1. Introduction

In his influential book, Laws and Symmetry (Reference van Fraassen1989), Bas C. van Fraassen dedicates an entire chapter to the theme of symmetry arguments in science and metaphysics. For van Fraassen (Reference van Fraassen1989, 216) symmetry is “the primary clue to the theoretically constructed world.” He regards symmetry arguments as the most impressive application of symmetry; although such arguments give the appearance of an a priori claim, they have far-reaching consequences for our understanding of the physical world. What is the nature of these arguments? According to van Fraassen, “There are two forms of argument which reach their conclusion ‘on the basis of considerations of symmetry'. One, the symmetry argument proper, relies on a meta-principle: that structurally similar problems must receive correspondingly similar solutions. A solution must ‘respect the symmetries' of the problem. The second form, rather less important, assumes a symmetry in its subject, or assumes that an asymmetry can only come from a preceding asymmetry. Both exert a strong and immediate appeal, that may hide substantial tacit assumptions” (van Fraassen Reference van Fraassen1989, 233). Indeed, one often finds in scientific literature the expression, “for reasons of symmetry,” some variant of it, or an implicit appeal to this expression. For example, David Hilbert (1862–1943), one of the leading mathematicians at the turn of the last century and a driving force in pure, as well as applied, mathematics in Göttingen, published a paper in 1904 entitled, “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen.” In it we find a certain mathematical argument whose frame is the following: “Due to the symmetry [wegen der Symmetrie] of the expression on the left-hand side, it follows by interchanging [Vertauschung] x with y.”Footnote ¹ We need not dwell on the mathematical content of the argument; rather, we draw attention to the connection Hilbert makes between symmetry (Symmetrie) and interchange (Vertauschung): due to the symmetry of a certain expression, x may be interchanged with y without changing the content of the equation. For examples in physics, we may refer in the first place to the case which Ernst Mach (1838–1916) examines in his influential book, Die Mechanik (Reference Mach1883). Mach begins the discussion of Archimedes’ equilibrium postulate by analyzing an ideal situation in which the bar of the balance, whose weight may be neglected, rests on a fulcrum in such a way that at equal distances from the fulcrum two equal weights are hung. Mach then remarks: “That the two weights, thus circumstanced, are in equilibrium, is the assumption from which Archimedes starts. We might suppose that this was self-evident entirely apart from any experience; … that in view of the symmetry of the entire arrangement there is no reason why rotation should occur in the one direction rather than in the other.”Footnote ² Mach suggests the possibility of justifying the postulate by appealing to the symmetry of the arrangement; we may paraphrase Mach's argument in the following way: “for reasons of symmetry no rotation should occur.” To consider a more complex example, we turn to Henri Poincaré (1854–1912), who, in 1899, drew an explicit distinction between symmetry and isotropy in the context of discussing the Zeeman effect, as interpreted according to Lorentz's theory. He analyzes isotropy in plane waves and invokes the term symmetrical to designate mirror symmetry. Poincaré argues: “The medium is not just isotropic, it is symmetrical; therefore, our equations ought not to change when one replaces our system of axes by a symmetrical system (the plane of symmetry being the xz-plane, for example).”Footnote ³ Symmetry allows then for a physical argument. Poincaré constructs the argument by appealing to the consequential term, “therefore” (donc); this encodes an implicit appeal to “for reasons of symmetry.” Poincaré determines that the medium is symmetrical and then proceeds to infer consequences for the equations. He refers to the relation of pairs of coordinate axes, or of positions with respect to a coordinate system, under whose transformation the properties of the system are preserved, and this physical feature is not the same as isotropy (for further discussion, see note 15 below). Another instructive example that links geometry with physics in the spirit of van Fraassen's claim that “a solution must ‘respect the symmetries’ of the problem” may be found in the work of one of the pioneers of the application of symmetry in modern physics, Woldemar Voigt (1850–1919). In his book, Die fundamentalen physikalischen Eigenschafter der Krystalle in elementarer Darstellung (Reference Voigt1898), a treatise on the physics of crystals, Voigt offers definitions of geometrical and physical symmetry; in the first chapter he assesses the relation between these two kinds of symmetry. He first sets up the geometrical meaning of symmetry with respect to crystals by defining Symmetriecentrum, Symmetrieebene, Symmetrieaxe and, finally, Spiegeldrehaxe (Voigt Reference Voigt1898, 7).Footnote ⁴ After explaining the geometrical meaning of these terms, he poses the question: “From what features can we infer the laws of the physical symmetry of crystals? [Aus welchen Merkmalen können wir die Gesetze der physikalischen Symmetrie der Krystalle erschliessen?]” He thus makes it clear that he expects some relation to hold between the geometry of the crystal and its physical properties. In particular, “in the great majority of cases the symmetry of the crystalline form can be directly equated with that of the [crystal's] physical behavior. Hence, the directions that are of equal value crystallographically are also of equal value physically.”Footnote ⁵ Here we have an unambiguous connection between geometrical symmetry and physical symmetry. That is, the symmetry of the physics must “respect” the symmetry that holds in the geometry. Finally, for an explicit usage of the expression, “for reason of symmetry,” we turn to August Föppl (1854–1924), who developed in his book, An Introduction to Maxwell's Theory (Reference Föppl1894), a brilliant synthesis of the works of Oliver Heaviside (1850–1925) and Heinrich Hertz (1857–1894). It is now well accepted that Föppl's Introduction was instrumental in making the young Einstein aware of the outstanding issues in electrodynamics, notably the issue of relative and absolute motion (see, e.g., Norton Reference Norton2004, 54; Hon and Goldstein Reference Hon and Goldstein2005a, 521). Following Heaviside, Föppl argues that D, the dielectric displacement, and E, the electrostatic force, are arranged in space in the same way (gleich gerichtet). He then justifies this claim by adding the argumentative expression: “for reasons of symmetry [aus Symmetriegründen].”Footnote ⁶

Van Fraassen has singled out an important aspect of the application of symmetry in science. He has distinguished between symmetry as a property and symmetry as an argument. For van Fraassen (Reference van Fraassen1989, 258) the essence of symmetry arguments in all their varieties is encapsulated in “the great Symmetry Requirement,” which is based on the crucial triad of symmetry, transformation, and invariance. It is the principle which underlies the methodology that generates symmetry arguments by establishing the maxim that “problems which are essentially the same must have essentially the same solution” (van Fraassen Reference van Fraassen1989, 259, italics in the original). The stated goal of van Fraassen's analysis is to support the claim that there are “intimate connections between symmetry and generality,” where laws are the classical expressions of generality in science. “To be truly general, a statement must be covariant, it must have this logical status: it is either true in all frames of reference or true in none. Equivalently: its truth value must remain invariant under all admissible transformations” (van Fraassen Reference van Fraassen1989, 287, italics in the original). To be sure, “admissible” depends on the theoretical context; hence, what is general to a classical physicist may not be so for a contemporary physicist. All the same, in van Fraassen's view “the symmetries of time, space, and motion determine the structure of modern science to a surprisingly large extent.” He then concludes that symmetry rigorously shapes theory. To use van Fraassen's figurative language (Reference van Fraassen1989, 262), “symmetry takes the theoretician's hand and runs away with him, at great speed and very far, propelled solely by what seemed like his most elementary, even trivial, assumptions.”

We take our cue from van Fraassen's innovative study of symmetry and expand the perspective he has offered. As indicated, van Fraassen focused his discussion on two types of argument. The first is characterized by the demand that the solution must “respect the symmetries” of the problem, while the second expresses conservation: an asymmetry can only come from a preceding asymmetry. We build on van Fraassen's important distinction between symmetry as a property and symmetry as a form of argument and begin by turning our attention to the general structure of any argument. We will then examine rules of inference in an argument and identify features that may be described as symmetrical. Moreover, we will note the impact of this type of reasoning on arguments in the papers of Albert Einstein (1879–1955) in Reference Einstein1905. We hold that the application of symmetry arguments in physics is relatively new, the product of several ingenious studies in the nineteenth century that served as background to Einstein's revolutionary work in that year. In fact, in our view, symmetry as a scientific concept is a nineteenth-century invention (see Hon and Goldstein Reference Hon and Goldstein2005a, Reference Hon and Goldstein2005b, Reference Hon and Goldstein2005c).

2. Symmetry as a Property

There is a general agreement that the set of lectures on symmetry which Hermann Weyl (1885–1955) delivered at Princeton University in the early 1950s, published in Reference Weyl1952, constitutes a milestone in the analysis and application of the concept of symmetry as a property. In this book Weyl does not address symmetry as argument.Footnote ⁷ This profound treatment of the concept came after a century and a half of growing interest in, and usage of, the modern concept of symmetry, and it consists of two principal moves: methodological and metaphysical.

Weyl begins with a delineation of two meanings of symmetry. In one sense symmetry means “something like well-proportioned, well-balanced.” Symmetry then denotes “concordance of several parts by which they integrate into a whole.” This is how beauty, in Weyl's view, is bound up with symmetry. In this sense harmony may be considered a synonym of symmetry (Weyl Reference Weyl1952, 3). However, for Weyl symmetry in another sense is a precise geometrical notion conveying an operation that carries a body of spatial configuration into itself. “The image of the balance provides a natural link to the second sense in which the word symmetry is used in modern times: bilateral symmetry, the symmetry of left and right, which is so conspicuous in the structure of the higher animals, especially the human body” (Weyl Reference Weyl1952, 4, italics in the original). In his opening remarks, Weyl does not adhere to a strict distinction between the two meanings. “Symmetry,” he claims, “as wide or as narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection” (Weyl Reference Weyl1952, 5). As the book progresses, Weyl demonstrates persuasively that the concept of symmetry is ultimately an application of group theory. He begins with the notion of congruence, for it captures a structural feature of space: “any … congruent transformation … is a similarity or an automorphism. … It is evident,” Weyl continues (Reference Weyl1952, 43), “that the congruent transformations form a group, a subgroup of the group of automorphisms.” Congruence may, for example, be the result of an operation of rotation or of translation; in each such operation the point p of a certain space V which is occupied by some rigid body, is mapped onto a point p′ in space V′ such that the rigid body remains invariant under this operation (see Weyl Reference Weyl1952, 43). Weyl then poses a question, “What has all this to do with symmetry?” to which he replies: “[Symmetry] provides the adequate mathematical language to define [a congruent transformation]. Given a spatial configuration F, those automorphisms of space which leave F unchanged form a group Γ, and this group describes exactly the symmetry possessed by F. … The symmetry of any figure in space is described by a subgroup of that group” (Reference Weyl1952, 45, italics in the original).

Thus, on methodological grounds, Weyl proceeds from considerations of a vague concept to one that is precise, gradually reaching greater generality, and guided—as he puts it—“more by mathematical construction and abstraction than by the mirage of philosophy” (Weyl Reference Weyl1952, 6).

In this process of generalization, Weyl intentionally discards the aesthetic appeal of the concept and concentrates on precision, ultimately aiming at determining its unifying power. This approach coheres with the second move. Weyl's metaphysical claim is that mathematics underlies the concept of symmetry. He thinks that mathematics is the common origin of symmetry in nature and in art: “the mathematical laws governing nature are the origin of symmetry in nature, the intuitive realization of the idea in the creative artist's mind its origin in art” (Weyl Reference Weyl1952, 6–8). Weyl's book, Symmetry, is the crystallization of this conception, whose unifying power originates in its mathematical foundation.

According to Weyl, there are three distinct domains to which symmetry may be applied: (1) material art(ifacts), (2) nature, and (3) physical theories. They are seen from a single point of view which the modern concept of symmetry offers. This overarching analysis has been very influential in pointing to a link between the arts, nature, and the sciences which is ultimately grounded in mathematics: “From art, from biology, from crystallography and physics I finally turn to mathematics, which I must include all the more because the essential concepts, especially that of a group, were first developed from their applications in mathematics” (Reference Weyl1952, 135, italics in the original). Thus, the modern concept of symmetry gains its strength not only from the fruitful application of a mathematical property in physics but also from its ability to link a variety of domains, revealing common patterns in them. This observation, which Weyl stressed, made symmetry the powerful concept that it has become in domains well beyond mathematics and physics.

Symmetry is a property of objects or elements that form a group and, to form this group, there must be a transformation with an invariance that meets four fundamental criteria: closure, associativity, identity, and inversion (Klein Reference Klein[1926] 1979, 315–316; Klein Reference Klein1926, 335; Weyl Reference Weyl1952, 41–43). The invariance (what stays the same under the transformation) is the symmetrical property.Footnote ⁸ For Weyl, then, symmetry as a property can be always recast in group theoretic terms.

In 1996, about half a century after Weyl's essay, David J. Gross, one of the chief architects of the fundamental theory of the strong force—quantum chromodynamics—a Nobel laureate in physics (2004) and a notable advocate of string theory, reaffirmed this definition of Weyl. Gross reviewed the fundamental role of symmetry in physics and explicitly identified symmetry with invariance, arguing that symmetry provides structure and coherence to the laws of nature.

He goes on to emphasize the role of symmetry in recent physics: “With the development of quantum mechanics in the 1920s symmetry principles came to play an even more fundamental role. In the latter half of the 20th century symmetry has been the most dominant concept in the exploration and formulation of the fundamental laws of physics. Today it serves as a guiding principle in the search for further unification and progress” (Gross Reference Gross1996, 14256). Gross thus reaffirms the position of Weyl in the context of modern physics in the closing years of the twentieth century.Footnote ⁹

3. Symmetry Arguments

While acknowledging the profound contribution of Weyl and Gross, we suggest, as van Fraassen did, that the application of symmetry to arguments needs further elaboration. The expression “symmetry argument” is a relatively recent coinage, and it is a useful analyst's tool. Like any argument, a symmetry argument has premises and rules of inference which lead to a conclusion. By symmetry argument we mean that symmetry, as a property, enters at least one of the three components that comprise an argument.

In the “Introduction” to their edited book, Symmetries in Physics: Philosophical Reflections (Reference Brading and Castellani2003b), Katherine Brading and Elena Castellani discuss symmetry arguments and identify the cases of Buridan's ass, Archimedes’ equilibrium law of the balance, and Anaximander's argument for the immobility of the Earth as typical examples of such arguments. Brading and Castellani then claim that these arguments are essentially applications of the Leibnizean Principle of Sufficient Reason (PSR): “if there is no sufficient reason for one thing to happen instead of another, the principle says that nothing happens (the initial situation does not change).” They observe that in each of the above cases the principle “is applied on the grounds that the initial situation has a given symmetry: in the first two cases, bilateral symmetry; in the third, rotational symmetry. The symmetry of the initial situation implies the complete equivalence between the existing alternatives (the left bundle of hay with respect to the right one, and so on). If the alternatives are completely equivalent, then there is no sufficient reason for choosing between them and the initial situation remains unchanged” (Brading and Castellani Reference Brading and Castellani2003a, 9). The idea that Brading and Castellani propound—which we hasten to add is widespread—is that “arguments of the above kind—that is, arguments leading to definite conclusions on the basis of an initial symmetry of the situation plus PSR—have been used in science since antiquity (as Anaximander's argument testifies). The form they most frequently take is the following: a situation with a certain symmetry evolves in such a way that, in the absence of an asymmetric cause, the initial symmetry is preserved. In other words, a breaking of the initial symmetry cannot happen without a reason, or an asymmetry cannot originate spontaneously” (Brading and Castellani Reference Brading and Castellani2003a, 9–10, italics in the original). A brief examination of the original formulation of PSR by Gottfried Wilhelm Leibniz (1646–1716) will suffice to show that there is a misleading step in linking symmetry to PSR.

Central to Leibniz's mode of reasoning, as developed in his Theodicy (Reference Leibniz[1710] 1720) and Monadology (Reference Leibniz[1714/1840] 1954), is the idea that there is always, in principle, an analytical path for validating any proposition that is indeed true, be it necessary or contingent. For necessarily true propositions the principle of contradiction suffices, but in the case of contingently true propositions sufficient reason is provided via the perfection of God, that is, sufficient reason of all contingent truth lies in God's choice of the best. The motivation is clear: to have an explicit rationale for the truth to be so rather than otherwise (see Rescher Reference Rescher1991, 118). In Leibniz's formulation in his Monadology, the Principle of Sufficient Reason means that “no fact can be real or actual, and no proposition true, without there being a sufficient reason for its being so and not otherwise, although most often these reasons just cannot be known by us.”Footnote ¹⁰ This is the principle of the determinant reason, as Leibniz had originally called it in his Theodicy. The principle states that “nothing ever comes to pass without there being a cause or at least a reason determining it, that is, something to give an a priori reason why it is existent rather than non-existent, and in this wise rather than in any other.”Footnote ¹¹

For our analysis it is important to note the essential role of an agent, be it God or a rational mind, who brings the PSR to completion; it assesses the situation in order to choose a course of action to attain a value, for example, goodness, beauty. This evaluative assessment has no algorithm; it requires a judgment and, hence, an agent. Linking symmetry considerations with an agent poses a problem. We ask, Does symmetry involve an agent? We say, No! We claim that symmetry is a unique scientific concept in that it has epistemological and ontological aspects. Clearly, when symmetry expresses the latter feature, that is, when it is related to mathematical entities (both geometrical and algebraic) or to physical objects (both natural and man-made)—in a word, when it expresses a property—no agent whatsoever is involved.Footnote ¹² In the former case, where symmetry expresses an argument, one may suspect that an agent is necessary. But here too this is not the case. What seems to be the action of an agent is in fact an algorithm of an operation that transforms the system in question in a specific way while leaving some elements invariant. No agent, benevolent or intelligent, is required to carry out this operation. But above all, the modern scientific concept of symmetry has nothing to do with value judgments associated with beauty and the good. In its group theoretic sense, symmetry is a technical, scientific term that does not express a value. In sum, we think it misleading to associate symmetry argument with PSR. We therefore look anew at forms of argument that may be said to be symmetrical in order to determine their nature.

The application of symmetry arguments in physics had not been proposed explicitly by anyone prior to 1894, the year Pierre Curie (1859–1906) put forward the idea of applying group theory in this domain (Curie Reference Curie[1894/1908] 1984).Footnote ¹³ According to Castellani (Reference Castellani2003, 323), Curie was persuaded by his studies of the phenomena of pyro- and piezo-electricity of crystals that the symmetry of the phenomena and that of the physical medium are mutually dependent. By applying a group theoretic analysis, he arrived at the following general principle: “When certain causes produce certain effects, the symmetry elements of the causes must be found in their effects. … In practice, the converse … is not true, i.e., the effects can be more symmetric than their causes” (Curie Reference Curie[1894/1908] 1984, 127, cf. 119; Rosen and Copié Reference Rosen and Copié1982, 20, cf. 17). Although Curie did not use the term “symmetry argument,” the principle he formulated underpins an argument that may be described as “considerations of symmetry.” In this argumentative scheme the physical relation between the cause and the effect—based as it is on symmetry considerations—can be recast in terms of premises and a conclusion which is drawn by applying specific rules of inference. Inspired by Curie's analysis, we discern two different moves, a forward move from the cause (“premises”) to the effect (“conclusion”), and the reverse. Based on the distinction between these two moves, we identify two different categories of symmetry argument.

4. Case Studies

We turn first to the axiomatic category. In his dissertation, “A New Determination of Molecular Dimensions,” Einstein presents us with such a case. Einstein explicitly refers in this work to his source, a work by Gustav Kirchhoff (1824–1887). Thus we have reliable evidence for a text that Einstein consulted (Einstein Reference Einstein1905a, 189).Footnote ¹⁴ Although Einstein does not cite Curie (and probably had not read the relevant paper), he uses the expressions, wegen der Symmetrie and aus Symmetriegründen, thereby explicitly appealing to a symmetry argument (Reference Einstein1905a, 188, 191, 196).

Einstein considers the motion of a solvent in the vicinity of a suspended solid body which plays the role of the dissolved molecule. At issue are the hydrodynamic equations of the motion of the solvent in these circumstances. As is customary in mathematical physics, several presuppositions are introduced to simplify the calculations: Einstein assumes that the solid body is spherical in shape and that the liquid is homogenous so that its own molecular structure need not be taken into consideration (for a complete list of the presuppositions, see Pais Reference Pais1982, 90). The next standard step in hydrodynamic analysis is to treat the motion of the liquid as a superposition of three different kinds of motion: (1) a parallel displacement without a change in the relative positions of the liquid particles; (2) a rotation without a change in the relative positions of the liquid particles, and (3) a dilatational motion in three mutually perpendicular directions (Kirchhoff Reference Kirchhoff1883, 123–124, 369–374; Einstein Reference Einstein1905a, 187; Beck Reference Beck1989, 106).

With reference to the three fundamental kinds of motion discerned in hydrodynamics Einstein says:

Due to the symmetry [wegen der Symmetrie] of the motion of the liquid, it is clear that the sphere can perform neither a translation nor a rotation during the motion considered, and we obtain the boundary conditions

(Beck Reference Beck1989, 107, boldface added; Einstein Reference Einstein1905a, 188, boldface added)

It should be clarified that “the motion considered” is the third one, dilatation, the only motion that gets modified by the presence of the rigid spherical body. Note further that u, v, and w are the three functions of the velocity components of the dilatational motion of the liquid; P is the radius of the sphere and ρ expresses the principal axes of dilatation, being equal to the square root of the sum of the squares of the three components of the coordinate system whose axes are parallel to the principal axes of the dilatation.

Symmetry here expresses the feature that there is no privileged direction: the three equations of motion are exactly analogous for each of the three orthogonal directions. This feature is usually called isotropy. According to William Thomson (1824–1907) and Peter G. Tait (1831–1901), isotropy is defined by two elements: resistance to compression and resistance to distortion. Thomson and Tait identify the quality of homogeneity as a prerequisite for isotropy: “A body is called homogeneous when any two equal, similar parts of it, with corresponding lines parallel and turned towards the same parts, are indistinguishable from one another by any difference in quality.” They then emphasize the physical nature of this quality: “The substance of a homogeneous solid is called isotropic when a spherical portion of it, tested by any physical agency, exhibits no difference in quality however it is turned.”Footnote ¹⁵ Thomson and Tait consider indifference (which they also call “indistinguishability”) as a way to depict this feature of homogeneity. It is significant that physical agencies are involved in determining this feature; in other words, according to this definition, isotropy is physical rather than geometrical. This is, then, the feature that exhibits equal physical properties or actions in all directions, for example, refraction of light, elasticity, and conduction of heat or of electricity. To return to Einstein's dissertation: given the conditions of the problem, that is, the presuppositions presented at the outset of the work and the geometry of the situation which Einstein laid down, the isotropic feature restricts the kind of motions that the rigid spherical body can undergo. The limitation thus imposed allows Einstein to obtain the boundary conditions he needed for proceeding with the calculation of the dilatational motion of the liquid.

Einstein follows very closely the text of Kirchhoff, who explicitly applied symmetry to hydrodynamic calculations, although not in the lecture which Einstein cites.Footnote ¹⁶ In his essay of Reference Kirchhoff1869 on the motion of a rotational body immersed in a fluid whose rotational axis is not fixed in one plane, Kirchhoff begins with a very complex system of equations of motion. He stipulates that the liquid is without friction, incompressible, and homogenous, and it stands to reason that he regards the fluid as isotropic (Kirchhoff Reference Kirchhoff[1869] 1882, 376). Having set the problem, he proceeds to apply symmetry to simplify the equations. The forces, according to Kirchhoff's argument, vanish due to the symmetrical distribution of the mass of the rotating body with respect to the axis of rotation (Kirchhoff Reference Kirchhoff[1869] 1882, 377). In another step of the calculation he sought to reduce the number of constants of the equations of motion when the body in question exhibits symmetry properties, that is, geometrical symmetries concerning physical properties, namely, the distribution of mass (Kirchhoff Reference Kirchhoff[1869] 1882, 388). And then he makes another crucial move when he notes that if the surface of the body is symmetrical in relation to some plane, say, the xz-plane, that is, when some point on the body whose coordinates are x, y, z, corresponds to another point on the body whose coordinates are x, −y, z, then the distribution of mass may be calculated on the basis of this presupposition with the result that the number of equations of motion is reduced substantially (Kirchhoff Reference Kirchhoff[1869] 1882, 389; see also 390–391). Kirchhoff concludes the analysis with a general claim that the resultant equation is not restricted to bodies of rotation; rather, it applies to all bodies as long as they are symmetrical in relation to two or more pairs of points located in two orthogonal planes with respect to the x-axis (Kirchhoff Reference Kirchhoff[1869] 1882, 392). In other words, while the mass-density is presumed to be the same, the geometrical requirements for the shape of the body have been relaxed. Joseph Larmor (1857–1942), Lucasian Professor of Mathematics at Cambridge from 1903 to 1932, singled out this result of Kirchhoff as remarkable (Larmor Reference Larmor[1884] 1929, 77).

Kirchhoff applies different notions of symmetry to the fluid as the medium and to the bodies which are immersed in it. The fluid is isotropic, while the symmetry which the bodies exhibit determines the way the masses are distributed in them with respect to the axis of rotation, and this physical feature fixes the number of equations required for the description of the motion of the bodies. However, in both cases, Kirchhoff's usage of symmetry is essentially geometrical. To be sure, the claim is physical: the forces, Kirchhoff argues, cancel each other. This is one of the ways Kirchhoff's analysis of Reference Kirchhoff1869 succeeds in reducing the number of equations. He elaborated this result in “Lecture 19” of his Mechanik—the same book to which Einstein refers—where in section 3 he demonstrates the symmetrical property of a solid of revolution (Kirchhoff Reference Kirchhoff1883, 243). This result of Kirchhoff indicates that symmetry as a geometrical property was assumed to have physical consequences for the motion of a body in a fluid, thus establishing a link between the geometry of the body and its physical properties, captured by the term isotropy.

The usage of this physical sense of symmetry became fairly common in physics circa 1900; hence, no innovation is involved in appealing to it. Thus Einstein's application of this kind of symmetry may not have originated in his reading of Kirchhoff. We note that Einstein appeals to symmetry in an argument which follows Kirchhoff closely despite the fact that Kirchhoff did not apply symmetry to these basic hydrodynamic equations. Furthermore, we suggest that Einstein made the assumption of isotropy explicit—calling it symmetry—a move which Kirchhoff had not taken. In other words, Einstein extended the meaning of symmetry to include isotropy, thereby articulating the presupposition that the equations of motions for the three orthogonal components of the motion of the solvent are the same.

In his analysis, Kirchhoff invariably refers to symmetry as a property of the initial state of the physical problem: what the premises of the argument convey. After he establishes the symmetrical properties, he draws conclusions that are implicitly based on a rule of inference that transforms the initial description of the physical state into a simplified form, amenable to calculation, which retains the physics of the problem. By contrast, Einstein renders such arguments explicit. He sets up the conditions of symmetry and then states the nature of the argument: “for reasons of symmetry.”

In physics, as in algebra, one is entitled to eliminate elements that cancel each other. As we have seen, Kirchhoff simplifies hydrodynamic equations when there is a symmetrical distribution of masses—in which case certain terms of the hydrodynamic equations vanish (verschwinden). He means that the sum of the quantities is zero. This makes no sense geometrically, where the elements remain in place. What then warrants the cancellation? It is the symmetry argument. The presentation of the physical arrangement in algebraic terms facilitates this summation to zero. To recapitulate: Kirchhoff uses the term symmetry to describe the physical conditions of the arrangement; in other words, he is setting the initial conditions of the physical arrangement in such a way that they can be transformed by a rule of inference (a certain algorithm) into a different set of conditions, much simpler than in the initial state which, despite the formal change, retain the same physical properties. This is an axiomatic symmetry argument.

In his paper of 1894, Curie remarked that physicists had been speaking “prose” all the time: “Physicists often utilize symmetry conditions, but generally neglect to define the symmetry in a phenomenon, for sufficiently often these symmetry conditions are simple and quite obvious a priori [simples et presque évidentes a priori]” (Curie Reference Curie[1894/1908] 1984, 118–119; Rosen and Copié Reference Rosen and Copié1982, 17). Indeed, Kirchhoff is not as explicit as Curie about the application of a symmetry argument; moreover, unlike Curie, Kirchhoff does not treat the general case. For Kirchhoff there is after all no “metaphysical principle” embedded in symmetry considerations. Although Weyl does not cite Curie in his study of symmetry, he seems to echo this view in his claim, “as far as I see, all a priori statements in physics have their origin in symmetry” (Weyl Reference Weyl1952, 126).

Curie explicitly pointed to symmetry as a powerful tool in physical arguments which he turned into a metaphysical principle. He then demonstrated its application in the domain of electromagnetism. In the opening section of his famous paper of Reference Curie1894, Curie called attention to the fact that “in the study of electricity, for example, one should state almost immediately the characteristic symmetry of the electric field and of the magnetic field. One can then use these notions to simplify proofs” (Curie Reference Curie[1894/1908] 1984, 119; Rosen and Copié Reference Rosen and Copié1982, 17). The characteristic symmetry of these two fields had been discussed in the physical literature of the time, beginning with Henry Rowland (1848–1901), followed by Hertz with several other illustrious protagonists in the background (for details, see Hon and Goldstein Reference Hon and Goldstein2006). But Curie departs from this tradition in turning from symmetry as a property of the state of the fields to symmetry as an argument whose rules of inference facilitate proofs. Curie was then conscious of his move when he began his paper, “Sur la symétrie dans les phénomènes physiques” (Reference Curie[1894/1908] 1984), by noting that symmetry arguments were not invoked in physics before 1894 and that he himself was importing the idea of symmetry from crystallography—where it was well developed at the time—into physics: “I think that there is an interest in introducing into the study of physical phenomena the symmetry arguments [considérations] familiar to crystallographers.”Footnote ¹⁷

We now turn to the heuristic category. Murray Gell-Mann's prediction in 1962 of the existence of the elementary particle, omega minus (Ω⁻), and its subsequent discovery, illustrates a clear case of the heuristic usage of a symmetry argument. Facing an increasing number of strongly interacting fundamental particles—including stable and semi-stable particles as well as the very short lived ones (resonancesFootnote ¹⁸)—Gell-Mann searched for a scheme of classification that would introduce order into the ever-growing “collection.” For this purpose, he established a new fundamental characteristic of a multiplet which he called “hypercharge.”Footnote ¹⁹ He then proposed a new rule: elementary particles can be transformed into others by the strong and the electromagnetic interactions only if the total hypercharge is conserved. This rule is reminiscent of the law of conservation of electric charge. Yuval Ne'eman (Reference Ne’eman, Gell-Mann and Ne’eman1964, 3) aptly described the situation: “This was symmetry in yet another aspect of its uses: the determination of selection rules for reactions. … With the new crop of particles, one had to postulate a symmetry first and then check it, because there was no easy way to observe the over-all forces ratio.” Ne'eman candidly revealed the methodology he and Gell-Mann had applied. They were facing experimental results that required theoretical underpinning. In a sense, they were looking at the solution, the “conclusion,” and they were trying to reconstruct the problem: the unknown premises of the problem that under certain constraints result in the known “collection” of particles. Gell-Mann made a theoretical proposal on the basis of growing experimental data that lacked order; symmetry considerations allowed him to predict empty places in the arrangement of the baryon multiplets that indicated the existence of two additional baryons, one of which was soon discovered and the other was found six years later (cf. Waller Reference Waller1972).Footnote ²⁰

Gell-Mann (Reference Gell-Mann1962a, 1067) states: “we shall suggest a particular symmetry group as the one most likely to underlie the structure of the system of baryons and mesons.” Gell-Mann applied to the physics of elementary particles a certain mathematical model in group theoretic form: SU(3). At the same time he pointed out that some particles were missing which were needed to complete the group; the identification of missing elements made him conjecture that these missing particles exist (Gell-Mann Reference Gell-Mann1962b). After they were experimentally confirmed, he was awarded the Nobel Prize in 1969 for this prediction. Gell-Mann's argument depends on a rule of inference that invokes symmetry: starting from the model (i.e., the group, SU(3), taken as a constraint on the premise), he sought its conclusion (the existence of these particles).

5. Conclusion

Einstein's usage of symmetry arguments, marked as wegen der Symmetrie and aus Symmetriegründen, has not been noted in the literature. His use of these expressions is an indication that he clearly discerned in symmetry arguments the distinction between the properties of the premises and the rule of inference that facilitates the drawing of the conclusion. Einstein works in the tradition of Kirchhoff but, unlike Kirchhoff, he makes it clear that in symmetry arguments (of the axiomatic kind) there has to be a rule of inference which transforms the form of the premises while retaining the physical properties of the initial state.Footnote ²¹ Such an inference also facilitates the appeal to an argument in the reverse direction, from the conclusion to the premises, thereby endowing symmetry arguments with great heuristic power.

The metaphysical principle of Curie, by which the symmetry of the premises in an argument is linked to that of the conclusion, underlies the rule of inference in symmetry arguments. By connecting the symmetry of the cause with that of the effect, this rule demonstrates that there is a difference between symmetry as a property and symmetry as an argument. The seminal work of Weyl from half a century ago addresses symmetry as a property in a fundamental way, but in that work he neither considers the role of symmetry in rules of inference nor does he refer to Curie's paper of Reference Curie1894 which laid the foundations for symmetry as an argumentative procedure in science. There seems to be something unique in this concept that makes it such an effective tool in modern science and especially in contemporary physics. It may well be the case that this unique feature of symmetry has to do with the fact that it is both an expression of a property and a powerful form of argument as well.