• [deleted]

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

  • [deleted]

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

    • [deleted]

    Hi Tom,

    You wrote:

    "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." "

    Tom, quoting you: "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."

    Your statement appears to me to be dependent upon interpretation. The equation f=ma before interpretation says only that 'something' equals 'what' times acceleration.

    Quoting you: " "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." "

    In other words the 'what' from my statement above is identical to a different 'what'? What is either 'what' without interpretation?

    Acceleration has no need for interpretation.

    Quoting you: "We call that increase of mass-energy by the name "force." "

    In other words: We call that increase in 'what_1' dash 'what_2' by the name 'something'. Or perhaps you are saying that the results of interpretation can be used to explain a 'something' without needing to interpret that 'something'? :)

    Can you please say more about your view of the meanings of 'not interpreting' and 'interpreting'?

    James

    Dear Readers,

    While I'm trying to answer your posts, a propose a challenge: try to read unambiguously the following equation:

    [math]\frac{5\big((a^2+3 b + 5)^{12} + 1\big) + x}{3+a}[/math]

    This exercise shows some limitations of natural language. It is very hard if not impossible to read this equation unambiguously.

    However, the real focus is not translating mathematical language to natural language, but the opposite, translating natural language to math. This can only be done when you have a theory and an interpretation. For example, everybody can talk about money and finances in natural language, but you can only say the same thing in mathematical language when there is a theory that allows you modeling the situation. This the great challenge of all sciences. At the time of Aristotle they could talk about the movement of particles, interactions and other physical notions, but they couldn't say the same things in math; classical mechanics was what allowed us to do so.

    The great challenges are for example what it means mathematically to say that QM is realist? What are the equations that must be satisfied for this to happen? What it means mathematically to say that QM violates classical logic? What it means mathematically to say that QM is complete or incomplete?

    One we can find a consensual answer to all these question, the philosophical problems vanish. Then all you have to do is to prove that QM satisfies or not an equation. The philosophical problem is finding the equation, the rest is simply theorem proving.

    Best Regards

    Frederico

      Dear Stefan,

      Every opinion is important for me. I found that there are two different things that we must try to make them equal: what I say and what you understand. Clarity is what makes them equal. All them mathematical concepts are defined within the paper, but most of them are not necessary for understanding what matters. I am not a mathematician too, that's why I try to keep the math as simple and elegant as possible. This is my email and google: pfrimer.physics@gmail.com. do you have goolge ? Let's keep talking and I'll try to explain you all my work.

      Well, an idempotent element is element such that x^2 = x, for example: 0 and 1.

      Best Wishes

      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

        • [deleted]

        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

        • [deleted]

        :) 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

        • [deleted]

        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

        • [deleted]

        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.

        Hi Tom,

        It is is very hard to find ambiguities in mathematics because good mathematics is always unambiguous. But I found some examples in Ambiguities in Mathematics. A simple example extracted from there:

        "Certain functions, particularly trigonometric functions like sin and cos, are often written without parentheses: "sin x" instead of "sin(x)". So what does the expression "sin ab" mean? It can mean either "sin(ab)" or "(sin a)b". Generally, it'll mean the former. However, it can sometimes mean the latter! For example, I'm looking at some lecture notes right now which uses implicit differentiation to find the derivative of arcsine: you let y=arcsin x, which means that sin y=x, then you differentiate both sides and get: "cos y dy/dx = 1″. In this context, "cos y dy/dx" means "(cos y)dy/dx"!"

        Well, the solutions of a polynomial are not ambiguous, they are a set of number. For example, a more rigorous way of expressing the solutions of x^2-3 is

        [math]\{x \in \mathbb{R}:\; x^2-3 = 0 \} = \{\sqrt{3}, - \sqrt{3} \}[/math].

        Now you can see that there is no ambiguity.

        Well, but that's it. In the end we depend heavily on mathematics. I'm not sure that the mathematics of today is already capable of providing us with all the tools. The problems are foundations of math itself are not really established. Math is the best language we have, but we are always limited by our language. Math allow us advancing further than natural language, but it still have limitations.

        About the continuous, a large part of quantum theory (i.e. information and computation theory) is done with discrete quantities. 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.

        Best regards

        Frederico

        P.S.: I'm taking a look on your presentation.

        • [deleted]

        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

        Dear Amanda,

        Thanks for reading and commenting on my essay. I also think to say that the logic is non-Boolean is not enough. I have an arXiv paper, "On the Nature of Reality" where I develop a new interpretation and formulation of QT. Most of it is related to the logic of quantum world. There I think it is clear why the logic is non-Boolean and how it actually is. There, I extend QT for having a logic about truth, knowledge, necessity and possibility, what is much more that pure propositional logic. This is how quantum logic becomes no classical: it becomes a kind of modal logic where truth is a modality, that is, a proposition can be true, false, known to be true, known to be false, necessarily true, necessarily false, possibly true... I suggest you reading it but outside the scope of this contest. I'll read your essay and rate it without judging from my own viewpoints. Please rate mine too!

        Well, I try to answer why the quantum and why non Boolean logic from the following approach: it is the simplest closed theory possible for describing reality. In my paper what I develop is a closed theory in the sense of this essay, and you will see how simple all the axioms and the definitions are. It is almost impossible to make them simpler.

        Best Regards,

        Frederico

        Dear Yuri,

        Great point. It fits perfectly at this situation. Also it shows something very deep: it is very simple to know what the opposite of a trivial truth represents, but to know what the opposite of a great truth represents is a great challenge. In this case, for example, we could say that quantum logic is non-classical or maybe non-boolean, but then what it really means? To answer it we must perfectly understand how is the logic of the quantum world...

        Best Regards,

        Frederico

        • [deleted]

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

        Yuri

        Dear Benjamin

        Thanks for your comments. For sure mathematics is our upper bound. We can go just as far as math allow us, and there will be times we'll need to first extend math; but that's not new in physics.

        Well, you talked about the incompleteness theorem. Some days ago I found something very interesting about it, and I was willing to discuss about it. A theory is complete in the sense of this theorem when

        "if it is consistent, and none of its proper extensions is consistent." [Wikipedia on Complete Theory]

        A closed theory must be consistent, but ideally, a closed theory should allow extensions! Every time you create a new theory that extends another one or uses the framework of another theory you are creating a proper extension! Therefore, a closed theory that is complete in this sense is useless! You cannot extend it, you cannot apply it! Every model of a theory is a proper extension because a model adds new assumptions and so allows new results! If a theory is complete, then you cannot add a new assumption to it! Because either the assumption or its negation is already part of the theory!!! Complete theories are useless for physics!!!

        About item 1

        You have shown a great point: if a physical theory is based on mathematical elements that are not clear and "closed" enough then all the theory runs in trouble and cannot be really closed. Set theory is really problematic when you analyze it in depth; however, there are parts of physics that only requires results of set theory which are really established. This parts I would say they are safe...

        About item 2

        Well, for me when interpretation mathematically works in on situation while another on doesn't work, then they are not just different interpretations. In this case the formalism or something else is different, and we need other criterions to choose which one is the best.

        About item 3

        Well, it is not impossible for a mathematical system to be contradictory, but it is very hard, or at least much harder than if the system as not described mathematically. But, when you system is very simple and clear, it is very different for it to have a contradiction, and if you have applied it successfully, than is even harder. But, even so you may never prove mathematically that it is consistent. But math evolved too much after Russell. Many of his viewpoints are not followed anymore, and, his approach is not good enough for physics. In fundamental physics everything is becoming more simple, unified and elegant, but Russell did the opposite with math! he took thousand pages to define the simplest mathematical notions! If physics started using his theory, one would have to understand his thousands pages before understanding the simplest closed theory!

        For sure there is a gap to be filled between our current theories and the best we can do. And I've been working at this program. In my arXiv paper I've tried to provide a closed formulation of part of a large part of quantum theory. It is a new interpretation and formulation, but also a closed theory. That's what I think. I hope others would point me what are the missing points for it being a completely closed theory, it is not perfect yet. However, the purpose of finding a closed theory is exactly what gives value to my work, if not, it is just another interpretation of QM. Well, I saw that you are interested in modern algebra, so you will like my new formulation: numbers, bras, kets and operators are all included in the same mathematical structure. I'll take the time to read and rate you essay, please rate mine too!

        Best Regards

        Frederico

        P.S.: what do you think about this view on godel's theorem?

          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