HAL Id: hal-00637572 https://hal.archives-ouvertes.fr/hal-00637572v1 Preprint submitted on 2 Nov 2011 (v1), last revised 20 Dec 2013 (v2) HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Reflections on the four facets of symmetry: how physics exemplifies rational thinking Amaury Mouchet To cite this version: Amaury Mouchet. Reflections on the four facets of symmetry: how physics exemplifies rational thinking. 2011. hal-00637572v1
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HAL Id: hal-00637572https://hal.archives-ouvertes.fr/hal-00637572v1Preprint submitted on 2 Nov 2011 (v1), last revised 20 Dec 2013 (v2)
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Reflections on the four facets of symmetry: how physicsexemplifies rational thinking
Amaury Mouchet
To cite this version:Amaury Mouchet. Reflections on the four facets of symmetry: how physics exemplifies rationalthinking. 2011. �hal-00637572v1�
In contemporary theoretical physics, the powerful notion of symmetry stands for a web of intricatemeanings among which I identify four clusters associated with the notion of transformation, comprehension,invariance and projection. While their interrelations are examined closely, these four facets of symmetryare scrutinised one after the other in great detail. This decomposition allows us to examine closely themultiple different roles symmetry plays in many places in physics. Furthermore, some connections withothers disciplines like neurobiology, epistemology, cognitive sciences and, not least, philosophy are proposedin an attempt to show that symmetry can be an organising principle also in these fields.
pacs: 11.30.-j, 11.30.Qc, 01.70.+w,Keywords: Symmetry, transformation group, equivalence, classification, invariance, symmetry breaking,reproducibility, objectivity, realism, underdetermination, inference, complexity, emergence, science laws,representationsRemark: In this manuscript, I have not displayed Figures 1 and 2 for copyright reasons.
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Contents
1 Introduction 2
2 Symmetries, classifications and hierar-chies 5
2.1 The four facets of a symmetry . . . . . 52.2 Symmetry as a classification tool and
3.4.2 Operational differences be-tween active and passivetransformations . . . . . . . . . 12
3.5 Perception of space-time transforma-tions: following Henri Poincaré’s intu-itions . . . . . . . . . . . . . . . . . . . 13
4 Comprehension and classification 16
4.1 From taxonomy to predictability . . . 164.2 Inference and the problem of change . 174.3 Stability and abstraction . . . . . . . . 174.4 Different levels of Darwinian selection 194.5 Symmetry-assisted research of simplicity 21
4.5.1 Simplicity as a general evalua-tion against underdetermination 21
4.5.2 Simplicity from an algorithmicpoint of view: conjectures andrefutations . . . . . . . . . . . 22
For the International Conference on TheoreticalPhysics that took place in 2002 at the Unesco con-ference centre in Paris, Chen Nin Yang entitled hisspecial plenary lecture Thematic melodies of twen-tieth century theoretical physics: quantisation, sym-metry and phase factor. At that time, nobody inthe audience must have been surprised that sym-metry was presented as one of the major conceptthat extensively irrigated physics since 1905; but,as Yang(2003b; 2003a) reminds us, it was not be-fore the 1930’s that the ideas of symmetry, invari-ance and more generally the “unfamiliar sophisti-cated mathematical concepts” borrowed from grouptheory started to be widely accepted by physicists.More than ten years after Emmy Noether obtainedher deep results, even the conservation laws—in par-ticular the conservation of energy at microscopicscales like in the β-decay—seem to have been notas firmly established as they are now, according toByers (1994, § III) in an attempt to explain the rar-ity of the references to Noether’s work in the physicsliterature four decades after 1918 (for the recep-tion of the now famous theorems, see also Kosmann-Schwarzbach, 2010).
During the twentieth century, the development ofquantum physics and relativity came with a seman-tic shift of the word “symmetry” in accordance withthe increasing power of such a conceptual instrument.Mathematicians and theoretical physicists have en-larged its meaning far beyond the one it had in theprevious centuries (Hon & Goldstein, 2008). Among
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the multiple uses of symmetry in theoretical physicsand more generally in science, I propose to discernfour clusters associated with the notion of transfor-mation, comprehension, invariance and projection.After having defined these four facets in the nextsection, I will discuss them individually with moredetails in sections 3, 4, 5 and 6 respectively. Thesefour notions are more or less explicit and fully recog-nised in the natural philosophy since its very begin-ning but, as far as I know, addressed with very looserelations between them, if any. Contemporary theo-retical physics point of view on symmetry offers a newand very precise way to tight bind these notions to-gether and cross-fertilise them. To make a long storyshort, originally being an aesthetic criterion it evolvesin science into an efficient method of classification be-fore it becomes a powerful theoretical tool to guideand even dictate our way of constructing models andtheories. More precisely, symmetry allows
1◦. to build (linear) algebraic representations1;
2◦. to constrain yet to unify interactions in the mod-els of theoretical physics;
3◦. to extract universal properties in statisticalphysics and in non-linear dynamics;
4◦. to solve dynamical equations by reducing thenumbers of degrees of freedom and establishingthe bridge between integrable and chaotic dyna-mics;
5◦. to predict selection rules.
Examples of 2◦ are the unification of inertial andgravitational forces in the theory of general relati-vity and the unification of electromagnetic and weak
1I use the epithet “algebraic” to reinforce the distinctionbetween a) the precise mathematical meaning of “representa-tion” in the present context (see also § 3.4.1, Fig. 4 below),b) the “internal” meaning it takes in cognitive science (mentalrepresentation) and c) the usual “external” meaning (model,symbol, image, spoken word, written sign, play, etc.). All thethree significations share the common point of being associatedto a correspondence and can refer to the target, the domain ofa mapping, one of their elements or the mapping itself. Someconfusion usually raises when the domain and the target arenot clearly distinguished.
interactions in the Standard Model of quantum fieldtheory. Conservation laws are particularly manifestin 4◦ and 5◦. Above all, the selection rules concernquantum processes but, as constraints on the evolu-tion of a system, they can be found in classical dyna-mics as well: the conservation of the angular momen-tum of a particle evolving freely in a circular billiardexplains the impossibility of the inversion of the senseof its rotation around the centre. In this essay I willhave to say more about 1◦ in § 3.4.1 and I will focuson 3◦ in § 4.5.3.
The roles 1◦–5◦ were discovered/invented by physi-cists essentially along the twentieth century (for adeep insight of these issues see the synthesis byBrading & Castellani, 2003, in particular § 1). Thesepowerful functions of symmetry, far from erasing it,strengthen what is certainly the most important one:the fact that symmetry allows
6◦. to classify.
Performing classification concerns domains broaderthat the one of crystals, quantum particles, living or-ganisms, historical periods, languages and philoso-phical camps: any law of nature can also be seen as atraduction of an objective regularity and I will try toshow how the modern interpretations of symmetry al-low to give a precise and, above all, coherent meaningof the terms “objective” and “regularity”. It providesa rigorous ground for the signification of universality.It also concerns the quantitative approach of infer-ence and understanding. Some connections with thedual enquiry for emergence and reduction can alsobe drawn. Last but not least, these conceptions ofsymmetry shed light on a (the ?) characteristic ofintelligence, namely modeling, and on an elementaryprocess necessary for rational thinking, namely ab-straction. Since,
The whole of science is nothing more than arefinement of every day thinking. It is for thisreason that the critical thinking of the physicistcannot possibly be restricted to the examina-tion of the concepts of his own specific field.He cannot proceed without considering criti-cally a much more difficult problem, the prob-lem of analyzing the nature of everyday think-ing (Einstein, 1936, § 1, p. 349),
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this feedback of the concept of symmetry fromphysics to science and from science to rational think-ing is quite natural after all.
In this perspective, I believe that the notion ofsymmetry revitalises and clarifies the long-standingphilosophical debate between realists and their op-ponents. The notions of “reality”, “being”, “existence”or even “self” have always been tricky because onecannot define them without introducing some tautol-ogy. In the endless controversial discussions aboutontology, these notions are, at best, acknowledged asprimary concepts (Einstein, 1949, pp. 669 and 673,for instance) and are, therefore, not necessarily un-derstood or shared in the same way by all the differentschools. Even more, and not only in epistemology butin the whole philosophy as well, it is quite commonto find arguments for or against realism that presup-pose the “existence” of what they are supposed toaccept or deny. What sense can be indeed given toan affirmation like “reality does (not) exist” ? In lesscaricatured sentences, like “an object is independentof the subject” or “objective facts are an illusion”,the meaning of “object”, “subject”, “fact” or even “il-lusion” and the use of the verb “to be” reflect somekind of reality (otherwise we fall in a vicious circle,paved with liar-like paradoxes where illusion is itselfan illusion, etc. This argument is also approved byFine, 1996, chap. 7, § 3, first paragraph).
Aside from practical (“fitted” as Darwin wouldhave said, “convenient” as Poincaré should have pre-ferred) everyday life, even an ultra anti-realist likea solipsist must accept the conclusion of the famousDescartes’credo “I think, therefore I am”2. Nomina-lists must accept the existence of equivalence classesthat each word denotes. Agnostic empiricists, phe-nomenologists, positivists, pragmatists, idealists, etc.must deal with the existence of facts, sensations,mental representations and so on. Post-modernistsof every kind use the existence of equivalence classeslike historical events, social structures and groups,cultural beliefs, etc. that, whether contingent ornot, may become true objects of scientific studies assoon as they contribute to a rational schema (the
2I leave as an exercise for the reader to explain, from thehints given throughout the present article, why it is preferableto invert the implication and say “I am, therefore I think”.
social or human sciences). As far as rationality isconcerned—and, I would say, by definition of whatrationality means—the best we can do is to remainat least non self-contradictory, in a virtuous circleso to speak. My preference in this matters inclinesto distribute the notion of existence into a sort ofholistic web where entities have a more or less highdegree of reality according to how tightly boundedthey are by coherent and logical relations. I will tryto show how the four facets of symmetry, throughthe cardinal concept of equivalence classes, help usto gain in coherence or, at least, help us to anchorand place in this circle. Anyway, wherever we stand,an Ouroboros loop remains unavoidable because we,as intelligent subjects, are fully part of the world andmay be, therefore, considered as objects of knowledgeas well.
Figure 1: Escher’s lithography “Prentententoon-stelling” (1956) can illustrate the virtuous circle ofreality. Note the role of the representations andthe presence of observers (N.B: copyright must stillbe asked to http://www.mcescher.com/ or [email protected]).
The two oil paintings by René Magritte entitled“La condition humaine” (the human condition) couldhave been chosen to illustrate the complex relationbetween real and its representation, but its circularityis even more admirably rendered by Maurits CornelisEscher’s lithography “Prentententoonstelling” (printgallery, fig. 1) see also (de Smit & Lenstra, 2003).
Of course, I can take on safely all the self-referenceaspects of the present reflections since, as being hope-fully rational, they are therefore partly recursive (ra-tional thinking must obviously be broad enough inscope to embrace the process of rational thinking it-self): to play with the multiple senses of the word“reflection” and to use a metaphor borrowed fromlaser physics, I hope that, through this paper, somelight will be produced or at least coherently ampli-fied, like the beam in a resonant optical cavity. Thissomehow risky and ambitious attempt to enlightendifferent other fields by extrapolating some notionswhose most precise meaning can only be offered by
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mathematics and physics was already a strategy pro-posed by Helmholtz in 1868, according to Cassirer(1944); the latter analyses carefully the precautionsthat must me taken to achieve this goal precisely withthe help of the notion of symmetry.
2 Symmetries, classifications
and hierarchies
The physical meanings of the notion of symmetry willbe our guidelines throughout this essay. However, Iwill start by rather formal considerations where thephysical motivations remain in the background. It isonly from the next sections that I will develop thephysical interpretations and induce to a broader do-main the matter of the present section. After the firstversion of this work was composed I became awareof Brading & Castellani’s (2003) book. The deepconnection between symmetries, classifications andgroups that will be shown in § 2.2 is already presentin Gordon Belot (2003, § 5) and Elena Castellani’s(2003, § 2) contributions. Some considerations onthe three first facets have already been proposed by(van Fraassen, 1989, § X.3).
2.1 The four facets of a symmetry
The oldest known trace of symmetry involved ina thought process are the engraved pieces of redochre found at Blombos Cave in South Africa (fig. 2)and are estimated to be older than 70,000 years(Henshilwood et al., 2002). Most likely, we will neverknow whether these engravings had a symbolic roleor were a pure aesthetical game. However, histori-cally, the concept of symmetry was first employed torefer to an aesthetic harmony made of unity, rhythmor balanced proportions emanating from internal re-lations, for instance in a sense exquisitely expressedby Charles Baudelaire in his famous poem Correspon-dences.
In a scientific context these associations are for-mulated with some mappings T , defined on a set E
of elements. A symmetry associated with the set ofmappings T encapsulates four significations that aremore or less clearly distinguished in the literature:
Figure 2: Figure 2 of (Henshilwood et al. (2002)).The engraved pieces of red ochre found in BlombosCave in South Africa presents some pattern wheremirror symmetry and spatial translation symmetryclearly appear (this piece is about 5 cm large) . Wewill probably never know how our remote ancestorsuse of these remarkable pieces. Their discoverers(Henshilwood et al. (2002)) suggest it is an ab-stract representation but it may be also a pure aes-thetic game as well.
(i). A transformation x 7→ T (x) where we retain the
image xTdef= T (x) of an element (or a set of
elements) x by one mapping T belonging to T .
(ii). A comprehension: We retain all the images of xobtained when applying all the mappings in Tand collect them in the set
σ(x)def= {T (x)}T∈T . (1)
When applied to many (and ideally all) elementsof E we can see the mapping x 7→ σ(x) as beinga classification.
(iii). An invariance: Applying any mapping T of Ton any subset of σ gives again a subset of σ. Inthe case of a mirror symmetry for instance, theaction of T on the pair σ(A) = {A,AT} where Ais a set of points of space does not modify σ(A).
(iv). A projection3 or symmetry breaking: Ratherthan dealing with the invariant sets (also calledglobally invariant or symmetric sets) σ1, σ2, . . .we retain only one element for each σ i.e.x1, x2, . . . such that x1 ∈ σ1, x2 ∈ σ2, . . . For-mally, we can see the projection as a map-ping σ 7→ x where an x is chosen to be a rep-resentative of each σ.
Projection constitutes the inverse operation ofcomprehension and rather than the widely acknowl-edged expression symmetry breaking, it should havebeen more precise to talk about invariance breaking.
3Mathematicians would certainly prefer the term “section”,keeping the term projection to exclusively qualify a mappingequal to its iterates. They would also talk of “orbit” ratherthan of “comprehension”.
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2.2 Symmetry as a classification tool
and vice versa
Recall that any classification requires the fundamen-tal mathematical concept of equivalence relation: itis a binary relation, hereafter denoted by ≡, be-tween elements of a set E which is, by definition(Bourbaki, 1968, § II.6.1 p. 113 to quote one of theparagons of structuralist mathematical books),
a) reflexive : every x in E is related to itself, x ≡ x;
b) symmetric4: for all (x, y) in E2, if x ≡ y then
y ≡ x;
c) transitive (Euclide’s first common notion): forall (x, y, z) in E 3 if x ≡ y and y ≡ z then x ≡ z.
From a set of mappings T acting on E we can define
x to be related to y if and only if there existsa mapping T in T such that y = T (x). (2)
Saying that this relation is reflexive means that thereexists a transformation in T for which x is invariant.A sufficient condition is of course thata’) T includes the identity mapping.The symmetry property is fulfilled ifb’) Every mapping is invertible and its inverse mustbelong to T .Finally, a sufficient condition to have the reflexivityproperty of the binary relation is the closure propertyfor T :c’) The composition T1◦T2 of two mappings belongingto T remains in T .Properties a’), b’), c’) are rather intuitive when usingthe language of transformations:a’) Doing nothing is a (somehow trivial) transforma-tion;b’) Every transformation can be undone;c’) Making two successive transformations is a trans-formation.Together with the associativity of the compositionlaw—we always have (T1 ◦ T2) ◦ T3 = T1 ◦ (T2 ◦T3)—they define the set T as a being group. The set σ(x)defined by (1) is called the equivalence class of x and
4It is not a coincidence if we recover this term here!
collects all the elements that are equivalent to x i.e.related by the equivalence relation.
Summing up, we have seen how symmetry and clas-sification are intimately related and why the groupstructure arises naturally .
Remark: It is conceptually interesting to remarkthat not only working with a group of transfor-mations is sufficient for having a classification, butit is also a necessary condition in the sense that,given a classification, we can always construct anad hoc group of transformations T such that (2) forthe a priori given equivalence relation (see also vanFraassen, 1989, § X.3). For example we can choose Tas the group generated by all one-to-one mappings(bijections) that are the identity everywhere but onone class. There is also the mathematical possibil-ity of working with a set T that is not a group butfor which (2) is still an equivalence relation. For ins-tance take for T the set of all mappings Tσ whoserestriction to one class σ is a bijection and that sendsany element belonging to σ′ 6= σ to one chosen repre-sentative element of σ′. Except for degenerate caseswhere there is only one class or when all but one ofthem have exactly one element, T has generally notthe closure property, does not contain the identityand the Tσ are not invertible5. In physics I do notknow any relevant examples of such a situation.
2.3 Hierarchies
The procedure of constructing new mathematical ob-jects as being equivalence classes is omnipresent inmathematics. The set of equivalence classes in E
constructed from the relation ≡ constitutes the so-called quotient set denoted by E / ≡. Just to retainone fundamental example: a hierarchy of numberscan precisely be built with equivalence classes definedat each step with an appropriate equivalence rela-tion: the rational numbers being equivalence classesof ordered pairs of integers6, the real numbers being
5I am grateful to Emmanuel Lesigne for helping me in clar-ifying these points.
6By the way, in the Frege construction, even natural num-bers are equivalence classes, namely the equivalence classes ofsets related by one-to-one mappings (Bourbaki, 1968, §§ III.3.1p. 157 and III.4.1 p. 166). As far as physics is concerned, I will
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equivalence classes of (Cauchy) sequences of ratio-nal numbers (and appear, therefore, as classes con-structed from classes), the unit-modulus numbers canbe seen as classes of real numbers considered as beingequivalent if they differ by an integer multiple of 2π(classes from classes from classes). More generally,we first define a new set E ′
1 of elements constructedfrom the elements of E1 and then we classify themin E2 = E ′
1/ ≡ using an equivalence relation ≡ definedin E ′
1. Another standard example is given in Euclid-ian geometry: E1 is the set of points, E ′
1 = E1 × E1
is the set of ordered pairs of points and, eventually,vectors appear as equivalence classes of ordered pairsof points, two pairs of points being equivalent if andonly if they form a parallelogram.
The objects formed this way gain a sort of au-tonomy with respect to the primitive elements fromwhich they are built. The different operations or re-lations that may exist in E1 can be used to defineoperations in E2 provided that they do not dependon the choice of the representative elements in theclasses. Therefore, computations in E2 can often (butnot always, see the last paragraph of § 6.1) be donewithout any reference to computations at the “lower”level in E1. To keep working with the same previousexample, in a d-dimensional space, one can add twovectors using their 2 × d coordinates, not using the2 × 2 × d numbers that encode the position of eachpoints of the two representative pairs. Then, whenconstructing the upper level E2 the information thatencodes the distinction between all the elements ofa class at the lower level is erased. In other wordsthe “internal position” of an element in a class hasbeen made irrelevant or superfluous (Ismael & vanFraassen, 2003; Castellani, 2003, § 3) as far as themanipulations of the elements of E2 are concerned.
Moreover, the main motivation of constructingequivalence classes is to obtain a richer structure.One can therefore truly speak of mathematical emer-gence in the sense that we can draw out properties
always stay at the level of “naive” set theory without taking thesubtle precautions which avoid paradoxes à la Russell, in par-ticular by distinguishing between the so-called proper classesand sets. Perhaps, category theory should provide a more suit-able framework, specially if one is akin to structural realism.As far as logical considerations are concerned, I mention thatI will also take for granted the axiom of choice that allows theoperation of projection.
in E2 that are not relevant for E1. The emergent no-tions (e.g. continuity or Borel measure) concern theelements of E2 (e.g. the real numbers) rather the el-ements of E1 (e.g. the integers).
According to the modern conception advo-cated by Klein, the characteristic properties ofa multiplicity must not be defined in terms ofthe elements of which the multiplicity is com-posed, but solely in terms of the group to whichthe multiplicity is related.[. . . ] The real foun-dation of mathematical certainty lies no longerin the elements from which mathematics startsbut in the rule by which the elements are re-lated to each other and reduced to a “unity ofthought” (Cassirer, 1944, § II, pp. 7–8).
3 Transformations
3.1 Active and passive physical inter-
pretations of a transformation
Let us now sew some physical flesh onto the mathe-matical bones I have introduced in the previous sec-tion. Let us begin with the first facet. As far as Iknow the distinction between passive and active in-terpretations of a transformation was made explicitlyfor the first time by Houtappel et al. (1965, § 2.4a).Furthermore I will follow Fonda & Ghirardi (1970,§ 1.7) and will consider that two active points ofview can be discriminated. I will emphasise that thethree physical interpretations of a transformation re-quire the division of the universe in three parts, nottwo: the system S (the object), the measurer M (thesubject, the observer) who describes quantitatively Swith the help of measuring devices, an environment Emade of everything else and that is necessary to de-tect if effectively S or M have been transformed ornot (figure 3). In a passive transformation, S re-mains anchored to E and the measurer is transformedto MT (for instance the measuring instruments, thereference frame, the standards are changed). In theactive points of view the system is changed with re-spect to E (in cosmology since, in principle, we con-sider S = E , the active points of view are meaning-less). Unlike in the first active point of view where
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p) Passive a1) Active 1 a2) Active 2
MT
MM
EE
MMT
STE
ST
S SS
Figure 3: Schematic view of the three physical interpretations of a transformation: p) In a passive trans-formation the same system S is described by two different measurers M and MT . For instance, they usetwo kinds of measuring instruments, different gaugings or standards and two different frames of reference tomap the space-time around S. In order to give a sense to “same” and “different” used above, it is necessaryto use an environment E as a reference. In the active points of view S is turned into ST and either M istransformed accordingly (a1) or remains unchanged (a2).
both M and S are changed, in the second active pointthe observer M remained unchanged. In the latter,
[The observations are] of the same ob-server, using one definite language to char-acterize his observations (Houtappel et al.,1965, § 2.4a, p. 601).
One can find an echo of such different points of viewat the mathematical level. For instance, in differen-tial geometry, (active) transformations, like paralleltransport, of geometrical objects such as vectors orforms defined on a manifold are conceptually diffe-rent from (passive) coordinates changes. However,in physics (and in all empirical sciences), active andpassive transformations have much stronger concep-tual differences (§§ 3.2 and 3.3) as well as importantoperational differences (§§ 3.4.2 and 3.5).
3.2 Moving the boundaries
Since S and M necessarily interact, the thresholdbetween them can be a matter of convention: in theMichelson-Morley experiment, we may decide freelyif the mirrors of the interferometer are included in Sor in M. As noted by von Neumann (1932/1955,§ VI.1), this convention is linked to the choice of whatfacts are considered to be observed, registered andacknowledged by M. For instance during an obser-vation of the Sun, M can consider that he countsthe sunspots directly or, rather, that he counts thedark spots that appear in the projection screen heuses; he can measure the degree of excitation of theretina cells of another human being looking throwa telescope or look on a PET scan screen the acti-vity of the corresponding brain areas during the ob-servation. Solving the a priori ambiguity of what a“direct” observable phenomenon or a “sense datum”is consists in choosing where we place, in a wholechain of processes, the border between S and M that
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defines the so-called “external world” (implicitly ex-ternal to M, for a comment on this expression seeSankey,2001, note 3). Consequently, before the inter-pretation in terms of correlations between data andin terms of concepts forged to cement them, this ope-ration of cutting the chain constitutes an unavoidabletask of science whose difficulty measures how remotewe stand from our human sensations and from ouroriginal evolutionary-shaped abilities (the so-calledintuition or common sense).
The important consequence is that, so far,we are left without criteria which would enableus to draw a non-arbitrary line between “ob-servation” and “theory”. Certainly, we will of-ten find it convenient to draw such a to-some-extent-arbitrary line; but its position will varywidely from context to context [. . . ] But whatontological ice does a mere methodologicallyconvenient observational-theoretical dichotomycut? Does an entity attain physical thinghoodand/or “real existence” in one context only tolose it in another? (Maxwell, 1962, pp. 7–8).
In sections §§ 4.2 and 4.3 below, I will propose anattempt, based on equivalence classes, to clarify thisontological point.
Besides, quantum physics renders these issues moreproblematic: since no generally well-accepted quan-tum theory has been proposed where the unicity ofthe result of one measurement emerges from purelyquantum processes, we still do not know how exactlythe transition between quantum systems and classicalmeasuring apparatus occurs. In the absence of a moresatisfactory solution, we are forced to keep the ortho-dox approach where Schrödinger equation is replacedby a discontinuous transition at the very moment ofmeasurement. In what follows, since I will not spe-cially be concerned by these quantum measurementissues (except in § 6.2), I will not distinguish betweenthe apparatus that prepares the system and the de-tectors. Within the very general level of the presentdiscussion, it is not important to specify which partof the degrees of freedoms belongs to S or to M andwe will not be concerned by their entanglement if any(the environment E may have disentangled them ac-cording to the decoherence process). Furthermore, asnoticed by Russell:
There has been a great deal of specula-tion in traditional philosophy which might havebeen avoided if the importance of structure, andthe difficulty of getting behind it, had beenrealised. For example, it is often said thatspace and time are subjective, but they haveobjective counterparts; or that phenomena aresubjective, but are caused by things in them-selves, which must have differences inter se cor-responding with the differences in the phenom-ena to which they give rise. Where such hy-potheses are made, it is generally supposed thatwe can know very little about the objectivecounterparts. In actual fact, however, if thehypotheses as stated were correct, the objec-tive counterparts would form a world having thesame structure as the phenomenal world, andallowing us to infer from phenomena the truthof all propositions that can be stated in abstractterms and are known to be true of phenom-ena.[. . . ] In short, every proposition having acommunicable significance must be true of bothworlds or of neither: the only difference mustlie in just that essence of individuality which al-ways eludes words and baffles description, butwhich, for that very reason, is irrelevant to sci-ence (Russell, 1919, chap. VI, p. 61).
I will therefore not consider the separation between Sand M to be of metaphysical origin and I will not en-dorse any kind of genuine dualism nor pluralism butprefer a realistic (materialistic) monism: the separa-tion between S and E (that demarcates the pertinentparameters from the irrelevant ones) together withthe separation between M and S consist not onlyon blurred but above all movable boundaries thatmust be moved to check the consistency of our lineof reasoning (for an interesting proposal that trendsto reinforce the conception of a unified ontology, seeClark,2008). For instance, when the effects of thecoupling to the environment are under the scope, thismay be implemented by splitting S in two: a subsys-tem coupled to a bath that furnishes a reduced modelof the “exterior”.
Where does that crude reality, in which theexperimentalist lives, end, and where does theatomistic world, in which the idea of realityis illusion and anathema begin ? There is, ofcourse, no such border; if we are compelled to
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attribute reality to the ordinary things of every-day life including scientific instruments and ma-terials used in experimenting, we cannot ceasedoing so for objects observable only with thehelp of instruments. To call these subjects realand part of the external world does not, how-ever, commit us in any way to any definite de-scription: a thing may be real though very dif-ferent from other things we know. [. . . ]
The boundary between the action of thesubject and the reaction of the object is blurredindeed. But this does not prohibit us from us-ing these concepts in a reasonable way. Theboundary of a liquid and its vapour is also notsharp, as their atoms are permanently evapo-rating and condensing. Still we can speak ofliquid and vapour7 (Born, 1953).
Indeed, since Samuel Johnson’s refutation of Berke-ley’s extreme idealism—obtained by merely “strikinghis foot with mighty force against a large stone, tillhe rebounded from it”(Boswell, 1791/2008, 6 august1763)—, the possibility of action on the world hasalways been an insight of what reality means. Activeand passive transformation just help to extend, pre-cise and formalise this common sense (by the way, theuse of the word “common” in this expression alreadybears the notion of invariance).
I will frequently adopt an Archimedean point ofview and look reality as a unique coherent blockwhere some part tends to explore the whole (the vir-tuous circle). In this perspective the self-consistentcondition can be seen as what is often called an an-thropic principle, at least in its weak form (Barrow& Tipler, 1986; Bostrom, 2010).
3.3 Reproducibility and objectivity
On the one hand, active points of view are at workwhen one wants to put to the test the reproducibil-ity of an experiment. Galilei’s ship(1632/1981, 2ndday (317))—see also its inspiring predecessors like
7No doubt that Born was also aware that one can bypassthe coexistence curve and circumvent the critical point in thepressure-temperature diagram by continuously connecting theliquid phase with vapour while remaining at the macroscopicscale all the way around.
Bruno’s ship thought experiment(1584/1995, thirddialogue) borrowed from an older argument used byCusa (1440/1985, II.12, p. 111)—is probably themost famous and, in a historical perspective, cer-tainly the most crucial illustration of a (first) activetransformation. An example, among many others, ofthe second active point of view can be found in theGibbs’interpretation of a thermodynamical ensemblewhere we consider “a great number of independentsystems” (Gibbs, 1902/2010, chap. I, p. 5 and thepreface). As usual in classical pre-twentieth centurytradition, the reference to any subject M is implicitbecause it is considered as being irrelevant (one no-ticeable exception being discussions on the Maxwell’sdemon).
Of course, from one experiment to another, timemay have evolved. This is reflected in (or equiva-lent to) many transformations that occurred in theenvironment E :
In this matter of causality it is a great incon-venience that the real world is given to us onceonly. We cannot know what would have hap-pened if something had been different. Wecannot repeat an experiment changing just onevariable; the hands of the clock will have moved,and the moons of Jupiter (Bell, 1977).
Whether the experiment is actually reproducible ornot will be a hint on the relevance of the part of theuniverse we have decided to consider as the system S.
On the other hand, the passive point of view putsto the test the intersubjectivity of an observation andtherefore paves the road to objectivity. An invarianceunder a passive transformation reveals or defines anentity as being (at least approximately) independentof M. To which extent this autonomy appears, andmore generally the degree of reality of S, is reflectedin the nature of the group of transformations T forwhich invariance occurs.
Before I specify T in the next subsection, fromwhat precedes we can already see how a bridge can beestablished between two main principles of the scien-tific method, reproducibility and objectivity: for thispurpose, symmetry is the appropriate tool (we shallcome back on this point below) .
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3.4 Relevant groups of transforma-
tions
Up to now, the strong affinity between classificationand transformations which was shown in § 2.2 mayappear of shallow interest to the physicist. Actually,one can classify anything with anything or even trans-form anything to anything else. When a property P isgiven a priori “Having the same property P as” definesautomatically an equivalence relation and the corre-sponding equivalence classes for which P is an invari-ant; reversely, if a partition of a set is given, the pro-perty P can always be identified with the membershipto a subset and the equivalence relation defined as“belongs to the same subset as”. The transformationsthat are most pertinent are those that can be appliedto a wide range of entities while keeping simple rules,that is, defined with less information than the set ofelements they act on (we will come back in § 4.5.3 tothe reduction of information provided by any classi-fication and therefore any symmetry). A reversibletransformation that turns a pumpkin into a coach,whether it concerns the three-dimensional objects ora morphing between two-dimensional images, mustencapsulate the huge amount of information than en-codes the pumpkin and the coach; it can hardly beapplied to anything else without artificially introduc-ing some additional ad hoc information. This is why,scientifically speaking, such a transformation is farless interesting than a rotation or a dilatation.
3.4.1 Algebraic representations
Indeed, the relevant groups of transformations thatare considered in physics have been abstracted in-dependently of the elements that are transformed.For instance, the usual transformations like space-translations or rotations have their roots in our intu-itive geometrical conception of the three-dimensionalspace; yet, in mathematical physics, they have nowacquired the status of an “abstract” group SO(3) thatcan be represented in many ways: they do not ap-ply to three-dimensional geometrical objects only butalso to a huge collection of mathematical objectsliving in much more abstract spaces (specially thequantum states and operators). Figure 4 provides
a schematic view of what an algebraic representa-tion of an abstract group is. The theory of abstractgroup and their representations constitutes a wholedomain of algebra and mathematical physics (amongmany treatises, see for instance (Cornwell, 1984),(Sternberg, 1994) or (Jones, 1998)).
G EE Tg
g
Figure 4: Schematic view of the algebraic representa-tion of an abstract group G in a set of elements E . Toeach element g in G is associated a transformation Tg
acting on the elements of E .
Many abstract symmetry groups have been shownto be relevant for the six purposes 1◦–6◦ listed in§ 1. Let us direct our attention on several substantialextensions that go far beyond the set of pure spatialisometries:
Evolution: First, the time evolution must be con-sidered as a transformation8. This strengthens the in-terpretation of the transformation in terms of repro-ducibility since the latter concerns space-translationsas well as time-translations. This is also in accor-dance with Einsteinian relativity that treats on analmost equal footing time and space.
Transformations mixing space and time: Sec-ond, in accordance to Galileo’s ship argument, we can
8Technically, if we are to describe irreversible processes wecan relax b’) in § 2.2 and work with a so-called semigroup.
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also include dynamical transformations correspond-ing to translations at constant speed (the transfor-mation of spatial positions depends on time). Gene-ral relativity incorporates any smooth transformationmixing space and time. Canonical formalism allowsto consider even more general transformations invol-ving mixture of spatio-temporal coordinates and mo-menta.
Exchange of particles: As suggested first byHeisenberg(1926; 1927, in particular eq. 8) and Dirac(1926), the permutations (or “substitutions”) of afinite number of elements (for which the term of“group” was originally coined by Galois in 1830) ap-peared of crucial importance for describing quan-tum collective effects of indistinguishable particles.Later on, Heisenberg (1932) considered the exchangeof non-identical particles, namely protons and neu-trons, to describe the spectral properties of nuclei.Another important example of transformations invol-ving changes of particles goes back to Dirac’s (1930)seminal work that led to the charge conjugation sym-metry (for instance, see Yang, 1994) where particlesmay be substituted with their antiparticles. Eventhough there is still no experimental evidence of itsrelevance, since the 1970’s, the transmuting of bosonsinto fermions and conversely have been introduced inseveral particle and string models. This concept ofsupersymmetry is an attempt to tackle some prob-lems raised by the Standard Model (Weinberg, 2000,§ 24.2), (Binétruy, 2006, chap. 1).
Gauge transformations: Field theories that de-scribe local interactions in condensed matter or inhigh energy physics, may require the introduction ofsome mathematical ingredients whose choice, in a fi-nite portion of space-time, is, to a large extent, a mat-ter of convention. A gauge transformation representsa change of convention and is, by definition of whata convention means, not expected to have any obser-vable influence. Hunting and discriminating system-atically the effects of these transformations on themathematical ingredients of a model provides suffi-ciently constrained guidelines to select the physicallyrelevant descriptions. There is no need to insist any
longer on the efficiency of such constructions through-out the twentieth century, from the elaboration of ge-neral relativity and Weyl’s first attempts starting in1918, up to Yang & Mills (1954) non-commutativegauge model and beyond. Among the vast literatureon the subject see, for instance, Moriyasu (1983) fora nice introduction.
3.4.2 Operational differences between activeand passive transformations
All the transformations described above have beenupgraded into abstract groups and can be alge-braically represented within the different speciesof the mathematical ingredients of various physicalmodels. However, if some transformations can beindifferently interpreted in both passive and activepoint of view, this is far from being the case for allof them.
Precisely because gauge transformations fulfil theirconstructive role when they have no detectable con-sequences, they can only be interpreted as passivetransformations (Brading & Brown, 2004, and itsreferences). The same situation occurs in classicalmechanics: after a relevant canonical transforma-tion that interweaves them, the dynamical variablesmay loose the empirical interpretation they had be-fore being transformed (say, the positions and veloc-ities of the planets when blended into action-anglevariables). Therefore, such transformations appearclearly as a change of description of the dynamicsnot as an active transformation of the system.
Examples of active (resp. passive) transformationsthat cannot naturally be considered from the passive(resp. active) point of view are given by transforma-tions associated with an exchange, addition or sub-traction of some constituents of the system (resp. themeasuring device). If we are ready to accept a per-mutation between quantum particles as a candidatefor being in T , why should not we consider also thereplacement of a spring in a mechanical apparatus orthe modification of the wavelength of a probe? Eventhough, these kinds of transformations are never (asfar as I know) considered when talking about symme-try, this is precisely invariance with respect to them,as being discussed in § 5 below, that allows eventually
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to establish stable and universal laws. In practise,when attempting to reproduce an old or a remoteexperiment, the experimental set up is rarely movedbut rather completely rebuilt from new materials.
3.5 Perception of space-time trans-
formations: following Henri
Poincaré’s intuitions
In fact, a way of understanding which mathemati-cal transformations can be interpreted both passivelyand actively was presented more a century ago byPoincaré(1895; 1903, § 5) when analysing carefullyour intuition of three-dimensional space (his consid-erations can be applied to a large extend to the four-dimensional space-time). He introduced a distinctionthat Einstein would sum up four decades after him:
Poincaré has justly emphasized the fact thatwe distinguish two kinds of alterations of thebodily object, “changes of state" and “changesof position." The latter, he remarked, are alter-ations which we can reverse by arbitrary mo-tions of our bodies (Einstein, 1936, § 2, pp. 354–355).
However, what does not appear clearly in this re-port (but was criticised by Einstein elsewhere, forinstance in his (1949, p. 677) reply) is that, accordingto Poincaré, this is a matter of definition or conven-tion:
Thence comes a new distinction among ex-ternal changes: those which may be so cor-rected we call changes of position; and theothers, changes of state (Poincaré, 1903, § 5,I underline).
For Poincaré, the possible compensation of an ac-tive transformation by a passive one is therefore in-timately linked with our perception of space.
For a being completely immovable therewould be neither space nor geometry; in vainwould exterior objects be displaced about him,the variations which these displacements wouldmake in his impressions would not be attributedby this being to changes in position, but tosimple changes of state; this being would have
no means of distinguishing these two sorts ofchanges, and this distinction, fundamental forus, would have no meaning for him (Poincaré,1903, § 5).
However, following Galileo’s conception of uni-form speed, Poincaré was aware that transforma-tions could not be considered absolutely and, as soonas 1902, he clearly formulated in chap. V, § V, ofScience and Hypothesis a “relativity law”. The trans-formations must be anchored somewhere: this isprecisely the main reason why we must introducethe environment E as a background and that Ein-stein (1936, § 2, p. 354) would call a “bodily ob-ject”. Otherwise, the first active point of view wouldbe undetectable and the discrimination between thepassive and the second active point of view wouldbe meaningless. Extending an old argument goingback to Laplace (according to Jammer, 1994, chap. 5,pp. 168–169), Poincaré not only discusses the case ofa uniform expansion of the universe but also consi-ders any continuous change of coordinates:
But more; worlds will be indistinguishablenot only if they are equal or similar, that is, ifwe can pass from one to the other by changingthe axes of coordinates, or by changing the scaleto which lengths are referred; but they will stillbe indistinguishable if we can pass from one tothe other by any ‘point transformation’ what-ever. [. . . ] The relativity of space is not ordi-narily understood in so broad a sense; it is thus,however, that it would be proper to understandit (Poincaré, 1903, § 1).
Poincaré considers that only topological structures—being unaffected by homeomorphisms—are physi-cally relevant. For him the choice of differentialstructure, in particular through the metric, is a mat-ter of convention similar to a choice of units or achoice of language. The imprint of curvature in thesum of the angles of a triangle or on the ratio betweenthe perimeter of a circle and its radius can always berectified empirically.
Think of a material circle, measure its ra-dius and circumference, and see if the ratio ofthe two lengths is equal to π . What have wedone? We have made an experiment on the
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properties of the matter with which this round-
ness has been realized, and of which the mea-sure we used is made (Poincaré, 1952, chap. V,§ 2).
and also
The same question may also be asked in an-other way. If Lobatschewsky’s geometry is true,the parallax of a very distant star will be finite.If Riemann’s is true, it will be negative. Theseare the results which seem within the reach ofexperiment, and it is hoped that astronomicalobservations may enable us to decide betweenthe two geometries. But what we call a straightline in astronomy is simply the path of a ray oflight. If, therefore, we were to discover nega-tive parallaxes, or to prove that all parallaxesare higher than a certain limit, we should havea choice between two conclusions: we could giveup Euclidean geometry, or modify the laws ofoptics, and suppose that light is not rigorouslypropagated in a straight line. It is needless toadd that every one would look upon this solu-tion as the more advantageous. Euclidean geo-metry, therefore, has nothing to fear from freshexperiments (Poincaré, 1891).
Albeit historical development of the theory of grav-itation has refuted Poincaré’s opinion that
Euclidean geometry is, and will remain, themost convenient: 1st, because it is the sim-plest, and it is not so only because of our men-tal habits or because of the kind of direct in-tuition that we have of Euclidean space; it isthe simplest in itself, just as a polynomial ofthe first degree is simpler than a polynomial ofthe second degree; 2nd, because it sufficientlyagrees with the properties of natural solids,those bodies which we can compare and mea-sure by means of our senses (Poincaré, 1891).
when gravitation was formulated as a gauge theoryby Utiyama (1956) then working either with Euclid-ian/Minkowskian or Riemann geometry appeared in-deed to be a conventional choice. See Carnap (1966,chaps. 15–17), Stump (1991) or Hacyan (2009) for amore detailed discussion on these issues. (We putaside the topological properties of space-time, like
worm-holes at large scales or any kind of topologi-cal defect that may have physical observable conse-quences which would definitely rule out Minkowskiangeometry).
Even if we keep sticking to the Riemann geome-try, we can remark that a confusion remains thatobscures the debate between realists and convention-alists (or their relativist post-modern successors). Arepresentation (here a choice of one coordinate map)is always, obviously, a matter of convention (it fixesa gauge); however, when quotienting all the equiva-lent descriptions obtained one from each other by atransformation (the change of coordinates), we canabstract geometrical objects with intrinsic properties(the Riemannian manifold itself and the tensors de-fined on it). Formally from a set of quantities indexedby coordinates labels, say vµ, whose values changewhen changing the coordinates from (xµ)µ to (xµT )µ,
vµT =∂xµT
∂xνvν , (3)
an abstract object can be defined, the tangent vec-tor v, which is independent of the precise choice ofcoordinates. While the coordinates (3) are said tobe co-variant under the (passive) transformation thatchanges the coordinates, the vector v remains insen-sitive to it. Besides, to emphasise the intrinsic char-acter of the construction, rather than starting fromthe transformation rule (3), a vector at a point xis preferably defined in differential geometry, as be-ing an equivalence class of tangent differential curveson the manifold passing through x (Choquet-Bruhat& DeWitt-Morette, 1982, chap, III, B.1, pp. 120–121). This geometrisation procedure was the core ofKlein’s Erlanger program(1893), (see Cassirer, 1944,for an historical account of the connections between agroup of geometrical transformations, perception, in-variance and objectivity. See also the excerpts givenabove in § 2.3 and below in § 5) and one of the fa-thers of the modern notion of symmetry was alreadyaware of its physical connection with objectivity (fora recent survey, see Kosso, 2003, § 2):
Without claiming to give a mechanically ap-plicable criterion, our description bears out theessential fact that objectivity is an issue de-cidable on the ground of experience only.[. . . ]
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Whereas the philosophical question of objectiv-ity is not easy to answer in a clear and definitefashion, we know exactly what the adequatemathematical concepts are for the formulationof this idea.[. . . ] The pure mathematician willsay: Given a group G of transformations in afield of symbols, a geometry is established byagreeing to study, and consider as objective,only such relations in that field as are invariantunder the transformations of G (Weyl, 1949,§ 13, pp. 72 and 77).
Perhaps the philosophically most relevantfeature of modern science is the emergence ofabstract symbolic structures as the hard coreof objectivity behind—as Eddington puts it—the colourful tale of the subjective storytellermind (Weyl, 1949, Appendix B, § 1, p. 237).
and also
But I preferred the name of automorphisms[. . . ], defining them with Leibniz as those trans-formations which leave the structure of spaceunchanged.[. . . ]
We found that objectivity means invariancewith respect to the group of automorphisms.Reality may not always give a clear answer tothe question what the actual group of automor-phisms is, and for the purpose of some investi-gations it may be quite useful to replace it by awider group. For instance in plane geometry wemay be interested only in such relations [that]are invariant under parallel or central projec-tions; this is the origin of affine and projec-tive geometry. The mathematician will preparefor all such eventualities by posing the generalproblem, how for a given group of transforma-tions to find its invariants (invariant relations,invariant quantities, etc.), and by solving it forthe more important special groups—whetherthese groups are known or are not known to bethe groups of automorphisms for certain fieldssuggested by nature. This is what Felix Kleincalled "a geometry" in the abstract sense. Ageometry, Klein said, is defined by a groupof transformations, and investigates everythingthat is invariant under the transformations ofthis given group (Weyl, 1952, chap. II, p. 42and chap. IV, pp. 132–133).
Poincaré’s strong conventionalism far from beingantithetic to objectivity appears to be a necessarycondition for the latter:
Now the possibility of translation impliesthe existence of an invariant. To translate isprecisely to disengage this invariant.[. . . ]
What is objective must be common to manyminds and consequently transmissible from oneto the other, and this transmission can onlycome about by [a] “discourse” [. . . ] we are evenforced to conclude: no discourse, no objectivity.[. . . ]
Now what is science? I have explained[above], it is before all a classification, a mannerof bringing together facts which appearancesseparate, though they were bound together bysome natural and hidden kinship. Science, inother words, is a system of relations. Now wehave just said, it is in the relations alone thatobjectivity must be sought; it would be vain toseek it in beings considered as isolated from oneanother.
To say that science cannot have objectivevalue since it teaches us only relations, this isto reason backwards, since, precisely, it is rela-tions alone which can be regarded as objective.
External object, for instance, for which theword object was invented, are really objects
and not fleeting and fugitive appearances, be-cause they are not only groups of sensations,but groups cemented by a constant bond. Itis this bond, and this bond alone, which isthe object in itself, and this bond is a relation(Poincaré, 1902, §§4 and 6, I underline).
The concept of symmetry allows to give a quan-titative meaning to the terms used in this remark-able text: “Relations” and “classification” have al-ready been discussed in § 2.2. “Translation” (nottaken here to synonymous of a shift!) appears to be apassive transformation in the sense considered in sec-tion § 3.1, and its relation with objectivity has beenemphasised in § 3.3; “object” being considered as anequivalence class, an “invariant”. Whereas “discourse”(we have seen above that Houtappel et al. also usethe term “language”) corresponds to a choice of con-vention (but, of course, Poincaré is far from endorsingnominalism, see § 4 of his1902 reflexions), a coordi-nate chart in differential geometry or more generallya representative of an equivalence class, that “breaks”the symmetry (or, rather, the invariance).
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4 Comprehension and classifica-
tion
Poincaré’s last remark on the importance of classifi-cation and its role to identify invariants leads us tofocus on the second facet of the concept of symmetry.
Comprehension is an old scholastic notion and itis not by chance that the same word is used to meanthe act of comprehending and the act of understand-ing. The latter requires to find analogies9, correla-tions, regularities between a priori distinct entities orphenomena; these connections are the most preciselyexpressed by equivalence relations (incidentally, re-latives constitute the fourth Aristotelian category;Aristotle1938, § IV, pp. 16–17) which are synony-mous, as we have seen, to classification and thereforeto unification. The common qualities shared by theelements of one class are the invariants. The dua-lity between a class and its elements reactivates, touse a terminology with some medieval taste, the oldopposition between universals and individuals.
4.1 From taxonomy to predictability
We do not have to probe very deeply to acknowledgethe major role of taxonomy in science. Far from beinga purely descriptive tool, sorting things out is aboveall relevant when it reveals some objective proper-ties, as Poincaré explains in the last quotation givenabove. In biology, for instance, the hierarchical clas-sification initiated by Carl Linnaeus which were origi-nally based on phenotype only, allowed to reveal somecorrelations that could be explained within the gene-ral scheme of the theory of evolution10. The degree ofsimilarity between some physical or even behaviouralcharacteristics of living organisms have been a rele-vant indicator of the chronology of the divergence be-tween species. Even before the genetic distance waseven conceived, these correlations could be associated(through correlations of correlations, so to speak) to
9See Kant (1781/1998, I, 2nd part Transcendendal logic,1st division, book II, chap. II, § III.3, Analogies of experience)and Cournot (1851/1956, chap. IV § 49).
10More generally, the role played by symmetries is of thegreatest importance in all hierarchy-based philosophies, parti-cularly in Cournot’s (1861, book IV, chap. XIV, § 514).
the history of our planet (climate fluctuations, atmo-sphere composition, continental drifts, etc).
Another example is provided by the appropri-ate way of classifying chemical elements which al-lowed Mendeleev to some successful predictions onthe properties of elements that were yet unknown in1869. Without knowing the atomic structure (at anepoch where only few physicists accepted the exis-tence of atoms), this periodic classification exhibitssome universal characteristics that were not under-mined by the subsequent improvements of the phy-sical theories and the numerous models of the atomsthat came with them (first classical, like the Kelvin’svortex atoms in 1867 and later on, of course, thequantum models).
During the second half of the twentieth century,symmetry has strengthened its classificatory and pre-dictive role. Originated in the 1930’s by Wigner—who received in 1963 the Nobel price in physicsprecisely “for his contributions to the theory of theatomic nucleus and the elementary particles, parti-cularly through the discovery and application of fun-damental symmetry principles"—our modern classi-fication of quantum particles, whether there are sup-posed to be elementary or not, relies on the classifica-tion of the (linear) algebraic representations (in thesense defined in § 3.4.1) of groups like the Poincarégroup (Wigner’s philosophical reflections on symme-try are collected in his1995 book. For a modernpresentation of technical details see Weinberg, 1995,chap. 2). Unlike taxonomy of macroscopic objects,the classification of quantum particles relies on avery few parameters only, like the mass, the elec-tric charge, the spin. The reason is that in quan-tum physics only very few observables are compati-ble with all the others (technically, those associatedwith operators that commute with all the others,which are associated with the so-called superselectionrules). Otherwise, an observed property is genericallycompletely erased by the measurement of an incom-patible observable: quantum particles have no his-tory nor accidents and quantum field theory naturallyincorporates the indistinguishability of the particlesbelonging to the same class. Anyway, the hadronclassification using the quark model or the predictionof the existence of particles like the W±, Z0 or the
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Higgs boson come from the Standard Model whichwas built on symmetry arguments. These providesanother illustration, like in Mendeleev’s contribution,that symmetry is a signature of an objective reality,the latter being taken in the sense given by Poincaréin the last quotation.
4.2 Inference and the problem of
change
Since any time-translation can be considered as aparticular case of symmetry transformation (§ 3.4),the foresight skills conferred by evolution equations(whether classical or quantum) appear as the mostimportant aspects of the predictive power providedby symmetry. Obviously, this ability of anticipationcovers fields much broader than science. There isno need to solve differential equations to successfullyforecast that the sun will rise tomorrow morning. Af-ter having closed your eyes, you know with almostcertainty that you will find the words you are cur-rently reading in the same order once you look againon this page. There is no miracle here (Putnam, 1975,chap. 4, p. 73), just a consistency for rational think-ing, a raison d’être: no thoughts, no language couldbe possible if we were unable to discern some “cons-tant bond” as Poincaré wrote. These stable proper-ties (at least for a moment long enough to be no-ticeable) reveal what we call the existence of things,building up together what we call the real world(including ourselves). A macroscopic object like acherry can be considered as an equivalence class asso-ciated with properties (shape, taste, colour, size, etc.)that remain invariant despite the change of light, theendless adsorption or desorption of molecules (ac-tive transformations in the sense explained in § 3.1)or the (passive) transformations of the spatial po-sition of the measurer or of his neural states (seealso Bell’s quotation given in § 3.3). As remarked byHeisenberg (1958, chap. IV) or Popper (1969, chap. 5,§ IX), this “problem of change” (or of becoming) hasalready been settled by pre-Athenian philosophersamong which Heraclitus and Parmenides, who pro-posed the two most extreme solutions along the wholecontinuous spectrum of possible answers.
We will come back in section 5 to the notion of
invariance, which constitutes the third facet of sym-metry. However, at this stage, we can usefully noticethat symmetry as a predictive tool provides an in-termediate support for the inductive/deductive infer-ence. Induction, as a reasoning from a part to a wholeis formally described by the operation x 7→ σ(x)while deduction, processing from general to partic-ular instances, offers a second look at the fourthfacet of symmetry, formally described by the opera-tion σ 7→ x.
4.3 Stability and abstraction
As analysed by Taine 150 years ago, any intelli-gent being is characterised by his ability of cons-tructing a representation of the world (see alsoCournot, 1851/1956, chap. VII, § 109):
Thus, every normal sensation correspondsto some external fact which it transcribes withgreater or less approximation, and whose in-ternal substitute it is. By this correspondence,internal events agree with external, and sensa-tions, which are the elements of our ideas, findthemselves naturally and beforehand adjustedto things, which adjustment will, further on, en-able our ideas to be in conformity with things,and consequently true. On the other hand, wehave seen that images are substitutes for sensa-tions, past, future, or possible, that individualnames are substitutes for images and sensationsmomentarily absent, that the more simple gene-ral names are substitutes for images and impos-sible sensations, that the more complex generalnames are substitutes for other names, and soon. It seems then that nature has undertakento provide in us representatives of her events,and has effected her purpose in the most eco-nomical way. She has provided, first, the sen-sation which translates the fact with more orless precision and delicacy; then, the survivingsensation capable of indefinite revival, that is tosay, the image, which repeats the sensation, andconsequently translates the fact itself; then, thename, a sensation or image of a particular kind,which, by virtue of its acquired properties, rep-resents the general character of many similarfacts, and replaces the impossible images andsensations which would be necessary to trans-
17
late this isolated character. By means of thiscorrespondence, of this repetition, and this re-placement, external facts, present, past, future,special, general, simple, or complex, have theirinternal representatives, and this mental rep-resentative is always the same internal event,more or less compounded, repeated, and dis-guised (Taine, 1872, end of book III, chap. II,§ V, pp. 141–142).
This correspondence between the world and a men-tal representation11 is necessarily partial not only be-cause a brain, as a tiny part of the universe, canonly encode far less information than the universeitself (James, 1880), but also for the sake of theefficiency that enforces its own stability as an or-ganised structure, as needed by Darwinian compet-itive selection (Popper,1979, § 2.16; Dawkins,1976,chap. 4; Dennett,1996, chap. 4). The processing ofinformation—memorising, learning, anticipating—byany intelligent entity cannot be done without a dis-crimination between relevant and irrelevant parame-ters (by the way, the famous Wisconsin card sortingtest allows neuropsychologists to quantify the adop-tion of classification rules and measure its flexibilityby trial and error, see for instance § 6.5, p. 412 ofBaars & Gage, 2010).
Ireneo Funes, the character dreamt by Jorge LuisBorges, whose memory was rigorously infallible,
was almost incapable of general, platonicideas. It was not only difficult for him to un-derstand that the generic term dog embraced somany unlike specimens of differing sizes and dif-ferent forms; he was disturbed by the fact thata dog at three-fourteen (seen in profile) shouldhave the same name as the dog at three-fifteen(seen from the front) (Borges, 1942, Funes, thememorious, p. 114).
11Not surprisingly there is an abundant literature that dis-cusses the congruence between objects and beliefs since it de-termines the nature of truth. As old as western philosophyitself, this subject is omnipresent in logic, in linguistic, in psy-chology, in jurisprudence, in epistemology etc. If we are readyto move the boundaries as suggested in § 3.2, the traditionalopposition between coherence and correspondence theory oftruth should be softened (see also the discussion in Weyl, 1949,§ 17, p. 117).
He who cannot forget cannot generalise; he whocannot generalise cannot think. Classification ap-pears therefore as a fundamental thought process andhence has been the subject of a wide field of researchin cognitive sciences where the synonym “categorisa-tion” is usually preferred (Rosch,1978, Smith,1998;Murphy,2002, chap. 7; Machery,2009, chap. 6, forrecent surveys on this matter). Abstraction, forg-ing concepts, using a language, constructing mod-els and eventually building up theories are varia-tions on the same theme, namely, the second facetof symmetry. Observation itself appears to bean act of abstraction—actually the lowest level ofabstraction— inconceivable without entities, the so-called observable entities, that are indeed concepts(Duhem, 1906/1954, part 2, chap. IV, §§ 1,2 andchap. VI, § 1) emerging when pruning the informationout of a bunch of data extracted from a magma of oc-currences. Following Hanson (1958, chap. 1, § C) andtalking about theory-laden observations is thereforeawkward, unless we discriminate multiple levels of ab-straction, because it supposes a theory-independentcore to be loaded and, thus, this leads immediatelyto a self-contradiction since this core appears to bean abstract notion in itself. In that sense, in my crit-icism of empiricism, I am going further than Hansonand according to Sellars (1956) I would rather prefertalking about the “myth of the given”.
In sum, facts are facts, and if it happens
that they satisfy a prediction, this is not an ef-
fect of our free activity. There is no precisefrontier between the fact in the rough and thescientific fact; it can only be said that suchan enunciation of fact is more crude or, onthe contrary, more scientific than such another(Poincaré, 1902, § 3).
De facto, the degree of precision of any classifica-tion depends on the criteria we retain. Fruits can beclassified according to the kind of desserts you canmake with or according to the genome of its plant.Stars can be grouped in constellations or in galaxies(Cournot, 1851/1956, chap. XI, § 161). The crite-ria are always moving, are often arbitrary or contin-gent and maybe difficult to define without referringto the category itself: the most efficient way to ex-plain to a child what a cherry is remains to show him
18
several examples of this fruit, not to give a defini-tion (Poincaré, 1903, § 4). This intuitive approach isall the more natural than brains are instinctively in-clined to think in terms of associations that nurtureits creativity (Hadamard, 1954; Weil, 1960, for ins-tance). Unlike computers, brains are analogue andselectionist rather that digital and instructionist de-vices (Popper,1978, p. 346; Dehaene,1997, speciallythe chap. 9). The reluctance of being classified of-fered by certain elements requires reflection, testifiesthe weakness of our comprehension and give evidenceof the rich variety and of the impermanence of theworld. A thing or a word can come with many varia-tions of significations, without which there cannot benuances, ambiguities, metaphors and even poetry.
But unlike art, rational thinking and, consequently,its most elaborated form that science represents,tends to sharpen and perennialise the boundaries ofthe equivalence classes; it brings out criteria of truthbeyond cultural barriers and socio-historical context.This strategy allows to reach provinces far away fromour natural (i.e. adequate from an evolutionary pointof view) narrow county and, as we have seen inthe previous section, it allows to make predictions(Llinás, 2001, specially chap. 2). I have already men-tioned (§ 2.3) the hierarchy of numbers in mathema-tics; it extends the concept of numbers much furtherthan the primitive mental representations of integers,genetically encoded with a broad brush in highestmammal brains (Dehaene, 1997; Butterworth, 1999).In physics, the both prominent frameworks of gene-ral relativity and quantum theory constitute chartsat scales considerably much wider than our every-daylife environment. If we adopt the adequation I pro-pose between existence and equivalence classes, thequestions of knowing if we invent or rather discoverthese provinces, if symmetry unveils or sculpturesreality, appear to be unessential (see also Margolis& Laurence, 2007); it is a matter of taste depen-ding from where we prefer to observe the virtuouscircle (for a particularly interesting examination ofthis old subject, namely the “reality of abstractions”,see Cournot, 1851/1956, chap. XI, specially the ar-guments in §§ 152 and 155). In addition, this orien-tation also allows to soften the question of the ade-quacy between the mathematical world and the phy-
sical world: if we merge these two worlds then theeffectiveness of mathematics does not appear as anunreasonable miracle (Wigner, 1960) but it is stilltrue that it is puzzling to see how far this fitness canspread.
4.4 Different levels of Darwinian se-
lection
The segmentation of the world in equivalence sym-metry classes, and the drawing of more or less fuzzyboundaries between them (the most fundamental be-ing the partition in M, S and E as explained in §§ 3.1and 3.2), can be seen as a selection process. JacquesHadamard (1954, first note of chap. III, p. 29) re-calls that the Latin original meaning of intelligenceis “selecting among” (inter-legere) and echoes thePoincaré’s statement:
What, in fact, is mathematical discovery?It does not consist in making new combina-tions with mathematical entities that are al-ready known. That can be done by any one, andthe combinations that can be so formed wouldbe infinite in number, and the greater part ofthem would be absolutely devoid of interest.Discovery consists precisely in not constructinguseless combinations, but in constructing thosethat are useful, which are an infinitely smallminority. Discovery is discernment, selection(Poincaré, 1908).
It is of course Darwin’s major contribution to haveshown how differential survival governs the evolutionof living species. This natural selective stabilisationis a (non-linear) mechanism at work in many otherdynamical systems made of different entities whosepopulations are susceptible of attenuation and am-plification (via reproduction, replication, combina-tions, etc.). It would be too long a digression, andbeyond my competence, to examine the subtle dis-tinctions of how Darwinian selection is implementedin so various fields as immunology, cognitive sciences,computer science, sociobiology and culture. Even inevolutionary biology, the details of the picture arestill objects of scientific controversies between spe-cialists like Richard Dawkins, Stephen Jay Gould,
19
William D. Hamilton, Richard Lewontin, John May-nard Smith, Edward Osborne Wilson, to name somepopular contemporary figures among many others(Segerstråle, 2000; Shanahan, 2004).
To remain within the scope of the present paper, Ijust wanted to point out that the formation of someequivalence classes I denoted by σ can be seen as a na-tural process à la Darwin not only when σ denotes aliving species, a pool of genes or of memes (elementsof material or immaterial culture, including knowl-edge, Dawkins, 1976, chap. 11), but also if we adopt aneurobiologist point of view. Any class σ, consideredas an object i.e. a part of the physical world (a preda-tor, a cherry, a pixel, an atom) may be representedin terms of some neural configuration distributed inseveral intertwined areas of the brain. Since moreabstract concepts (
√2, a quantum phase, entropy, a
blue tiger, Prometheus, beauty, freedom, etc.) and,more generally, ideas are also encoded in brains viathe neural material support, the insurmountable dif-ficulty of delimiting what reality is (see § 1) can beunderstood here, when we stand in the virtuous circleat the neurophysiological level (the recursive charac-ter appears here through consciousness, i.e. when themodel of the world that characterises intelligence in-cludes a representation of itself). Some concepts (acherry, the Moon, an electron, a number, Kepler’slaw, a rotation, yourself) are strongly reinforced andstabilised by being intricately bound within a largehierarchy, including observations or sense-data, whilesome others (a blue tiger, a tachyon, chess rules, thepumpkin-coach transformation, Homer), being lessfirmly established, have a smaller degree of reality. Asany practitioner of quantum physics knows it (evenbefore the modulus of some electronic wave-functionin quantum corrals has been imaged by scanning mea-surement), the concept of a quantum state generallyrefers to a more real entity that the toy concept of asolid sphere made of gold with a radius of 1 km.
Recent investigations in neuroscience seem togive a neural basis of an old suggestion made byJames (1880) and to confirm the existence of Dar-winian (epigenetic) selective mechanisms betweenvarious pre-representations generated by combinato-rial games throughout neuron networks. Many se-lectionist processes may be distinguished that are
characterised by transformations of neural states atdifferent time-scales; compared to the fast dynamicsof, say, decision making, the neural Darwinism in-volved in learning and memorisation may concernslower developments of neural circuitry, starting be-fore birth soon after conception (dendritic and ax-onal generation or regression, synaptic reinforcementor inhibition, etc.). The “survival of the fittest" rep-resentation would be determined by complex testsof evaluation, reentrance and reinforcements proce-dures (driven by pleasure and pain for instance) thatinvolve the internal dynamics of the brain as well asits coupling to its external environment (includingother brains). We still know very few about how thisdynamics is physiologically implemented; however, itshares many common points with selectionism at an-other level, namely the evolution of scientific knowl-edge. Since the first proposals of an “evolutionaryepistemology” by Campbell (1959, 1960 and their nu-merous references that anticipated the subject even ifJames is not mentioned) and Popper (1979) (see alsoToulmin, 1967; Toulmin, 1972; Campbell, 1974), thisapproach has known many outgrowths (Radnitzky &Bartley, 1987; Hull, 2001). For a recent review ofthe neural Darwinism introduced by Edelman(1978;1987), see Seth & Baars (2005), Platek et al. (2007)and their references. For a less specialised presenta-tion see, for instance, Changeux(1985; 2002) or Edel-man (2006). The emergence of a pattern of suc-cesses from an “overwhelming background of failures”(Fine, 1996, chap. 7, § 1, note 5) in science is there-fore explained in an analogous way as the formationof stable and viable combinations of genes from theoverwhelming possibilities of deadly mutations (as faras the cumulative progress of knowledge is concerned,we have not been able to make up the memetic ana-logue of such an efficient process like sexuality forgenerating reliable combinations. However, in thesematters, from the association of ideas, both at themental level and at the human collaboration level,we are closer to the memetic equivalent of meiosisthan to agamogenesis, see however the conservativetrends described by Toulmin,1967, § VI).
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4.5 Symmetry-assisted research of
simplicity
4.5.1 Simplicity as a general evaluationagainst underdetermination
Most obviously, one may fit a given discrete set ofregistered events with many different laws (Cournot,1851/1956, chap. IV). Still, this so-called underde-termination, acknowledged under one form or an-other since Epicurus (in the Letter to Pythocles, seeLaertius, 1925, chap. X, mainly §§ 86–88) and care-fully examined by Hume (1758/1999), remain abun-dantly discussed in epistemological debates on induc-tion (Laudan, 1990; Psillos, 1999, chap. 8). Notwith-standing, underdetermination appears to be less se-rious than some philosophers think it is, providedthat we take into account the notion of simpli-city as a selection criterion (see Poincaré’s 1891 ci-tations given above; Cournot,1851/1956, chap. IV,§ 41; Jeffreys,1931, chap. IV; Popper,1959, chap. VIIand1969, chap. 1, point (6) of the appendix, chap. 10,§ XVIII; Reichenbach1938, § 42). Following the samelines of thought drawn above, the aim of this sectionis to suggest how the second facet of symmetry is atwork in this Darwinian-like selective operation.
The problem of simplicity is of central im-portance for the epistemology of the naturalsciences. Since the concept of simplicity ap-pears to be so inaccessible to objective formu-lation, it has been attempted to reduce it tothat of probability, which has already been in-corporated to a large extent into mathematicalthought (Weyl, 1949, § 21, pp. 155–156).
Here we join back the aesthetical appeal of asymmetry we briefly recalled in the very beginningof § 2.1. The harmony of a theory, the beautyof an argument or the elegance of an experimentare frequently evoked by scientists (Poincaré,1908;Russell,1917, chap. IV; Hardy,1940; Weyl,1949,§ 21; Heisenberg,1974, chap. XIII; Penrose,1974; Lipscomb,1980; Yang,1980; Dirac,1963,1982;Chandrasekhar1987; Weinberg,1992, chap. VI), someof them considering it as a constructive principle ifnot a criterion of truth. For a curious attempt toquantify aesthetics using a, somehow vague, notion
of complexity see Birkhoff (1933); for a more seriousand interesting study of the role of aesthetics in sci-ence see McAllister (1996) where Haldane’s text be-low is partially quoted (chapter 7 provides an actu-alised discussion on simplicity, see also the diversecontributions on this subject in Zellner et al.,2001).One reason of this attractiveness is the gain in sim-plicity, a parsimony of descriptive means with respectto the wide range of their scope.
In scientific thought we adopt the simplesttheory which will explain all the facts underconsideration and enable us to predict new factsof the same kind. The catch in this criterion liesin the word ‘simplest’. It is really an aestheticcanon such as we find implicit in our criticismsof poetry or painting. The layman finds such
a law as ∂x∂t
= κ∂2x
∂y2 much less simple than ‘itoozes’, of which it is the mathematical state-ment. The physicist reverses this judgement,and his statement is certainly the more fruitfulof the two, so far as prediction is concerned. Itis, however, a statement about something veryunfamiliar to the plain man, namely, the rateof change of a rate of change. Now, scientificaesthetic prefers simple but precise statementsabout unfamiliar things to vaguer statementsabout well-known things. And this preferenceis justified by practical success (Haldane, 1927,Science and Theology as Art Forms p. 227).
Broadening the scope of a theory and reducingthe number of laws and primitive concepts are twofaces of a single coin. The tension between thesetwo antagonistic tendencies is the mainspring of theprogress of scientific knowledge. Focusing on justone of those opposite poles leads to ridiculous po-sitions: on the one hand, if we consider only veryfew particular events, we can explain their occur-rence by arbitrary simple or rather simplistic lawsand concepts (this is one of the main characteristicof superstitions and pseudo-sciences) and the com-plication comes from the multiplication of such adhoc explanations as the number of observations in-creases (see for instance Popper,1969, chap. 1, § Ipoint (7) and its appendix or Worrall,1989, p. 114).On the other hand, as Feynman puts it with hisinimitable style (Feynman et al., 1970, vol. II, § 25-6), from any physical law (not necessarily discovered
21
yet!) labelled by l, once written in the (dimension-less) form Ql = 0, we can always introduce the phy-sical concept of “unworldliness” associated with thislaw, namely Ul
def
=|Ql| where | · | denotes the modulusor the norm, and write the “equation of everything”as
U = 0 (4)
where the total unworldliness is given by U =∑
l Ul.Of course this trick to hide the complexity under thesimplest conceivable equation remains sterile becauseof the artificial and shallow interpretation of U . AsFeynman explains it, this stratagem is not motivatedby any symmetry arguments: because it does not takeinto account any transformation rules, its nature isvery different from the grouping of the scalar quan-tities vµ into one compact geometrical object v (see§ 3.5 p. 14).
4.5.2 Simplicity from an algorithmic point ofview: conjectures and refutations
Since the 60’s theoretical computer science have pro-vided us with clues of how to give a quantified andobjective measure of complexity (Li & Vitányi, 2008,for an upgraded account on this vivid field and forthe original references). Most notable are the notionsof Solomonoff-Kolmogorov-Chaitin complexity C(x)and Bennett’s logical depth D(x) of an element x be-longing to a countable set. Aside from technicalitiesthat allow stable definitions (i.e. reasonably indepen-dent on the way computations are done; objectivityagain !), C(x) is given by the length of the short-est description of x and D(x) essentially counts thenumber of steps that is required to recover x from atypical minimum description of x. In more intuitiveterms, C(x) expresses quantitatively the amount ofrandomness in x (in other terms, the lack of order,the lack of redundancy) while D(x) expresses quan-titatively how this superabundance of information isencoded in x (it measures the degree of organisationof x). These two interpretations of the informationcontent of x rely on a principle of economy, respec-tively of description and of elaboration of x, thatechoes the notion of simplicity mentioned above. Itis therefore tempting to extrapolate these two inter-pretations of the information content of x to physicsand even to epistemology. As far as the problem of
inductive inference—which was the original motiva-tion of Solomonoff in his seminal 1964 paper—maybe concerned, it has been proposed to model scientificdiscovery with the help of a computabilist approach(Jain et al., 1999) and, within this scope, to satis-factorily tackle the problem of underdetermination.Whether this extrapolation to the physical world or,rather, to the description of the physical world, issound or not is the matter of a wide-open debate.
Even though a precise connection has been madebetween the average algorithmic information C(x)contained in an individual x and the Shannon en-tropy associated with a statistical ensemble of x (Li& Vitányi, 2008, § 2.8.1), there is a long ontologi-cal path to travel when considering x to be a phy-sical state or a description of a physical object (andeven longer, I would say, if x is considered as a phy-sical law or a scientific theory). As far as I know,the measure of complexity that C and D encapsu-late have been proved to be universal provided thatwe keep working with algorithms and discretised setsof objects (digitalised sequences of symbols, Turingmachines, computable functions, finite steps, etc.).Solomonoff-Kolmogorov-Chaitin complexity seems tobe partially applicable in purely countable models ofuniverses only; because of the Gödel-Turing undecid-ability it would require these models to be finite forthe complexity to be completely applicable. Further-more, if we want to avoid the support of intuitionor subjective ingredients, a satisfactory definition ofrandomness has been proposed by Martin-Löf to be“non-computability by any finite process” (for a sum-mary of all the conceptual difficulties that paved thislong road, see Li & Vitányi, 2008, § 1.9). Indeter-minism is then incompatible with a notion of simpli-city directly inspired from computability theory. Wehave therefore identified two obstructions for usingcomputability arguments in physics, namely continu-ity (or, even, the less demanding countably infinity)and randomness (Delahaye, 1999, chap. 7).
One of the most difficult lessons that quantumphysics taught us is that, unlike its traditional inter-pretation in classical statistical physics, randomnessdoes not appear in the physical world as the testi-mony of a lack of information. The renunciation oflocality seems a too high price to recover an entirely
22
deterministic theory. As far as the second obstacleis concerned, I have briefly played a Devil’s advocateand proposed a sketchy plea against a pure arithmeti-sation of physical world in the appendix.
However, despite I have not been able to find inthe literature how the Kolmogorov complexity canbe applicable to define the complexity of one elec-tron, of one phonon or of the Hydrogen atom—justto retain individual elementary systems12—some at-tempts have been proposed to use C (Zurek, 1989)or D (Lloyd & Pagels, 1988) to measure the com-plexity of discretised physical systems. Extendingthese notions with the help of quantum informationtheory has been proposed (Li & Vitányi, 2008, § 8.7and its references). However, from my point of view,this requires drastic restrictions on the manipulationsof qbits; not surprisingly only systems described bydiscrete chains of qbits can be handled, see the ap-pendix to understand how limited this approach re-mains hitherto.
A fair discussion around the computable natureof the universe, seriously supported by prestigioustheoreticians like Wheeler (1990), Feynman (1982, inparticular the last answer in § 9) or ’t Hooft (1988,a return to continuity is discussed in a later pa-per published in 1999) (maybe also Gell-Mann andLloyd 2004), goes beyond the scope of this paper andwould remain largely on speculative grounds. Nev-ertheless, what I wanted to defend by making thepresent digression is the following thesis. In order tocope with the underdetermination problem, one cantentatively specify a selective criterion, among others,that is based on simplicity. To estimate the simpli-city of a concept or of a theory, a direct extrapolationof complexity borrowed from standard computabil-ity theory, though seductive13, raises many unsolvedprofound difficulties. Whether this strategy will befruitful or not, I suspect that the second facet of sym-metry has to play a more explicit role in any attempt
12For classical waves Pour-El & Richards (1981) have exhib-ited a non computable solution of the discretised d’Alembertequation evolving from a computable initial condition.
13It is amusing to remark one recursive loop (simplicity isitself a concept that should be made as simple as possible)and a self-referencing question (we deal with the problem ofthe inductive inference of a concept, simplicity, precisely toanswer to an induction problem).
to quantify what simplicity is. The following providesan esquisse of how this may happen.
4.5.3 Symmetry as a pruning tool
One of the major flaws of the attempts to definephysical complexity/simplicity from algorithmic in-formation theory is that it relies on information com-pression without any loss. In fact, to gain in sim-plicity we must be allowed to irreversibly cut somesuperfluous hypothesis or initial conditions with anOckham’s razor. Moreover, we have seen in § 4.3that information erasing is essential because classifi-cation is primordial, ubiquitous and necessary14. Wecannot avoid working with some equivalence classes,some of which we nonetheless take as primitive enti-ties by not considering the distinct properties that al-low to discriminate its elements, ones from the others.By quotienting a set in equivalence classes, the sec-ond facet of symmetry is more a mean of eliminationthan a mean of compression. Here we stick to theoriginal meaning of abstraction, considered as a syn-onymous of removal (Cournot 1851/1956, chap. XI,§§ 147-149 discusses several processes of such a dis-entanglement). Algorithmic reduction appears onlyto be a special case of labelling each class by oneprivileged element x∗ (specifically, one of the short-est) and, then, the algorithmic depth of x quantifiesthe transformation T that maps x∗ to x = T (x∗).However, this algorithmic interpretation can hardlyembrace the physical world without substantial newfoundations; as we have seen, because of continuityand randomness, whether they are fundamental orjust convenient, most, if not all, physical transfor-mations cannot be reduced in a discrete sequence ofbinary operations.
Here we join back the thermodynamical (or Shan-non) conception of information, where entropy mea-
14A parallel point of view is proposed by Landauer (see, forinstance his1967 paper and the collection of papers on infor-mation erasure edited by Leff and Rex,2003, specially chap. 4)and leads to a modest realism different from the one proposedhere. Landauer adopts a strong ontological difference betweenmathematical entities and reality by denying the existence of π,for instance (Landauer, 1999, p. 65). The proposal I prefer todefend here is that what π and a cherry designate differ onefrom the other by their degree of reality on a continuous scale.
23
sures the loss of information that occurs when con-sidering a statistical ensemble rather than the indi-vidual systems (or messages, if we are concerned byinformation transmission). The latter is describedin terms of a huge number of variables ϕ (say, themicroscopical variables like the positions and the ve-locities of classical atoms) while the former is char-acterised by few variables Φ (say, the average ve-locity of the atoms, their total energy, etc.). Theart of statistical physics consists precisely in iden-tifying quantitatively the relevant variables Φ thatallow a tractable account of which ϕ’s are compati-ble with Φ (reduction) and how Φ can be obtainedfrom ϕ (emergence). We often speak of macroscopicvariables (resp. of a macrostate) when considering Φ(resp. the ensemble), by contrast to the microscopicvariables (resp. micro or pure state) that refer(s) toone element. This is often justified because the spa-tial extension usually allows to intuitively discrim-inate both systems (fluid/molecules, galaxies/stars,heavy nucleus/nucleons, etc.). It may also happen,when studying dynamical systems, that we may takeadvantage of a separation of time scales between afast evolution of ϕ and a slow, secular, evolutionof Φ. If we want to circumvent the favouritismof space and/or time, it is preferable to see themapping ϕ 7→ Φ as the erasing (the decimation asKadanoff would say, 2000, chaps. 13 and 14) of ir-relevant degrees of freedom. This is the quantitativeformulation of the classification x 7→ σ when the ele-ments x (resp. σ) can be labelled with the numericalvariables ϕ (resp. Φ); the irrelevant parameters beingthose, discrete and/or continuous, that allow to labelthe different transformations T of T .
One given statistical ensemble contains microstatesthat differ one from the other by transformationsof ϕ compatible with the macroscopic constraintsfixed by Φ. Building out the effective model con-sists in coarse-averaging that is, integrating out, saywithin the partition function, the variables that allowto discriminate the microstates (in quantum statisti-cal physics this represents the computation of par-tial traces of the density operator) and retaining thedependence in Φ only. Some of the individual prop-erties of the particles—typically the ones involved inthe interactions like the charges but not their indivi-
dual energy for instance—are unaveraged and remainas such among the Φ. How some properties of the ef-fective system depend on such variables reflects thedegree of universality of the construction. It is one ofthe great achievements of statistical theory to haveindeed exhibited some quantities (the critical expo-nents) that depends only on the grossest features ofthe system (the dimension of space, etc.) and not onthe precise form of the interactions, as does the crit-ical temperature of a second order phase transition.Less universal but much simpler examples of Φ arethe pressure of a fluid or the absorption coefficient αof a material. Both have their value depend on the de-tails of the microscopic interactions but, surprisingly,they can be used to characterise a wide variety ofsubstances whose microscopically structures stronglydiffer one from the other (for instance α is involvedin the Beer-Lambert law to describe the light trans-mission through a liquid, a gas, a plastic, a glass, adiamond).
More generally, beyond statistical physics, from allthe preceding considerations presented in this section,we can understand that picking up a small numberof relevant variables Φ out of a wide collection of ϕremains an indispensable mechanism of any cognitiveprocess.
4.6 Emergence
The last line of thought conveys us into the core of thedebate about reduction/emergence issues (Bedau &Humphreys, 2008). Since, recently, there have beenmany discussions on these matters (Batterman, 2010;Butterfield, 2010), I will not venture too long into thiswide realm (see also for instance Castellani, 2002).At the level of abstraction I adopt in this article, Iformally take up the stance that emergence standsfor constructing the variables Φ describing σ fromthe variables ϕ describing x. Reduction, appears tobe the opposite operation σ 7→ x (the fourth facetof symmetry, I will have to say more about it be-low). Thus, we have identified the two dual elemen-tary mechanisms from which are built the hierarchiesthat structure the physical world and shape ratio-nal thinking. We have encountered an example ofmathematical emergence in § 2.3. In § 4.5.3, I have
24
sketchily recalled how statistical physics, within theframe of renormalisation theory, offers what is per-haps the most precise quantitative illustration of therearrangement and elimination of the degrees of free-doms that allow to establish the relation between xand σ. In every-day thinking, this is intuitively ac-complished by the categorisation capacity of whatwe call the common-sense that make a cherry or adog to emerge from a bunch of interweaved organicmolecules (or cells if we want to add some intermedi-ate structure).
Identifying what properties are relevant or con-venient at one level and establishing quantitativelytheir connection with the properties at other levelsremains a formidable challenge, the nub of scienti-fic research I would say. There is no general methodfollowing well-established instructions to increase ourcomprehension (again, our individual minds and theweb of our socially interconnected minds are rathermore selectionist than instructionist, more Darwinianthan Lamarkian so to speak). This is obvious whenfollowing the reductionist direction; however, it is notless true when climbing up the stairs in the other way.Digging out the emergent properties by picking upa small number of relevant variables among a widecollection can seldom be done explicitly. Above, Iused the word “art” to qualify statistical physics not,of course, to deny the standards of rational think-ing which I fully endorse, nor to indicate a direc-tion of increasing beauty15, but precisely to empha-sise that there were no general rule of how to elab-orate an effective theory, and even more to compute(specially when infinities are to be dealt with), theΦ from the ϕ. The large number of appropriate pa-rameters Φ and the difficulties to establish protocolsof their measurements mainly explains why medicine,social sciences, psychology, economics, ecology, agri-culture, and even the technical aspects of cookingetc. are less exact (i.e. predictive with less preci-sion) that more quantitatively supported branches ofscience like chemistry or physics. The objects of in-vestigation of the latters are not only relatively much
15As Weinberg (1992, chap. VI) writes it, general relativitymay appear more beautiful than Newton’s theory though in-volving far more quantities, namely the infinitely many degreesof freedom of the gravitational field.
simpler but also the statistical fluctuations are muchmore reduced by the huge number of elements in asample (as far as I know it is only in physics thatself-averaging quantities can be exhibited). Even inevery-day rational thinking, the variables Φ remainmostly qualitative rather than quantitative and naiveattempts of their formalisation often appear as dan-gerous oversimplifications.
5 Invariance
In the introduction of § 3.4, I have shown how thesecond facet of symmetry allows to identify classifi-cation with property attribution. I have explained in§ 4.2 why these two equal cognitive capacities, inhe-rent to any kind of measurement (including percep-tion), can be viewed as a stabilisation16. A sine quanon condition of intelligibility is that some physicalproperties must remain unchanged under transforma-tions like some space or time translations, rotations,etc. I precise “some” because this stability necessarilyexists in a finite portion of space-time only: The cor-related bundle of attributes that allow to characteriseBorges’dog lasts more than one minute but certainlyno more than 30 years. The norm and direction ofthe terrestrial acceleration ~g remain approximatelyuniform at laboratory scales but change accordingto where we are on Earth; moreover, of course, it isonly meaningful in the immediate neighbourhood ofEarth, which in its turn has a finite space-time ex-tension. Some quantum properties like those of thekaon K± last 10−8 s while others like the electroncharge (more precisely the fine-structure “constant”)seem not to vary on cosmological scales more thanone part in 106 (as far as we can safely measure upto now)17.
16Or the acknowledgement of some kind of stability. Again,the issues regarding whether the stabilisation is acknowledgedor created/defined by intelligence is just a matter of conventionof where we prefer to stand in the virtuous circle. In any case,stability “exists somewhere”.
17By the way, while we construct the equivalence classeswe consider as physical objects, whether we prefer to imple-ment or not invariance under some time-translations remainsa matter of convenience. Both approaches are admissible andbroadly equivalent in a non-relativistic approach but, of course,
25
It is the great merit of Noether’s theorems to pro-vide a conserved quantity associated with each inde-pendent one-dimensional continuous family of trans-formations under which the functional driving thedynamics (more precisely, the action together withappropriate boundary conditions) is invariant. I willnot devote more attention to this specific aspect ofinvariance since the implications and the interpre-tations of Noether’s work have already scrutinisedelsewhere (Kosmann-Schwarzbach, 2010; Brading &Brown, 2004, notably). What I want to proposein § 5.1 is a more general perspective where invari-ance, viewed as the third facet of symmetry, concernsnot only formulations of variational principles but, infact, any law of science. Then, in §§ 5.2 and 5.3,I will open this interpretation to even broader epis-temic considerations.
This rudimentary tendency toward “objecti-fication” reappears in conceptual, in particularmathematical, thought, where it is developedfar beyond its primitive stage. When we deter-mine the size of an object by measurement, itis owing to such “objectification” that we suc-ceed in transcending the accidental limits of ourbodily organisation. It enables that elimina-tion of “anthropomorphic elements” which is,according to Planck, the proper task of scien-tific natural knowledge. To geometrical inva-riants have to be added physical and chemicalconstants. It is in these terms that we formulatethe “existence” of physical objects, the objectiveproperties of things.[. . . ] Hering speaks [. . . ]the language of the scientist, i.e., of realism.He assumes the empirical reality of the objectsabout which our senses have to inform us. Buta critical analysis of knowledge must go farther.Such an analysis reveals that the “possibility ofthe object” depends upon the formation of cer-tain invariants in the flux of sense-impressions,no matter whether these be invariants of per-ception or of geometrical thought, or of physi-cal theory. The positing of something endowed
the time extension cannot be avoided as soon as relativity istaken into account since then, simultaneity becomes a relativeproperty. The philosophical numerous discussions between en-durantists and perdurantists seem therefore neither obscurenor very interesting once it is illuminated by physics argu-ments (see, for instance Butterfield’s (2005) point of view andthe references provided there).
with objective existence and nature depends onthe formation of constants of the kinds men-tioned. It is, then, inadequate to describe per-ception as the mere mirroring in consciousnessof the objective conditions of things. The truthis that the search for constancy, the tendencytoward certain invariants, constitutes a charac-teristic feature and immanent function of per-ception. This function is as much a condition ofperception of objective existence as it is a con-dition of objective knowledge (Cassirer, 1944,end of § III, pp. 20–21).
Born is also very clear:
I think the idea of invariant is the clue to arational concept of reality, not only in physicsbut in every aspect of the world.
[. . . ] not every concept from the domain ofscientific constructs has the character of a realthing, but only those which are invariant in re-gard to the transformation involved. [. . . ]
This power of the mind to neglect the differ-ences of sense impressions and to be aware onlyof their invariant features seems to me the mostimpressive fact of our mental structure.[. . . ]
Thus we apply analysis to construct whatis permanent in the flux of phenomena, the in-variants. Invariants are the concepts of whichscience speaks in the same way as ordinarylanguage speaks of “things," and which it pro-vides with names as if they were ordinary things(Born, 1953).
I perfectly agree except that I would nuance andwould prefer to say that “not every concept sharesthe same degree of reality” because, as I will recallbelow, invariance of a “real thing” always appears tobe an approximation.
5.1 Science laws as quantitative man-
ifestations of invariance
Whatever school of thinking one belongs to, what-ever status one attributes to a science law18, it ought
18Among the overwhelming literature on this subject, Ishould mention van Fraassen’s (1989) influential attempt todeny laws of nature while, strangely enough, keeping the notionof symmetry. If I am not mistaken, what van Fraassen crit-
26
to be uncontroversial that most, if not all, sciencelaws yields to relations between quantities x that canalways be written as19
R(x) = 0 . (5)
Frequently, these relations are written in the mostexplicit form
q = Q(y) (6)
by privileging among the x’s some quantities, namelyq, against others, namely y. I use here two differentfonts to discriminate between mappings, here Q andR, and the physical quantities, here x, y and q. Thesemathematical relations (often taken by many to besynonymous to the laws themselves) have a physicalmeaning only when they are embedded in a theorywhere the quantities x or (q, y), have been relatedto observations, via a model incorporating more orless intermediate other quantities. This does not im-ply that all the x’s should be uniquely determinedby measurements; some of them may be unobserv-able, superfluous or irrelevant. Moreover, when talk-ing about laws, relations (5) or (6) should not beconfused with definitions. “All ravens are black” canrepresent either a partial definition of what a ravenis or a law if the class of ravens and the blacknesshave been defined by other ways. Both status aremutually exclusive but remain a matter of choiceas we keep these instances in a coherent networkof significations. History of science is full of exam-ples where theorems/laws/synthetic judgements haveswitched to axioms/definitions/analytic judgementsand vice versa. See, for instance, the discussion byWeyl (1949, § 17, p. 114) on the status of the elec-tric field ~E in the equation giving the electrostaticforce ~F = e ~E; or Feynman’s explication (1970, § 12-1) of the meaning the force as its appears in Newton’ssecond law ~F = m~a.
icises is the dogmatic conceptions of laws, specially when weforget that their quantitative formulations should come withtheir conditions of validity.
19Inequalities, like Heisenberg’s or the second law of ther-modynamics can also be turned to this form with the help ofHeaviside step function. The mathematical relation betweenthe x’s is of course not unique. For instance, E − mc2 = 0and exp(1 − E/mc2) − 1 = 0 are strictly equivalent expres-sions. However, here also, a principle of parsimony shouldhelp to select a convenient form.
A transformation T that is performed on the vari-ables x defines a transformation of (5) according tothe rules
RT (xT ) = R
(
x(xT ))
(7)
where, in the right hand side, x are considered asfunctions of the transformed variables xT . Then, inthe second case, the transformed function QT is de-fined by
qT = QT (yT ) . (8)
Relation (5) will be said invariant if the same relationexists between the transformed quantities xT :
R(x) = 0 ⇐⇒ R(xT ) = 0 . (9)
If q stands for just one scalar quantity with respectto T i.e. that by definition qT = q (this not an essen-tial point, the argument can be adapted if we were towork with non trivial linear or even non-linear alge-braic representations of T ), then invariance of (6) isformally equivalent to the equality between the twofunctions Q
T and Q, i.e. for all y
QT (y) = Q(y). (10)
In some circumstances, using the tools developedin catastrophe theory (see point (h) of the appendix),the structural stability of relations (5) or (6) is rea-sonably guaranteed. Anyway, the general notion ofinvariance I want to consider here comes closer toPoincaré’s conception that pervades in his 1902 quo-tation in § 3.5. Let us just retain two simple exam-ples, involving neither integrals nor derivatives andonly scalars, not tensors, not fields, nor operators:The Kepler’s third law connecting the orbital pe-riod P of a planet whose semi-major axis length isa:
P 2
a3− 4π2
GM= 0 (11)
(M is the mass of the Sun, G stands for the grav-itational constant); and the equation giving the pe-riod P of the harmonic oscillations of a mass m at-tached to an ideal spring of strength κ:
P = 2π
√
m
κ. (12)
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First, these equations manifest some invariance withrespect to the usual transformations that constitutethe Galilean group and that can be both interpretedpassively or actively. Indeed, equation (11) remainsas correct now as it was 350 years ago. Equation (12)is valid in any laboratory on Earth, even if embarkedon a boat with constant speed. However, the raisond’être of a law is an even broader notion of invariance:if we actively transform the mass-spring system byvarying the mass m 7→ mT and/or substituting thespring by a different one κ 7→ κT , we may wonder ifthe transformed period of the oscillator will still begiven by the same law
P T = 2π
√
mT
κT. (13)
Kepler’s third law is valuable not only because it isstill accurate for Mars since Kepler’s century, butalso because we can keep the same relation if wejump from Mars to Venus (then, formally, a and Pare transformed into aT and P T but neither M norG are altered) or if we change the centre of attrac-tion by considering the orbital periods of the Galileanmoons around Jupiter (now, formally M 7→ MT aswell). Even more, G is qualified as a genuine uni-versal constant precisely because in a given system ofunits (that encapsulates connections with other quan-tities conventionally considered as standard ones), itsvalue remains unchanged when performing the previ-ous transformations.
This considerations can be transposed without dif-ficulties to any field mature enough to be described byreasonably quantitative laws; for instance to Mendel’sstatistical laws of heredity or to equations governinga Darwinian dynamics. As we recalled in the intro-duction, it is in modern quantum theories of fieldsand in relativity that invariance has been first setup as a constructive principle. What is postulatedis the a priori invariance of q (a Lagrangian density,an action, a partition function, a transition ampli-tude, etc.) with respect to some transformations or,in other words, the independence of q with respectto some quantities among the y’s that appear to besuperfluous (the absolute spatial position, the gaugefields, etc.). It happens that these symmetry prin-ciples may impose, on the choice of Q, constraints
stringent enough to select the models drastically andcompel a specific dynamics on the interactions (gaugesymmetries dictate the properties the photon or thegraviton for instance). This line of pursuit yieldsto a feeling of deep inevitability (Weinberg, 1992,chap. VI) and I conjecture that, ultimately, this ne-cessity comes from the constraints imposed on con-tingency by internal coherence.
5.2 Modest truth and humble univer-
sality
I hasten to add that relations like (5) or (6) remainalways conditionally true. Ideally, in mathematics,theorems should be formulated with all their hypo-thesis. In the same way, when faced to empirical veri-fication and to numerical tests, laws should come withsome error bounds, whether of statistical or system-atical origin, even if their conditions of validity cannever all be explicited (see also the point (a) of theappendix). In other words, as far as predictivity isat stake, we should include an evaluation of expectedrisk or loss.
If we look at any particular law, we may becertain in advance that it can only be approxi-mative. It is, in fact, deduced from experimen-tal verifications, and these verifications wereand could be only approximate. We should al-ways expect that more precise measurementswill oblige us to add new terms to our formu-las; this is what has happened, for instance, inthe case of Marriotte’s law.
Moreover the statement of any law is ne-cessarily incomplete. This enunciation shouldcomprise the enumeration of all the antecedentsin virtue of which a given consequent can hap-pen. I should first describe all the conditions ofthe experiment to be made and the law wouldthen be stated: If all the conditions are fulfilled,the phenomenon will happen.
But we shall be sure of not having forgot-ten any of these conditions only when we shallhave described the state of the entire universeat the instant t; all the parts of this universemay, in fact, exercise an influence more or lessgreat on the phenomenon which must happenat the instant t+ dt.
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Now it is clear that such a description couldnot be found in the enunciation of the law; be-sides, if it were made, the law would become in-capable of application; if one required so manyconditions, there would be very little chance oftheir ever being all realized at any moment.
Then as one can never be certain of not hav-ing forgotten some essential condition, it cannot be said: If such and such conditions arerealized, such a phenomenon will occur; it canonly be said: If such and such conditions arerealized, it is probable that such a phenomenonwill occur, very nearly (Poincaré, 1902, § 5).
Indeed (see § 3.2), any modelisation requires an ab-straction of the relevant quantities describing the sys-tem S from the irrelevant degrees of freedom that areswept away in the environment E . The mass m ofthe planet does not appear in Kepler’s third law butthis is true up to terms of order m/M . The lattermay be not negligible any longer if we want to de-scribe a binary star. If damping is at issue, amongthe infinitely many external parameters that have notbe considered in the spring-mass model, the air vis-cosity must be incorporated in the right hand sideof (12) together with a reinterpretation of P as apseudo-period. For transformations such that mT istoo large, we may also quit the linear regime; then,the period will depend on the initial amplitude (butstill not on the position of the moons of Jupiter!).
One famous criticism against a realistic concep-tion of science is the pessimistic (meta-)inductionaccording to which the falsity of the past scienti-fic theories should lead us to conclude that presenttheories are not reliable either (Putnam, 1978; Lau-dan, 1981; Laudan, 1984). However, this argumentis fallacious (another reason was recently given byLewis, 2001) because it presupposes for science a tooambitious, unreachable and eventually meaninglessgoal. Following Zénon, the Renaissance protagonistof Marguerite Yourcenar’s novel The Abyss,
I have refrained from making an idol oftruth, preferring to leave to it its more modestname of exactitude (Yourcenar, 1976, p. 123, Aconversation in Innsbruck).
Indeed, realism should remain “modest” (Bricmont& Sokal, 2001) since the reduction of information re-
quested by any modelisation implies that the truth ofa statement, of a prediction, can be conditional only.The most we can ask to a science law is to come withthe knowledge of its limitation and hopefully with aquantitative control of the errors; but, of course, thiscan be achieved once the theory is mature enoughand/or embedded in a wider one as a so-called effec-tive theory. If we are to use a softer meta-inductionargument (qualifying it as pessimistic or optimistic isa litmus test to determine whether you think that thescience cup is half empty or half full), I am inclinedto follow the lessons of history of science and con-sider that there cannot be but effective theories. The“Dreams of a final theory” (Weinberg, 1992) remainan act of faith not far from the idealistic, thereforeunrealistic, desire of Platonic truth (for a fair and re-cent clarification on the subject see Castellani, 2002).Nevertheless we can argue, for instance, that Newto-nian mechanics remains universally true for macro-scopic objects with small enough masses and veloci-ties. Albeit we know relativity and quantum physics,computation of the location and the dates of eclipseswill still be correctly done within the old classicalframework; the main improvements on the precisionof the predictions and retrodictions of eclipses areexpected to come from a refinement of the planetarymodels (possibly with post-newtonian amendmentsbut certainly not with full general relativity and evenless quantum corrections) and are known to be lim-ited in time by the chaotic character of the many-body (classical) problem. In that sense, Pythagoras’theorem or Kepler’s law have not been underminedby the development of non-euclidean geometry, gene-ral relativity or quantum physics.
The thing essential is that there are pointson which all those acquainted with the experi-ments made can reach accord.
The question is to know whether this accordwill be durable and whether it will persist forour successors. It may be asked whether theunions that the science of today makes will beconfirmed by the science of tomorrow. To af-firm that it will be so we can not invoke any a
priori reason; but this is a question of fact, andscience has already lived long enough for us tobe able to find out by asking its history whetherthe edifices it builds stand the test of time, orwhether they are only ephemeral constructions.
Now what do we see? At the first blush it29
seems to us that the theories last only a dayand that ruins upon ruins accumulate. Todaythe theories are born, tomorrow they are thefashion, the day after tomorrow they are clas-sic, the fourth day they are superannuated, andthe fifth they are forgotten. But if we look moreclosely, we see that what thus succumb are thetheories, properly so called, those which pre-tend to teach us what things are. But there isin them something which usually survives. Ifone of them has taught us a true relation, thisrelation is definitively acquired, and it will befound again under a new disguise in the othertheories which will successively come to reignin place of the old (Poincaré, 1902, § 6)20.
For the same essential reasons that lead us to es-timate the degree of reality with a continuous bal-ance, I prefer not to consider truth to be a binaryquantity. I will not follow Kuhn (1970, speciallychap. IX, X, XII, XIII) neither Feyerabend (1962)on their emblematic fields: talking about incommen-surability between two theories is often an overstate-ment and I would keep the term to qualify logicallycontradictory assertions or dogmatic ideologies only(to play with the words, recall that etymologically‘symmetry’ is the Greek form of the latin ‘commensu-rable’). Thus the notion of scientific revolution is justa convenient tool, at best, to simplify the vivid evo-lutionary river basin of science made of bifurcationsas well as fusions of many streams. As usual, thereis no fundamental cleavage but a (seemingly conti-nuous) gradation that separates entities different indegree but not in nature.
5.3 Epistemic invariance
Extending the notion of symmetry transformationsoutside the usual corral of space-time transforma-tions gives us a bird’s-eye view of what science lawsrepresent. In particular, the perspective I proposedabove allows to reinforce the connection between in-variance and objectivity (see also Nozick’s2001 gene-ral reflexions on this point where a non-binary degreeof truth is proposed). Of course, see fig. 2, we have
20See also Poincaré’s quotation given by Worrall (1989,p. 103).
preserved elements of human culture much older thanPythagoras’ theorem, Archimedes’ or Kepler’s lawsbut, unlike science laws (Sokal & Bricmont, 1998, theintermezzo), most are crucially dependent on theircontextual signification and appear to be fragmentspartially detached from their contemporary culturalbackground; we appreciate their value with criteriathat are now completely different from the ones thatwere in vogue at those ancient times. It is preciselya cornerstone of the structuralism method applied tovarious fields of human science (including linguisticand anthropology) to find out structural invariantsthat are shared by the whole mankind (and even fur-ther, the functional invariants that are shared by sub-strates of variable structure, likewise the brain func-tions in regards to the neuroanatomical organisation,see Changeux, 2002, § VI.7):
Any classification is superior to chaos andeven a classification at the level of sensibleproperties is a step towards rational order-ing. It is legitimate, in classifying fruits intorelatively heavy and relatively light, to beginby separating the apples from the pears eventhough shape, colour and taste are unconnectedwith weight and volume. This is because thelarger apples are easier to distinguish from thesmaller if the apples are not still mixed withfruit of different features. This example al-ready shows that classification has its advan-tages even at the level of aesthetic perception(Lévi-Strauss, 1968, chap. 1).
At a philosophical level, as suggested by Weyl inthe conclusion of his philosophical book, despite theapparently irreducible divisions between numerousphilosophical schools, we can identify some commondenominators (the invariants) that reveal some sortof unity beyond what is often—but of course notalways—merely a semantic variation, a matter of sub-jective preference or a pure intellectual game.
The more I look into the philosophical liter-ature the more I am impressed with the generalagreement regarding the most essential insightsof natural philosophy as it is found among allthose who approach the problems seriously andwith a free and independent mind rather than in
30
the light of traditional schemes—or if not agree-ment then at least a common direction in theirdevelopment. Whether one talks about spacein the language of phenomenology like Husserlor ‘physiologically’ like Helmholtz is less impor-tant, in view of their substantial concordance,than it appears to the ‘standpoint philosophers’who swear by set formulae (Weyl, 1949, end of§ 23.D, p. 216).
I hope to have shown that symmetry is actuallya good candidate for being such an invariant asfar as rationalists are concerned, among whom, atleast, Poincaré’s heirs like Worrall (1989) (see alsomore recent approaches like the one proposed byEsfeld, 2006) and his successors—who belong tothe multiple subdivisions of the structural realistbranch (Bokulich & Bokulich, 2010)—but, maybe,also the supporters of Fine’s natural ontological atti-tude (Fine, 1996, chap. 7) or even those who preferto endorse Van Fraassen’s constructive empiricism.
It would certainly be too presumptuous to para-phrase Dawkins’expression (1976, chap. 3) and con-sider the scientific laws to be “immortal” meme com-plexes; however, it is sure that they constitute the farmost stable ones. History shows that they have beenable to resist to the most dramatic memocides (likethe multiple destructions of the library of Alexandria)or to the most insidious eumemisms (like the selectivecopy of texts during the western Middle Ages thatalmost completely stemmed the flow of greek mate-rialist philosophy initiated by the ancient atomistsLeucippus and Democritus, then adapted by Epicu-rus). Recall,
If, in some cataclysm, all of scientific knowl-edge were to be destroyed, and only one sen-tence passed on to the next generations ofcreatures, what statement would contain themost information in the fewest words? I be-lieve it is the atomic hypothesis (or the atomicfact, or whatever you wish to call it) that all
things are made of atoms—little particles that
move around in perpetual motion, attracting
each other when they are a little distance apart,
but repelling upon being squeezed into one an-
other. In that one sentence, you will see, thereis an enormous amount of information about
the world, if just a little imagination and think-ing are applied (Feynman et al., 1970, vol I.§ 1-2).
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6 Projection
6.1 Invariance breaking
We have already encountered the fourth facet of sym-metry at multiple places.
From an epistemic point of view, this is preciselythe reductionist strategy. The choice of one elementin an equivalence class constitutes the basic operationof decomposing a class in its constituents: real num-bers enclose rational numbers, the class of fruits com-prises cherries, the class of dogs includes Borges’dog,atoms contain electrons, one-phonon states are builtfrom condensed atomic states, etc. Together with theelaboration of effective models and the identificationof emergent properties discussed in § 4.6, reductionties the hierarchical web of interlaced concepts.
From a logical point of view, we talk about de-duction in the sense that we infer some properties onan x as soon as it is identified as an element of class σ(e.g., “Hume is mortal because humans are”). In thatsense, x may be considered as a prototype (I preferto use this word, following Rosch,1978, rather thanthe too pejorative “stereotype” or the too much con-noted by idealistic flavour “archetype”) of the classand a systematic choice of one representant in eachclass can be used to label the classes, specially if a na-tural (canonical) rule can be proposed (for instance,rational numbers are represented and can even safelybe identified to a fraction of coprime integers). Here,when talking about the representation of σ by x, bothacceptions b) and c) in footnote 1 fit in this scheme.
There is an alternative way to interpret the pro-jection operation σ 7→ x. The identification of oneelement x among all the others is possible only ifwe are able to attribute properties to x that are notcharacteristic of σ but, rather, whose values allow todiscriminate its elements. The differences are encap-sulated in the set T of transformations connectingtwo elements. Because of these distinctive proper-ties, the global invariance of σ under T is brokensince x 6= xT in general. As acknowledged by PierreCurie in a famous paper, assymmetry is essential tocharacterise the physical phenomena:
What is necessary is that certain symmetry
elements do not exist21. Assymmetry is whatcreates the phenomenon (Curie, 1894, § IV,p. 400).
For more modern variations on this theme, see theinteresting essay by Morrison (1995, On Broken sym-metries. pp. 99-114).
In practise, the manipulation of concepts often re-quires the use of one of its representant. For instance,in theoretical physics computing scalar products re-quires the choice of a representation of each vectorby introducing a specific basis. In differential geo-metry a coordinate chart is most often required toproceed with numerical computations. When makingmeasurements, some frame, gauge and units must beused and, more generally, a measurer M maps somepart of the universe that includes the physical systemat issue (see § 3.1 and figure 3). Let us hand over toWeyl
A typical example of this is furnished by abody whose solid shape constitutes itself as thecommon source of its various perspective views(Weyl, 1949, § 17, p. 113).
and Born, who uses an analogy with the differentelliptical shapes of the projected shadows of one cir-cular cardboard (this simple but acute example isalready in Cournot, 1851/1956, chap. XIII, § 198),
This root of the matter is a very simplelogical distinction, which seems to be obviousto anybody not biased by a solipsistic meta-physics; namely this: that often a measurablequantity is not a property of a thing, but a pro-perty of its relation to other things. [. . . ] Mostmeasurements in physics are not directly con-cerned with the thing which interest us, butwith some kind of projection, this word takenin the widest possible sense. The expression co-ordinate or component can also be so used. Theprojection (the shadow in our example) is de-fined in relation to a system of reference (thewalls, on which the shadow may be thrown)(Born, 1953, p. 143).
21I translate literally the original French. Rosenand Copié’s1982 English translation interprets the words“n’existent pas” in a more appropriate and careful way andpropose “are missing” instead of “do not exist”.
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In any case some superfluous variables are introducedthat mask the objectivity by breaking the invari-ance of the equivalence classes. This is, I guess, themain source of confusion that obscures some ontolog-ical or epistemological controversies. One should notconfuse the invariance/objectivity/universality of theclass with the assymmetry/subjectivity/conventionalchoice of one of its representation. As Debs andReadhead write
[. . . ] the claim that these representationsare inescapably based on conventional choiceshas been taken by many as a denial of their ob-jectivity. As such, objectivity and convention-ality in representation have often been framedas polar opposites.[. . . ]
[However], a view of science that emphasizesthe role of conventional choice need not be inconflict with a realist account of representationthat allows for objectivity. One may maintaina cultural view of science and still be commit-ted, as most realists are, to the existence of asingle real ontology that humans inhabit (Debs& Redhead, 2007, Introduction, pp. 3 and 4).
I do not know if Born, when writing the abovetext, had also in mind Plato’s famous allegory ofthe cave; however Weyl, in pursuing his philosophi-cal reflections, clearly endeavours to reconcile realismwith idealism (p. 117). Then, in almost exactly thesame terms we discussed in § 3.5, the distinction be-tween the vector v (“analogue of the objects in thereal world”) and its components vµ (“analogues ofthe subjective phenomena") obtained by projectionon a specific coordinate basis (“analogues of real ob-servers”), Weyl concludes
Hence the model is the world of my phe-nomena and the absolute basis is that distin-guished observer ‘I’ who claims that all phe-nomena are as they appear to him: on thislevel, object, observer and appearance all be-long to the same world of phenomena, linkedhowever by relations among which we can dis-tinguish the ‘objective’ or invariant ones. Realobserver and real object, I, thou, and the ex-ternal world arise, so to speak, in unison andcorrelation with one another by subjecting thesphere of ‘algebraic appearances’ to the view-point of invariance. [. . . ] The analogy renders
the fact readily intelligible that the unique ‘I’of pure consciousness, the source of meaning,appears under the viewpoint of objectivity asbut a single subject among many of its kind(Weyl, 1949, § 17, pp. 123–124).
6.2 Three open problems
As far as contemporary physics is concerned, thereexists of course an enormous literature on the sub-ject of symmetry breaking. In addition, part III ofBrading and Castellani’s compilation(2003) exploresinteresting epistemological and philosophical implica-tions of this concept. I will therefore not say moreabout these matters. Before we reach the conclu-sion, I would like to end this section by listing threeopen problems, that I consider to be crucial and morescientific than metaphysical (I believe that any kindof dualist approach cannot provide a satisfactory so-lution but just shifts the emphasis of the problems).All of three seem pertained to the fourth facet ofsymmetry.
(PQ) The problem of quantum measurement: Howdoes the interaction between a system S and a mea-surer M selects just one observed property or oneobserved event among the class of observable onesthat a quantum state encompasses ? A satisfactoryanswer must come with a consistent and unified de-scription of both S, M and possibly E unlike theorthodox contemporary quantum theory which stillattributes classical properties to M.
(PN) The problem of now: In physics, all the pointsof the world line of a system are treated on an equalfooting. In the space-time zone where intelligence hasemerged, what does select a special thin slice of timewe call present ?
(PS) The problem of self: Among all the consciousminds, what does favour the special one you call yourself ?
As Born recalls about a quantum measurement
Expressed in mathematical terms the wordprojection is perfectly justified since the mainoperation is a direct generalisation of the geo-metrical act of projecting, only in a space ofmany, often infinitely many, dimensions (Born,1953).
33
Some connections between these problems havebeen proposed. If Darwinian selection has not shapedour brains like a digital device, it has neither preparedus for a natural apprehension of the world at smallscales. After one century of controversy on the quan-tum measurement issues, we still do not know if wewill be able to obtain, within a consistent theory, thereduction of a (possibly diagonalised by decoherence)density operator to a pure state or if this problem isirreducible any further because of our intrinsic limi-tations. This last line of pursuit follows a traditionultimately stem from Bohr and also endorsed lateron by Wigner (1995, part III), and now defended byPenrose (1997), who proposed to relate PQ to PS.
By the way, if we consider PS the opposite way byreversing its terms, viz. if we ask how we can infer theexistence of other selves, we just face the old problemof other minds (Avramides, 2001). Nevertheless, Ifind the problem of self much more interesting thanthe problem of other minds because the latter findsnaturally its answer in the necessity of escaping froma sterile kind of noological solipsism.
Even though quantum theory of fields offers themost brilliant and convenient way to reconcile spe-cial relativity and quantum theory, it leaves un-changed the problem of measurement. There seemsto be a deep incompatibility between objective prob-abilities and a four-dimensional space-time. Accor-ding to Everett1957’s multiworld interpretation, a bi-furcation between universe branches occurs at eachmeasurement. Had this attempt come with a pro-posal that clarifies the special status of measurementagainst pure quantum events, it would have ruled PQ
out but at the cost of reinforcing the importance ofPN and especially PS
22.
22It is noteworthy that to try to solve another old problemof choice, namely the problem of free will, Boussinesq has pro-posed in 1877 to connect it with the existence of bifurcations ofthe solutions of dynamical equations, letting the choice of thebranch not to an extreme sensitivity to “a very small changein the initial conditions” (Poincaré, 1959, chap. IV, p. 76, seealso p. 68) but to an external “guiding principle” (principe
directeur) that establishes a fundamental clear-cut betweeninanimate and animate beings. If, on the contrary, we wantto remain self-consistent within a unique materialistic contin-uum, the “guiding principles” that break the symmetry mustdefinitely remain internal.
Many physical and philosophical discussions havebeen devoted to the arrow of time, but relativelyfew concern attempts to physically define the present(Hartle, 2005). Maybe questions PN and PS are twoaspects of the same question, the first being formu-lated in physical terms while the other in neuropsy-chological terms.
A signature of a final theory should precisely tobe free of parameters with contingent value. Thosewho believe in the existence of such a theory shouldtherefore add a fourth problem, possibly connectedto the others with the help of anthropic selectionistarguments, namely,
(PC) The problem of constants : How are selectedthe dimensionless constants like the coupling con-stants at low energy scales, some cosmological cha-racteristics or even the dimension of space-time?
However, if we deny such an ultimate goal of get-ting rid of contingency, the importance of PC isweaken because it is the usual statement of how todigest an effective theory into the stomach of a widerone.
7 Conclusion
Throughout the twentieth century, the notion of sym-metry has acquired extraordinary scope and depth inmathematics and, especially, in physics. In this pa-per I have proposed to recognise in this wide domainfour intricately bound clusters each of them beingscrutinised into some fine structure. First, this de-composition allowed us to examine closely the mul-tiple different roles symmetry plays in many placesin physics. Second, I have tried to unveil some rela-tions with others disciplines like neurobiology, episte-mology, cognitive sciences and, not least, philosophy.These excursions and the connections they reveal,that should gain to be investigated more thoroughly,offer, I hope an intellectually rewarding transversaljourney.
To put in a shell many themes we encountered, Icould not find a better shell than Valéry’s; I cannotresist the temptation of offering it as the last words:
Like a pure sound or a melodic system ofpure sounds in the midst of noises, so a crys-
34
tal, a flower, a sea shell stand out from thecommon disorder of perceptible things. For usthey are privileged objects, more intelligible tothe view, although more mysterious upon re-flection, than all those which we see indiscrimi-nately. They present us with a strange union ofideas: order and fantasy, invention and neces-sity, law and exception. In their appearance wefind a kind of intention and action that seemto have fashioned them rather as man mighthave done, but as the same time we find evi-dence of methods forbidden and inaccessible tous. We can imitate these singular forms; ourhands can cut a prism, fashion an imitationflower, turn or model a shell; we are even ableto express their characteristics of symmetry ina formula, or represent them quite accurately ina geometric construction. Up to this point wecan share with “nature”: we can endow her withdesigns, a sort of mathematics, a certain tasteand imagination that are not infinitely differentfrom ours; but then, after we have endowed herwith all the human qualities she needs to makeherself understood by human beings, she dis-plays all the inhuman qualities needed to dis-concert us. . . (Valéry, 1964).
Appendix : a plea for continuity
This appendix presents some (non-independent) ob-jections that ought to be addressed by any defender,like Wheeler (1990, specially in the somehow sloppydefence of a “no continuum” position in § 3) or Lan-dauer (1999), of a purely discretised version of ouruniverse.
(a) The platitude of saying that any set of mea-surements is finite and that numerical data consist ina finite sequence of (q)bits, does not imply that theobjects they represent are finite or discrete. Here wemust not confuse the domain and the image of a rep-resentation (see note 1) albeit they live in the sameontologic virtuous circle; otherwise we are trapped ina solipsist-like pitfall if we just deny all but the finitenumber of data one or many minds can grasp (solip-sism originally coined for one mind, is not less inde-fensible when extended to one collectivity of minds).
(b) As any experimentalist knows, quantitative
measurements should, in principle, be reported witha confidence interval. If we accept the modest real-istic attitude I propose in § 5.2, even at the onto-logic level, most of the physical quantities are asso-ciated with certain amount of uncertainty and, forinstance, all the scalars may be represented by areal interval as well (more pedantically, a neighbour-hood in a topological space). Since the first pro-posal of a “law of continuity” about the continuousnature of perception by Leibniz (1765/1996, Preface),many successors have advocated this opinion accor-ding to which “nature never makes leaps”. Some morearguments have been developed in that directionby Cournot (1851/1956, chap. XIII) and Poincaré(1905; 1893; 1903, § 3). The quantum theory ofcourse strengthens this argument, via Heisenberg’sinequalities, notably by introducing the distinctionbetween observable and observed quantities. Half-joking I often say to my undergraduate students that“zero does not exist in physics” and explain that theyshould always keep in mind that when writing theequation R = 0, it means that the quantity R is neg-ligible; improving the model, modifying the order ofapproximations or working with a better (or a worse!)precision of measurements may reveal a finite valueof R.
(c) We can truly consider that some neighbourhoodof infinity exists in physics. For my camera, infin-ity starts beyond 10 meters. In condensed matter,not only we can safely define a sample to be macro-scopic when it contains an infinite numbers of atoms,but also we can choose the relevant quantities des-cribing it to be continuous. The same can be saidof course to any theory involving fields or waves (in-cluding quantum wavefunctions if we consider thattheir ontological status is not very different from theclassical electromagnetic waves for instance).
(d) Following the same vein, actually very few phy-sical quantities seem to me to be fundamentally dis-crete (up to now): Some topological indexes in con-densed matter, some eigenvalues of some quantumoperators like the angular momentum, the chargeconjugation, the parity, the time reversal operatoror the number of particles in a Fock space are suchcandidates but not the eigenvalues of a Hamiltoniansince the coupling with the environment turns the
35
discrete quantum energies into bands. I would alsoadd the dimension of space-time to this short list23.However, the dimensional regularisation trick d − ǫintroduced by ’t Hooft & Veltman (1972), thoughavoidable and empty of physical interpretation, pro-vides another example of the pragmatic convenienceof continuity.
(e) If we look at the virtuous circle of realitythrough the neural gate, it seems (Dehaene, 1997,specially chap. 4) that the general tendency to con-struct “sharp numbers” (for instance “natural” inte-gers) from “round numbers” is preferred rather thanthe commonly admitted inverse way. This internal“fuzzy number sense” may be extended to the scienti-fic method as far as large numbers are concerned sinceI do not know of any experiment that would allow todiscriminate between 1023 and 1023 + 1 entities.
(f) The classification of quantum particles fromwhich discrete quantities like spin, the flavour, etc.,relies on the algebraic representations of Lie groupswhose continuous structure is essential. Even a sim-ple qbit is characterised by a continuous relativephase.
(g) In quantum field theory, interactions provokethe condensation of bare particles into dressed par-ticles (formally, as the interactions change we moveinside a class of Fock spaces; this Hilbert space asa whole is non-separable, i.e. that a continuousindex is necessary to label every dense set, seeStreater and Wightman, 2000, § 2-6). This phasetransition is obviously observed in condensed mat-ter physics (phonons, superfluidity, superconductiv-ity, Bose-Einstein condensate, etc.) but may alsohave occurred at a cosmological scale: the appar-ently discrete electric charges or masses of the ele-mentary particles described by the Standard Modelare involved in the coupling constants of interactionsand, as renormalised quantities, may change contin-uously.
(h) Even though we can construct fundamentalunits of length and time (the Planck length and mass)this does not imply the discretised nature of space-time. Up to now we have found no experimental clues
23Shouldn’t a complete theory of quantum gravity give anaccount of this observed quantity by introducing a dimensionoperator as an observable ?
that space-time is discrete; without any satisfactoryquantum theory of gravity, we cannot but speculateabout the ultraviolet cut-offs as being, say, given bythe Planck length.
(i) Continuity and even smoother properties arealso ubiquitously relevant in physics because they al-low to take into account structural stability. Anyequation R0(x) = 0 can be deformed into R0(x) +ǫR1(x) = 0 if R1(x) denotes a physical expres-sion having the same dimensions as R0. If ǫ issmall enough not only this deformed equations re-main experimentally indistinguishable from the origi-nal one (this is one way to formulate underdetermina-tion) but also, theoretically speaking, it may remaincompletely equivalent to the first by an appropriatesmooth redefinition of the variables x. Catastrophetheory (Poston & Stewart, 1978; Arnold, 1984; De-mazure, 2000, as three remarkable introductions tothis subject) helps us to formulate the conditionson R1 for which such generic perturbations are arm-less and the search of simplicity, here, means lookingfor the normal forms where all the irrelevant vari-ables have been put to zero, keeping only those thatencapsulates the geometrical essential properties ofthe manifold defined by R0(x) = 0.
Acknowledgements: It is a pleasure to thanksXavier Bekaert, Emmanuel Lesigne and Karim Nouifor precious discussions that allowed to selectively im-prove the formulation of some ideas presented here.
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