Hilbert space as a conceptual space .The semantic and conceptual interpretation of wave function

Motivation:

Mathematical spaces (such as Hilbert space) sharing topological and metric (geometric) structures are introduced to modeled sets of concepts or words in some theory or language (and even human knowledge and mental capabilities) thus representing them thoroughly quantitatively and interpreting by mathematical (topological, geometrical, set-theoretical) notions such as points, vectors, distances, operators, transformations, limits, neighborhoods, domains, subspaces, etc.

This can in turn provoke the reverse problem: Given a certain mathematical space such as the complex Hilbert space (designated as “CHS” further) even eventually having a well-elaborated and established interpretation in quantum mechanics. Might both it and that interpretation to be reinterpreted in terms of conceptual space? Once done, what should the meta-interpretation of the relation of quantum-mechanics and conceptual-spaces interpretation be? How might those two quite different interpretations to assist each other and extend their range?

Additional motivation to be chosen just the interpretation of CHS by quantum mechanics:

The kind of knowledge in quantum mechanics is featured to be unique being a meta-knowledge, namely the cognition of knowledge: our recognition of the indications of macroscopic apparatuses measuring some properly quantum and thus microscopic systems. Furthermore, that interpretation in quantum mechanics claims to be both fundamental and universal facilitating philosophical generalizations and thus a series of new secondary reinterpretations of CHS as a conceptual space.

Prehistory and background: Mathematical logic shows that a kind of mathematical structures, namely lattices, supplied by relevant algebras is able absolutely successfully to model and generate almost all logical systems including the most utilized ones. The close connection of mathematical logic to formal and cognitive semantics addresses the problem whether a certain class of relevant mathematical structures can analogically model those kinds of semantics. P. Gärdenfors’s works (in “References”) demonstrate convincingly that vector spaces, each naturally supplied by some topology, seem to fit. Unlike the lattices with algebras, the vector and topological spaces have been used a long time ago to model processes and phenomena in physics including its fundamental and universal subdomains such as quantum mechanics or relativity. In turn, this opens questions about the relation between semantic and physical interpretations of vector spaces including the background of onto-logical and onto-semantic philosophical tradition(s).

Thesis: CHS can be always interpreted as a conceptual space. Any element (both “point” and “vector”) in it is a state of some possible or actual quantum system, and not less, a conceptual state meaning an element (both “point” and “vector”) in the corresponding conceptual space. Thus 

the wave function can identify one-to-one and consequently absolutely a physical state with a conceptual state, after which our knowledge (about the former) represented by the latter coincides with the former.

A few arguments:

1 CHS is a generalization of the usual 3D Euclidean space of our experience, which is the most utilized mathematical structure for conceptual spaces. Being infinitely dimensional in general, it is able to include all finitely dimensional particular conceptual spaces as subspaces therefore considerable as universal and even as the single universal one.

2 The following principles of the interpretation of CHS as a conceptual space are able to satisfy the conditions of the thesis:

2.1 Any predicate corresponds to just one axis “n” ( of the complex Hilbert space. The one-to-one mapping between predicates and axes is conventional, but the interpretation is independent of the chosen convention (whatever be) as only self-adjoint (Hermitian, or “hypermaximal”) operators (transformations) of CHS are considered, and they conserve the ordinal number of each axis.

2.1.1 Any quantitative predicate constitutes a subspace, in which any axis corresponds to just one numerical value of it.

2.2 Any subject corresponds to an element in Hilbert space, after which each axis is supplied by a complex number (coefficient). Three basic cases of coefficients are to be interpreted accordingly:

2.2.1 The coefficient is zero: the corresponding predicate is irrelevant to that subject.

2.2.2. The coefficient is uncertain, i.e. it is a variable: the corresponding predicate should be included into the extension of the subject.

2.2.3. The coefficient is certain, i.e. it is a constant: the corresponding predicate should be included into the intension of the subject. The set of those predicates should be finite and it constitutes the definition of the subject.

2.2.4 (Note): That predicate, which is variable in a specified and thus restricted range, is a case being able to be decomposed to both 2.2.2 and 2.2.3.

2.2.5 The real and imaginary part of any coefficient are interpreted as follows: the former is the number of cases (or probability) of the subject where the predicate in question has been observed; the latter is the expected number of cases (or probability) where the predicate is expected to be observed. Thus the module of the coefficient corresponds to the probability (both objective and subjective) for the predicate in question to be observed as to the subject, and its phase to the ratio of observed cases to expected cases for that predicate to belong to the subject.

2.3 Any relation or motion is represented by a set of self-adjoint operators transforming a set of subjects into another.

3. Thus any subject is a particular case of relation or motion, and any predicate, that of subject.

References:

P. Gärdenfors:

(1993) "Induction and the evolution of conceptual spaces," in: Charles S. Peirce and the Philosophy of Science, University of Alabama, 72-88

(1995) “Meanings as conceptual structures,” Lund University cognitive studies, 40.

(1996) "Conceptual spaces as a framework for cognitive semantics." In: Philosophy and Cognitive Science, Kluwer, 159-180.

(1996) “Language and the evolution of cognition," In: Penser l’Esprit: Des Sciences de la Cognition à une Philosophie Cognitive, Université de Grenoble, 151-172.

(2000) Conceptual Spaces: The Geometry of Thought, MIT

(2004) “Conceptual Spaces as a Framework for Knowledge Representation,” Mind and Matter, 2(2), 9–27.

(2005) “The role of expetations in reasoning,” In: Knowledge Representation and Reasoning Under Uncertainty, Lecture Notes in Computer Science, 808, Springer, 1-16.

(2014) Geometry of Meaning: Semantics Based on Conceptual spaces, MIT.



Appendix:  A joint model of causality in both brain and mind by Hilbert space

Motivation: The mind-body problem might be reduced to that of the relation of the two partly independent causalities in the brain and mind. That approach addresses philosophical phenomenology in Husserl’s or Heidegger’s sense: One postulates identical elements in both brain and mind, just “phenomena”, in which the signifier coincides with its signified and thus with their common and joint sign.

However, those identical elements are differently rearranged and reordered in the brain and in the mind. Reducing to causality means reducing only to reordering. In other words, a single fundamental sequence (supposedly attached to the construct of “transcendental subject”) is differently encoded in the mind or in the brain and thus the mind-body problem is reduced to that of decoding.

Thesis: The complex Hilbert space is able to model exhaustedly that approach formally and mathematically.

Arguments:

1 One can postulate one and the same “Ψ-function” (i.e. an element of the complex Hilbert space) as to the “transcendental subject” where the brain is formally identical to the mind. Furthermore, that Ψ-function being therefore the function of the state of the single united system “mind-brain” is differently rotated qubit by qubit correspondingly in the two partly independent and partly entangled subsystems: “mind” and “brain”. One can start as from the brain as from the mind and then sees the other subsystem as an encoded image of the former. Formally this will mean that the sequence of qubits of the former are reordered into that of the latter.

2 Causality generates a well-ordered series by means the recursive relation “cause – effect”: Thus in the model, the system “brain – mind” and both subsystems “brain” and “mind” share ones and the same elements, namely the “phenomena”, but reordered in three different ways in general. They can be interpreted as three different causalities: neurophysiological, chemical, and physical one (in the brain); mental and psychological one (in the mind) and even one third: transcendental and phenomenological (in the “transcendental subject”), though redundant in the practical applications.  

3 That approach allows of delinking events occurring in different time as to the brain and mind. This is not paradoxical for the mind is gifted by the fundamental capabilities of memory and forecast (expectation) thus involving past or future moments in the present therefore corresponding to the actual brain state.

4 Nevertheless the model has unexpected predictions, which can serve to be refuted or confirmed: The relation of brain and mind in it is symmetrical. Thus the brain, though being a material system, should share the capability of forecasting: Future states of the brain might influence the present mental state. 

5. Penrose is who paid attention (The Emperor’s New Mind: “The time delays of consciousness”) to experiments interpretable as the influence of future brain states into present mind states therefore corroborating the Hilbert space model.    

Bonus: Ψ-function in quantum mechanics is interpretable as the state of a brain-mind (e.g. of the experimenter(s) or of the universe). 

Appendix II: Equating the transcendental to the naturalistic: The unity of mind-brain as a quantum system

Prehistory and background:

Husserl, both mathematician and philosopher, was who offered a new reading of transcendentalism, mathematical in essence. The transcendental might be understood as the collection of all possibilities therefore interpreting the “condition of possibility” in thus. Mathematics accepts consistency seen as the possibility of existence as mathematical existence as well. The collection of all possibilities might be defined as a certain invariant shared by all possibilities at issue, obtainable by “eidetic reduction”, which is phenomenological in the sense of Husserl’s psychology, or transcendental in his philosophy. One might say that eidetic, phenomenological and transcendental reduction are only different senses (or contexts) of one and the same meaning mapping all possibilities of a kind into their shared invariant. Then Husserl’s opposition of the phenomenological (transcendental) to the naturalistic might be further thought as the opposition of the set of all possibilities, defined by their invariant, to an arbitrary and therefore random element of the same set.

However, the system of mind-brain unifies somehow both aspects allowing to be described as both mind (i.e. phenomenologically and transcendentally) and brain (i.e. naturalistically). One might even postulate that kind of duality as the essential feature of that system, necessary for its relevant definition. If that is the case, and Husserl’s approach to the transcendental and naturalistic is used, one would need a certain equation of the transcendental and the naturalistic to define relevantly the system of mind-brain.         

Thesis: The interpretation of the mind-brain system as a quantum system satisfies the condition for an element of a set to be equated to the set, and therefore that of the reduction whether eidetic or phenomenological, or transcendental in Husserl’s sense.

A few comments on the thesis:

Quantum mechanics being only an exemplification and interpretation of a much more general set including it shares the same property, namely, the equivalence of a set to its element. Then, the term “quantum system” means it in the sense of both quantum mechanics and generalization definable by that equivalence of ‘set’ and ‘element’.

No finite and constructive element can satisfy that kind of equivalence. Even more, that equivalence is interpretable as a version of Dedekind’s definition of infinity. However if the axiom of choice is attached, a finite, though unknown in principle, set equivalent one-to-one to each one infinite set should exist “purely” and mathematically, i.e. only possibly, but not ever actually. That paradoxical corollary is implied by Skolem’s consideration of the “relativeness of ‘set’” (1922). Thus infinity is decomposable to finiteness and randomness if randomness be equated to “pure” (never actual) possibility.

Then by interpreting in terms of mind-brain, a random element of the one half of that duality would correspond to each one element of the other. This is equivalent to the suggested by Niels Bohr conception about mind-brain complementarity as a generalization of complementarity in quantum mechanics.