The Final Theory and the Language of Physics by Frederico Pfrimer

Dear Frederico,

I greatly admire your essay. You ambitiously tackle issues that some of history's greatest scientists, from Liebniz to the founders of quantum theory, have wrestled with. Your general approach is relevant to the whole practice of science. For mathematical reasons (principally Godel's incompleteness theorem), I think that the achievement of a "perfectly closed theory" may not be possible, but I see from your comment thread that you have already considered this, and presumably the intent of your program is to achieve a theory as "closed as mathematically possible." In any case, I think that the approach you suggest should be followed as far as mathematics will allow.

Hence, you may have already thought about many of the following considerations. Please don't interpret them as criticism; rest assured that I rate your contribution very highly!

1. I am not quite sure how far one can go in the requirement that a theory be "closed." For example, general relativity invokes a four-dimensional Lorentzian manifold interpreted as "spacetime." But what is a manifold? Well, first of all, it is a set. What is a set? Well, one might use the Zermelo-Frankel axioms. However, this immediately leads to Godel-type issues. Is the question of what statements are "true" in the theory included in its "meaning?" If so, then there is immediate trouble because of Godel's incompleteness theorem.

2. To some extent, I agree with those among the quantum theorists who believe something along the lines of the statement that "quantum theory should provide us with a new worldview." However, it seems that this line of reasoning can also be dangerous, because it can lead one to dismiss as meaningless issues of "interpretation" which are actually significant after all. For instance, the Hilbert space/operator algebra version of quantum field theory and Feynman's sum-over-histories version are indeed equivalent for ordinary flat spacetime, but these versions generalize in very different ways and apply to different physical models, for instance, in quantum gravity. If a model corresponding to one version turns out to "work," while all models involving the other version fail, then it really does matter what interpretation one takes. Of course, this does not disagree with anything you are saying, since it would merely narrow the choices of "interpretation" (i.e. "worldview"), and move one towards a more "closed" theory.

3. Regarding Heisenberg's definition of a closed theory, the ghost of Godel rises again to frown on the phrase "non-contradictory fashion," and the sentence "The mathematical image of the system ensures that contradictions cannot occur in the system." Heisenberg may not have known this at the time, but mathematical formalism is no refuge from contradiction. In general, it is not possible to prove such a system noncontradictory. Leibniz's dream of a "characteristica universalis," is what Bertrand Russell and company were trying to do with their Principia Mathematica when they ran into Russell's paradox. Later Godel wrecked the whole program with his undecidability theorem.

However, regardless of whether mathematical perfection of this sort is possible, there is a vast gulf between our current physical theories and the "best that could be done" in developing a closed theory. Hence, I feel the idea and the program are well-worth pursuing.

I congratulate you for a deep and insightful contribution, and wish you the best of luck in the contest. Take care,

Ben Dribus

Dear Frederico,

i am interested in your explanation. I don't have google (i even don't know what that is), but i will send you an email after the community rating has finished, so we can have further discussions about our topics.

For now, i try to read and vote the essays i have promised to do.

Best wishes,

Stefan

Hi Frederico,

Okay, I understand better what you're saying now. Really, though, what does "ambiguous" mean in terms of mathematics? After all, by the fundamental theorem of algebra, a polynomial equation has as many solutions as the equation has degrees. Does that mean the solutions are ambiguous? -- to say so would imply that there is one "real," or true, solution to the equation, yet such is not the case. All solutions are true for the given degree.

Your approach may shed light on something very important, however, about the form of mathematics we use to model physical reality. If the fundamentally algebraic rules of quantum mechanics leave us with the questions you ask: -- ... what it means mathematically to say that QM is realist? ... equations that must be satisfied for this to happen? ... QM violates classical logic? ... QM is complete or incomplete? -- then just maybe algebra (i.e., the mathematics of discrete functions) can't tell us what lies at the foundation of reality.

Problem is, that our thinking equipment *is* designed for discrete decisions, not continuous functions. How the quantum-mechanical brain connects with the stately motion of the cosmos will be as science-changing a model as Newton's explanation of why the moon falls toward the Earth without hitting it.

You might be interested in this powerpoint I did for ICCS 2007 that takes advantage of Gregory Chaitin's research into the uncertainty of arithmetic that we talked about earlier.

Best,

Tom

Hi James.

I thought I was clear that it is correspondence between the symbol and the object that obviates interpretation, and begs the primacy of theory.

The physical reality of acceleration is identical to Newton's explanation of identity between a falling apple and the falling moon. This in turn generalizes Galileo's earlier principle, that objects falling in a straight line (the apple) fall at the same rate as those in a curved trajectory (the moon). Because the moon has to maintain a trajectory constantly changing in time, however, to avoid colliding with the Earth, we have to calculate the difference between that curved path of the accelerating object (curvilinear acceleration), and a uniformly straight path of unaccelerated motion. All these insights are necessary to arrive at relativity.

(Your questions give me a lot of understanding of the difficulty in communication we were having in Vesselin Petkov's forum.)

To the question of "force," and its measurement, when you write " ... perhaps you are saying that the results of interpretation can be used to explain a 'something' without needing to interpret that 'something'? :)..." you imply that interpretation imparts meaning. I am not saying that at all -- I am saying that "force" derives its meaning by correspondence between the theoretical prediction and the measured result.

Tom

:) in all humility of course,This theory has been found by a young belgian of 37 years old.

Of course it is irritating for a lot of scientists. Just due to the potential of this theory of spherization. Indeed I have found dear sciences community. Of course there are a lot of jealousy and envy. Just due I am repeating to this potential at shot middle and long term. Of course several academicians are going to try just for the funds. But in fact.These persons are not really academicians but simply false scientists.Indeed they do not imrpove, they just decrease the real evolution. Their hates in general are proportional with their frustrations. Of course their credibility is on a sad road if I am recognized. So of course probably that they are obliged to be very bad in their strategies. The irony at its paroxysm above the cries of desesperated frustrated. I suggest that they learn real searchers and real generalsits. But of course their vanity and their taste of opulences imply that they have no times for the real learning , general and foundamental.Me I am a real generalist. Them, no !

ps the incompleteness is rational and the serie is finite and precise. We are just far of our walls separating this light without motion and the light with motions, so the physical universal sphere in 3D and its 3D quantum spheres and cosmological spheres. If the systems of informations is encoded , it rests rational.Not need of extradimensionalities and multiverses where we have different laws. I beleive that a lot of scientists become very irrational there !!! The problem is that we cannot utilize monney without wisdom and universal consciousness. The hour is serious, we have real global probelms and we are near several chaotical exponentials.We must harmonize this earth with a pure harmony. The rest is vain. If the scientists are not able to solve the problems, so they are not scientists. If they loose their time with pseudos extrapolations, so it is very bizare.We return about a simple evidence, this money and the vanity and the frontiers. It desreases the speed of evolution spherization. Each governments must take its responsabilities. The china must take its responsabilities, the USA also, the Russia and europa and the others. The scientists , rational must have responsabilities in their countries. The solutions exist in respecting the sciences and the road of optimization.We have the tools, so what is the problem? the monney, the unconsciousness, what? the stupidity? what ? it is time to create a global earthian commission of quick optimization. the governments of big countries must act quickly together !!! China: take the 2500 best scientists, general and universal of your country and give them powers of acting. USA make the same, take your 1000 best scientists with solutions and act also.Europa :make the same, africa also ....it is essential for our earth , it is now or never you know dear responsibles of governments. We are in a very bad global situation. The solutions exist.

Regards

Tom,

I arrived in the middle of your conversation. I have read the essay and the comments now. I see no weakness in your statement:

"Sure. However, I *can* prove that it is possible to express every mathematical statement in natural language, even though it is impractical, unnecessary and exceedingly tedious."

In my view, I would say that every mathematical statement is formed from natural language. The shorthand use of simple symbols to substitute for the more complex symbols of formal language does not change this. The form of any mathematical statement is purely symbolic and derives all meaning from the same source to which natural language is also only symbolically pointing us toward.

I find your conversation with Frederico very interesting and intellectually stimulating.

James

Ben

You see the common point between my essay and Frederico´s? A ''closed theory'' for Frederico is something very close to what I called semantically complete in our discussions. And just to remember, by trying to produce a closed classical theory, the outcome is Machian philosophy which gives rise to GR almost uniquely via Barbour´s arguments. But the classical theory is not yet completely closed. I feel this could be made rigourous.

Dear James, Frederico, Tom,

i too think maths that refers to our classical world can be expressed by our classical language. But what about the maths of QM? Here it depends on the interpretation if one does say "the particle is in a superposition of positions" or "the particle does take all the paths at the same time" or "the particle isn't a particle but a wave that interferes with its parts".

As i understood it, this is the issue Frederico was referring to in his essay.

Best wishes,

Stefan

I do not understand why my cоmmunity rating becomes lower....

Yuri

Hi Frederico,

I make a distinction between abuse of notation and ambiguous results. Bad notation, as your examples show, can lead to misinterpretation. The same is true of bad grammar in natural language, as well as ambiguities of syntax and semantics and other linguistic failings (such as Chomsky's well known example, "Colorless green ideas sleep furiously").

Unlike natural language, though, mathematics gets its meaning from logical judgments which are never purposely ambiguous -- if they are, one charges the result with an error. Internal consistency of the language is independent of the correspondent meaning.

"Time flies like an arrow; fruit flies like a banana" shows the dependence of natural language on context, where concrete terms (flies) and abstract terms (time) relate in entirely different ways to an object corresponding to the meaning.

Mathematical symbols are limited in the same way. The Greek letter pi can stand for the transcendental number that describes the relation between the radius and circumference of a circle; pi can also represent a discrete prime integer. Results from use of these terms, however, cannot be confused by one who speaks the language -- just as in the natural language above.

You write, " ... a large part of quantum theory (i.e. information and computation theory) is done with discrete quantities."

Actually, all of it is done with discrete quantities. That's where "quantum" gets its meaning.

"I think we should first solve the foundational problems for the discrete, then we extend it to the continuous. But I agree with what you said, but we must find ways to overcome it."

We have, actually. Topological quantum field theory -- among other topological methods -- incorporates the global meaning of quantum events. We can't get a non-arbitrary continuum from discrete quantities; we can, however, derive discrete events from the continuum.

All best,

Tom

Stefan,

You write, "But what about the maths of QM? Here it depends on the interpretation if one does say 'the particle is in a superposition of positions' or 'the particle does take all the paths at the same time' or the particle isn't a particle but a wave that interferes with its parts'."

This is the baggage that comes with a probabilistic description of reality. For most all cases of probability -- i.e., excepting those cases where we have perfect knowledge of the outcomes (such as the six sides of a die) -- very little is really known of probability theoretically.

(James -- thanks.)

Tom

Frederico

A brilliant theory of theory, and a helpful eye opener for me. I agree translation and interpretation between words and maths is crucial and that we're very short of 'words' to discern meanings. This seems equivalent to mathematical abstraction being very limited in 'bits' compared to nature. But do you agree the brain can work best and find important results without either, then it's only communication with other brains that limits us!

I also consider Quantum Logic in my essay, and agree; "This is how quantum logic becomes classical: it becomes a kind of modal logic where truth is a modality," and; "...why the quantum and why non Boolean logic...: it is the simplest closed theory possible for describing reality." I'm really glad I found your essay, and think it should be much higher up. I also hope you'll now read mine, and hope you may be able to give me useful input on it.

I've conceived a new and apparently more consistent fundamental theory deriving Classical and CSL 'commensurably' from a quantum mechanism, identifying the issues. I found this had a pattern similar to Propositional Dynamic Logic (PDL, or Quantum/Modal Logic) and the precise hierarchical structure of Truth Functional Logic TFL. If we consider an inertial frame as being the exact analogy of a proposition, we find a possibly infinite structure of compound propositions being part of compound propositions. In this case each can only be resolved LOCALLY.

The PDL analogy is the interleaved kinetic states, which are separate from and only relate to the next LOCAL state (or 'mode') up or down. The analogy may not be perfect but the actual theory, or rather ontological construction, also seems far more empirically consistent than current theory, being constructed from freshly reviewed and 'logicised' epistemological elements. All paradoxes 'evaporate'!

It is not 'The Final Theory', but appears to help us out of a 300yr deep rut, opening new doors as an open not closed theory. Perhaps you could comment on how I may better formulate the model as a theory. (I do have a whole string of postulates). I look forward to reading your 'On the Nature of Reality' as soon as I've recovered from the essays!

Best of luck

Peter

Dear Frederico,

I am interested to see your "closed formulation" of part of quantum theory. I take it that is your reference 1?

Regarding Godel, I'm not an expert on undecidability, but I think that Godel's result applied to systems like the natural numbers is rather clear intuitively. The following isn't a proof, but it shows intuitively what an absolute miracle it would be if Godel's theorem were wrong. Take the natural numbers. You have a system which is defined by means of a finite set of postulates, yet it has an infinite number of elements, and you can make an infinite number of different statements about it, each of which may be true or false. (I am assuming, for the sake of the theorem, that it is consistent.)

Now suppose you have a statement that IS true, meaning that whenever you substitute natural numbers into it, you will get an identity. How could you PROVE this? Well, you could test it number by number, but you would never finish because there are an infinite number of numbers. So you have to fall back on the axioms. A certain chain of reasoning using the axioms might prove the result true for a certain infinite subset of natural numbers (for instance, multiples of 3), but might not work for others. So you could use a different chain of reasoning for another subset, but there might still be numbers left out. Once again, you might never arrive at a proof.

You can see that to prove this true statement, you would have to be very lucky: a finite number of finite chains of reasoning would have to suffice to prove a statement about an infinite number of numbers. This statement might be true for some numbers for entirely different reasons than for others. "Most" true statements about the natural numbers are undecidable in this sense. Intuitively, this is similar to the fact that "most" real numbers are not rational; to be rational, the decimal expansion must terminate or repeat in a finite number of steps. You have to be very lucky for this to occur.

I agree with what you said about item 2; that's why I prefer to call these "versions" rather than "interpretations." They do generalize differently.

I haven't rated most of the essays yet because I have not finished reading them all. But I won't forget to rate yours! Take care,

Ben

P.S. For the view of Godel's theorem you mentioned; it's a good way to think about it, but one must be sure that the word "extension" means the same thing you intend it to mean in the context of physics!

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is [math]R_1 [/math] and [math]N_1 [/math] was the quantity of people which gave you ratings. Then you have [math]S_1=R_1 N_1 [/math] of points. After it anyone give you [math]dS [/math] of points so you have [math]S_2=S_1+ dS [/math] of points and [math]N_2=N_1+1 [/math] is the common quantity of the people which gave you ratings. At the same time you will have [math]S_2=R_2 N_2 [/math] of points. From here, if you want to be R2 > R1 there must be: [math]S_2/ N_2>S_1/ N_1 [/math] or [math] (S_1+ dS) / (N_1+1) >S_1/ N_1 [/math] or [math] dS >S_1/ N_1 =R_1[/math] In other words if you want to increase rating of anyone you must give him more points [math]dS [/math] then the participant`s rating [math]R_1 [/math] was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process.

Sergey Fedosin

Frederico,

Great essay! Voted. I do agree very much with your essay. I just put up a Powerpoint for my essay, and am curious how you would use FTP to describe the Poisson equation. Specifically slide 18. (Let me know if you catch any goofs on my part, I put it together for hopefully before voting ends.)

Thanks

Jeff