Just a word about the jump from "exists x such that x = 2" to "x = 2". The first formal statement contains a variable x which is bounded by a quantifier. So the statement is closed, it needs no supplimentary information to be understood. The second statement, "x=2", is open, because it speaks about an x that depends of the context. It may contain more information in the presence of (say) "x is the number of € Mary owns" and less Information in the presence of (say) "let x be a number". Other said, "x=2" is an open formula. How do you want to solve this difficulty? Very often, one adds constants to the language, so the second statement must be "c=2", and no "x=2". c=2 is a statement and not an open formula, it says something about two constants in the language.

The point is that the normal rules of proof in logic (the deduction rules) are Modus Ponens and Generalization. Generalization is sometimes expressed like this: from A(c) derive "exists x A(x)". What you say  is the other way around!... OK, you want this maybe only for "measurements" so formulae like "x = value", but this is still very non-standard. Question: would it suffice if you add a deduction rule like """ "exists x, x = value" implies "c=value" """"? I mean, just build an axiom containing what you need, and then study the resulted system: if it is consistent, sound, complete, etc...

Thank you Charles.  I'll have a look through your paper.  My quantum theory is formal arithmetic, consisting only of arithmetical propositions.  I do not invoke any postulates of quantum theory.  My theory treats information strictly.  In order to write a quantum theory for the free particle, five items of information are needed.

That is the information I have in the theory.  A principle of this treatment is that the mathematician is not allowed to introduce new information.  I must solve my measurement problem with the information that is already there.  Or, formally, discover additional information that forces  "there exists an x such that x=2 " to become  " x=2 ".  That is: a proposition asserting existence to one asserting equality.

Steve,

my opinion is that pure logic contains no information exept it's own postulates. Therefore physical theory can not be derived only from logic. But, nevertheless, to my mind You found the key point in logical structure of QM. Surprisingly, non-proffesional Manuel Morales from the thread "Is mathematics a human contrivance or is it innate to nature?" also stopped on this point. But he has negative conclusion, that all science is wrong. I can recommend You Marcel Van der Vel from the same thread as a good logician. May be he can help You.

Regards,

Eugene.

Steve,

remark of non-professional.

And if You introduce two levels of existence: existence as a possibility and existence as a fact? The last corresponds to constructive mathematics.

Regards,

Eugene.

Steve,

arithmetics, words, information are all descrete world. But reality may be continueous.

Regards,

Eugene.

Descrete or continuous is fundamental question, Charles. But in Math may be no difference (see p-adic numbers).

Regards,

Eugene.

Steve, Charles,

my opinion is that mearsurements (observations), arithmetics and so on (observable and codified reality, we call it Nature) are descrete. But reality itself (true existence, world of ideas) is continuous or has not this dilemma at all. QM stopped on the first. From this is it's incompleteness. Human being evidently belongs to both.

Regards,

Eugene.

Charles and Eugene,

Thank you both for your contributions to this.  The axioms of arithmetic I am using are a modified form of the Field Axioms.  The modification to these is a sequence of axioms that prevents modulo addition, because I am not interested in that.  Models of my axioms of arithmetic are the infinite fields.  This includes the complex plane, the real line and the rational line.  There may be many others too;  I don't know. Under these axioms of arithmetic, the rational field is the only one whose elements (scalars) are logically dependent on axioms.  All other scalars are logically independent of axioms. This is well known and provable using Soundness and Completeness Theorems, together. Of particular interest is the fact that existence of rational scalars my be proved (caused) by axioms, yet existence of complex scalars may not.  Nevertheless, existence of complex scalars does not contradict axioms and is consistent with them.  In contrast to rational scalars that are caused, the complex scalars are permitted, but not caused.  I therefore have a theory of two qualities of existence.

All that said, my quantum theory, based on this arithmetic, does accommodate continuous fields of scalars. 

Charles, what I am saying is: my theory is Hilbert space for the wave packet. Though we all learnt quantum mechancis is on Hilbert space, nevertheless, only the part which needs orthogonality is.  For the rest of quantum theory Hilbert space is non-essential.  It is the logical difference between non-Hilbert space (the Banach space) and Hilbert space that is missing from standard theory.  The reason Hilbert space is generally believed to be needed throughout the whole theory is that one of the Postulates of quantum theory is that any theory should be unitary or that it should be self-adjoint or Hermitian.  That unitarity by Postulate hides the fact that unitarity is implied in the wave packet anyway.  I don't much care what my theory is called.  But I think most people will best understand what job it does if it's referred to as a quantum theory.

Dear Han and Charles,

Originally this question was directed to Logicians.  "Logicians, please help with The Measurement Problem from quantum mechanics?"   But the moderator changed my question to what you see now.  I have worked on this research for 8 years solid.  I know this subject very very very well.  I have a paper almost ready to release.  I am not looking for any answers with reference to that paper.  That paper is about the logic we see in experiments.  It does not attempt to cover The Measurement Problem.  I asked that question simply to consider work that will follow on from what I'm doing now.

I invite you Charles to attempt building a quantum theory with vectors not in Hilbert space.  You will find that Hilbert space is not needed throughout the theory except in the wavepacket.  Maybe you take the view that there is nothing wrong with quantum theory.  Many people do.  If you think it's not perfect logically, then you should be able to accept there is something wrong somewhere.  And that whatever is wrong needs to be looked for.  By that reasoning, none of the standard doctrine can be accepted without challenge.

I invite you to derive the Canonical Commutation Relation from first princiuples.  You will remember this is the Lie algebra deriving from the symmetry: homogeneity of space.  In standard theory, this Relation has unitarity or self-adjointness imposed on it. If you do the derivation you see that homogeneity does not demand unitarity.   But unitarity is imposed in order to ensure vectors have a scalar product.  That is they support a probability amplitude.  So in standard theory, unitarity is imposed from the outset.  If you continue with your derivations, you will find that your theory works.  This includes eigenvalue equations whose eigenvalues are real, but whose eigenvectors have no scalar product.  However, when those vectors form up into superpositions, self-reference is unpreventable (by axioms), after which, unitarity is implied, and vectors transform into Hilbert space and the scalar product comes into existence.  The blanket cover unitarity, imposed by Postulate is redundant.  I hope you can see I have done my work.

Logical independence of the self-reference, unitarity, Hilbert space and indeed the wave packet is proved by showing these all rely on existence of the square root of miinus one. And that can be proved by showing that the square root of minus one is logically independent of the rest of this theory.  The logical independence of the square root of minus one with axioms of arithmetic is well-known to logicians and is proved using Soundness and Completeness theorems (from Model Theory, a branch of mathematical Logic) in conjunction.  In fact, from the fact that arithmetic is incomplete, in the Gödel sense, it's looking highly likely that quantum indeterminacy derives ultimately from incompleteness of arithmetic.

I invite you read through all that I've written in my replies to you, above, and see that the story of my claims is self consistent and consistent with experiments.  And that it alters standard theory in no way that affects theory quantitatively.

I am saying that standard theory needs change.  And so, there is no point in referring me to textbooks which expound the status quo.  That said, if you can show me any flaw or inconsistency in my claims, I shall be very very pleased to have you explain them.

Please remember, this is actually the question to which I am looking for answers:

Starting from the proposition: " there exists an x such that x=2 ", what impetus might make that information jump to: " x=2 "?

Thank you.

Have you considered the many worlds interpretation of the wavefunction and its collapse? It solves the indeterminacy problem since every possible value of the wavefunction is expressed by a branching of the universe (into higher dimensions?).

The standard model of Quantum Theory is still based on the Copenhagen interpretation. It was Bell's theorem that proved that hidden variable cannot exist. However the proof must assume the Copenhagen interpretation to be valid. Determinism is not completely lost. The fact is the mechanism for the collapse of the wavefunction is one of the biggest questions in the field, and I don't think any honest physicist would claim they understood it, or were satisfied with the Copenhagen interpretation. There is certainly a theory that supersedes Quantum Theory just as QM superseded Newtonian Mechanics (which also uses mathematics which are subject to Godels theorem). Looking forward to your paper.

Steve,

QM is not about existence, but only about possibilities.

Regards,

Eugene.

Charles,

Many worlds does not use classical probability. It use the probability function based on Schrodinger's equation. It directly addresses the issues of quantum theory, and modern supersymmetry and M theory suggest that this is the correctly interpretation. There are empirical measurements which can test if something like many worlds is the correct interpretation of the collapse of the wave function. It will be some years before we have the accuracy to resolve these differences.

Bell's theorem also does not use classical mechanics. It uses Quantum Mechanics expressions for the wavefunction (again Schrodinger's time dependent equation) and it's collapse. You could not even pose Bell's theorem in classical mechanics. Bell's theorem is dependent on the Copenhagen interpretation. In string theory there are more rigorous proofs that there can be no hidden variables (but string theory is still not able to reconcile Quantum mechanics with General Relativity, so something is still missing.

I can't speak for every physicist in the world, but I have never met one who didn't admit that the collapse of the wavefunction has no know mechanism. It's a mystery. It is never derived in Quantum Theory. It is just stated. Ok, so the wavefunction will change over time in a predictable manner based on Schrodinger's equation, but when a measurement is made, poof it suddenly becomes a delta function. No physical or mathematical justification. Just take it on faith, as ridiculous as it is, and strangely enough the consequences that follow match experimental results to an unusually high degree. It is basically impossible to understand the collapse of the wavefunction because it has never been explained, even when it was introduced. At this point as we have no way of empirically probing the mechanism, it resides on the metaphysics side of the line for now.

Please can we stick to a fruitfull, open-minded, open-hearted, friendly exchange of thoughts.

I'm sorry Chris, but I don't agree with what you are saying. I have read the Bell paper. It is quite clear. Again standard probability theory is not compatible with the Copenhagen interpretation which says that a particle is not in an unknown state before it is measured (in the classical sense), it is in a superposition of all states with non-zero values of the wavefunction. I'm not sure what you mean by collapse does not require a mechanism. It would not occur if there were not a mechanism. Even an indeterminate process has a mechanism.

I should also point out that Bell's theorem relies on the assumption that non-locality is not possible. The EPR experiment was eventually performed in the 1980's and it was found that it happened exactly as the thought experiment predicted. Locality was violated. The EPR paradox was meant to disprove the Copenhagen interpretation of Quantum Mechanics since it resulted in the impossible process of information traveling faster than the speed of light. Turns out non-locality is possible. If hidden variables have been disproved, then the only explanation for the EPR results is that locality is violated. Pick your poison.

Lawrence, Charles,

collapse of wave function takes place not in real world, but in world of possibilities, as Steve pointed. We have to distinguish between these two worlds. QM deals only with the world of possibilities.

Regards,

Eugene.

P.S. When mearsurement is done, the whole world of possibilities changes.

Steve,

Future is possibilities, Past is facts. Think about the sense of Time. This is the resolution of your paradox. QM tries to predict, therefore it deals with possibilities.

Regards,

Eugene.

You can see the following papers:

Anatolij Dvurecenskij and Sylvia Pulmannova, Difference Posets, Effects, and Quantum

Measurements, International Journal of Theoretical Physics, Vol. 33, No. 4, 1994

C. A. Drossos, G. Markakis, Boolean Powers and Stochastic Spaces, Mathematica Slovaca, 43, 1-19, 1994.

I take the view that at collapse a decision is made where a choice is made from a selection of possible outcomes.  I do not believe that the decision made before and that collapse is the process of gaining knowledge of that decision.

I'm going to repeat again.  Hilbert space is necessary in the wave packet. I actually worked through to establish that. That evidence contradicts standard theory.  My Hilbert space stems from Banach spaces. It's the same Hilbert space as Hilbert space in standard theory.  It's just that it is a consequence of my theory and not postulated.  I shall release my paper on this as soon as I can.  Presently it is not ready.  Then others will be able to see in full, what I am saying.  I acknowledge that it's unfair of me to argue about this without giving the full story.    But then, what I am actually interested in is my original question:

Starting from the proposition: " there exists an x such that x=2 ", what impetus might make that information jump to: " x=2 "?

Charles, I'll try and elaborate a little.

The reason for the wording of the original question I pose is that my theory gives me something like all these:  " there exists an x such that x=1 ",  " there exists an x such that x=2 ",  " there exists an x such that x=3 ", " there exists an x such that x=4 ", etc, etc..  This may be interpreted as " a field whose variable I call x exists at every number value.   This statement asserts existence of the field for all those values; it does not assert equality of x with those numbers.  So this statement may be seen as asserting existence of every position x, for example.

Directly after measurement, a new statement is true, in which x=3 (say) which excludes x=1, x=2, x=4 and every other number.  This new statement does assert equality. And so I am speculating that the measurement process forces some sort of transition from: " there exists an x such that x=1 ", " there exists an x such that x=2 ", " there exists an x such that x=3 ", " there exists an x such that x=4 ",..............    to   " x=3 (say)".

I am looking for some mechanism in Mathematical Logic or combination of statements that will force that new statement.

Steve,

to my mind such mechanism does not exist in Mathematical Logic. It is not logical problem. It is about existence. Time is absent in logic.

Regards,

Eugene.

Time can be represented in logic. For Steve's example it would be an operation that produces a single value for x. This is a change in the values of a system, which is the fundamental condition for time. If the state of a system never changed it would render time at best as meaningless or non existent. There is also Temporal logic developed by Arthur Prior. It has some very useful implications for computation.

Lawrence,

Logic deals with statements, not operations. It is the great difference.

Who is Arthur Prior?

I don't think there is a real measurement problem in Quantum Mechanics without collapse. This follows from an understanding of the work of Everett and Gleason. You need to abandon the idea of the unicity of the outcome of a measurement, but the appearance of unicity can be explained entirely from QM (without collapse). My own work shows that classical logic + arithmetic + the computationalist hypothesis in the cognitive science (basically "my body is a machine") entails a strong from of indeterminism, local non locality, etc. Computationalism enforces the existence of a personal-internal "many-dreams/worlds/states" interpretation of elementary arithmetic, and Everett apparent proliferation of physical realities, alias the measurement problem, is something to be expected. Eventually QM itself must be derived from classical computation theory together with modalities due to our distributed self-embedding in arithmetic, enforced by the computationalist hypothesis.

The passage from Ex(x = 2) to x = 2 ?  This is ambiguous. In general you can jump from ExP(x) to P(n), for some n, if the "E" is a constructive quantifier, but that is not generally the case in classical logic. 

I think that nobody took my opinion seriously!

Lawrence,

I guess, that Temporal logic is one of the many valued logics. Am I right?

Costas,

we simply had no time to read your paper.

Regards,

Eugene.

Just a word about the jump from "exists x such that x = 2" to "x = 2". The first formal statement contains a variable x which is bounded by a quantifier. So the statement is closed, it needs no supplimentary information to be understood. The second statement, "x=2", is open, because it speaks about an x that depends of the context. It may contain more information in the presence of (say) "x is the number of € Mary owns" and less Information in the presence of (say) "let x be a number". Other said, "x=2" is an open formula. How do you want to solve this difficulty? Very often, one adds constants to the language, so the second statement must be "c=2", and no "x=2". c=2 is a statement and not an open formula, it says something about two constants in the language.

The point is that the normal rules of proof in logic (the deduction rules) are Modus Ponens and Generalization. Generalization is sometimes expressed like this: from A(c) derive "exists x A(x)". What you say  is the other way around!... OK, you want this maybe only for "measurements" so formulae like "x = value", but this is still very non-standard. Question: would it suffice if you add a deduction rule like """ "exists x, x = value" implies "c=value" """"? I mean, just build an axiom containing what you need, and then study the resulted system: if it is consistent, sound, complete, etc...

Thank you Milhai for your thoughts on this.  I am beginning to think I need to write a short article on this, to show exactly to others, what problem I need to solve, with suggestions of the kind of solution I might expect.  When I posted this question, I thought possibly the answers might already be known.  Currently I am working on a paper.  After that, I'll try and get something written for you.  Again thanks.