Philosophical accounts of biological functions $

Review of: Justin Garson (2016) A critical overview of biological functions. Springer International Publishing, Dordrecht. ISBN: 978-3-319-32020-5, 113 pages, price: $ 54.99 (paperback). The book A critical overview of biological functions is a short monograph by J. Garson, which provides a survey of the views on biological functions in the analytic tradition of philosophy. The notion of function is ubiquitous in biology and all of its subfields. Behind the notion of biological functions lurks the shadow of final causes. Overcoming this shadow is a challenge that has stimulated many philosophers and the literature on this topic is very rich. In the analytic tradition, researchers focus on providing naturalized accounts of functions. To do so, the main difficulty is to provide accounts of functions that exclude the use of final causes. The outcome of this collective work is a diversity of accounts of functions. Some of these accounts are fairly recent while others have been proposed several decades ago and are the object of many discussions.

Overall, the author provides an impressive and concise review of the debate on the various accounts of functions that are held by recent and current philosophers. The focus is on biological functions, and artifacts are discussed only inasmuch as some accounts aim to theorize functions in both domains together. The core chapters of the book, chapters 2 to 6, focus on the different accounts of functions. The first kind of accounts is discussed in chapter 2 and defines functions on the basis of goal-directedness, either from a behaviorist perspective in the case of Sommerhoff and Braithwaite or from a mechanistic perspective in the case of Nagel and the cyberneticists. Both families of accounts pertain to the history of philosophy in the sense that they have been the object of severe criticism and are no longer at the forefront of current debates. A second approach is developed in chapter 3 and is called the selective account of function. This account builds on the idea that a trait is functional when it has been selected in a population. This account has been introduced independently by Neander and Millikan and has been further developed by many philosophers such as Godfrey-Smith and Griffiths. In chapter 4, the author presents a third kind of accounts, which aims to define the function of a trait on the basis of its contribution to present day fitness. Boorse's account of function typically falls in this category. The fourth kind of accounts, in chapter 5, grounds functions on their causal role in a system. This view has been introduced by Cummins and further developed by Craver and Davies among others. Last, chapter 6 presents three accounts of functions that are more recent. In particular, the author discusses the organizational account of functions sensu Moreno, Mossio and Saborido, an account that this reviewer contributed to develop. In a nutshell, this account states that particular theoretical entities that we call constraints are functional when they are interdependent parts of an organized whole.

In providing this survey, the author adopts a critical stance and he "make[s] no attempt to conceal [his] own viewpoint or to pretend to neutrality (p.11)." In spite of his defense of pluralism, the author clearly favors the selective account of functions and the corresponding chapter is more than twice longer than the other chapters. This bias is not an issue per se, especially because it is clearly stated in the beginning of the book. Nevertheless, we find that the motivations, the backgrounds or the strengths of other accounts are not as developed as their limitations. Sadly, this restricts the interest of the book when considered as a survey. In spite of this weakness, this book should be very helpful as a reference map of the accounts defended in current literature and as an introduction to the field.

In this context, the author discusses pluralism where pluralism means that different accounts of function may be simultaneously acceptable. He criticizes "'betweendiscipline pluralism,' which seeks to restrict the applicability of the selected effect theory to some branches of evolutionary biology (p.109)." We share the idea that functions as selected effects are relevant to all fields of biology including, for example, physiology and development. Let us illustrate this point with examples that are not covered in the book. The notion of selective effect function is necessary to several biophysical models, which goes against the idea that these models would be only about proximal causes and would be independent or at least disjoint from ultimate causes. Let us mention two examples. The first example comes from the work of West, Brown and Enquist who developed models of lung and vascular physiology to understand allometric relationships of the metabolism. In these models, the authors assume that the function of lungs is to exchange oxygen meaning that oxygen ex-change rate have been optimized by natural selection. This assumption is required for the model to make predictions and reach its explanatory aim. To understand the second example, let us recall that two unrelated parameters, describing for instance the activation energy of different molecules, cannot be assumed to be equal without a reason. When developing a model of chromatin published in 2006, Lesne and Victor require such an equality for the system to perform its function. Since this equality cannot be considered as the result of randomness, it is justified on the ground of selection.

In spite of the relevance of selective effects functions in other fields than evolutionary biology, cross-disciplinary pluralism on accounts of functions raises questions that are not clearly solved. In particular, proving that a trait clearly comes from selection is a difficult task and leads to a heavy epistemological burden for fields which do not focus on this very question. In the examples above, selection is used to justify mathematical assumptions but the validity of this rational is not proven empirically. The author acknowledges this difficulty but does not really hint at solutions for practitioners.

The book uses a few running examples that are biologically sound and help understand the discourse without adding unnecessary complexities. However, the commitment of the author to the selective account of functions can sometimes be misleading in the treatment of these examples. For example, the author considers absurd the notion that the function of the nose is to support glasses. The article "the" makes sense in the selective account because this account focuses on the historical origin of a trait. But from other points of view, such as the organizational account of functions, a constraint may very well have many functions, and it is fair to say that a function of the nose is to support glasses for some humans. Moreover, such statements are at home, for example, in Lotka's view that a proper aspect of human evolution is a heavy trend towards exosomatisation, that is to say, the development of inorganic "organs" such as glasses.

We think that philosophical accounts of functions in biology depend strongly on their articulation to a theoretical framework. For example, the emphasis on selection stems from (Neo-) Darwinism. Similarly, the organizational account of function is based on a series of work from Kauffman, Varela, Rosen, among others, which aim to provide a theoretical account of the relationship between the part and the whole, a central notion to physiology. In this account, it is ultimately the circularity in the interdependence of constraints that grounds functionality. Then, the functionality holds with respect to this circularity: a part is functional for the larger entity defined by the circularity considered (Montévil & Mossio, 2015, Biological organisation as closure of constraints). We fear that the author missed this rationale when designing counter-examples. For example, he considered that obesity or panic attacks are part of causal loops that ultimately maintain themselves and as a result would be functional. This should be absurd because these situations are pathological. However, these circularities do not involve the "bulk" of the organism and, as such, it is not sound to call them functional for the organism in the organizational account. Moreover, it is not clear whether they are constraints sensu Montévil & Mossio.

Overall this book provides a good introduction to the debate on functions in analytic philosophy, and our criticism should be mitigated by the breadth of the literature considered and the short size of the book. We advise it mostly for a philosophical readership, as a map of the field. It is also possible for the philosophically-inclined biology teacher to use it as a reference when aiming to dispel the specter of final causes in her teaching, although this readership is clearly not the main target of this book.

Maël

  Montévil a,b a Laboratoire "Matière et Systèmes Complexes" (MSC), UMR 7057 CNRS, Université Paris 7 Diderot, Paris, France b Institut d'Histoire et de Philosophie des Sciences et des Techniques (IHPST) -UMR 8590, 13, rue du Four, 75006 Paris, France


domain: info math phys sdv shs.hisphilso

text:
Giuseppe Longo

Maël Montévil

M Dorato

L Mag- Nani

T Bertolotti

Springer

Comparing Symmetries in Models and Simulations *

Keywords: computer simulation, symmetries, randomness, theoretical framework, biology, equational modeling

Computer simulations brought remarkable novelties in knowledge construction. In this paper, we first distinguish between mathematical modeling, computer implementations of these models and purely computational approaches. In all three cases, different answers are provided to the questions the observer may have concerning the processes under investigation. These differences will be highlighted by looking at the different theoretical symmetries of each frame. In the latter case, the peculiarities of Agent Based or Object Oriented Languages allow to discuss the role of phase spaces in mathematical analyses of physical vs. biological dynamics. Symmetry breaking and randomness are finally correlated in the various contexts where they may be observed.

Introduction

Mathematical and computational modeling have become crucial in Natural Sciences, as well as in architecture, economics, humanities, ….

Sometimes the two modeling techniques, typically over continuous or discrete structures, are conflated into or, even, identified to natural processes, by considering nature either intrinsically continuous or discrete, according to the preferences of the modeler.

We analyze here the major differences that exist between continuous (mostly equational) computational (mostly discrete and algorithmic) modeling, often referred to as computer simulations. We claim that these different approaches propose different insights into the intended processes: they actually organize nature (or the object of study) in deeply different ways. This may be understood by an analysis of symmetries and symmetry breakings, which are often implicit but strongly enforced by the use of mathematical structures.

We organize the World by symmetries. They constitute a fundamental "principle of (conceptual) construction", in the sense of [START_REF] Bailly | Mathematics and the natural sciences; The Physical Singularity of Life[END_REF], from Greek geometry, to XXth century physics and mathematics. All axioms by Euclid may be understood as "maximizing the symmetries of the construction" (see [START_REF] Longo | Critique of Computational Reason in the Natural Sciences[END_REF]). Euclid's definitions and proofs proceed by rotations and translations, which are symmetries of space.

Symmetries govern the search for invariants and their preserving transformations that shaped mathematics from Descartes spaces to Grothendieck toposes and all XXth century Mathematics (see [START_REF] Zalamea | Synthetic Philosophy of Contemporary Mathematics[END_REF]. Theoretical physics has been constructed by sharing with mathematics the same principle of (conceptual) construction. Among them, symmetries, which describe invariance, and order, which is needed for optimality, play a key role from Galileo's inertia to the geodetic principle and to Noether's theorems (see [START_REF] Van Fraassen | Laws and symmetry[END_REF][START_REF] Kosmann-Schwarzbach | Les théorèmes de Noether: invariance et lois de conservation au XXe siècle[END_REF][START_REF] Longo | From physics to biology by extending criticality and symmetry breakings[END_REF]. The fundamental passage from Galileo's symmetry group, which describes the transformation from an inertial frame to another while preserving the theoretical invariants, to Lorentz-Poincaré group characterizes the move from classical to relativistic physics. The geodetic principle is an extremizing principle and a consequence of conservation principles, that is of symmetries in equations (Noether).

Well beyond mathematics and modern physics, the choice of symmetries as organizing principle is rooted in our search for invariants of action, in space and time, as moving and adaptive animals. We adjust to changing environment by trying to detect stabilities or by forcing them into the environment. Our bilateral symmetry is an example of this evolutionary adjustment between our biological structure and movement: its symmetry plane is given by the vertical axis of gravitation and the horizontal one of movement2 . In this perspective, the role we give to symmetries in mathematics and physics is grounded on pre-human relations to the physical world, well before becoming a fundamental component of our scientific knowledge construction.

By this, we claim that an analysis of the intended symmetries and their breaking, in theorizing and modeling, is an essential part of an investigation of their reasonable effectiveness and at the core of any comparative analysis.

Approximation?

Before getting into our main theme, let's first clarify an "obvious" issue that is not so obvious to many: the discrete is not an approximation of the continuum. They simply, or more deeply, provide different insights. Thus, in no way we will stress the superiority of one technique over the other. We will just try to understand continuous vs. discrete frames in terms of different symmetries.

It should be clear that, on one hand, we do not share the view of many, beautifully expressed by Réné Thom, on the intrinsically continuous nature of the World, where the discrete is just given by singularities in continua. On the other hand, many mythical descriptions of a Computational World or just of the perfection of computational modeling seem to ignore the limits of discrete approximation as well as some more basic facts, which are well-known, since always, in Numerical Analysis (the first teaching job, for a few years, of the first author). There is no way to approximate long enough a continuous non-linear dynamics by an algorithm on discrete data types when the mathematical description yields some sensitivity to initial/border conditions. Given any digital approximation, the discrete and the continuous trajectories quickly diverge by the combination of the round-off and the sensitivity. However, in some cases (some hyperbolic dynamics), the discrete trajectory may be indefinitely approximated by a continuous one, but not conversely. The result is proved by difficult "shadowing theorems", see [START_REF] Pilyugin | Shadowing in dynamical systems[END_REF]. Note that this is the opposite of the "discrete approximating the continuum", which is given for granted by many.

We are hinting here just to a comparison between mathematical techniques that provably differ, but which, a priori, says nothing about the actual physical processes that are not continuous nor discrete, as they are what they are. Yet, it is very easy to check an algorithmic description of a double pendulum against the actual physical device (on sell for 50 euros on the web): very soon the computational imitation has nothing to do with the actual dynamics. The point is that there is no way to have a physical double pendulum to iterate exactly on the "same" initial conditions (i.e. when started in the same interval of the best possible measurement), as this device is sensitive to minor fluctuations (thermal, for example), well below the unavoidable interval of measurement. By principle and in practice, instead, discrete data types allow exact iteration of the computational dynamics, on exactly the same initial data. Again, this is a difference in symmetries and their breaking.

In conclusion, on one side, a mathematical analysis of the equations allows to display sensitivity properties, from "mixing", a weak form of chaos, to high dependence on minor variations of the initial conditions (as well as topological transitivity, a property related to the density of orbits, etc). These are mathematical properties of deterministic chaos. We stress by this that deterministic chaos and its various degrees are a property of the mathematical model : by a reasonable abuse one may then say that the modeled physical process is chaotic, if one believes that the mathematical model is a good/faithful/correct representation of the intended process. But this is an abuse: the dice or a double pendulum know very well where they will go: along a unique physical geodetics, extremizing a Lagrangian action, according to Hamilton principle. If we are not able to predict it, it is our problem due to the non-linearity of the model, which "amplifies fluctuations", and due to our approximated measurements.

As it happens, the interval of measurement, the unavoidable approximated interface between us and the World, is better understood by continua than over discrete data types (we will go back to this) and, thus, physicists usually deal with equations within continuous frames.

On the other side, the power of discrete computations allows to …compute, even forever, and, by this, it gives fantastic images of deterministic chaos. As a matter of fact, this was mathematically described and perfectly understood by Poincaré in 1892, yet it came to the limelight only after Lorentz computational discovery of "strange attractors" (and Ruelle's work, [START_REF] Ruelle | On the nature of turbulence[END_REF]. As deterministic chaos is an asymptotic notion, there is no frame where one can better see chaotic dynamics, strange attractor or alike than on a computer. Yet, just push the restart button and the most chaotic dynamics will iterate exactly, as we observed and further argue below, far away from any actual physical possibility. And this is not a minor point: it is "correctness of programs" a major scientific issue in Computer Science. Of course, one can artificially break the symmetry, by asking a friend to change the 16th decimal in the initial conditions. Then, the chaotic dynamics will follow a very different trajectory on the screen, an interesting information, per se. However, our analysis here is centered on symmetry breaking intrinsic to a theory, that is on changes which have a physical meaning. This control, available in computer simulations, is thus an artifact from a physical perspective.

3 What do equations and computations do?

Equations

In physics, equations follow symmetries, either in equilibrium systems, where equations are mostly derived from conservation properties (thus from symmetries, see below), or in far from equilibrium sys-tems, where they describe flows, at least in the stationary cases -very little is known in non stationary cases. This is the physical meaning of most equational descriptions.

Then one "computes" from equations and, in principle, derives knowledge on physical processes, possibly by obtaining and discussing solutions -or the lack of solutions: a proof of non-analyticity, such as Poincaré's Three Body Theorem for example, may be very informative. But these derivations are not just formal: they are mostly based on proofs of relevant theorems. The job of mathematical deductions, in physics in particular, is to develop the consequences of "meaningful" writings. Mathematics is not a formal game of signs, but a construction grounded on meaning and handled both by formal "principles of proofs" and by semantically rich "principles of constructions" Bailly and Longo 2011. Typically, one reasons by symmetries, uses order, including wellordering, the genericity of the intended mathematical object or generalized forms of induction that logicians analyze by very large cardinals, an extension of the order of integer numbers obtained by alternating limits and successor operations [START_REF] Barwise | Handbook of Mathematical Logic[END_REF]. Once more, theoretical symmetries and meaning step in while proving theorems and solving/discussing equations; also the passage from Laplace's predictability of deterministic process, to Poincaré's proof of deterministic though unpredictable processes is a breaking of the observable symmetries (see below for more).

As a matter of fact, in order to solve equations, or discuss their solvability, we invented very original mathematical structures, from Galois' groups to differential geometry. The use of enriched construction principles, often based on or yielding new mathematical meaning, has been constantly stimulated by the analysis of equations. This is part of the common practice of mathematical reasoning. However, well beyond the extraordinary diagonal trick by Gödel, it is very hard to prove that "meaningful" procedures are unavoidable in actual proofs, that is to show that meaning is essential to proofs. An analysis of some recent "concrete" incompleteness result is in Longo 2011: meaning, as well-ordering, a geometric judgment, provably and inevitably steps in proofs even of combinatorial theorems (of Arithmetic!). Or, very large, infinite cardinals may be shown to be essential to proofs [START_REF] Friedman | Finite functions and the necessary use of large cardinals[END_REF]. In this precise sense, formal deductions as computations, with their finitistic principles of proof, are provably incomplete.

In particular, physico-mathematical deductions, used to discuss and solve equations, are not just formal computations, i.e. meaningless manipulations of signs. They transfer symmetries in equations to further symmetries, or prove symmetry changes or breaking (non-analyticity, typically). In Category Theory, equations are analyzed by drawing diagrams and inspecting their symmetries.

From Equations to Computations

The mathematical frame of modern computers was proposed within an analysis of formal deductions. Actually, Gödel, Kleene, Church, Turing …invented computable functions, in the 1930's, in order to disprove the largely believed completeness hypothesis of formal/axiomatic systems and their formally provable consistency3 . Turing, in particular, imagined the logical Computing Machine, imitating a man in the least action of sign manipulation according to formal instructions (write or erase 0 and 1, move left or right of one square in a "child's notebook"), and invented by this the modern split between software and hardware. He then wrote an equation that easily defines an incomputable arithmetic function. Turing's remarkable work for this negative result produced the modern notion of program and digital computer, a discrete state machine working on discrete data types. As we said, computing machinery, invented as an implementation of formal proofs, are provably incomplete even in arithmetic, let alone in proper extension of it, based on principles richer that arithmetic induction (well-ordering, symmetries, infinite ordinals …).

Thus, beyond the limits set by the impossibility of approximation mentioned above, there is also a conceptual gap between proving over equations and computing solutions by algorithms on discrete data. The first deals with the physical meaning of equations, their symmetries and their breaking, transfers this meaning to consequences, by human reasoning, grounded on "gestures" (such as drawing a diagram) and common understanding. It is based on the invention, if needed, of new mathematical structures, possibly infinitary ones, from Galois' groups to Hilbert Spaces to the modern fine analysis of infinitary proofs Rathjen 2006. These, in some cases such as for wellordering or the large infinite cardinals mentioned above, may even be proved to be unavoidable, well beyond computations and formalisms (see the reference above). Do algorithms transfer "physical meaning" along the computation? Do they preserve symmetries? Are those broken in the same way we understand they are in the natural process under scrutiny?

Our claim is that algorithmic approaches (with the notable exception of interactive automated formal calculus, within its limits) involve a modification of the theoretical symmetries used to describe and understand phenomena in physics, in particular by continua. This means that algorithmic approaches usually convey less or a different physical meaning than the original equational approaches. In other words, the modification of the equations needed for a completely finitary and discrete approach to the determination of a phenomenon leads to losses of meaningful aspects of the mathematization and to the introduction of arbitrary or new features.

As far as losses are concerned, the most preeminent ones probably stem from the departure from the continuum, an invention resulting from measurement, from Pythagoras' theorem to the role of intervals in physical measurement. As we already hinted, in the computing world, deterministic unpredictability does not make sense. A program determines and computes on exact data: when those are known, exactly (which is always possible), the program iterates exactly, thus allows a perfect prediction, as the program itself yields the prediction. The point is that deterministic unpredictability is due to the nonlinearity, typically, of the "determination" (the equations) and triggered by non-observable fluctuations or perturbation, below the (best) interval of measure-ment. Now, approximation is handled, in mathematics, by topologies of open intervals over continua, the so called "natural topology" over the real numbers.

At this regards, note that a key assumption, bridging mathematics of continua and classical physics, is that any sequence of measurements of increasing, arbitrary precision converge to a well defined state. This is mathematically a Cauchy condition of completeness, which implies that the rational numbers are not sufficient to understand the situation. Cantor's real numbers have been invented exactly to handle this kind of problems (among other reasons, such as the need to mathematize rigorously the phenomenal continuum in its broadest sense, the continuum of movement, say). Also, the fundamental relation between symmetries and conservation properties exhibited by Noether's theorems depend on the continuum (e.g. continuous time translations), so that these results can no longer be derived on a discretized background. In short, these theorems rely on the theoretical ability to transform states continuously along continuous symmetries in equations (of movement, for example) since the intended conserved quantity cannot change during such a transformation. With a discrete transformation the observed quantities can be altered (and it is the case usually in simulations) because there is no continuity to enforce their conservation.

Reciprocally, the changes due to the discretization introduce features that are arbitrary from a physical perspective. For example a basic discretization of time introduces an arbitrary fundamental time-scale. In Numerical Analysis, the methodology is to have the (differential) equations as the locus of objectivity and to design algorithms that can be shown to asymptotically converge (in a pertinent mathematical sense, and hopefully rapidly in practice) towards the mathematical solutions of the physically meaningful equations. In these frames, the theoretical meaning of the numerical (or algorithmic) approaches is entirely derivative: such numerical approaches are sound only with respect to, and inasmuch as there are mathematical results showing a proximity with the original equations and the trajectories determined by them. The mathematical results (convergence the-orems) define the nature of this proximity, and are usually limited to specific cases, so that entire research communities develop around the topic of the simulation of a specific family of equations (Navier-Stokes or alike for turbulence, Schrödinger in Quantum Physics, …). As a result, the methods to approach different (non-linear) equations by computing rely on specific discretizations and their close, often ad hoc, analysis.

Computations

As we said, we are just singling-out some methodological differences or gaps between different modeling techniques. On the "side of algorithms", the main issue we want to stress here is that equational approaches force uniform phase spaces. That is, the list of pertinent observables and parameters, including space and/or time, of course, must be given a priori. Since the work by Boltzmann and Poincaré, physicists usually consider the phase spaces made out of (position, momentum) or (energy,time) as sufficient for writing the equational determination. By generalizing the Philosopher's (Kant) remark on Newton's work, the (phase) space is the very "condition of possibility" for the mathematical intelligibility of physics. Or, to put it as H. Weyl, the main epistemological teaching of Relativity Theory is that physical knowledge begins when one fixes the reference system (that is to say, the way to describe the phase space) and the metrics on it. Then Einstein's Invariantentheorie allows to inspect the relevant invariants and transformations, on the grounds of Lorentz-Poincaré symmetry groups, typically, within a pregiven list of observables and parameters. Now, there exists a rich practice of computational modeling, which does not need to pass through equations, skips this a priori. Varenne nicely describes the dynamic mixture of different computational contexts as a "simulat", a neologism which recalls "agrégat" (an aggregate) Varenne 2012. This novelty has been introduced, in particular, by the peculiar features of Object Oriented Programming (OOP), but other "agent oriented systems" exist.

As a matter of fact, procedural languages require all values to share the same representation -this is how computer scientists call names for observables and parameters4 . "Objects" instead may interact even with completely different representations as long as their interfaces are compatible5 . Thus, objects behave autonomously and do not require knowledge of the private (encapsulated) details of those they are interacting with. As a consequence, only the interface is important for external reactions [START_REF] Cook | Object-oriented programming versus abstract data types[END_REF][START_REF] Bruce | Comparing object encodings[END_REF].

In biological modeling, aggregating different techniques, with no common a priori "phase space", is a major contribution to knowledge construction. Organisms, niches, ecosystems may be better understood by structuring them in different levels of organization, each with a proper structure of determination, that is phase space and description of the dynamics. For example, networks of cells are better described by tools from statistical physics, while morphogenesis, e.g. organ formation, are currently and mostly modeled by differential equations in continua. Each of these approaches requires pregiven phase spaces, which may radically differ (and the communities of researchers in the two fields hardly talk to each other). In a computer, by its high parallelism, one may mix these different techniques, with some more or less acceptable approximations, in spite of their differences. Even more so, ad hoc algorithms may describe specific interactions, independently of a unified equational description that may be impossible. Then "objects", in the sense above, may interact only on the grounds of the actual interface, both within a level of organization and between different levels, without reference to the proper or internal (to the object, to the level), causal structure.

In other words, OOP allows independent objects' dynamics, reminiscent of individual cell dynamics. Then, proliferation with variation and motility, the default state of life (see [START_REF] Longo | In search of principles for a Theory of Organisms[END_REF]) may be added to the models of morphogenesis that usually consider cells as inertial bullets, which they are not; that is, their proliferation, changes and motility are not entailed by physical forces that contribute to shape organs (in particular, when organs function for the exchange of energy and matter). By the computational power of modern computers, agent or object based programming styles (such as OOP) may implement autonomous agency for each cell, have them simultaneously interact within a morphogenetic field shaping the dynamics or a network ruled by statistical laws.

In summary, in computer simulation, one may "put together" all these techniques, design very complex "simulat" as aggregation of algorithms, including stochastic equations, probabilities distributions and alike. In particular, OOP allows the simulation of discrete dynamics of individual cells in an organism or of organisms in an ecosystem. And this with no need to write global first equations: one directly goes to algorithms in their changing environment.

However, let the process, or images on a computer, run …then push the restart button. Since access to discrete data is exact, as we said and keep stressing, the computer will iterate on the same initial conditions, exactly, with the same discrete algorithms. Thus, it will go exactly along the same computation and produce exactly the same trajectories, images and genesis of forms. This has no physical meaning as an unstable or chaotic system would never "iterate identically". It is even less biologically plausible, as biology is, at least, the "never identical iteration of a morphogenetic process" (see [START_REF] Longo | In search of principles for a Theory of Organisms[END_REF]. Observe now that exact iteration is a form of (time-shift/process-identity) symmetry; while non identical iteration is a symmetry breaking (see below for more on randomness vs. symmetry breaking). Noise, of course, may be introduced artificially, but this makes a deep conceptual difference, at the core of our analysis.

Note finally, that stochastic equations, probability values and their formal or algorithmic descriptions, are expressions and measurement of randomness, they do not implement randomness. And this is a key issue.

Randomness in Biology

Theoretical Physics proposes at least two forms of randomness: classical and quantum. They are separated by different probability theories and underying logic: entanglement modifies the probability correlations between quantum events Belavkin 2000. Even the outcome of the mesurement of generic states is contextual which means that this outcome depends on the other measurements performed and cannot be assumed to be predefined [START_REF] Abbott | Value-indefinite observables are almost everywhere[END_REF][START_REF] Cabello | Experimentally Testable State-Independent Quantum Contextuality[END_REF], and this situation departs from classical ones which are not contextual. A new form of randomness seems to be emerging from computer networks; or, at least, it is treated, so far, by yet different mathematics Longo, Palamidessi, and Paul 2010. In particular, some analysis of randomness are carried without using probabilities.

In the same way that we said that the world is neither intrinsically continuous or discrete, randomness is not in the world: it is in the interface between our theoretical descriptions and "reality" as accessed by measurement. Randomness is unpredictability with respect to the intended theory and measurement. Both classical and quantum randomness, though different, originate in measurement.

The classical one is present in dynamics sensitive to initial or border conditions: a fluctuation or perturbation below measurement, which cannot be exact by physical principles (it is an interval, as we said), is amplified by the dynamics, becomes measurable and …"we have a random phenomenon" Poincaré 1902. This amplification is mathematically described by the non-linearity of the intended equations or evolution function, with a subtle difference though. If a solution of the non-linear system exists, then the analysis of the Lyapounov exponents, possibly, yields some information on the speed of divergence of trajectories, initially indistinguishable by measurement: a non measurable fluctuation is amplified and produces an unpredictable and measurable event, yet the amplification is computable. In the case of nonexistence or non-analyticity of solutions of the given differential equations, one may have bifurcations or an unstable homoclinic trajectories (i.e. trajecto-ries at the intersection of stable and unstable manifolds). The choice at bifurcation or the physical trajectory is then highly unpredictable, thus random, and may be also physically ascribed to fluctuations or perturbations below measurement. However, in this case, one does not have, in general, a criterium of divergence, such as Lyapounov exponents. The fluctuation or perturbation "causes" the unpredictable event, thus Curie's principle is preserved: "a physical effect cannot have a dissymmetry absent from its efficient cause" -a symmetry conservation principle, or "symmetries cannot decrease". Yet, at the level of measured observables one witness a symmetry breaking, as the causing dissymmetry cannot be observed.

Quantum randomness is grounded on noncommutativity of the measurement of conjugated variables (position and momentum or energy and time), given by a lower bound -Planck's ℎ. It is represented by Schroedinger's equation that defines the of a probability amplitude (or law), in a very abstract mathematical space (a Hilbert space). As hinted above, measurement of entangled particles gives probabilities that are different from the classical contexts (Bell inequalities are not respected, see [START_REF] Aspect | Bell's inequality test: more ideal than ever[END_REF].

In quantum physics, though, there is another fundamental difference: in classical and relativistic mechanics, from Aristotle, to Galileo and Einstein, it is assumed that "every event has a cause". As mentioned above in reference to Curie's principle, the unpredictable, but measurable, classical event is "caused" by the (initial or border) undetectable fluctuation. Instead, in current interpretations of QM, random events may be a-causal -the spin up / spin down of an electron, say, is pure contingency, it does not need to have a cause. This radically changes the conceptual frame -and many still do not accept it and keep looking, in vain, for hidden variables (hidden causes), along the classical paradigm.

Surprisingly enough, a quantum event at the molecular level may have a phenotypic effect, in biology. This is the result of recent empirical evidence, summarized and discussed in Buiatti and Longo 2013. Thus, a phenotype, that is a structural property of an organism, possibly a new organism, may result from an a-causal event, happening at a completely different level of organization (molecular vs. organs or organisms). This micro event may be amplified by classical dynamics of molecules, including as their enthalpic oscillations and their Brownian motion. Brownian motion is omnipresent in cells' proteome, where macromolecules are very "sticky" and their chemical interactions are largely stochastic -though canalized by strong chemical affinities and cell compartmentalization. So, quantum and classical randomness may "superpose" in a highly constrained environment. Moreover, it is increasingly recognized that gene expression is mostly stochastic, see [START_REF] Elowitz | Stochastic Gene Expression in a Single Cell[END_REF][START_REF] Arjun | Stochastic gene expression and its consequences[END_REF] This leads to the fully general fact that: macromolecular interactions and dynamics are stochastic, they must be described in terms of probabilities and these probabilities depend on the context. This context includes the global proteomic composition, the torsion and pressure on the chromatin Lesne and Victor 2006, the cell activity in a tissue [START_REF] Bizzarri | Fractal analysis in a systems biology approach to cancer[END_REF][START_REF] Barnes | From Single Cells to Tissues: Interactions between the Matrix and Human Breast Cells in Real Time[END_REF], the hormonal cascades…up to the ecosystem, as containing fundamental constraints to biological dynamics. The up and down interactions between different levels of organization yield a proper form of biological randomness, a resonance between levels, called bio-resonance in Buiatti and Longo 2013. Bio-resonance destabilizes and stabilizes organisms; it both yields and follows from variability, as correlated variations contribute also to the changing structural stability of organisms. Note that variability produces adaptation and diversity, at the core of biological dynamical stability: an organism, a population, a species is "biologically stable", while changing and adapting, also because it is diverse. Both stability and diversity are also the result of randomness. "Also", because, as we said, randomness is highly canalized in biology, by cellular compartments of molecules, tissues tensegrity, organismal control (hormones, immune and neural systems …) and the ecosystem may downward influence these constraints (methylation and demethylation, which may regulate gene expression, can be induced by the environment), [START_REF] Gilbert | Ecological developmental biology: integrating epigenetics, medicine, and evolution[END_REF] Variability and diversity are constrained by history as well: phenotypes are the result of an evolutionary history that canalizes, but does not determine (at least in view of quantum events) further evolution. For example, as for historical "canalization" there are good reasons to believe that we, the vertebrates, we will never get out of the "valley" of tetrapodes -at most we may lose, and some of us have lost, podia and keep just traces of them.

In conclusion, randomness has a constitutive role in biology, as variability and diversity contribute to structural stability, beginning with gene expression. We developed above a comparative analysis in terms of symmetries of physical processes with respect to their equational and computational modeling. We now hinted to the different ways randomness is understood in various physical and biological frames. In biology, this later issue becomes particularly relevant, in view of the organizing role of randomness, including for small numbers (a population of a few thousands individuals is biologically more stable when diverse). Further on, we will propose a 'general 'thesis" relating randomness and symmetry breaking.

Symmetries and information, in

physics, in biology.

Turing, Discrete State Machines and Continuous Dynamics

We already stressed the key role of invariants and invariant preserving transformations in the construction of mathematical and physical knowledge. The sharing of construction principles in these two disciplines, first of all, symmetry principles and order principles, are the reason of the reasonable, though limited, effectiveness of mathematics for physics: these disciplines have been actually co-constituted on the grounds of these common construction principles, see Bailly and Longo 2011. However, since so few physical processes can be actually predicted -frictions and many-body interactions, i.e. nonlinearity, are everywhere -, the effectiveness of mathematics stays mostly in the reasonable intelli-gibility we have of a few phenomena, when we can organize them in terms of invariants and their transformations, thus of symmetries, well beyond predictability.

In the account above, changing fundamental symmetries produced the change from one theoretical frame to another, such as from classical to relativistic physics. Further useful examples may be given by thermodynamics and hydrodynamics. The irreversibility of time, a symmetry breaking, steps in the first by the proposal of a new observable, entropy; the second assumes incompressibility and fluidity in continua, two symmetries that are irreducible to the quantum mechanical ones, so far.

There is a common fashion in projecting the sciences of information onto biological and even physical processes. The DNA, the brain, even the Universe would be (possibly huge) programs or Turing Machines, sometimes set up in networks -note that the reference to networks is newer, it followed actual network computing by a many years delay.

We do not discuss here the Universe nor the brain. It may suffice to quote the inventor of computing by discrete state machines, Turing: " …given the initial state of the machine and the input signal it is always possible to predict all future states. This is reminiscent of Laplace's view that from the complete state of the universe at one moment of time, as described by the positions and velocities of all particles, it should be possible to predict all future states. The prediction which we are considering is, however, rather nearer to practicability than that considered by Laplace. The system of the 'universe as a whole' is such that quite small errors in the initial conditions can have an overwhelming effect at a later time. The displacement of a single electron by a billionth of a centimeter at one moment might make the difference between a man being killed by an avalanche a year later, or escaping. It is an essential property of the mechanical systems which we have called 'discrete state machines' that this phenomenon does not occur. Even when we consider the actual physical machines instead of the idealized machines, reasonably accurate knowledge of the state at one moment yields reasonably accurate knowledge any number of steps later" (A. M Turing 1950, p. 440)6 .

As for the brain, Turing continues: "The nervous system is certainly not a discrete-state machine. A small error in the information about the size of a nervous impulse impinging on a neuron, may make a large difference to the size of the outgoing impulse" (A. M Turing 1950, p. 451). As a matter of fact, the notions of spontaneous symmetry breaking, "catastrophic instability", random fluctuations…are at the core of Turing's analysis of continuous morphogenesis, A. M. Turing 1952, far remote from his own invention of the elaboration of information by the "Discrete State Machine" (DSM, his renaming in 1950 of his Logical Computing Machine of 1936).

It is worth stressing here the breadth and originality of Turing's work. He first invented the split hardware/software and the DSM, in Logic. Then, when moving to bio-physics, he invented a continuous model for morphogenesis, viewed just as physical matter (hardware) that undergoes continuous deformations, triggered by (continuous) symmetry breaking of an homogeneous field, in a chemical reactiondiffusion system. The model is given by non-linear equations: a linear solution is proposed, the nonlinear case is discussed at length.

A key property of Turing's continuous model is that it is "a falsification" (his words on page 37) of the need for a (coded) "design". This clearly appears from the further comments on the role of genes, mentioned below. In discussions reported by [START_REF] Hodges | Turing: a natural philosopher[END_REF], Turing turns out to be against Huxley's "new synthesis", which focused on chromosomes as fully determining ontogenesis and phylogenesis [START_REF] Huxley | Evolution. The Modern Synthesis[END_REF]. He never refers to the already very famous 1944 booklet by [START_REF] Schrödinger | What Is Life? Cambridge U[END_REF], where Schrödinger proposes to understand the chromosomes as loci of a coding, thus as a Laplacian determination of embryogenesis, as he says explicitly ("once their structure will be fully decoded, we will be in the position of Laplace's daemon" says Schrödinger in chapter 2, The hereditary code-script). As a matter of fact, in his 1952 paper, Turing quotes only Child, D'arcy Thompson and Waddington as biologists, all working on dynamics of forms, at most constrained (Waddington), but not determined nor "predesigned" by chromosomes. Indeed, Turing discusses the role of genes, in chromosomes, which differ from his "morphogenes" as generators of forms by a chemical action/reaction system. He sees the function of chromosomal genes as purely catalytic and, says Turing, "genes may be said to influence the anatomical form of the organism by determining the rates of those reactions that they catalyze …if a comparison of organisms is not in question, the genes themselves may be eliminated from the discussion", page 38 (a remarkable proposal, in the very fuzzy, ever changing notion of "gene", see Fox [START_REF] Keller | The century of the gene[END_REF]. No (predefined) design, no coded or programmed Aristotelian homunculus in the chromosomes (the myth of the chromosomes as a program), for Turing, the man who invented coding and programming. This is science: an explicit proposal of a (possibly new) perspective on nature, not the transfer of familiar tools (the ones he invented, in this case!) on top of a different phenomenology.

Note finally that, when comparing his DSM to a woman's brain in A. M Turing 1950, Turing describes an "imitation game", while he talks of a "model" as for morphogenesis. This beautiful distinction, computational imitation vs. continuous model, is closely analyzed in Longo 2009.

Classifying information

Let's further analyze the extensive use of "information" in biology, molecular biology in particular. Information branches in at least two theories:

• elaboration of information (Turing, Church, Kleene and many others, later consistently extended to algorithmic information theory: Martin-Loef, Chaitin, Calude, see Calude 2002 and

• transmission of information (Shannon, Brillouin, see SHANNON 1948).

In [START_REF] Longo | Is information a proper observable for biological organization?[END_REF], we stressed the key differences between these two theories that are confusedly identified in molecular biology, with unintelligible consequences in the description of the relationship of information to entropy and complexity …two relevant notions in biology 7 .

As scientific constructions, both information theories are grounded on fundamental invariants. And this is so since at least Morse practical invention, with no theory, of information transmission. Information is independent of the specific coding and the material support. We can transmit and encode information as "bip-bip", by short and long hits, as flashes, shouts, smoke clouds …by bumping on wood, metal, by electricity in cables or whatever and this in a binary, ternary, or other code …. Information is the invariant with respect to the transformation of these coding and material supports: this is its fundamental symmetry. Up to Turing's fundamental invention: distinguish software from hardware. So, a rich Theory of Programming was born, largely based on Logic, Typed and Typed-free languages, term rewriting systems etc, entirely independent of the specific encoding, implementation and hardware. The computer's soul is so detached from its physical realization that Descartes dualism is a pale predecessor of this radical and most fruitful split. And when the hardware of your computer is dying, you may transfer the entire software, including the operating system, compilers and interpreters, to another computer. This symmetry by transfer is called "metempsychosis", we think. Now, it does not apply in biology, nowhere.

The DNA is not a code, carrying information. There is no way to detach a soft content from it and transfer it to another material structure: it cannot be replaced by metal bullets, or bumps on a piece of wood. What gets transferred to RNA and then proteins is a chemical and physical structure, a most relevant one, as the DNA is an extraordinary chemical trace of an history. And it transmits to other chemicals an entirely contingent physico-chemical confor-7 See Smith 1999), where Turing-Kolmogorov's elaboration theory is quoted as well as Shannon's theory. The author considers the second as more pertinent for biology. Then a notion of complexity as amount of information is given that is actually based on the first theory and it is described as co-variant to entropy. Finally, in the paper, Shannon's theory pops out again -the more pertinent theory, according to the author, where complexity is contravariant to entropy, it is negentropy. mation. If a stone bumps against other stones in a river and de-forms them (in-forms them, would say Aristotle), there is no meaning to speak of a transmission of information, in the scientific invariant sense above, unless in reference to the Aristotelian sense. No informational invariant can be extracted, but the ones proper to the physico-chemical processes relative to stone bumping. Life is radically contingent and material: no software/hardware split. The pre-scientific reference to information, sometimes called "metaphorical", has had a major misleading role. First, it did not help to find the right invariants. The physico-chemical structure of cellular receptors, for example, has some sort of generality, which yields some stereospecificity Kuiper et al. 1997. Yet, this is still strictly related to a common chemistry that has nothing to do with an impossible abstract information theoretic description. The proposal of a too abstract and matter independent invariant did not help to find the right scientific level of invariance. Or, more severely so, it forced exact stereospecifity of macromolecular interaction, as a consequence of the information theoretic bias.

Monod, one of the main theoreticians of molecular biology, claims that the molecular processes are based on the "oriented transmission of information …(in the sense of Brillouin)". In [START_REF] Monod | Le hasard et la nécessité[END_REF], he derives from this that the "necessarily stereospecific molecular interactions explain the structure of the code …a Boolean algebra, like in computers" and that "genes define completely the tridimensional folding of proteins, the epigenetic environment only excludes the other possible foldings". Indeed, bio-molecular activities "are a Cartesian Mechanism, autonomous, exact, independent from external influences". Thus, the analysis based on the search for how information could be transmitted, forced an understanding inspired by the Cartesian exactness proper to computers as well as the Laplacian causal structure, Turing would say, proper to information theories. It induced the invention of exact stereospecificity, which is "necessary" to "explain" the Boolean coding! That is, stereospecificity was logically, not empirically, derived, while, since 1957 Novick and Weiner 1957, robust evidence had already shown the stochasticity of gene expression (see [START_REF] Kupiec | A probabilistic theory of cell differentiation, embryonic mortality and DNA C-value paradox[END_REF][START_REF] Kupiec | Ni Dieu ni gène: pour une autre théorie de l'hérédité[END_REF][START_REF] Arjun | Stochastic gene expression and its consequences[END_REF][START_REF] Arjun | Stochastic gene expression and its consequences[END_REF][START_REF] Heams | Randomness in Biology[END_REF] for a recent synthesis).

We now know that the protein folding is not determined by the coding (yet, Monod did consider this possibility). Macromolecular interactions, including gene expression, are largely random: they must at least be given in probabilities, as we said, and these probabilities would then depend on the context. No hardware independent Boolean algebra governs the chemical cascades from DNA to RNA to proteins, also because these cascades depend, as we already recalled, on the pressure and tensions on the chromatin, the proteome activities, the intracellular spatial organization, the cellular environment and many other forms of organismal regulations, see for example [START_REF] Weiss | Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells[END_REF][START_REF] Lesne | Chromatin fiber functional organization: Some plausible models[END_REF] In summary, the informational bias introduced a reasoning based on Laplacian symmetries, far away from the largely turbulent structure of the proteome, empowered also by chaotic enthalpic oscillations of macromolecules. This bias was far from neutral in guiding experiments, research projects and conceptual frames. For example, it passed by the role of endocrine disruptors of the more than 80,000 molecules we synthesized and used in the XXth century, an increasingly evident cause of major pathologies, including cancer, [START_REF] Zoeller | Endocrine-Disrupting Chemicals and Public Health Protection: A Statement of Principles from The Endocrine Society[END_REF][START_REF] Soto | Environmental causes of cancer: endocrine disruptors as carcinogens[END_REF][START_REF] Demeneix | Losing Our Minds: How Environmental Pollution Impairs Human Intelligence and Mental Health[END_REF]. These molecules were not supposed to interfere with the exact molecular cascades of key-lock correspondences, a form of stereospecificity. The bias guided the work on GMO, which have been conceived on the grounds of the "central dogma of molecular biology" and of Monod's approach above: genetic modifications would completely guide phenotypic changes and their ecosystemic interactions (see [START_REF] Buiatti | Functional dynamics of living systems and genetic engineering[END_REF].

One final point. Information theories are "code independent", or analyze code in order to develop general results and transmission stability as code insensitive (of course cryptography goes otherwise: but secrecy and code breaking are different purposes, not exactly relevant for organisms). Information on discrete data is also "dimension independent": by a polynomial translation one may encode discrete spaces of any finite dimension into one dimension. This is cru-cial to computing, since it is needed to define Turing's Universal Machine, thus operating systems and compilers.

Biology instead is embedded in a physical world where the space dimension is crucial. In physics, heat propagation and many other phenomena, typically field theories, strictly depend on space dimension. By "mean field theories" one can show that life, as we know it, is only possible in three dimensions (see [START_REF] Bailly | Mathematics and the natural sciences; The Physical Singularity of Life[END_REF]. Organisms are highly geometric in the sense that "geometric" implies sensitivity to coding and dimensions. In this sense, continuous models more consistently propose some intelligibility: in "natural" topologies over continua, that is when the topology derives from the interval of physical measurement, dimension is a topological invariant, a fundamental invariant in physics, to be preserved in biology, unless the reader believes that he/she can live encoded in one dimension, just exchanging information, like on the tape of a Turing Machine. A rather flat Universe …yet, with no loss of information. But where one has only information, not life.

Missing the right level of invariance and, thus, the explanatory symmetries, is a major scientific mistake. Sometimes, it may seem just a "matter of language", as if language mattered little, or of informal metaphors, as if metaphors were not carrying meaning, forcing insight and guiding experiments. They actually transfer the conceptual structure or the intended symmetries of the theory they originate from, in an implicit, thus more dangerous and un-scientific way. Just focusing on language, consider the terminology used when referring to DNA/RNA as the "universal code for life", since all forms of life are based on it. This synchronic perspective on life -all organisms yield these molecules and the basic chemical structure of their interactions, thus there is a universal code -misses the historical contingency of life. There is no universality in the informational sense of an invariant code with respect to an independent hardware. Life is the historical result of contingent events, the formation somewhere and somehow of DNA or RNA or both, sufficiently isolated in a membrane, which occurred over that hardware only. Then, the resulting cell reproduced with varia-tion and diversified, up to today's evolutionary diversity. One contingent material origin, then diversification of that matter, of that specific hardware and no other. Invariance, symmetries and their breaking are different from those proper to "information", in this strictly material, evolutionary perspective.

Theoretical symmetries and randomness

In this section, we would like to elaborate on a "thesis", already hinted in Longo and Montévil 2014. In physical theories, where the specific trajectory of an object is determined by its theoretical symmetries, we propose that randomness appears when there is a change in some of these symmetries along a trajectory and reciprocally that changes of symmetries are associated to randomness.

Intuitively theoretical symmetries enable to understand a wide set of phenomenal situations as equivalent. In the end of the day, the trajectory that a physical object will follow, according to a theory, is the only trajectory which is compatible with the theoretical symmetries of a given system. Symmetries, in this context, enable to understand conservation properties, the uniqueness of the entailed trajectory and ultimately the associated prediction, if any. Now, what happens when, over time or with respect to a pertinent parameter, a symmetry of the system is broken? A symmetry corresponds to a situation where the state or the set of possible states and the determination of a system does not change according to specific transformations (the symmetries). After the symmetry breaking, the state(s) becomes no longer invariant by these transformations; typically, the trajectory goes to one of the formerly symmetric states and not to the others (a ball on top of a mathematical hill falls along one of the equivalent sides). Since the initial situation is exactly symmetric (by hypothesis), all the different "symmetric" states are equivalent and there is no way to single out any of them. Then, in view of the symmetry breaking, the physical phenomena will nevertheless single out one of them. As a result we are confronted with a nonentailed change: it is a random change.

This explanation provides a physico-mathematical meaning to the philosophical notion of contingency as non-necessity: this description of randomness as symmetry breaking captures contingency as a lack of entailment or of necessity in an intended theory. Note that usually the equivalent states may not be completely symmetric as they may be associated to different probabilities, nevertheless they have the same status as "possible" states.

For now, we discussed the situation at the level of the theoretical determination alone, but the same reasoning applies mutadis mutandis to prediction. Indeed, we access to a phenomenon by measurement, but measurement may be associated to different possible states, not distinguishable individually. These states thus are symmetric with respect to the measurement, but the determination may be such that these (non-measurably different) states lead to completely different measurable consequences. This reasoning is completely valid only when the situation is such for all allowed measurements, so that randomness cannot be associated to the possible crudeness of an arbitrary specific measurement.

Reciprocally, when we consider a random event, it means that we are confronted with a change that cannot be entailed from a previous observation (and the associated determination). When the possible observations can be determined (known phase space), this means that the different possibilities have a symmetric status before the random event (precisely because they are all pre-defined possibilities) but that one (or several) of them are singled out by the random event in the sense that it becomes the actual state. We recognize in this statement the description of a symmetry that is broken during the random event.

Let us now review the main physical cases of randomness.

• Spontaneous symmetry breaking in quantum field theories and theories of phase transitions (from a macroscopic viewpoint) are the most straightforward examples of the conjecture we describe. In these cases, the theoretical deter-mination (Hamiltonian) is symmetric and the change of a parameter leads the systems equilibrium to shift from a symmetric state to an asymmetric one (for example isotropy of a liquid shifting to a crystal with a specific orientation). Randomness stems then just from the "choice" of a specific orientation, triggered by fluctuations in statistical mechanics.

• Classical mechanics can, in spite of its deterministic nature, lead to unpredictability as a consequence of the symmetrizing effect of measurement on one side (there are always different states which are not distinguished by a measurement), and a determination that leads those states to diverge (which breaks the above symmetry). This reasoning applies to chaotic dynamics but also to phase transitions where, from a strictly classical viewpoint, fluctuations below the observation determine the orientation of the symmetry changes.

• In classical probabilities, applied to "naive" cases such as throwing a dice or to more sophisticated framework such as statistical mechanics, our reasoning also applies. When forgetting about the underlying classical mechanics, the probabilistic framework is a strict equivalence between different possibilities, except for their expected frequencies which may differ: those are given by the associated probabilities. In order to define theoretically these probabilities, some underlying theoretical symmetries are required.

In our examples, the symmetries are the symmetry between the sides of a dice and for statistical mechanics, the symmetry between states with the same energy for the microcanonical ensemble. From a strictly classical viewpoint, these symmetries are assumed to be established on average by the properties of the considered dynamics. In the case of dice, it is the rotation, associated to the dependence on many parameters which leads to a sufficient mixing, generating the symmetry between the different sides of the dice. In the case of statistical mechanics, it is the property of topological mixing of chaotic dynamics (a property met by these systems by definition). This property is assumed in order to justify the validity of statistical mechanics from the point of view of classical mechanics. In both cases, a specific state or outcome corresponds to a breaking of the relevant symmetry.

• In quantum mechanics, the usual determination of the trajectory of a state is deterministic, randomness pops out during measurement.

The operator corresponding to the measurement performed establishes a symmetry between its different eigen vectors, which also correspond to the different outcomes corresponding to the eigen values. This symmetry is partially broken by the state of the system, which provides different weights (probabilities) to these possibilities.

The measurement singles out one of the eigen vectors which becomes the state of the system and this breaks the former symmetry.

We can conclude from this analysis and these examples that randomness and symmetry breaking are tightly associated. We can put this relationship into one sentence:

A symmetry breaking means that equivalent "directions" become no longer equivalent and precisely because the different directions were initially equivalent (symmetric) the outcome cannot be predicted.

As discussed elsewhere [START_REF] Longo | Critique of Computational Reason in the Natural Sciences[END_REF][START_REF] Longo | From physics to biology by extending criticality and symmetry breakings[END_REF], we assume that theoretical symmetries in biology are unstable. It follows that randomness, understood as associated to symmetry breaking, should be expected to be ubiquitous; however, this approach leads also to propose a further form of randomness. In order to show that randomness can be seen as a symmetry breaking, we needed to assume that the set of possibilities was determined before the event.

In biology, the instability of the theoretical symmetries does not allow such an assumption in general. On the opposite, a new form of randomness appears through the changes of phase spaces, and this randomness does not take the form of a symmetry breaking stricto sensu inasmuch as it does not operate on a pre-defined set. In other words, these changes cannot be entailed but they cannot even be understood as the singling out of one possibility among others -the list of possibilities (the phase space) is not pre-given.

In brief, theoretical symmetries in physics enable to single-out a specific trajectory in a phase space, formed by a combination of observables. Thus, a symmetry breaking corresponds to the need of one or several supplementary quantities to further specify a system on the basis of already defined quantities (which were formerly symmetric and thus not useful to specify the situation). In biology, instead, the dynamic introduces new observable quantities which get integrated to the determination of the object as the latter is associated to the intended quantities and symmetries. This dynamics of the very phase space may be analyzed a posteriori as a symmetry breaking. Thus, randomness moves from within a phase space to the very construction of a phase space, a major mathematical challenge.

The Burgess fauna, some 520 millions years ago Gould 1989, seems to present many cases of "asymmetric" beasts among these early multicellular organisms, later negatively selected.

It is not by chance that an immense mathematical physicist, H. Weyl, was one of the few who claimed that the formalist /computational project was trivializing mathematics and conjectured incompleteness, already in 1918,[START_REF] Weyl | Das kontinuum[END_REF], see also[START_REF] Bailly | Mathematics and the natural sciences; The Physical Singularity of Life[END_REF]

Technically, an existential quantifier is opened at the beginning of the program and then everyone shares all private informa-

tion.5 The existentials are opened only at the point of performing the operation

In popular references to unstable or chaotic dynamics, instead of quoting the famous "Lorentz's butterfly effect", proposed in 1972 on the grounds of Lorentz' work of 1961, one should better refer the "Turing's electron effect", published in 1952.

Acknowledgments

We would like to thank Kim Bruce for reminding and updating us on the Foundations of Object Oriented Languages, an area which he contributed to trigger, in part by joint work with GL, and by starting the FOOL series conferences, twenty years ago.

1 G. Longo's work was supported in part by Marie Curie FP7-PEOPLE-2010-IRSES Grant RANPHYS. M. Montévil's Work is supported by région Île-de-France, DIM ISC


Maël Montévil
email: mael.montevil@gmail.com

Giuseppe Longo
email: giuseppe.longo@ens.fr

David Ceccarelli

Giulia Frezza

Big Data and Biological Knowledge 12

Keywords:

Some authors assert that the analysis of huge databases could replace the scientific method. On the contrary, we argue that the best way to make these new technologies bear fruits is to frame them with theories concerning the phenomena of interest. Such theories hint to the observable that should be taken into account and the mathematical structures that may link them. In biology, we argue that the community urgently needs an overarching theory of organisms that would provide a precise framework to understand lifecycles. Among other benefits, such a theory should make explicit what we can and cannot predict in principle.

Introduction

Biology is a domain where variation has a fundamental theoretical role. Biological variation is profound and qualitative, and we have defended elsewhere the idea that variation justifies that biology requires its own epistemology. Notably, this variation is the basis of the historicity and the contextual nature of living things (G. LONGO AND M. MONTÉVIL 2014; M. MONTÉVIL et al. 2016) and it is at the core of the adaptivity and diversity of life. Variation is, in part, due to random phenomena at diferent levels of organization, and to the many forms of interaction between these levels (bio-resonance, see M. [START_REF] Buiatti | Randomness and Multi-level Interactions in Biology[END_REF]), yet it is always canalized by constraints and contexts and may be induced by the context to an extent (M-J. [START_REF]Deevelopmental plasticity and evolution[END_REF]MONTÉVIL et al. 2016;[START_REF] Longo G | How Future Depends on Past Histories and Rare Events in Systems of Life[END_REF]. In our perspective, variation is thus an integral component, but not the only component, of diversity and adaptation, both in phylogenesis and ontogenesis, up to having a crucial role in the etiology of cancer (A. SOTO, G. LONGO and D. NOBLE, 2016). It finally leads to a peculiar form of unpredictability, proper to biological dynamics, since variation is largely based on random phenomena, at all levels of organization (M. [START_REF] Buiatti | Randomness and Multi-level Interactions in Biology[END_REF]. As for this issue, note that randomness is not an absolute notion, but it means "unpredictability w.r. to the intended theory" (C. CALUDE and G. LONGO, 2016b). And biological randomness deserves its proper treatment as related to the changing phase space (the pertinent observables and parameters or the space of all possible dynamics) and to the role of rare events, in particular along evolution (MONTÉVIL et al. 2016;G. LONGO, 2016).

Biologists are thus confronted with the evolutionary diversity and adaptivity of the living. Moreover, organisms possess an internal heterogeneity which corresponds to their diferent organs (and organites, in the case of cells): "correlated variations" in the terms used by Darwin, depends both on the internal coherence of each organisms and on the changing eco-systemic conditions.. Faced with these two dimension of biological complexity, the human mind sometimes seems disarmed. In this context, the contemporary possibility of developing immense digital databases in collaborative frameworks is regarded as a major opportunity. But this opportunity is not without peril -and analyses lacking biological meaning is not the least of these perils.

All fields of biological sciences are not equally equipped to use these growing databases. Some fields build on robust theoretical thinking. For example, phylogenetic analyses rely on the conceptual framework of the theory of evolution, extensively enriched in the XXth century. Tis theory frames the production of knowledge on the basis of data by relying on nontrivial theoretical structures, in particular Darwin's principles ("descent with modification" and "selection"). By contrast, there is no wellestablished, unified theory to understand organisms, their physiology and their development, in spite of recent advances (see A. MINELLI and T. PRADEU, 2014;A. SOTO, G. LONGO, and D. NOBLE, 2016). Despite decades of informal use, the traditional notion of a genetic program has never acquired a real theoretical status, for a lack of both biological pertinence and of referenrence to a rigorous actual scientific notions (PA. MIQUEL; G. LONGO, 2012). Tis lasting tradition leads to a causal priority assigned to the molecular level, a priority that is embodied in the nature of the data obtained by high throughput techniques. By contrast, many relevant quantities are neglected by the use of Big Data in biology. For example, the modeling of an organ like the heart requires taking simultaneously into account several levels of organization [START_REF] Noble | The Music of Life: Biology beyond the Genome[END_REF]. Similarly, many physicists and biologists emphasize the importance of physical quantities in the determination of biological phenomena. Here physical quantities refer informally to the forces and fields of classical mechanics. For example, the stifness of a tissue or the forces exerted by cells are fundamental determinant of a tissue. However, these quantities are not associated with high throughput experimental methods. For example, the interplay of forces in a morphogenetic dynamics is neither measured in genomics, nor proteomics or metabolomics. As a result, we can see that the choice of a theoretical framework impacts directly the quantities that should be measured and analyzed.

Beyond the choice of the quantities relevant to understand a given phenomenon, theoretical frameworks also matter for the analysis of data. Statistical analyses are based on mathematical hypotheses that, in general, correspond to theoretical hypotheses, albeit the latter are sometimes informal or even implicit. Te capacity of databases to contribute to the comprehension of phenomena depends on the theoretical view that frames the use of these data and confers meaning to them, as well as on the pertinence of these data in relation to a theoretical frame. In short, there is always a choice, sometimes considered to be "obvious" if not unique, of observables to be measured, of a metric, of criteria of numerical approximation: this choice needs to be made and explicitly so.

Te application of Big Data to cancer, for example, is developed in a particular theoretical frame, the somatic mutation theory, where the process of carcinogenesis is conceived as the appearance of cancerous cells by the accumulation of somatic, genetic mutations: "Te story of cancer is a story of how the body's complex coding systems go awry through the creation of self-perpetuating errors in cellular replication and growth" (A.R. [START_REF] Shaikh | Cuollaborative Biomedicine in the Age of Big Deata: The Cuase of Cuancer[END_REF]. However, this theoretical point of view encounters major conceptual and empirical difculties. Tese difculties manifest themselves in translational researches and explain the limited medical outcomes of cancer biology despite significant investments. For example, changes in the proportion of deaths due to cancer are not large except in cases which can be interpreted in terms of prevention (R.L. SIEGEL, K.D MILLER and A. JEMAL 2015). One of the most infuential advocates of the somatic mutation theory of carcinogenesis acknowledges the difculties of this genocentric approach and stresses that we are once again faced with the "endless complexity" of these phenomena (R.A. WEINBERG 2014).

Several scholars analyze the situation as the manifestation of a theoretical problem and propose alternative viewpoints about the nature of carcinogenesis (C. [START_REF] Sonnenschein | The Society of Cuells: Cuancer and Cuontrol of Cuell Proliferation[END_REF]2011;[START_REF] Baker | TOFT better explains experimental results in cancer research than SMT[END_REF]. Tese theoretical viewpoints also come with diferent research strategies, consider diferent levels of organization and relevant quantities (BERTOLASO M. 2016). However, most of the community stick to the somatic mutation theory. From their perspective, it is then appealing to consider Big Data analysis as a solution permitting the treatment of cancer while keeping the focus on molecular and more specifically genomic data. Tis technological solution is called personalized medicine or precision medicine. Precision oncology is advocated by groups such as the Personalized Medicine Coalition and is supported by the US government through the Precision Medecine Initiative program.

More generally, the absence of a theoretical framework for organisms makes particularly seductive a certain rhetoric that goes beyond -if not against -the rational use of data. The omnipotence and autonomy of database analysis is at the center of a contemporary myth. For a decade, several successful articles, including one by Chris [START_REF] Anderson | The end of theory: The data deluge maees the scientiic method obsolete[END_REF], maintain that the figures speak for themselves: "We can throw the numbers into the biggest computing clusters the world has ever seen and let statistical algorithms find patterns where science cannot ... Correlation supersedes causation, and science can advance even without coherent models, unified theories … No semantic or causal analysis is required." Te idea is that "data miners" are capable of detecting correlations and orienting decisions without having to perform any theoretical discussion. So it is no longer a matter of enriching the "obsolete" scientific method but instead of replacing it, in particular by bypassing theoretical thinking. Tis point of view is associated with the slogan that the larger the database, the easier it is to find relations on the basis of which to act.

Immense databases, prediction, and chance

Te rhetoric that defend the replacement of the scientific method by the analysis of big databases can be assessed by the use of Mathematics. Teorems enable us to demonstrate the limits of these purely algorithmic methods by showing the impossibility of replacing the scientific quest for meaning by pure "data mining". Teorems at the crossroads of ergodic theory and Ramsey's Teory, a combinatory theory of numbers born in the 1920s and well-developed since then, permit to contradict this use of Big Data (C. CALUDE and G. LONGO 2016; H. HOSNI and A. VULPIANI 2017).

The deluge of spurious correlations

Let us first consider "Ramsey-type" theorems, used in [START_REF] Calude | The Deeluge of Spurious Cuorrelations in Big Deata[END_REF]. Tese theorems show that for any correlation between numbers in a database, there exists a number (let us say m) such that any database having at least m elements contains the demanded correlation. Terefore, it is just a matter of size, and it is possible to compute a threshold beyond which all databases (sequence of numbers) will contain a regularity with the stipulated characteristics. In other words, be as precise as you wish about the criterion for correlating pairs, triplets, etc., as well as the minimal number of times that you want to observe them, in what space and over what duration and the manner in which you will divide up your database (for example, by correlating proximate values, even iterated … according to the preferred criterion). Ten the theorems mentioned will tell you how many data to gather in order to achieve those criteria, that is to find some correlation realizing them. More precisely, a regularity in an ensemble of numbers may be established by fixing three parameters, or even more ("arity" of the relation, cardinality of the threshold of interest -how many you wish to have, and the partition of the database...). On the basis of these parameters, we can then calculate a number m, such that any ensemble of numbers A that contains at least m elements will satisfy the required regularity.

We should observe that A is any ensemble and that the only requirement is that A must be "sufciently large", enormous in fact, since m is growing very rapidly as a function of the given parameters. But being arbitrary, A may be engendered by … dice throwing, measurements of an electron's spin-up/spin-down, a random quantum phenomenon, or random phenomena of any kind (physical, biological, social)… Te bigger the database the better, the credulous propagandists of Big Data tell us. Is this number m too big to be encountered in our Universe for a correlation between a sufcient number of elements? Ten not all sets of numbers of a cardinality below the Ramsey threshold need to contain the pre-given regularity, yet … lots of them will.

In summary, these results tell us that any A that is sufciently big contains arbitrary, thus potentially "spurious" correlations; moreover, if we ask merely that these correlations appear in a high percentage, but lower than 100% of the ensembles, that is "only" in a reasonably high percentage of ensembles, then we would obtain an m attainable by our databases. In short, this hazard in the huge quantities of numbers is by no means rare. Let us explain.

A finite ensemble of numbers may be considered (algorithmically) "random" when it cannot be engendered by a program smaller than its number of elements. Tis is a notion of "incompressibility" for sequences of numbers, that may be extended to matrices or other organizations of data in finite dimension. It does not correspond exactly to randomness, yet it is a good "symptom" of randomness: that is an incompressible sequence has a high chance to be random; moreover, asymptotically (for sequences tending to infinite length), it does yield a robust notion of randomness for infinite sequences (C. CALUDE and G. LONGO 2016b). Now, the percentage of ensembles of random numbers in this weak sense (incompressibility) tends toward 100 % (measure 1, to be more precise) when their cardinality grows toward infinity. Infinity is big, even for "data miners" who are the richest in data, yet as soon as we are dealing with ensembles of numbers that are expressed with 2000 bits, for example -which is not out of reachwe approach 80 % of incompressible ensembles [START_REF] Calude | The Deeluge of Spurious Cuorrelations in Big Deata[END_REF]. So good luck making any kind of use in terms of prediction or action of data that may derive from chance! In every case where chance dominates, it is out of the question for the regularities found by clever dataexploration programs to be of any help at predicting if not acting, precisely because they are the fruit of chance, and they, therefore, may not be reproduced in time and in space, or derived from any causal relation. Tus, it is due to chance that one finds spurious correlations as illustrated in the eponymous book by T. VIGEN (2015, see also the associated website http:// www.tylervigen.com/spurious-correlations). Picturesque examples include the correlation between the US spending on science, space and technology which correlates with the suicides by hanging, strangulation and sufocation (r=00.99, from 1999 to 2009) or the number of Japanese passenger cars sold in the US which correlates with the suicides by crashing of motor vehicle (r=00.93, from 1999 to 2009). We leave the causal relevance of these correlations to the reader's appreciation. In (C. Calude and G. Longo 2016), we gave the mathematical arguments that justify these spurious correlations and their high chances to appear.

Data, prediction and dynamical systems

Te analysis of prediction is a central question in meteorology. [START_REF] Hosni | Forecasting in light of big data[END_REF] present an introductory survey of the problems encountered in this scientific area written by two insiders. Te first point is that too many data may kill information and forecasting. Te issue was understood by von Neumann and Charney since the 1950s. For example, it follows from the nature of hydrodynamic (and thermodynamics) equations that knowledge and description of data concerning waves of too high or too low frequencies may distort the analysis. So data, possibly implicit in the data bases, concerning nonpertinent phenomena, may incorrectly afect the forecast. Moreover, the larger the database, the larger the physical space required to organize them; that is, the data may belong to description spaces (the spaces of the pertinent observables and parameters) of large or even huge dimension. If the dynamics happens to generate some "attractors" (a precise mathematical notion5 ), then the dimension of the attractors also matters, since the relative unpredictability of future evolutions of the intended dynamical system grows exponentially with both the phase space and attractors' dimensions [START_REF] Cecconi | Predicting the future from the past: An old problem from a modern perspective[END_REF]). Finally, CECCONI et al. (2012) give a further mathematical argument against the abuses of Big Data rhetoric. In linear and non-linear dynamics, in bounded phase spaces, regularities may appear under the form of "recurring phenomena". Tat is, patterns of the dynamics such as series of observable values that go very close to already traveled paths, may be proven to recur. Tat is to say they may -and actually will -appear again, a famous theorem by Poincaré, 1892. Yet, as later intuited by Boltzmann and proved by KAC (1947), the recurrence times are immense. If the acritical Big Data proponents claim that they do have sufciently large sets of numbers to accommodate recurrence and thus "predict", then they surely fall under the case analyzed in section 2.1. Tat is, their database must be so huge as to exceed the cardinality limits given by Ramsey theorems, beyond which one finds a "deluge of spurious correlations" in any database. Te conditions necessary to use Big Data strategies for these dynamics are exactly the ones which lead to the appearance of spurious correlations. As a result, their use for prediction and action is not a valid strategy: a correlation does not need to recur (i.e to continue in time) nor to be due to any "causal" structure -beyond certain large sizes, today accessible to Big Data, they are "meaningless" or due only to the size of the database.

A few remarks on biological unpredictability

In the introduction, we hinted to the idea that biological variation plays a fundamental theoretical role in biology. Te principle of variation that we have proposed entails that biological objects cannot be defined theoretically like in physics (MONTÉVIL et al 2016).

In physics, objects are assumed to follow stable equations which can be found on the basis of quantitative transformations (symmetries) and invariants under these transformations. Tese transformations define the space of possibilities. Changes are then quantitative changes of state in this predefined state space. By contrast, in biology, we defend the notion that changes also impact these invariants and symmetries [START_REF] Longo | Perspectives on Organisms: Biological Time, Symmetries and Singularities[END_REF]. As a result, variation is also a variation of the relevant equations and a biological object cannot be defined by its invariants and symmetries. Accordingly, the space of possibilities is not a biological invariant, instead it can change over time. Methodologically, it is not possible to assume the existence of an invariant mathematical structure underlying the biological object of interest and to probe this mathematical structure by experiments.

Nevertheless, there are elements endowed with a restricted stability in biological objects. We call "constraints" these relatively stable elements which play a causal role on the processes that they constrain. Constraints are only stable for a limited time and can only be used as invariants at a given time scale. In an organism, constraints mutually stabilize and reconstruct each other so that the organism can maintain itself over time. With M. Mossio, we call this idea closure of constraints (M. MONTEVIL AND M. MOSSIO 2015) and we have proposed the principle of organization which states that closure of constraints is a hallmark of biological organisms (M. MOSSIO, M. MONTEVIL and G. LONGO 2016). In line with previous work of Rosen, Varela, Kaufman, etc., the principle of organization is a way to understand the mutual dependencies in an organism and to interpret biological functions. A constraint is a part of the closure of an organism when it is maintained by a process under another constraint of the organism and at the same time contributes to maintain at least another constraint of the organism, thus contributing to maintaining the whole and ultimately itself through the whole.

Let us now discuss a few consequences of this framework when considering Big Data approaches. Following the principle of organization, the relations between the parts of the individual is a fundamental notion. Following the principle of variation, the set of relevant constraints and their mutual dependencies may undergo variations. Te ubiquity of variations is precisely why we can talk of an individual and not of generic organisms which would all have exactly the same organizations. In this context, data analysis cannot unravel a stable structure that would be instantiated in all the data points corresponding to diferent individuals. Instead, these diferent data points correspond to individuals that are diferent to an extent: the constraints involved and their relations are slightly diferent for diferent individuals. Of course, data analysis may still help when focusing on a few constraints that are stable enough among the individuals considered. However, analyzing jointly the organization of many individuals leads to mixing diferent organization together and leveling down their specificity.

Conclusion

Te results cited in section 2 are technical: they belong to the combinatory theory of numbers and to the theory of algorithms or involve non-trivial aspects of dynamical systems theory and ergodic theory. Te defenders of what we define here as "Big Data without Teory" and of data-mining algorithms without analyses of meaning aim to disregard questions pertaining to theoretical frameworks. Another way to look at their aim is to say that they defend the idea of a generic theoretical framework that would apply in all kinds of empirical contexts without the need of a specific elaboration of meaning, from physics to social sciences.

In this context, recall that the Teory of Computability was invented in the 1930s by Gödel, Church and Turing in order to prove the existence of undecidable propositions and incomputable functions. More particularly, in our case, variants of results of Ramsey's Teory are situated in the difcult space of "what is computable" (the set of decidable propositions and computable functions), but such that its "computability cannot be proven" within formal number theory. Tat is, they allow defining functions that are computable but cannot be proven to be computable within the proper Teory of Computability (Arithmetic) (G. LONGO 2011) -one needs to step outside this theory and use infinitary or geometric tools in the proofs. Tese methods and objects are totally extraneous to efective computability and discrete Data Types. Tus, as a non obvious consequence of these results, even checking that a correlation is spurious is highly undecidable for a machine. Instead, it happens that we can generally detect the spurious correlations as in the examples above, whenever we have reasonably good, meaningful theories of many aspects of the world: one can give good reasons why the relation between the number of Japanese passenger cars sold in the US and the number of suicides by crashing of motor vehicle are spurious, in principle (or if it applies, search for a meaningful correlation...).

Mathematical theories such as computability demonstrate their own limits in the possibilities of computations and prediction by "negative results" that are present at the origin of scientific knowledge and characterize it. Once we have grasped the importance of the limits of the myths that "all is algorithmic" or "all is computable", we may make a better use of these immense quantities of data that computer networks make available, which is a great chance for science in every domain, including biology. Once we clarify the hypotheses that make us choose certain observables and not others, and choose measures suitable to the objectives of the knowledge that we are adopting, then digital information can help conjecture or corroborate a theory or a sketch of it, even produce new understanding. Whether it precedes or is propelled by data analysis, it seems urgent and necessary to develop theoretical frameworks for understanding organisms. In this context, we are engaged in a collaborative and interdisciplinary efort whose latest results are contained in a special issue of Progress in Biophysics and Molecular Biology: From the century of the genome to the century of the organism: New theoretical approaches (G. LONGO, A.M. SOTO and D. NOBLE 2016).

An attractor describes the asymptotic behavior of a dynamical system, that is to say its behavior after the disappearance of short terms behaviors. For example, the attractor of a dynamics which converge to a single state is this state. More complex situations include limit cycles for dynamics which converge towards an oscillatory behavior and strange attractors in the case of chaotic dynamics.