The Final Theory and the Language of Physics by Frederico Pfrimer

Frederico,

I think it's a rather bold conjecture that "The quest for giving a precise meaning to our fundamental concepts cannot be accomplished using natural language" ... without giving an example of a mathematical statement that cannot in principle be translated to natural language.

So I'm in the uncomfortable position of agreeing with your conclusions while disagreeing with the way that you got there. It is surprising to me that you venture into meta-mathematics in your commentary without mentioning Chaitin's leading-edge research (particularly since he is a Brazilian Professor). For example, Chaitin's Omega gives a clear example of an algorithmic compression of a number that is uncompressible (the halting probability of a Turing machine). This example would seem to stand your conjecture on its head: an example of a meta-mathematical result whose natural language translation is straightforward: The number is algorithmically compressible IFF the halting probability of a universal Turing machine asymptotically approaches zero, regardless of the machine language by which such probability is calculated. With this extreme example, I am doubtful that your conjecture is true.

Chaitin has extended his meta-mathematical program to life itself, with the recent publication of *Proving Darwin: Making Biology Mathematical.*

Nevertheless, as a separate subject, I have to agree that what you call a closed theory (and which I would characterize as a closed logical judgment of a mathematically complete theory) is sine qua non -- not only to a final theory -- but to any scientific theory. Closed logical judgements, such as those in relativity, correspond to physical results that make the theory mathematically complete. I like your discussion of quantum theory interpretation, because it clearly exposes why anti-systematic analyses cannot impose a logically objective meaning. Quantum theory is mathematically incomplete. (I don't understand how you reconcile Wittgenstein's anti-systematic philosophy to your philosophy of science, though I would be interested to know.)

Anyway, thanks for a great read and all best wishes in the contest. If you would like to see an information-theoretic take on the Schrodinger's cat problem, and which explicitly uses the excluded middle, please visit my essay ("The Perfect First Question.")

Tom

Minor corrections to my latest post:

instead

"by assuming an extended pure state for every mixed state by assuming the mixed state"

please read

"by assuming an extended pure state for every mixed state"

i wrote

"Extrapolating the Schroedingers' wave function to be universally valid only leads to many-worlds; they may be complete in a certain sense and consistent, but the measurement problem for me seems to be not solved (the problem why the mathematical description of Schroedinger does "collapse" at the "moment" of measurement)."

i would add

It's not only the moment of a "collapse" that is problematic (in many worlds the "collapse" does not exist and therefore is not a problem), but also the fact that for every binary decision i make, another "me" realizes the other alternative. The question here is if this alternative "me" existed before *i* made my binary decision or is it somewhat "cloned" by the Schroedinger equation? I assume that the two branches arising out of *my* decision aren't symmetrical in the sense that the other branch (the alternative "me") is forced to decide between *my* binary decision and gains exactly the "opposite" of *my* decision. Here the question arises why there should be multiple quantum experimentalists within a Schroedinger equation that branches those experimentalists in a non-deterministic manner. Not enough, where is the borderline between a classical decision and a quantum decision? A quantum decision surely seems to be random, but the Schroedinger equation says it is deterministic. So even *my* own decisions (for example my decision to write this post) should seem to be as random as single quantum events seem. An interpretation of QM as strictly deterministic does not alter the mystery why *i* got entangled such that i had to write down this comment. It further does not elucidate that human behaviour in most cases is consistent and makes some sense. Although this could be due to past entanglements, but nothing in the whole QM theory and the Schroedinger equations neccessitates that human behaviour should make sense and shouldn't be non-sensical when averaged over the whole human population.

Dear Frederico,

i now read your arXiv-paper. Although i must confess that i did not check every part of your statements to be reasonable or not, my main interest was to look at what you have to say in your plain text.

You wrote

"The importance of the distinction between pure and mixed states, and the fact that only pure states can be associated with a vector (ket) is dismissed. The distinction show totally different mathematical properties and meaning of each of them. We say that the pure state [ψ] is something objective, actually that it represents reality itself. And the mixed states ρ represents something

epistemic, that they are the state of knowledge of some observers. This two elements are not, however, completely independent, there is a condition for a density operator to be considered a possible state of knowledge. However, the state of knowledge is not totally defined by reality, and that's why we say that it is subjective. We can say that this theory is a theory of knowledge and truth, and this the explanation of why there are elements of this two natures: epistemic and ontic."

As i understand this, it is not possible to unambigiously assign some reality to the "time" between two measurements?

I think this would be difficult, because in the case of Schroedingers Cat (if one leaves decoherence processes out of the discussion) you write

"Note that we have propositions that are neither true, neither false; and that is something new, because according to classical logic, a proposition is either true or false."

In the time in between the measurement of the Cat's state, there must be some "time" elapsing, i guess. In my own interpretation, with the help of some ideas of decoherence, i tried to interpret what's the meaning of the "time" between those two measurements.

It is true that classical logic leads to either true or false values, but if we consider the opposite of the classical logics to return neither true or false values, we cut out a huge part of reality, if we take this new logics to be a passive negation of the classical logic. We can interpret that as the subjective lack of knowledge (the epistemologic part), but this interpretation in my opinion is not sufficient to understand why QM is the way it is, this interpretation is only sufficient to illustrate why QM *itself* cannot give the answer why it is how it is. That's an important difference.

My own interpretation assumes a meta-physical (meta-physical in the sense that classical logic is only a special case of a logic that isn't anymore bound to causal thinking - but nonetheless to reasons - and to physical time) level of reality where the *reasons* for the non-classical logic reside. In my opinion this level has no classical timeframe, but only conistent interdependencies. Therefore no assumption are needed in this realm and all answers to questions are rendered senseless because questions in a timless, interconnected realm don't make anymore sense.

Maybe we are not so far apart from each others point of views, except that i speculate about what's beyond the classical logic and epistemological knowledge, and you seem to not do this.

Thanks again for an exciting essay and an interesting arxiv-paper!

Best wishes,

Stefan

Frederico Pfrimer wrote:

"I have means to say that the main wrong assumption of physics is not

a physical assumption, but a millenary logical assumption: the principle of excluded middle1."

Niels Bohr talking about it

"There are trivial truths and the great truths. The opposite of a trivial truth is plainly false. The opposite of a great truth is also true."

Read more at http://www.brainyquote.com/quotes/authors/n/niels_bohr.html#u2LwwTk8pFUyOzeb.99

Few peoples understand it unfortunately.

Dear Frederico,

o.k., now i understand your approach better. I will take a closer look at the mathematical part and if i can say anything helpfull, i will post it here. Unfortunately i am not a mathematician, so i first must research the meaning of some expressions like idempotent and a few others.

Thanks again for explaining your work.

Best wishes,

Stefan

Hi Frederico,

I enjoyed reading your interesting essay. I agree that the non-Boolean logic inherent to quantum mechanics is precisely what gives rise to all the weirdness that conflicts with our classical intuitions. Personally, however, I'm not sure it's enough to say that quantum logic is non-Boolean and leave it at that - that's merely a description, not an explanation. I think it remains important to ask, in Wheeler's famous phrase, why the quantum? Why non-Boolean logic?

I take a speculative stab at that question in my essay, suggesting that non-Boolean quantum logic expresses a radical frame-dependence in the nature of reality, one most strongly argued for by holography and Lenny Susskind's notion of "horizon complementarity".

Great work.

Regards,

Amanda

Hi Frederico,

You write, "Try to read the equation F=ma. In the context of classical mechanics you would say the total force equals to the product of the mass and the acceleration. But out of context, or better, without any interpretation, you cannot say it."

If you mean only that I have to expand the shorthand symbols to natural language, such that I should write, "Force equals mass times acceleration," then I don't really think that constitutes either context or interpretation. The symbols, after all, mean the same things whether one's native language is Portuguese or English, or any other. Mathematical symbols are universal; we learn this "alphabet" of symbols in learning the artificial language of math, yet the symbols are themselves derived from natural language.

Suppose you mean, though, that to understand how to interpret the way in which these symbols correspond to personal experience, such that we are assured the symbols are indeed universal -- we resort to comparing the symbols to objects, exactly the way one learns natural language. "Mass" one can understand as identical to weight, by balance measurement. "Acceleration" is exactly why Newton invented the calculus, to explain acceleration as the rate of change of the rate of change -- can this be explained by direct comparison to an object? -- yes, if one sees the difference between uniform motion described by a straight line, and acceleration described by a curved line, which is the visual basis of the calculus. Then corresponding experience informs one that a hammer head "weighs" more when accelerated toward the nail head, than when resting on it uniformly. We call that increase of mass-energy by the name "force."

"The problem of natural language is that it is not as precise as math, and there are too undefined concepts or open concepts. For example, define reality? Define proposition? Define truth? These concepts are very hard if not impossible to define in natural language. I cannot prove you it is impossible to define these concepts in natural language, but if you look to a dictionary you will see that some definitions are at most circular and you cannot remove the circularity.'

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. Formal proofs rely on logical judgments derived from a given set of axioms -- they do not necessarily tell us what is true; in fact, the common way of conveying the meaning of Godel's theorem is in the statement: "Truth is stronger than proof." In other words, there exist true statements derived from any set of axioms that cannot be proven from that set, no matter how many or how few axioms the set comprises.

A dictionary is no help here, and in fact we can prove it! Your set of questions above asking to define terms can be reduced to "Define definition." The answer: "Definition is defined by the set of all definitions in the dictionary." Is that useful? -- it is, if one is a mathematical realist (Platonist) as Godel was. Roger Penrose is another example of a modern Platonist. To such a mathematician, there exists a universal "dictionary" -- pure perfect mathematics, in fact, is only a set of self consistent statements. In fact, Godel used to refer to proofs he found particularly elegant as having come straight "from the Book."

Science as a whole, though, conventionally follows the logic of Tarski (correspondence theory of truth) adapted by Popper to correspondence between logically closed mathematical judgments (theory) and experimental results. That's an even longer discussion.

"Really, I was not aware of Chaitin's work." Try his site . I know you'll be interested.

"I liked the way you characterized a closed theory, although we need to take with the meaning of mathematically complete, because if it is understood in the sense Godel then we have a problem with his incompleteness theorem."

I hope I covered that sufficiently above. "Mathematically complete" I take from the EPR definition: every element of the mathematical theory corresponds to every element of the physical reality.

"I've read his book but I'm not a specialist on Wittgenstein philosophy, and so I cannot really answer how my ideas are in relation to his philosophy. If you have something more specific ..."

This is a longer discussion still. Maybe later.

"I'll visit you essay. Thanks for visiting mine, and pleas rate me if you haven't! I wish you all the best in this contest, and that we could keep our talks beyond this scope."

I'd be delighted to engage further ... and of course will award your essay a deservedly high rating.

All best,

Tom

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.