Dear Alfredo,

    We are mixing many thing whose result is confusion. Let me try to synthesize some fundamental points trying to answer you:

1. The answer to your question is yes. The set theory can account of the quantum theory, as you can see in the paper that I have attached you.

2. Let me take one of the sentences of the paper: "The states of a quantum system can still be thought of as forming a set. However, we do not take the product of these sets to be the set of states for a joint quantum system. Instead, we describe states of a system as unit vectors in a Hilbert space, modulo phase. We define the Hilbert space for a joint system to be the tensor product of the Hilbert spaces for its parts. "

3. The importance of this formalism is only conceptual, showing that classical states distinguish from quantum ones, mainly because the first are topologically trivial while the seconds no. This can have applications to the concept of entanglement or the transmition of information.

   Thus you can see that it is not trivial to formalize, in fact using functors instead of tensors as algebraic objects, but is possible indeed.

Philadelphia, PA

Dear Pereira, 

It strikes me that this is, in some sense or degree, a question about the relationship of set theory to the mathematics of probabilities. It is generally held that set theory is fundamental to mathematics in the sense that any development of mathematics can be expressed (or paraphrased) in set-theoretic terms. I would be very surprised if this generalization failed to hold up for the mathematical probability theory used in quantum mechanics. (I suspect that "probabilities" is more appropriate here than "possibilities."  Probability is quantifiable.)  

On the other hand, I doubt that there would be much problem in a set-theoretic expression of any measured eigenstates of quantum systems.  

What remains, then, is to understand the ontology of the wave-function in quantum mechanics. There has always been some tendency to understand this as something of a device for calculations without worrying too much about what a probability wave might be in ontological terms. ("Shut up and calculate!" as the saying goes.) On the other hand, there is also some tendency to understand the wave function as a physical wave. I suspect that is the nub of the present question. If so, then we might leave set theory aside and ask instead:

What sort of thing is a quantum-mechanical wave-function? 

H.G. Callaway

Dear H.G. Callaway:

You asked: what is the wave-function? The answer that we often find is that it is a mathematical function (probabilistic or deterministic). What is a mathematical function? It is a univocal relation between elements of sets. However, there are no such elements in quantum theory. 

Should we conclude that the quantum-mechanical wave-function is not a mathematical function? I would be happy with this conclusion, but Feynman would be not. And his views are far more important than mine!

Another remark is that set theory seems to be inspired on a mechanical and atomist world view. I wonder why nobody (as far as I know) has questioned this inspiration! 

Philadelphia, PA

Dear Pereira, 

(Simply "Callaway" will do, if you want to address your remarks to me.) 

I am somewhat skeptical where our answers regarding physical phenomena are excessively dependent on (or conflated with) the mathematical descriptions of them. The mathematical descriptions are no doubt needed, but there is some importance, as I see it, to distinguishing what is described from the means of description. 

So, in particular, the quantum mechanical wave-function is, presumably or on the evidence, a physical phenomenon in nature. The Schrodinger equation is used to describe it. (The same phenomenon can be described by means of matrix mechanics.) It is called the quantum mechanical wave-function, basically, because the mathematics used to describe it borrows from or resembles the mathematics used to describe waves in water or other liquids. In a sense, to speak of the wave function is to engage in an extended metaphor--which is not a reason to be doubtful about it. Its a point of history. But the QM wave function is not itself a mathematical function --which is usually considered an abstract object. The mathematics used to describe the wave function is a mathematical function and can be viewed as a construction on sets. (If, with Feynman, you are willing to "Shut up and calculate!" then once you have the right formula, your question is answered.) 

To get much further, I suspect we'd need to be a bit clearer about what is deterministic and what is probabilistic. The evolution of the wave function is said to be deterministic, but what this means is that how the probabilities of eigenstate outcomes of measurements evolve over time is deterministic. Its a matter of a deterministic evolution of probabilities. I don't think you have shown that either the deterministic evolution or the probabilities are incapable of a set-theoretic treatment. It is perhaps worth noting in this connection that specialized, non-standard quantum logics have not been very persuasive, as I understand the matter; but instead of questioning set theory, the larger tendency has been to question standard logic. 

There may be ever so many reasons to be skeptical of set theory or of set-theoretic foundations of mathematics. But the related developments in mathematics and in logic and philosophy of mathematics are rather massive in their import and not to be questioned incidentally and by the way. So I would think. 

Thanks for your further thoughts.

H.G. Callaway

Alfredo,

''In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac,[17] David Hilbert,[18] John von Neumann,[19] and Hermann Weyl,[20] the possible states of a quantum mechanical system are symbolized[21] as unit vectors (called state vectors). Formally, these reside in a complex separable Hilbert space—variously called the state space or the associated Hilbert space of the system—that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can attain only those discrete eigenvalues.

https://en.wikipedia.org/wiki/Quantum_mechanics

This is the basic mathematic. Like all modern mathematic, this whole formulation, including the Hilbert space, the complex number, etc are formulated in terms of set theory.   Do you have any problem with the above formulation?   

Usually physical theories are not falling because their mathematical formulation is intrinsically wrong. 

Dear Callaway,

You wrote: "The mathematics used to describe the wave function is a mathematical function and can be viewed as a construction on sets."

Alfredo: What are the elements of this set? 

You: "I don't think you have shown that either the deterministic evolution or the probabilities are incapable of a set-theoretic treatment.""

Alfredo: Yes, I did not show it. I am just asking: what are the elements of the sets?

Dear Louis, you asked after the Wiki quote:

(Wiki) "the possible states are points in the projective space of a Hilbert space, usually called the complex projective space....Each observable is represented by a maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate."

(Louis) "This is the basic mathematic. Like all modern mathematic, this whole formulation, including the Hilbert space, the complex number, etc are formulated in terms of set theory.   Do you have any problem with the above formulation? "  

Alfredo: Yes I have. What are the elements that compose the sets in which the operators, vectors, eigenstates and eigenvalues are defined? It seems we have functions without elements being related!

Alfredo,

An operator is always function defined on a vector space into itself.  It transforms any vector into another vector of the vector space.  A 4x4 square matrix is an operator of R4.  Hilbert spaces are vector spaces whose member are continuous functions.  In quantum mechanic, the state of a system is a point/vector of a complex Hilbert space, i.e. a certain continuos function.  The scalar space is the set of complex number and so the inner product of a continous function with another one give a complex number.  In linear algebra, any linear operator has eigenvectors and each of them has an eigenvalue. The operator transforms any component of a vector in the direction of an eigenvector by the multiplication of that vector component by the eigenvalue.  It is linear in each of the eigenvector directions.  Fourier transform of continuous function are the eigenvalues of the function (the eigenfunctions being the circular function of each corresponding frequency).  I hope it help.  I do not often review stuff.

Dear all,

Set theory is the ubiquitous and building block of any mathematical or physical discourse. Everything in the universe exists in terms of relations of objects which are simply sets, in which most of these relations are functions that are well behaved to study nature. First quantum mechanics is a subset of physics that studies nature at the subatomic label. Particles at this level do not behave in the way matter in a grand scale behaves, which classical physics studies. The probabilistic functions  or states of subatomic particles and their behaviors are still described in terms of relations which are sets again. The only place where set theory has some difficulty is when we try to include sets of arbitrary objects  such as  the set of all sets that do not contain themselves as elements which lead to some paradoxes such as Russel's paradox. 

    Dear Alfredo,

   The set that you ask about is simple to find: the set of states. Their number is infinite as elements of a functional space (Hilbert, Fock, etc). But your question can go deeper if you try to "visualize" such states. In such a case you have some basic remarks to do:

  1. The states don't exists physically and they are indetermined mathematically because they belong to Abelian gauge unitary group U(1).

  2. As elements of the U(1) group they are vectors (with infinite dimensions if we consider acting with the space-time). But they can be also consider functions as solutions of the Schrodinger equation and without the constraint of Born interpretation.

  3. The states depend of the of the statistic of particle which is within them. If we have fermions then the wave functions must be purely antisymmetric (compatible with the fractional spin) while if there are bosons then we have symmetric functions (compatible with integer spins).

   Thus the set on what quantum mechanics works is clear and well defined, although (at least for me) it is impossible to have a realistic picture of them.

First of all, let us first see, what is the ontology of set theory. Set Theory may be construe as a formalization of Plato forms and ideas. Set Theory and first order bivalent logic, have as their basis the low of excluded middle. We have “white” and “black”. In Set Theory there is no “motion”! Even the concept of function, is expressed as a SET of ordered pairs! This way there is no variability. Categorical Set Theory is just the sheaves over a singleton! Thus set theory is inappropriate for modellling dynamical systems, like the quantum mechanics. Instead Category and Topos Theory, are made for these situations. Lately, together with Algebraic Geometry and especially the work of Grothendieck, we have a proposal of Mumford, reasoning against set Theory and instead choose  probability as foundation for all mathematics, see “The Dawning  of the Age  of  Stochasticity.”

(https://www.google.gr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjBpraqjd_PAhVL6xoKHVwsCasQFggaMAA&url=http%3A%2F%2Fwww.stat.uchicago.edu%2F~lekheng%2Fcourses%2F191f09%2Fmumford-AMS.pdf&usg=AFQjCNGrPp3T1ZiKwqD94siJyLfExsIWow&sig2=pOpYABCZfk589iabifOmMA&bvm=bv.135974163,d.d2s)

and A Unified Theory of Randomness | Quanta Magazine. A high order project would be to construct a sythesis of all these with the work of Grothendieck. This will open new roads to Quantum Mechanics and in general to mathematics.

Finally, see Cecilia Flori A First Course in Topos Quantum Theory  2013.

@all,

This is an interesting discussion since it illustrates the wide-spread difficulties in finding a helpful coexistence of mathematical notions and notions as used in physics.

The role of set theory as a foundational system for mathematics should not be overestimated. There are many other systems of basic ideas on which mathematics can be built (e.g. type theory, formal languages, programming languages, ...). Even within set theory the correspondence between sets and mathematical concepts is not free of ambiguities. For instance both {{}} and {{},{{}}} can be chosen as a representation of the number 2 --- although they have different lists of elements. So all questions related to the role of sets and their elements should be kept within mathematics and should not spoil the clarity of notions in physics. (One of my univerity professors became very angry when one of his students used the element sign of set theory (\in of LATEX) in his writings. Needless to say that all his excellent lecture notes didn't make use of any set notation).

When we speak of states in physics, we may gratefully acknowledge that the world is made in a way that presents us with a multitude of good amenable examples: atoms and molecules in their ground state. (Due to their capability of emitting and absorbing photons they attain their ground state 'automatically' if held in isolation). Also the state of colliding atoms/molecules and of atoms/molecules irradiated by laser-beams are amenable and so the quantum mechanical description of the micro cosmos can be tested in many ... many cases. It is an iteresting observation that all these tests confirm the quantum mechanical description although the latter is manifestly incomplete in not taking into account the many particles which are arround and which increase the dimension of the 'actual' state space up to un-manageable values.

So, the world is not only made in a way that it gives many examples of well-defined states. It is also made such that it allows us to learn from the behavior of these states by simple local experiments which are not significantly disturbed by the complexities of the world arround.

Mathematics is designed in a way that it allows to describe the simple local experiments by a multitude of related formalisms. Although Hilbert spaces play a role in most of them one has to remember that all separable Hilbert spaces are isomorphic and that we need many additional structures if we try to predict what happens if two Helium atoms collide.

   Dear Ulrich,

   Please, what do you mean by "the world is not only made in a way that it gives many examples of well-defined states"? Could you explain it a little bit more, over all what is the meaning of well-defined. Thanks.

Dear All:

Many thanks for your very rich and clarifying thoughts on the question!

It is interesting to note that there is no convergence in the answers, but everyone seems to have a part of the truth.

Louis, my problem with your short review is that you did not address the elements of the sets. Vectors and values seem to be in a Platonic world without elements of reality under them!

Dejenie, you mention "particles" as the default elements of the sets. However, there are so many particles! In chemistry, protons and electrons are relevant. In electromagnetism, photons are the particles (that compose the fields). There are so many other subatomic particles that the whole picture seems to be a pandemonium, not a mathematical set! M-string theorists try to order the pandemonium, imposing super-symmetry, at the cost of postulating unobserved particles.

Daniel, in my understanding states are not elements of sets, but (groups of) values assumed by the elements. For instance, in classical statistical mechanics, macro states are defined by the values assumed by the molecules (the elements).

Costas, I think you are completely right. You gave substance to my suspicions!

Ulrich, many thanks for your well-thought short essay on the topic! You showed that, in spite of Costas' criticism of set theory, it may be successfully applied to physics - but not naively, as so many people do!

Dear Daniel,

more explicitely I could have written '...many examples of quantum mechanical systems in a well-defined state'. Well defined is a state if one can describe how to prepare a system in the same state. (Notice that you may prepare any number of gold atoms in their ground state despite the 'no-cloning theorem'.)

   Dear Ulrich,

  As you know well, the states are not well defined. They are gauge undefined (belonging to the U(1) continous group) up to a phase: usual dynamic and Berry phase, for instance. Even the ground state of a system is undetermined as the use of density functional theory (DFT) shows everyday by allowing to substitute Schrodinger equation by Kohn-Sham equations employing exchange-correlation.

   It is true that, in practice, this underdetermination is not too important because we can choose one representation and to act as if they were in trivial topology ( for instance, the Berry curvature being zero). But nowadays there are all a kind of new materials (topological insulators, topological superconductors) which use these phases to obtain special physical behaviours. Therefore, let me to finish saying that the states are not well defined physically and only its square has a well defined value, avoiding to enter in how the experimentalists or the technicians might be using some representation of them.

    Dear Alfredo,

    What is the problem for having a set of solutions of Schrodinger equations? What is your definition of a set?

Dear Daniel:

Sets are made of elements. Solutions of equations are not elements. The elements are those basic components that combine to generate the state space; the state space is generated by all possible combinations of the elements. A solution is a point or region of the state space defined by values attributed to the elements. Attributing values without defining the elements is nonsense - or not?

     Dear Alfredo,

     Pick up pardon and sorry, but:   

    The elements are those basic components that combine to generate the state space; the state space is generated by all possible combinations of the elements.     

     Is this not a tautology?

   A solution is a point or region of the state space defined by values attributed to the elements.

    Please, could you tell me what are you saying in this sentence?

     Dear Alfredo,

     Let me just to show you one example of the use of "set of solutions". What is wrong?

     http://tutorial.math.lamar.edu/classes/de/fundamentalsetsofsolutions.aspx

Dear Daniel:

An example might clarify the concepts. Please correct me if I am wrong.

In classical statistical mechanics:

1) Elements = molecules

2) Sets = a quantity of molecules having a common property "belong" to a set

3) System = a set of molecules having a common property of being inside the same closed recipient

4) State of an Element = given by values such as kinetic energy, direction of velocity, position in 3D space (all relative to spatiotemporal references)

5) State of a System = a group of states of the elements that compose the system

6) State Space = an abstract space containing all possible combinations of values of the elements of a system; each group of values define a state of the system

7) (Dynamical) Functions = relations between the states of elements of a system in time 1 and time 2

8) Solutions (of Equations composed of Dynamical Functions) = points or regions of the state space, describing a group of values of the elements in a given time

Dear Daniel,

In regard to your question about what is wrong with the expression "set of solutions": it is not wrong, but it is an usage of the word "set" that is different from the Set Theory that is at the Foundations of Mathematics.

For this reason, I used the expression "group of values"instead of "set of values" in my last reply (above).

    Dear Alfredo,

    Do you prefer a more formal point of view? Is this the formalism?

    http://www.math.ucr.edu/home/baez/quantum/node4.html

Philadelphia, PA

Dear Pereira & readers, 

Basically set theory is a branch of mathematics, and, of course, it is often regarded as being capable of expressing all the other varieties of mathematics. Since you seem to want to put that idea is question, I would think that you would want to go back to the history and major milestones of the related discussion in the foundations of mathematics and see what has been said about the capacity to express the mathematics needed for quantum mechanics. Instead, in my impression, the present thread seems to involve a great deal of hand-waving about supposed problems and little detailed explanation of the alleged problems.

You wrote, for example:

Louis, my problem with your short review is that you did not address the elements of the sets. Vectors and values seem to be in a Platonic world without elements of reality under them!

---End quotation

No doubt, at some level of analysis, one would want to get down to physical elements of sets. But, on the other hand, it is often quite acceptable to get along with set-theoretic simulations of the ultimate physical elements of sets --so long as the needed mathematics is not disrupted. As a general matter, the question of the relationship between abstract, mathematical objects and whatever the mathematics may be applied to is not a particularly crucial question of interpretation --so long as the mathematics involved serves the extra-mathematical purposes of scientific theory.

Vectors and values, I would assume, could be represented set-theoretically, just as functions can be represented set-theoretically. Your quoted comment above certainly does not show the contrary.

It is a general sort of failing of on-line discussions, including discussions of scientific questions on RG that the mere posing of a question or positing of possible alternatives is allowed to take equal standing with complex results of inquiry established over decades. It looks to me that the present thread is falling into that kind of trap. Its an astounding development! 

I see, as yet, no convincing demonstration of the inability of set theory to encompass the mathematics needed for QM. 

H.G. Callaway

Dear Callaway,

They are into a platonic mathematical world.  Where else?  If I define with a set of set theory what is number 2, then where is number 2.  In that abstract world where it has been defined.  Nowhere else.  I will elaborate this huge question a bit later.  But I thought that Alfredo question is only a basic mathematical question and has nothing to do with the reality quantum theory is pointing (Eddington) for.

Dear Daniel:

The Baez essay you posted the link seems to provide a good answer to my question.

The author discusses why in quantum theory the joint state of two systems is not the cartesian product of the states of their parts (parts = elements). 

He also uses the expression "set of states" in the (I think) wrong way, but the reasoning about the conflict between set theory and quantum ontology is a straightforward answer to my question!

Is Baez well accepted among theoretical physicists today?

Dear Callaway, you wrote: 

"It is a general sort of failing of on-line discussions, including discussions of scientific questions on RG that the mere posing of a question or positing of possible alternatives is allowed to take equal standing with complex results of inquiry established over decades. It looks to me that the present thread is falling into that kind of trap."

Alfredo: It may be, but the kind answers offered by our colleagues reveal that there is a deeper problem. Of course, as in other scientific problems, many researchers prefer the conservative "results of inquiry established over decades". 

Philadelphia, PA

Dear Pereira and readers, 

Consider that neutrinos on their way from the sun to a particular experimental device may oscillate from one flavor to another and stand in a superposition of flavors or masses somewhere along the way. No doubt, this seem a very peculiar sort of particle --revealed in the light of quantum mechanics. 

However, I suppose there may be nothing at all problematic about our taking the set of all neutrinos in superposition of masses or flavor on their way from the sun to our detector at time T.  Right?

However, curious the objects, they may still enter into proper sets of an orthodox mathematical sort. In fact, the physicists calculate how many neutrinos of a given sort there should be, given known nuclear processes of their solar production; and they calculate how many of a given sort should be detected, given the limitations of their experimental device. That is how the problem of the "missing solar neutrinos" arose. Apparently, there is no problem with sets and subsets of neutrinos.  

Regarding proper scientific conservatism, recall that no one would have taken Einstein's new theory of gravity quite seriously, if he had not matched Newtonian results in the massive number of cases where Newtonian results had been confirmed. There is acceptable place for scientific conservatism, just as there is a proper place for scientific innovations. The innovations have the larger burden of proof. So it is with the question of the adequacy of set-theoretic conceptions of higher mathematics. 

H.G. Callaway

Dear Callaway:

I agree that there is no problem in taking neutrinos as elements of a mathematical set and elaborating on functions and equations that describe their movements, including superposition and entanglement. If there is a problem in this task, it concerns to quantum logics, not set theory.

My worries concern the (also) classical issue of quantum-to-classical transition (also called the Decoherence Problem). Assuming that a classical system is composed of quantum elements, what are the elements and what is the function that describes the process of decoherence? Or, as a friend has asked me (and I was not able to answer), how to depart from the wavefunction of a system and arrive at its macroscopic state (and respective properties)? My impression is that the wavefunction is floating in thin Platonic air, not tied to basic elements (except for photons, in Feynmann´s Quantum Field Theory). This may be caused by my ignorance on these matters, but in any case the question is helping me to learn from the experts in the field.

    Dear Alfredo,

    We are mixing many thing whose result is confusion. Let me try to synthesize some fundamental points trying to answer you:

1. The answer to your question is yes. The set theory can account of the quantum theory, as you can see in the paper that I have attached you.

2. Let me take one of the sentences of the paper: "The states of a quantum system can still be thought of as forming a set. However, we do not take the product of these sets to be the set of states for a joint quantum system. Instead, we describe states of a system as unit vectors in a Hilbert space, modulo phase. We define the Hilbert space for a joint system to be the tensor product of the Hilbert spaces for its parts. "

3. The importance of this formalism is only conceptual, showing that classical states distinguish from quantum ones, mainly because the first are topologically trivial while the seconds no. This can have applications to the concept of entanglement or the transmition of information.

   Thus you can see that it is not trivial to formalize, in fact using functors instead of tensors as algebraic objects, but is possible indeed.

Dear Daniel:

It is OK for me that you agree with Baez, but - in a very critical analysis of the use of words and concepts - I would not refer to "unit vectors" as elements of a set. The elements are - in my view - the "parts" of the Hilbert space. Vectors are values that apply to some parts at some times, describing their dynamical states. Am I wrong?

Alfredo,

A vector space is a set and the elements of that set are called vectors or simply points.  

    Dear Alfredo,

    The Hilbert space is also a vector space. The main difference is that we can have the distance between two vectors in a Hilbert space and the inner product which allows it usually is defined by one integral of functions instead of the Euclidean vectors that usually employ.

    Thus concepts as continuity (necesary for working with integrals), diferenciality, singularity, etc... tell us that there is an extra topology in the Hilbert space (mainly defined on functions). These functions in quantum theory as the states given as solution of Schrodinger if we are in non relativistic states. For relativistic you have Dirac equations and in general you need to go to quantum field states where the degrees of freedom is infinite for the states.

Daniel,

I thought that all Hilbert spaces are infinite dimensional spaces and so also the Hilbert spaces in normal quantum theory are infinite dimensional.  

    Dear  Louis,

    A Hilbert space is a vector space possessing the structure of an inner product of two square integrable functions within a closed interval, which includes the ordinary inner (or scalar) product of vectors with finite components . Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Notice that the ordinary vectors (arrows in the 3-D Euclidean space) can be considered functions (in fact they are antisymmetric 1-rank tensors although we have a very intuitive picture of them) are well included in this definition. For instance the electric field E(r,t) is one example of this.

   Therefore the infinity is not necessary at all as a condition for being a Hilbert space and the ordinay vector spaces are Hilbert spaces too simple.

I'd like to add the following excerpt from Hughes, whose remarkably clear explanation was particularly helpful to me.

"A second and related mathematical correspondence presents itself, one which is both wonderfully simple and which goes to the heart of our use of vectors in physical theory generally. The example of classical mechanics shows us that there are possible representations of physical theories, which do not involve Hilbert spaces. Of course, this doesn’t mean that classical mechanics could not be reformulated in this way. In fact, our strategy for providing a partial answer to the question, “Why Hilbert spaces?” will be to show that the theory of vectors has very general application. We will take as an example a particular physical situation and model it mathematically. The situation will be paradigmatically of the kind with which physical theory deals, but our description will be general enough to leave open the question of what sorts of processes, deterministic or indeterministic, are involved. Similarly, its representation, in terms of vector space, will be general enough to be employed for a variety of physical theories; the particular features of quantum mechanics on the one hand, or classical mechanics on the other, will then appear as additional constraints on these mathematical structures, as proposed by Feynman."

__________________

What follows is a bit of simple mathematics, which this forum doesn't seem to support; that bit can be found here, under 3) Vectors: 

http://bit.ly/7Lutv5

______________

Here's a link to the book, Structure & Interpretation of QM.

http://bit.ly/2edhVRn

As you know, I've been investigating M(atrix) theory. Been making a little progress, lately, and came across a nice article by Dijkgraaf. Here's an excerpt which echoes remarks made by others in the field, and which I hope may be of interest to you.

"Recently there has been much progress in understanding a more fundamental description of the theory that has become known as M-theory. M-theory seem to be the most complex and richest mathematical object so far in physics. It seems to unify three great ideas of twentieth century theoretical physics:

(1) General relativity – the idea that gravity can be described by the Riemannian geometry of space-time.

(2) Gauge theory – the description of forces between elementary particles using connections on vector bundles. In mathematics this involves K-theory and index theorems.

(3) Strings, or more generally extended objects, as a natural generalization of point particles. Mathematically this means that we study spaces primarily through their (quantized) loop spaces.

At present it seems that these three independent ideas are closely related,

and perhaps essentially equivalent. To some extent physics is trying to build a dictionary between geometry, gauge theory and strings.

It must be said that in all developments there have been two further ingredients that are absolutely crucial. The first is quantum mechanics —the description of physical reality in terms of operator algebras acting on Hilbert spaces. In most attempts to understand string theory quantum mechanics has been the foundation, and there is little indication that this is going to change."

____________

As I may have mentioned once or twice, this is all very exciting to me for a number of reasons, but especially so since NNs are also aptly modeled by matrix operators acting on vectors.

http://www.hup.harvard.edu/catalog.php?isbn=9780674843929&content=reviews

http://wordassociation1.net/FieldWork.html

http://wordassociation1.net/spectra1.html

https://www.math.uni-bielefeld.de/~rehmann/ECM/cdrom/3ecm/pdfs/pant3/dijkgr.pdf

Dear Brian and All:

In spite of the kind help provided by all of you, I still suspect that quantum ontology cannot be fully captured by set theory. Unfortunately I do not have competence to formulate the objection properly.

I only add that - for my narrow philosophy - physical states cannot be elements of sets, because states change every fraction of time. Elements have to be time invariants (at least, in our models). Their different combinations in space and time compose physical states. Vectors (in Hilbert space) represent physical states, not the elements of the states.

Philadelphia, PA

Dear Pereira, 

Though you seem to be left with some reasons to doubt your initial thesis or hypothesis, this thread of discussion seems to me to have been a useful exercise. I suspect the question might be better formulated as a matter of the possibility of the set-theoretical formulation of the mathematics of quantum mechanics. After all, why should we suppose that the ontology of QM does not come out or is not expressed in its formal, mathematical theory? 

You wrote:

In spite of the kind help provided by all of you, I still suspect that quantum ontology cannot be fully captured by set theory. Unfortunately I do not have competence to formulate the objection properly.

---End quotation

Well, of course, we may be left to suspect that it cannot be cogently formulated! But that is itself a useful kind of result --leastwise in an open discussion.

But let me go a bit further. Consider your claim that "physical states cannot be elements of sets." Why not? Well, you say that it is because physical states change all the time. But take a simple example, say a traffic light. Let us say that it has three physical states of interest for our particular study--red, yellow and green. Can these three states of the traffic lights, say the traffic lights in São Paulo, be elements of sets? Well, since their states are changing from time to time, we have to specify a time of interest (or any number of times of interest, depending on why we want this account of the states of traffic lights.)

One way to think of this is to take the traffic lights themselves at a particular time T as a proxy for the state of the same traffic lights at that time T. A certain number of them x will be in state red, at T, another number of them y will be in state yellow at time T, and some number of them y will be in state green at time T. So, if the traffic lights themselves can be elements of sets, then we can use them to represent the traffic lights in particular states at a given time. Won't the various sets of traffic lights in a particular set do proxy for the set of states of the traffic lights? (For some purposes, surely, for others perhaps not?) 

More basically, your supposition seems to be faulty. If states can't be elements of sets, because they are changing from time to time, then neither can the things which have these states be members of sets, since they are also changing from time to time. But once we specify a reference time, the objection seems to dissipate. Otherwise, you would have to hold that anything which changes can't be an element of a set. Notice the multitude of counter-examples, e.g., the set of all human beings alive at this moment. Surely, we can make various cogent claims about this set, e.g., that it is larger than the set of human beings alive at exactly 12:00 pm on December 31, 1900. The varieties of definitions of sets makes them very pliable. 

H.G. Callaway

Dear Callaway,

The elements of sets can change, but in this case the set also changes, and then the mathematical functions of quantum theory are built on sand!

Why M-string theory can be better than functions built on sand? Because the fundamental strings are always the same. What changes? Their combinations.

     Dear Alfredo,

    The states have their energy fixed in time as an eigenvalue of the Hamiltonian.

    Dear Alfredo,

    Let me to complete the above post for showing that there are no problem to make a set of states defined by their energies and also by operators which relate them.

   If you have fermions (for instance electrons) you have even a countable set associated to the states. This is made within an special functional space as it is a Fock space, where creation and annihilation operators acts on them (or, more intuitive, the particles which belong to them).

  For bosons, it is also possible to employ a Fock space but without exclusion Pauli's principle to act and therefore the countability is much more difficult to do. 

Philadelphia, PA

Dear Pereira, 

I sense a simple product-process ambiguity in you usage of "change." (The related phenomenon is very common in English.) What has changed and perhaps will change may still, at a time between, be unchanging --though its state at T may be a product of a process of change. 

But even changing things can be elements of sets. Consider again the set of traffic lights just now changing from, say, yellow to red. What's wrong with that?

H.G. Callaway

Dear Daniel and Callaway:

I understand that what both of you are arguing for is well established in quantum theory, and works well for probabilistic predictions. However, as a philosopher, what I would like to have is:

1) A well-defined hetararchy:

a) (Eternally) Invariant elements = strings, or what I call "elementary energy forms";

b) First-order sets = particles;

c) Second-order sets = chemical elements;

d) Third-order sets = chemical substances; ... ...

e) Nth-order sets = observables of human experience (including the use of all available technology)

f) Nth + 1 sets = conscious contents.

2) Each function being defined at the adequate level of analysis.

Am I asking too much from science?

By the way, thanks for your patience with me!

    Dear Alfredo,

    Sorry, but what is the sense of your characterization? For me, as physicist, I don't understand what are sets that you mention. For instance,

   b) First-order sets = particles;

If you refers to the set of fundamental particles, then following the Standard Model, at present we have only 17 fundamental particles which creates all the matter.

-6 quarks (u, d, c, s, t, b).

-6 leptons (electron, muon, tau, electron neutrino, muon neutrino, tau neutrino).

-4 gauge interaction bosons ( photon, gluon, Z boson, W boson).

-1 Higgs boson.

    The rest of non fundamental particles is a very huge set if you consider what is called as nanoparticles, etc.

    c) Second-order sets = chemical elements;

    d) Third-order sets = chemical substances; ... .

    By chemical elements I suppose that you refer to the ones of the periodic table ( a set of 118 elements which are physically atoms) and by chemical substances, do you mean molecules, crystals, amorphous, alloys, etc... For a physicist these categories of matter are absolutly different.

    Please, how a philosopher can use these concepts? How do you can obtain further information or what is your aim with using your classification?

Dear Daniel:

In the discipline of Ontology, philosophers attempt to understand the structure of reality, with its elements and levels of organization. Set theory inherited this kind of framework that comes from Plato and Aristotle.

Assuming that the elements are strings, what is the lower order set generated by the combination of strings? I imagine that these sets are the particle. Each particle is kind of combination of the strings. I have no idea of how many particles exist; if you physicists conclude that they are 17, then I agree.

When you classify the types of combinations of particles (e.g. the hydrogen 1 proton, the oxigen atom 2 protons and 2 electrons, the calcium atom 18 protons and 18 electrons ,etc.), the result is the periodic table. If so, the set of 17 particles have 118 kinds of combination, which are traditionally called "atoms" (although they are no more considered to be the elements of reality, but a second-order set).

Yes, by chemical substances I mean the types of combination of chemical 'atoms'.

You wrote that for a physicist "the categories of matter are absolutely different". Could you please briefly describe them, as I did (in my 19th-century framework)?

Let me make another try to understand the problem at hand. Set Theory, is based on an "analysis" of objects into "elements", and then everything is determined be the relation "x\in A". Having a quantum framework, the notions of "element" or "point" is very ill defined! This is also connected with the concept of "analysis". Should we analyse up to particles, leptons, etc.? But more important, is any notion of analysis, legitimate when everything is in motion? And how we go from one level of reality to other level of reality?Because of all these problems, what we need is a "holistic" approach. This approach is provided by Category Theory. An object in a Category is determined not using analysis but knowing the social behaviour of the particular object with the other objects. So an object in a Category can be defined by knowing either the ingoing or outgoing arrows to the object. This is a consequence of the Yoneda lemma. Thus we do not need any analytic concept. Having said all the above we are ready to apply all of what is usually called "real mathematics" (algebraic topology and algebraic geometry,etc.) to express the quantum framework in a physically intuitive way! The concussion then is that we need soem kind of holistic and not analytic mathematics for quantum theory.  

    Dear Afredo and Costas,

    First of all thank you for your answers. We are working in very different areas of knowledge and it is not trivial to translate one in the other. What I am sure is that you are serious people who tries to understand certain subjects as well as I try to do.

    For me in quantum mechanics or in quantum field theory, the particles are always very clear of objects which doesn't change in time if they are fundamental: electrons or quarks are examples. Particles which are no fundamental ,as the neutrons, they decay if you put them in vacuum out of a nucleous (around 15 minutes). The particles are classified using certain quantum numbers as electric charge, color, flavour, etc...and this classification includes them in certain set or not. This is independent of the space-time (horrible for the unification of physics because puts gravitation out the other three fundamental interactions).

    Thus my questions were only for saying that I don't understand what is the meaning of your classification of particles or matter in general. I suppose that this is reasonable and useful, at least for you.

Dear Daniel and Costas,

If there is a classification of particles that are stable in quantum theory, the issue becomes why is there no classification at the bottom? String theorists attempt to provide this level of analysis by means of the very concept of string. If they are able of identifying the fundamental types of strings and to derive particles from them, quantum ontology may be fully compatible with set theoretical approaches. However, I tend to agree with Costas that this kind of project may be not the case. I do not know the differences of formalism between Set Theory and Category Theory, but the relational approach seems to be very interesting. Daniel's remarks about superposition and entanglement in Hilbert spaces seem to be compatible with Category Theory!

Alfredo,

I do not think we should be guided by string theorists.  The ''string theoristic episode'' into the history of physics show clear signs of decays and seem to be one of these dead ends of evolution. It is one of these Franklin expeditions; they got lost into the mathematical world and they never found the Northwest passage of the shore of empirical ground. It is not even wrong. We should applaude their spirit of aventure and courage. But take the lost of this expedition on the mathematical ocean as remenberance of the danger of saling to far from shore.  Refering to a branch of physics that is not even a theory yet and which has no empirical grounding is not a great way to get clue of where to go or a way to clarified other domain of inquiries.  But it is a very good way to muddle and create mysteries where that are none.

Dear Daniel,

I think I learn something that always thought differently! When I have a rigid body of macrocosm, I was always thinking that particles was not constant but moving, and the internal of the rigid body is represented essentially as a function space! You say "the particles are always very clear of objects which doesn't change in time if they are fundamental: electrons or quarks are examples." If one can have "a photograph" of the "particle level" then we may apply set theory to that "photography". I am trying to understand your statement, but really I am confused, because of some miss of knowledge of Quantum Mechanics. I will try to clear up the situation and I may come back. Thank you ant way!

Dear Costas,

I do not think, I may be wrong, that how mathematics is logically build at its base, really do matter at higher level and even less matter in physics.  I think that it really matter of an mathematician which is concerned for the best way his field should be expressed.  I also think that a more beautifully express mathematics would ripe benefit for all in the long terms.  

When I look at Quantum Theory, and Quantum Field Theory , I am not seeing a description of individual particle but I am seeing statistic of interaction.  This is what most contrasted between that physics and the physics of gravitation which is about individual mass in interaction.  The other contrast between the two descriptions is that the Quantum is not a God'Eye view, it is not a description of the world as it is but a description of the specific interaction the observer is engaging in, while the physics of gravity is description of the world without interference of the point of view of observation.  What is I think is being oppose here is the old tension between the theorizing which always attempting to remove as much as possible of what is a particular observer related, the senses, and that nothing would be known without the senses/measurement.  I am hearing the prophecy of Democritus: '' "Foolish intellect! Do you seek to overthrow us, while it is from us that you take your evidence?"   We are using fixed words and fixed equation in our communication about the world.  We are necessarily sending fixed short finite message about a world of fluxes and it cannot only be about fixed patterns in that fluxes.  How could we recognized on this communication world that is fixed, the flux realities behind these patterns?  The expressions are fixed, what it express is fixed, how could we within these limitations, acknowledge, not describe, the flux or what cannot be reduced? The mathematical language of probability and statistic is a first step in this direction.

Dear Louis,

I agree with you, and you restore the picture I had! I like also you are saying "Quantum is not a God'Eye view". This means that Quantum mechanics cannot be modeled using set theory, which essentially IS a kinf of God's view. If you want to model things "locally" and perhaps using "sheaf Theory" a kind of mathematical video, then Category and Topos  theory should be appropriate. any way this matter needs a lot of discussion and interactions! 

Dear Louis,

I am interested in the string theory approach, extending David Bohm's attempt to complete quantum theory. Bohm's approach did not lead immediately to empirical results, but after some time it did! Of course this kind of speculation should be viewed with caution, but your criticism that it is "not even wrong" is probably exaggerated, since there is no 'a priori' reason to believe that their project cannot be accomplished.

I diistinguish their assumption of supersymmetry - that leads to the postulation of particles we do not have evidence for - from their concept of elementary strings - a valid attempt to build physical theory on a a well established conceptual basis, affording an explanation of the qualities of conscious experience (as macro manifestations of high order combinatory properties of the subquantum strings). This approach is not for the faint of heart...

The classical concept of probability is that it measures our ignorance, not that it is an expression of interacting fields. Quantum (non-classical) probability is a concept to be explained, not an explanation of quantum reality!

Alfredo,

The critic of String Theory as not even wrong is not new and stem from the fact that here a domain of theorizing that is already old and which has to manage even to produce a theory o or even a partial theory that could  be tested. The sign of decays are multiple , number of yearly publication, number of new theoretician in the field, rate of publication of major scientists, etc all the indicators are down.  None of that are really a proof that one day something positive may come from it.  That day will have to come before the last string theorist die.  I just wish them good luck.   Probability was first introduced to intelligently bet based on peripheral.

''The classical concept of probability is that it measures our ignorance''

It  measure certainty of prediction given a set of evidence.

''Probability has a dual aspect: on the one hand the likelihood of hypotheses given the evidence for them, and on the other hand the behavior of stochastic processes such as the throwing of dice or coins.

... Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptions and half-proof to deal with the uncertainties of evidence in court.[2] In Renaissance times, betting was discussed in terms of odds such as "ten to one" and maritime insurance premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums.[3]

The mathematical methods of probability arose in the correspondence of Gerolamo Cardano, Pierre de Fermat and Blaise Pascal (1654) on such questions as the fair division of the stake in an interrupted game of chance.''

I like this paper:

http://philsci-archive.pitt.edu/12051/1/Quantum_Mechanics_Generalised_Probabilities.pdf

Quantum Mechanics as Generalised Theory of Probabilities

Michel Bitbol

Prologue

The thesis I will defend here comprises two propositions: Firstly, quantum

mechanics is not a physical theory that happens to make use of probability calculus; it

is itself a generalised form of probability calculus, doubled by a procedure of

evaluation that is probabilistic by way of its controlled usage of symmetries.

Secondly, quantum mechanics does not have merely a predictive function like other

physical theories; it consists in a formalisation of the conditions of possibility of any

prediction bearing upon phenomena whose circumstances of detection are also

conditions of production.

...

''Pascal, for whom ‘the end of things and their

beginning are hopelessly hidden from [man] in an impenetrable secret’.

6 If man must

content himself, according to Pascal, with ‘perceiving the appearance of the middle of

things, in an eternal despair of knowing either their beginning or their end’7

, he

cannot content himself with denigrating the appearances in favour of an ungraspable

backworld governed by principles. Man must learn to inhabit his milieu; he must

know how to focus his attention upon the play of his experimental manipulations and

the phenomena that result from them; he must admit the inconsistency of cutting up

the world into separate and intrinsically-existing objects, since phenomena are so tied

one to another that it is impossible to know how to grasp one without grasping all; he

must understand, also, that no cognition can free itself from the nexus of

interrelations, but can only situate itself within it, remaining cognisant of the perspective from which it derives. Finally, man must consent to make the effort to

domesticate the uncertainty that is his lot, by mathematizing directly the relations

between antecedents and expectations, and between expectations and observations''

Thank you for your answer Louis, very interesting. To Alfredo I recomend the book of Lee Smoling The Trouble with Physics, I think the tittle could be generalized to the \the Trouble With science in general

Dear Louis:

I was talking about the concept of probability in Physics; more precisely, in Statistical Mechanics.

About Bitbol: his theory does not answer my question(s). The first claim (quantum theory is not a physical theory, but a branch of probability calculus) is weird and wrong, and the second (about conditions of possibility and conditions of production) brings the worst of Kant to a central discussion in the epistemology of physics.

Dear Vera Maura:

How could Smolin`s book help to answer my question?

Alfredo,

Bitbol is extending Kant transcendental philosophy and proposed the name: ''Kantum Physics''.  Did you come to the conclusion that it is weird and wrong based on gut feeling or after reading the paper?  

Dear Louis:

After reading the abstract you posted. In the history of quantum physics, nobody intended it to be about probability. Therefore, the first claim is historically wrong - and weird, because it does not derive from any legitimate claim in the field.

The second claim - that leads to "Kantum Physics" - only helps to increase the confusion in the epistemology of physics, by adding the obscure philosophy of Kant - who is not understood even by his followers. For instance, please tell me what YOU understand by "conditions of possibility" and "conditions of production" in the context of quantum theory!

Alfredo,

Bitbol has degree in medecine, ph.D in bio chemistry, Ph.D. in physics, Ph.D. in philosophy and top scholar in Shrodinger and philosophy of science, theory of mind, etc.

He is top notch authority on the philosophy of Quantum Mechanics.  I am not saying to you should agree with what he is saying but you can be sure that he is backing up his claim.  Personally, this is the first time I find an interpretation of Quantum Mechanics that is simple and where there is no weird paradox.  If the world become weird because of an interpretation, it is more likely that the interpretation is weird.  Kant philosophy is not obscured at all and quantum mechanics fits naturally into transcendentalism philosophy as extended by Bitbol.  

You published a RG question on the separation of quantity from quality.  Read what Bitbol says on this topic in this paper.  Very interesting.

''tell me what YOU understand by "conditions of possibility" and "conditions of production" in the context of quantum theory!'' 

I will answer later.

Dear Alfredo

First of all  , I would like to thank you for this Question ! Sorry for being late to join   ,  I  had been away for a long time  for a science philosophy Dialog .. 

So far my present  understanding of quantum mechanics is concerned ,  I will answer your question ( Can Set Theory account for Quantum Ontology? ) with a straightforward ' No ' !

   Let me quote something very interesting from  Yuri Manin -

      "    ...   I would like to  point out that it is  rather  an extrapolation of common-place physics  , whether we can  distinguish things , count them  , put them in some order , etc .. New quantum physics  has shown us models  of entities with quite different behaviour.  Even  sets of photons in a looking -glass box , or of electrons in a nickel piece are  much less Cantorian  than the sets of grains or sand. "

The twentieth  century return  to Middle Age scholasticism  taught us  a lot about  formalisms . Probably it is time to look outside again . Meaning is what really matters . "       [  "Problems in present day Mathematics :  I  ( Foundations )" , in Browder, F. E .( Ed.) Proceedings of Symposia in Pure  Mathematics 28    Mathematical Society  , Providence , Page 36  ]

             This is still an open issue  , but  the Philosophers of mathematics / Physics  , rather than the Physicists themselves  ,  worry  much about this issue  . 

     In fact the whole issue   is linked with the amazing question of  Classical mathematical origin of  standard Quantum mechanics ..!

However , Quantum mechanics is still a successful Theory  ; not just a fanciful account of the human enquirer  committed 'psychologically' to Standard Set theoretical ontology !

So subsequent to your Question , I would like to  ask , that  to what extent Nature tolerate  the extrapolation of classical mathematics ( based on SET ONTOLOGY ) to talk about the domain of quantum  in which the  features of the classical domains allegedly fail !

Paradoxical .... ! Isn't it  ?

More soon 

Very Best

Debajyoti 

Dear Alfredo,

As a theorectical physicist that contributed to string theory, he criticizes it, better demolhishes it. In doing that reflects about contemporary science.

    General comment to the question.

   Of all the physical theories, quantum mechanics is the most phenomenological one in the sense that it was close to the experiment and despising the common sense or physical mental picture of the reality. It is the outcome of many brilliant minds facing measurements and trying to predict accurate results. Nobody could doubt nowadays its success as a progress of our knowledge on nature.

   Unfortunately the despise of the common sense has a price to pay: there are many concepts difficult to explain in simple words. For instance, spin, wave function, entanglement, etc... Even if quantum mechanics provides clear answers for physicists, there are good reasons to think that an adequate understanding of the quantum world will result in a radical reshaping of our classical world-view in some way or other. Whatever, the world is not like the behaviour of the fundamental particles at the atomic scale, it is almost certainly not the swarm of particles pushed around by forces that is often presupposed. This makes the task of metaphysicians very difficult indeed if they want to obtain a clear ontological picture of quantum world.

   My answer to the interesting question made by Alfredo, is clear: yes it is possible that set theory can account for quantum ontology. In fact the Fock spaces can do it by means of creation and anhilation operators working in countable states. The problem is that it is necessary to invent a new "common sense" and physical-mathematical background to assimilate so huge amount of concepts that we do not employ in our macroscopic (no-atomic or even lower) scale.

Excellent, Daniel! 

Your General Comment focuses on the conflict between, at one side, commonsense and conventional philosophical wisdom, and, at the other side, quantum ontology. 

Two variation on the question:

Is Set Theory a part of the commonsense/conventional philosophy side?

Is Category Theory fully compatible with the quantum ontology side?

     Dear Alfredo.

     what is your answer to the above questions?

Yes and Yes, Daniel.

I still need to study contemporary Category Theory as I studied Set Theory (when I was a graduate student 30 y.a.). Now I am based on Costas Drossos' claims. My impression is that Set Theory is more Platonic and Category Theory is more Aristotelian. 

    Dear Alfredo

   My knowledge of philosophy is quite limited and Plato or Aristotle are both very far of what I understand by "common sense" which can be considered within our macroscopic scale. For instance, Aristotle believed that one body needed a force for being in motion or his celestial spheres were contradicted by Galileo or Newton in a deeper form, but all of them where into classical mechanics "pictures". But quantum mechanics is absolutly out of the scope of all the greeks philosophers, it is the motion without trajectories ( usual idea in our minds) or the imposibility to know where one particle is and what velocity has in a given point. Obviously there are much more examples which makes difficult to assimile them in our educated minds within actions much much higher than the one of the Planck constant. 

   For me, a layman in philosophy, what would be very worthy is if the logic associated to the quantum word could be understand properly. For instance, nowadays we use a pure boolean (only binary information 1 or 0) in computers but I am sure that in the future the information is going to use a fuzzy logic as it happens in quantum mechanics. This is only one of the possible applications that I can see for this question. What do you think about?

Dear Daniel:

I was not making a reference to Aristotle's Physics, but to his Theory of (conceptual) Categories developed in the book Organon.

To be sincere, I think that any (ancient or contemporary) mathematical and logical thinking requires some kind of temporally invariant "atoms" or "particles" to be the units that belong or do not belong to a category. If the logics is fuzzy, the 'belonging" relation is not exact, but falls within a range of possibilities. Zadeh distinguished "possibility" from "probability" (see the Wiki article linked below). 

In sum, I think the main paradigmatic change with quantum theory seems to be in the concept of mathematical relation. Instead of functions, we have non-functional relations between the invariant elements. I would not eliminate the elements, because mathematical relations need them.

https://en.wikipedia.org/wiki/Possibility_theory

Dear Alfredo & Daniel,

I have suggested Category Theory as a framework for QM. There is however some non-standard models of Set Theory that can be used for this purpose. I mean Boolean-valued models. Boolean algebras essentially incorporates macroscopic reasoning, whereas Boolean-valued introduce B-sets, i.e "functions" which can modelled variability. Based on Boolean-valued models, Takeuti introduced "Boolean Analysis". In this framework, if one chooses as Boolean algebra, a Boolean algebra of the lattice of subspaces of a Hilbert space, then the "Real Numbers" in this model are self-adjoined operators. So one can imitate the classical mechanics, which uses the standard R as the model of measurements, by using R^B, the Boolean-valued model of the reals as the model of measurements. If we change the Boolean algebra and take a Measure algebra instead, then "the Real Numbers" became now "random variables"! So this models might be used for QM purposes.  I attached a paper of the second kind.

   Dear Costas,

   Although I didn't read your paper in detail, it is quite interesting because you follow what I was thinking. You work with Hermitian Hilbert space where real eigenvalues are obtained for the operators and also you employ a B-distribution to extend the Boolean algebra.

   I think that this is a very good line of work for formalizing the logic structure behind Quantum Mechanics.