and |1> in a certain dressed state configuration can be said to be in the undecided logical state if the total state is singlet or triplet or |ψ> = c(|0> + e^{iφ}|1>), c = 1/sqrt{2}. This undecidability is relative to a certain basis, which is unitarily equivalent to any other basis. By this it means that a basis with

|ψ> = c(|0> + e^{iφ}|1>), |ψ'> = c(|0> + e^{i(φ+π)}|1>),

are related by a π pulse or ½ pulse of the laser (or what ever system adjusts these states), and if the system is in one of these states it is decided within that basis. What is then decidable is relative to the choice of basis uses.

Using the superposed nature of quantum states as a measure of decidability might be relevant for einselection problems. The measurement outcome or classical einselected states might in some formal sense have a level of undecidability that is described in this manner.

Cheers LC

Hi Steve,

I'm glad to see your essay here. Are you familiar with Penrose and Hameroff's work? Your quantum consciousness discussion, although different in the points it brings up, reminds me of their work. I'd be interested to hear a comparison of their work and some of your ideas.

In your essay, you talk about two half pulses equaling a NOT, and one half pulse equaling SQR(NOT). I do like this interpretation better than SQR(-1), but I am still a little confused... but this may be due to my lack of familiarity with many technical aspects of physics. You say that two half pulses (Is this the same thing as a half or quarter cycle?) will definitely put the rubidium electron into the excited state. Would a 3rd half pulse put it back into superposition and a 4th put it back into the ground state? If so (and my understanding may be wrong), why would a "continuous" wave polarized laser keep the electron only in the excited state? Wouldn't it be changing states according to your analysis?

Other parts of your essay reminded me of some of the questions I posed at the end of my essay. Questions such as...

B) How quickly could a tape be processed through a Turing machine and is this constraint physical or informational in nature?

...among others related to Gödel, formal systems, the multiverse, etc. may interest you.

I am still pondering your statement:

"Now, consider again the self referential situation. It is only when the state is considered with respect to itself that it is no longer itself. That's the paradox."

It reminds me of this Zen analysis of "non-self", as well as a discussion I was having with another FQXi member on the objective nature of color only existing in some relational sense..

Overall, this is a very enjoyable and thought-provoking essay. I like how it does not get bogged down in the details, yet it is technical enough to be taken seriously. I think it is the kind of foundational questioning that can lead to real leaps in understanding. It also reminds me of your "It from Bit" essay in that it takes some of these philosophical ideas and ties them into the physical world. I hope it scores well in the essay contest, and I hope you discuss it some more in the discussion thread here.

Jon

Hello Steven,

I enjoyed reading your essay and I agree with much of what you write. But one thing that peaked my interest is your discussion of Planck's Law for blackbody radiation. The very beginnings of Quantum Theory. You write,

"Five years later Einstein rederived Planck's [blackbody spectrum] results by changing the physical assumptions of the cavity oscillations of the electromagnetic field. He proposed they were quantized themselves, and thus light and all electromagnetic radiation were quantized."

Einstien's Quantum Hypothesis used to derive Planck's Law is not, however, needed to mathematically derive this result! But it was the "first" explanation. And in Physics the "first" successful explanation is often taken as "physically true"!

In my FQXi 2010 Contest essay, "A World Without Quanta?", I show how Planck's Law can be derived using continuous methods and not needing the physical assumption of "quanta". In fact, I show Planck's Law is actually an exact Mathematical Truism (like the Pythagorean Theorem) and is not really a Physical Law. (see also, "The Thermodynamics in Planck's Law")

Further, Einstein's Constant Speed of Light hypothesis contradicts his Quantum Hypothesis! Since it can be mathematically proven that "if CSL, then light is a wave"! (Proposition 11 in "The Thermodynamics in Planck's Law")

Often, the remarkable fit of the experimental curve for Cosmic blackbody radiation with the theoretical curve using Planck's Law is used to argue for the existence of "quanta"! But such confirmation is not proof. What can only explain why we have such a remarkable fit between the experimental spectra and Planck's Law is my proof that Planck's Law is an exact Mathematical Identity. Such remarkable fit we would also expect for a finely drawn circle with the Pythagorean Theorem.

In my current Contest essay, "The 'man-made' Universe", I argue all Basic Laws of Physics are Mathematical Truisms. But I also argue we cannot know "what is" the physical Universe. But can only know our measurements and understanding of "what is". Thus, the only Universe we can know is the Universe we conceive! To that end, we should be guided by an Anthropocentric Principle: Our Knowledge of the Universe is such as to make Life possible.

I welcome your thoughts on this!

Constantinos

(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

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