Multidimensional Representation of Semantic Relations between Physical Theories, Fundamental Constants and Units of Measurement with Formal Concept Analysis

Abstract

We propose several hierarchical graphs that represent the semantic relations between physical theories, their fundamental constants and units of measurement. We begin with an alternative representation of Zel’manov’s cube of fundamental constants as a concept lattice. We then propose the inclusion of a new fundamental constant, Milgrom’s critical acceleration, and discuss the implications of such analysis. We then look for the same fundamental constants in a graph that relates magnitudes and units of measurement in the International System of Units. This exercise shows the potential of visualizing hierarchical networks as a tool to better comprehend the symmetries, interrelations and dependencies of physical magnitudes, units and theories. New regimes of application may be deduced, as well as an interesting reflection on our ontologies and corresponding theoretical objects.

1. Introduction

The first study dedicated to the natural units system made up of the three fundamental constants, named c, G and ℏ, was written as a humorous present to a female student that young George Gamow, Dmitrii Ivanenko and Lev Landau courted back in 1927 [1,2]. They hypothesized what would happen if any of these values were different from what we measure [3]. George Johnstone Stoney in 1874 and Max Planck in 1899 had already proposed defining metric standards in terms of fundamental constants: a basic system of units for distance, time, mass and temperature from which all the other quantities—based on the constants $c,G,h,k$ or e—can be normalized to 1. Some words of Max Planck are as follows [4]:

“These quantities retain their natural significance as long as the law of gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always to be the same, when measured by the most widely differing intelligences according to the most widely differing methods”.

Those ideas may have found no practical use or meaning and attracted little attention at the time, but were shared by M. Bronstein, a well respected mate of the Jazz Band (Landau, Ivanenko and Gamow), who had talked about the limits of theories and was deeply concerned by the general physical picture of the world as a whole. He was one of the few that continued research on gravitation during the 1930s, talking about a unified theory that would marry the field theory of quanta, the electromagnetic field and gravitation. This unified theory would be a cosmological theory. Einstein too was looking for a unified theory, but by trying to deduce quanta from his generalized theory of relativity, while Bronstein recognized both theories as equally fundamental that needed some kind of synthesis [2]. In [5], he published two diagrams, reproduced in Figure 1 and Figure 2, where he maps the world of physics. In Figure 1, the areas of applicability of classical mechanics, quantum mechanics and special relativity are delimited by intersecting rectangles. The dotted lines correspond to a broader area for the not yet well-defined quantum relativistic theory that would contain the other three areas. Such theories are characterized by the presence of the two fundamental constants c and ℏ. Figure 2 is not about areas of applicability but relations between the theories themselves, including general relativity theory and the theory of quanta-electromagnetic field-gravitation, which would in turn be the “cosmological theory”. The dotted lines correspond to unsolved problems. Notice that this representation does not include Newton’s gravitation, nor a path from G-theory to $cG$-theory.

According to [2], to represent Bronstein’s ideas in three dimensions, in 1967 A.L. Zel’manov [6] drew the cube of physical theories (Figure 3), which was reproduced by Okun [7] and Barrow [4]. This cube consists of three orthogonal axes $(G,1/c,\hslash )$. The origin (0,0,0), corresponds to the limits $G\to 0$, $1/c\to 0$ and $\hslash \to 0$, representing classical particle mechanics in the absence of extreme fields. In other words, the speed of light $c\to \infty$—or more formally the speed of propagation of interactions—is infinite, and Planck’s constant does not describe the nature of classical physics, so $\hslash \to 0$ and, given that there is no gravitational description of nature, $G\to 0$. Note that the fundamental constants of physics $c,\hslash$ and G are independent of each other. This principle about fundamental constants then implies that there is no mathematical formula in which one of them can be obtained by the other two.

Figure 3 is a translated copy from the original diagram in Russian of Zel’manov’s cube (also called Okún’s cube, “physical theories cube” or “hypercube of physical units”). There are two empty vertices, no Newtonian Mechanics, no TOE and no QFT. Instead, there is a General Relativistic Quantum Theory. This situation was changed by Okun in the left side of Figure 4. Newtonian gravitation is then located at the vertex $(G,0,0)$, while special relativity is at $(0,1/c,0)$ and general relativity corresponds to $(G,1/c,0)$. Non-relativistic quantum mechanics is located at $(0,0,\hslash )$, quantum field theory at $(0,1/c,\hslash )$ and the Theory of Everything (TOE) is located at the corresponding corner $(G,1/c,\hslash )$ (see also [9]). This representation by means of axes can take the reader to interpret a smooth and continuous gradient in the values of the constants, which is incorrect because the values of each fundamental constant of physics is unique and, in principle, they do not have a spatial or temporal change. However, it should be noted that there are proposals for temporal variations in the gravitational constant [10], in the speed of light [11] and even in the fine structure constant cf. [4,12].

The interest in the relations between fundamental constants was also expressed in the western world by Dirac [13,14,15], who proposed his large number hypothesis, observing relations between ratios of fundamental constants and size scales of the Universe, such as the gravitational constant being inversely proportional to the age of the universe $G\propto 1/t$, suggesting THAT constants could be subjected to the proportions of the Universe’s scales.

The Boltzmann constant is not considered fundamental since it serves only to transform temperature to energy values. However, Huggett et al. [16] proposed a hypercube with a fourth direction, labeled N for the number of degrees of freedom of a system, but related to Boltzmann’s constant as a way of linking information theory and its entropy.

As we will show, lattices are an alternative and more powerful multidimensional representation. Physical theories can be classified on the basis of many different approaches such as their axioms, their models or their parameters, fundamental constants and symmetries using different formalisms such as set theory or category theory. These classifications can also be represented using diagrams. Recent work by Espinosa [17,18] show the benefits of using concept lattices and computing technologies to create complex hierarchical networks that represent relations between models of physical theories. (See also Méndez and Casanueva [19] for a similar study on genetics). In this work, more than a dozen theoretical grids were generated that relate more than 44 gravitational and space-time theories. (These interactive networks can be found on the following web page: http://remo.cua.uam.mx/vis/Exploratorium, accessed on 1 July 2024). These conceptual maps allow students to locate their working models in a sea of concepts; identify the foundations of theoretical models and theoretical branches; and differentiate classes of theories, classes of concepts as well as sub-specialties. By studying these networks, it was possible to refine our definition of theoretical specialization and the theorizing relationship, as well as to identify a certain parsimony in scientific progress. These networks, called lattices, are obtained using Formal Concept Analysis (FCA), a mathematical theory based on Garrett Birckoff’s lattice theory and Galois’ connection that relates concepts within a formal context, see [20,21].

In the present article, we focus on the presence of fundamental constants in law statements of the most relevant physical theories such as quantum theory, gravitation and relativity. Under the structuralist view of the philosophy of science, these constants are considered distinguished elements. These are variables that always take an exact value, under certain conditions, either as quantities of some physical variable or product numbers of ratios between quantities. Dimensionless constants, like the fine-structure constant $\alpha \approx 1/137$ that quantifies the strength of electromagnetic interactions between charged particles, take the same value regardless of the system of units, and in that sense they are universal. Instead, the dimensional constants refer to properties of classes of physical systems whose value, which is constant in time, depends on the system of selected units. As we know, the value of the gravitational constant in the SI is G = 6.67384 × 10⁻¹¹ N · m²/kg². In astrophysics, where distances are measured in parsecs $\left(pc\right)$, velocities in km/s² and the masses in solar units $M_{\odot }$, the value is G = 4.3009 × 10⁻³ $pc/M_{\odot }$(km/s²), while in the system of natural Planck units, the constants are normalized, where $G=1$.

Not all referential constants are universal, and there are constants specific to particular systems; the speed of light depends on the medium and the speed of sound even more. The specific heat is proper to the material, and charge acts only upon specific particles. (See [22,23] for other discussions on the nature of fundamental constants and their graphic representations.) Constants are characteristic of certain regimes. The extension of a particular regime tells us how more fundamental or less fundamental a constant is. For example, the charge of the electron e belongs to micro physics. Others, like Boltzmann’s constant, belong to macro physics, and others like G and c are assumed to be independent of the scale. G is a universal constant since it does not depend on the medium, the material or the scale. Under some theories like Modified Newtonian Dynamics (MoND), Milgrom’s acceleration is a fundamental constant [24] that defines the low-acceleration regime [25]. A first study on the incorporation of Milgrom’s acceleration as a fundamental constant in a diagram that semantically relates it to the other three fundamental constants, c, h and G, can already be found in [17].

The article is organized as follows. Section 2 explains in a nutshell the mathematical theory called Formal Concept Analysis (FCA), which helps us relate concepts hierarchically and present them as lattices, showing how it can be applied to Zel’manov’s cube. Section 3 continues to pinpoint some relevant aspects of structuralism, which is our theoretical framework and comes in handy for our analysis, mostly presented in Section 4. Section 5 takes a small detour around the international system of units and their relation to fundamental constants in order to present an alternative representation using FCA. We then take the dual graph to include some 47 derived magnitudes as objects. In this multidimensional space, we can also locate the fundamental constants as distinguished objects and evaluate their semantic hierarchical relations. Finally, we share our insights and possible follow-ups in Section 6.

2. Formal Concept Analysis

Lattices are tree-like, hierarchical networks that have the peculiarity that for any pair of nodes there is one and only one supreme and infimum node. Conceptual lattices are hierarchical networks whose nodes represent formal concepts. The meaning of a formal concept $C:<B,A>$ is given by a subset of certain attributes $A\subseteq M$ called the intention of the concept, and by all those objects $B\subseteq G$ that have those attributes, the extension of the concept. Formal concepts are dependent on the context in which they are found. A formal context $(G,M,I)$ is a binary table with an incidence relation $I\subseteq G\times M$, where a set of objects G is related to a set of attributes M. Formal Concept Analysis (FCA) is a mathematical theory aimed at data analysis and classification, developed by Rudolf Wille and his group since the 1980s at the Technical University of Darmstadt, Germany. It is a theory derived from the lattice theory of Birkhoff [26] incorporating the monotone Galois connection [20] in order to obtain formal concepts and their hierarchical relationships, levels, sub-concepts and supra-concepts; assign labels; explore objects and attributes; obtain implications and associations; calculate meanings; and obtain indexes of stability, conceptual probability, etc. FCA has developed many techniques and applications that nowadays are widely used in knowledge representation and reasoning, information retrieval, concept classifiers, knowledge processing, knowledge discovery, ontology engineering, pattern structures recognition, machine learning, science communication and in the analysis of theoretical structures and semantic networks (cf. [21,27,28,29,30,31,32]).

Two derivation operators represented by ≪′≫ have the function of assigning sets of objects to sets of attributes or sets of attributes to sets of objects. If a set of objects $B=(g_1,\dots g_n)$ has (shares) certain attributes $B^{\prime }=(m_1,m_2,\dots m_m)$, and the corresponding objects to such attributes—say $B^{″}$— are exactly B, then $〈B^{″},B^{\prime }〉$ is a formal concept from the context. In a dual way, let A be a set of attributes and $A^{\prime }$ the set of all objects that share such attributes. If after applying again the operator ′ to $A^{\prime }$ we obtain $A=A^{″}$, then $〈A^{\prime },A^{″}〉$ is a formal concept of the context. This double operation ″ is known as derivation. So, within a formal context we can find formal concepts $〈B_1,A_1〉$, $〈B_2,A_2〉$, $〈B_i,A_i〉$, within all the pairs of sets from the context $(G,M,I)$.

Formal concepts are then ordered by comparing their extensions and intentions:

$$\left(A_i\subseteq A_j\right)⟺\left(〈B_i,A_i〉\ge 〈B_j,A_j〉\right)⟺\left(B_i\supseteq B_j\right)$$

This is a partial order relation with the properties of reflexivity, antisymmetry and transitivity. We can draw this hierarchical relation between the concepts as links between nodes, assigning in this manner a level to each concept starting from the bottom node, which is always unique. For any lattice, any pair of nodes has a unique common supreme node and a unique common infimum node. Thus, the top concept contains, if any, the attributes shared by all objects, while the bottom concept contains all the attributes and generally no objects.

The analysis of the formal context and lattice visualization is performed using Concept Explorer (ConExp 1.3 available on https://conexp.sourceforge.net/, accessed on 15 March 2024), an easy-to-use Java native application that also allows exploring different dependencies that exist between attributes [33]. A table is offered to be filled as a formal context, naming objects and attributes, and an interactive lattice is built upon the calculation of concepts.

3. Structuralism

The structuralist view of philosophy of science follows a set–model–theoretic approach to analyze theories, using three units of analysis: theoretical models, theoretic elements and theoretical networks [34,35,36,37]. Theoretical models are concepts too, with intentions and extensions. Intentions are formed by n-tuples of classified attributes that comply with certain axioms of the theory, while extensions are exemplary physical models that represent them as objects. The attributes include domains, relations between domains, constraints and their interpretations under a context. Classes of models such as previous models, potential models, actual models and specialized models can be identified in a lattice configuration. In physics, previous models may be models of space-time theories, protophysics, topology, geometry, etc. (e.g., [38], on the notion of protophysics). Potential models include new physical domains such as mass or electric charge, as well as certain distinguished elements. Actual models enter the physical laws that relate the given domains, while specialized models introduce constraints, conditions and other peculiarities of a given scenario. It is in the actual models where the laws of movement and field equations are found. The fundamental constants are present in such equations, carrying nature’s impositions that we measure, observe or hypothesize; as such distinguished elements found in several theories, they can represent families of models that contain them. Law statements may occupy several of these constants; in that sense, they share regimes of applications.

4. Physical Theories Lattice

Following the idea that distinguished elements such as fundamental constants may represent classes of potential models of physical theories, and using FCA to classify theories and construct theoretical networks, we proceed to interpret Zel’manov’s cube as a concept lattice (Figure 4). In the right side representation of Figure 4, the positions of the nodes are reflected on the horizontal plane: Newtonian Mechanics (NM) is no longer at the origin but at the top node, while the Theory Of Everything (TOE) has moved to the bottom node. We have called “non-Relativistic Quantum Physics” (nRQPh) and “Relativistic Quantum Physics” (RQPh) to all classes of physical theories that require ℏ and c in their mathematical description. As such, QM and non-relativistic QFT enter a general classification scheme called nrQPh and QFT enters RQPh. This allows the inclusion of other possible theories that are not QM or QFT into the proper class.

Figure 5 incorporates Milgrom’s acceleration $a_0\approx 1.2\times {10}^{-10}$ m/s² as a fundamental constant [24], generating four additional nodes that suggest the possible existence of new application regimes. These regimes can be delimited using phase state diagrams or conceptual spaces. Over the past 40 years, dozens of alternative theories of gravitation that consider Milgrom’s acceleration $a_0$ as a relevant parameter have been proposed, both relativistic and non-relativistic. Researchers such as Milgrom [39], Bekenstein and Milgrom [40], Mendoza et al. [41], Famaey and McGaugh [42], Skordis and Złośnik [43] have tested their extended theories across all astrophysical scales with promising results and predictions.

The introduction of Milgrom’s acceleration constant $a_0$ as a fundamental constant in physical theories has been considered since it was first introduced as a modification or extension of Newtonian gravitation to understand different astrophysical phenomena instead of using non-baryonic dark matter. See, e.g., [39,40,44]. As explained by Bernal et al. [24], the constant $a_0$ has a strong fundamental nature in exactly the same form as the physical constants $G,c$ and ℏ. These authors also showed that the very well-known relation between $a_0$ and the Hubble constant $H_0$ at the present epoch and the cosmological constant $\Lambda$ given by $a_0\approx c{H}_0\approx c\sqrt{\Lambda }$ [39,45] is a coincidental relation and only occurring at the present epoch in the evolution of the Universe. This fundamental fact was also found in relativistic extensions of MOND, as described by Barrientos et al. [46,47]. In other words, to account for different astrophysical and cosmological phenomenology without postulating the existence of dark matter and/or energy by using an extended gravitational theory, the introduction of the constant $a_0$ is required to understand a new regime of gravitation at low acceleration scales $\lesssim a_0$.

We can also envision theories that would attempt to describe microscopic interactions in a low-acceleration regime or even a relativistic quantum low-acceleration regime. This would represent a new TOE, requiring the consideration of all relevant attributes. Adding another constant to the set of distinguished elements may counter some expectations of reducing the number of fundamental constants needed to achieve a TOE or reducing them to dimensionless constants, as was Einstein’s intention during his later research period. Nevertheless, exploring the consequences of such addition, including constants like e, k or ${\mu }_0$ as shown in Figure 6, is a worthwhile endeavor.

5. The International System of Units

According to the International System (SI) of units, there are seven basic units of measurement: kilogram (kg), second (s), meter (m), ampere (A), Kelvin degree (K), candela (cd) and mole (mol). These units are defined from our ability to count and from combinations and ratios of measured quantities of length (L), time (t), mass (M), electric current (I), luminous intensity (J) and temperature (T). The charge of the electron e is measured in Coulombs or Ampere per second: ($\left[e\right]=\left[C\right]=\left[\mathrm{As}\right]$); Avogadro’s number $N_A$ is measured in moles: ($\left[N_A\right]=\left[\mathrm{mo}{\mathrm{l}}^{-1}\right]$). One mole of a substance is the amount of it that contains a number specific to elemental entities. Although it is an amount of substance, we cannot express it as a mass (for example, grams) until the elementary entity is specified. Those entities can be either atoms, molecules, ions, electrons, other particles or specific groups of such particles [48]. The proportionality factor between the kinetic energy of a gas and its temperature is Boltzmann’s constant $k_B$, where ($\left[k_B\right]=\left[M{L}^2/T{t}^2\right]$). For Planck’s constant h, the units are ($\left[h\right]=[M{L}^2/t]$); for the speed of light c, ($\left[c\right]=[L/t]$); while the oscillation frequency of a Caesium atom $\Delta {\nu }_{Cs}$ in its ground state defines the second ($[\Delta {\nu }_{Cs}]=\left[{\mathrm{s}}^{-1}\right]$).

Other unit systems do not include $k_B$ or $N_A$ between their axiomatic components, since it is considered that they are derivable from the base set and do not describe properties of the universe but factors of proportionality.

The SI went through a revision during the second decade of the 21st century by the International Committee for Weights and Measures, achieving that all units, including the kilogram, will be defined from fundamental constants that can be obtained by experiment and not from exemplary physical objects. In Figure 7, two representations of the relationships between the units are shown for the SI. The figure on the left side, by Pisanty [49], and published by Lockwood and Rowell [50], presents a circular or heptagon configuration with some disconnected nodes. The figure on the right side is an alternative representation obtained with FCA, which emphasizes the hierarchical level of the constants as attributes and units as objects. The lattice was constructed using the same relations indicated in Pisanty’s diagram and simply adding a top and bottom nodes. It is presented with the same colors for easy comparison. In the lattice, the attributes are the fundamental constants and the basic units are objects. The resulting concepts are as follows: Time, Space, Mass, Temperature, Electric Intensity, Amount of substance and Luminous Intensity. At this point, we draw our attention to the absence of G in the definition of the basic units.

In this diagram, our ontology is based on phenomenology. Our units are artificial objects that we create from the constancy of our observations. We can now build our world based on those objects. In Figure 8, we built a network taking basic units as attributes and magnitudes as objects—relating derived magnitudes such as force, energy, power—and represent their hierarchical placement using two different kinds of relation: multiplication (red arrows) and division (blue arrows). Figure 8 also gives nodes a combination of colors according to their parents’ color. There is still a hierarchy and top-down directionality, but different arrows represent different relations between units. We can find out the basic units of any magnitude just by following the arrows backwards, adding the red ones and subtracting the blue ones.

In this new space, we can also locate the fundamental constants such as G, c, h, k and $a_0$. Figure 9 shows a clarified network where only attributes relevant to the fundamental constants are shown. Here, constants are derived from basic units, showing how some of them do not have a magnitude properly associated.

6. Insights and Follow Ups

This paper presents a novel hierarchical multidimensional representation of physical theories, fundamental constants and units of measurement using Formal Concept Analysis (FCA). By re-examining Zel’manov’s cube and introducing Milgrom’s critical acceleration as a new fundamental constant, we aim to provide a more nuanced understanding of theoretical structures and their interrelations.

Graphic representations of relations between concepts are surely powerful aids to a better communication of ideas, as well as aids in the scientific thinking process and search for symmetries. Here, we define a formal context and identify formal concepts, which are not random combinations of objects and attributes but intentions and extensions. Researchers in physics and metatheoretic studies as well as the public may benefit from this new algorithmic technique of FCA, which increases our confidence on a certain network of concepts: a concept lattice. This choice of methodology to represent multidimensional inter-theoretical relations, in a historical discussion that clearly has not been solved, may give a refreshing perspective to the problem of classifying theories.

We have seen that such networks suggest the possible existence of other theories that would be applied to new regimes (i.e., Mondian quantum theories). Drawing phase-space diagrams of such regimes and finding their concept spaces would be a suitable follow-up for this research.

Certainly, the construction of formal contexts depends on the selection of our premises as fundamental, as well as the reconstruction method, which can be subtractive or additive. Algebraic formulations with global variables or differential formulations also exhibit different aspects of the theoretical structure. Notations may also take a part of our representations. This is why we have to state which approach we are taking: the semantic-set-model-theoretic approach of the meta structuralist view developed in [36].

Focusing on a distinguished element allows us to reduced the kind of attribute to the minimum, leaving out domains, fields, special conditions and equations, etc. Fundamental constants as representatives of potential models embrace a huge amount of actual and specialized classes of models. We are here presenting just the most general classes of models and some or their relations that contain our to-date physics.

There is an evident decision on the selection of a proper ontology when we decide whether basic units or basic constants are to be our attributes and the abstractions (objects) that can be produced with them. It is evident that the use of artificial units allows a simple derivation method and layout of relations between magnitudes.

Future research could focus on integrating other fundamental constants and refining the lattice structure to accommodate more specialized models. Developing computational tools to automate the generation and analysis of these lattices will be crucial in enhancing their practical utility and accessibility.

In such a direction, we can further analyze in the search for possible loops, application regimes and visual aid in theoretical studies. In our way, we have valued the interplay that the relation between quantities and units have and also considered the space where fundamental constants live in relation to basic and derived quantities.