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.

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  • Post #47

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

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  • Post #50

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

    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 Peter,

      Thanks for your comments. It looks like you could understand a bit deeper the content of my essay. That's really a theory of theory, and curiously the only possible theory of theory is a theory about closed theories, because you cannot be more explicit giving the form of an open theory. Almost anything can be an open theory.

      As you like quantum and other logics you might like my arXiv paper. There you'll see three logics unified and applied to quantum theory: propositional, modal, and epistemic. You can talk about truth, knowledge, necessity and possibility in quantum theory! I'll read and rate your essay, and if you haven't, please rate mine! I would like to keep discussing beyond the scope of this essay! We have close interests, and you might improve my ideas...

      Wish you all the best!

      Frederico

      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

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      • Post #58

      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

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      • Post #59

      Sergey G Fedosin is bombing entrants' boards with the same "why your rating has dropped" message. They are all dated Oct. 4... same message.

      WTH? I've seen one fine essay drop 89 (eighty-nine) positions, in "Community Rating" in the past 24 hours, and "Sergey's note" came BEFORE it plummeted. Hmm.

      The vote/scaling of this contest is quite nebulous.

      "Hackers Rule!", I suppose!

      Well??? What else is one to think? The General Public is... Watching...

      Hi Frederico,

      Please check this link and find how five essays, including yours, were removed from the 35 finalists. I posted some messages with attachments containing the page and screenshots at 0:01.

      Good luck,

      Cristi Stoica

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