Semiotics and Epistemology of Physics: Reflections on Language and the Interpretation of Quantum Mechanics

Abstract

When a physical theory is in its early stages of development, it presents many concepts and constructs that may not be necessary for its interpretation, or may simply be equivocal. As the theory is developed by the physics community, it hopefully passes through a depuration process that washes away many of these constructs and introduces others not initially devised. This process can be viewed, and modeled, as a semiotic process by which the suggested interpretations of the theory, initially dispersed in the space of concepts, taper off in a way that leaves only a small number of possibilities, ideally only one. However, to qualify this process and impose semiotic and epistemological constraints on the depuration process, it seems natural to consider a physical theory as an excerpt of a language and its suggested interpretations as texts, endowed with syntactics and semantics. In this paper we present this framing of general physical theories and apply the resulting semiotic and epistemological constraints we uphold to the special case of quantum mechanics, which shows particular resistance to interpretation tapering. We then show that the findings of this paper are especially important for allowing one to form a hierarchy of interpretations of the same formal structure of a physical theory, even in the case of an experimental underdetermination of these interpretations, which is precisely the case for quantum mechanics. This result is particularly important for more modern physical theories, which are becoming increasingly more abstract and difficult to interpret.

1. Introduction

The present paper is a follow-up of a recently published paper [1] on the interpretation of quantum mechanics. It is well-known that quantum mechanics has many incompatible interpretations. In fact, their number does not cease to increase. In [1], a specific epistemological principle was advanced. Simply stated, this principle maintains that any physical theory should have its semantics (the interpretation) be as close as possible to its syntactics (the formal structure). The authors argue in that paper that this epistemological principle can drastically reduce the number of interpretations of any physical theory, in general, and quantum mechanics, in particular.

The objective of this paper is to fully develop the epistemology on which that paper [1] was based. Although quantum mechanics is our implicit interest, the elements presented here can be applied to any physical theory and can also be used to understand other historical situations where some physical formalism faced two or more different and incompatible interpretations, such as the phlogiston and atomistic interpretations of heat in thermodynamic processes.

Thus, this paper investigates the linguistic, metaphysical, and methodological elements that provide any physical theory with the conditions for the emergence of different interpretations and their acceptance by the scientific community. In the context of quantum mechanics, this may help one understand how wild interpretations, which could be considered the mere result of active imaginations, are introduced and accepted by a substantial part of the physicist community.

To this end, in Section 2, we adopt an approach that takes existing interpretations of some physical theory and characterizes this theory in linguistic terms by showing that it has a semantic and syntactic structure. This means that we consider physical theories as a technical human-made language with a syntactic apparatus and semantic interpretation that are specific to physics. We explain this approach in Section 3. In Section 4 we begin our descent into the semiotic elements that will allow us to understand the differences in epistemological strength regarding the propositions of different interpretations (different semantics) to the same formal structure (same syntactic). To this end, we characterize physical theories as a very specific type of text. We then present, in Section 5, a widely used semiotic scheme (involving syntax and semantics) adapted for physical theories that may produce a hierarchy for competing interpretations, particularly in the context of experimental and formal underdetermination, which can be used in physics more broadly, and not only in quantum mechanics. In Section 6, we use our characterization of physical theories as language and their interpretations as text to establish the relationship they build with the ontological and syntactic (formal) dimensions by means of symbolic connections in the context of experimental situations. In Section 7, we formulate the semiotic and epistemological criterion for the analysis and hierarchy of rival interpretations of any physical theory by discussing the role of experimentation in physical theories. In Section 8 we apply our semiotic and epistemological criterion to four examples of interpretations. In this section, we are interested in the interpretations as they were proposed originally, and not in the eventual developments through which they must have gone after their proposition. In Section 9, we present our conclusion.

2. Physical Theories as Language

A physical theory can be characterized as an interpreted calculus in relation to natural observations, experiments, and theoretical models. As such, this characterization establishes that (at least) one interpretation must correspond to a mathematical formalism whose formal interrelations are capable of describing behaviors of physical entities of the natural world.

This interpretation can be local or global (systemic). A local interpretation establishes precisely the referential elements that, in a more restricted phenomenal context, coordinate the construction of specific experimental arrangements, capable of making explicit the behaviors of physical entities and their relations, captured by measuring apparatuses, to confirm or refute what the physical theory affirms in propositional terms, such as the proposition that states that bodies fall with the acceleration of gravity.

Thus, it is because one can link a set of physical symbols to certain natural entities (and link these to the behavior of elements of an experimental arrangement, such as pointers of certain measuring devices) that a given descriptive or predictive scheme of a physical theory can be connected to an experimental arrangement and thus be empirically tested. Through these relationships, certain variables of the formal structure underlying the physical theory can be considered important or irrelevant for testing and, therefore, can be considered or disregarded, respectively, in the experimental setup. This is the process by which this experimental arrangement is “isolated” from the rest of the world and, thus, manipulated to find the description of the quantitative correlation between the variables considered pertinent, such as distance and time in a free fall in which a vacuum was achieved.

However, these local interpretations, as they refer to more restricted testing contexts, are not sufficient to produce a global or systemic interpretation of a physical theory, although they suggest one. The Galilean description of free fall and the Newtonian gravitational theory are examples of that.

This systemic and broader semantic dimension characterizes the total set of calculations and procedures, now interpreted as an actual physical theory. It is no longer a question of having a set of islands of interpretation, not necessarily connected, but one of precisely forming an interpretation of the entire formal system that, in localized cases, is capable of attributing the natural correlations and correlative formal variables to the relevant elements of the phenomenon in question. This global or systemic interpretation can be called the “world specified by the physical theory” or, simply, the “World”, although always relative to the application context of the elements of the theory (e.g., electromagnetic phenomena to an electromagnetic theory). In general, this interpretation proposes a particular ontology by specifying the presupposed natural elements that constitute such a World.

This is a general feature of the maturation process of physical theories. In the case of Newtonian mechanics, for example, it took approximately two thousand years (from Aristotle and his physics and cosmology to Newton) to develop the construction of a calculus and its interpretation, as well as its underlying ontology. An example of the process of construction of this ontology is the maturation, partly gradual, partly erratic, of a notion of emptiness, connected with the notion of an infinite universe, as the only one capable of receiving that formal and conceptual system [2]. Newton’s three laws exemplify the global systematization referred to above [3] when some of these ontological elements are coupled with these laws, such as the existence of a void (strictly speaking, an ether), but in the sense of allowing a corpuscular theory of atomistic flavor and the constitution of the material world as corpuscles (and not a continuum).

The case of electromagnetic theory was no different. In the nineteenth century, various experiments and concepts (such as the induction experiments carried out by Faraday, Oersted, and others, and the proposition of the concept of lines of force developed by Faraday) were developed and forged as a determination of particular natural contexts until, in 1873, James Clerk Maxwell proposed a set of equations, called Maxwell’s equations, named in his honor, capable of incorporating all the phenomena known at the time involving electric charges and currents (stationary or not), in addition to those related to geometric or physical optics.

The ontology that resulted from this maturation process is given in terms of electric and magnetic fields that oscillate transversely, also suggesting the presence of a material medium (called ether), as well as corpuscles charged with two opposite types of electric charge.

The theory of relativity, on the other hand, eludes the scheme presented because it is an extension of the scope of existing theories. At that time, it was found that the transformations of spacetime coordinate systems that kept the electromagnetic theory invariant (Lorenz transformations) were incompatible with those that kept Newtonian mechanics invariant (Galileo transformations). Still, the postulation that all physics (and not only Newtonian mechanics) must be invariant by a single transformation imposed changes in the latter. The change in inertial systems required an adjustment, made at the time in Newtonian mechanics, engendering (special) relativistic mechanics as a sub-area of the special theory of relativity. The special theory of relativity stems formally from the assumption that the speed of light is constant (and has the same value) in any inertial frame of reference and, with that, corrects Newtonian mechanics in the sense of making it now invariant by the same spacetime transformations of electromagnetic theory. On the other hand, the theory of general relativity does something similar, in general terms, to Newtonian gravitation through the principle of equivalence. Therefore, in this sense, the theories of relativity (special and general) are corrections of existing theories, with their conceptual fields already specified (but extended—as, for example, with the concepts of inertial mass and spacetime curvature, among others).

At this point, it would not be superfluous to understand what this paper conceives as a “physical interpretation”, either of theories (systemic) or parts of a theory (local). In logic, as is well known, an interpretation $\mathfrak{I}$ is a function that takes a proposition in the set of truth values, i.e., $\mathfrak{I}:p\to \{V,F\}$. In physics, however, this notion becomes much more complex. A physical interpretation of a physical theory (${\mathfrak{I}}_T$), generally speaking, is given by the establishment (explicit or not) of a background ontology and a (preferably) minimal set $\mathfrak{R}$ of mathematical laws and equations. The best examples of this are axiomatic physical theories in which the fundamental equations of the theory are found, such as Newton’s equations in Newtonian mechanics, Maxwell’s equations in electromagnetic theory, Einstein’s equations in general relativity, or some set of axioms [1] that allow one to derive the Schrödinger equation in non-relativistic quantum mechanics. For such cases, as we will argue, the referents of the symbolic elements $\mathfrak{E}$ are already correlated with the relevant natural elements, that is, referentially interpreted precisely due to the underlying ontology.

Some examples related to interpretations of physical theories:

• In Newtonian mechanics, we have a background corpuscular ontology ${\mathfrak{R}}_N$ with the attribution of the referents of m (as the mass of these corpuscles or their grouping), x (as the position of these corpuscles or the center of mass of their grouping), v (as the velocity of these corpuscles or of the center of mass of their assemblage), a (as the acceleration of these corpuscles or of the center of mass of their grouping) and F (such as the force acting between these corpuscles or between their groupings) and Newton’s laws ${\mathfrak{E}}_N$, with Newton’s equation (second law) already established. We have something like ${\mathfrak{I}}_N=\langle \mathfrak{R}(m,x,v,a,F),{\mathfrak{E}}_N\rangle$. This interpretation allows, when applied to specific elements of oblique throwing, for example (which is a specification of the Newtonian theoretical field made from a model), one to characterize the phenomenon as referring to a particle (or body) being thrown from a certain point $P_0=(x_0,y_0)$, with velocity $\overrightarrow{v}$, and being subject to acceleration by gravity $\overrightarrow{g}=(0,-g)$ (vertically down), so that, applying Newton’s second law, it follows that the particle will have a parabolic trajectory. From the general interpretation of the theory follow the interpretations of concepts such as “equilibrium”, “energy”, “work”, etc., as derived concepts.
• In electromagnetic theory, we have an ontology of fields that assigns the symbol E to an electric field, the symbol B to a magnetic field, and the symbols q and i to electric charges and currents. The goal of electromagnetic theory is to establish the configurations of the fields, as given in Maxwell’s equations, from the elements of charge and current. Such elements of charge and current are generally considered given in electromagnetic theory, and from them the fields can be obtained. The analysis of the behavior of charge and current elements in the presence of fields is an intersection between electromagnetic theory and Newtonian mechanics, modified by the special theory of relativity through the Lorentz force. That is, we have something like (in the integral version of the fundamental equations) ${\mathfrak{I}}_{EM}=\langle {\mathfrak{R}}_{EM}(E,B,q,i),{\mathfrak{E}}_{EM}\rangle$. Such fields are interpreted as wave-like because one can mathematically derive and interpret the resulting equation as a wave equation from ${\mathfrak{E}}_{EM}$. Later, we further specify this point, that is, how the overall interpretation of the theory also depends on the consequences of its basic interpretation. The interpretation of the integrals that appear in Maxwell’s equations is part of the mathematical structure that physics absorbs, as with the notion of a derivative in the context of Newtonian mechanics.

On the other hand, when it comes to interpretations of specific phenomena (of a physical theory that may or may not already exist), other elements must appear in the interpretation processes, such as a particular ontology, which is specified in the model $\mathfrak{M}$ chosen to represent the phenomenon, and any physical set of hypotheses $\mathfrak{H}$ related to this model, as well as equations $\mathfrak{E}$ (usually already existing, related to the underlying ontology), eventually modified by the physical hypotheses: something like $\mathfrak{I}=\langle \mathfrak{R},\mathfrak{M},\mathfrak{H},\mathfrak{E}\rangle$. If the theory exists, $\mathfrak{R}$ is already given (as in our earlier characterization of the oblique throw in the context of Newtonian mechanics), and it is necessary to specify $\mathfrak{M}$ and $\mathfrak{H}$ [4,5]. If the theory does not yet exist (as it does in the initial moments of its creation), we hope that the elements mobilized for the interpretation of the specific phenomenon will, in the process of the construction of the theory, be used to construct its general ontology. Such elements are a particular ontology related to a specific model and hypotheses that allow the correct interpretation of the phenomenon being analyzed.

Thus, as an example more related to the theme of this work, in the first maturation process of quantum mechanics, which took place from the year 1900 to the year 1927, we find several local interpretations referring to specific experimental schemes. We can mention, for example, the following:

• The problem of blackbody radiation (the ontology of electromagnetic fields internal to the body in interaction with the atoms of the walls of the body) led Max Planck to suggest considering the electromagnetic field as a set of harmonic oscillators (model) with quantized energy (hypothesis) to avoid the problem of the ultraviolet catastrophe, i.e., extending the electromagnetic theory to fields with quantized energy.
• The photoelectric effect that led Albert Einstein to propose the characterization of electromagnetic radiation as consisting of corpuscles (ontology), called photons, which transmitted their energy in a quantized way (hypothesis) through shocks (model), thus avoiding the problem of transmitting energy continuously, as predicted by the classical electromagnetic theory, that is, extending it.
• The Compton effect used Einstein’s idea of photons (ontology) to characterize the scattering of these photons by electrons attached to atoms (ontology) through elastic shocks (model) and the relationship between the photons and the momentum $p=h/\lambda$ (hypothesis), where h is a constant that explains the appearance of two wavelengths (or momentum) in the scattered beam.
• The description of the hydrogen atom by Niels Bohr aimed to obtain the spectral emission sequences of this atom, in which a corpuscular ontology was used in a Rutherford model of the atom, with the hypothesis of quantization of the angular momentum.
• The diffraction of X-rays by a crystal was modeled in a corpuscular way through crystalline planes made up of atoms, but with the hypothesis of the relationship between momentum and wavelength for the incident radiation of the form $p=h/\lambda$ (h is a constant) (hypothesis), which allowed the use of the wave model for incident radiation. This allowed, through the notion of reciprocal space, the explanation of the appearance of a diffraction figure, characteristic of wave behavior.
• The interference produced by beams of entities that were hitherto considered particles now seemed to behave like waves (ontology) in the same experiment. The ontology was wave-like, and the model was essentially that of wave interference. The hypothesis was the same as that used in X-ray diffraction by a crystal, described earlier for the incident beam.

Each of these experiments and many others suggested a set of ontological possibilities to be coordinated or systematized in a physical theory. However, they were not yet able to establish this systematization on their own, encompassed by the proposition of an underlying general ontology. They often proposed ontological sets that seemed to contradict each other, as presented in all the experiments mentioned before related to spectroscopy, in which fields interact with (supposed) corpuscles and which, moreover, seemed to have been unavoidably distant from the two known ontological sets until then (exclusively corpuscular or undulatory) (but see [1]).

A general systematization appeared, as is well known, at the Solvay Conference in 1927, with the proposition of the first Copenhagen Interpretation for quantum mechanics (there are numerous “interpretations of Copenhagen” that vary on specific points but maintain a relatively large body of common constructs [6]). This interpretation was only possible because, in 1925, Heisenberg proposed a general formal scheme, the so-called matrix calculus developed by Heisenberg, Born, and Jordan in the drei manner arbeit [7], and in 1925/1926, Schrödinger introduced the fundamental equation of his formalism—both were capable of embracing all the particular phenomena known up to then, onto which a global interpretation could now be superimposed (which, for the reasons mentioned above, should be based on a dualistic wave–particle principle from the ontological point of view).

It is worth mentioning that the appearance of Schrödinger’s formalism, applicable to the same physical phenomena, was treated by Heisenberg’s formalism and was of great historical importance, since this new calculus suggested constructs, an ontology and, in general, interpretations different from those originating from the matrix calculus. This issue closely touches on the theme of this work. The appearance of an equation that presented the construct “wave function” generated endless discussions among physicists at that time [7] related to the eventual referents of this function.

Hence, in the case of quantum mechanics, the maturation process towards a global (systemic) interpretation, such as those that occurred for Newtonian mechanics, electromagnetic theory, and classical statistical physics, has not fully occurred or has not yet been concluded. In this context, it was realized that its underlying formal framework and its experimental results could correspond to a relatively large set of possible interpretations. In fact, at present, we find a myriad of interpretations of quantum mechanics, such as: Copenhagen; Quantum Information; Relational; Bayesian; Many Worlds; Consistent Histories; De Broglie–Bohm or Causal; Quantum Darwinism; Transactional; Objective Collapse; von Neumann–Wigner (consciousness causes collapse); Quantum Logic; Modal; Stochastic; and Statistical or Ensemble.

In addition to those mentioned above, there are many others, since many of these interpretations can be subdivided into slightly modified derivatives. The fact is that each of the interpretations presented provides a different ontology for quantum mechanics while retaining its formal and experimental backgrounds.

Such a situation is, in fact, unprecedented in the history of physics. Each physical theory had, in its beginnings, alternative interpretations and ontological sets that competed. An example of this is the alternative theory to Maxwell’s electromagnetism and Newtonian mechanics called relational electrodynamics and mechanics [8,9], derived from Weber’s electrodynamics, which, however, currently occupies marginal positions in the literature. Equally important were the theories of imponderable fluids (e.g., caloric and phlogiston), which were replaced by theories of heat based on the corpuscular constitution described by the kinetic theory of gases. In the specific case of quantum mechanics, however, the incredibly high number of alternatives draws attention. Such interpretative options use similar equations but very different ontological sets of constructs to explain the same phenomena, whose value of truth or adequacy in the relationship between the formal apparatus and the experimental results is not in question.

One might imagine, perhaps, that the elimination of alternative interpretations could be done solely by experiments that will be able to select one (or at least a small number) of the alternative interpretations of the same formal structure.

It happens that, in physics, there are plenty of examples in which experimentation does not present such a capacity, that is, in which to the same formal corpus and the experimental achievements that underlie it, there corresponds more than one interpretation, thus being an underdetermination of interpretation by the experimental dimension.

A trivial historical example is the one that has been established around geometric optics, interpreted by Newton [10] as related to a corpuscular ontology (for the “corpuscles of light”) and interpreted by Huygens as related to a wave ontology. The experiments were equally open to interpretation from both the wave perspective and the corpuscular perspective. The nineteenth century selected a wave-like interpretation of light in what could be said to be a process of purification. However, the principle of wave–particle duality seems to have given a new and substantially different configuration to the problem of this interpretation.

Thus, since the empirical content of a physical theory, at least at a certain historical moment, does not provide formal or experimental elements for selecting a given interpretation, it may be relevant to present one or more methodological principles capable of eliminating interpretative possibilities, eventually reducing their set to a relatively low number.

Again, the History of Science can help us. Copernicus’s calculations for celestial orbits were originally based on the use of epicycles (after all, they assumed that such orbits were circular). However, by placing the Sun at the center of such orbits, the calculations used a much smaller number of epicycles. This result alone would not be in the least capable of altering the geocentric perspective of that time, as expressed in Andreas Osiander’s preface [11] to Copernicus’ The revolutions of heavenly spheres [12]. This was only possible with the advent of a metaphysics of mathematics, introduced by Galileo, which supported the content of reality for the heliocentric hypothesis, based on the meta-discursive notion of mathematical simplicity, among others, an ontological principle regarding interrelated formal calculation (mathematics) and the notion of truth or adequacy.

In the context of quantum mechanics, we may cite two examples:

• The de Broglie–Bohm interpretation argues for the ontological existence of particles that are guided by a wave function (ontology). Thus, such a wave function acts as a concrete physical field, which makes the particle move as required by Schrödinger’s equation. In this interpretation, there is no need to impose the collapse of the wave function. The approach is of an ensemble type; i.e., the result advocated by the wave function is reconstructed by the analysis of an ensemble of particles with well-defined trajectories, based on the establishment of the initial conditions. The theory is deterministic, but there is action at a distance (therefore, it is non-local). The simultaneous experimental determination of the position and momentum of particles obeys the principle of uncertainty or indeterminacy, but only concerning the initial conditions, given that the approach advocates the existence of trajectories. From the fact that it is a non-local interpretation, it follows that there is no way to individualize the phenomena, which are all interconnected, constituting an undivided universe [13]. The approach assumes that there are “hidden variables” responsible for the non-classical behavior of particles [14] and argues that there is a “quantum potential” whose source is unknown.
• The Many Worlds interpretation argues for the existence of a universal wave function that always obeys deterministic and reversible laws, because Schrödinger’s equation imposes such characteristics on the dynamics of the wave function. Thus, it rejects the idea of the collapse of the wave function (which would imply indeterminacy and irreversibility). From this arises the need to deal with the problem of the superposition of states in their relation to the phenomenon of measurement (which always measures a single result, rather than a superposition of both). The measure, therefore, was due to a process in which the diverse experimental results (which would be the result of statistical indeterminacy in other theories) are explained by a continuous unfolding of the universe into a multiplicity of mutually unobservable histories, generating the notion of the Multiverse [15,16]. In each universe, only a single experimental output would be observed. It is a local and deterministic theory (with respect to the wave function). This interpretation of quantum mechanics is crucially dependent on the linear character of Schrödinger’s equation, due to the central role played by the superposition of states.

It is striking, just from the brief summary of these two interpretations cited above, that not only is there an incredibly high number of interpretations of quantum mechanics, but many (or almost all) interpretations also seem unusual, far from physical common sense (for other interpretations, see [1]). However, the fact that they are unusual does not seem to have considerably influenced the success they achieved in terms of acceptance by a large part of the physicist community [17].

3. Linguistics and Physical Theories

The study of discourse theories is an established field of research in linguistics. Those who work in this field show that there exists a grammar that drives the construction of any text. Knowing how to recognize this grammar helps to interpret texts adequately. To achieve such an objective, it is necessary to make explicit some implicit elements that allow us to recognize the structure and interpretation of the text, while taking into consideration physical theories.

This perspective takes us into the broad field of semantics, a term that was used at the end of the 19th century by Michel Bréal to name the field of the study of meaning [18] when linguists became interested in investigating changes in the meaning of words over time and the mechanisms that govern these changes. These developments constituted what is called Diachronic Semantics. In the first half of the 20th century, a series of investigations began using a synchronic approach. For those linguists, such as J. Trier, involved in this line of research, semantics aimed to investigate the “semantic fields”, which were assumed to be a set of lexical units that are kept together based on some underlying structure. G. Matoré gave this kind of study the name Lexicology, since its unit of analysis is the word [18].

Around the 1960s, in parallel to logical semantics, there appeared structural semantics, which assumed the postulate of parallelism between the plane of expression and the plane of content. Despite all the criticism it suffered, it had the undeniable merit of reintroducing concerns about meaning within linguistic studies. Structural semantics developed principles and methods for studying meaning from the work of L. Hjelmslev, who proved it was possible to examine the plane of content separately from the plane of expression, as phonology had done with the plane of expression [19].

Thus, it was assumed that the plane of expression is made up of differential distinctions that should correspond to distinctions in the plane of content, which are considered meaning distinctions. In this article, this means that some discourse considers an interpretation of a physical theory to be structured in distinctions (e.g., ontological distinctions) that are introduced into its superficial expressions as speech or text. A set of binary semic categories can be the origin of thousands of combinations that allow us to distinguish one interpretation from another. The difficulty in establishing these semic unities, except in very restricted lexical fields, led linguists to pursue another approach. Structural semantics faced many difficulties and was unable to go beyond the limits of the phrase.

The change in position regarding language facts led to the emergence of diverse theoretical proposals that conceive the text, and no longer the sentence, as a unit of meaning and that, therefore, consider the meaning of the sentence to depend on the meaning of the text. Semiotics has text as its object and seeks to describe and explain what the text says and how it goes about saying what it says.

For linguists such as Greimas [19], semantics must be

• Generative, in the sense of presenting models that apprehend ever-growing invariant levels of meaning, such that it allows one to understand that different elements in the surface level are related to the same original meaning (in the context of an axiomatic field of physics, we may assume that the axioms represent this original meaning, together with mathematics, that has the generative structure);
• Syntagmatic, in the sense that its unit of explanation should be the production and interpretation of a discourse;
• General, in which it is assumed as a postulate that the unity of meaning can be presented by different planes of expression (different sets of axioms must lead to the same overall interpretation, as was presented in [1] for quantum mechanics).

In this approach, in the analysis of a text, for instance, one goes from the more concrete—the text itself—to the more abstract. In its production, however, the path is inverted. These processes are related to the notion of a generative path of meaning, which is represented as a succession of steps on a ladder, each capable of receiving an adequate description showing how meaning is produced and interpreted.

The approach that will be used in this work is the one proposed by Greimas and presents three levels of analysis, as explained further. At this point, what is important to note is that a relational structure connects these levels of a generative path. The syntax and semantics of these levels are interwoven, not in a sense that renders the syntactic elements meaningless, but that the syntax is a more autonomous plane since the same syntactic relation can receive a bunch of distinct semantic investments, as nowadays we experiment in quantum mechanics, given the numerous interpretations of the theory (semantics) based on the same formalism (syntax).

Before discussing Greimas’ approach in detail, let us clarify what we understand as an interpretation in physics—and what we assume to be its syntactic structure.

4. Interpretation in Physics

As noted above, physics can be understood as a science that uses mathematics as an underlying formal language for the understanding of natural phenomena. Among different perspectives, physics itself can be considered a language structured based on specific methodological requirements. In these statements, there is a double interpretative key, one of historical character and another of epistemological character, which, however, are intertwined. Thus, it is important to address the semiotics involved in their relationships in the context of a physical theory. Such an approach is necessary since we are interested in raising essentially semantic questions (of interpretation) about physical theories in general (and about quantum mechanics, in particular), and the role of mathematics in this process, whether as grammar or as the syntax, cannot be ignored.

To clarify this point, consider an example from electromagnetic theory. The equation that is interpreted as representing the conservation of energy is given by

$${\displaystyle \frac{\partial u}{\partial t}}+\nabla ·\left({\displaystyle \frac{\overrightarrow{E}\times \overrightarrow{B}}{{\mu }_0}}\right)=-\overrightarrow{E}·\overrightarrow{J}.$$

(1)

That interpretation requires that $\overrightarrow{E}$ be interpreted as the electric field vector, $\overrightarrow{B}$ be interpreted as the magnetic field vector, and u be the energy density (given in terms of the fields $u={\displaystyle \frac{{ϵ}_0}{2}}{E}^2+{\displaystyle \frac{1}{2{\mu }_0}}{B}^2)$, while $\overrightarrow{J}$ is the current density vector, and the term $-\overrightarrow{E}·\overrightarrow{J}$ represents the eventual dissipation of energy if there are free currents (in addition to ${\mu }_0$ and ${ϵ}_0$, which are the vacuum constants related to the speed of light).

Thus, using this example, it is easy to understand that the concepts, when linked by (1), can be interpreted as conservation of electromagnetic energy in the absence of dissipative effects ($\overrightarrow{E}·\overrightarrow{J}\equiv 0$). To this end, it should be noted that it is also necessary to understand the semantic–mathematical dimension of the physical syntax. In the example in question, the aforementioned conservation is interpreted from the interpretation of the derivative symbol (partial derivative with respect to time) as a rate of change (in this case, exclusively in time), while the formal symbols in $\nabla ·\overrightarrow{P}$ (with $\overrightarrow{P}$ the Poynting vector defined as $\overrightarrow{P}=(\overrightarrow{E}\times \overrightarrow{B})/{\mu }_0$) represent a flow ($\nabla ·$) of an electromagnetic momentum density ($\overrightarrow{P}$).

Thus, consider the equation

$${\displaystyle \frac{\partial R}{\partial t}}+\nabla ·\overrightarrow{T}=0,$$

(2)

where R and $\overrightarrow{T}$ are any scalar and vector quantities, respectively, written in a purely mathematical context. This equation implies that the quantity R is conserved. That is, it implies that

$${\displaystyle \frac{d}{dt}}\int R(\overrightarrow{r},t)d\overrightarrow{r}=0.$$

(3)

This represents the conservation of electromagnetic energy only when R and $\overrightarrow{T}$ are physically interpreted as the electromagnetic energy density and energy flux, respectively, in the way we have briefly indicated in (1). For the purpose of this work, we assume this mathematical semantics to be given and accepted, and we place our focus on the semantics of physical theories when mathematics becomes their syntax.

It is worth noting that not all formal symbols of a physical theory have a referent in the world as the electromagnetic potentials in electromagnetic theory do. However, they can help define an interpretation function that assigns a referent (in the world) to some of the symbolic terms of the theory, that is, $\mathfrak{E}\to \mathrm{Electric} \mathrm{field}$, for example.

As mentioned previously, physics can be understood as an interpreted calculus [20] in the sense of having, on the syntactic plane, its equations, symbols, and interrelations formally established (e.g., as the interrelations between the electric field and the magnetic field with energy, in the context of electromagnetic theory) and, at the same time, fulfilling the requirement for a physical theory to even to exist, namely, that this syntactic plane (the equations and symbols relevant to the field) is coupled with a semantic dimension.

Thus, we have the perception of a physical theory as a realization of semantics. However, we do not approach it as in [21], thinking of it as a mere collection of models (see [22], p. 2) that can be formalized from set theory. We do not even intend to formalize this idea. In our case, the syntax of a physical theory is given by mathematics, not logic. Suppes, for his part, states that “A scientific theory consists of two parts. One part is an abstract logical calculus. In addition to the logic vocabulary, this calculus includes the primitive symbols of the theory, and the logical structure of the theory is fixed by stating the axioms or postulates of the theory in terms of its primitive symbols.” (See [22], p. 56, italics added). If one changes “logic” to “mathematics” in this citation, one gets our perspective. Hence, there is an interpretation that provides part of the terms of the formalism with a referent in the world, either as an object or as a behavior. It is this interpretation that, in a virtuous cycle, allows one to understand an experiment but that may also be derived from experimental situations in the context of their diachrony, as explained in what follows.

From a historical point of view, physics becomes an autonomous field of knowledge when it selects for itself its criterion for understanding the natural world through empirical adequacy, considering the metaphysics of mathematics—which assumes that mathematics “follows nature” in some sense [23]—as well as a specific approach to seeking reality based on the scientific method, distinct from other ways of discussing reality, e.g., through philosophy and psychology.

This historical process had its origins in the early modern period, with the work of Francis Bacon and others on the role that experimentation should play in the search for natural truth (as a manipulation of certain physical arrangements), which goes beyond the mere empirical approach (as a way of merely recording the observed phenomena) and the establishment, in new terms, of the relationship between logic and mathematics in this process. To this last point, one has the essential postulations of Galileo Galilei [24], who revitalized the Neo-Platonic (and Neo-Pythagorean) idea of a harmony in nature that is essentially mathematical, which could be recovered by the human mind, something already expressed seminally in Timaeus [25,26]. Therefore, a new metaphysics of mathematics was created in the context of physics [23], strictly related to its notion of truth.

Since Galileo, experimentation has been responsible for the contextual verification of the general statements contained in the physical laws, expressed quantitatively through mathematical symbology. On the other hand, it is up to mathematics to develop the formal consequences of such laws, according to its own syntactic criteria. Such a characteristic is, for example, essential for the possibility of the emergence of discovery in physics, even though mathematics is, a fortiori, an essentially closed symbolic system. Discovery often plays a highly relevant role in the context of epistemological abduction processes [27] associated with the reception of physical theories. This is one of the points at which historical determinations intersect with epistemological determinations, and if discovery does not engender meanings, it is responsible for the reliability of those established meanings of the existing physical theory.

This historical moment changed the roles of logic and mathematics in the context of physical theories. Indeed, logic began to be assigned a dual role, quite different from the one it fulfilled in the Aristotelian approach to the description of nature, in which logic was seen as a privileged means of accessing the natural world in the search for a reliable description of its phenomena [28].

In modern times, it has been determined, first, that logic (both formal and informal [29]) plays the role of a structuring element of the argumentation through which a text is constructed, thus being capable of articulating the link between mathematical expressions and elucidating its interpretative dimension (and therefore, its semantics). Secondly, and less easily perceptible, classical formal logic is the basis for the very construction of the mathematical apparatus that underlies a physical theory. In the structure of a physical theory, therefore, the first function of logic, as mentioned above, is essential, given that it provides, together with the mathematical apparatus, an interpretation of physical theories. Thus, a language is created through which we can associate mathematics with the syntactic apparatus of a physical theory while associating the semantic dimension of the theory with interpreting this syntactic apparatus. In this semantic dimension, physics establishes a set of propositions and meanings that support a specific perception of what the world is and what it cannot be.

Thus, the epistemological basis of physics presupposes the use of logical–formal thinking, establishing propositions that refer to the relations between the abstract symbols and also providing elements that produce an overflow from such symbols. Thus, there is a dual reference: on the one hand, there is the dialogue between physics and mathematics, in which one frequently summons the other, and on the other hand, there is the dialogue between the linguistic codes and the universe of meaning, which are articulated in the production of the latter. There is, hence, a pairing between two distinct systems of signification that are autonomous but interdependent. Physics would be the construction of this interdependence a posteriori. For this reason, in physics, to represent the complexity of operations that have the function of denoting their events, based essentially on a previous codification of the phenomena, a congeniality between mathematics and the world is generally assumed, constituting part of its deepest metaphysics [23].

Examples of this hermeneutic overflow in relation to the syntactic dimension are certain statements about the world that are, in fact, conditions of possibility, syntactically represented in the underlying calculus, for the physical theory to make sense. In general, one must have a clear interpretation of the referents of mathematical symbols to construct a worldview—quantum mechanics, in that matter, is a particularly interesting case in which the straightforward interpretation of the referents of mathematical symbols is not finalized, which is most likely the basis for the myriad of general interpretations (worldviews) that derive from it.

As an example, we can cite the assumption present in Newton’s physics of a concept of a void (or space), as opposed to the Aristotelian concept of Place [28], without which the idea of conservation of linear momentum, fundamental to Newtonian mechanics, does not even arise or make sense. This point is particularly representative because it shows the profound semantic differences between the Aristotelian and Newtonian systems.

Thus, for Aristotle, emptiness could not exist, as this would imply an infinite space. According to the Stagirite, in addition to the fact that the existence of something infinite in Act is not possible (see [28], 216^b:20), an empty space would imply the absolute identity of its points (and, therefore, the impossibility of natural places). Thus, as described in his book Physics, Aristotle says that if the void existed, a body, once set in motion, could not stop at a definite point x, since such a point would be in no way distinct from another point $x+a$ (any a), so the body thus set in motion would remain in uniform rectilinear motion indefinitely—for which an infinite space in Act would be required. Thus, Aristotle just inferred, based on his notion of place, the opposite of Newton’s first law, and, with respect to the notion of the existence of a void (as a homogeneous space), also the possibility of linear momentum conservation, based on his notion of space.

The very exercise of experimentation in physics depends, to a large extent, on an interpretation of its mathematical formalism; since experimentation deals with manipulating—in the process of construction of experimental apparatuses and their arrangements—the referents of those mathematical symbols, so too must the exercise of experimentation necessarily precede a semantic stage, since the process of referencing is essentially semantic, particularly in a perspective that embraces the notion of truth by correspondence.

In this sense, in addition to their origin from articular socio-historical constructions, physics and mathematics establish surprising and intrinsic relationships in the formal organization of the universe [30].

From this perspective, we consider the interpretations of physical theories as text. In the following section, we introduce a methodology for evaluating the characteristics of texts related to interpretations of physical theories based on the approach of Greimas and Courtés [31], but it also considers other developments in the area.

The Interpretation of Physical Theories as a Text

Fiorin [18], based on the proposal of Greimas and Courtés [31], understood that, for the description of the production and comprehension of discourses and their textual manifestations, it is possible to conceive the generation of meaning as a path that goes from the simplest and most abstract to the most complex and concrete—as in physical theories, particularly those that can be written in terms of axioms. In this way, and in line with what we have already stated, the generative path of the constitution of meaning proposed by Greimas and Courtés [31] helps us understand not only the effects of ontology on the meaning conveyed by interpretations of physical theories but also the epistemology behind these effects of meaning. Thus, the generative path constitutes a methodological simulacrum that represents the marks of the discourse in its textual manifestations because, from what is said, one infers how it is said and why it is said.

In the case of semiotic representations, of which physics is an example [32], there is simultaneously a relationship of formulation and communication of a symbolic structure [33]. The framework that we propose, therefore, aims to approach the underdetermined interpretations of physical theories from a semiotic perspective, detached from the truth value (given the underdetermination that equalizes them), but still capable of articulating a hierarchy between such interpretations based on their modes of generating meaning, as presented in what follows.

Thus, when we talk about the generative path of meaning, we are referring to the plane of content. However, there is no linguistic content that departs from a linguistic expression. This plane of expression can be made concrete in different forms: verbal, gestural, pictorial, etc. All these forms are considered texts. The generative path of meaning represents a linguistic model that simulates the production and interpretation of the meaning of some content (of a physical theory, for example). It does not emulate the concrete way by which the discourse was fabricated, but represents a methodological simulacrum that allows us to get a text (in the extended perspective just mentioned) in its most profound structural dimensions. It reinforces, then, that the meaning of a text cannot be reduced to the words that it contains, nor even the articulations in which the words appear, but comes from a structural articulation of the elements that form it: a syntax and a semantics of the discourse. Interpreting a physical theory is producing a discourse about what one supposes it is saying about the world. Thus, the plane of content of the physical theory becomes a text when it becomes linked with a plane of expression: when a discourse is uttered in any plane of expression, one gets a text. In short, a text is defined in two complementary ways: by an organization or a structure that makes it a “whole of meaning” and as an object of communication that is established between an emissary and a recipient. The first conception of a text, understood as an object of meaning, causes its study to be confused with the examination of the procedures and mechanisms that structure it and weave it as a “meaningful totality”. This type of description has been called an internal analysis or a structural text. Different theories focus on this analysis of the text based on principles and with other methods and techniques [34]. The semiotics that we will adopt here is but one of them. We are not interested in viewing the text as a unit of communication in itself but only in considering its structural manifestation in the plane of content.

It must be clear that the final objective of the communication act is not necessarily to inform but to persuade others to accept what is being communicated. The act of communication is a complex interplay of manipulations that aim to make others believe in something. Language, in this sense, is related to persuasion through the production of meaning.

5. Semiotic Perspectives

Semiotics is the science that studies signs and the languages they compose. Its theoretical foundations are based on the work of Ferdinand de Saussure (1857–1913) and Charles Sanders Peirce (1839–1914). As Coelho, Costa and Fontanari [35] explain, there are many different theories that address meaning in the field of Human Sciences and language. What differentiates the semiotic perspective from other lines of research is that semiotics strives to constitute a discipline that can account for the effects of meaning of and in texts. From this perspective, different schools of thought can be found in Paris [31], Moscow [36], and the Anglo-Saxon world [37], among others.

Important for semiotics is the distinction between signified and signifier proposed by Saussure [38], which have an intrinsic dialectical relationship. Although the signified is related to the content (which can be given by physical models; for example, the signifier is related to the material form of the sign (e.g., extracts from the World or experimental situations, among others)), “meaning” expresses the relationship between the two. Interpretation, therefore, is an exercise in the attribution of meaning.

From the perspective of French semiotics, which deepens these interrelations between signifier and signified, we have, as stated above, the notion of a generative path of meaning. This path, according to Costa [39] and Greimas and Courtés [31], is tripartite at the fundamental, narrative and discursive levels, which can be adapted to the description of physical theories (see

Table 1. Generative path of meaning and its three levels. Adapted from [18].

Generative Path Structure	Level	Syntactic Component	Semantic Component
Semio-narrative	Fundamental	Fundamental syntax (laws in their mathematical format, especially in axiomatic form)	Fundamental semantics (referential relationships between physical quantities and variables)
	Narrative	Surface narrative syntax (general formal consequences of laws)	Narrative semantics (determination of the world by semantic inheritance from the fundamental level)
Discursive	Discursive	Discursive syntax and theme construction	Discursive semantics (determination of the world using constructs not present at the two other levels)

), which, in fact, forms a central point of this work.

The fundamental level is the simplest and most abstract; it is at this level that we find the fundamental oppositions of meaning, which we consider here as it applies to physical theories, as constituted by the processes of establishing the referential elements of the theory of interest and attributing their symbolic counterparts in the syntactic apparatus, in addition to their conditions of possibility. Also at this level are the fundamental laws of the theory, particularly in the case of axiomatic theories, such as in Newtonian mechanics or electromagnetic theory.

The fundamental semantics of electromagnetic theory would be associated with the assignment of the variables $\overrightarrow{E}$ and $\overrightarrow{B}$ to the electric and magnetic fields, respectively, with the interpretation of the integrals that appear there as flows or circulations; with the symbols ${ϵ}_0$ and ${\mu }_0$ as the dielectric constant of the vacuum and the magnetic permeability of the vacuum, respectively; with c as the speed of light; with q as the electric charge (or its density), with i as the electric current (or its density); and so on. Note that it is the fundamental semantics that gives the condition of possibility for any construction of an experimental setup.

At the narrative level, from the point of view of the syntactic component, we have the formal unfolding of the fundamental laws. Since these developments connect interpreted elements of the fundamental semantics, they establish the narrative semantics through which a worldview is constituted. In general, it is at this level that the empirical findings (or refutations) of the theory are constituted.

Following the previous example, in electromagnetic theory, we would have, for example, the formal operations that lead to the wave equations for the fields and the appearance of the speed of light in such equations, which determine the interpretation that the fields propagate with the speed of light and that they have a transversal character, among other properties. In this sense, it is interpreted, for example, that the luminous phenomenon is the propagation of these fields, which is, furthermore, oscillatory, and that electromagnetic theory encompasses the whole field of optics.

At the discursive level, we have strategies for specializing the emission contexts and justifying the underlying theory, which was constructed at the fundamental and narrative levels. Still using electromagnetic theory as an example, at this level, one can present the concept of an ether, which never appears in the deep syntactic structure, nor can it be obtained operationally in the narrative dimension by articulating elements of the deep syntax. The notion of an ether was assumed because of the transversal character of light propagation; the fact that it is articulated at the discursive level helps us understand why this notion could be eliminated without changing the electromagnetic theory itself.

Thus, as we have seen from this quick example, and according to this slightly modified version of Greimas and Courtés’ approach [31] to encompassing physical theories, this generative path of meaning is capable of representing the marks of meaning construction processes in texts/interpretations, providing each of the three stages of meaning production with different epistemological values.

As can be inferred from Table 1, each stage of the generative path of meaning has its own semantic and syntactic components. At the deepest, simplest, and most abstract level, there is a fundamental syntax and semantics. At the narrative level, there is also a surface narrative syntax and a narrative semantics. At the discursive level, the most complex and concrete is the process of discursivization, in which the processes of figurativization and thematization occur.

From what has been said, it is essential to understand the relationship between physics and mathematics in the production of meaning, together with the historical processes that give rise to such production. But how can a syntactic apparatus be a generator of meaning—a semantic aspect? We return to this issue in what follows.

Interpretation is the process of signification that is constitutive of theoretical systems, that, without it, remain devoid of meaning. It is therefore an essential element in the construction of a worldview (systematic semantics) that physical theories bring with them. Signification is in no way contained in signs or in symbology, precisely because it is not determined by any of them, as Benveniste [33] and Saussure [38] have argued. This, after all, allows the same theory—as well as the same sign—to be interpreted in different ways.

There is, between signified and signifier, a symmetric independence whose a posteriori relinking is essentially arbitrary (in principle, as the precursors of linguistics stated). It is this a priori arbitrariness, which permeates any attribution of meaning, as noted by Benveniste [33], that constitutes the fundamental effort of physics in the production of meaning in an a posteriori, but non-arbitrary, way: a production permeated by experimental procedures, but also of a textual nature. The expectation of physics (necessarily of metaphysical or methodological character) is that a supposed ontological unity of the world ought to engender, a posteriori, an interpretative unity of human-made physical theories.

6. The Semantics of Physics and Natural Languages

Considering that physics is a human construct, it has its own semantics at the deep and narrative levels, which is not reduced to that of natural languages, being characterized by elements that are essential to it; to a large extent, it is worth noting, for its use of mathematics. Thus, it is interesting at this point to establish certain relations between the semantics of natural languages and the semantics of physics, since the latter come into contact with the former (sometimes even in conflict) in the discursive dimension.

Like any field of study, the semantics of natural languages has different schools of thought, in which the notion of meaning changes substantially. Thus, we have the following:

• In the representational or mentalist approach, “meaning is essentially a way in which we mentally represent to ourselves the content of what is said. There are several ways in which this idea can be articulated. One possibility is in terms of mental images” (see [40], pp. 40–41, our translation).
• The pragmatic–social approach, in turn, “qualifies meaning as a social praxis, assimilating it to the way expressions are used” (see [40], pp. 43–44, our translation).
• The denotational approach imagines language as consisting of a set of words and rules for combining them. “Words are associated by convention with objects (i.e., they denote them). By virtue of this association, we can employ sequences of lexical elements to encode the situations in which the objects are.” ([40], p. 45, our translation).

In the context of natural languages, each of these approaches says something relevant about meaning and language. However, from the last approach emerged a way of relating semantics and logic (for natural languages).

Since Frege’s work and the resurgence of logic, semantics has come to be treated, by some scholars, in strict relation to logic (see, for example, Bertrand Russell’s treatment of the definite article [41]). Logic has semantics of truth values, i.e., it establishes an interpretation function $\mathfrak{I}$ that takes propositions as a domain and takes as the image set $\{V,F\}$ of logical constants. It is, moreover, a denotational approach to language, since it treats the referents of first-order predicates as sets of objects, and we understand “logic” as a syntax (with an alphabet), plus a proof method, plus a denotational semantics. In parallel ways, the denotational approach to natural language assumes that, on the basis of language, there should be:

A logic language of a certain kind that allows us to do this: the sentences of language must have a logical form from which we can derive their consequences. This makes it possible to integrate the most compelling aspects of traditional approaches to meaning for the following reasons. Firstly, logic is a system of representations, as in the representational approach. Secondly, these representations are equipped with a set of rules of inference, constituting their use conditions—as in the pragmatic-social approach. Third, logical representations must be interpretable in terms of truth conditions, as in the denotational approach.

(see [40], p. 57, our translation)

What is critical for issues on the relations between semantics and logic is that logic allows us to introduce a structure that, in principle, underlies ordinary speech. “To attribute a logical form means specifying a symbolic medium on the basis of which the conditions of truth and inferential procedures are defined (…) so that the expressions of languages are associated with their denotations through mediation of logical forms” (see [40], p. 71, our translation).

Thus, it happens that the logical form related to natural languages is exactly what is sought in the study of semantics for adherents of the denotational approach because “if languages have logic, it is appropriate to ask what are the representations that this logic uses. What language do we use to mentally calculate the connections between the things we say?”

In this regard, it becomes important to distinguish physics from natural languages since this “logic” substratum that the semantics of natural languages exhaustively seeks is, for physics, precisely mathematics, which is already given or constructed. It is through mathematics that we can proceed to calculate the connections between the things we say in physics (passing from the fundamental to the narrative level). This does not, of course, exclude pragmatic or social elements, as Thomas Kuhn’s research clearly shows, but these elements interact beyond mathematics and are articulated on the discursive plane.

Mathematics is the symbolic medium through which the conditions of truth and inferential procedures are defined, even though objective truth is determined by the concurrence of experimentation and serves to eventually contest the attributions made at the fundamental and narrative levels. It is from mathematics and the concrete denotational connections that it establishes between physically interpreted symbols in the constitution of the physical theory that one can interpret objectively, and not only in terms internal to the language, the phenomenal data. Mathematics also provides, through the notion of operation, the compositional principle present in virtually all semantic theories since it is essential for the explanation of complex phenomena in terms of their simplest elements (in the narrative level, for instance, it can be the explanation of experiments in terms of the axioms and denotations made at the fundamental level). From this element, axiomatic approaches receive their utmost epistemological force. For these approaches, from a very limited set of symbolic and ontological attributions, one gets, by syntactical calculus and semantic heritage, almost all the other elements of the theory, except, eventually, those concocted at the discursive level.

In this sense, mathematics constitutes a supporting but essential element in the production of meaning in the context of physics. With the semantic determinations of a fundamental level combined with the compositional character of mathematics, the mathematical formalism of the theory spreads, at the narrative level, a virtually infinite set of derived meanings (such as the proposition about the conservation of electromagnetic energy or the wave character of fields). Such derived meanings, many of them originally unsuspected, may also provide criteria for the rejection of the underlying theory, since they may lead to blatantly absurd descriptions of the world or simply descriptions incompatible with the results of a set of experiments.

Moreover, mathematics, which functions as the universal grammar of physics, has a generative character (in the sense of Chomsky’s generative grammar). From this we can infer the inestimable importance of Galileo for the replacement of logic by mathematics at the fundamental and narrative levels and for the constitution of modern physics, which in these terms has permeated all physics since then [42]. There is no physics, in the classical sense or not, apart from the Galilean metaphysics of mathematics and experimentation [23].

Interpretations of physical theories can be understood as semiotic relationships between mathematics and certain sets of natural phenomena, duly selected by the theory in question, with mathematics being expressed by means of symbols and their relations, usually presented as formal operations, such as derivations, integrations, etc., made symbolic in the context of the equations of the theory.

However, we emphasize that the point is not the relationship between mathematical language and the concrete “objects” of the world, but between objects of the world whose relations are mathematically expressed. In other words, it is about “relationships” between properties of objects. Thus, the constitution of meaning initially takes the referential elements that give each symbol an abstract referent linked to characteristics possible within the theory, for example, mass, specific heat, and charge, among others. This endows physics with natural classes under which concrete objects can be referenced a posteriori in the experiments as mere instantiations of the theory. This constitution of meaning is completed with the organization, at the mathematical–operational level, of relations that are intended to substitute, in the linguistic dimension, the physical phenomena themselves—thus, mathematics, symbolic in the fundamental and narrative dimensions, becomes symbolic and indexical in the discursive dimension.

Physics, as a linguistic and epistemic action, is constituted in the construction of theories that aim to subsume broad experimental contexts into a relatively small set of fundamental laws, mathematically expressed and physically interpreted. The interpretations of physical theories can be understood as semiotic relationships between mathematics and certain sets of natural phenomena, duly selected by the theory in question, in which such phenomena are mathematically expressed through relations between symbols. In these semiotic relations, as already mentioned, the trust placed in metaphysics, which attributes to phenomena an organization that can be grasped by mathematical language, plays an essential role.

7. The Role of Experimentation in the Process of Meaning Generation

In the previous sections, we have analyzed a purely textual dimension related to the interpretation of physical theories. However, it is evident that experimental physics participates in this process, not only in the context of testing the symbolic relations that appear at the narrative level but also in the very suggestion of symbolic articulations that appear in the attributions of referents for the fundamental instances of a physical theory.

Faraday’s lines of force, which would later become Maxwell’s fields, are an example of this. The very notion of “electric charge” can be considered another example. Thus, in the process of experimentation, there is a hierarchic connection between the fundamental and narrative levels, mainly in the sense of turning the referents of the basic symbols of the theory (fields, charges, flows, etc.) into concrete experimental entities apt for manipulation. The experimentation at the narrative level is also important for testing the relationships that arise at this level from the articulations developed from the syntax and semantics of the fundamental level.

There must exist a sort of trade-off between the fundamental and narrative levels because experiments already have predetermined theoretical elements, which allow us to interpret the movements of their parts (pointers, panels, digital numbers, etc.) in terms of the presence or absence of elements of the physical theory under scrutiny and their interrelationship in the context of experimentation—the very idea that all experimentation carries some theoretical elements.

The Millikan Oil Drop Experiment [43], to cite a concrete example, only makes sense in a context in which symbolic elements, such as gravitational force, electrical force, static equilibrium, mass, charge, electric field, ionization, and electrons (as a hypothesis), allow us to understand what it means for the charged oil droplet to remain static in the experimental apparatus. From these theoretical elements, the electron charge can be obtained (since, at the time, only the value of the charge relationship between charge and mass was known). We consider that it is this constant interplay, promoted by experiments, between the fundamental and narrative levels that gives the experimental approach its epistemic force (in its predictive dimension), allowing it to become a crucial element in the criterion of truth or adequacy of physical theories, from the end of the Middle Ages to the present.

The experimental dimension, which produces and consumes references to the symbols of the theory and its articulations, in a context of truth by correspondence, is an extralinguistic element of the process that generates meaning. It is one of the major epistemological constraints of physics. However, it loses its epistemological relevance if one is focused on understanding different interpretations of the same physical theory in a context of experimental underdetermination, as seems to be the case with the myriad of interpretations of quantum mechanics.

In this sense, we move away in part from the perspective of Greimas and Courtés, from which everything is text. Even if it is possible to “textualize” the production of meaning of an experimental activity, this would not only bring little information to the analysis but also eventually distort it in the face of the idea of having truth by correspondence, a perspective that necessarily includes textual and non-textual elements. As such, this would have a negative impact on the question of the non-arbitrary character of the establishment of this truth since it involves a dimension (that of natural systems) that escapes the type of flexibility that one has in the process of usual textual creation, transforming truth by correspondence into mere internal logical consistency. In some sense, that was the main characteristic of Aristotelian physics, which lacked the experimental element in favor of logic made to “save the appearances”.

This is the difference between a scientific text and any other text. A scientific text is subject to methodological ties that are specific to its field of articulation (e.g., reproducibility of experimental findings) and are not part of the usual context of natural languages. Physics, then, is a technical language artificially constructed by humans based on mathematics, instead of a usual natural language, given that a physical theory must comply with specific epistemological constraints.

Experimentation and Discursiveness

Thus, experimentation, as an epistemic process endowed with epistemological weight, happens only at the narrative level, despite being symbolically fed by the fundamental level, at which one should not expect any role for experimentation, given its abstract nature. For instance, one does not perform experiments to verify Maxwell’s equations, which are at the fundamental level, but we can perform experiments to investigate the flux of energy and the transversal nature of light.

If some physical theory comes from a set of interpreted axioms at the fundamental level (the fundamental syntax and semantics) and remains only at the fundamental and narrative levels, there would be no differences in interpretation, since the formal results and interpretations at the narrative level are just inherited from the interpretations at the fundamental level.

This is hardly how physical theories historically grow. Historically, theories were not initially proposed in an axiomatic way, with clearly interpreted axioms and their symbols, but with a set of results, generally of an experimental nature, that called for interpretation. Newtonian physics, which is axiomatic, took more than fifteen thousand years to surface from the genius of Galileo and Newton, among others. Electromagnetic theory took approximately seventy years to become axiomatic through the work of Maxwell. These initial moments are full of semantic proposals that are not equivalent in regard to the world they imply.

Quantum mechanics is only one more example of this initial maturation process, but with its own characteristics. After twenty-five years of varied proposals and uses of principles, its formal apparatus was constructed from Heisenberg’s matrix calculus or from Schrödinger’s equation. However, these syntactic developments were not able to induce an exactly unified interpretation because, to continue with Schrödinger’s equation, the referent of its main symbol, the probability amplitude, was not known.

Roughly speaking, from the moment of the proposal of the Schrödinger equation to the present, the interpretations advanced for the formal apparatus of quantum mechanics have differed in how they interpret this symbol: for instance, as a descriptor that sets potentialities for a single system in Heisenberg’s Copenhagen interpretation, as a descriptor of the statistics of ensemble of systems in Ballentine’s statistical interpretation [44], as a field that sets particle trajectories in Bohm’s hidden variables interpretation, as a source for branching universes in Everett’s late Many Worlds interpretation, and so on. From this basic disagreement, occurring at the fundamental level, enormous ontological differences developed, leading to the appearance of new constructs in each one of them that are usually incompatible with those appearing in other interpretations.

For example, there is no place for the notion of “observer” in Ballentine’s statistical interpretation, although it is a major semantic notion in Heisenberg’s Copenhagen interpretation. On the other hand, the notions of “implicate order” or “wholeness”, which come from the non-locality of Bohm’s interpretation, are irrelevant for Heisenberg’s Copenhagen interpretation and Ballentine’s statistical interpretation. In a previous paper [1] we showed that we can take a “step back” and provide quantum mechanics with two simple (and easily interpreted) axioms, and we argued that this move may give quantum mechanics its unit of interpretation, at least to the extent that such a unit is expected for physical theories.

Our main point here is that all these constructs proposed in previous interpretations of quantum mechanics are articulated at the discursive level, and not as the result of an agreement on the interpretation of the symbols at the fundamental level, as we exemplified with the issue on the interpretation of the probability amplitude (or wave function). In the end, this is why the Schrödinger equation cannot be part of the axioms of quantum mechanics if we want to search for some unit of its interpretation.

The main point here is that the discursive level, in opposition to the narrative level, is not constrained by the syntactic apparatus developed at the fundamental level. The constructs proposed at this discursive level generally have no bearing on the fundamental syntactics. For example, one cannot find a syntactic descriptor for “observers”, “consciousness”, or “wave-packed collapse”.

On other occasions, the concepts at the discursive level develop from dissent at the fundamental level, as with quantum mechanics, when one uses the Schrödinger equation (a syntactic element that is put at the fundamental level) with different interpretations for the probability amplitude. Thus, for example, “implicate order” and “wholeness” are developed from Bohm’s rewriting of the Schrödinger equation and his assumption of the probability amplitude as a true physical field (with unknown source), and the notion of “branching”(worlds) is developed from Everett’s assumption of Heisenberg’s Copenhagen interpretation without the notion of collapse.

All these considerations and examples in the context of quantum mechanics show that discursive constructs must have lower epistemic and epistemological weight. The proliferation of incompatible constructs among the interpretations relying on these discursive constructs should be considered close to proof of that lower epistemological weight.

With physical theories becoming more abstract, it may be important to bear this in mind when advancing interpretations of them. In addition to that, the epistemological principle that interpretations should adhere to the first two semiotic levels as much as possible is a key to forming a hierarchy of interpretations (we present this in [1]).

Thus, assuming the semiotic reconstruction that we do in this paper, it is not because the interpretations of a physical theory may be empirically underdetermined that they are semiotically equivalent or have the same epistemic and epistemological weights.

8. Four Examples

In previous sections, we presented a tripartite division that can be applied to the construction of any physical theory, considered a language with its specific syntactic apparatus and semantics. From this approach, we put forward the epistemological rule that any interpretation (semantics) of a physical theory should keep itself as close as possible to the underlying formal results (syntactic structure). This rule is based on the conclusion that all parts of an interpretation that disagree with this rule should be put at the discursive level and thus have less epistemic weight or no weight at all. We now present four examples of the use of this epistemological rule to assess interpretations of quantum mechanics. We do not intend to be exhaustive, but only present these interpretations to the extent that they can help us understand the relevance of the epistemological rule mentioned before.

8.1. Heisenberg’s Copenhagen Interpretation

It is known that there are many versions of the Copenhagen interpretation that vary in some specific assumptions. Bohr and Heisenberg, for example, disagreed on a number of elements of the interpretation of quantum mechanics [45]. We will limit ourselves to Heisenberg’s perspective.

Thus, at the syntactic level, there is the Schrödinger equation (or Heisenberg’s matrix calculus) and an operation formation technique (quantization rules). The following elements are present in Heisenberg’s Copenhagen semantics:

• The interpretation of the operation $\widehat{\mathrm{\Theta }}\psi (x,t)$ as the measurement of the physical element related to the dynamical variable $\mathrm{\Theta }$ and the role of the observer;
• The principles of wave–particle duality, complementarity, and indeterminacy;
• The state vector as related to a single system and the principle of reduction of the wave packet;
• The active role of observers;
• The reduction of the wave packet.

We analyze each of them in the following.

8.1.1. The Operationalist Interpretation and Observers

The interpretation of $\widehat{\mathrm{\Theta }}\psi (x,t)$ as a measurement of the physical element related to $\mathrm{\Theta }$ ($\widehat{H}\psi (x,t)$ represents the measurement of energy, for example) is the most basic semantic element of the theory, since it is the direct interpretation of the symbols presented at the fundamental syntactic level (the Schrödinger equation). This interpretation comes from the operationalist perspective that was very common at the time [46]. Thus, it seems that it must be put at the semantic fundamental level.

Indeterminacy, an important concept in this approach, is derived by semantic inheritance from the preceding interpretation. It comes from the interpretation of dispersions $\mathrm{\Delta }\xi$ (where $\xi$ represents, e.g., position x or momentum p) as experimental errors, which is consistent with item 1 above.

Note that, since $\widehat{\mathrm{\Theta }}\psi (x,t)$ represents the act of measuring the dynamical element represented by $\mathrm{\Theta }$, the theory requires an observer to implement this act of measuring $\mathrm{\Theta }$ in some specific experimental situation, which can vary in its arrangements and measurement precision. However, the theory does not allow one to specify the actual experimental setup used (there are no “apparatus” variables in the formalism, let alone those related to an “observer” [47]). This means that there is no syntactic element related to the semantic assumption of an observer. This is important, as we shall see when we present the interpretation of Heisenberg’s indeterminacy relation.

Since there is at least one element of the interpretation that has no syntactic counterpart, this assumption related to $\widehat{\mathrm{\Theta }}\psi (x,t)$, representing the act of measuring the dynamical element represented by $\mathrm{\Theta }$, must be placed at the discursive level, and we should be at least suspicious of it.

8.1.2. The Three Principles

Furthermore, in Heisenberg’s version of the Copenhagen interpretation, there are three principles that should be satisfied: (a) the principle of “indeterminacy”, connected to an interpretation of Heisenberg’s inequalities

$$\mathrm{\Delta }x\mathrm{\Delta }p\ge ħ/2,$$

(4)

as related to experimental errors; (b) the “duality” principle, which states that quantum-mechanical entities are some sort of dual particle–wave-like entities; (c) the “complementarity” principle, which states that our limited macroscopic language is not equipped with the semantic elements needed to cope with the duality that emerges from the experiments.

Following Heisenberg, these principles should be applied to the description of any quantum-mechanical phenomenon describable by the Schrödinger equation, given in terms of the Hamilton operator by

$$\widehat{H}\psi (x,t)=iħ{\displaystyle \frac{\partial \psi (x,t)}{\partial t}}.$$

(5)

These principles (a-c) are, of course, of a semantic nature. Being principles, they should be placed, at first glance, at the fundamental level. In what follows, we show that this is not the case.

First, it is interesting to qualify in what sense duality, indeterminacy relations, and complementarity are semantic principles. It is certainly not in a logical sense, as Popper argued [48], that is, not as axioms from which the theory can be obtained (they are not syntactic principles). On the one hand, they may be considered semantic principles, since they form the basis of the Copenhagen spirit [45]; on the other hand, they may be considered in the context of empirical principles, in what Einstein called principles theories [49], meaning theories endowed with experimental principles—principles that seem to inexorably come from the experiments. For instance, the fact that experiments can reveal a wave nature as their outcome in some instances and a corpuscular nature in other instances seems to call for the assumption of some sort of duality (but see [1]).

At the syntactic level, one formal principle is that the Schrödinger equation itself must be placed at the fundamental level of the proposed interpretation. Another syntactic principle is the way in which one forms operators from classical dynamical variables, which we discuss later on.

However, note that the dispersion relations are mathematically derived from the Schrödinger equation as a formal result. That is, they are, syntactically, at the narrative level. From the interpretation presented in item 1, Heisenberg’s relation (4) should be interpreted in terms of the way in which experimental errors regarding non-commuting variables behave in experimental situations, but should be placed, in principle, at the narrative level, not the fundamental one, since it is the interpretation of a syntactic element present at the narrative level. On the other hand, since the interpretation in item 1 was placed at the discursive level because of the notion of an observer, this derived interpretation of Heisenberg’s inequalities should also be placed at the discursive level.

Moreover, this interpretation is clearly inadequate. Indeed, it is based on the assumption of the presentation of Heisenberg’s relation in (4) as an inequality. However, the inequality symbol appears because, when deriving the relation, one uses only the commutator $[x,p]$, abstracting from the state being considered. However, if one assumes that the system is in any state ${\psi }_n\left(x\right)$ (for example, the pure state of the harmonic oscillator), one gets the equality sign. Indeed, for the harmonic oscillator, one gets

$${\left(\mathrm{\Delta }x\right)}_n{\left(\mathrm{\Delta }p\right)}_n=\left(n+{\displaystyle \frac{1}{2}}\right)ħ.$$

(6)

Now, if we try to impose the previous interpretation on (6), we must conclude that the error in measuring the position x, multiplied by the error in measuring the momentum p, both for the pure state labeled by n, is always the same, no matter how the system is observed and which apparatus or technique is used to observe it. This is blatantly absurd, since one can use an experimental setup that is terribly inadequate when making both x and p measurements, as should be obvious. At this point, one may try to assume that (6) refers to ideal experiments, but the point in question, of course, is the meaning of the word “ideal”. Saying that (6) refers to ideal experiments means that it must represent an objective feature of the physical system, from which actual measurements, with different measurement techniques, will give different results for the experimental dispersions and whose measurement quality will be judged.

One could try to avoid this conundrum by appealing to the notion of a minimum error for the product in (6). However, this assumption would deconstruct the appeal to observers and experimental setups since this “minimum” should now be related to an objective property of the system, not something related to some (unqualified) observer.

The conclusion must be that the interpretation (that $\widehat{\mathrm{\Theta }}\psi (x,t)$ represents a measurement of the physical element related to $\mathrm{\Theta }$), which is made at the discursive level, has no epistemic weight and must be rejected, together with its derived interpretations.

The duality principle comes from experimental situations in which the outcomes seem to be sometimes of a wave nature and sometimes of a corpuscular nature. Of course, experimental principles, to use Einstein’s nomenclature, must be placed at the discursive level (see [1] for an interpretation without the duality principle).

The complementarity principle is not even a physical principle, but one of a linguistic nature that is concocted only to give consistency to the notion of duality. It is clearly a discursive element.

8.1.3. The State Vector as Related to a Single System and Collapse

Another main element of Heisenberg’s version of the Copenhagen interpretation is the assumption that the descriptor $\psi (x,t)$, the wave function, is related to single systems, not to ensembles; that is, it references single systems. This must be qualified; of course, Heisenberg knows that any essential statistical theory must be related to ensembles; that is, within Heisenberg’s Copenhagen spirit, $\widehat{\mathrm{\Theta }}\psi (x,t)$ must be related to a set of measurements of the dynamical variable $\mathrm{\Theta }$. However, what Heisenberg is assuming is a specific interpretation of the equation

$$\widehat{\mathrm{\Theta }}\psi (x,t)=\theta \psi (x,t),$$

(7)

where the representation refers, mathematically, to the state vector $\psi$ as an eigenfunction of the operator $\mathrm{\Theta }$, with eigenvalue $\theta$. Thus, he assumes (interprets) that, in these cases, $\widehat{\mathrm{\Theta }}\psi (x,t)$ measures the dynamical variable Θ, giving the value θ, and leaves the system in the same state $\psi (x,t)$ after the measurement.

This completes the interpretation of the Schrödinger equation, which is Equation (7), written for the Hamiltonian operator and the energy eigenvalue. Thus, Heisenberg’s Copenhagen interpretation singles out the complete set of commuting operators, endowing it with particular significance. Needless to say, this interpretation means, at the syntactic level, that the whole measured ensemble is dispersionless, since

$${\widehat{\mathrm{\Theta }}}^2\psi (x,t)=\widehat{\mathrm{\Theta }}\widehat{\theta }\psi (x,t)={\theta }^2\psi (x,t),$$

(8)

and, thus, $\mathrm{\Delta }E=0$. Thus, a single $\psi (x,t)$ represents the whole measured dispersionless ensemble, which qualifies Heisenberg’s assumption that the state vector represents a single system.

This must be placed at the fundamental level since it is a direct interpretation of the symbols present at the fundamental syntactic level.

The first problem with this interpretation appears when one applies it to a superposition of states

$$\mathrm{\Psi }(x,t)=\sum _i{c}_i{\psi }_i(x,t),$$

(9)

where $c_i$ denotes constants and ${\psi }_i(x,t)$ represents the eigenvectors of the operator $\widehat{\mathrm{\Theta }}$ with the eigenvalues ${\theta }_i$. Since each state vector ${\psi }_i(x,t)$ of the sum is a solution of the eigenvalue equation and the eigenvalue equation is linear, each state vector ${\psi }_i(x,t)$ is also a solution of this equation. In such cases, following the interpretation just advanced, the measurement of superposition (9) must result in the measurement of the system in all superposed states (and leave the system in this state). However, the interpretation assumes that the measurement of such a superposition must select one of the states ${\psi }_i$. The only way to make these assumptions coherent is to postulate that the measurement (as an act of a concrete observer) causes the superposition in (9) to collapse to a single state ${\psi }_i$, for some i.

This is the principle of the reduction of the wave packet of the wave function (collapse). However, one cannot find, in any part of the syntactic apparatus of Heisenberg’s quantum mechanics, the formal (syntactic) representation of this collapse process, let alone the issue that it uses the notion of an active observer, already analyzed. Thus, there is no other option than to put this collapse principle at the discursive level. It was concocted just to make the theory consistent with the assumption of dispersionless ensembles.

Moreover, Heisenberg assumes that if we have a classical dynamical variable $\mathrm{\Theta }(x,p)$ and its square ${\mathrm{\Theta }}^2(x,p)$ and apply quantization to both $\mathrm{\Theta }(x,p)\to \widehat{\mathrm{\Theta }}(\widehat{x},\widehat{p})$ and ${\mathrm{\Theta }}^2(x,p)\to {\widehat{\mathrm{\Theta }}}^2(\widehat{x},\widehat{p})$, we will have

$${\widehat{\mathrm{\Theta }}}^2(\widehat{x},\widehat{p})=\widehat{\mathrm{\Theta }}(\widehat{x},\widehat{p})\widehat{\mathrm{\Theta }}(\widehat{x},\widehat{p}),$$

(10)

which seems natural at first sight. Thus, Heisenberg assumes von Neumann’s operator formation approach, which is wrong when non-commuting dynamical variables compose the operator $\widehat{\mathrm{\Theta }}$, as with the Hamiltonian for systems in the presence of a potential force depending on the position, for instance [50]. It is wrong because, by not explicitly dealing with the order in which these non-commuting dynamical variables will appear as operators in ${\widehat{\mathrm{\Theta }}}^2$ (${\widehat{H}}^2$), it furnishes two or more operators in the quantization process [50]; that is, it is ambiguous. Simply stated, the quantization of the dynamical variable ${\mathrm{\Theta }}^2$ is generally not equivalent to the product of the quantization of the dynamical variable $\mathrm{\Theta }$, as Heisenberg assumed.

Other elements of Heisenberg’s interpretation can be analyzed in the same fashion, and the same can be said of other interpretations of quantum mechanics, but this would take us too far afield (but see [51]). In essence, the previous arguments show that the semantics of Heisenberg’s Copenhagen interpretation pertains completely to the discursive level and has a very small epistemic weight.

8.2. Everett’s Relative State Interpretation

With respect to forming a hierarchy of interpretations, we can extend the previous example using Everett’s approach “relative state” interpretation [15]. Everett [15] assumes all the previous elements of Heisenberg’s interpretation but wants to eliminate the collapse principle, and he makes it explicit in [15] that he wants this because it has no syntactic counterpart.

However, Everett assumes all other interpretation issues of Heisenberg’s Copenhagen interpretation, especially the one regarding the assumption that the probability amplitude refers to single systems (and throughout his paper [15] he uses von Neumann’s operator formation rule, which we have already criticized). Everett avoids the collapse principle, substituting it with the notion of “branches” that he later interprets in terms of “many worlds” [16].

Thus, with respect to only the formal result regarding “branches”, Everett’s interpretation supersedes Heisenberg’s interpretation, even assuming all other interpretation elements of the latter. However, all these formal developments are not at the fundamental level, since they depart from a previous, unsubstantiated interpretation of the Schrödinger equation (mainly with respect to the “observers” that he uses throughout his paper [15]).

This means that all the syntactic developments advanced by Everett are at the discursive level, since the construct of “observers” is at this level. In fact, his approach provides a very good example of a syntactic apparatus developed at the discursive level.

On the other hand, his late semantic interpretation of these branchings in terms of “many worlds” [16] is, of course, a purely discursive element that is not mandatory, given his development of the syntactic structure. In fact, another interpretation with the same syntactic structure compares “branching” to “many minds”, proposing that the distinction between worlds should be made at the level of the mind of an individual observer [52].

The fact that Everett assumes the semantic construct of active observers and that the state vector refers to single dispersionless quantum mechanical systems reduces the epistemic weight of his approach, according to our previous analysis. This should be considered an internal criticism of his approach, since he explicitly assumes the same principle that we develop in this paper: that no semantic element should be advanced without a syntactic counterpart.

8.3. Bohm’s Hidden Variable Interpretation

In 1952, David Bohm published two ground-breaking papers [14,53] on the foundations of quantum mechanics. He was mainly unsatisfied with Heisenberg’s assumption that the physical state of an individual system is completely specified by a wave function that determines only the probabilities of actual results that can be obtained in a statistical ensemble of similar experiments, in the qualified sense described above.

He presented another interpretation that involves additional elements or parameters, permitting a detailed causal and continuous description of all processes and not requiring one to forego the possibility of conceiving the quantum level in precise terms. He called these additional elements or parameters “hidden variables”.

Bohm’s approach is interesting for the present paper because Bohm developed a new formal presentation of the Schrödinger equation, leading to an equivalent mathematical structure with, however, a quite different interpretation from Heisenberg’s Copenhagen interpretation. We show how he did this in what follows.

8.3.1. The Mathematical Structure of the Approach

To accomplish his objective, presented in the last section, Bohm found a way to rewrite the Schrödinger equation in a new form that allowed him to introduce his hidden variable interpretation of quantum mechanics.

Mathematically, for the wave function (which is a complex function), Bohm introduced the notation

$$\psi (x,t)=R(x,t)e^{iS(x,t)/ħ},$$

(11)

and substituted it into the Schrödinger equation to find

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \rho (\overrightarrow{x},t)}{\partial t}}+\nabla ·\left(\rho (\overrightarrow{x},t){\displaystyle \frac{\nabla S(\overrightarrow{x},t)}{m}}\right)=0\\ \hfill {\displaystyle \frac{\partial S(\overrightarrow{x},t)}{\partial t}}+{\displaystyle \frac{{(\nabla S(x,t))}^2}{2m}}+V\left(x\right)-{\displaystyle \frac{{ħ}^2}{2mR(\overrightarrow{x},t)}}{\nabla }^{2}R(\overrightarrow{x},t)=0\end{array},$$

(12)

where $\rho (\overrightarrow{x},t)=R{(\overrightarrow{x},t)}^2$ is just the probability density function in the configuration space.

At this point, Bohm took his main step towards his ontological interpretation by establishing a relationship between the second equation in (12) and the Hamilton–Jacobi equation

$${\displaystyle \frac{\partial S(\overrightarrow{x},t)}{\partial t}}+H\left(\overrightarrow{x},{\displaystyle \frac{\partial S(\overrightarrow{x},t)}{\partial t}},t\right)=0,$$

(13)

which, by the way, is known in classical mechanics as a different way to formalize Newton’s equations in terms of a single partial differential equation. Of course, this formulation in classical mechanics is also equivalent to Hamilton’s set of differential equations. The important point here is that the Hamilton–Jacobi equation presents the deterministic movement of particles in terms of waves that describe the movement of an ensemble of particles.

From this relationship, Bohm concluded that quantum mechanics is related to an ensemble of particles that follow deterministic trajectories in the phase space, given by $p=\nabla S(\overrightarrow{x},t)$.

However, note that this relationship is just an analogy for the classical case. As an analogy, it should be placed at the discursive level. Indeed, in the classical counterpart of the Hamilton–Jacobi theory, the function $S(\overrightarrow{x},t)$, sometimes called Hamilton’s principal function, establishes a set of lines of flow perpendicular to the surface, given by $S=const.$, representing the movement of particles with different initial conditions. However, the relation $p_i=\partial S(\overrightarrow{x},t)/\partial q_i$, where $q_i$ is some generalized coordinate, is established between two non-statistical variables. Indeed, one can derive the Hamilton–Jacobi equation from a type-2 canonical transformation using the function $G(\overrightarrow{p},\overrightarrow{P},t)$, from which one gets

$$\overrightarrow{p}={\displaystyle \frac{\partial G_2}{\partial \overrightarrow{x}}};\overrightarrow{X}={\displaystyle \frac{\partial G_2}{\partial \overrightarrow{P}}};K(\overrightarrow{X},\overrightarrow{P},t)=H(\overrightarrow{x},\overrightarrow{p},t)+{\displaystyle \frac{\partial G_2}{\partial t}},$$

(14)

where $K,P,Q$ are the new Hamiltonian and phase-space variables. All variables $p_i,q_i,P_i,Q_i$ are related to a single particle, and K is set equal to zero. Thus, for instance, for a one-particle problem, $S(\overrightarrow{x},t)$ in the classical approach refers to the wave-like behavior of the ensemble of one-particle systems having coordinates $\overrightarrow{x}$ for the particle of each ensemble’s system related to different initial conditions. It is a deterministic function that can be made statistical by creating a statistical distribution in the initial conditions.

From this identification, Bohm adopts his notion of a “quantum potential”, which will have a central importance in his approach. To make this identification, Bohm (see [14], p. 170) argues that, in the limit $ħ\to 0$, one recovers the classical Hamilton–Jacobi equation, but this is not correct, as becomes obvious for stationary systems. Some properties of this quantum “potential” can be used to reinforce that Bohm’s analogy should be placed at the discursive level.

8.3.2. The Quantum “Potential”

The formal relation between the Hamilton–Jacobi theory and Newton’s equations lets Bohm write

$$m{\displaystyle \frac{d^2\overrightarrow{x}}{d{t}^2}}=-\nabla \left(V\left(\overrightarrow{x}\right)-{\displaystyle \frac{{ħ}^2}{2m}}{\displaystyle \frac{{\nabla }^{2}R(\overrightarrow{x},t)}{R(\overrightarrow{x},t)}}\right),$$

(15)

with the last term being the one previously obtained.

Needless to say, we now have the term

$$Q(\overrightarrow{x},t)=-{\displaystyle \frac{{ħ}^2}{2mR(\overrightarrow{x},t)}}{\nabla }^{2}R(\overrightarrow{x},t)$$

(16)

playing a role that seems to be identical to the physical potential $V\left(\overrightarrow{x}\right)$. Bohm thus called this new term the quantum potential.

Thus, all trajectories must now be calculated considering these two potentials: one that reflects physical causality in the equation, and another that introduces all quantum effects that supervene in any classical context.

One can easily verify that the properties of this quantum potential are as follows:

• The wave function of an individual electron (or a set of electrons) is a concrete natural field (see [14], p. 170).
• The equation for this $\psi$-field is the Schrödinger equation.
• The $\psi$-field does not have a known physical source, such as a charge, a mass, or an electric current.
• In a one-particle system, the behavior of one particle of one of these systems depends on the behavior of other systems of the ensemble, despite the fact that each system of an ensemble must be independent of all the other systems that compose it.
• Stationary states give a rather strange result. For example, the electron at the fundamental state of a hydrogen atom must be static at some point of the configuration space due to the balance between the true electric force and the one coming from the quantum potential.
• It is non-local, and one of its properties is that its “strength” does not depend on the amplitude of the wave function—quite different from actual physical fields.

Thus, we are left with the interpretation that quantum mechanics is a deterministic corpuscular non-local theory based on a sourceless concrete field $\psi (x,t)$. Furthermore, there is no role for the notion of collapse of the state vector in this interpretation, nor does it refer to observers. Heisenberg’s dispersion relation is a constraint on the way in which we can construct the initial conditions of the ensemble.

Thus, Bohm’s approach downsizes the number of semantic constructs that must be placed at the discursive level and, according to our epistemological principle, is a better interpretation of quantum mechanics than Heisenberg’s Copenhagen interpretation and Everett’s relative state approach.

The main problem, thus, is that there are many other ways to interpret the function $Q(\overrightarrow{x},t)$. For example, in [1] it is mathematically shown that this term represents fluctuations that appear from the separation of the physical system into a corpuscular subsystem and a field-like subsystem, the latter functioning as a thermal reservoir. The resulting Equation (15) is thus interpreted as the result of a Langevin equation (not a Hamilton–Jacobi equation) in the context of a stochastic theory (not a deterministic theory). This latter interpretation is consistent with $Q(\overrightarrow{x},t)$ being sourceless, for instance. It is also consistent with the fact that it represents non-local connections in terms of statistical correlations, which are “non-local” by definition. Thus, the assumption, by analogy, that the function $Q(\overrightarrow{x},t)$ is a concrete potential, in the same sense that the electrostatic potential is, must be placed at the discursive level. Another issue is the appeal to “hidden variables”, which, of course, must be placed at the discursive level. Semantic notions such as “wholeness”, “implicate order” and others [13] that emerge from this discursive analogy must be considered equally epistemologically weightless.

8.4. Ballentine’s Ensemble Interpretation

Ballentine formulated his ensemble interpretation in an influential paper in 1970 [44]. The paper follows an axiomatic approach that includes the Schrödinger equation as one of its postulates. The syntactic apparatus is the one we are accustomed to and is presented in Section 1.1 of that paper [44]. Then, Ballentine proceeds to the semantics of his approach, that is, its interpretation (which he calls “correspondence rules” in Section 1.2). As expected, there are two constructs that need interpretation: the construct of an observable and the construct of a state, which provide “a more specific interpretation of the averages and probabilities” (see [44], p. 3).

The interpretation of the construct “observable” is somewhat direct as “a dynamical variable whose value can, in principle, be measured” (see [44], p. 3). Of course, in the context of quantum mechanics, this is not that simple, given the interpretation of Heisenberg’s inequalities. Ballentine discusses these issues in Section 1.3 of his paper. Our interest here is focused on his interpretation of a pure state, which defines the ensemble interpretation more broadly [44].

Thus, Ballentine interprets a pure state (and a general state) as the formal construct that “provides a description of certain statistical properties of an ensemble of similarly prepared systems, but need not provide a complete description of an individual system” (see [44], p. 3). Ballentine’s ensemble interpretation is a corpuscular interpretation that assumes that the wave nature embodied in the state refers only to the ensemble. Because of that, Ballentine’s ensemble interpretation does not assume the existence of well-defined trajectories (as observables), since the state vector does not provide a complete description of an individual system.

The ensemble interpretation is acknowledged to permit an enormous decrease in the number of constructs necessary to understand quantum phenomena. In principle, this should be considered a very important issue, considering the approach we are assuming in this paper. However, it may be asked whether this reduction in the number of constructs renders the interpretation inadequate for some actual physical situations. This seems to be the case and, in fact, is the most criticized characteristic of this interpretation [54].

Ballentine, as a possible counterargument to this criticism, argues that “the averages of quantum theory (…) are realized by performing the same measurement on many similarly prepared systems (or equivalently by performing the measurement many times on the same system which must be resubmitted to the same state preparation before each measurement” (see [44], p. 7—our italics).

The text in italics assumes that some ergodic principle is valid for quantum mechanics, meaning that ensemble and single-system measurements would be equivalent. However, such a result is not the result of the syntactic apparatus of the theory. Thus, in this interpretation, this statement must be placed at the discursive level, removing much of its epistemic weight. In this case, it seems, the reduction of constructs has gone too far (but see [1]).

9. Conclusions

The problem of interpretations is statutory in physics. It exists in all its fields, but it becomes particularly acute in quantum mechanics, given the respective levels of abstraction, the dependence on advanced mathematical methods, and the sensibility of experimental techniques. Conceived as a technical language, an epistemological and semiotic analysis (at the syntactic level, given by mathematical formalism, and at the semantic level, by the way symbols and equations are associated with the world of experience), linguistic structures, and the interpretations of a theory, all of them may be reified as texts that compete for the attribution of meaning to a formal set of structures that compose its lexicon.

A key to understanding this exegesis is Greimas’ semiotic model, which assumes that the production of meanings occurs at three levels: fundamental—basic laws and preliminary meanings; narrative—formal consequences and inherited interpretations; and discursive—constructions of complementary meanings, not predicted at the fundamental and narrative levels, from which most of the non-convergent interpretations arise.

This arrangement has allowed us to propose an epistemological criterion: the closer an interpretation is to the fundamental and narrative levels, and therefore the less constrained by external constructions, the greater its epistemological strength. Indeed, this criterion makes it possible to establish a hierarchy among interpretations, even in cases of experimental underdetermination, as is the case in quantum mechanics.

The proliferation of “wild” interpretations stems from excessive distancing between formal syntax and posited semantics, something that the semiotic approach makes it possible to measure and compare. In fact, experimentation is the element that limits discursive content and re-establishes links with the narrative interpretative level. The conclusion thus obtained is that physics is not merely a logical system but an engineering of meaning production oriented by methodological constraints—above all, experimental confrontation.

It is no small matter that the clarity afforded by the semiotic analysis developed in this article makes it possible to more precisely understand why certain formal developments produce numerous, mutually incompatible interpretations. By recognizing that a theory is not merely a set of equations but a technical text with distinct levels of meaning production, it becomes possible to distinguish in which aspect and in what manner interpretative divergences arise.

Based on this perspective, we suggest that proximity between syntax and semantics be taken as an epistemological criterion: interpretations that depend heavily on discursive constructions external to the formalism tend to be less robust than those that emerge directly from the fundamental structures of the theory. This proposal does not eliminate interpretative plurality but provides an analytical tool to assess it under its own constitutive bases as a corpus. Instead of considering all interpretations equivalent to experimental underdetermination, we show that it is possible to construct a rational hierarchy among them, based on the degree of detachment between mathematical symbolism and additional semantic elements.

It should be pointed out that, despite the fact that we have adopted in this paper the assessment of whole interpretations, their independent issues can also be considered in the context of our approach. These issues may be assumed by one or more interpretations and should be considered within the context of each one, but the epistemological constraint can be applied in each context. As an example, let us examine the debate on the ontic versus epistemic nature of the wave function. Whatever the interpretations may be that give this debate some context, the assumption of the present epistemological constraint would demand that one be able to find mathematical formulas in which the formal presence of the wave function could not be referred to by its related probability density—some in which the phase of the wave function plays an essential role, for instance. If this cannot be found in the formalism, the debate should be deemed baseless. If this situation can be found, the debate should be deepened by asking whether the wave function is only a theoretical variable (without a reference in the world) or whether it does have a clear reference—a discussion that is often somewhat difficult and theoretically driven, if we recall the role of the electromagnetic potentials within classical electromagnetic theory and quantum mechanics in the context of the Aharonov–Bohm effect [55].

The proposed semiotic application is not a definitive solution to the problem of interpretations, but a method that re-establishes links between language, experiment, and scientific ontology. Especially in quantum mechanics, where the multiplicity of interpretations remains an open question, the framework presented here makes it possible to restrict arbitrariness and promote greater epistemological coherence.

The core semiotic levels of a physical theory are the fundamental and narrative ones. From the axioms of the basic equations of a physical theory, its fundamental syntax, and the attribution of referents and meaning to them—its fundamental semantics—one gets, by the compositional feature of mathematics, a virtually infinite set of relations at the narrative level, strictly connected to the fundamental level with respect to both the syntax and the semantics. The narrative level “carries on” the syntax and semantics of the fundamental level in such a way that it is possible to perform experiments with the entities of the theory. This process is what we have called semantic inheritance.

This strict connection, however, is not usually taken to the discursive level, where one may introduce, for instance, ontological entities or even other syntactic and semantic structures, which may not have any corresponding syntactic and/or semantic counterpart at the fundamental or narrative levels—and this is generally done when physical theories are in their earlier stages.

This is generally done to make the physical theory meaningful, since, at this point, its fundamental and/or narrative levels are not developed enough to restrict these proposals. This is why physicists must, at some point, develop an epistemological process for collecting rejected views, in which physical theories purge unnecessary constructs from themselves a posteriori, much along the lines of Occam’s razor, but now made explicit in its semiotic realization in the present paper. This should be considered the normal “cycle of life” of physical theories.

In fact, only when we have the semantic elements of the theory anchored in its syntactic counterparts are we able to provide actual falsification [56] of the interpretation, since we can use the formalism to obtain the outcome provided by the theory for the semantic element and compare it with the experimental result—and we can use the formal apparatus to change the context in which this element appears, furnishing other, different outcomes. If one does not have this semantic–syntactic connection, the semantic element may float above the formal apparatus in whatever way we want it to.

Thus, from a diachronic perspective, there are two movements related to the construction of a physical theory that take place in opposite directions: In its infancy, the discursive level is important for filling in the blanks in the theory that the fundamental and narrative levels cannot fill at that moment. Ontological entities, connections, and wild guesses are plentiful.

However, after some time, when the fundamental and narrative levels are sufficiently developed, the direction (hopefully) inverts, and this filling must be done by the results obtained at these levels, in a process that washes away much of the proposed unnecessary ontological entities and, more importantly, much of the wild guesses. This “cleansing” process is generally performed at the discursive level. Much of this process is exemplified in [1] in the context of interpretations of quantum mechanics.