Extended Criticality, Phase Spaces and Enablement in Biology

Giuseppe Longoa, Maël Montévilb

aCentre Cavaillès, CNRS – École Normale Supérieure, ParisbTufts University Medical School, Dept. of Anatomy and Cell Biology

Abstract

This paper analyzes, in terms of critical transitions, the phase spaces of biological dynamics. The phase space is thespace where the scientific description and determination of a phenomenon is given. We argue that one major aspectof biological evolution is the continual change of the pertinent phase space and the unpredictability of these changes.This analysis will be based on the theoretical symmetries in biology and on their critical instability along evolution.

Our hypothesis deeply modifies the tools and concepts used in physical theorizing, when adapted to biology. Inparticular, we argue that causality has to be understood differently, and we discuss two notions to do so: differentialcausality and enablement. In this context constraints play a key role: on one side, they restrict possibilities, on theother, they enable biological systems to integrate changing constraints in their organization, by correlated variations,in un-prestatable ways. This corresponds to the formation of new phenotypes and organisms.

Keywords: Conservation properties, symmetries, biological causality, phase space, unpredictability, phylogeneticdrift, enablement

1. Introduction

As extensively stressed by H. Weyl and B. vanFraassen, XXth century physics has been substituting tothe concept of law that of symmetry. Thus, this conceptmay be “considered the principal means of access to theworld we create in theories”, [VF89].

In this text1, we will discuss the question of biologicalphase spaces in relation to critical transitions and symme-tries. More precisely, we will argue, along the lines of[Kau02, BL08, LMK12], that in contrast to existing phys-ical theories, where phase spaces are pre-given, in biol-ogy these spaces need to be analyzed as changing in un-predictable ways through evolution. This stems from thepeculiar biological relevance of critical transitions and the

Email address: longo@ens.fr (Giuseppe Longo)URL: http://www.di.ens.fr/users/longo (Giuseppe

Longo), http://www.montevil.theobio.org (Maël Montévil)1This paper revises and develops early joint work with Stuart Kauff-

man [LMK12].

related role of symmetry changes.

In order to understand the peculiarities of biologicaltheorizing, we will first shortly recall the role, in physics,of “phase spaces”. A phase space is the space of the perti-nent observables and parameters in which the theoreticaldetermination of the system takes place. As a result, toone point of the phase space corresponds a complete de-termination of the intended object and properties that arerelevant for the analysis.

Aristotle and Aristotelians, Galileo and Kepler closelyanalyzed trajectories of physical bodies, but without amathematical theory of a “background space”. In a sense,they had the same attitude as Greek geometers: Euclid’sgeometry is a geometry of figures with no space. It isfair to say that modern mathematical physics (Newton)begun by the “embedding” of Kepler and Galileo’s Eu-clidean trajectories in Descartes’ spaces. More precisely,the conjunction of these spaces with Galileo’s inertia gavethe early relativistic spaces and their invariant properties,as a frame for all possible trajectories — from falling bod-

Preprint submitted to Chaos, solition and fractals; accepted March 2013. March 10, 2013

ies to revolving planets2. In modern terms, Galileo’s sym-metry group describes the transformations that preservethe equational form of physical laws, as invariants, whenchanging the reference system.

Along these lines, one of the major challenges for a(theoretical) physicist is to invent the pertinent space or,more precisely, to construct a mathematical space whichcontains all the required ingredients for describing thephenomena and to understand the determination of itstrajectory, if any. So, Newton’s analysis of trajectorieswas embedded in a Cartesian space, a “condition of pos-sibility”, Kant will explain, for physics to be done. Bythis, Newton unified (he did not reduce) Galileo’s analy-sis of falling bodies, including apples, to planetary orbits:Newton derived Kepler’s ellipsis of a planet around theSun from his equations. This is the astonishing birth ofmodern mathematical-physics as capable of predicting ex-actly the theoretical trajectory, once given the right spaceand the exact boundary conditions. But, since Poincaré,we know that if the planets around the Sun are two ormore, prediction is impossible due to deterministic chaos.Even though their trajectories are fully determined byNewton-Laplace equations their non-linearity yields theabsence almost everywhere of analytic solutions and for-bids predictability, even along well determined trajecto-ries at equilibrium.

As a matter of fact, Poincaré’s analysis of chaotic dy-namics was essentially based on his invention of the so-called Poincaré section (analyze planetary orbits only bytheir crossing a given plane) and by the use of momen-tum as a key observable. In his analysis of chaoticity,stable and unstable trajectories in the position-momentumphase space, nearly intersect infinitely often, in “infinitelytight meshes” and are also “folded upon themselves with-out ever intersecting themselves”, (1892). Since then,in physics, the phase space is mostly given by all pos-sible values of momentum and position, or energy andtime. In Hamiltonian classical mechanics and in Quan-tum Physics, these observables and variables happen tobe “conjugated”, a mathematical expression of their per-

2The Italian Renaissance painting invented the mathematical “back-ground” space by the perspective, later turned into mathematics byDescartes and Desargues, see [Lon11].

tinence and tight relation3. These mathematical spacesare the spaces in which the trajectories are determined:even in Quantum Physics, when taking Hilbert’s spaces asphase spaces for the wave function, Schrödinger’s equa-tion determines the dynamics of a probability density andthe indeterministic aspect of quantum mechanics appearswhen quantum measurement projects the state vector (andgives a probability, as a real number value).

It is then possible to give a broader sense to the notionof phase space. For thermodynamics, say, Boyle, Carnotand Gay-Lussac decided to focus on pressure, volume andtemperature, as the relevant observables: the phase spacefor the thermodynamic cycle (the interesting “trajectory”)was chosen in view of its pertinence, totally disregardingthe fact that gases are made out of particles. Boltzmannlater unified the principles of thermodynamics to a par-ticle’s viewpoint and later to Newtonian trajectories byadding the ergodic hypothesis. Statistical mechanics thus,is not a reduction of thermodynamics to Newtonian tra-jectories, rather an “asymptotic” unification, at the infinitetime limit of the thermodynamic integral, under the novelassumption of “molecular chaos” (ergodicity). In statisti-cal mechanics, ensembles of random objects are consid-ered as the pertinent objects, and observables are derivedas aspects of their (parameterized) statistics.

It should be clear that, while the term phase space isoften restricted to a position/momentum space, we use ithere in the general sense of the suitable or intended spaceof the mathematical and/or theoretical description of thesystem. In this sense the very abstract Hilbert space ofcomplex probability densities is a phase space for the statefunction in Quantum Mechanics, very far form ordinaryspace-time.

Now, in biology, the situation is more difficult. Ourclaim here, along the lines of [Kau02, BL11, LMK12]is that, when considering the biologically pertinent ob-servables, organisms and phenotypes, no conceptual normathematical construction of a pre-given phase space ispossible for phylogenetic trajectories. This constitutes amajor challenge in the study of biological phenomena.We will motivate it by different levels of analysis. Ofcourse, our result is a “negative result”, but negative re-

3One is the position and the other takes into account the mass andthe change of position.

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sults may open the way to new scientific thinking, in par-ticular by the very tools proposed to obtain them, [Lon12].Our tools are based on the role of symmetries and critical-ity, which will suggest some possible ways out.

2. Phase Spaces and Symmetries

We understand the historically robust “structure of de-termination of physics” (which includes unpredictability)by recalling that, since Noether and Weyl, physical lawsmay be described in terms of theoretical symmetries inthe intended equations (of the “dynamics”, in a generalsense, see below). These symmetries in particular expressthe fundamental conservation laws of the physical observ-ables (energy, momentum, charges . . . ), both in classi-cal and quantum physics. And the conservation proper-ties allow us to compute the trajectories of physical ob-jects as geodetics, by extremizing the pertinent function-als (Hamilton principle applied to the Langrangian func-tionals). It is the case even in Quantum Mechanics, asthey allow to derive the trajectory of the state function ina suitable mathematical space, by Schrödinger equation.

As we said, only with the invention of an (analytic) ge-ometry of space (Descartes), could trajectories be placedin a mathematically pre-given space, which later becamethe absolute space of Newtonian laws. The proposal ofthe more general notion of “phase space” dates of thelate XIX century. Then momentum was added to spa-tial position as an integral component of the analysis of atrajectory, or energy to time, in order to apply the corre-sponding conservation properties, thus the correspondingtheoretical symmetries. In general, the phase spaces arethe right spaces of description in the sense that they allowone to soundly and completely specify “trajectories”: ifone considers a smaller space, processes would not havea determined trajectory but would be able to behave arbi-trarily with respect to the elements of the description (forexample, ignoring the mass or the initial speed in classicalmechanics). Adding more quantities would be redundantor superfluous (for example, considering the color or fla-vor, in the usual sense, in classical mechanics).

In other words, in physics, the observables (and pa-rameters), which form the phase space, derive from the(pertinent/interesting) invariants / symmetries in the tra-jectories and among trajectories. More exactly, they de-rive from the invariants and the invariant preserving trans-

formations in the intended physical theory. So, Poincaré’smomentum is preserved in the dynamics of an isolatedsystem, similarly as Carnot’s product pV is preserved atconstant temperature while p and V may vary. Again, oneuses these invariants in order to construct the “backgroundspace” where the phenomena under analysis can be ac-commodated. That is, the conceptual construction of thephase space follows the choice of the relevant observablesand invariants (symmetries) in the physico-mathematicalanalysis.

In summary, the historical and conceptual developmentof physics went as follows:

• analyze trajectories

• pull-out the key observables as (relative) invariants(as given by the symmetries)

• construct out of them the intended phase space.

Thus, physical (phase) spaces are not “already there”,as absolutes underlying phenomena: they are our remark-able and very effective invention in order to make physicalphenomena intelligible [Wey83, BL11].

As H. Weyl puts it, the main lesson we learn from XXcentury physics is that the construction of scientific ob-jectivity (and even of the pertinent objects of science) be-gins when one gives explicitly the reference system (or thephase space with its symmetries) and the metric (the mea-surement) on it. This is why the passage from the symme-tries of Galileo group to Lorentz-Poincaré group framesRelativity Theory, as it characterizes the relevant physi-cal invariants (the speed of light) and invariant preservingtransformations (Poincaré group) in the phase space.

In summary, the modern work of the theoretical physi-cist begins by setting the phase space and the measurein it, on the grounds of the observables he/she consid-ers to be essential for a complete description of the in-tended dynamics — in the broadest sense, like in Quan-tum Physics, where quanta do not go along trajectories inordinary space-time, but the wave or state function does,in a Hilbert space.

As for the formal foundation, from Descartes’ spacesup to the later more general phase spaces (Hilbert spacesor alike), all these spaces are finitistically (axiomatically)describable, because of their symmetries. That is, their

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regularities, as invariants and invariant preserving trans-formations in the intended spaces (thus their symmetries),allow a finite description, even if they are infinite. Con-sider, say, a tri- (or more) dimensional Cartesian space,since Newton our preferred space for physics. It is infi-nite, but the three straight lines are given by symmetries(they are axes of rotations) and their right angles as well4

(right angles, says Euclid, are defined from the most sym-metric figure you obtain when crossing two straight lines).When adding the different groups of transformations (thesymmetries) that allow to relativize the intended spaces,one obtains the various physical theories that beautifullyorganize the inert matter, up to today.

Hilbert and Fock’s spaces require a more complex butconceptually similar definition, in terms of invariants andtheir associated transformations. These invariants (sym-metries) allow to handle infinity formally, possibly in theterms of Category Theory. Note that symmetries, in math-ematics, have the peculiar status of being both invariant(structural invariants, say) and invariant preserving trans-formations (as symmetry groups). Symmetries thus allowto describe infinite spaces and mathematical structures,even of infinite dimension, in a very synthetic way, by thefinitely many words of a formal definition and of a fewaxioms.

We will argue for the intrinsic incompressibility of thephase space of intended observables in biology: no wayto present it a priori, as a time invariant system, by finitelymany pre-given words.

2.1. More lessons from Quantum and Statistical Mechan-ics

As we observed, quantum mechanics takes as statefunction a probability density in possibly infinite dimen-sional Hilbert or Fock spaces. More generally, in quan-tum mechanics, the density matrix allows to deal alsowith phase spaces which are known only in part. In suchcases, physicists work with the part of the state space thatis known and the density matrix takes into account thatthe system can end up in an unknown region of the statespace, by a component called “leakage term”. The point isthat this term interferes with the rest of the dynamics in a

4More generally, modern Category Theory defines Cartesian prod-ucts in terms of a symmetric commuting diagrams.

determined way, which allows us to capture theoreticallythe situation in spite of the leakage term.

In Quantum Field Theory (QFT) it is even more chal-lenging: particles and anti-particles may be created spon-taneously. And so one uses infinite dimensional Hilbert’sspaces and Fock spaces to accommodate them. Of course,quanta are all identical in their different classes: a newelectron is an electron . . . they all have the same observ-able properties and underlying symmetries. Also, theanalysis by Feynman diagrams allows us to provide theparticipation in the quantum state of each possible spon-taneous creation and annihilation of particles (and, ba-sically, the more complex a diagram is, the smaller itsweight). The underlying principle is that everything thatcan happen, for a quantum system, happens, but only alimited number of possibilities are quantitatively relevant.

In statistical mechanics one may work with a randomlyvarying number n of particles. Thus, the dimension ofthe state space stricto sensu, which is usually 6n, is notpre-defined. This situation does not, however, lead toparticular difficulties because the possibilities are known(the particles have a known nature, that is relevant observ-ables and equational determination) and the probabilitiesof each phase space are given5. In other terms, even if theexact finite dimension of the space may be unknown, ithas a known probability — we know the probability it willgrow by 1, 2 or more dimensions, and, most importantly,they are formally symmetric. The possible extra particleshave perfectly known properties and possible states: thepertinent observables and parameters are known, one justmisses: how many? And this becomes a new parame-ter . . . (see for example [Set06], for an introduction).

In these cases as well, the analysis of trajectories orthe choice of the object to study (recall the role given tomomentum or the case of the thermodynamic cycle or theprobability density for QM) lead to the construction of thepertinent phase space, which contains the proper observ-ables and parameters for the trajectories of the intendedobject. Then, as mentioned above, the symmetries of thetheories allowed synthetic, even axiomatic, definitions of

5In general, n changes either because of chemical reactions, and itis then their rate which is relevant, or because the system is open, inwhich case the flow of particles is similar to an energetic flow, that isthe number of particles plays the same role than energy: they are bothfluctuating quantities obeying conservation laws.

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these infinite spaces, even with infinite or fluctuating di-mensions. In other words, the finite description of thesespaces of possibly infinite dimension, from Descartes toQuantum spaces, is made possible by their regularities:they are given in terms of mathematical symmetries. And,since Newton and Kant, physicists consider the construc-tion of the (phase) space as an “a priori” of the very intel-ligibility of any physical process.

2.2. Criticality and Symmetries

Certain physical situations are particularly interestingwith respect to phase spaces and their symmetries. We re-fer by this to critical transitions, where there is a changeof global behavior of a system, which may be largely de-scribed in terms of symmetries changes.

For example, spin lattices phase transitions are under-stood, from a purely macroscopic point of view, as achange of phase space: a parameter (the order parame-ter which is the global field in this example) shifts frombeing degenerate (uniformly null) to finite, non zero quan-tities. In other words, a new quantity becomes relevant.From a microscopic point of view, this quantity, however,is not exactly new: it corresponds to aspects used for thedescription of the microscopic elements of the system (afield orientation, for example). In the equational deter-mination of the system as a composition of microscopicelements, there is no privileged directions for this observ-able. In the disordered phase (homogeneous, in terms ofsymmetries), the order parameter, that is the average ofthe field, is 0. However, in the ordered phase, the stateof the system has a global field direction and its averagedeparts from 0.

The appearance of this observable at the macroscopiclevel is understood thanks to an already valid observableat the microscopic level, and by changing macroscopicsymmetries. That is, at the critical point, the point of tran-sition, we have a collapse of the symmetry of the macro-scopic orientations of the field (the symmetry is verifiedwhen the field is null). This change corresponds to the for-mation of a coherence structure which allows microscopicfluctuations to extend to the whole system and in fine leadto a non null order parameter, the global field, after thetransition. The system at the transition has a specific de-termination, associated to this coherence structure. De-pending on the dimension of space, this physical process

can require a specific mathematical approach, the renor-malization method, which allows to analyze the charac-teristic multi-scale structure of coherence, dominated byfluctuations at all scales, proper to critical situations. Inall cases, this situation is associated to a singularity in thedetermination of the system, which stem from the orderparameter changing from a constant to a non-zero value.

On the basis of physical criticality, the concept of ex-tended critical transition has been first proposed to ac-count for the specific coherence of biological systems,with their different levels of organization [Bai91], see also[LMP12]. The notion of different levels of organization israther polysemic. It refers usually to the epistemic struc-turing of an organism by different forms of intelligibility,thus, a fortiori and if mathematically possible, by differ-ent levels of determination or mathematical description(molecular cascades, cells’ activities and interactions, tis-sues’ structures, organs, organisms . . . ). In the context ofextended criticality, however, we propose to objectivizelevels of organization and especially the change of levelby the mathematical breaking of the determination of thefirst level, by singularities. This approach sheds an orig-inal light on the notion of level of organization, as thenew level correspond to a coupling between scales andnot simply to a higher scale [LMP12].

The core hypothesis of extended criticality is that, whenphysical systems have a mainly point-wise criticality6, or-ganisms have ubiquitous critical points (dense in a viabil-ity space, for example). Note that the interval of extendedcriticality may be given with respect to any pertinent pa-rameter. Its main properties along this line are given in[BL08, BL11]. The different levels of organization in thiscontext are presented by fractal or fractal-like structuresand dynamics, as proposed by Werner and others, see[Wes06, Wer10]. More recent applications of this con-cept may be found in [LAG+12]. Note also that criticalityenables a multi-scale heterogeneity to take place, that theconstraints of a normal state prevent, which is of interestfor biological symmetry changes [Wer10, MPSV11].

A crucial aspect of extended criticality is given by therole of symmetries and symmetry changes in biologicaldynamics, developed in [LM11a]. The density of criti-

6This critical point can be an attractor: this is the paradigm of self-organized criticality.

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cal points leads to omnipresent symmetry changes. Now,this has consequences for the very constitution of the sci-entific object. Physical objects are generic inasmuch dif-ferent objects with the same equational determination willbehave in the same way, and this way is determined by thespecific trajectory provided precisely by the equations, ageodetic in the intended phase space. It is these specifictrajectories, possibly after some transformations, whichallow to state that objects behave the same, both in thetheory and in experiments (i.e. they have invariant prop-erties). The trajectory is thus obtained by using theoret-ical symmetries (conservation principles, see above) andin fine it allows to define physical objects as generic be-cause they are symmetric (they behave the same way).

In contrast to this core perspective in physics, we pro-pose that biological objects do not have such stable sym-metries, and, thus, that their trajectories are not specific:there are no sufficiently stable symmetries and corre-sponding invariants, as for phenotypes, which would al-low to determine the evolutionary dynamics of the object.On the contrary, the object follows a possible evolutionarytrajectory, which may be considered generic. Conversely,the biological object is not generic but specific, [BL11].And it is so, since it is determined by a historical cascadeof symmetry changes, [LM11a]. In our approach, the in-version of generic vs. specific is a core conceptual dualityof biological theorizing vs. physical one. It deeply modi-fies the status of the object.

The starting assumption in this approach to evolution-ary trajectories is based on Darwin’s first principle (anddefault state for biology, [SS99]: Descent with modifi-cation. Darwin’s other principle, selection, would makelittle sense without the first.

Notice that Darwin’s first principle is a sort of non-conservation principle as for phenotypes (see 6): any re-production yields (some) changes. It is crucial for us thatthis applies at each individual cellular mitosis. As a mat-ter of fact, each mitosis may be seen as a critical transi-tion. In a multicellular organism, in particular, it is a bi-furcation that yields the reconstruction of a whole coher-ence structure: the tissue matrix, the collagen’s tensegritystructure, the cells’ dialogue in general. And this besidesthe symmetry breakings due to proteome and DNA varia-tions, which we will further discuss. In short, in view ofthe “density" of mitoses in the life interval of an organism,we may already consider this phenomenon at the core of

its analysis in terms of extended criticality.In this context, the mathematical un-predefinability of

biological phase space we discuss below will follow bycomparing the physico-mathematical constructions to theneeds of biology, where theoretical symmetries are notpreserved. Let us recall that we work in a Darwinianframe and consider organisms and phenotypes as the per-tinent observables.

3. Non-ergodicity and quantum/classical randomnessin biology.

We will discuss here the issue of “ergodicity” as wellas the combination of quantum and classical random phe-nomena in biology. By ergodicity, we broadly refer toBoltzmann’s assumption in the 1870’s that, in the courseof time, the trajectory of a closed system passes arbitrar-ily close to every point of a constant-energy surface inphase space. This assumption allows to understand a sys-tem without taking into account the details of its dynamic.

From the molecular viewpoint, the question is the fol-lowing: are (complex) phenotypes the result of a randomexploration of all possible molecular combinations andaggregations, along a path that would (eventually) exploreall molecular possibilities?

An easy combinatorial argument shows that at levels ofcomplexity above the atom, for example for molecules,the universe is grossly non-ergodic, that is it does not ex-plore all possible paths or configurations. Following anexample in [Kau02], the universe will not make all pos-sible proteins length 200 amino acids in 10 to the 39thtimes its lifetime, even were all 10 to the 80th particlesmaking such proteins on the Planck time scale. So, their“composition” in a new organ, function or organism (thus,in a phenotype) cannot be the result of the ergodicity ofphysical dynamics7.

The point is that the lack of ergodicity presents an im-mediate difficulty for the (naive) reductionist approach tothe construction of a phase space for biological dynamics,

7Notice here that this argument only states that ergodicity in themolecular phase space does not help to understand the biological dy-namics of phenotypes. The argument does not preclude the trajectoriesfrom being ergodic in infinite time. We can then say that ergodicity isbiologically irrelevant and can take this irrelevance as a principle.

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as given in purely molecular terms. In order to under-stand this, let’s consider the role of ergodicity in statisti-cal mechanics. A basic assumption of statistical mechan-ics is a symmetry between states with the same energeticlevel, which allows to analyze their probabilities (on therelevant time scales). This assumption is grounded on ahypothesis of ergodicity as for the dynamics of the parti-cles: at the limit of infinite time, they “go everywhere” inthe intended phase space, and they do so homogeneously(with a regular frequency). In this case, the situation isdescribed on the basis of energetic considerations (energyconservation properties, typically), without having to takeinto account the Newtonian trajectory or the history of thesystem.

In biology, non-ergodicity in the molecular phase spaceallows to argue that the dynamic cannot be describedwithout historical considerations, even when taking onlyinto account molecular aspects of biological systems. Afortiori, this holds when considering morphological andother higher scale biological aspects (the phenotypes inthe broadest sense). In other terms, non-ergodicity in bi-ology means that the relevant symmetries depend on a his-tory even in a tentative phase space for molecules, whichis in contrast with (equilibrium) statistical mechanics.

To sum the situation up, non-ergodicity prevent us tosymmetrize the possible. With respect to a Darwinianphase space, most complex things will never exist anddon’t play a role, [Kau02]. The history of the system en-ters into play and canalizes evolution.

Note that some cases of non-ergodicity are well stud-ied in physics. Symmetry breaking phase transitions is asimple example: a crystal does not explore all its possibleconfigurations because it has some privileged directionsand it “sticks” to them. The situation is similar for themagnetization of a magnet (see [Str05] for a mathematicalanalysis). A more complex case is given by glasses. De-pending on the models, the actual non-ergodicity is valideither for infinite time or is only transitory, yet relevantat the human time scales. Crucially, non-ergodicity cor-responds to a variety of possible states, which depend onthe paths in the energetic landscape that are taken (or nottaken) during the cooling. This can be analyzed as an en-tropic distance to thermodynamic equilibrium and corre-sponds to a wide variety of “choices”. However, the var-ious states are very similar and their differences are rela-tively well described by the introduction of a time depen-

dence for the usual thermodynamic quantities. This corre-sponds to the so-called “aging dynamics” [JS07]. The ex-ample of glassy dynamics show that the absence of a rel-evant ergodicity is not sufficient in order to obtain phasespace changes in the sense we will describe, because inthis example the various states can be understood in an apriori well-defined phase space and are not qualitativelydifferent.

Note, finally, that an ergodic trajectory is a “random”,yet complete, exploration of the phase space. However,ergodicity does not coincide with randomness, per se: astep-wise random trajectory (i.e. each step at finite timeis random), does not need to be ergodic, since ergodicity,in mathematical physics, is an asymptotic notion.

Now, biological dynamics are a complex blend of con-tingency (randomness), history and constraints. Our the-sis here is that biological (constrained) randomness is es-sential to variability, thus to diversity, thus to life.

The most familiar example is provided by meiosis, asgametes randomly inherit chromosomes pairs from theparents. Moreover, chromosomes of a given pair may ex-change homologous portions and, so far, this is analyzedin purely probabilistic terms. It is a well established factthat DNA recombinations are a major contribution to di-versity. However, all aspects of meiosis depend on a com-mon history of the mixing DNA’s and viable diversity isrestricted by this history.

A finer analysis can be carried on, in terms of ran-domness. In a cell, classical and quantum randomnessboth play a role and “superpose”. Recall first that, inphysics, classical and quantum randomness differ: dif-ferent probability theories (thus measures of randomness)may be associated to classical events vs. (entangled) quan-tum events. Bell inequalities distinguishes them (see[AGR82]).

Some examples of biologically relevant quantum phe-nomena are electron tunneling in cellular respiration[GW03], electron transport along DNA [WGP+05], quan-tum coherence in photosynthesis [ECR+07, CWW+10].Moreover, it has been shown that double proton trans-fer affects spontaneous mutation in RNA duplexes[CCRPM09]. The enthalpic chaotic oscillations of macro-molecules instead have a classical nature, in physicalterms, and are essential to the interaction of and withDNA and RNA. Quantum randomness in a mutation istypically amplified by classical dynamics (including clas-

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sical randomness), in the interaction between DNA, RNAand the proteome (see [BL13] for a discussion). This kindof amplification is necessary in order to understand thatchanges at the nanometer scale impact the phenotype ofthe cell or of the organism. Moreover, it may be sound toconsider the cell-to-cell interactions and, more generally,ecosystem’s interactions as classical, at least as for theirphysical aspects, yet affecting the biological observables,jointly with quantum phenomena.

Poincaré discovered the destabilizing effects of plane-tary mutual interactions, in particular due to gravitationalresonance (planets attract each other, which cumulateswhen aligned with the Sun); by this, in spite of the de-terministic nature of their dynamics, they go along unsta-ble trajectories and show random behavior, in astronomi-cal times (see [LJ94]). In [BL13], by analogy, the notionof “bio-resonance” is proposed. Different levels of orga-nization, in an organism, affect each other, in a stabiliz-ing (regulating and integrating), but also in a destabilizingway.

A minor change in the hormonal cascade may seriouslydamage a tissue’s coherence and, years later, cause or en-able cancer. A quantum event at the molecular level (amutation) may be amplified by cell to cell interaction andaffect the organism, whose changes may downwards af-fect tissues, cells, metabolism. Note that Poincaré’s res-onance and randomness are given at a unique and homo-geneous level of organization (actually, of mathematicaldetermination). Bio-resonance instead concerns differentepistemic levels of organization, thus, a fortiori and ifmathematically possible, different levels of determinationor mathematical description (molecular cascades, cells,tissues, organs, organisms ...).

In evolution, when a (random) quantum event at themolecular level (DNA or RNA-DNA or RNA-protein orprotein-protein) happens to have consequences at the levelof the phenotype, the somatic effects may persist if theyare inherited and compatible both with the ever changingecosystem and the “coherence structure” of the organism,that is, when they yield viable Darwin’s correlated varia-tions. In particular, this may allow the formation of a newfunction, organ or tool or different use of an existing tool,thus to the formation of a new properly relevant biologicalobservable (a new phenotype or organism). This new ob-servable has at least the same level of unpredictability asthe quantum event, but it does not belong to the quantum

phase space: it is typically subject to Darwinian selectionat the level of the organisms in a population, thus it in-teracts with the ecosystem as such. Recall that this is thepertinent level of observability, the level of phenotypes,where biological randomness and unpredictability is nowto be analyzed.

We stress again that the effects of the classical / quan-tum blend may show up at different levels of observabilityand may induce retroactions. First, as we said, a mutationor a random difference or expression in the genome, maycontribute to the formation of a new phenotype8. Second,this phenotype may retroact downwards, to the molecu-lar (or quantum) level. A molecular activity may be ex-cluded, as appearing in cells (organs / organisms) whichturn out to be unfit — selection acts at the level of or-ganisms, and may then exclude molecular activities as-sociated to the unfit organism. Moreover, methylationand de-methylation downwards modify the expression of“genes”. These upwards and downwards activities con-tribute to the integration and regulation of and by thewhole and the parts. They both contribute to and con-strain the biological dynamics and, thus, they do not allowto split the different epistemic levels of organization intoindependent phase spaces.

We recall that our choice of the biologically pertinentobservables is based on the widely accepted fact that noth-ing makes sense in biology, if not analyzed in terms ofevolution. We summarized the observables as the “phe-notype”, that is, as the various (epistemic) componentsof an organism (organs, tissues, functions, internal andecosystemic interactions . . . ).

Thus, evolution is both the result of random events at alllevels of organization of life and of constraints that canal-ize it, in particular by excluding, by selection, incompati-ble paths — where selection is due both to the interactionwith the ecosystem and the maintenance of a possibly re-newed internal coherent structure of the organism, con-structed through its history. So, ergodic explorations arerestricted or prevented both by selection and by the historyof the organism (and of the ecosystem). For example, thepresence and the structure of a membrane, or a nucleus,in a cell canalizes also the whole cellular activities along

8In some bacteria, the lac-operon control system may be inherited atthe level of proteome, see [RPC+10].

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a restricted form of possible dynamics9.In conclusion, the “canalizing” role of history and se-

lection, which excludes what is incompatible with theecosystem and/or with the internal coherence of the or-ganism, coexists with the various forms of randomnesswe mentioned. We find it critical that neither quantummechanics alone, nor classical physics alone, accountfor evolution. Both seem to work together. Mutationsand other molecular phenomena may depend on random,acausal, indeterminate quantum events. Thus they mayinterfere or happen simultaneously to or be amplified byclassical dynamics, as well as by phenotype - phenotypeinteraction. In this amplification, evolution is also notcompletely random, as seen in the similarity of the oc-topus and vertebrates’ camera eye, independently evolved(see below). Thus evolution is both strongly canalized(or far from ergodic) and yet indeterminate, random andacausal. Our key point is then that random events, in biol-ogy, do not “just” modify the (numerical) values of an ob-servable in a pre-given phase space, like in physics. Theymodify the very phase space, or space of pertinent biolog-ical (evolutionary) observables, the phenotypes.

4. Symmetries breakings and randomness

We propose that in all existing physical theories eachrandom event is associated to a symmetry change. This isa preliminary, still conjectural remark, yet it may turn outto be important when stressing the role of randomness inbiology.

A random event is an event where the knowledge abouta system at a given time does not entail its future descrip-tion. In physics though, the description before the eventdetermines the complete list of possible outcomes: theseare numerical values of pre-given observables — mod-ulo some finer considerations as the ones we made as forQM and statistical physics, on the dimensions of the phasespace, typically. Moreover, in most physical cases, thetheory provides a metric (probabilities or other measures)which determines the observed statistics (random or un-predictable, but not so much: we know a probability dis-

9See [MPSV11] for an analysis of the molecular spatial heterogene-ity in the membrane as enabled by the coupling of phase transition fluc-tuations and the cytoskeleton.

tribution). Kolmogorov’s axiomatic for probabilities workthis way and provide probabilities for the outcomes. Thedifferent physical cases can be understood and comparedin terms of symmetry breaking.

• In Quantum Mechanics, the unitarity of the quantumevolution is broken at measurement, which amountsto say that the quantum state space assumes privi-leged directions (a symmetry breaking).

• In classical probabilities, the intended phase spacecontains the set of all possibilities. Elements of thisset are symmetric inasmuch they are possible, more-over the associated probabilities are usually given byan assumption of symmetry, for example the sidesof a dice (or the regions of the phase space withthe same energy). These symmetries are broken bydrawing, which singles out a result.

• Algorithmic concurrency theory states the possibil-ities but do not provide, a priori, probabilities forthem. These may be added if the physical event forc-ing a choice is known (but computer scientists usu-ally "do not care" — this is the terminology they use,see [LPP10]).

In physical theories, we thus associated a random eventto a symmetry breaking. In each case, we have severalpossible outcomes that have therefore a symmetrical role,possibly measured by different probabilities. After therandom event, however, one of the “formerly possible”events is singled out as the actual result. Therefore, eachrandom event that fits this description is based on a sym-metry breaking, which can take different yet precise math-ematical forms, depending in particular on the probabilitytheory involved (or lack thereof).

Let’s now more closely review, in a schematic way, howthe random events are associated to symmetry breakings:

Quantum Mechanics: the projection of the state vec-tor (measurement); non-commutativity of measure-ment; tunneling effects; creation of a particle . . . .

Classical dynamics: bifurcations, for example, corre-spond typically to symmetric solutions for periodicorbits. Note that in classical mechanics, “the knowl-edge of the system at a given time” involve the mea-surement (inasmuch it limits the access to the state)and not only the state itself.

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Critical transitions: the point-wise symmetry changelead to a “choice” of specific directions (the orienta-tion of a magnet, the spatial orientation of a crystal,etc.). The specific directions taken are associated tofluctuations. Also, the multi-scale configuration atthe critical point is random, and fluctuating.

Thermodynamics: the arrow of time (entropy produc-tion). This case is peculiar as the randomness andthe symmetry breaking are not associated to an eventbut to the microscopic description. The time rever-sal symmetry is broken at the thermodynamic limit.Also, the evolution is towards a symmetrization ofthe system, since it tends towards the macroscopicstate to which correspond the greatest number of mi-croscopic states (they are symmetric from a macro-scopic viewpoint), that is the greatest entropy, com-patible with other constraints.

Algorithmic concurrency: The choice of one of the pos-sible computational paths (backtracking is impossi-ble).

If this list is exhaustive, we may say that random events,in physics, are correlated to symmetry breakings. Thesesymmetry changes and the associated random events hap-pen within the phase space given by the intended physicaltheory.

The challenge we are facing, in biology, is that random-ness, we claim, manifests itself at the very level of the ob-servables: randomness breaks theoretical symmetries andmodifies the very phase space of evolution. Critical transi-tions are the closest physical phenomenon to the needs ofthe theoretical investigation in biology. Yet, the change ofphysical observables is given within a uniform theoreticalframe. The new macroscopic observable, after the transi-tion, is already used for the description of the elements ofthe system. At the critical point, the system may be de-scribed by a cascade of parametrized models, whose pa-rameter, the scale, lead the sequence to converge to a new,but predictable coherent structure, with a specific scalesymmetry. Moreover, the process may be sometimes re-versed, always iterated; when iterated, fluctuations closeto transition may at most give quantitative differences inthe symmetries obtained at the transition.

The perspective that we advocate in biology differs thenfrom the physical cases, and takes care of dynamics where

the possible outcomes, as defining properties of biologi-cal observables, cannot be entailed from the knowledge ofthe system. There is no such situation to our knowledge inphysics: the so frequently claimed “emergence" of waterproperties from quantum properties, [LMSS13], or of rel-ativistic field from the quantum one, confirms our claim.In these cases, physicists, so far, change symmetries soradically that they need to change theory. Hydrodynam-ics, for example, deals with observable properties of watersuch as fluidity and incompressibility in continua, whichare symmetries far away from those of QM. And the chal-lenge is to invent a third theory, framing the existing in-compatible ones, a “unification" as people say in field the-ory10. Note that a common way to go from a theory toanother is to use asymptotic reasonning, that is to say, toconsider that some quantity goes to infinity, which altersthe symmetry of a situation [BL11, Bat07].

5. Randomness and phase spaces in biology

As hinted above, we understand randomness in fullgenerality as unpredictability with respect to the intendedtheory. This is of course a relativized notion, as the prac-tice of physics shows, for example in the quantum vs.classical randomness debate. In either case, randomnessis “measurable” and its measure is given by probabilitytheory. As a matter of fact, in pre-given spaces of pos-sibilities (the pertinent phase spaces), modern probabilitytheory may be largely seen as a specific case of LebesgueMeasure Theory.

More precisely, the measure (the probabilities) is givenin terms of (relative) probabilities defined by symmetrieswith respect to the observable in a prestated phase space,as for the 6 symmetric faces of a fair dice. A more sophis-ticated example is the microcanonical ensemble of sta-tistical mechanics, where the microstates with the sameenergy have the same probability (are symmetric or inter-changeable), on the grounds of the ergodic hypothesis. In

10Einstein tried very hard to reduce the Quantum Field to the rela-tivistic one, or to have the first theoretically “emerge" from the second,by claiming its incompleteness, if not completed by relativistic (hidden)variables. And he deduced and discovered quantum entanglement, bythe mathematical investigation in [EPR35]; a positive consequence of anegative result.

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either case, the random event results in a symmetry break-ing: one out of the six possible (symmetric) outcomes fora dice; the random exploration of a specific microstate instatistical mechanics (see section 4 above).

Recall that, by “theoretical symmetries”, in biology, werefer both to the phenomenal symmetries in the pheno-type and to the “coherence structure” of an organism, aniche, an ecosystem, in the broadest sense. In some cases,these symmetries may be possibly expressed by balanceequations, at equilibrium or far from equilibrium, like inphysics, or just by the informal description of its work-ing unity as balanced processes of functions, organs andglobal autopoietic dynamics [VMU74, MM10]. Under allcircumstances, a permanent exploration and change is atthe core of biology, or, as Heraclitus and Stuart Kauffmanlike to say: “Life bubbles forth”. Yet, it does so whilestruggling to preserve its relative stability and coherence.

We need to understand this rich and fascinating inter-play of stabilities and instabilities. Extended critical tran-sitions in intervals of viability, the associated symmetrychanges and bio-resonance are at the core of them: theyyield coherence structures and change them continually,through epistemic levels of organization. Bio-resonanceintegrates and regulates the different levels within an or-ganisms, while amplifying random effects due to transi-tions at one given level. At other levels of organization,these random events may yield radical changes of symme-tries, coherent structures and, eventually, observable phe-notypes.

In biology, randomness enhances variability and diver-sity. It is thus at the core of evolution: it permanentlygives diverging evolutionary paths, as theoretical bifurca-tions in the formation of phenotypes. We also stressed thatvariability and diversity are key components of the struc-tural stability of organisms, species and ecosystems, aloneand together. Differentiation and variability within an or-ganism, a species and an ecosystem contribute to their di-versity and robustness, which, in biology, intrinsically in-cludes adaptiveness. Thus, robustness depends also onrandomness and this by low numbers: the diversity ina population, or in an organ, which is essential to theirrobustness, may be given by few individuals (organisms,cells). This is in contrast to physics, where robustness bystatistical effects inside a system is based on huge num-bers of elementary components, like in thermodynamics,in statistical physics and in quantum field theory [Les08].

Actually, even at the molecular level, the vast majorityof cell proteins are present in very low copy numbers, sothe variability due to proteome (random) differences aftera mitosis, yields new structural stabilities (the new cells)based on low but differing numbers.

Moreover, there exists a theoretical trend of increas-ing relevance that considers gene expression as a stochas-tic phenomenon. The theory of stochastic gene expres-sion, usually described within a classical frame, is per-fectly compatible, or it actually enhances our stress onrandomness and variability, from cell differentiation toevolution11. In those approaches, gene expression mustbe given in probabilities and these probabilities may de-pend on the context (e. g. even the pressure on an embryo,see [BF04]). This enhances variability even in presenceof a stable DNA.

Besides the increasingly evident stochasticity of geneexpression, contextual differences may also force verydifferent uses of the same (physical) structure. For ex-ample, the crystalline in a vertebrate eye and the kid-ney and their functions use the same protein [MOF+06],with different uses in these different context. Thus, ifwe consider the proper biological observable (crystalline,kidney), each phenotypic consequence or set of conse-quences of a chemical (enzymatic) activity has an a prioriindefinite set of potential biological uses: when, in evolu-tion, that protein was first formed, there was no need forlife to build an eye with a crystalline. There are plenty ofother way to see, and animals do not need to see. Simi-larly, a membrane bound small protein, by Darwinian pre-adaptation or Gould’s exaptation, may become part of theflagellar motor of a bacterium, while originally it had var-ious, unrelated, functions [LO07]. Or, consider the bonesof the double jaw of some vertebrates that evolved intothe bones of the middle ears of mammals (one of Gould’spreferred examples of exaptation), see [All75]. A newfunction, hearing, emerged as the “bricolage” (tinkering)of old structures. There was no mathematical necessityfor the phenotype nor for the function, “listening”, in thephysical world. Indeed, most complex things do not existin the Universe, as we said.

Evolution may also give divergent answers to the same

11A pioneering paper on this perspective is [Kup83]: recent surveysmay be found in [Pal03, AvO08, Hea13].

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or to similar physical constraints. That is, the same func-tion, moving, for example, or breathing, may be biologi-cally implemented in very different ways. Trachea in in-sects versus vertebrates’ lungs (combined with the vascu-lar system), are due both to different contexts (differentbiological internal and external constraints) and to ran-dom symmetry changes in evolutionary paths. Thus, verydifferent biological answers to the “same” physical con-text make phenotypes incomparable, in terms of physicaloptima: production of energy or even exchanging oxygenmay be dealt with in very different ways, by organisms inthe “same” ecosystem.

Conversely, major phenomena of convergent evolutionshape similarly organs and organisms. Borrowing the ex-amples in [LMK12], the convergent evolutions of the oc-topus and vertebrate eye follow, on one side, random, pos-sibly quantum based acausal and indeterminate mutations,which contributed to very different phylogenetic paths.On the other, it is also "not-so-random" as both eyes con-verge to analogous physiological structures, probably dueto physical and biological similar constraints — actingas co-constituted borders or as selection. The conver-gent evolution of marsupial and mammalian forms, likethe Tasmanian wolf (a marsupial) and mammalian wolfare other examples of convergent, not-so-random compo-nents of evolution, in the limited sense above.

In conclusion, randomness, in physics, is "constrained"or mathematically handled by probabilities, in generalwith little or no relevance of history, and by possiblydecorrelating events from contexts. In biology, historiesand contexts (sometime strongly) canalize and constraintrandom evolutions.

That is, randomness may be theoreticaly constrained,in physics, by probability values in a pre-given list of pos-sible future events; in biology, it is constrained by the pasthistory and the context of an event.

5.1. Non-optimality

Given the lack of ordered or orderable phase spaces,where numbers associated to observables would allowcomparisons, it is hard to detect optimality in biology, ex-cept for some local organ construction. In terms of phys-ical or also biological observables, the front legs of anelephant are not better nor worst than those of a Kanga-roo: front podia of tetrapodes diverged (broke symmetries

differently) in different biological niches and internal mi-lieu. And none of the issuing paths is “better” than theother, nor followed physical optimality criteria, even lessbiological ones: each is just a possible variation on anoriginal common theme, just compatible with the internalcoherence and the co-constituted ecosystem that enabledthem.

In general, thus, there is no way to define a real val-ued (Lagrangian) functional to be extremized as for phe-notypes, as this would require an ordered space (a realvalued functional), where “this phenotype” could be saidto be “better” than “that phenotype”. The exclusion ofthe incompatible, in given evolutionary context, in noways produces the “fittest” or “best”, in any physico-mathematical rigorous sense. Even Lamarckian effects, ifthey apply, may contribute to fitness, not to “fitter”, evenless “fittest”. Only “a posteriori” can one say that “this isbetter than that” (and never “best” in an unspecified par-tial upper semi-lattice): the a posteriori trivial evidenceof survival and successful reproduction is not an a pri-ori judgment, but an historical one. Dinosaurs dominatedthe Earth for more than 100 millions years, leaving littleecological space to mammals. A meteor changed evolu-tion by excluding dinosaurs from fitness: only a posteri-ori, after the specific consequences of that random event,mammals may seem better — but do not say this to themammals then living in Yucatan. The blind cavefish, an“hopeful monster" in the sense of Goldschmidt, a poste-riori seems better than the ascendent with the eyes, onceit adapted to dark caverns by increasing peripheral sen-sitivity to water vibrations. This a priori incomparabil-ity corresponds to the absence of a pre-given partial orderamong phenotypes, thus of optimizing paths, simply be-cause their space is not pre-given. At most, sometimes,one can make a pair-wise a posteriori comparisons (whichis often associated to experimental situation, with con-troled, simple conditions). This incomparability is alsodue to the relative independence of niches, which are co-constituted by organisms.

More generally, conservation or optimality propertiesof physical observables (the various forms of physical en-ergy, for example) cannot help to determine the evolution-ary trajectory of an organism. No principle of “least freeenergy” (or “least time consumption of free energy”, if itapplies) can help to predict or understand completely theevolution of a proper and specific biological observable,

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nor of an organism as object of selection. A given phys-ical ecosystem may yield very different organisms andphenotypes. A Darwin says, reproduction always impliesvariation, even without being prompted by the environ-ment.

Physical forces may help to determine the dynamicsonly locally, for example the form of some organs, whereexchange of matter or energy dominates (lungs, vascu-lar system, phyllotaxis . . . ). Their forms partly followoptimality principles (dynamical branching, sprouting orfractal structures or alike, see [Jea94, Fle00, BGM88]). Inthese cases, physical forces (the pushing of the embryonalheart, respiration . . . , tissue matrix frictions . . . ) are fun-damental dynamical constraints to biology’s default state:proliferation with variation and motility. Then selectionapplies at the level of phenotypes and organisms. Thus theresult is incompletely understood by looking only at thephysical dynamical constraints, since variability and di-versity (the irregularity of lungs, of plants organs in phyl-lotaxis . . . ) contribute to robustness. They are not “noise"as in crystals’ formation.

6. A non-conservation principle

The phylogenetic drift underlying evolution must beunderstood in terms of a “non-conservation principle” ofbiological observables. Darwin proposed it as a principle,to which we extensively referred: descent with modifi-cation, on which selection acts. This is the exact oppositeof the symmetries and conservation properties that governphysics and the related equational and causal approaches.There is of course structural stability, in biology, whichimplies similar, but never identical iteration of a morpho-genetic process. Yet, evolution requires also and intrin-sically this non-conservation principle for phenotypes inorder to be made intelligible. In particular, one needs tointegrate randomness, variability and diversity in the the-ory in order to understand phylogenetic and ontogeneticadaptability and the permanent exploration and construc-tion of new niches.

In a sense, we need, in biology, a similar enrichmentof the perspective as the one quantum physicists dared topropose in the ’20th: intrinsic indetermination was intro-duced in the theory by formalizing the non-commutativityof measurement (Heisenberg non-commutative algebra of

matrices) and by Schrödinger equation (the determinis-tic dynamics of a probability law). We propose here ananalysis of indetermination at the level of the very forma-tion of the phase space, or spaces of evolutionary possi-bilities, by integrating Darwin’s principle of reproductionwith modification and, thus, of variability, in the intendedstructure of determination.

As a further consequence, the concept of randomness inbiology we are constructing mathematically differs fromphysical forms of randomness, since we cannot apply aprobability measure to it, in absence of a pre-given spaceof possible phenotypes in evolution (nor, we should say, inontogenesis, where monsters appear, sometimes hopefulfrom the point of view of evolution). The lack of prob-ability measures may resemble the “do not care" princi-ple in algorithmic concurrency, over computer networks,mentioned above (and networks are fundamental struc-tures for biology as well). However, the possible compu-tational paths are pre-given and, moreover, processes aredescribed on discrete data types, which are totally inade-quate to describe the many continuous dynamics presentin biology. Indeed, the sequential computers, in the nodes,are Laplacian Discrete State machine, as Turing first ob-served [LMSS12], far away from organisms.

In summary, in biology, the superposition of quantumand classical physics, bio-resonance, the coexistence ofindeterminate acausal quantum molecular events, with so-matic effects, and of non-random historical and contextualconvergences do not allow to invent, as physicists do, amathematically stable, pre-given phase space, as a “back-ground” space for all possible evolutionary dynamics.

Random events break symmetries of biological trajec-tories in a constitutive way. A new phenotype, a new func-tion, organ . . . organism, is a change (a breaking and a re-construction) of the coherence structure, thus a change ofthe symmetries in the earlier organism. Like in physics,symmetry changes (thus breakings) and randomness seemto coexist also in life dynamics, but they affect the dynam-ics of the very phase space.

Our approach to the biological processes as extendedcritical transitions fits with this understanding of bi-ological trajectories as cascades of symmetry changes[LM11a]. Of course, this instability goes together withand is even an essential component of structural stability:each critical transitions is a symmetry change and it pro-vides variability, diversity, thus adaptivity, at the core of

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biological viability. Even an individual organism is adap-tive to a changing ecosystem, thus biologically robust, bythe ever different re-generation of its parts. The sensi-tivity to minor fluctuations close to transition, which is asignature of critical phase transitions, enhances adaptivityof organisms (DNA methylation may affect even adaptivebehavior, [KMFM08]).

It seems thus impossible to extract relevant invariantsconcerning the specific structure of phenotypes and con-struct by them a space of all possible phenotypes. It maybe even inadequate as variability is (one of) the mainmathematical invariant in biology, beginning with indi-vidual mitoses. This does not forbid to propose somegeneral invariants and symmetries, yet not referring tothe specific aspects of the phenotype, as form and func-tion. This is the path we followed when conceptualizingsufficiently stable properties, such as biological rhythms[BLM11], extended criticality and anti-entropy (a tool forthe analysis of biological complexity as anti-entropy, see[BL09, LM12]).

We follow by this physics’ historical experience of “ob-jectivizing” by sufficiently stable concepts. In biology,these must encompas change and diversity. As a mat-ter of fact, our investigations of biological rhythms, ex-tended criticality and anti-entropy are grounded also onvariability. In a long term perspective, these conceptsshould be turned all into more precisely quantified (andcorrelated) mathematical invariants and symmetries, inabstract spaces. This is what we did as for the two dimen-sional time of rhythms and as for anti-entropy, by imitat-ing the way Schrödinger defined his equation in Hilbertspaces, far away from ordinary space-time. Abstractproperties such as extended criticality and anti-entropy donot refer to the invariance of specific phenotypes, but theyare themselves relatively stable, as they seem to refer tothe few invariant properties of organisms. Their analysis,in a quantified space of extended criticality, may give us abetter understanding of objects and trajectories within theever changing space of phenotypes.

7. Causes and Enablement

We better specify now the notion of enablement, pro-posed in [LMK12] and already used above. This notionmay help to understand the role played by ecosystemic

dynamics in the formation of a new observable (mathe-matically, a new dimension) of the phase space. Examplesare given below and we will refine this notion throughoutthe rest of this paper.

In short, a niche enables the survival of an otherwiseincompatible/impossible form of life, it does not cause it.More generally, niches enable what evolves, while evolv-ing with it. At most, a cause may be found in the “differ-ence" (a mutation, say) that induced the phenotypic vari-ation at stake, as spelled out next.

This new perspective is motivated, on one side, by ourunderstanding of physical “causes and determinations” interms of symmetries, along the lines above of modernphysics, and, on the other side, by our analysis of bio-logical “trajectories” in phylogenesis (and ontogenesis),as continual symmetry changes. Note that, in spite of itsmodern replacement by the language of symmetries, thecausal vocabulary still makes sense in physics: gravitationcauses a body to fall (of course, Einstein’s understandingin terms of geodetics in curved spaces, unifies gravitationand inertia, it is thus more general).

In biology, without sufficiently stable invariances andsymmetries at the level of organisms, thus (possibly equa-tional) laws, “causes” positively entailing the dynamics(evolution, typically) cannot be defined. As part of thisunderstanding, we will discuss causal relations in a re-stricted sense, that is, in terms of “differential causes”.In other words, since symmetries are unstable, causalityin biology cannot be understood as “entailing causality”as in physics and this will lead us to the proposal thatin biology, causal relations are only differential causes.If a bacterium causes pneumonia, or a mutation causes amonogenetic diseases (anemia falciformis, say), this is acause and it is differential, i.e. it is a difference with re-spect to what is fairly considered “normal“, “healthy” or“wild” as biologist say as for the genome, and it causes ananormality in the phenotype.

A classical mistake is to say: this mutation causesa mentally retarded child (a famous genetic disorder,phenylketonuria), thus . . . the gene affected by the muta-tion is the gene of intelligence, or . . . here is the gene thatcauses/determines the intelligence [Wei92] or encodes for(part of) the brain. In logical terms, this consists in deduc-ing from “notA implies notB” that “A implies B” (or from“not normal A implies not normal B”, that “normal A im-plies normal B”): an amazing logical mistake. All that we

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know is a causal correlation of differences12.We then propose to consider things differently. The ob-

served or induced difference, a mutation with a somaticeffect, say, or a stone bumping on someone’s head, ora carcinogen (asbestos), does cause a problem; that is,the causal dictionary is suitable to describe a differentialcause - effect relation. The differential cause modifies thespace of possibilities, that is the compatibility of the or-ganism with the ecosystem. In other terms, it modifies the“enablement relations". This is for us, [LMK12], the wayan organism, a niche, an ecosystem may accommodate aphenotype, i.e. when the modified frame becomes viablefor a new or different phenotype (a new organ or function,a differentiated organism).

We are forced to do so by the radical change of the de-fault state in biology. Inertial movement, or, more gen-erally, conservation principles in physics, need a forceor an efficient cause to change (see the revitalizationof the Aristotelian distinction efficient/material cause, in[BL11]). In biology, default states guaranty change:reproduction with variation and motility, [LMSS13].Causes “only" affect the intrinsic (the default) dynamicsof organisms. More precisely, in our view, the differentialcauses modify the always reconstructed coherence struc-ture of an organism, a niche, an ecosystem. So enable-ment is modified: a niche may be no longer suitable for anorganism, an organism to the niche. Either selection mayexclude the modified organism. Or a change in a niche,due to a differential physical cause (a climate change, forexample), may negatively select existing organisms or en-able the adaptive ones, since the enablement relations dif-fer.

Differential analysis are crucial in the understanding ofexisting niches. Short descriptions of niches may be givenfrom a specific perspective (they are strictly epistemic):they depend on the “purpose” one is looking at, say. Andone usually finds out a feature in a niche by a difference,that is, by observing that, if a given feature goes away, theintended organism dies. In other terms, niches are com-

12Schrödinger, in his 1944 book, was well aware of the limits of thedifferential analyses of the chromosomes and their consequences: “Whatwe locate in the chromosome is the seat of this difference. (We call it, intechnical language, a ’locus’, or, if we think of the hypothetical materialstructure underlying it, a ’gene’.) Difference of property, to my view, isreally the fundamental concept rather than property itself.”, p.28.

pared by differences: one may not be able to prove thattwo niches are identical or equivalent (in enabling life),but one may show that two niches are different. Oncemore, there are no symmetries organizing over time thesespaces and their internal relations.

In summary, while gradually spelling out our notion ofenablement, we claim that only the differential relationsmay be soundly considered causal. Moreover, they ac-quire a biological meaning only in presence of enable-ment. In other words:

1. In physics, in presence of an explicit equational de-termination, causes may be seen as a formal symme-try breaking of equations. Typically, f = ma, a sym-metric relation, means, for Newton, that a force, f ,causes an acceleration a, asymmetrically. Thus, onemay consider the application of a Newtonian force asa differential cause13. This is so, because the inertialmovement is the “default“ state in physics (”nothinghappens” if no force is applied). This analysis cannotbe globally transferred to biology, inasmuch symme-tries are not stable and, thus, one cannot write equa-tions for phylogenetic trajectories (nor break theirsymmetries). Moreover, the default state is far frombeing inertia (next point).

2. As just mentioned the default state in physics is in-ertia. In biology instead, the default state is ”activ-ity“, as proliferation with variation and motility. Asa consequence, an organism, a species, does not needa cause to be active, e.g. to reproduce with modifica-tios and possibly occupy a new niche14. It only needsto be enabled in order to survive by changing. More-over, in our terms, this default state involves contin-ual critical transitions, thus symmetry changes, up tophase space changes.

Consider for example an adjacent possible emptyniche, for example Kauffman’s example of the swim blad-

13In a synthetic/naive way, one may say that Einstein reversed thecausal implication, as a space curvature “causes“ an acceleration that“causes“ a field, thus a force (yet, the situation is slightly more compli-cated and the language of symmetries and geodetics is the only rigorousone).

14Energy or matter, of course, is needed in order to reproduce, but itis not a cause. As we spell out in [BL11, BLM11], in biology energy isa parameter, like in allometric equations, it is not an “operator”, like inphysics.

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der (see for example [Kau02, Kau12, LMK12]), formedby Gould’s exaptation from the lung of some fishes. Is ita boundary condition? Not in the sense this term has inphysics, since the swim bladder may enable a (mutated)worm or a bacterium to live and evolve, according to un-predictable enabling relations. That is, the observablefeatures of the swim bladder to be used by the new organ-ism to achieve functional closure in its environment maybe radically new, possibly originating for both in a quan-tum based acausal/indeterminate molecular event and bycorrelated variations: the niche and the bacterium func-tionally shape each other. As discussed above, the combi-nation of various forms of (physical and biological) ran-domness modify the set of observables (the new organ, thenew bacterium), not just the values of some observables.

Once more, in physics, energy conservation propertiesallow us to derive the equations of the action/reaction sys-tem proper to the physical phenomenon in a pre-givenphase space. Random event may modify the value ofone of the pertinent observable, not the very set of ob-servables. Typically, a river does co-constitute its bor-ders by frictions, yet the observables and invariants to bepreserved are well-know (energy and/or momentum), thegame of forces as well. It may be difficult to write allthe equations of the dynamics and some non-linear effectsmay give the unpredictability of the trajectory. Yet, weknow that the river will go along a unique perfectly deter-mined geodetics, however difficult it may be to calculate itexactly (to calculate the exact numerical values of the dy-namics of the observables). Yet, a river never goes wrongand we know why: it will follow a geodetics. An onto- orphylogenetic trajectory may go wrong, actually most ofthe time it goes wrong. We are trying to theoretically un-derstand “how it goes”, between causes and enablement.

In summary, enablement and proliferation with varia-tion and motility as default states are at the core of lifedynamics. They conceptually frame the development oflife in absence of a pre-definable phase space.

As we recalled, niches and phenotypes are co-constituted observables. Typically, the organism adjustingto / constructing a new niche may be a hopeful monster,that is the result of a “pathology" [Die03, Gou77]. Now,notions of “normal” and “pathological” makes no sensein physics. They are contextual and historical in biology;they are contingent yet fundamental.

These differing notions may also help to distinguish be-

tween enablement and causality, as the latter may be un-derstood as a causal difference in the “normal” web ofinteractions. In evolution, a difference (a mutation) maycause a “pathology”, as hopeful monster. That is, thismonster, which is such with respect to the normal or wildphenotype, may be killed by selection or may be enabledto survive by and in a new co-constituted niche. A darkcavern may be modified, also as a niche for other forms oflife, by the presence of the blind fish. And the contingentmonster becomes the healthy origin of a speciation.

Thus, besides the centrality of enablement, we maymaintain the notion of cause — and it would be a mistaketo exclude it from the biological dictionary. As a mat-ter of fact, one goes to the doctor and rightly asks for thecause of pneumonia — not only what enabled it: find andkill the bacterium, please, that is the cause. Yet, that bac-terium has been enabled to grow excessively by a weaklung, a defective immune system or bad life habits . . . So,the therapy should not only concern the differential cause,the incoming bacteria, but investigate enablement as well.And good doctors do it, without necessarily naming it so,[Nob09].

Finally, following [SS99], by our approach we un-derstand cancer as being enabled by a modified “soci-ety of cells” (the concerned tissue, organ, organism). Acarcinogen affecting the organism (typically, the epithe-lial stroma, [SS99]) deferentially modifies the “normal”tissue-niche for the cells and its coherence structure. Theless controlled cells’ default state, proliferation with vari-ation, may then lead to the abnormal proliferation, possi-bly with increasing variation (as an elementary example,a teratoma has a larger number of cell types than a normaltissue).

8. Structural stability, autonomy and constraints

Organisms withstand the intrinsic unstability / unpre-dictability of the changing phase space, by the rela-tive autonomy of their structural stability. They havean internal, permanently reconstructed autonomous co-herent structure, Kantian wholes (in Kant’s sense, see[Kan81, LP13]), or Varela’s autopoiesis, that gives theman ever changing, yet “inertial” structural stability —where inertia for organisms must be understood in theterms of biological protension, in [LM11b]. They achievea closure in a functional space by which they reproduce,

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and evolve and adapt by changing alone or together out ofthe indefinite and unorderable set of functions, or by find-ing new uses of pre-existing components to sustain theiractivity in the ongoing co-evolution in the ecosystem.

The niche is indefinite in features prior to prolifera-tion with variation and selection revealing what will co-constitute “task closure” for the organism. The niche al-lows the tasks’ closure by which an organism survives andreproduces.

Organisms and ecosystems are structurally stable, alsobecause of their constrained autonomy, as they perma-nently and non-identically reconstruct themselves, theirinternal and external constraints. They do it in an al-ways different, thus adaptive, way. They change the co-herence structure, thus its symmetries. This reconstruc-tion is thus random, but also not random, as it heavily de-pends on constraints, such as the proteins types imposedby the DNA, the relative geometric distribution of cellsin embryogenesis, interactions in an organism, in a niche.Yet, the autopoietic activity is based also on the oppo-site of constraints: the relative autonomy of organisms.In other words, organisms transform the ecosystem whiletransforming themselves and they can stand this continualchanges because they also have an internal preserved co-herent structure (Bernard’s “milieu interieur”). Its stabil-ity is maintained also by slightly, yet constantly changinginternal symmetries, which enhance adaptivity, beginningwith individual cellular mitosis in a mulitcellular organ-isms.

As we said, autonomy is integrated in and regulated byconstraints, within an organism itself and of an organismwithin an ecosystem. Autonomy makes no sense withoutconstraints and constraints apply to an autonomous unity.So constraints shape autonomy, which in turn modifiesconstraints, within the margin of viability, i.e. within thelimits of the interval of extended criticality.

A way to understand the impossibility of a completea priori description of actual and potential biological or-ganisms and niches may be the following. Recall first therole of observable invariants and conservation propertiesin establishing physical phase spaces, since Galileo’s in-ertia as a symmetry group. Then, recall how this allowedfinite definitions, in terms of symmetries, of the most ab-stract infinite phase spaces. As a consequence of our anal-ysis in terms of symmetry breakings, any given, possiblycomplete description of an ecosystem is incompressible,

in the sense that any linguistic description may requirenew names and meanings for the new unprestatable func-tions. These functions and their names make only sense inthe newly co-constructed biological and historical (evenlinguistic) environment. There is no way to define them apriori with finitely many words. The issue then is not in-finity, but incompressibility by the lack of invariant sym-metries, which we described in relation to extended criti-cality.

9. Conclusion

We stressed the role of invariance, symmetries and con-servation properties in physical theories. Our prelimi-nary aim has been to show that the powerful methodsof physics that allowed to pre-define phase spaces on thegrounds of the observables and the invariants in the ”tra-jectories” (the symmetries in the equations) do not applyin biology.

An immense literature has been tackling ”emergence”in life phenomena. Yet, in the technical analyses, thestrong and dominating theoretical frames inherited frommathematical physics (or even computing) do not seemto have been abandoned. From Artificial Life, to Cellu-lar Automata and various very rich analysis of dynamicalsystems, the frame for intelligibility is a priori given, un-der the form, often implicitly, of one or more pre-definedphase spaces, possibly to be summed up by adequatemathematical forms of products (Cartesian, tensorial ...).

A very interesting and motivated frame for these per-spectives is summarized in [DFZG07]. Well beyond themany analysis which deal with equilibrium systems, aninadequate frame for biology, these authors deal with in-teractions between multiple attractors in dissipative dy-namical systems, possibly given in two or more phasespaces15. Then, according to [DFZG07], two determin-istic, yet highly unpredictable and independent systems,which interact in the attractor space, may ”produce persis-tent attractors that are offsprings of the parents . . . . Emer-gence in this case is absolute because no trajectories existlinking the child to either parent (p. 158) ... [The] source

15The notion of attractor is a deep mathematical notion; in principle,in order to be soundly presented, it requires explicit equations or evolu-tion functions – solutions with no equations – in pertinent phase spaces.

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[of emergence] is the creation, evolution, destruction, andinteraction of dynamical attractors (p. 179)”.

This analysis is compatible with ours and it may enrichit by a further component, in pre-given interacting phasespaces. Yet, we go somewhat further by a critical perspec-tive, which, per se, is a tool for intelligibility, and, below,we will hint again to further possible (and positive) work.

In our approach, the intrinsic unpredictability of thevery Phase Space of biological processes is due, in sum-mary, to:

1. extended criticality, as a locus for the correlation be-tween symmetry breakings and randomness;

2. cascades of symmetry changes in (onto-) phyloge-netic trajectories;

3. bio-resonance, due to interacting levels of organiza-tion, as a component both of integration and regu-lation, in an organism, as well as of amplificationof random fluctuations in one level of organizationthrough the others;

4. enablement, or the co-constitution of niches and phe-notypes, a notion to be added to physical determina-tion.

These phenomena are crucial also in order to under-stand life persistence, as they are at the origin of variabil-ity, thus of diversity and adaptability, which essentiallycontribute to biological structural stability. Our theoret-ical frame, in particular, is based on reproduction withmodification and motility, as proper default states for theanalysis of phylo- and ontogenenesis. This justifies therole of enablement, in particular.

More precisely, in biology, symmetries at the pheno-typic level are continually changed, beginning with theleast mitosis, up to the “structural bifurcations” whichyield speciations in evolution. Thus, there are no biologi-cal symmetries that are a priori preserved, except and forsome time, some basic structures such as bauplans (stillmore or less deeply modified during evolution). There areno sufficiently stable mathematical regularities and trans-formations, to allow an equational and law like descrip-tion entailing the phylogenetic and ontogenetic trajecto-ries. These are cascades of symmetry changes and thusjust cumulative historical dynamics. And each symmetrychange is associated to a random event (quantum, classi-cal or due to bio-resonance), at least for the breaking of

symmetries, while the global shaping of the trajectory, byselection say, is also due to non-random events. In thissense biological trajectories are generic, that is just pos-sible ones, and yield a historical result, an individuated,specific organism (see [BL11, LM11a]).

As a consequence, this sum of individuals and individ-ualizing histories, co-constituted within an ever changingecosystem, does not allow a compressed, finite or formaldescription of the space of possibilities, an actual bio-logical phase space (functions, phenotypes, organisms):these possibilities are each the result of an unpredictablesequence of symmetry breakings, associated to randomevents, in contrast to the invariant (conservation) prop-erties which characterize physical “trajectories”, in thebroad sense (extended to Hilbert’s spaces, in QuantumMechanics).

By the lack of mathematically stable invariants (stablesymmetries), there are no laws that entail, as in physics,the observable becoming of the biosphere. The geode-tic principle mathematically forces physical objects neverto go wrong. A falling stone follows exactly the grav-itational arrow. A river goes along the shortest path tothe sea, it may adjust it by nonlinear well definable inter-actions as mentioned above, but it will never go wrong.These are all geodetics. Living entities, instead, go wrongmost of the time: most organisms are extinct, almost halfof fecundations, in mammals, do not lead to a birth, anamoeba does not follows, exactly, a curving gradient —by retention it would first go along the tangent, then cor-rect the trajectory, in a protensive action. In short, lifegoes wrong most of the time, but it “adjusts” to the en-vironment and changes the environment, if possible. Itmaintains itself, always in a critical transition, at eachmitosis, that is within an extend critical interval, whoselimits are the edge of death. It does so by changing theobservables, the phenotypes and its niche, thus the verynature and space of the living object.

Then, we must ask new scientific questions and inventnew tools, for this co-constitution by organisms as theyco-evolve and make their worlds together. This must beseen as a central component of the biosphere’s dynamics.The instability of theoretical symmetries in biology is not,of course, the end of science, but it sets the limits of thetransfer of physico-mathematical methods, as taught usfrom Newton onward, to biology. In biological evolutionwe cannot use the same very rich interaction with math-

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ematics as it has been constructed at the core of physicaltheories. However, mathematics is a human adaptive con-struction: an intense dialogue with biology may shape forit new scientific paths, concepts, structures, as it did withphysics since Newton.

By providing some theoretical arguments that yieldthis “negative result”, in terms of symmetries and criti-cal transitions, we hope to have provided also some toolsfor a new opening. Negative results marked the begin-ning of new sciences in several occasions: the thermody-namic limit to energy transformation (increasing entropy),Poincaré’s negative result (as he called his Three BodyTheorem), Gödel’s theorem (which set a new start to Re-cursion Theory and Proof Theory) all opened new ways ofthinking, [Lon12]. Limits clarify the feasible and the nonfeasible with the existing tools and may show new direc-tions by their very nature, if this has a sufficiently precise,scientific content.

The scientific answer we propose to this end of thephysicalist certitudes, is based on our analysis of symme-try changes in extended critical transitions and on the no-tion of “enablement” in evolution (and ontogenesis). En-ablement concerns how organisms co-create their worlds,with their changing symmetries and coherence structures,such that they can exist in a non-ergodic universe.

Following [LMK12], our thesis then is that evolutionas a “diachronic process” of becoming (but ontogenesisas well) “enables”, but does not cause, unless differen-tially, the forthcoming state of affairs, in the sense spec-ified above. Galileo and Newton’s entailed trajectoriesmathematized Aristotle’s “efficient cause” only. Instead,in our view, such entailed causal relations must be re-placed by “enablement” relations, plus differential, oftenquantum indeterminate, causes, in biological processes.

Life is caught in a causal web, but lives also in a web ofenablement and radical emergence of life from life, whoseintelligibility may be largely given in terms of symmetrychanges and their association to random events at all lev-els of organization.

We based our analysis on the Darwinian default stateand key principles: reproduction with modification (plusmotility) and selection. Selection shapes this bubblingforth of life by excluding the unfit. Our approach is just afurther theoretical specification along these lines.

As hinted in 6, a long term project would be to bet-ter quantify our approaches to two dimensional time for

rhythms, to extended criticality and to anti-entropy (seethe references), in order to construct from them an ab-stract phase space based on these mathematically stableproperties. The dynamical analysis should follow the na-ture of Darwin’s evolution, which is an historical sci-ence, not meant at all to “predict”, yet giving a remark-able knowledge of the living. Thus, the dynamics of ex-tended criticality or anti-entropy should just provide theevolution of these state functions, or how these abstractobservables may develop with respect to the intended pa-rameters, including time. And this, without being “pro-jectable” on specific phenotypes, even not in probabilities,as it is instead possible for Schrödinger’s state functionsin Quantum Mechanics. To this purpose, one should givea biologically interesting measure for extended critical-ity, as we did for anti-entropy [BL09], and describe in aquantitative way, in the abstract space of extended criticaltransitions, the qualitative evolution of live.

Now, going back to brain dynamics, the general frame-work we discussed has a number of consequences.

The first consequence concerns mathematical modelsfollowing usual physical methodology, for example themodel of the primary visual cortex as an implementationof a sub-riemannian geometry, corresponding to the vi-sual field [Pet03]. This model is biologically very rele-vant since it enables scientists to make sense of the func-tional architecture of the primary visual cortex. However,its geometry is only valid, stricto sensu, outside of an ac-tive brain, and outside of a living organism, as this cortexis closely connected and dependent with the rest of theorganism in vivo [GHB99], and also because of the un-derlying variability in its intrinsic activity. In our moregeneral vocabulary, the biological symmetries that allowto write this model are not stable and they are broken inmany ways in vivo by the activity of the organism.

The second consequence, along the line of the work byGerhard Werner, see for example [Wer10, Wer11], con-cerns the change of level of organization associated tocritical-like phenomenon in the brain and more generallyin the organism. This viewpoint emphasizes the relation-ship between scales, and the new level is not simply ata higher scale but across scales. In this frame, with re-spect to the question of consciousness and the classicalMind-body dichotomy, where “body” more or less meansspatial extension, it is noteworthy that the renormalizationviewpoint provides an original approach: phenomena are

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no longer understood as the combination of spatially ex-tended interactions, as this combination may diverge (forsmall and/or large scales).

The third consequence concerns the changes of sym-metry of the biological object. If we consider animal be-haviours as a macroscopic aspect, the diversity in theirpattern, both intra- and interspecific, is precisely an aspectof these symmetry changes that should not vanish fromany analysis. This kind of considerations applies also toparts of the brain. Now, from a theoretical viewpoint,the key question is that of the objectivation of phenom-ena with such unstable symmetries. In particular, notethat the usual physical renormalization relies mainly onan asymptotic symmetry of the equations by the changeof scale, a symmetry that doesn’t seem completely stablein biology. Note that a way to stabilize partially biolog-ical symmetries is to have an experimental setup that doso, this point will be discussed in a forthcoming article.

Acknowledgments

We would like to thank Stuart Kauffman for the verystimulating joint work that preceded this paper and theeditors for their interest in "extended criticality". Longo’s(and co-authors’) papers are downloadable from http://www.di.ens.fr/users/longo/.

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