Unpacking "For Reasons of Symmetry": Two Categories of Symmetry Arguments

Philosophy of Science, 73 (October 2006): 419–439. 0031-8248/2006/7304-0004$10.00Copyright 2006 by the Philosophy of Science Association. All rights reserved.

419

Unpacking “For Reasons ofSymmetry”: Two Categories of

Symmetry Arguments*

Giora Hon and Bernard R. Goldstein§‡

Hermann Weyl succeeded in presenting a consistent overarching analysis that accountsfor symmetry in (1) material artifacts, (2) natural phenomena, and (3) physical theories.Weyl showed that group theory is the underlying mathematical structure for symmetryin all three domains. But in this study Weyl did not include appeals to symmetryarguments which, for example, Einstein expressed as “for reasons of symmetry” (wegender Symmetrie; aus Symmetriegrunden). An argument typically takes the form of a setof premises and rules of inference that lead to a conclusion. Symmetry may enter anargument both in the premises and the rules of inference, and the resulting conclusionmay also exhibit symmetrical properties. Taking our cue from Pierre Curie, we distin-guish two categories of symmetry arguments, axiomatic and heuristic; they will bedefined and then illustrated by historical cases.

1. Introduction. In his influential book, Laws and Symmetry (1989), BasC. van Fraassen dedicates an entire chapter to the theme of symmetryarguments in science and metaphysics. For van Fraassen (1989, 216) sym-metry is “the primary clue to the theoretically constructed world.” Heregards symmetry arguments as the most impressive application of sym-metry; although such arguments give the appearance of an a priori claim,they have far-reaching consequences for our understanding of the physicalworld. What is the nature of these arguments? According to van Fraassen,“There are two forms of argument which reach their conclusion ‘on thebasis of considerations of symmetry’. One, the symmetry argument proper,

*Received May 2006; revised January 2007.

§To contact the authors, please write to: Giora Hon, Department of Philosophy, Uni-versity of Haifa, Haifa 31905, Israel; e-mail: [email protected]; Bernard R.Goldstein, Faculty of Arts and Sciences, University of Pittsburgh, 2604 Cathedral ofLearning, Pittsburgh, PA 15260, USA; e-mail: [email protected].

‡We are grateful to Michael Dickson, John Norton, and an anonymous referee forhelpful suggestions.

420 GIORA HON AND BERNARD R. GOLDSTEIN

relies on a meta-principle: that structurally similar problems must receivecorrespondingly similar solutions. A solution must ‘respect the symme-tries’ of the problem. The second form, rather less important, assumes asymmetry in its subject, or assumes that an asymmetry can only comefrom a preceding asymmetry. Both exert a strong and immediate appeal,that may hide substantial tacit assumptions” (van Fraassen 1989, 233).Indeed, one often finds in scientific literature the expression, “for reasonsof symmetry,” some variant of it, or an implicit appeal to this expression.For example, David Hilbert (1862–1943), one of the leading mathema-ticians at the turn of the last century and a driving force in pure, as wellas applied, mathematics in Gottingen, published a paper in 1904 entitled,“Grundzuge einer allgemeinen Theorie der linearen Integralgleichungen.”In it we find a certain mathematical argument whose frame is the follow-ing: “Due to the symmetry [wegen der Symmetrie] of the expression onthe left-hand side, it follows by interchanging [Vertauschung] x with y.”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 acertain expression, x may be interchanged with y without changing thecontent of the equation. For examples in physics, we may refer in thefirst place to the case which Ernst Mach (1838–1916) examines in hisinfluential book, Die Mechanik (1883). Mach begins the discussion ofArchimedes’ equilibrium postulate by analyzing an ideal situation in whichthe bar of the balance, whose weight may be neglected, rests on a fulcrumin such a way that at equal distances from the fulcrum two equal weightsare hung. Mach then remarks: “That the two weights, thus circumstanced,are in equilibrium, is the assumption from which Archimedes starts. Wemight suppose that this was self-evident entirely apart from any experi-ence; . . . that in view of the symmetry of the entire arrangement thereis no reason why rotation should occur in the one direction rather thanin the other.”2 Mach suggests the possibility of justifying the postulate byappealing to the symmetry of the arrangement; we may paraphrase Mach’sargument in the following way: “for reasons of symmetry no rotationshould occur.” To consider a more complex example, we turn to HenriPoincare (1854–1912), who, in 1899, drew an explicit distinction between

1. Hilbert 1904, 55: “Hieraus folgt wegen der Symmetrie des Ausdrucks linker Handbei Vertauschung von x mit y.” Cf. Vahlen 1899, 450–462.

2. Mach [1883/1893/1960] 1974, 14–15; Mach [1883/1912] 1988, 34–35: “Daß diesejetzt im Gleichgewicht sind, ist eine Voraussetzung, von der Archimedes ausgeht. Mankonnte meinen, dies sei . . . , abgesehen von aller Erfahrung, selbst verstandlich, essei bei der Symmetrie der ganzen Vorrichtung kein Grund, warum die Drehung eherin dem einen als in dem andern Sinne eintreten sollte.”

UNPACKING “FOR REASONS OF SYMMETRY” 421

symmetry and isotropy in the context of discussing the Zeeman effect, asinterpreted according to Lorentz’s theory. He analyzes isotropy in planewaves and invokes the term symmetrical to designate mirror symmetry.Poincare argues: “The medium is not just isotropic, it is symmetrical;therefore, our equations ought not to change when one replaces our systemof axes by a symmetrical system (the plane of symmetry being the xz-plane, for example).”3 Symmetry allows then for a physical argument.Poincare constructs the argument by appealing to the consequential term,“therefore” (donc); this encodes an implicit appeal to “for reasons ofsymmetry.” Poincare determines that the medium is symmetrical and thenproceeds to infer consequences for the equations. He refers to the relationof pairs of coordinate axes, or of positions with respect to a coordinatesystem, under whose transformation the properties of the system are pre-served, and this physical feature is not the same as isotropy (for furtherdiscussion, see note 15 below). Another instructive example that linksgeometry with physics in the spirit of van Fraassen’s claim that “a solutionmust ‘respect the symmetries’ of the problem” may be found in the workof one of the pioneers of the application of symmetry in modern physics,Woldemar Voigt (1850–1919). In his book, Die fundamentalen physikal-ischen Eigenschafter der Krystalle in elementarer Darstellung (1898), a trea-tise on the physics of crystals, Voigt offers definitions of geometrical andphysical symmetry; in the first chapter he assesses the relation betweenthese two kinds of symmetry. He first sets up the geometrical meaning ofsymmetry with respect to crystals by defining Symmetriecentrum, Sym-metrieebene, Symmetrieaxe and, finally, Spiegeldrehaxe (Voigt 1898, 7).4

After explaining the geometrical meaning of these terms, he poses thequestion: “From what features can we infer the laws of the physical sym-metry of crystals? [Aus welchen Merkmalen konnen wir die Gesetze derphysikalischen Symmetrie der Krystalle erschliessen?]” He thus makes itclear that he expects some relation to hold between the geometry of thecrystal and its physical properties. In particular, “in the great majority ofcases the symmetry of the crystalline form can be directly equated withthat of the [crystal’s] physical behavior. Hence, the directions that are ofequal value crystallographically are also of equal value physically.”5 Here

3. Poincare [1899] 1954, 453: “Le milieu n’est pas seulement isotrope, il est symetrique;nos equations ne doivent donc pas changer quand on remplace notre systeme d’axespar un systeme symetrique (le plan de symetrie etant le plan des xz, par exemple).”

4. Voigt builds on his earlier analysis of the symmetry elements that a crystal possesses,as expressed in physical laws: see Voigt 1895, 129ff.

5. Voigt 1898, 9: “lasst sich in den bei weitem meisten Fallen die Symmetrie derKrystallform direct derjenigen des physikalischen Verhaltens gleichsetzen, sodass alsokrystallographisch gleichwerthige Richtungen auch physikalisch gleichwerthig sind.”


we have an unambiguous connection between geometrical symmetry andphysical symmetry. That is, the symmetry of the physics must “respect”the symmetry that holds in the geometry. Finally, for an explicit usage ofthe expression, “for reason of symmetry,” we turn to August Foppl (1854–1924), who developed in his book, An Introduction to Maxwell’s Theory(1894), a brilliant synthesis of the works of Oliver Heaviside (1850–1925)and Heinrich Hertz (1857–1894). It is now well accepted that Foppl’sIntroduction was instrumental in making the young Einstein aware of theoutstanding issues in electrodynamics, notably the issue of relative andabsolute motion (see, e.g., Norton 2004, 54; Hon and Goldstein 2005a,521). Following Heaviside, Foppl argues that D, the dielectric displace-ment, and E, the electrostatic force, are arranged in space in the sameway (gleich gerichtet). He then justifies this claim by adding the argu-mentative expression: “for reasons of symmetry [aus Symmetriegrunden].”6

Van Fraassen has singled out an important aspect of the applicationof symmetry in science. He has distinguished between symmetry as aproperty and symmetry as an argument. For van Fraassen (1989, 258)the essence of symmetry arguments in all their varieties is encapsulatedin “the great Symmetry Requirement,” which is based on the crucial triadof symmetry, transformation, and invariance. It is the principle whichunderlies the methodology that generates symmetry arguments by estab-lishing the maxim that “problems which are essentially the same must haveessentially the same solution” (van Fraassen 1989, 259, italics in the orig-inal). The stated goal of van Fraassen’s analysis is to support the claimthat there are “intimate connections between symmetry and generality,”where laws are the classical expressions of generality in science. “To betruly general, a statement must be covariant, it must have this logicalstatus: it is either true in all frames of reference or true in none. Equivalently:its truth value must remain invariant under all admissible transforma-tions” (van Fraassen 1989, 287, italics in the original). To be sure, “ad-missible” depends on the theoretical context; hence, what is general to aclassical physicist may not be so for a contemporary physicist. All thesame, in van Fraassen’s view “the symmetries of time, space, and motiondetermine the structure of modern science to a surprisingly large extent.”He then concludes that symmetry rigorously shapes theory. To use vanFraassen’s figurative language (1989, 262), “symmetry takes the theore-tician’s hand and runs away with him, at great speed and very far, pro-pelled solely by what seemed like his most elementary, even trivial, as-sumptions.”

We take our cue from van Fraassen’s innovative study of symmetry

6. Foppl 1894, 91: “wie schon aus Symmetriegrunden hervorgeht, D mit E gleichgerichtet.”


and expand the perspective he has offered. As indicated, van Fraassenfocused his discussion on two types of argument. The first is characterizedby the demand that the solution must “respect the symmetries” of theproblem, while the second expresses conservation: an asymmetry can onlycome from a preceding asymmetry. We build on van Fraassen’s importantdistinction between symmetry as a property and symmetry as a form ofargument and begin by turning our attention to the general structure ofany argument. We will then examine rules of inference in an argumentand identify features that may be described as symmetrical. Moreover,we will note the impact of this type of reasoning on arguments in thepapers of Albert Einstein (1879–1955) in 1905. We hold that the appli-cation of symmetry arguments in physics is relatively new, the product ofseveral ingenious studies in the nineteenth century that served as back-ground to Einstein’s revolutionary work in that year. In fact, in our view,symmetry as a scientific concept is a nineteenth-century invention (seeHon and Goldstein 2005a, 2005b, 2005c).

2. Symmetry as a Property. There is a general agreement that the set oflectures on symmetry which Hermann Weyl (1885–1955) delivered atPrinceton University in the early 1950s, published in 1952, constitutes amilestone in the analysis and application of the concept of symmetry asa property. In this book Weyl does not address symmetry as argument.7

This profound treatment of the concept came after a century and a halfof 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 onesense symmetry means “something like well-proportioned, well-balanced.”Symmetry then denotes “concordance of several parts by which they in-tegrate into a whole.” This is how beauty, in Weyl’s view, is bound upwith symmetry. In this sense harmony may be considered a synonym of

7. Weyl refers very briefly to quantum mechanics in his Symmetry, explicitly informingthe reader that he will “refrain from giving . . . a more precise account of this difficultsubject.” He remarks further that “here symmetry once more has proved the clue toa field of great variety and importance” (Weyl 1952, 135). In this book Weyl did notreport his own contributions to quantum mechanics, which are based in part on ap-pealing to symmetry arguments: see, e.g., Weyl 1928. In his study of classical groups,Weyl appeals to symmetry as a precisely defined technical term which does not functionas an argument: see Weyl [1946] 1966. Moreover, in his philosophical works on math-ematics and physics, Weyl points out the well-known use of symmetry arguments indeveloping probability theories (Weyl and Helmer [1927] 1949, 197). Thus, Weyl, themathematical physicist, applied symmetry arguments in his scientific works, but Weyl,the philosopher-analyst, had little to say about them. We also note that symmetry inprobability theory is discussed in van Fraassen 1989, 292ff. We do not pursue theseissues in this paper for they deserve separate and detailed treatment.


symmetry (Weyl 1952, 3). However, for Weyl symmetry in another senseis a precise geometrical notion conveying an operation that carries a bodyof spatial configuration into itself. “The image of the balance provides anatural link to the second sense in which the word symmetry is used inmodern times: bilateral symmetry, the symmetry of left and right, whichis so conspicuous in the structure of the higher animals, especially thehuman body” (Weyl 1952, 4, italics in the original). In his opening re-marks, Weyl does not adhere to a strict distinction between the two mean-ings. “Symmetry,” he claims, “as wide or as narrow as you may defineits meaning, is one idea by which man through the ages has tried tocomprehend and create order, beauty, and perfection” (Weyl 1952, 5). Asthe book progresses, Weyl demonstrates persuasively that the concept ofsymmetry is ultimately an application of group theory. He begins withthe notion of congruence, for it captures a structural feature of space:“any . . . congruent transformation . . . is a similarity or an automorph-ism. . . . It is evident,” Weyl continues (1952, 43), “that the congruenttransformations form a group, a subgroup of the group of automorph-isms.” Congruence may, for example, be the result of an operation ofrotation or of translation; in each such operation the point p of a certainspace V which is occupied by some rigid body, is mapped onto a pointp! in space V ! such that the rigid body remains invariant under this op-eration (see Weyl 1952, 43). Weyl then poses a question, “What has allthis to do with symmetry?” to which he replies: “[Symmetry] provides theadequate mathematical language to define [a congruent transformation].Given a spatial configuration F, those automorphisms of space whichleave F unchanged form a group G, and this group describes exactly thesymmetry possessed by F. . . . The symmetry of any figure in space isdescribed by a subgroup of that group” (1952, 45, italics in the original).

Thus, on methodological grounds, Weyl proceeds from considerationsof a vague concept to one that is precise, gradually reaching greater gen-erality, and guided—as he puts it—“more by mathematical constructionand abstraction than by the mirage of philosophy” (Weyl 1952, 6).

In this process of generalization, Weyl intentionally discards the aes-thetic appeal of the concept and concentrates on precision, ultimatelyaiming at determining its unifying power. This approach coheres with thesecond move. Weyl’s metaphysical claim is that mathematics underliesthe concept of symmetry. He thinks that mathematics is the commonorigin of symmetry in nature and in art: “the mathematical laws governingnature are the origin of symmetry in nature, the intuitive realization ofthe idea in the creative artist’s mind its origin in art” (Weyl 1952, 6–8).Weyl’s book, Symmetry, is the crystallization of this conception, whoseunifying 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 the-ories. They are seen from a single point of view which the modern conceptof symmetry offers. This overarching analysis has been very influentialin pointing to a link between the arts, nature, and the sciences which isultimately grounded in mathematics: “From art, from biology, from crys-tallography and physics I finally turn to mathematics, which I must includeall the more because the essential concepts, especially that of a group,were first developed from their applications in mathematics” (1952, 135,italics in the original). Thus, the modern concept of symmetry gains itsstrength not only from the fruitful application of a mathematical propertyin physics but also from its ability to link a variety of domains, revealingcommon patterns in them. This observation, which Weyl stressed, madesymmetry the powerful concept that it has become in domains well beyondmathematics 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 invariancethat meets four fundamental criteria: closure, associativity, identity, andinversion (Klein [1926] 1979, 315–316; Klein 1926, 335; Weyl 1952, 41–43). The invariance (what stays the same under the transformation) is thesymmetrical property.8 For Weyl, then, symmetry as a property can bealways recast in group theoretic terms.

In 1996, about half a century after Weyl’s essay, David J. Gross, oneof the chief architects of the fundamental theory of the strong force—quantum chromodynamics—a Nobel laureate in physics (2004) and anotable advocate of string theory, reaffirmed this definition of Weyl. Grossreviewed the fundamental role of symmetry in physics and explicitly iden-tified symmetry with invariance, arguing that symmetry provides structureand coherence to the laws of nature.

Although . . . conservation laws [in classical physics], especiallythose of momentum and energy, were regarded to be of fundamentalimportance, these were regarded as consequences of the dynamicallaws of nature rather than as consequences of the symmetries thatunderlay these laws. Maxwell’s equations, formulated in 1865, em-bodied both Lorentz invariance and gauge invariance. But these sym-metries of electrodynamics were not fully appreciated for over 40years or more. This situation changed dramatically in the 20th cen-tury beginning with Einstein. Einstein’s great advance in 1905 wasto put symmetry first, to regard the symmetry principle as the primary

8. Furthermore, there is an important distinction between discrete (or finite) groupsand continuous groups which has a considerable impact on the domains of application(Weyl 1952, 106–107, 119–120).


feature of nature that constrains the allowable dynamical laws. Thusthe transformation properties of the electromagnetic field were notto be derived from Maxwell’s equations, as Lorentz did, but ratherwere consequences of relativistic invariance, and indeed largely dic-tate the form of Maxwell’s equations. This is a profound change ofattitude.

He goes on to emphasize the role of symmetry in recent physics: “Withthe development of quantum mechanics in the 1920s symmetry principlescame to play an even more fundamental role. In the latter half of the20th century symmetry has been the most dominant concept in the ex-ploration and formulation of the fundamental laws of physics. Today itserves as a guiding principle in the search for further unification andprogress” (Gross 1996, 14256). Gross thus reaffirms the position of Weylin the context of modern physics in the closing years of the twentiethcentury.9

3. Symmetry Arguments. While acknowledging the profound contribu-tion of Weyl and Gross, we suggest, as van Fraassen did, that the ap-plication of symmetry to arguments needs further elaboration. The ex-pression “symmetry argument” is a relatively recent coinage, and it is auseful analyst’s tool. Like any argument, a symmetry argument has prem-ises and rules of inference which lead to a conclusion. By symmetry ar-gument we mean that symmetry, as a property, enters at least one of thethree components that comprise an argument.

In the “Introduction” to their edited book, Symmetries in Physics: Phil-osophical Reflections (2003b), Katherine Brading and Elena Castellanidiscuss symmetry arguments and identify the cases of Buridan’s ass, Ar-chimedes’ equilibrium law of the balance, and Anaximander’s argumentfor the immobility of the Earth as typical examples of such arguments.Brading and Castellani then claim that these arguments are essentiallyapplications of the Leibnizean Principle of Sufficient Reason (PSR): “ifthere is no sufficient reason for one thing to happen instead of another,the principle says that nothing happens (the initial situation does notchange).” They observe that in each of the above cases the principle “isapplied on the grounds that the initial situation has a given symmetry:in the first two cases, bilateral symmetry; in the third, rotational symmetry.

9. Gross’s claim (1996, 14256) that in 1905 Einstein “put symmetry considerationsfirst” depends on recasting what Einstein actually wrote: on Einstein’s usage in 1905of the term symmetry, see Hon and Goldstein 2005a. In fact, evidence of symmetry inthe Lorentz transformations was explicitly introduced soon after Einstein’s initial paperon relativity theory: see, e.g., Minkowski 1908, 65: “um bestimmte Symmetrien inEvidenz zu setzen.”


The symmetry of the initial situation implies the complete equivalencebetween the existing alternatives (the left bundle of hay with respect tothe right one, and so on). If the alternatives are completely equivalent,then there is no sufficient reason for choosing between them and the initialsituation remains unchanged” (Brading and Castellani 2003a, 9). The ideathat Brading and Castellani propound—which we hasten to add is wide-spread—is that “arguments of the above kind—that is, arguments leadingto definite conclusions on the basis of an initial symmetry of the situationplus PSR—have been used in science since antiquity (as Anaximander’sargument testifies). The form they most frequently take is the following: asituation with a certain symmetry evolves in such a way that, in the absenceof an asymmetric cause, the initial symmetry is preserved. In other words,a breaking of the initial symmetry cannot happen without a reason, or anasymmetry cannot originate spontaneously” (Brading and Castellani 2003a,9–10, italics in the original). A brief examination of the original formulationof PSR by Gottfried Wilhelm Leibniz (1646–1716) will suffice to show thatthere is a misleading step in linking symmetry to PSR.

Central to Leibniz’s mode of reasoning, as developed in his Theodicy([1710] 1720) and Monadology ([1714/1840] 1954), is the idea that thereis always, in principle, an analytical path for validating any propositionthat is indeed true, be it necessary or contingent. For necessarily truepropositions the principle of contradiction suffices, but in the case ofcontingently true propositions sufficient reason is provided via the per-fection of God, that is, sufficient reason of all contingent truth lies inGod’s choice of the best. The motivation is clear: to have an explicitrationale for the truth to be so rather than otherwise (see Rescher 1991,118). In Leibniz’s formulation in his Monadology, the Principle of Suf-ficient Reason means that “no fact can be real or actual, and no prop-osition true, without there being a sufficient reason for its being so andnot otherwise, although most often these reasons just cannot be knownby us.”10 This is the principle of the determinant reason, as Leibniz hadoriginally called it in his Theodicy. The principle states that “nothing evercomes to pass without there being a cause or at least a reason determiningit, that is, something to give an a priori reason why it is existent ratherthan non-existent, and in this wise rather than in any other.”11

10. Rescher 1991, 116; Leibniz [1714/1840] 1954, 89, § 32: “ET CELUI DE LA RAISON

SUFFISANTE, en vertu duquel nous considerons qu’aucun fait ne scauroit se trouvervrai, ou existent, aucune Enonciation veritable, sans qu’il y ait une raison suffisantepour quoi il en soit ainsi et non pas autrement. Quoi que ces raisons le plus souventne puissent point nous etre connues.” See also Theodicee, §§ 44, 196.

11. Huggard [1951/1985] 1990, 147; cf. Rescher 1991, 116; Leibniz [1710] 1720, 101 §44: “C’est que jamais rien n’arrive sans qu’il y ait une cause ou du moins une raison


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 assessesthe situation in order to choose a course of action to attain a value, forexample, goodness, beauty. This evaluative assessment has no algorithm;it requires a judgment and, hence, an agent. Linking symmetry consid-erations with an agent poses a problem. We ask, Does symmetry involvean agent? We say, No! We claim that symmetry is a unique scientificconcept in that it has epistemological and ontological aspects. Clearly,when symmetry expresses the latter feature, that is, when it is related tomathematical entities (both geometrical and algebraic) or to physical ob-jects (both natural and man-made)—in a word, when it expresses a prop-erty—no agent whatsoever is involved.12 In the former case, where sym-metry 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 agentis in fact an algorithm of an operation that transforms the system inquestion in a specific way while leaving some elements invariant. No agent,benevolent or intelligent, is required to carry out this operation. But aboveall, the modern scientific concept of symmetry has nothing to do withvalue judgments associated with beauty and the good. In its group the-oretic sense, symmetry is a technical, scientific term that does not expressa value. In sum, we think it misleading to associate symmetry argumentwith PSR. We therefore look anew at forms of argument that may besaid to be symmetrical in order to determine their nature.

The application of symmetry arguments in physics had not been pro-posed explicitly by anyone prior to 1894, the year Pierre Curie (1859–1906) put forward the idea of applying group theory in this domain (Curie[1894/1908] 1984).13 According to Castellani (2003, 323), Curie was per-

determinante. C’est-a-dire quelque chose qui puisse servir a rendre raison a priori,pourquoi cela est existant plutot que toute autre facon.”

12. In the case of a man-made object, clearly an agent is involved in its production.But the fact that it has a symmetrical property in the modern sense (if it has one) isestablished by verifying that it conforms to a set of mathematical criteria, independentof the motives of the maker or its perception by the viewer. See Weyl 1952, where—as we have indicated—symmetry is illustrated throughout the book by examples takenfrom man-made and natural objects without distinction; in both cases they are recastin mathematical terms in the same way.

13. For Curie’s explicit introduction of the notion of a symmetry argument, see Bradingand Castellani 2003a, 10, and 2007, 1332–1336. In his relativity paper of 1905 Einsteinindicates in a “throwaway” remark that a certain set of transformations forms a group,but he does not cite the paper of Curie (see Einstein 1905b, 907; cf. Hon and Goldstein2005a, 468–475). At roughly the same time as Curie, Sophus Lie also suggested applyinggroup theory to mathematical physics: “Ayant vu combien les idees de Galois [sur latheorie des groupes] se sont peu a peu montrees fecondes dans tant de branches del’analyse, de la geometrie et meme de la mecanique, il est bien permis d’esperer que


suaded by his studies of the phenomena of pyro- and piezo-electricity ofcrystals that the symmetry of the phenomena and that of the physicalmedium are mutually dependent. By applying a group theoretic analysis,he arrived at the following general principle: “When certain causes pro-duce certain effects, the symmetry elements of the causes must be foundin their effects. . . . In practice, the converse . . . is not true, i.e., theeffects can be more symmetric than their causes” (Curie [1894/1908] 1984,127, cf. 119; Rosen and Copie 1982, 20, cf. 17). Although Curie did notuse the term “symmetry argument,” the principle he formulated underpinsan argument that may be described as “considerations of symmetry.” Inthis argumentative scheme the physical relation between the cause andthe effect—based as it is on symmetry considerations—can be recast interms of premises and a conclusion which is drawn by applying specificrules of inference. Inspired by Curie’s analysis, we discern two differentmoves, a forward move from the cause (“premises”) to the effect (“con-clusion”), and the reverse. Based on the distinction between these twomoves, we identify two different categories of symmetry argument.

3.1. The Axiomatic Category. The key to the axiomatic form of sym-metry arguments is that the propositions of the premises convey symmetryproperties of the initial state of the given physical situation. Accordingto Weyl’s analysis, symmetry in the premises refers to the invariance ofa certain group. Here the rule of inference is applied as an algorithm tothe symmetrical properties of the initial state such that the premises aremathematically transformed into a form which is amenable to some op-eration from which the conclusion follows. While the conclusion is distinctin form from the premises, physically they are equivalent to a very closeapproximation. The application of a rule of inference (in the form of analgorithm) transforms the initial group into another group while main-taining the same (physical) invariance. In this category one moves in theforward mode from cause (“premises”) to effect (“conclusion”), that is,the symmetry of the cause is also in the effect; in sum, the symmetry isconserved, and so the move from the premises to the conclusion is de-termined. We call this category “axiomatic” since setting the premisesdetermines the invariance.

3.2. The Heuristic Category. In this category the property of symmetryneed not apply to the initial state of the physical system as conveyed bythe premises. In fact, the initial state may lack the characteristic of agroup and thus no invariance may be found. Rather, the application of

leur puissance se manifestera egalement en physique mathematique” (Lie [1895] 1989,9).


a rule of inference augments the premises so that properties of symmetrymay be identified and the initial state becomes a group; only then doesthe conclusion flow from the premises—the conjectured invariance beingnow secured. Typically, the way to augment the premises is suggested bythe conclusion, that is, the effect. If one were to turn the effect into thepremise and, furthermore, the rule of inference is symmetrical, then onecould reverse the rule of inference (“reflection”) and reach the cause inthe form of a conclusion. This may be a general characteristic of rules ofinference in symmetry arguments. In the specific case of heuristics onemoves backward from the effect to the cause, that is, the symmetry ofthe effect suggests looking for symmetry in the cause. Hence the argumentis contextually dependent and is not determined by the premises. This iswhy we call it “heuristic.”

4. Case Studies. We turn first to the axiomatic category. In his disser-tation, “A New Determination of Molecular Dimensions,” Einstein pres-ents us with such a case. Einstein explicitly refers in this work to hissource, a work by Gustav Kirchhoff (1824–1887). Thus we have reliableevidence for a text that Einstein consulted (Einstein 1905a, 189).14 Al-though Einstein does not cite Curie (and probably had not read the rel-evant paper), he uses the expressions, wegen der Symmetrie and aus Sym-metriegrunden, thereby explicitly appealing to a symmetry argument(1905a, 188, 191, 196).

Einstein considers the motion of a solvent in the vicinity of a suspendedsolid body which plays the role of the dissolved molecule. At issue arethe hydrodynamic equations of the motion of the solvent in these cir-cumstances. As is customary in mathematical physics, several presup-positions are introduced to simplify the calculations: Einstein assumesthat the solid body is spherical in shape and that the liquid is homogenousso that its own molecular structure need not be taken into consideration(for a complete list of the presuppositions, see Pais 1982, 90). The nextstandard step in hydrodynamic analysis is to treat the motion of the liquidas a superposition of three different kinds of motion: (1) a parallel dis-placement without a change in the relative positions of the liquid particles;(2) a rotation without a change in the relative positions of the liquidparticles, and (3) a dilatational motion in three mutually perpendicular

14. Einstein refers again to the same text of Kirchhoff in footnotes to both the paperon the movement of small particles suspended in stationary liquids (received by theAnnalen der Physik, May 11, 1905), and the paper on Brownian motion (received bythe Annalen der Physik, December 19, 1905): see Stachel et al. 1989, 230, 342. Thusit is evident that “Lecture 26” in Kirchhoff ’s Mechanik was an important source forEinstein. See Kirchhoff 1883, 369–387.


directions (Kirchhoff 1883, 123–124, 369–374; Einstein 1905a, 187; Beck1989, 106).

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

Due to the symmetry [wegen der Symmetrie] of the motion of theliquid, it is clear that the sphere can perform neither a translationnor a rotation during the motion considered, and we obtain theboundary conditions

u p v p w p 0 for r p P, . . .

(Beck 1989, 107, boldface added; Einstein 1905a, 188, boldfaceadded)

It should be clarified that “the motion considered” is the third one, di-latation, the only motion that gets modified by the presence of the rigidspherical body. Note further that u, v, and w are the three functions ofthe velocity components of the dilatational motion of the liquid; P is theradius of the sphere and r expresses the principal axes of dilatation, beingequal to the square root of the sum of the squares of the three componentsof the coordinate system whose axes are parallel to the principal axes ofthe dilatation.

Symmetry here expresses the feature that there is no privileged direction:the three equations of motion are exactly analogous for each of the threeorthogonal directions. This feature is usually called isotropy. Accordingto William Thomson (1824–1907) and Peter G. Tait (1831–1901), isotropyis defined by two elements: resistance to compression and resistance todistortion. Thomson and Tait identify the quality of homogeneity as aprerequisite for isotropy: “A body is called homogeneous when any twoequal, similar parts of it, with corresponding lines parallel and turnedtowards the same parts, are indistinguishable from one another by anydifference in quality.” They then emphasize the physical nature of thisquality: “The substance of a homogeneous solid is called isotropic whena spherical portion of it, tested by any physical agency, exhibits no dif-ference in quality however it is turned.”15 Thomson and Tait considerindifference (which they also call “indistinguishability”) as a way to depictthis feature of homogeneity. It is significant that physical agencies are

15. Thomson and Tait 1867, 517–518; see also 519. For the second edition, see Thom-son and Tait 1883, 217; see also 216, 218–219. The opposite term is eolotropy (sometimescalled anisotropy): “a substance which . . . exhibits differences of quality in differentdirections” (ibid., 217). Thus, in the case of eolotropy, electric, optical, or other physicalqualities are affected by change of position, as when the refractive property of atransparent body is not the same in all directions.


involved in determining this feature; in other words, according to thisdefinition, isotropy is physical rather than geometrical. This is, then, thefeature that exhibits equal physical properties or actions in all directions,for example, refraction of light, elasticity, and conduction of heat or ofelectricity. To return to Einstein’s dissertation: given the conditions of theproblem, that is, the presuppositions presented at the outset of the workand the geometry of the situation which Einstein laid down, the isotropicfeature restricts the kind of motions that the rigid spherical body canundergo. The limitation thus imposed allows Einstein to obtain the bound-ary conditions he needed for proceeding with the calculation of the dil-atational motion of the liquid.

Einstein follows very closely the text of Kirchhoff, who explicitly ap-plied symmetry to hydrodynamic calculations, although not in the lecturewhich Einstein cites.16 In his essay of 1869 on the motion of a rotationalbody immersed in a fluid whose rotational axis is not fixed in one plane,Kirchhoff begins with a very complex system of equations of motion. Hestipulates that the liquid is without friction, incompressible, and homog-enous, and it stands to reason that he regards the fluid as isotropic (Kirch-hoff [1869] 1882, 376). Having set the problem, he proceeds to applysymmetry to simplify the equations. The forces, according to Kirchhoff ’sargument, vanish due to the symmetrical distribution of the mass of therotating body with respect to the axis of rotation (Kirchhoff [1869] 1882,377). In another step of the calculation he sought to reduce the numberof constants of the equations of motion when the body in question exhibitssymmetry properties, that is, geometrical symmetries concerning physicalproperties, namely, the distribution of mass (Kirchhoff [1869] 1882, 388).And then he makes another crucial move when he notes that if the surfaceof 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 thispresupposition with the result that the number of equations of motion isreduced substantially (Kirchhoff [1869] 1882, 389; see also 390–391).Kirchhoff concludes the analysis with a general claim that the resultantequation is not restricted to bodies of rotation; rather, it applies to allbodies as long as they are symmetrical in relation to two or more pairsof points located in two orthogonal planes with respect to the x-axis(Kirchhoff [1869] 1882, 392). In other words, while the mass-density ispresumed to be the same, the geometrical requirements for the shape ofthe body have been relaxed. Joseph Larmor (1857–1942), Lucasian Pro-

16. Kirchhoff also applied symmetry extensively in his discussion of elasticity: seeKirchhoff 1883, 389–390, 395.


fessor of Mathematics at Cambridge from 1903 to 1932, singled out thisresult of Kirchhoff as remarkable (Larmor [1884] 1929, 77).

Kirchhoff applies different notions of symmetry to the fluid as themedium and to the bodies which are immersed in it. The fluid is isotropic,while the symmetry which the bodies exhibit determines the way themasses are distributed in them with respect to the axis of rotation, andthis physical feature fixes the number of equations required for the descrip-tion of the motion of the bodies. However, in both cases, Kirchhoff ’s usageof symmetry is essentially geometrical. To be sure, the claim is physical:the forces, Kirchhoff argues, cancel each other. This is one of the waysKirchhoff ’s analysis of 1869 succeeds in reducing the number of equations.He elaborated this result in “Lecture 19” of his Mechanik—the same bookto which Einstein refers—where in section 3 he demonstrates the sym-metrical property of a solid of revolution (Kirchhoff 1883, 243). Thisresult of Kirchhoff indicates that symmetry as a geometrical property wasassumed to have physical consequences for the motion of a body in afluid, thus establishing a link between the geometry of the body and itsphysical properties, captured by the term isotropy.

The usage of this physical sense of symmetry became fairly commonin physics circa 1900; hence, no innovation is involved in appealing to it.Thus Einstein’s application of this kind of symmetry may not have orig-inated in his reading of Kirchhoff. We note that Einstein appeals to sym-metry in an argument which follows Kirchhoff closely despite the factthat Kirchhoff did not apply symmetry to these basic hydrodynamic equa-tions. Furthermore, we suggest that Einstein made the assumption ofisotropy explicit—calling it symmetry—a move which Kirchhoff had nottaken. In other words, Einstein extended the meaning of symmetry toinclude isotropy, thereby articulating the presupposition that the equationsof motions for the three orthogonal components of the motion of thesolvent are the same.

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

In physics, as in algebra, one is entitled to eliminate elements that canceleach other. As we have seen, Kirchhoff simplifies hydrodynamic equationswhen there is a symmetrical distribution of masses—in which case certainterms 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 arrangementin algebraic terms facilitates this summation to zero. To recapitulate:Kirchhoff uses the term symmetry to describe the physical conditions ofthe arrangement; in other words, he is setting the initial conditions of thephysical arrangement in such a way that they can be transformed by arule 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 argu-ment.

In his paper of 1894, Curie remarked that physicists had been speaking“prose” all the time: “Physicists often utilize symmetry conditions, butgenerally neglect to define the symmetry in a phenomenon, for sufficientlyoften these symmetry conditions are simple and quite obvious a priori[simples et presque evidentes a priori]” (Curie [1894/1908] 1984, 118–119;Rosen and Copie 1982, 17). Indeed, Kirchhoff is not as explicit as Curieabout the application of a symmetry argument; moreover, unlike Curie,Kirchhoff does not treat the general case. For Kirchhoff there is after allno “metaphysical principle” embedded in symmetry considerations. Al-though Weyl does not cite Curie in his study of symmetry, he seems toecho this view in his claim, “as far as I see, all a priori statements inphysics have their origin in symmetry” (Weyl 1952, 126).

Curie explicitly pointed to symmetry as a powerful tool in physicalarguments which he turned into a metaphysical principle. He then dem-onstrated its application in the domain of electromagnetism. In the open-ing section of his famous paper of 1894, Curie called attention to the factthat “in the study of electricity, for example, one should state almostimmediately the characteristic symmetry of the electric field and of themagnetic field. One can then use these notions to simplify proofs” (Curie[1894/1908] 1984, 119; Rosen and Copie 1982, 17). The characteristicsymmetry of these two fields had been discussed in the physical literatureof the time, beginning with Henry Rowland (1848–1901), followed byHertz with several other illustrious protagonists in the background (fordetails, see Hon and Goldstein 2006). But Curie departs from this traditionin turning from symmetry as a property of the state of the fields to sym-metry as an argument whose rules of inference facilitate proofs. Curie wasthen conscious of his move when he began his paper, “Sur la symetriedans les phenomenes physiques” ([1894/1908] 1984), by noting that sym-metry arguments were not invoked in physics before 1894 and that hehimself was importing the idea of symmetry from crystallography—whereit was well developed at the time—into physics: “I think that there is an


interest in introducing into the study of physical phenomena the symmetryarguments [considerations] familiar to crystallographers.”17

We now turn to the heuristic category. Murray Gell-Mann’s predictionin 1962 of the existence of the elementary particle, omega minus (Q!),and its subsequent discovery, illustrates a clear case of the heuristic usageof a symmetry argument. Facing an increasing number of strongly in-teracting fundamental particles—including stable and semi-stable par-ticles as well as the very short lived ones (resonances18)—Gell-Mannsearched for a scheme of classification that would introduce order intothe ever-growing “collection.” For this purpose, he established a newfundamental characteristic of a multiplet which he called “hyper-charge.”19 He then proposed a new rule: elementary particles can betransformed into others by the strong and the electromagnetic inter-actions only if the total hypercharge is conserved. This rule is reminiscentof the law of conservation of electric charge. Yuval Ne’eman (1964, 3)aptly described the situation: “This was symmetry in yet another aspectof its uses: the determination of selection rules for reactions. . . . Withthe new crop of particles, one had to postulate a symmetry first andthen check it, because there was no easy way to observe the over-allforces ratio.” Ne’eman candidly revealed the methodology he and Gell-Mann had applied. They were facing experimental results that requiredtheoretical underpinning. In a sense, they were looking at the solution,the “conclusion,” and they were trying to reconstruct the problem: theunknown premises of the problem that under certain constraints resultin the known “collection” of particles. Gell-Mann made a theoreticalproposal on the basis of growing experimental data that lacked order;symmetry considerations allowed him to predict empty places in thearrangement of the baryon multiplets that indicated the existence of two

17. Rosen and Copie 1982, 17; Curie [1894/1908] 1984, 118: “Je pense qu’il y auraitinteret a introduire dans l’etude des phenomenes physiques les considerations sur lasymetrie familieres aux cristallographes.” Cf. Brading and Castellani 2003a, 10.

18. Resonances are short-lived particles which “decay before they have a chance toleave a track [e.g., in a bubble chamber], but they leave an indelible impression on theenergy and momenta of their decay products” (Wick 1972, 60).

19. Particles which possess identical quantum numbers are called multiplet; thus,the particles that comprise the nucleus, the proton and the neutron, are consideredmultiplet since they have the same spin, the same intrinsic parity as well as the samestrangeness. In this case, the only difference between the two particles is their electriccharge; they are then called “charge multiplet.” Hypercharge is a quantum numberthat is akin to strangeness; it establishes a connection between stable particles andresonances. Thus, each charge multiplet has a characteristic hypercharge (see Wick1972, 81, 92–93).


additional baryons, one of which was soon discovered and the otherwas found six years later (cf. Waller 1972).20

Gell-Mann (1962a, 1067) states: “we shall suggest a particular symmetrygroup as the one most likely to underlie the structure of the system ofbaryons and mesons.” Gell-Mann applied to the physics of elementaryparticles a certain mathematical model in group theoretic form: SU(3).At the same time he pointed out that some particles were missing whichwere needed to complete the group; the identification of missing elementsmade him conjecture that these missing particles exist (Gell-Mann 1962b).After they were experimentally confirmed, he was awarded the NobelPrize in 1969 for this prediction. Gell-Mann’s argument depends on a ruleof inference that invokes symmetry: starting from the model (i.e., thegroup, SU(3), taken as a constraint on the premise), he sought its con-clusion (the existence of these particles).

5. Conclusion. Einstein’s usage of symmetry arguments, marked as wegender Symmetrie and aus Symmetriegrunden, has not been noted in theliterature. His use of these expressions is an indication that he clearlydiscerned in symmetry arguments the distinction between the propertiesof the premises and the rule of inference that facilitates the drawing ofthe conclusion. Einstein works in the tradition of Kirchhoff but, unlikeKirchhoff, he makes it clear that in symmetry arguments (of the axiomatickind) there has to be a rule of inference which transforms the form ofthe premises while retaining the physical properties of the initial state.21

Such an inference also facilitates the appeal to an argument in the reversedirection, from the conclusion to the premises, thereby endowing sym-metry arguments with great heuristic power.

The metaphysical principle of Curie, by which the symmetry of thepremises in an argument is linked to that of the conclusion, underlies therule of inference in symmetry arguments. By connecting the symmetry ofthe cause with that of the effect, this rule demonstrates that there is adifference between symmetry as a property and symmetry as an argument.The seminal work of Weyl from half a century ago addresses symmetryas a property in a fundamental way, but in that work he neither considersthe role of symmetry in rules of inference nor does he refer to Curie’s

20. In addition to the nucleons (protons and neutrons), other members of the baryonfamily include the D, L, S, Y, and Q particles. The name alludes to the fact that these“heavy” particles experience the strong nuclear force.

21. Einstein continues to apply symmetry arguments in his relativity paper (1905b),completed about two months after the dissertation. Once again he invokes the ex-pression, aus Symmetriegrunden and this time stipulates the rule of inference (cf. Honand Goldstein 2005a).


paper of 1894 which laid the foundations for symmetry as an argumen-tative procedure in science. There seems to be something unique in thisconcept that makes it such an effective tool in modern science and es-pecially in contemporary physics. It may well be the case that this uniquefeature of symmetry has to do with the fact that it is both an expressionof a property and a powerful form of argument as well.

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