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The Expressional Limits of Formal Language in the Notion of Quantum Observation

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Abstract

In this article I deal with the notion of observation, from a phenomenologically motivated point of view, and its representation mainly by means of the formal language of quantum mechanics. In doing so, I have taken the notion of observation in two diverse contexts. In one context as a notion related with objects of a logical-mathematical theory taken as registered facts of phenomenological perception (Wahrnehmung) inasmuch as this phenomenological idea can also be linked with a process of measurement on the quantum-mechanical level. In another context I have taken it as connected with a notion of temporal constitution basically as it is described in E. Husserl’s texts on the phenomenology of temporal consciousness. Given that mathematical objects as formal-ontological objects can be thought of as abstractions of perceptual objects by means of categorial intuition, the question is whether and under what theoretical assumptions we can, in principle, include quantum objects in abstraction in the class of formal-ontological objects and thus inquire on the limits of their description within a formal-axiomatical theory. On the one hand, I derive an irreducibility on the level of individuals taken in formal representation as syntactical atoms-substrates without any further content and on the other hand a transcendental subjectivity of consciousness objectified as a self-constituted temporal unity upon which it is ultimately grounded the possibility of generation of an abstract predicative universe of discourse.

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Notes

  1. The Life-World in Husserlian terminology can be roughly described to a non-phenomenologist as the physical world in its ever receding horizon including in intersubjective sense all reduction performing subjects in a special kind of presence in the World. More on this in Husserl’s The Crisis of European Sciences and Transcendental Phenomenology (1962).

  2. Regarding as phenomenal objects those given primordially in perception, then objects like atoms, electrons, etc. can be also regarded as given by experience and thus considered as real to the extent that their ‘reality’ is based upon the interpretation of sensible signs in an experimental situation (see Heelan 1988, I, p. 5).

  3. Formal ontology deals with the formal structures of facts as registered (by intentionality of experience) whereas apophantic logic deals with the formal structures of facts or state of affairs as supposed, that is, it deals with categorial meanings associated with judgements, theories, etc. (Sokolowski 1974, Append., p. 289).

  4. Intentionality is not to be understood as a relation of a psychological character. By intentionality, which is a phenomenological notion, it is meant something fundamentally deeper and a priori. To a non-expert in Phenomenology it can be roughly described as grounding the a priori necessity of orientation of a subject’s consciousness to the object of its orientation.

  5. The notion of impredicativity of the objective unity of temporal consciousness should be taken as referring to the impossibility to formally describe the continuous whole in any other way but in terms of parts belonging to the same genus as the whole. Consider, for example, the mathematical continuum and the circularities produced in definitions where the definiens cannot be defined but in terms of the definiendum e.g. in the definition of an open interval of the real line.

  6. In E. Husserl’s view, perception by virtue of perceptual acts provides the concrete, immediate and non-reflective basis for all our experiences and thus provides the basis for any intuition of abstract objects. Concerning mathematical acts and contents, in particular, Husserl’s view was that they are too founded on immediate perceptual acts and contents which could anyway exist even in the absence of mathematical activity. For details, see resp. R. Tieszen’s (1984, pp. 412–15, 2005, pp. 54–55).

  7. Ultimate substrates are classified in the following two categories: the ultimate material substance or the eidetic material singularity with regard to a content and the eidetic formal singularity with regard to a form; both are considered as pure individual singularities with no syntactical content. The latter singularity, termed ‘Dies-da’ by Husserl (close to the meaning of Aristotelian τó\(\delta\varepsilon\,\tau\iota\)), is called an individuum inasmuch as it can be instantiated as bearing a concrete ‘thingness’ substance (sachhaltiges Wesen) (Husserl 1995, pp. 33–35). To this last category belong, as ‘state-of-things’, the syntactical individuals of logical formulas within a formal mathematical theory in the sense of ‘empty substrates’; on the syntactical level they can be thought of as modifications of an ‘empty-something’ (Leeretwas).

  8. For instance the forms of transversal intentionality, that is, retention and protention, are purely phenomenological notions and can be roughly communicated as respectively a kind of spontaneous conservation of immediate past (retention) and a spontaneous expectation of original impression (protention); they are of an a priori character in the process of constitution of a sequence of original impressions together with their retentions in the flux of consciousness. For more details, see Husserl (1966a).

  9. In Husserl’s view the content of an intentional act is thought of as the meaning of the act by virtue of which consciousness refers to an object or state of affairs as its own, (see Tieszen 2005, p. 53).

  10. This is a direct implication of the Indistinguishability Postulate which states that there is no way of distinguishing those quantum states differing by a permutation of particles only, (French 1989, pp. 440–441).

  11. This homogenous space-time continuity must fulfill the three minimal conditions set for the trajectory of re-identifying individuals in French (1989, p. 434). In short: (1) the trajectory must be spatio-temporally continuous, (2) the trajectory must be qualitatively continuous in the sense that any individual stage on the trajectory must be qualitatively similar to the neighboring one and (3) there is a sortal term S such that the succession is a succession of S-stages or the trajectory underlies such a succession.

  12. With regard to this consciousness-related orientation I specifically point to its ‘psychological’ version where the quantum measurement process is reduced to a ‘splitting’ of a single consciousness before interaction to several afterwards yet retaining by a certain psychological mode its unity through time. See, Everett (1957a, b).

  13. Based on the continuity of the wave function of a free particle at the boundaries \(x= \pm \frac{a}{2}\) of a potential well V we get the equations \(\psi(\frac{a}{2})= \psi(-\frac{a} {2})\) and \(\psi^{\prime}(\frac{a}{2})= \psi^{\prime}(-\frac{a}{2})\). For \(x< -\frac{a}{2}\) or \(x>\frac{a}{2}\), in the limit to infinity it holds that: \(\mathop{\hbox{lim}}\limits_{x\longrightarrow \pm \infty}\,\psi(x)=0\) for the wave function ψ of a free particle of energy E (with E < V) which in this region of the plane takes the form of a descending exponential function \(\psi(x)=C \exp(k_{1}x)+D \exp(-k_{1}x)\) (see any QM textbook).

  14. The vector potential A is linked to the magnetic induction B by the well-known formula B = curlA.

  15. In their exposition of the experiment in 1993, Bohm and Hiley propose an interpretation which implies a physical effect for the vector potential A on the quantum level by means of a mathematical formalism which reduces again to a peculiarity of the mathematical model of the configuration space; this is almost bizarre if one thinks in classical terms as, indeed, the physical effect of A becomes negligible in classical limit. But as they point out, a common mistake is to take the classical idea about the vector potential to hold quantum mechanically (Bohm and Hiley 1993, pp. 50–54).

  16. A prototype of a compound (EPR-correlated) system experimentally confirmed is the compound system S of spin-singlet pairs. It consists of a pair (S 1, S 2) of spin \(\frac{1}{2}\) particles in the singlet state

    $$ W= \frac{1}{\sqrt{2}}\,\left\{\mid \psi_{+}> \otimes \mid\phi_{-}> - \mid \psi_{-}> \otimes \mid \phi_{+}>\right\}, $$

    where \(\{\mid \psi_{\pm}>\}\) and \(\{\mid \phi_{\pm}>\}\) are orthonormal bases of the two dimensional Hilbert spaces H 1 and H 2 associated with states S 1 and S 2 respectively. In such a situation, it is theoretically predicted and experimentally confirmed that the spin components of S 1 and S 2 have always opposite orientations. As a matter of fact, this set-up may be a compound two-fermions system used to illustrate S. Saunders’ position concerning the weak discernibility and, for that reason, the particular kind of individuality of quantum objects; see relevant discussion on p. 9.

  17. In a certain sense this approach is formally related to H. Everett’s ‘Relative State Interpretation’ of Quantum Theory with regard to a decoupling to world components ψ(R),  ψ(L) of a certain superposition state \(\psi(t)= e^{iHt} \phi (\varphi_{R}\pm \varphi_{L})\) corresponding to a localization of consciousness not only in space and time but also along certain Hilbert space components (see Zeh 1970, pp. 73–74).

  18. In this connection, I refer to M.L. dalla Chiara’s view of the measurement problem of quantum mechanics as a characteristic question of the semantical closure of a theory, in other words as to ‘what extent a consistent theory (in this case QM) can be closed with respect to the objects and the concepts which are described and expressed in its metatheory’. According to dalla Chiara, quantum mechanical theory as a consistent theory satisfying some standard formal requirements, turns out to be the subject of some limitations due to purely logical reasons concerning its capacity to completely describe and express certain physical objects and concepts.

  19. A form of J. von Neumann’s Projection Postulate is the following: Let the mathematical translation τ of the physical system s at time t be: τ((s)(t)) = ∑ c j ψ j where ψ j are eigenvectors of ρ(Q i ),  ρ(Q i ) being the mathematical interpretation of operationally defined quantity Q i . Suppose someone carries out a first-kind measurement (i.e. one in which the measured system described by s is taken to interact with the measuring apparatus described by quantum state ϕ, so that the total wave function before the interaction is \(s\cdot \phi\)) for the quantity Q i in state s(t) getting as a result the interval \(r_{\kappa}\pm \epsilon_{Q_i}\) where r κ is an eigenvalue of ρ(Q i ) with corresponding eigenvector ψκ. Then, soon after the measurement (at t 1 > t) the translation of s(t 1) will still be ψκ, that is, τ((s)(t 1)) = ψκ. See, dalla Chiara (1977, p. 334).

  20. The implicit assumption of an underlying continuous unity may be also noted in the bid for a properly defined probability density (at detection time) in the continuum limit within the frame of a consistent histories approach to quantum mechanics; see C. Anastopoulos, C. J. Isham, papers in, respectively, (Anastopoulos 2001; Anastopoulos and Savvidou 2006; Isham 1994; Isham and Linden 1994).

  21. This kind of abstraction is not meant as a free variation in content leaving common traits as invariants but as a complete evacuation of all traits relative to a content. This abstraction leads to the form of a mathematical object in general, an eidetic form whose specific instances bearing a content are just a fulfillment within temporality. This is, moreover, related with the meaning of empty-substrates, as abstract forms lacking any material content, which are taken as intentionalities towards certain ‘states of affairs’ (mentioned on p. 3). These empty eidetic forms are considered to be the class of objects of Mathematical Logic, see Husserl (1995, section 14, p. 33).

  22. This is in part reflected in the proof of the independence of actual infinity principles such as the Continuum Hypothesis and the Axiom of Choice from the other axioms of the Zermelo-Fraenkel Set Theory. As a matter of fact, there is an ongoing theoretical discussion on the possibility of a non-analytical character of the Continuum Hypothesis question in the foundations of mathematics; on this account, I refer to S. Feferman’s claim in 1999 that ‘the Continuum Hypothesis is an inherently vague problem that no new axiom will settle in a convincingly definite way’. A more thorough analysis of mathematical objects as objects of intentional observation will be given in a forthcoming paper.

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Livadas, S. The Expressional Limits of Formal Language in the Notion of Quantum Observation. Axiomathes 22, 147–169 (2012). https://doi.org/10.1007/s10516-011-9168-6

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