New geometries and dynamics at the heart of nature and the living? Towards a renewal of the philosophy of nature

Towards a new philosophy of nature?

1 In this article I will defend the idea that science needs a philosophy of nature. In order to do so, we must first identify the problems and shortcomings encountered in current science, and then specify what is meant by a “new philosophy of nature.”

2 The idea is precisely that science alone cannot give rise to fruitful theoretical developments without the contribution of a fundamental philosophical reflection on the origin, the meaning, and the scope of the concepts it employs in order to construct those abstract representations that seem the most apt to interpret and explain real phenomena. Historical experience actually shows that the constant free circulation between scientific concepts and philosophical ideas constitutes the necessary condition for important experimental discoveries. This conceptual coming together of science and philosophy seems all the more necessary at a time when the fundamental sciences are experiencing a very strong double tendency which seems to lead to a serious stagnation of theoretical thought and, more generally, of knowledge.

3 On the one hand, they are becoming increasingly hyperspecialized in the methods and techniques used, thus irretrievably losing an overall view of the phenomena and problems, of their deep relationships and connections.

4 The great physicist Richard Feynman remarked in this regard that the separation between the different fields of knowledge comes from human activity and, in this sense, it is conventional and sometimes arbitrary. [1] Indeed, nature has often taught us that this separation is not justified, and that the most interesting (and apparently most remote) phenomena are precisely those that break down the barriers between these same fields by showing us unsuspected and new connections.

5 But Paul Valéry, in a very beautiful text, [2] already drew our attention to the fact that the same natural phenomenon can only result from the concurrence of several elements and materials at the same time. He tells us that very probably during the growth of the mollusk and its shell, according to the ineluctable theme of the spiral helix, “are composed indistinctly and indivisibly all the constituents that the no less ineluctable form of the human act has taught us to consider and to define distinctly: forces, time, matter, connections, and the various ‘orders of magnitude’ between which our senses impose us to distinguish.” He then continues: “life goes back and forth from the molecule to the micelle, and from these to the sensitive masses, without having regard to the compartments of our sciences, i.e. of our means of action.” From this text emerges a true plea for the theoretical and empirical interdisciplinarity of knowledge, a passionate critique of the compartmentalization of our sciences, but also a warning of the risk of reducing science to their pragmatic efficiency and technological applicability.

6 For his part, the eminent mathematician and philosopher René Thom has repeatedly insisted on the need to reduce the gap that has widened over the centuries between science and philosophy, which can be done by restoring an essential place to thinking and theorizing, which must take precedence over mere observation and purely experimental applications. [3] The term “theorizing” means precisely the search for images or models that allow us to establish the necessary relations between objects and phenomena that are at first sight diverse and sometimes disparate, that is to say, to encompass each of them in one and the same theory, thanks to which we generally succeed in providing the sought-after explanation of what unites them, and also of what possibly differentiates them.

7 However, it seems to us that the mode of thought most apt to allow the elaboration of such an explanatory theory is geometry, in which we see a kind of archetypal form of knowledge that best reflects the universal and dynamic character of certain spatial and temporal principles according to which nature is organized. Which is to say, basically, that geometry is both rooted in (and linked to) reality and capable of producing new structures and properties in phenomena. It should be noted moreover that there are results of geometry which, in spite of their precise and technical content, contain in nuce innumerable possibilities of discovering new and surprising entities and structures in mathematics as well as in other scientific fields.

8 It is for these reasons that a new philosophy of nature must be essentially geometrical and dynamic. This is, in our view, an extremely topical vision, full of possibilities of what science should be today, which we will have to deepen and reformulate in very precise terms.

9 On the other hand, science is less and less a form of creative, conjectural and disinterested knowledge, and it depends more and more on other interests, economic, social, and political, completely external to its theories and methods, to its vital “space” of thought. We need only think of the pressure exerted by the pharmaceutical industry on certain important sectors of medical research, or of the power that the military and financial lobbies have over space research. Everyone can see that science today, to a large extent, is deprived of its characteristics and qualities, dispossessed of its autonomy, because of the excessive power that technoscience exercises over it, and also over other forms of knowledge.

10 To this I would add, finally, that there is something quite superficial and unsatisfactory in the fact of claiming at all costs that science is a social “enterprise” in the same vein as an economic, political, cultural, or other enterprise, and that it is therefore ipso facto deprived of any conceptual and/or ontological specificity and autonomy. Here again, we witness a desire, widespread in very diverse intellectual circles, to hyper-socialize or even ideologize scientific thought and to drown it in a kind of pseudo-savant undifferentiated magma, which often results in a total confusion between their contents and their methods, between their sources and their results, between their concepts and the use that is then made of them in different economic and social contexts. The confusion is total when one mixes up the rational quest for concepts and theories, a task proper to science, and the question, of a completely different nature, of the technological applications to which such a quest can possibly give rise.

11 This “sociologizing” position has, in my opinion, the major flaw of overestimating the social matrix and scope of science, and ends up in fact underestimating or even ignoring its other, deeper motivations and meanings, which are inherent to its raison d’être. The result is that it is reduced to a single dimension, the sociological one, while all the other dimensions—theoretical, empirical, but also philosophical, aesthetic, and educational—which are much more fundamental, are considered to be less important or even marginal.

12 In fact, this is a biased way of evading the question of meaning in science, which is inseparable from taking into account the dimensions we just mentioned. As obsolete as it may seem to many, the main purpose of science remains the creation and then the formal consolidation of new theories and new concepts, and the search for intelligible explanations of the nature and behavior of phenomena. We study and practice science, just as we study and practice philosophy or literature, not because it serves a purpose or because it allows us to achieve a professional or other goal, but rather because its study and practice form us, “open our eyes” to many aspects and phenomena that are invisible at first glance, and above all because it makes us feel an unparalleled joy of the spirit that can be communicated and shared with others: it is therefore an essential and profoundly human experience of life.

13 In view of this situation, I think that a new project of philosophy of nature must first seek to provide explanations of phenomena without contenting itself with describing them; observation and description, while certainly important, do not exhaust the work of scientific discovery and creation. A new philosophy of nature must, in other words, try to understand the relations of interdependence and co-determination between these same phenomena, which often appear very disparate. This concern to bring together problems and concepts only makes sense if it leads to finding a common explanation for heterogeneous phenomena. The discovery of this connection is a first fundamental step and opens up the path to the elaboration of a basic formalism sufficiently general to highlight a substratum common to a great variety of phenomena, an essential structural and qualitative analogy between them. [4]

14 The main objective of a new philosophy of nature seems to me to be the following: to explain the principles underlying the spatial and temporal actualization of the unity of matter and form in processes of a physical, biological, morphological, physiological, psychological and other nature.[5]

Diversity, creativity and Penrose’s aperiodic tilings

15 What has just been said is not a critique of heterogeneity or an underestimation of the importance of diversity in its various forms, whether in that of differentiation (for example of cells but also of individuals and species), complexity (particularly of those phenomena which can exhibit complex properties and behaviors even though they can be defined by a simple system of equations comprising a small number of parameters) or spontaneous self-organization, that is to say not determinable by means of a finite set of instructions or rules, in other words, of a program or an algorithm. [6] Suffice to think of what are called aperiodic shapes or structures in very varied fields: “aperiodic tilings” (or Penrose tilings) in mathematics, [7] “aperiodic crystals” in crystallography, “aperiodic orbits” in celestial mechanics, “aperiodic cycles or rhythms” in biology or physiology.

16 Let us consider here the first of these examples. A Penrose tiling is an aperiodic tiling pattern. By tiling we mean a covering or partitioning of the plane by elements of a finite set called tiles (polygons or other figures) which do not overlap, and aperiodic means that when we move a finite distance a tile having one of these shapes, without rotation, we do not obtain the same tiling, but a new one which, while resembling it, is not identically the same. This is because a Penrose tiling has no translational symmetry, although it can have both reflective and rotational symmetries. One can obtain a very large number of different variations of Penrose tilings by taking tiles of different shapes. The original tiling discovered by Penrose can be achieved by using two tile shapes: either two different rhombs, or two different quadrilaterals called “darts” and “kites”. Penrose tilings are obtained by applying certain rules to how these two shapes can be composed together. [8] This can be done in different ways: for example, by substitution tiling or by finite subdivision rules. Even with these constraints, each variation makes it possible to obtain an infinity of different Penrose tilings. One of the most remarkable properties of Penrose tilings is their self-similarity, i.e. they can be converted into equivalent Penrose tilings with tiles of another size, by the application of two processes called inflation and deflation. The shape represented by each finite arrangement of tiles in a Penrose tiling can repeat itself a finite number of times across the tiling. These shapes correspond to quasi-crystals: [9] a type of physical structure that is analogous to the mathematical Penrose tilings in the physical world; quasi-crystals exhibit diffraction patterns with peaks (of the curve) of Bragg (phenomenon which indicates the evolution of the loss of energy of ionizing radiation during their journey in matter) and a pentagonal symmetry. The essential point is that the structure of the tiling reveals the repetition of the patterns and the permanence of the tiles.

figure im1

17 It is clear, with this brief description of Penrose tilings, that many forms in the mathematical world and phenomena in physical reality obey two fundamental properties which, far from being incompatible, combine to generate an infinite variation of the same shape. Indeed, there can exist an infinity of different Penrose tilings (up to translation), and even an uncountable infinity. However, they cannot be distinguished locally. And, as we have seen, they are non-periodic: none can be obtained by regularly repeating a finite pattern on a square grid; they are however all quasi-periodic: if a pattern appears somewhere in a tiling, then it reappears at a bounded distance from any point of this tiling (which is true for all periodic tilings). In other words, they are relatively regular but not too regular… They also have a local symmetry: if a finite pattern appears in a tiling, then its image by a 36° rotation appears in this same tiling. This symmetry, at the same time as a certain imperfection in the regularity, is not foreign to the beauty of these structures.

18 Similarity and diversity together will thus create something new, and this something new has a certain permanence (the continuity of the motif) while including change (a certain variation in the rules of composition). We thus find two philosophical themes dear to Gilles Deleuze, difference and repetition, sources of inspiration and food for thought in one of his most important works. [10]

19 Knowing how to tell when two objects (whether physical, abstract or perceptual), two classes of objects or, to progress still further in abstraction, two categories of classes of objects are homologous, that is to say equivalent from the point of view of their essential qualitative properties, these are the kind of fundamental questions to which a philosophy of nature, as I understand it, must seek to provide the most precise answer possible, without however claiming to answer them exhaustively. The problem presents to me an obvious relevance and generality. Basically, it is a matter of knowing how we proceed to distinguish an object from other objects “inhabiting” our ambient space or, when it comes to abstract objects, in a substrate space (Rn in the most general case): is it possible, on the other hand, to establish a relation of equivalence, a homology, between the objects of the first type and those of the second type? And if so, what is the nature and possible extension of this homology? In other words, what criteria should the individuation of objects obey (at least objects extended in three-dimensional space)? And, moreover, on what general principles do we base the differentiation of natural and organic beings (for example, cell differentiation in young embryos—one of the fundamental problems of contemporary biology, directly linked to the question of processes underlying the genesis of forms)? However, the possibility of arriving at a better understanding of the genesis and organization of physical phenomena and biological beings depends largely on the answer that we will be able to give to such questions.

The role of mathematics in changing the physical world: morphological approach, chaos theory and complex systems

20 The question that we can now ask is the following: what is the role of mathematical concepts in the diversification of natural phenomena and in the appearance of new levels of organization of these same phenomena?

21 First, the wide variety of phenomena and levels of organization that we observe in matter and in living things must be explained by a few fundamental scientific discoveries that have occurred in the last thirty years. Among these, we must undoubtedly count the resumption and renewal of morphological methods (introduced by Goethe then developed by a few embryologists at the beginning of the last century, in particular by Hans Spemann and his school, then by the naturalist D’Arcy Thompson), the rise of the qualitative theory of dynamic systems and deterministic chaos by Henri Poincaré and by the Russian school (Pontryagin, Kolmogorov, Anosov, Arnold), and finally, the development in the 1970s (notably thanks to the work of I. Prigogine and his school on dissipative structures and irreversibility) of the theories of self-organization and emergence in physical phenomena and in biological systems, closely related to what is called complex systems today. It is not a matter here of going into the deep mathematical and physical ideas that underlie these three theories. Let us limit ourselves to highlighting a few particularly significant points.

22 The morphological level, which includes everything that involves the emergence and appearance of new structures, properties, and behaviors in a given substrate space (physical, chemical or biological) on which such and such a spatio-temporal transformation is made to act as a symmetry or a continuous and invertible (topological) deformation, such and such a dynamic variable like temperature or pressure, appears more and more as a level of structural organization constitutive of physical reality, of living matter and of the phenomenal world. But, as a result, it is also a source of signifying systems such as sensory reconstructions and perceptual representations, pictorial and literary thought, or even abstract language. Since, according to our meaning given above, morphology is the study of the principles and processes leading to the emergence and appearance of new forms, embryogenesis and morphogenesis are then two morphological disciplines par excellence which appear essential to understand the processes which are responsible for the formation of both global and specific, and therefore individuated, structures and functions of any living organism, be it human, animal, or plant.

23 In general, we are witnessing a rediscovery of the interest of the morphology of structures in biology. [11] Indeed, molecular biologists, generally reluctant to recognize the importance of the morphological level in the structural organization of living organisms, often quite naturally lead to the observation of morphology during development, of the concrete formation of the organism. From this point of view, it is all the same quite extraordinary to see extremely “in-depth” articles, in the field of molecular biology, presenting images showing the shapes and structures of embryos that embryologists of the late of the 19th century would not deny! This is one of the reasons why it seems important to me to redirect research in biology towards a reunification of its theories and methods. From its finest molecular bases, it “goes up” more and more towards the complex three-dimensional morphology, from the macromolecular level to the cellular level, then towards the complete organism, completed or in the process of development. But it is just as conceivable and sometimes more enlightening to follow the inverse explanatory model, that is to say, starting from the observation and study of global morphological structures to try to understand how they could have retroacted on specific molecular and cellular mechanisms, in the case of the occurrence of a physiological dysfunction or neurological disorder. [12]

24 As for chaos theories, they have profoundly changed our conception of space and perhaps even more that of time. At the origin of this fundamental change, there was the development of a geometric theory of dynamical systems. In particular, the introduction of the concept of phase space, where, simply speaking, to each trajectory of a body or a system corresponds a particular phase of their behavior which evolves over time, acted as a catalyst for this change. The first point to emphasize is certainly that, in the theory of dynamical systems, time acquires the status of a real dynamic variable equally important as the space variable. Indeed, it turns out to be one of the essential constituents allowing a qualitative understanding of the way in which the behavior of several classes of natural phenomena and living systems evolves. The concept of dynamic system is therefore not a simple modeling tool, but rather a magnificent example showing, on the one hand, the intimate relationship that exists between space, time, and the different modes of change (called, in technical language, phase transitions) affecting organic and inorganic phenomena, on the other, the creative and immanent nature of time. In other words, time acts on phenomena, by creating a dynamic with varied properties and structures, and directs their future evolution. This is reflected in the fact that many phenomena, whether they are physical, biological or perceptual, have a history and even multiple histories, each unfolding according to its own temporality.

25 One of the most considerable consequences of this evolution is a conception of Nature which no longer separates “with an ax” and in an absolute manner the living from the so-called inert nature, and this is thanks to taking into account of spatial transformations, of the temporal dimension and of non-linearities—source of a great diversity of forms and behaviors. [13] We can think that a certain disorder can be the reflection of an underlying order (at first sight unsuspected), or that disordered systems can evolve, by self-organizing and acquiring new symmetries, towards ordered systems. The great variety of phenomena and apparent forms that one encounters in nature and in the sensible world can thus manifest a continual transformation and evolution to which beings are subjected under the action of some major spatial and temporal principles. Chaos and complexity theory clearly shows that even the most apparent disorder actually hides a sometimes very rich underlying (mathematical and physical) order, and that to imagine reality and its change without principles, or chaotic phenomena without a dynamically constituted substrate, seems inconceivable. This is why the purpose of this new natural philosophy is to clarify the nature of these principles and the properties of these substrates.

26 I would like to point out, with regard to chaos theory, that one of the most important scientific and philosophical results that it has enabled us to understand is that, even if the laws of physics are deterministic (as is indeed the case in classical mechanics and, to a certain extent, in relativistic physics), they do not make it possible to predict the future of the universe and of nature down to the smallest details, for the fundamental reason that these are very complex systems, exhibit the phenomenon known as sensitivity to initial conditions. It is indeed enough that unforeseen factors intervene in our knowledge of the initial state of the system, so that the real evolution of the system deviates from our predictions in a considerable way. Thus determinism does not imply predictability, and the rigor of physical laws is not in contradiction with the contingency of certain facts. This sensitivity to initial conditions, which many physical, biological, or social systems present, means that it suffices to modify them slightly for the subsequent evolution of the system to change considerably. In other words, the fact of amplifying certain microscopic disturbances quickly produces important macroscopic changes. This is one of the characteristic properties, perhaps the most important, of the behavior of chaotic systems.

27 A second property can be summarized by saying that trajectories from neighboring points in phase space can diverge exponentially and only stay close to each other for a very short time. This resembles an operation of a topological nature of folding and stretching in phase space: the folding, even local, of a space brings closer the points, the curves which are traced there, while its stretching expands the distances between the points and moves the curves away from each other. At least two cases are possible: either the curves (or trajectories) come together again after a period of separation, or they will end up diverging ad infinitum.

28 The third fundamental property is that the behavior of a system can no longer be analyzed in terms of that of the elements that constitute it. Even though there are physical systems composed of a very large number of atoms or molecules (for example, in statistical mechanics) which indeed behave as the sum of their constituents, the only possible description, in these cases, seems to be, for the moment, of statistical type, and the properties which one reveals by this method do not involve associated geometrical objects like bifurcations, strange attractors, fractals or catastrophic singularities. This requires a new conceptual framework for describing and explaining the qualitative properties of systems, because it is sufficiently clear that even if one were to arrive, for example, at a complete “map” of brain cells or genes contained in our body, we could not draw the slightest conclusion on their behavior and, a fortiori, on ours. For the same reason, it is illusory to believe that a detailed description of all the fundamental forces will make it possible to definitively understand all of physics (to have a “theory of everything”).

29 In reality, in many scientific fields, in physics as well as in biology or in psychophysiology, we realize more and more that the interaction of components, at a given scale, sometimes translates, at a higher scale, into a complex overall behavior that cannot be accounted for from knowledge of the individual elements. This question of difference of scale and level (which must not be confused, because on the same scale of physical or biological reality, we can observe different levels of organization) must be properly posed in order to determine which theorization or precise mathematical modeling we need in every scientific field.

30 We therefore see, with these brief remarks, that it is generally impossible to give a description and above all a satisfactory explanation of the qualitative and global behavior by confining ourselves to purely quantitative models which are entirely based on methods of analytical approximation and calculations of statistical averages. It became increasingly clear that to account for certain qualitative and global properties of dynamical systems, whether chaotic or non-chaotic, conservative or dissipative, it was necessary to introduce geometric and topological objects and concepts of quite a different nature which, instead of numerically simulating their behavior, often give a deep picture of the characteristic properties and processes at the very heart of the genesis and evolution of the various systems and phenomena. A model is all the more significant when it turns out to be a concrete realization of a given phenomenology.

31 Let us now briefly address the issue of complexity and self-organization. The purpose of a theory of complexity consists in finding the laws which govern the behavior of systems involving a very large number of constituents and of interactions between these same constituents: these are phenomenological laws which could not be, in most cases, easily deducible from the laws which “control” each of the components of such and such a system. For example, the behavior of the neurons contained in our brain is, at least in certain aspects and for the most part, relatively well understood today, but we are very far from having understood the reasons why ten trillion neurons, connected to each other by a hundred trillion synapses, form a thinking brain. The transition from a neural network to thought cannot be of a purely quantitative or statistical nature, but it must necessarily involve a profound qualitative change in the substrate and the associated dynamics.

32 It is therefore more than reasonable to suppose that there is the emergence of certain new “collective” behaviors. This is a phenomenon that is already quite well known in physics and in particular in the theory of phase transitions. When the state of a system reaches a certain threshold (or critical point), for example the temperature in the case of the transition of the type water→ ice or water → steam, its behavior changes giving rise to new qualitative effects. This transformation of the physical system can be induced by the action of a dynamic parameter, which could also be called the determination of a variation (which will be progressive or sudden depending on the case) in the process; above or below this threshold, the number no longer produces observable significant physical effects. However, in the case of many biological, neurophysiological or even cognitive systems, the overall behavior of the system is much more complex than it is, for example, in statistical physics.

33 Let us specify further. The theory of complex systems is based on the idea that, if it is important to know the nature of the interactions between the constituents of a system, it is in fact even more important to know the global laws contributing to the emergence of collective behaviors. This is because the collective behavior of a system can remain invariant by slight modifications of the laws to which its constituents obey. One of the characteristics of complex systems is therefore to admit a fairly large number of different equilibrium states, in the sense that what does not change and always remains identical to itself over time does not present complexity, whereas a system will be said to be complex if it can take several different forms (and admit several states), while retaining a certain fundamental stability.

34 It is clear, from this point of view, that a biological organism is a complex system par excellence, because it goes through several different forms during evolution, each of which corresponds to a precise stage of its development. These different stages, however, seem to follow a general plan of organization or rather of self-organization of the organism, the main function of which is to ensure, within certain precise biochemical and morphological limits and constraints, its metabolism and its regeneration. One could say, in other words, that it is a “teleonomically” channeled system. [14] It should be noted in this regard that there is a major difference between knowing the elementary biochemical reactions of a living being and understanding its overall metabolism. [15]

35 The above considerations conceal an epistemologically fundamental point, which had already been underlined in particular by René Thom and Freeman Dyson. They both agree in recognizing the importance of morphogenetic processes and the overall metabolism of organisms. Thom writes as follows:


Starting from the genome in order to build the whole organism and its temporal evolution falls within the realm of belief. The correspondence between genotype and phenotype is a pure and simple black box of which we know only a few articulations, all in the direction genotype → phenotype, because those which go in the opposite direction would clash with the anti-Lamarckian dogmatism which currently reigns. […] DNA does not have the exclusivity of all information concerning humans. Genes certainly contain the outline of their own structure and that of the proteins, but not the whole of the morphogenetic information. This means that the genome is not the overall metabolism. It is only the fixed part of the latter. It is therefore the result of metabolism and not the reverse. Forms are dynamic structures linked to invariants. Admittedly, genes participate in overall morphogenesis; however, they are stabilized by morphogenesis itself. […] There is an aura of mechanisms that surrounds all form. […] The gene therefore participates in a more global dynamic structure. This is the meaning of the synthetic relationship between genes and forms. Let us also note that there is no reason to think that force has in principle a more important ontological status than that of form. [16]

37 As for Dyson, he criticizes Schrödinger’s position developed in What is Life? [17] and shows the limits of its reductionism in particular. He puts forward another hypothesis by recognizing the important role that the property of homeostasis plays in vital processes. His criticism targets in particular the central dogma of molecular biology partly inspired by the ideas of Schrödinger and partly anticipates the epigenetic revolution:


The Central Dogma [of molecular biology] says that genetic information is carried only by nucleic acids and not by proteins. […] According to the model, the first cells passed genetic information to their offspring in the form of proteins. There is no logical reason why a population of proteins mutually catalyzing each other’s synthesis should not serve as a carrier of genetic information. The question, how much genetic information can be carried by a population of molecules without exact replication, is intimately bound up with the question of the nature of homeostasis. Homeostasis is the preservation of the chemical architecture of a population in spite of variations in local conditions and in the numbers of molecules of various kinds. Genetic information is carried in the architecture and not in the individual components. But we do not know how to define architecture or how to quantify homeostasis. Lacking a deep understanding of homeostasis, we have no way to calculate how many items of genetic information the homeostatic machinery of a cell may be able to preserve. It seems to be true, both in the world of cellular chemistry and in the world of ecology, that homeostatic mechanisms have a general tendency to become complicated rather than simple. Homeostasis seems to work better with an elaborate web of interlocking cycles than with a small number of cycles working separately.

39 Dyson concludes with some general thoughts that resonate interestingly with the approach developed here:


I have been trying to imagine a framework for the origin of life, guided by a personal philosophy which considers the primal characteristics of life to be homeostasis rather than replication, diversity rather than uniformity, the flexibility of the cell rather than the tyranny of the gene, the error tolerance of the whole rather than the precision of the parts. […] I hold the creativity of quasi-random complicated structures to be a more important driving force of evolution than the Darwinian competition of replicating monads. [18]

41 It has already been mentioned that a certain type of complexity can emerge from the disordered nature, on a certain scale, of such and such a physical or biological phenomenon, and that, moreover, a system governed by simple laws can give rise to very complex dynamics. This means two things in particular. First, that in nature (but this is also true, in many cases, of human or animal societies) there are systems whose overall behavior can be extremely complex, while their fundamental constituents (or individual elements) are simple. Complexity, for many physical and biological systems, or even ecosystems, can therefore emerge as an effect of the cooperation between a very large number of constituents. [19] The fact that all these elements begin to interact according to certain rules and to exchange energy (as in the case of atoms) or functional signals (as in the case of macromolecules) or even information (cell communication), allows the appearance of new configurations and global effects that induce a characteristic qualitative behavior in the system. Moreover, this will subsist over time even if the elementary constituents or the individual elements of which the system is composed undergo slight disturbances or modifications. This qualitative change corresponds to what is called, in condensed matter physics, a phase transition, which corresponds to a change in the regime of the fundamental properties of an object, a system or an organism. Secondly, many complex systems admit what is called the property of universality, [20] which says that a phase transition affecting these same systems presents a global macroscopic behavior that does not depend on the specific model adopted, and this, although such and such an individual parameter, such as the temperature at which the phase transition occurs, depends on the model.

The importance of the theory of forms (Gestalttheorie) in psychology

42 The essential idea of the program of Gestalttheorie, as we know, is to consider the perception of objects and their forms as that of a phenomenal world endowed with a certain geometric organization. To understand how these forms are constituted, it is therefore necessary to analyze the way in which the perceptual structures are organized, and to explain the role played by certain fundamental geometric properties of the physical world and its objects. Recent developments in the fields of psychophysics and neurophysiology show clearly in fact that these geometric properties, instead of being an admittedly important but altogether accidental element of the phenomenal world, play an important role in the processes of constitution of our perceptions. [21]

43 The thinking of theorists of forms can be summarized schematically in the following points: (i) perception bears, from the beginning, the characteristics of space and time as its essential attributes; (ii) the characteristics specific to perception cannot be understood independently of the objective physical properties and laws which characterize phenomena as such in the natural world; (iii) certain geometric properties, such as connectedness, curvature, degree of symmetry and orientation with respect to the surrounding space and/or the observer, underlie the formation of the perception of objects in the space as “objective” phenomena, which can be apprehended in particular during a dynamic and intentional interaction of the subjects and their bodies with a larger natural and emotional environment.

44 This means that these geometric properties are necessary for the perceptions to assert themselves as autonomous units and coherent wholes, and to acquire a certain permanence and stability in relation to an ever-changing environment as well as to the flow of time. In other words, they are necessary to structure the pre-spatial phenomenal world. To this, it should perhaps be added that these geometric properties are in a way the source of a large number of physical saliences which, by the very fact of investing different objects located in our ambient space, produce different types of meaningful forms and, therefore, highly significant psychic and cognitive phenomena.

figure im2

45 The program of the first theoreticians of form was taken up and developed, among others, by the Italian psychologist Gaetano Kanisza [22] within the framework of his theory of subjective or “modal” contours and amodal completion, following Wolfgang Metzger. [23] The famous experiments of “subjective contours” (combinations of incomplete figures give rise to contours, clearly visible although not physically existing, which are therefore a creation of our visual system) prove that the perceived image is literally rewritten, at the point that only a directed attention of the subject makes it possible to make the difference between a subjective contour and a physically realized contour. The “modal” contours are therefore the only real contours (i.e. they have a real ontological status), and it is on them that the theory of perception must be based to study the form of the objects of perception. It is this same method of the more geometrico type that makes it possible to explain the fundamental phenomenon of amodal completion as a creative and productive practice, as discovered by the perception of certain phenomenal qualities of the world and of physical space.

46 Let us specify what amodal completion is by giving a few examples taken from the cited work by Kanisza. First, the term “amodal” refers to phenomenal qualities of space, such as contraction, extension, enclosure, etc., induced by supplementing the physical space of stimulation with geometric elements that are not part of it. Let us think of the example of the three squares whose central square covered by the black band seems to most observers less wide than the other two squares, which are entirely visible, and even looks more like a rectangle. The figure shows that this construction deforms not only the regions which have the character of a figure, but also those which take on the role of background: indeed, the empty space between the two squares here also seems narrower when it is covered by the black band only when fully visible. Other figures show that the formation of anomalous surfaces must be explained by the tendency of our visual system to complete the often discontinuous and dispersed environment, so as to create perceptual configurations endowed with a certain simplicity and stability and also greater symmetry and regularity. This operation of completion intervenes much more in the case where the figures are open, because the geometric closure is in itself a necessary property of perceptual completeness.

47 Let us consider a rather significant situation, less complex than others, which each of us can observe every day. There is a relationship of interdependence between the shape of the outline of any surface-object and its mode of appearance. For example, if we compare a shape with jagged contours and another with straight contours, we will notice that the colors differ in luminosity, in saturation and especially in their mode of appearance. This shows that replacing a straight edge (continuous) with an edge made of broken lines (discontinuous) causes a kind of dilation, a loss of cohesion and density of the surface. The meaning that should be attributed to such a phenomenon is most likely the following: certain characteristics, although not substantially modifying the shape of the closed surface, influence the perception of chromatic qualities.

48 All of these visual phenomena are interesting because of the theoretical consequences that can be drawn from them on the nature of visual space. This is not a static geometric diagram, nor the simple transposition on the perceptual level of the topographic arrangement of the stimuli on the retina. On the contrary, it must be considered as a highly dynamic event. Under certain experimental conditions, the dimensions and the articulation of the phenomenal space are rigorously determined by the intensity and the energetic distribution which derive from the type of stimulus, and from its distribution. In particular, when the stimulus is not very intense and homogeneous, the visual space is restricted and tends to adopt the smallest dimensions allowed by the play of dynamic forces in action. The structure that is created therefore obeys a “principle of minimization.”

49 Let us return to the idea already suggested that space-time is in some way embodied in our organism. This theoretical idea, which takes up the perspective of a geometric interpretation of the phenomenological approach developed by Husserl, [24] Stumpf, Merleau-Ponty [25] and certain gestaltists, consists in showing that the perceptual system and the central nervous system organize themselves, through the information obtained through the interactions they have with the physical world, incorporating a certain type of geometry. Thus, geometry could be seen as an intrinsic property specific to the organization and functioning of the central nervous system. In this way, the neural correlates of perception could have evolved into true geometric processes having internalized specific spatial properties. The question is therefore less about knowing if space is already included a priori in our brain prior to any relationship with reality, or if the brain is only an epiphenomenon of geometric and physical space, than about understanding if the nature and function of the structures of the different sensory systems and of the global perceptual system do not imply that certain fundamental properties of differential geometry and topology are (ontogenetically, phylogenetically and cognitively) incorporated into perception.


  • [1]
    Feynman [2004].
  • [2]
    Valéry [1937].
  • [3]
    Thom [1983] and [1991].
  • [4]
    See H. Weyl [1947], Thom [1977], and Boi [2011].
  • [5]
    Boi [2000] and [2020]
  • [6]
    See Boi [2020]
  • [7]
    See Penrose [1974]. A Penrose tiling consists of two tiles which, in sufficient quantity, cover the entire plane. It is an aperiodic tiling (it is not invariant by any translation) but quasi-periodic (any pattern appearing in the tiling reappears regularly).
  • [8]
    Penrose [1974] et [1979]. Also see Robinson [1971].
  • [9]
    In 1982, a diffraction pattern, which physicists were able to see thanks to the beam of an electron microscope passing through a sheet of a few hundred nanometers of a rapidly solidified aluminum and manganese alloy, revealed the hidden existence of a pentagonal symmetry in matter. This observation made it possible to discover a new type of object, since called quasi-crystals, which differs from crystals precisely by the fact of possessing a symmetry of order five, absent in crystals. See Senechal [1996].
  • [10]
    Deleuze [1969] and also [1988].
  • [11]
    See Bouligand [1980].
  • [12]
    See Boi [2017]
  • [13]
    Simondon [2005] and Boi [2005].
  • [14]
    See Waddington [1961].
  • [15]
    See Noble [2016].
  • [16]
    Thom [1983].
  • [17]
    Schrödinger [1944]
  • [18]
    Dyson [1985].
  • [19]
    See Kauffman [1993].
  • [20]
    Sapoval [2001].
  • [21]
    Boi, Kerszberg, and Patras [2007].
  • [22]
    Kanisza [1998].
  • [23]
    Metzger [1975].
  • [24]
    Husserl [1989].
  • [25]
    Merleau-Ponty [1945].