E Postek

F Dubois

R Mozul

P Ca Ñadas

MODELLING OF A COLLECTION OF NON-RIGID PARTICLES WITH SMOOTH DISCRETE ELEMENT METHOD

Introduction

Usually, the models of generation of the cell colonies use the quasi-static discrete element approach to evaluate the contact forces between the cells [START_REF] Adra | Development of a three dimensional multiscale computational model of the human epidermis[END_REF]. The forces are necessary for evaluation of the mechanotransduction phenomena [START_REF] Postek | Parameter sensitivity of a monolayer tensegrity model of tissues[END_REF]. In contrast to this approach we use compliant particles. The use of the non-rigid particles changes the stresses distribution in the particles assembly. Even intuitively, it is closer to reality.

Cell colony

The cell colony stands for a piece of tissue. We employ a non-rigid model of a single particle of equivalent stiffness to the cell. The calibration is done following the paper [START_REF] Mcgarry | A three-dimensional finite element model of an adherent eukaryotic cell[END_REF].

Multibody approach

We apply the program LMGC90 [START_REF] Renouf | A parallel version of the Non Smooth Contact Dynamics Algorithm applied to the simulation of granular media[END_REF], [START_REF] Radjai | Discrete-element Modelling of Granular Materials[END_REF] with the possibility of modelling of contact of large number of compliant particles that are discretized with finite elements.

The governing equations are written in the framework of the approach that was proposed by Moreau and Jean [START_REF] Renouf | A parallel version of the Non Smooth Contact Dynamics Algorithm applied to the simulation of granular media[END_REF]. The set of equations of motion including the initial and the boundary conditions takes the form:

M( qi+1 -qi ) = t i+1 t i (F(q, q, s) + P(s))ds + p i+1 (1)
where M is the mass matrix, q is the vector of generalized displacements, P(t) is the vector of external forces, F(q, q, t) is the vector of internal forces including the inertia terms and p i+1 is the vector of impulse resulting from contacts over the time step.

The integration with the θ scheme of the above system of equations leads to the equation:

Mk ∆ qk+1 i = p k f ree + p k+1 i+1 (2)
The effective mass matrix Mk reads:

Mk = M + h 2 θ 2 K k (3)
where h is the time increment, θ is the integration coefficient [0.5, 1] and K is the tangent stiffness. The θ coefficient is taken as 1.0 yielding the Newton-Raphson integration rule. The effective vector of forces free of contact is of the form:

p k f ree = Mk qk i+1 + M( qi -qk i+1 ) + h[(1 -θ)(F i + P i ) + θ(F k i+1 + P i+1 )] (4)
Contact impulses are computed using the NSCD method implemented in the LMGC90 software platform. Firstly, it will perform contacts detection between cells. Then the previous dynamics equations will be expressed in term of contact unknowns (gap or relative velocity and contact impulse). Afterward a Non Linear Gauss-Seidel method computes the contacts impulses. Finally, the resulting impulse on cells nodes due to contacts impulses are added to the dynamics equation to compute the new velocities and positions. We use the Open-MPI parallel version of the program [START_REF] Postek | Parameter sensitivity of a monolayer tensegrity model of tissues[END_REF].

Concluding remarks

With the presented scheme we calculate contact forces between the particles. The scheme that is based on a coupling of LMGC90 software and the modeling of cells, has been joined to an agent model able to take into account the effect of the stress evolution in the growing tissue [START_REF] Postek | Concept of an Agent-stress Model of a Tissue[END_REF].

Fig. 1 .

 1 Single cell (a) Group of cells (b). The group of cell avatars in Fig. 1 (b) has been generated employing the agent model in the framework of the FLAME platform (Flexible Large-scale Agent Modelling Environment) [1]. It consists of the stem cells, the TA cells and the Committed cells. The avatars are replaced with the equivalent stiffness cells, Fig. 1 (a).


Carlo Rovelli

Michelangelo's Stone: an Argument against Platonism in Mathematics

If there is a 'platonic world' M of mathematical facts, what does M contain precisely? I observe that if M is too large, it is uninteresting, because the value is in the selection, not in the totality; if it is smaller and interesting, it is not independent of us. Both alternatives challenge mathematical platonism. I suggest that the universality of our mathematics may be a prejudice hiding its contingency, and illustrate contingent aspects of classical geometry, arithmetic and linear algebra.

Mathematical platonism [START_REF] Plato | The Republic[END_REF] is the view that mathematical reality exists by itself, independently from our own intellectual activities. 1 Many top level mathematicians hold this view dear, and express the sentiment that they do not "construct" new mathematics, but rather "discover" structures that already exist: real entities in a platonic mathematical world. 2 Platonism is alternative to other views on the foundations of mathematics, such as reductionism, formalism, intuitionism, or the Aristotelian idea that mathematical entities exist, but they are embodied in the material world [START_REF] Gillies | An Aristotelian Approach to Mathematical Ontology[END_REF].

Here, I present a simple argument against platonism in mathematics, which I have not found in the literature.

The argument is based on posing a question. Let us assume that a platonic world of mathematical entities and mathematical truths does indeed exist, and is independent from us. Let us call this world M, for Mathematics. The question I consider is: what is it reasonable to expect M to contain? I argue that even a superficial investigation of this question reduces the idea of the existence of M to something trivial, or contradictory.

In particular, I argue that the attempt to restrict M to "natural and universal" structures is illusory: it is the perspectival error of mistaking ourselves as universal ("English is clearly the most natural of languages".) In particular, I point out contingent aspects of the two traditional areas of classical mathematics: geometry and arithmetic, and of a key tool of modern science: linear algebra.

Michelangelo's stone and Borges' library

Say we take a Platonic stance about math: in some appropriate sense, the mathematical world M exists. The expressions "to exist", "to be real" and similar can have a variety of meanings and usages, and this is a big part of the issue, if not the main one. But for the sake of the present argument I do not need to define them-nor, for that matter, platonism-precisely. The argument remains valid, I think, under most reasonable definitions and usages.

So, what does M include? Certainly M includes all the beautiful mathematical theories that mathematicians have discovered so far. This is its primary purpose. It includes Pythagoras' theorem, the classification of the Lie groups, the properties of prime numbers and so on. It includes the real numbers, and Cantor's proof that they are "more" than the integers, in the sense defined by Cantor, and both possible extensions of arithmetic: the one where there are infinities larger than the integers and smaller than the reals, and the one where there aren't. It contains game theory, and topos theory. It contains lots of stuff.

But M cannot include only what mathematicians have discovered so far, because the point of platonism is precisely that what they will discover tomorrow already exists in the platonic world today. It would not be kind towards future mathematicians to expect that M contains just a bit more of what we have already done. Obviously whoever takes a Platonic view must assume that in the platonic world of math there is much more than what has been already discovered. How much?

Certainly M contains, say, all true (or at least all demonstrable) theorems about integer numbers. All possible true theorems about Euclidean geometry, including all those we have not yet discovered. But there should be more than that, of course, as there is far more math than that.

We can get a grasp on the content of M from the axiomatic formulation of mathematics: given various consistent sets A 1 , A 2 , ... of axioms, M will include all true theorems following from each A n , all waiting to be discovered by us. We can list many sets of interesting axioms, and imagine the platonic world M to be the ensemble of theorems these imply, all nicely ordered in families, according to the corresponding set of axioms.

But this is still insufficient, because a good mathematician tomorrow could come out with a new set of axioms arXiv:1508.00001v2 [math.HO] 6 Sep 2015 and find new great mathematics, like the people who discovered non-commutative geometry, or those who defined C * algebras did.

We are getting close to what the platonic world must contain. Let us assume that a sufficiently universal language exist, say based on logic. Then the platonic world M is the ensemble of all theorems that follow from all (non contradictory) choices of axioms. This is a good picture of what M could be. We have found the content of the platonic world of math.

But something starts to be disturbing. The resulting M is big, extremely big: it contains too much junk. The large majority of coherent sets of axioms are totally irrelevant.

Before discussing the problem with precision, a couple of similes can help to understand what is going on.

(i) During the Italian Renaissance, Michelangelo Buonarroti, one of the greatest artists of all times, said that a good sculptor does not create a statue: he simply "takes it out" from the block of stone where the statue already lay hidden. A statue is already there, in its block of stone. The artists must simply expose it, carving away the redundant stone [START_REF] Neret | [END_REF]. The artist does not "create" the statue: he "finds" it.

In a sense, this is true: a statue is just a subset of the grains of stone forming the original block. It suffices to take away the other grains, and the statue is taken out. But the hard part of the game is of course to find out which subset of grains of stone to leave there, and this, unfortunately, is not written on the stone. It is selection that matters. A block of stone already contained Michelangelo's Moses, but it also contained virtually anything else -that is, all possible forms. The art of sculpture is to be able to determine, which, among this virtual infinity of forms, will talk to the rest of us as art does.

Michelangelo's statement is evocative, maybe powerful, and perhaps speaks to his psychology and his greatness. But it does not say much about the art of the sculptor. The fact of including all possible statues does not confer to the stone the immense artistic value of all the possible beautiful statues it might contain, because the point of the art is the choice, not the collection. By itself, the stone is dull.

(ii) The same story can be told about books. Borges's famous library contained all possible books: all possible combinations of alphabet letters [START_REF] Borges | The Library of Babel[END_REF]. Assuming a book is shorter than, say, a million letters, there are then more or less 30 10 6 possible books, which is not even such a big number for a mathematician. So, a writer does not really create a book: she simply "finds" it, in the platonic library of all books. A particularly nice combinations of letters makes up, say, Moby-Dick. Moby-Dick already existed in the platonic space of books: Melville didn't create Moby-Dick, he just discovered Moby-Dick... Like Michelangelo's stone, Borges's library is void of any interest: it has no content, because the value is in the choice, not in the totality of the alternatives.

The Platonic world of mathematics M defined above is similar to Michelangelo's block of stone, or Borges library, or Hegel's "night in which all cows are black" 3 : a mere featureless vastness, without value because the value is in the choice, not in the totality of the possibilities. Similarly, science can be said to be nothing else than the denotation of a subset of David Lewis's possible worlds [START_REF] Lewis | On the Plurality of Worlds[END_REF]: those respecting certain laws we have found. But Lewis's totality of all possible world is not science, because the value of science is in the restriction, not in the totality.

Mathematics may be called an "ensemble of tautologies", in the sense of Wittgenstein. But it is not the ensemble of all tautologies: these are too many and their ensemble is uninteresting and irrelevant. Mathematics is about recognizing the "interesting" ones. Mathematics may be the investigation of structures. But it is not the list of all possible structures: these are too many and their ensemble is uninteresting. If the world of mathematics was identified with the platonic world M defined above, we could program a computer to slowly unravel it entirely, by listing all possible axioms and systematically applying all possible transformation rules to derive all possible theorems. But we do not even think of doing so. Why? Because what we call mathematics is an infinitesimal subset of the huge world M defined above: it is the tiny subset which is of interest for us. Mathematics is about studying the "interesting" structures.

So, the problem becomes: what does "interesting" mean?

Interest is in the eye of the interested Can we restrict the definition of M to the interesting subset? Of course we can, but interest is in the eyes of a subject. A statue is a subset of the stone which is worthwhile, for us. A particular combination of letters is a good book, for us. What is it that makes certain set of axioms defining certain mathematical objects, and certain theorems, interesting?

There are different possible answer to this question, but they all make explicit or implicit reference to features of ourselves, our mind, our contingent environment, or the physical structure our world happens to have.

This fact is pretty evident as far as art or literature are concerned. Does it hold for mathematics as well? Hasn't mathematics precisely that universality feel that is at the root of platonism? 3 Hegel utilized this Yiddish saying to ridicule Schelling's notion of Absolute, meaning that -like mathematical platonism-this included too much and was too undifferentiated, to be of any relevance [START_REF] Hegel | Phenomenology of Spirit[END_REF].

Shouldn't we expect -as often claimed-any other intelligent entity of the cosmos to come out with the same "interesting" mathematics as us?

The question is crucial for mathematical platonism, because platonism is the thesis that mathematical entities and truths form a world which exists independently from us. If what we call mathematics ends up depending heavily on ourselves or the specific features of our world, platonism looses its meaning.

I present below some considerations that indicate that the claimed universality of mathematics is a parochial prejudice. These are based on the concrete examples provided by the chapters of mathematics that have most commonly been indicated as universal.

The geometry of a sphere Euclidean geometry has been among the first pieces of mathematics to be formalized. Euclid's celebrated text, the "Elements" [START_REF] Euclid | [END_REF], where Euclidean geometry is beautifully developed, has been the ideal reference for all mathematical texts. Euclidean geometry describes space. It captures our intuition about the structure of space. It has applications to virtually all fields of science, technology and engineering. Pythagoras' theorem, which is at its core, is a familiar icon.

It is difficult to imagine something having a more "universal" flavor than euclidean geometry. What could be contingent or accidental about it? What part do we humans have in singling it out? Wouldn't any intelligent entity developing anywhere in the universe come out with this same mathematics?

I maintain the answer is negative. To see why, let me start by recalling that, as is well known, Euclidean geometry was developed by Greek mathematicians mostly living in Egypt during the Hellenistic period, building on Egyptians techniques for measuring the land. These were important because of the Nile's floods cancelling borders between private land parcels. The very name of the game, "geometry", means "measurement of the land" in Greek. Two-dimensional Euclidean geometry describes, in particular, the mathematical structure formed by the land.

But: does it? Well, the Earth is more a sphere than a plane. Its surface is better described by the geometry of a sphere, than by two-dimensional (2d) Euclidean geometry. It is an accidental fact that Egypt happens to be small compared to the size of the Earth. The radius of the Earth is around 6,000 Kilometers. The size of Egypt is of the order of 1,000 Kilometers. Thus, the scale of the Earth is more than 6 times larger than the scale of Egypt. Disregarding the sphericity of the Earth is an approximation, which is viable when dealing with the geometry of Egypt and becomes better and better as the region considered is smaller. As a practical matter, 2d Euclidean geometry is useful, but it is a decent approximation that works only because of the smallness of the size of Egypt. Intelligent beings living on a planet just a bit smaller than ours [START_REF] De Saint-Exupéry | [END_REF], would have easily detected the effects of the curvature of the planet's surface. They would not have developed 2d Euclidean geometry.

One may object that this is true for 2d, but not for 3d geometry. The geometry of the surface of a sphere can after all be obtained from Euclidean 3d geometry. But the objection has no teeth: we have learned with general relativity that spacetime is curved and Euclidean geometry is just an approximation also as far as 3d physical space is concerned. Intelligent beings living on a region of stronger spacetime curvature would have no reason to start mathematics from Euclidean geometry. 4 A more substantial objection is that 2d euclidean geometry is simpler and more "natural" than curved geometry. It is intuitively grasped by our intellect, and mathematics describes this intuition about space. Its simplicity and intuitive aspect are the reasons for its universal nature. Euclidean geometry is therefore universal in this sense. I show below that this objection is equally ill founded: the geometry of a sphere is definitely simpler and more elegant than the geometry of a plane.

Indeed, there is a branch of mathematics, called 2d "spherical" geometry, which describes directly the (intrinsic) geometry of a sphere. This is the mathematics that the Greeks would have developed had the Earth been sufficiently small to detect the effects of the Earth's surface curvature on the Egyptians fields. Perhaps quite surprisingly for many, spherical geometry is far simpler and "more elegant" than Euclidean geometry. I illustrate this statement with a few examples below, without, of course, going into a full exposition of spherical geometry (see for instance [START_REF] Todhunter | Spherical trigonometry[END_REF][START_REF] Harris | Spherical Geometry[END_REF]).

Consider the theory of triangles: the familiar part of geometry we study early at school. In Euclidean geometry, a triangle has three sides, with lengths, a, b and c, and three angles α, β and γ (Figure 1). We measure angles with pure numbers, so, α, β and γ are taken 4 It is well known that Kant was mistaken in his deduction that the Euclidean geometry of physical space is true a priori [START_REF] Kant | The Critique of Pure Reason[END_REF]. But even Wittgenstein bordered on mistake in dangerously appearing to assume a unique possible set of laws of geometry for anything spatial: "We could present spatially an atomic fact which contradicted the laws of physics, but not one which contradicted the laws of geometry". Tractatus, Proposition 3.0321 [START_REF] Wittgenstein | Tractatus Logico-Philosphicus[END_REF]. to be numbers with value between 0 and π. Measuring with numbers the length of the sides is a more complicated business. Since there is no standard of length in Euclidean geometry, we must either resort to talk only about ratios between lengths (as the ancients preferred), or to choose arbitrarily a segment once and for all, use it as our "unit of measure", and characterize the length of each side of the triangle by the number which gives its ratio to the unit (as the moderns prefer). All this simplifies dramatically in spherical geometry: here there is a preferred scale, the length of the equator. The length of an arc (the shortest path between two points) is naturally measured by the ratio to it. Equivalently (if we immerge the sphere in space) by the angle subtended to the arc. Therefore the length of the side of a triangle (a, b, c) is an angle as well. See Figure 2. Compare then the theories of triangles in the two geometries (Figure 1):

Euclidean geometry: (i) Two triangles are equal if they have equal sides, or if one side and two angles are equal. (ii) The area of the triangle is

A = 1 4 √ 2a 2 b 2 + 2a 2 c 2 + 2b 2 c 2 -a 4 -b 4 -c 4 .
(iii) For right triangles:

a 2 + b 2 = c 2 .

Spherical geometry:

(i) Triangles with same sides, or same angles, are equal.

(ii) The area of a triangle is

A = α + β + γ -π.
(iii) For right triangles: cos c = cos a cos b.

Even a cursory look at these results reveals the greater simplicity of spherical geometry. Indeed, spherical geometry has a far more "universal" flavor than Euclidean geometry.

Euclidean geometry can be be obtained from it as a limiting case: it is the geometry of figures that are much smaller than the curvature radius. In this case a, b and c are all much smaller than π. Their cosine is well approximated by cos θ ∼ 1 -1 2 θ 2 and the last formula reduces to Pythagoras' theorem in the first approximation. Far from being a structural property of the very nature of space, Pythagoras' theorem is only a first order approximation, valid in a limit case of a much simpler and cleaner mathematics: 2d spherical geometry. There are many other beautiful and natural results in spherical geometry, which I shall not report here. They extend to the 3d case: the intrinsic geometry of a 3sphere. A 3-sphere is a far more reasonable structure than the infinite Euclidean space: it is the finite homogeneous three-dimensional metric space without boundaries. The geometry may well be the large scale geometry of our universe [START_REF] Einstein | Cosmological Considerations in the General Theory of Relativity[END_REF]. 5 It shape is counterintuitive for many of us, schooled in Euclid. But it was not so for Dante Alighieri, who did not study Euclid at school: the topology of the universe he describes in his poem is precisely that of a 3-sphere [START_REF] Peterson | Dante and the 3-sphere[END_REF]. See Figure 3. What is "intuitive" changes with history.

These considerations indicate that the reason Euclidean geometry has played such a major role in the foundation of mathematics is not because of its universality and independence from our contingent situation. It is the opposite: Euclidean geometry is of value to us just because it describes-not even very wellthe accidental properties of the region we happen to inhabit. Inhabitants of a different region of the universe-a smaller planet, or a region with high space curvaturewould likely fail to consider euclidean geometry interesting mathematics. For them, Euclidean geometry could 5 Cosmological measurements indicate that spacetime is curved, but have so far failed to detected a large scale cosmological curvature of space. This of course does not imply that the universe is flat [START_REF] Ellis | Relativistic Cosmology[END_REF], for the same reason for which the failure to detect curvature on the fields of Egypt did not imply that that the Earth was flat. It only shows that the universe is big. Current measurements indicate that the radius of the Universe should be at least ten time larger than the portion of the Universe we see [START_REF] Hinshaw | Five-Year Wilkinson Microwave Anisotropy Probe Observations: Data Processing, Sky Maps, and Basic Results[END_REF]. A ratio, by the way, quite similar to the Egyptian case.

be a uninteresting and cumbersome limiting case.

Linear algebra

Every physicist, mathematician or engineer learns linear algebra and uses it heavily. Linear algebra, namely the algebra of vectors, matrices, linear transformations and so on, is the algebra of linear spaces, and since virtually everything is linear in a first approximation, linear algebra is ubiquitous. It is difficult to resist its simplicity, beauty and generality when studying it, usually in the early years at college. Furthermore, today, we find linear algebra at the very foundations of physics, because it is the language of quantum theory. In the landmark paper that originated quantum theory [START_REF] Heisenberg | Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen[END_REF], Werner Heisenberg understood that physical quantities are better described by matrices and used the multiplication of matrices (core of linear algebra) to successfully compute the properties of quantum particles. Shortly later, Paul Dirac wrote his masterpiece book [START_REF] Dirac | Principles of Quantum Mechanics[END_REF], where quantum mechanics is entirely expressed in terms of linear algebra (linear operators, eigenvalues, eigenvectors...).

It would therefore seem natural to formulate the hypothesis that any minimally advanced civilization would discover linear algebra very early and start using it heavily.

But it is not the case. In fact, we have a remarkable counterexample: a civilization that has developed for millennia without developing linear algebra-ours.

When Heisenberg wrote his famous paper he did not know algebra. He had no idea of what a matrix is, and had never previously learned the algorithm for multiplying matrices. He made it up in his effort to understand a puzzling aspect of the physical world. This is pretty evident from his paper. Dirac, in his book, is basically inventing linear algebra in the highly nonrigorous manner of a physicist. After having constructed it and tested its power to describe our world, linear algebra appears natural to us. But it didn't appear so for generations of previous mathematicians.

Which tiny piece of M turns out to be interesting for us, which parts turns out to be "mathematics" is far from obvious and universal. It is largely contingent.

Arithmetic and identity

The last example I discuss is given by the natural numbers (1, 2, 3, ...), which form the basis of arithmetic, the other half of classical mathematics. Natural numbers seem very natural indeed. There is evidence that the brain is pre-wired to be able to count, and do elementary arithmetic with small numbers [START_REF] Vallortigara | Cervelli che contano[END_REF]. Why so? Because our world appears to be naturally organized in terms of things that can be counted. But is this a feature of reality at large, of any possible world, or is it just a special feature of this little corner of the universe we inhabit and perceive? I suspect the second answer to be the right one. The notion of individual "object" is notoriously slippery, and objects need to have rare and peculiar properties in order to be countable. How many clouds are there in the sky? How many mountains in the Alps? How many coves along the coast of England? How many waves in a lake? How many clods in my garden? These are all very ill-defined questions.

To make the point, imagine some form of intelligence evolved on Jupiter, or a planet similar to Jupiter. Jupiter is fluid, not solid. This does not prevent it from developing complex structures: fluids develop complex structures, as their chemical composition, state of motion, and so on, can change continuously from point to point and from time to time, and their dynamics is governed by rich nonlinear equations. Furthermore, they interact with magnetic and electric fields, which vary continuously in space and time as well. Imagine that in such a huge (Jupiter is much larger than Earth's) Jovian environment, complex structures develop to the point to be conscious and to be able to do some math. After all, it has happened on Earth, so it shouldn't be so improbable for something like this to happen on an entirely fluid planet as well. Would this math include counting, that is, arithmetic? Why should it? There is nothing to count in a completely fluid environment. (Let's also say that our Jovian planet's atmosphere is so thick that one cannot see and count the stars, and that the rotation and revolution periods are equal, as for our Moon, and there are neither days nor years.) The math needed by this fluid intelligence would presumably include some sort of geometry, real numbers, field theory, differential equations..., all this could develop using only geometry, without ever considering this funny operation which is enumerating individual things one by one.

The notion of "one thing", or "one object", the notions themselves of unit and identity, are useful for us living in an environment where there happen to be stones, gazelles, trees, and friends that can be counted. The fluid intelligence diffused over the Jupiter-like planet, could have developed mathematics without ever thinking about natural numbers. These would not be of interest for her. I may risk being more speculative here. The development of the ability to count may be connected to the fact that life evolved on Earth in a peculiar form characterized by the existence of "individuals". There is no reason an intelligence capable to do math should take this form. In fact, the reason counting appears so natural to us may be that we are a species formed by interacting individuals, each realizing a notion of identity, or unit. What is clearly made by units is a group of interacting primates, not the world. The archetypical identities are my friends in the group. 6Modern physics is intriguingly ambiguous about countable entities. On the one hand, a major discovery of the XX century has been that at the elementary level nature is entirely described by field theory. Fields vary continuously in space and time. There is little to count, in the field picture of the world. On the other hand, quantum mechanics has injected a robust dose of discreteness in fundamental physics: because of quantum theory, fields have particle-like properties and particles are quintessentially countable objects. In any introductory quantum field theory course, students meet an important operator, the number operator N , whose eigenvalues are the natural numbers and whose physical interpretation is counting particles [START_REF] Itzykson | Quantum Field Theory[END_REF]. Perhaps our fluid Jovian intelligence would finally get to develop arithmetic when figuring out quantum field theory...

But notice that what moderns physics says about what is countable in the world has no bearing on the universality of mathematics: at most, it points out which parts of M are interesting because they happen to describe this world.

Conclusion

In the light of these consideration, let us look back at the development of our own mathematics. Why has mathematics developed at first, and for such a long time, along two parallel lines: geometry and arithmetic? The answer begins to clarify: because these two branches of mathematics are of value for creatures like us, who instinctively count friends, enemies and sheep, and who need to measure, approximately, a nearly flat earth in a nearly flat region of physical space. In other words, this mathematics is of interest to us because it reflects very contingent interests of ours. Out of the immense vastness of M, the dull platonic space of all possible structures, we have carved out, like Michelangelo, a couple of shapes that talk to us. From the immense vastness of M, the dull platonic space of all possible structures, we have carved out, like Michelangelo, a couple of shapes that speak to us.

There is no reason to assume that the mathematics that has developed later escapes this contingency. To the contrary, the continuous re-foundations and the constant re-organization of the global structure of mathematics testify to its non-systematic and non-universal global structure. Geometry, arithmetic, algebra, analysis, set theory, logic, category theory, and -recentlytopos theory [START_REF] Caramello | The unification of Mathematics via Topos Theory[END_REF] have all been considered for playing a foundational role in mathematics. Far from being stable and universal, our mathematics is a fluttering butterfly, which follows the fancies of inconstant creatures. Its theorems are solid, of course; but selecting what represents an interesting theorem is a highly subjective matter.

It is the carving out, the selection, out of a dull and undifferentiated M, of a subset which is useful to us, interesting for us, beautiful and simple in our eyes, it is, in other words, something strictly related to what we are, that makes up what we call mathematics.

The idea that the mathematics that we find valuable forms a Platonic world fully independent from us is like the idea of an Entity that created the heavens and the earth, and happens to very much resemble my grandfather.

-Thanks to Hal Haggard and Andrea Tchertkoff for a careful reading of the manuscript and comments.

FIG. 1 .

 1 FIG. 1. Flat and spherical triangles.

FIG. 2 .

 2 FIG.2. Two points on a sphere determine an arc, whose size is measured by the angle it subtends, or equivalently, intrinsically, by its ratio to an equator.

FIG. 3 .

 3 FIG.3. Dante's universe: the Aristotelian spherical universe is surrounded by another similar spherical space, inhabited by God and Angel's spheres. The two spheres together form a three-sphere.

For an overview, see for instance[START_REF] Linnebo | Platonism in the Philosophy of Mathematics[END_REF].

Contemporary mathematicians that have articulated this view in writing include Roger Penrose[START_REF] Penrose | The Road to Reality : A Complete Guide to the Laws of the Universe[END_REF] and Alain Connes[START_REF] Connes | A View of Mathematics[END_REF].

This might be why ancient humans attributed human-like mental life to animals, trees and stones: they were perhaps utilizing mental circuits developed to deal with one another -within the primate group-extending them to deal also with animals, trees and stones.


Valerio Astuti
email: valerio.astuti@roma1.infn.it

Marios Christodoulou
email: christod.marios@gmail.com

Carlo Rovelli
email: rovelli@cpt.univ-mrs.fr

Volume entropy

Building on a technical result by Brunnemann and Rideout on the spectrum of the Volume operator in Loop Quantum Gravity, we show that the dimension of the space of the quadrivalent states -with finite-volume individual nodes-describing a region with total volume smaller than V , has finite dimension, bounded by V log V . This allows us to introduce the notion of "volume entropy": the von Neumann entropy associated to the measurement of volume.

I. Introduction

Thermodynamical aspects of the dynamics of spacetime have first been pointed out by Bekenstein's introduction of an entropy associated to the horizon of a black hole [START_REF] Bekenstein | Black Holes and Entropy[END_REF]. This led to the formulation of the "laws of black holes thermodynamics" by Bardeen, Carter, and Hawking [START_REF] James M Bardeen | The Four laws of black hole mechanics[END_REF] and to Hawking's discovery of black role radiance, which has reinforced the geometry/thermodynamics analogy [START_REF] Hawking | Particle creation by black holes[END_REF]. The connection between Area and Entropy suggests that it may be useful to treat aspects of space-time statistically at scales large compared to the Planck length [START_REF] Jacobson | Thermodynamics of space-time: The Einstein equation of state[END_REF], whether or not we expect the relevant microscopic elementary degrees of freedom to be simply the quanta of the gravitational field [START_REF] Chirco | Spacetime thermodynamics without hidden degrees of freedom[END_REF], or else. Black hole entropy, in particular, can be interpreted as cross-horizon entanglement entropy (see [START_REF] Bianchi | Horizon entanglement entropy and universality of the graviton coupling[END_REF] for recent results reinforcing this interpretation, and references therein), or -most likely equivalently-as the von Neumann entropy of the statistical state representing a macrostate with given horizon Area. In the context of Loop Quantum Gravity (LQG), this was considered in [START_REF] Rovelli | Black Hole Entropy from Loop Quantum Gravity[END_REF] and later extensively analyzed; for a recent review and full references see [START_REF] Barbero | Quantum Geometry and Black Holes[END_REF][START_REF] Ashtekar | The Issue of Information Loss: Current Status[END_REF].

All such developments are based on the assignment of thermodynamic properties to spacetime surfaces. This association has motivated the holographic hypothesis: the conjecture that the degrees of freedom of a region of space are somehow encoded in its boundary.

In this paper, instead, we study statistical properties associated to spacetime regions. We show that it is possible to define a Von Neumann entropy for the quantum gravitational field, associated to the Volume of a region, and that this entropy is (under suitable conditions) finite. The existence of an entropy associated to bulk degrees of freedom of a spin network was already considered in [START_REF] Krzysztof | Eigenvalues of the volume operator in loop quantum gravity[END_REF].

To this aim, we prove a finiteness result on the num-ber of quantum states of gravity describing a region of finite volume. More precisely, we work in the context of LQG, and we prove that the dimension of the space of diffeomorphism invariant quadrivalent states without zero-volume nodes, describing a region of total volume smaller than V is finite. We give explicitly the upper bound of the dimension as a function of V . The proof is based on a result on the spectrum of the LQG Volume operator proven by Brunnemann and Rideout [START_REF] Brunnemann | Properties of the volume operator in loop quantum gravity. I. Results[END_REF][START_REF] Brunneman | Properties of the volume operator in loop quantum gravity. II. Detailed presentation[END_REF]. Using this, we define the Von Neumann entropy of a quantum state of the gravitational field associated to Volume measurements.

II. Counting spin networks

Consider the measurement of the volume of a 3d spacelike region Σ. The physical system measured is the gravitational field. In the classical theory, this is given by the metric q on Σ: the volume is V = Σ √ det q d 3 x. In the quantum context, using the LQG formalism, the geometry of Σ is described by a state in the kinematical Hilbert space H diff . The volume measurement of Σ are described by a volume operator V on this state space. We refer to [START_REF] Rovelli | Covariant Loop Quantum Gravity[END_REF][START_REF] Rovelli | Quantum Gravity[END_REF] for details on basic LQG results and notation.

We restrict H diff to four-valent graphs Γ where the nodes n have non-vanishing (unoriented) volume v n . The spin network states |Γ, j l , v n ∈ H diff , where j l is the link quantum number or spin, form a countable, orthonormal basis of H diff . (We disregard here eventual additional quantum numbers such as the orientation, that have no bearing on our result.) The intertwiner basis at each node is chosen so that the local volume operator Vn , acting on a single node, is diagonal and is labelled by the eigenvalues v n , of the node volume operator Vn associated to the node n.

Vn |Γ, j l , v n = v n |Γ, j l , v n (1)
The states |Γ, j l , v n are also eigenstates of the total volume operator V = N n=1 Vn , where N is the number of nodes in Γ, with eigenvalue

v = N n=1 v n , (2)
the sum of the node volume eigenvalues v n . We seek a bound on the dimension of the subspace H V spanned by the states |Γ, j l , v n such that v ≤ V . That is, we want to count the spin-networks with volume less than V . We do this by bounding the number N Γ of four valent graphs in H V , the number N {j l } of possible spin assignments, and the number of the volume quantum numbers assignments N {vn} on each such graph. Clearly

dim H V ≤ N Γ N {j l } N {vn} . (3)
Crucial to this bound is the analytical result on the existence of a volume gap in four-valent spin networks found in [START_REF] Brunnemann | Properties of the volume operator in loop quantum gravity. I. Results[END_REF][START_REF] Brunneman | Properties of the volume operator in loop quantum gravity. II. Detailed presentation[END_REF]. The result is the following. In a node bounded by four links with maximum spin j max all nonvanishing volume eigenvalues are larger than

v gap ≥ 1 4 √ 2 ℓ 3 P γ 3 2 j max (4)
Where ℓ P is the Planck constant and γ the Immirzi parameter. Numerical evidence for equation ( 4) was first given in [START_REF] Brunnemann | Simplification of the spectral analysis of the volume operator in loop quantum gravity[END_REF] and a compatible result was estimated in [START_REF] Bianchi | Discreteness of the volume of space from Bohr-Sommerfeld quantization[END_REF]. Since the minimum non-vanishing spin is j = 1 2 , this implies that

v gap ≥ 1 8 ℓ 3 P γ 3 2 ≡ v o (5)
From existence of the volume gap, it follows that there is a maximum value of N Γ , because there is a maximum number of nodes for graphs in H V , as every node carries a minimum volume v o . Therefore a region of volume equal or smaller than V contains at most

n = V v o (6)
nodes. Equation ( 4) bounds also the number of allowed area quantum numbers, because too large a j max would force too large a node volume. Therefore N {j l } is also finite. Finally, since the dimension of the space of the intertwiners at each node is finite and bounded by the value of spins, it follows that also the number N {vn} of individual volume quantum numbers is bounded. Then (3) shows immediately that the dimension of H V is finite. Let us bound it explicitely. We start by the number of graphs. The number of nodes must be smaller than n, given in [START_REF] Bianchi | Horizon entanglement entropy and universality of the graviton coupling[END_REF]. The number N Γ of 4-valent graphs with n nodes is bounded by

N Γ ≤ n 4n (7)
because each node can be connected to each other n n four times (n n ) 4 .

Equation ( 4) bounds the spins. Since we must have V ≥ v gap , we must also have

j ≤ j max ≤ 32 V 2 ℓ 6 P γ 3 = 1 2 n 2 (8)
In a graph with n nodes there are at most 4n links (the worst case being all boundary links), and therefore there are at most (2j max + 1) 4n spin assignments, or, in the large j limit, (2j max ) 4n . That is

N {j l } ≤ (2j max ) 4n ≤ n 8n (9)
Finally, the dimension of the intertwiner space at each node is bounded by the areas associated to that node:

dim K j1,j2,j3,j4 = = dim Inv SU(2) (H j1 ⊗ H j2 ⊗ H j3 ⊗ H j4 ) = min (j 1 + j 2 , j 3 + j 4 ) -max ((j 1 -j 2 ), (j 3 -j 4 )) + 1 ≤ 2 max(j l∈n ) + 1 ≤ 4 max(j l∈n )
with the last step following from max(j l∈n ) ≥ 1/2. Thus on a graph with n nodes, the maximum number of combination of eigenvalues is limited by:

N {vn} ≤ (4j max ) n = 2 n n 2n (10)
Combining equations ( 3), ( 7), ( 9) and ( 10), we have an explicit bound on the dimension of the space of states with volume less than

V = nv o : dim H V ≤ (cn) 14n ( 11
)
where c is a number. For large n we can write

S V ≡ log dim H V ≤ 14 n log n (12)
which is the entropy associated to Hilbert space. Explicitly

S V ≤ 14 V v o log V v o ∼ V log V. (13)
In the large volume limit, when the eigenvalues become increasingly dense, this corresponds to a density of states

ν(V ) ≡ d(dim H V )/dV similarly bounded ν(V ) < 14 [log(n) + C] (cn) 14n . ( 14
)

III. Von Neumann proper volume entropy

In the previous section, we have observed that the dimension of the space of (with four-valent, finite-volume nodes) quantum states with total volume less than V is finite. This results implies that there is a finite von Neumann volume entropy associated to statistical states describing to volume measurements. The simplest possibility is to consider the microcanonical ensemble describing the volume measurement of a region of space. That is, we take Volume to be a macroscopic (or thermodynamic, or"coarse grained") variable, and we write the corresponding statistical microstate that maximizes entropy. If the measured volume is in the interval I V = [V -δV, V ], with small δV , then the corresponding micro-canonical state is simply

ρ = P V,δV dim H V . ( 15
)
where P V,δV is the projector on

H V,δV ≡ Span{|Γ, j l , v n > : v ∈ I V }. ( 16
)
namely the span of the eigenspaces of eigenvalues of the volume that are in I V . Explicitly, the projector can be written in the form

P V,δV ≡ v∈IV |Γ, j l , v n >< Γ, j l , v n | (17)
The von Neumann entropy of ( 15) is

S = -T r[ρ log ρ] = log dim H V < S V ∼ V log V. (18)
It is interesting to consider also a more generic state where ρ ∼ p(V ), for an arbitrary distribution p(V ) of probabilities of measuring a given volume eigenstate with volume V . For this state, the probability distribution of finding the value V in a volume measurement is

P (V ) = ν(V )p(V ) (19)
and the entropy can be written as the sum of two terms

S = dV ν(V )p(V ) log(p(V )) = S1 + S2 (20)
where the first

S P = -dV P (V ) log(P (V )) (21)
is just the entropy due to the spread in the outcomes of volume measurements, while the second

S Volume ≡ S -S P = dV P (V ) log(ν(V )) (22)
can be seen as as a proper volume entropy. The bound found in the previous Section on ν(V ), which indicates that ν(S) grows less that V 2 , shows that this proper volume entropy is finite for any distribution P (V ) whose variance is finite. S Volume can be viewed as the irreducible entropy associated to any volume measurement.

IV. Lower bound

Let us now bound the dimension of H V from below. The crucial step for this is to notice the existence of a maximum δV in the spacing between the eigenvalues of the operator Vn . For instance, if we take a node between two large spins j and two 1 2 spins, the volume eigenvalues have decreasing spacing, with maximum spacing for the lowest eigenvalues, of the order v o . Disregarding irrelevant small numerical factors, let's take v o as the maximal spacing.

Given a volume V , let, as before, n = V /v 0 and consider spin networks with total volume in the interval

I n = [(n -1)v o , nv o ].
Let N m be the number of spin networks with m nodes that have the total volume v in the interval I n . For m = 1, there is at least one such spin network, because of the minimal spacing. For m = 2, the volume v must be split between the two nodes: v = v 1 + v 2 . This can be done in at least n -1 manners, with v 1 ∈ I p and v 1 ∈ I n-p and p running from 1 to n -1. This possibility is guaranteed again by the existence of the maximal spacing. In general, for m nodes, there are

N n,m = n -1 m -1 (23)
different ways of splitting the total volume among nodes. This is the number of compositions of n in m subsets. Finally, the number m of nodes can vary between 1 and the maximum n, giving a total number of possible states larger than

N n = n m=1 N n,m = n m=1 n -1 m -1 = 2 n-1 . (24)
From which it follows that

dim H V ≥ 2 n-1 . (25)
Can all these states be realised by inequivalent spin networks, with suitable choices of the graph and the spins? To show that this is the case, it is sufficient to display at least one (however peculiar) example of spin network for each sequence of v n . But given an arbitrary sequence of v n we can always construct a graph formed by a single one dimensional chain of nodes, each (except the two ends) with two links connecting to the adjacent nodes in the chain and two links in the boundary. All these spin networks exist and are non-equivalent to one another. Therefore we have shown that there are at least 2 n-1 states with volume between V -v o and V . In the large volume limit we can write

dim H V ≥ 2 n = 2 V vo .
(26) so that the entropy satisfies

cV ≤ S ≤ c ′ V log V. (27)
with c and c ′ constants.

V. Discussion

Geometrical entropy associated to surfaces of given Area plays a large role in the current discussions of the quantum nature of spacetime. Here we have shown that, under suitable conditions, it is also possible to compute a Von Neumann entropy associated to measurements of the Volume of a region of space. We have not discussed possible physical roles played by this entropy. A number of comments are in order:

(i) Since in the classical low energy limit Volume and area are related by V ∼ A (ii) The result presented above depends on the restriction of H diff to four-valent states. We recall that the discussion is currently open in the literature on which of the two theories, with or without this restriction, is physically more interesting, with good arguments on both sides. However, it might be possible to extend the results presented here to the case of higher-valent graphs. Indeed, there is some evidence that there is a volume gap in higher-valent cases too, see for instance [START_REF] Haggard | Pentahedral volume, chaos, and quantum gravity[END_REF]. The effect of zerovolume nodes on the Volume entropy will be discussed elsewhere.

(iii) Volume entropy appears to fail to be an extensive quantity. The significance of this conclusion deserves to be explored. This feature is usual for systems with long range interactions, and in particular for systems of particles governed by the gravitational interaction. It is suggestive that gravity could retain this feature even when there are no interacting particle, and the role of long range interactions is taken by "long range" connections between graph nodes1 . A final word on this behaviour, however, has to wait for a more precise computation of the entropy growth with volume.

(iv) It has been recent pointed out that the interior of an old black old contains surfaces with large volume [START_REF] Christodoulou | How big is a black hole?[END_REF][START_REF] Christodoulou | The (huge) volume of an evaporating black hole[END_REF] and that the large volume inside black holes can play an important role in the information paradox [START_REF] Ashtekar | The Issue of Information Loss: Current Status[END_REF][START_REF] Perez | No firewalls in quantum gravity: the role of discreteness of quantum geometry in resolving the information loss paradox[END_REF]. The results presented here may serve to quantify the corresponding interior entropy.

(v) A notion of entropy associated to the volume of space might perhaps provide an alternative to Penrose's Weyl curvature hypothesis [START_REF] Penrose | Before the big bang: An outrageous new perspective and its implications for particle physics[END_REF]. For the sec-ond principle of thermodynamics to hold, the initial state of the universe must have had low entropy. On the other hand, from cosmic background radiation observations, the initial state of matter must have been close to having maximal entropy. Penrose addresses this discrepancy by taking into consideration the entropy associated to gravitational degrees of freedom. His hypothesis is that the degrees of freedom which have been activated to bring the increase in entropy from the initial state are the ones associated to the Weyl curvature tensor, which in his hypothesis was null in the initial state of the universe. A definition of the bulk entropy of space, which, as would be expected, grows with the volume, could perhaps perform the same role as the Weyl curvature degrees of freedom do in Penrose's hypothesis: the universe had a much smaller volume close to its initial state, so the total available entropy was low -regardless of the matter entropy content -and has increased since, just because for a space of larger volume we have a greater number of states describing its geometry.

(vi) We close with a very speculative remark. Does the fact that entropy is large for larger volumes imply the existence of an entropic force driving to larger volumes? That is, could there be a statistical bias for transitions to geometries of greater volume? Generically, the growth of the phase space volume is a driving force in the evolution of a system: in a transition process, we sum over out states, more available states for a given outcome imply greater probability of that outcome. A full discussion of this point requires the dynamics of the theory to be explicitly taken into account, and we postpone it for future work.

3 2 , 3 2

 23 the Volume entropy we have considered S V ∼ V log V ∼ A log A may exceed the Bekenstein bound S < S A ∼ A. Volume entropy is accessible only by being in the bulk, and not necessarily from the outside, therefore it does not violate the versions of the Bekenstein bound that only refer to external observables.

Of course they are not really long range, in the sense that graph connections actually define locality.

Acknowledgments

MC and VA thank Thibaut Josset and Ilya Vilenski for critical discussions. MC acknowledges support from the Educational Grants Scheme of the A.G.Leventis Foundation for the academic years 2013-2014, 2014-2015, 2015-2016, as well as from the Samy Maroun Center for Time, Space and the Quantum. VA acknowledges financial support from Sapienza University of Rome.


come   L'archive ouverte pluridisciplinaire

Meaning = Information + Evolution

Carlo Rovelli

CPT, Aix-Marseille Université, Université de Toulon, CNRS, F-13288 Marseille, France.

Notions like meaning, signal, intentionality, are difficult to relate to a physical word. I study a purely physical definition of "meaningful information", from which these notions can be derived. It is inspired by a model recently illustrated by Kolchinsky and Wolpert, and improves on Dretske classic work on the relation between knowledge and information. I discuss what makes a physical process into a "signal".

I. INTRODUCTION

There is a gap in our understanding of the world. On the one hand we have the physical universe; on the other, notions like meaning, intentionality, agency, purpose, function and similar, which we employ for the like of life, humans, the economy... These notions are absent in elementary physics, and their placement into a physicalist world view is delicate [START_REF] Price | Naturalism without Mirrors[END_REF], to the point that the existence of this gap is commonly presented as the strongest argument against naturalism.

Two historical ideas have contributed tools to bridge the gap.

The first is Darwin's theory, which offers evidence on how function and purpose can emerge from natural variability and natural selection of structures [START_REF] Darwin | On the Origin of Species[END_REF]. Darwin's theory provides a naturalistic account for the ubiquitous presence of function and purpose in biology. It falls sort of bridging the gap between physics and meaning, or intentionality.

The second is the notion of 'information', which is increasingly capturing the attention of scientists and philosophers. Information has been pointed out as a key element of the link between the two sides of the gap, for instance in the classic work of Fred Dretske [START_REF] Dretske | Knowledge and the Flow of Information[END_REF].

However, the word 'information' is highly ambiguous. It is used with a variety of distinct meanings, that cover a spectrum ranging from mental and semantic ("the information stored in your USB flash drive is comprehensible") all the way down to strictly engineeristic ("the information stored in your USB flash drive is 32 Giga"). This ambiguity is a source of confusion. In Dretske's book, information is introduced on the base of Shannon's theory [START_REF] Shannon | A Mathematical Theory of Communication[END_REF], explicitly interpreted as a formal theory that "does not say what information is".

In this note, I make two observations. The first is that it is possible to extract from the work of Shannon a purely physical version of the notion of information. Shannon calls its "relative information". I keep his terminology even if the ambiguity of these terms risks to lead to continue the misunderstanding; it would probably be better to call it simply 'correlation', since this is what it ultimately is: downright crude physical correlation.

The second observation is that the combination of this notion with Darwin's mechanism provides the ground for a definition of meaning. More precisely, it provides the ground for the definition of a notion of "meaningful infor-mation", a notion that on the one hand is solely built on physics, on the other can underpin intentionality, meaning, purpose, and is a key ingredient for agency.

The claim here is not that the full content of what we call intentionality, meaning, purpose -say in human psychology, or linguistics-is nothing else than the meaningful information defined here. But it is that these notions can be built upon the notion of meaningful information step by step, adding the articulation proper to our neural, mental, linguistic, social, etcetera, complexity. In other words, I am not claiming of giving here the full chain from physics to mental, but rather the crucial first link of the chain.

The definition of meaningful information I give here is inspired by a simple model presented by David Wolpert and Artemy Kolchinsky [START_REF] Wolpert | Observers as systems that acquire information to stay out of equilibrium[END_REF], which I describe below. The model illustrates how two physical notions, combined, give rise to a notion we usually ascribe to the nonphysical side of the gap: meaningful information.

The note is organised as follows. I start by a careful formulation of the notion of correlation (Shannon's relative information). I consider this a main motivation for this note: emphasise the commonly forgotten fact that such a purely physical definition of information exists. I then briefly recall a couple of points regarding Darwinian evolution which are relevant here, and I introduce (one of the many possible) characterisation of living beings. I then describe Wolpert's model and give explicitly the definition of meaningful information which is the main purpose of this note. Finally, I describe how this notion might bridge between the two sides of gap. I close with a discussion of the notion of signal and with some general considerations.

II. RELATIVE INFORMATION

Consider physical systems A, B, ... whose states are described by a physical variables x, y, ..., respectively. This is the standard conceptual setting of physics. For simplicity, say at first that the variables take only discrete values. Let N A , N B , ... be the number of distinct values that the variables x, y, ... can take. If there is no relation or constraint between the systems A and B, then the pair of system (A, B) can be in N A × N B states, one for each choice of a value for each of the two variables x and y. In physics, however, there are routinely constraints between systems that make certain states impossible. Let N AB be the number of allowed possibilities. Using this, we can define 'relative information' as follows.

We say that A and B 'have information about one another' if N AB is strictly smaller than the product N A × N B . We call

S = log(N A × N B ) -log N AB , (1)
where the logarithm is taken in base 2, the "relative information" that A and B have about one another. The unit of information is called 'bit'. For instance, each end of a magnetic compass can be either a North (N ) or South (S) magnetic pole, but they cannot be both N or both S. The number of possible states of each pole of the compass is 2 (either N or S), so N A = N B = 2, but the physically allowed possibilities are not N A × N B = 2 × 2 = 4 (N N, N S, SN, SS). Rather, they are only two (N S, SN ), therefore N AB = 2. This is dictated by the physics. Then we say that the state (N or S) of one end of the compass 'has relative information'

S = log 2 + log 2 -log 2 = 1 ( 2
)
(that is: 1 bit) about the state of the other end. Notice that this definition captures the physical underpinning to the fact that "if we know the polarity of one pole of the compass then we also know (have information about) the polarity of the other." But the definition itself is completely physical, and makes no reference to semantics or subjectivity.

The generalisation to continuous variables is straightforward. Let P A and P B be the phase spaces of A and B respectively and let P AB be the subspace of the Cartesian product P A × P B which is allowed by the constraints. Then the relative information is

S = log V (P A × P B ) -log V (P AB ) ( 3
)
whenever this is defined. 1Since the notion of relative information captures correlations, it extends very naturally to random variables. Two random variables x and y described by a probability distribution p AB (x, y) are uncorrelated if

p AB (x, y) = pAB (x, y) (4)
where pAB (x, y) is called the marginalisation of p AB (x, y) and is defined as the product of the two marginal distributions

pAB (x, y) = p A (x) p B (y), (5)
in turn defined by

p A (x) = p AB (x, y) dy, p B (y) = p AB (x, y) dx. (6)
Otherwise they are correlated. The amount of correlation is given by the difference between the entropies of the two distributions p A (x, y) and pA (x, y). The entropy of a probability distribution p being S = p log p on the relevant space. All integrals are taken with the Luoiville measures of the corresponding phase spaces.

Correlations can exist because of physical laws or because of specific physical situations, or arrangements or mechanisms, or the past history of physical systems.

Here are few examples. The fact that the two poles of a magnet cannot have the same polarisation is excluded by one of the Maxwell equations. It is just a fact of the world. The fact that two particles tied by a rope cannot move apart more than the distance of the rope is a consequence of a direct mechanical constraint: the rope. The frequency of the light emitted by a hot piece of metal is correlated to the temperature of the metal at the moment of the emission. The direction of the photons emitted from an object is correlated to the position of the object. In this case emission is the mechanism that enforces the correlation. The world teams with correlated quantities. Relative information is, accordingly, naturally ubiquitous.

Precisely because it is purely physical and so ubiquitous, relative information is not sufficient to account for meaning. 'Meaning' must be grounded on something else, far more specific.

III. SURVIVAL ADVANTAGE AND PURPOSE

Life is a characteristic phenomenon we observe on the surface of the Earth. It is largely formed by individual organisms that interact with their environment and embody mechanisms that keep themselves away from thermal equilibrium using available free energy. A dead organism decays rapidly to thermal equilibrium, while an organism which is alive does not. I take this -with quite a degree of arbitrariness-as a characteristic feature of organisms that are alive.

The key of Darwin's discovery is that we can legitimately reverse the causal relation between the existence of the mechanism and its function. The fact that the mechanism exhibits a purpose -ultimately to maintain the organism alive and reproduce it-can be simply understood as an indirect consequence, not a cause, of its existence and its structure.

As Darwin points out in his book, the idea is ancient. It can be traced at least to Empedocles. Empedocles suggested that life on Earth may be the result of random happening of structures, all of which perish except those that happen to survive, and these are the living organisms. 2 The idea was criticised by Aristotle, on the ground that we see organisms being born with structures already suitable for survival, and not being born at random ([6] II 8, 198b35). But shifted from the individual to the species, and coupled with the understanding of inheritance and, later, genetics, the idea has turned out to be correct. Darwin clarified the role of variability and selection in the evolution of structures and molecular biology illustrated how this may work in concrete. Function emerges naturally and the obvious purposes that living matter exhibits can be understood as a consequence of variability and selection. What functions is there because it functions: hence it has survived. We do not need something external to the workings of nature to account for the appearance of function and purpose.

But variability and selection alone may account for function and purpose, but are not sufficient to account for meaning, because meaning has semantic and intentional connotations that are not a priori necessary for variability and selection. 'Meaning' must be grounded on something else.

IV. KOLCHINSKY-WOLPERT'S MODEL AND MEANINGFUL INFORMATION

My aim is now to distinguish the correlations that are are ubiquitous in nature from those that we count as relevant information. To this aim, the key point is that surviving mechanisms survive by using correlations. This is how relevance is added to correlations.

The life of an organisms progresses in a continuous exchange with the external environment. The mechanisms that lead to survival and reproduction are adapted by evolution to a certain environment. But in general environment is in constant variation, in a manner often poorly predictable. It is obviously advantageous to be appropriately correlated with the external environment, because survival probability is maximised by adopting different behaviour in different environmental conditions.

A bacterium that swims to the left when nutrients are on the left and swims to the right when nutrients are on the right prospers; a bacterium that swims at random has less chances. Therefore many bacteria we see around us are of the first kind, not of the second kind. This simple observation leads to the Kolchinsky-Wolpert model [START_REF] Wolpert | Observers as systems that acquire information to stay out of equilibrium[END_REF],.

A living system A is characterised by a number of variables x n that describe its structure. These may be numerous, but are in a far smaller number than those describing the full microphysics of A (say, the exact position 2 [There could be] "beings where it happens as if everything was organised in view of a purpose, while actually things have been structured appropriately only by chance; and the things that happen not to be organised adequately, perished, as Empedocles says". of each water molecule in a cell). Therefore the variables x n are macroscopic in the sense of statistical mechanics and there is an entropy S(x n ) associated to them, which counts the number of the corresponding microstates. As long as an organism is alive, S(x n ) remains far lower than its thermal-equilibrium value S max . This capacity of keeping itself outside of thermal equilibrium, utilising free energy, is a crucial aspects of systems that are alive.

Living organisms have generally a rather sharp distinction between their state of being alive or dead, and we can represent it as a threshold S thr in their entropy. Call B the environment and let y n denote a set of variables specifying its state. Incomplete specification of the state of the environment can be described in terms of probabilities, and therefore the evolution of the environment is itself predictable at best probabilistically.

Consider now a specific variable x of the system A and a specific variable y of the system B in a given macroscopic state of the world. Given a value (x, y), and taking into account the probabilistic nature of evolution, at a later time t the system A will find itself in a configuration x n with probability p x,y (x n ). If at time zero p(x, y) is the joint probability distribution of x and y, the probability that at time t the system A will have entropy higher that the threshold is

P = dx n dx dy p(x, y) p x,y (x n )θ(S(x n ) -S thr ), ( 7
)
where θ is the step function. Let us now define P = dx n dx dy p(x, y) p x,y (x n )θ(S(x n ) -S thr ). (8) where p(x, y) is the marginalisation of p(x, y) defined above. This is the probability of having above threshold entropy if we erase the relative information. This is Wolpert's model.

Let's define the relative information between x and y contained in p(x, y) to be "directly meaningful" for B over the time span t, iff P is different from P . And call

M = P -P (9)
the "significance" of this information. The significance of the information is its relevance for the survival, that is, its capacity of affecting the survival probability. Furthermore, call the relative information between x and y simply "meaningful" if it is directly meaningful or if its marginalisation decreases the probability of acquiring information that can be meaningful, possibly in a different context.

Here is an example. Let B be food for a bacterium and A the bacterium, in a situation of food shortage. Let x be the location of the nurture, for simplicity say it can be either at the left of at the right. Let y the variable that describe the internal state of the bacterium which determines the direction in which the bacterium will move. If the two variables x and y are correlated in the right manner, the bacterium reaches the food and its chances of survival are higher. Therefore the correlation between y and x is "directly meaningful" for the bacterium, according to the definition given, because marginalising p(x, y), namely erasing the relative information increases the probability of starvation.

Next, consider the same case, but in a situation of food abundance. In this case the correlation between x and y has no direct effect on the survival probability, because there is no risk of starvation. Therefore the x-y correlation is not directly meaningful. However, it is still (indirectly) meaningful, because it empowers the bacterium with a correlation that has a chance to affect its survival probability in another situation.

A few observations about this definition: i. Intentionality is built into the definition. The information here is information that the system A has about the variable y of the system B. It is by definition information "about something external". It refers to a physical configuration of A (namely the value of its variable x), insofar as this variables is correlated to something external (it 'knows' something external).

ii. The definition separates correlations of two kinds: accidental correlations that are ubiquitous in nature and have no effect on living beings, no role in semantic, no use, and correlations that contribute to survival. The notion of meaningful correlation captures the fact that information can have "value" in a darwinian sense. The value is defined here a posteriori as the increase of survival chances. It is a "value" only in the sense that it increases these chances.

iii. Obviously, not any manifestation of meaning, purpose, intentionality or value is directly meaningful, according to the definition above. Reading today's newspaper is not likely to directly enhance mine or my gene's survival probability. This is the sense of the distinction between 'direct' meaningful information and meaningful information. The second includes all relative information which in turn increases the probability of acquiring meaningful information. This opens the door to recursive growth of meaningful information and arbitrary increase of semantic complexity. It is this secondary recursive growth that grounds the use of meaningful information in the brain. Starting with meaningful information in the sense defined here, we get something that looks more and more like the full notions of meaning we use in various contexts, by adding articulations and moving up to contexts where there is a brain, language, society, norms... iv. A notion of 'truth' of the information, or 'veracity' of the information, is implicitly determined by the definition given. To see this, consider the case of the bacterium and the food. The variable x of the bacterium can take to values, say L and R, where L is the variable conducting the bacterium to swim to the Right and L to the Left. Here the definition leads to the idea that R means "food is on the right" and L means "food is on the left". The variable x contains this information. If for some reason the variable x is on L but the food happens to be on the Right, then the information contained in x is "not true". This is a very indirect and in a sense deflationary notion of truth, based on the effectiveness of the consequence of holding something for true. (Approximate coarse grained knowledge is still knowledge, to the extent it is somehow effective. To fine grain it, we need additional knowledge, which is more powerful because it is more effective.) Notice that this notion of truth is very close to the one common today in the natural sciences when we say that the 'truth' of a theory is the success of its predictions. In fact, it is the same.

v. The definition of 'meaningful' considered here does not directly refer to anything mental. To have something mental you need a mind and to have a mind you need a brain, and its rich capacity of elaborating and working with information. The question addressed here is what is the physical base of the information that brains work with. The answer suggested is that it is just physical correlation between internal and external variables affecting survival either directly or, potentially, indirectly.

The idea put forward is that what grounds all this is direct meaningful information, namely strictly physical correlations between a living organism and the external environment that have survival and reproductive value. The semantic notions of information and meaning are ultimately tied to their Darwinian evolutionary origin. The suggestion is that the notion of meaningful information serves as a ground for the foundation of meaning. That is, it could offer the link between the purely physical world and the world of meaning, purpose, intentionality and value. It could bridge the gap.

V. SIGNALS, REDUCTION AND MODALITY

A signal is a physical event that conveys meaning. A ring of my phone, for instance, is a signal that means that somebody is calling. When I hear it, I understand its meaning and I may reach the phone and answer.

As a purely physical event, the ring happens to physically cause a cascade of physical events, such as the vibration of air molecules, complex firing of nerves in my brain, etcetera, which can in principle be described in terms of purely physical causation. What distinguishes its being a signal, from its being a simple link in a physical causation chain?

The question becomes particularly interesting in the context of biology and especially molecular biology. Here the minute working of life is heavily described in terms of signals and information carriers: DNA codes the information on the structure of the organism and in particular on the specific proteins that are going to be produced, RNA carries this information outside the nucleus, receptors on the cell surface signal relevant external condition by means of suitable chemical cascades. Similarly, the optical nerve exchanges information between the eye and the brain, the immune system receives information about infections, hormones signal to organs that it is time to do this and that, and so on, at libitum. We describe the working of life in heavily informational terms at every level. What does this mean? In which sense are these processes distinct from purely physical processes to which we do not usually employ an informational language? I see only one possible answer. First, in all these processes the carrier of the information could be somewhat easily replaced with something else without substantially altering the overall process. The ring of my phone can be replaced by a beep, or a vibration. To decode its meaning is the process that recognises these alternatives as equivalent in some sense. We can easily imagine an alternative version of life where the meaning of two letters is swapped in the genetic code. Second, in each of these cases the information carrier is physically correlated with something else (a protein, a condition outside the cell, a visual image in the eye, an infection, a phone call...) in such a way that breaking the correlation could damage the organism to some degree. This is precisely the definition of meaningful information studied here.

I close with two general considerations.

The first is about reductionism. Reductionism is often overstated. Nature appears to be formed by a relative simple ensemble of elementary ingredients obeying relatively elementary laws. The possible combinations of these elements, however, are stupefying in number and variety, and largely outside the possibility that we could compute or deduce them from nature's elementary ingredients. These combinations happen to form higher level structures that we can in part understand directly. These we call emergent. They have a level of autonomy from elementary physics in two senses: they can be studied independently from elementary physics, and they can be realized in different manners from elementary constituents, so that their elementary constituents are in a sense irrelevant to our understanding of them. Because of this, it would obviously be useless and self defeating to try to replace all the study of nature with physics. But evidence is strong that nature is unitary and coherent, and its manifestations are -whether we understand them or not-behaviour of an underlying physical world. Thus, we study thermal phenomena in terms of entropy, chemistry in terms of chemical affinity, biology in terms functions, psychology in terms of emotions and so on. But we increase our understanding of nature when we understand how the basic concept of a science are ground in physics, or are ground in a science which is ground on physics, as we have largely been able to do for chemical bonds or entropy. It is in this sense, and only in this sense, that I am suggesting that meaningful information could provide the link between different levels of our description of the world.

The second consideration concerns the conceptual structure on which the definition of meaningful information proposed here is based. The definition has a modal core. Correlation is not defined in terms of how things are, but in terms of how they could or could not be. Without this, the notion of correlation cannot be constructed. The fact that something is red and something else is red, does not count as a correlation. What counts as a correlation is, say, if two things can each be of different colours, but the two must always be of the same colour. This requires modal language. If the world is what it is, where does modality comes from?

The question is brought forward by the fact that the definition of meaning given here is modal, but does not bear on whether this definition is genuinely physical or not. The definition is genuinely physical. It is physics itself which is heavily modal. Even without disturbing quantum theory or other aspects of modern physics, already the basic structures of classical mechanics are heavily modal. The phase space of a physical system is the list of the configurations in which the system can be. Physics is not a science about how the world is: it is a science of how the world can be.

There are a number of different ways of understanding what this modality means. Perhaps the simplest in physics is to rely on the empirical fact that nature realises multiple instances of the same something in time and space. All stones behave similarly when they fall and the same stone behaves similarly every time it falls. This permits us to construct a space of possibilities and then use the regularities for predictions. This structure can be seen as part of the elementary grammar of nature itself. And then the modality of physics and, consequently, the modality of the definition of meaning I have given are fully harmless against a serene and quite physicalism.

But I nevertheless raise a small red flag here. Because we do not actually know the extent to which this structure is superimposed over the elementary texture of reality by ourselves. It could well be so: the structure could be generated precisely by the structure of the very 'meaningful information' we have been concerned with here. We are undoubtably limited parts of nature, and we are so even as understanders of this same nature.

[ 6 ]FIG. 1 .

 61 FIG.1. The Kolchinsky-Wolpert model and the definition of meaningful information. If the probability of descending to thermal equilibrium P increases when we cut the information link between A and B, then the relative information (correlation) between the variables x and y is "meaningful information".

Here V (.) is the Liouville volume and the difference between the two volumes can be defined as the limit of a regularisation even when the two terms individually diverge. For instance, if A and B are both free particles on a circle of of size L, constrained to be at a distance less of equal to L/N (say by a rope tying them), then we can easily regularise the phase space volume by bounding the momenta, and we get S = log N , independently from the regularisation.

ACKNOWLEDGMENTS

I thank David Wolpert for private communications and especially Jenann Ismael for a critical reading of the article and very helpful suggestions.