{"id":2644,"date":"2023-10-12T19:45:09","date_gmt":"2023-10-12T19:45:09","guid":{"rendered":"https:\/\/cch.um6p.ma\/?page_id=2644"},"modified":"2025-08-18T13:08:17","modified_gmt":"2025-08-18T13:08:17","slug":"glossaire-scientifique","status":"publish","type":"page","link":"https:\/\/cch.um6p.ma\/?page_id=2644","title":{"rendered":"Scientific Glossary"},"content":{"rendered":"

\"press\"Version : English <\/a>– <\/strong>Fran\u00e7aise<\/strong><\/a><\/p>\n

Excerpts translated by Nicolas Sperry-Guillou from R\u00e9da Benkirane, La Complexit\u00e9, vertiges et promesses. Histoires de sciences.<\/em> Nouvelle \u00e9dition, Benguerir, UM6P-Press, 2023.<\/p>\n

<\/a>[Algorithm<\/a>, Algorithmic information theory<\/a>, Belouzof-Zhabotinsky chemical reaction<\/a>, Boolean automata networks,<\/a> Brownian motion<\/a>, Cellular automaton<\/a>, Combinatorial optimization<\/a>, Cybernetics<\/a>, Entropy<\/a>, Feynman\u2019s sum<\/a>, Fitness landscape<\/a>, Frankfurt School,<\/a> Gauss Carl Friedrich<\/a>, Growth problems<\/a>, Hilbert David<\/a>, Incompleteness theorem<\/a>, Koch von, Helge<\/a>, Kuhn Thomas<\/a>, Laplace Pierre-Simon de<\/a>, Learning curves<\/a>, Lorenz Edward<\/a>, Mandelbrot Beno\u00eet<\/a>, Moore\u2019s Law<\/a>, Occam’s razor<\/a>, Power law<\/a>, Randomness<\/a>, Red Queen<\/a>, Regulatory genetic network<\/a>, Riemann Bernhard<\/a>, Schr\u00f6dinger equation<\/a>, Self-organized criticality<\/a>, Spin networks<\/a>, String theory<\/a>, Symmetry<\/a>, Tabula rasa<\/a>, Thermodynamics<\/a>, Traveling Salesman<\/a>, Turing effect<\/a>, Turing Machine,<\/a> Uncertainty principle<\/a>, Weierstrass Karl<\/a>]\n

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<\/a>Algorithm<\/strong><\/a>: the set of operating rules whose application enables a problem involving calculation to be solved by means of a finite number of operations.  The word comes from the name of the 9th-century Persian astronomer and mathematician Muhammad ibn Musa Al-Khawarizmi (c. 780-850, Latinized as algorithmus), who is credited with introducing algebra, the rules of arithmetic, and trigonometric tables to Europe. Al-Khawarizmi, a scholar from Baghdad’s school of Wisdom, is also the author of the first book devoted to algebra (Kitab al jabr wal muqabala) and a treatise on arithmetic (Al jam’ wal tafriq bil hisab al Hind, translated into Latin in the 12th  century), which sets out all the rules of calculation from India. It was this work, which included a chapter on calculations related to trading activities, that was instrumental in introducing the Indo-Arabic numbering system to societies still unfamiliar with written calculation.<\/p>\n

<\/a>Algorithmic information theory<\/a><\/strong>: Gregory Chaitin was one of the key architects of algorithmic information theory, alongside mathematicians Kolmogorov and Solomonoff.  Developed in the 1960s, this theory aims to determine the degree of complexity of an object or a mathematical statement by measuring the minimum amount of information required to generate it. From this general principle, this approach has focused on information compression and the calculation problems that may or may not be feasible in computer science. It was subsequently generalized to measure the information content of formal logical systems, which consist of sets of axioms and theorems. From this perspective, scientific theories themselves are viewed as algorithms and compressions of information that describe the complexity of natural phenomena. In the span of a few decades, algorithmic information theory has emerged as a universal tool for measuring complexity.<\/p>\n

<\/a>Belouzof-Zhabotinsky chemical reaction<\/a><\/strong>: one of the most typical examples of dissipative structures, i.e. structures that undergo variations over time. Named after two Russian scientists who observed it in the 1950s and 1960s, it was the first manifestation of self-organization in chemistry. For a long time, however, it was ignored by chemists all over the world, as their discipline essentially dealt with chemical reactions whose products proceeded from a stable, monotonous evolution. It wasn’t until the early 1970s that scientists began to take an interest in the spatio-temporal patterns of this chemical reaction. These patterns were studied by computer simulation, in particular using cellular automata, and it was found that they were surprisingly reminiscent of several biological phenomena, such as the dissipative structures found in a medium of amoebae (Dictyostelium discoideum), which have the particularity of being both unicellular and multicellular during their life cycle, or the activity of heart cells, which oscillate independently of each other during fibrillation (desynchronization of the cardiac rhythm that can lead to death). What these various physico-chemical phenomena have in common is not their components, but the dynamics of their interactions.<\/p>\n

<\/a>Boolean automata networks<\/a><\/strong>: sets of automata which, when randomly connected, generate particular collective behaviors.<\/p>\n

<\/a>Brownian motion<\/strong><\/a>: describes the erratic movement of microscopic particles in a fluid, and is named after the British botanist Robert Brown (1773-1858), who observed in 1828 that pollen grains suspended in water were subject to disorderly motion. Brownian motion, which in fact reflects the thermal agitation of atoms and molecules, was first studied by Albert Einstein in 1905 and then by French physicist Jean Perrin, who described it in 1913 in his book Les Atomes. It was mathematically conceptualized in 1923 by the American mathematician Norbert Wiener, in order to study other random physical processes.<\/p>\n

<\/a>Cellular automaton<\/a><\/strong>: modeling of the interaction of a large number of elements, whose individual behavior has been simplified to the extreme for study purposes. A cellular automaton evolves in a diagram representing its space-time, its state depending at each instant t on the state of its neighbors, according to a rule that can be modulated at will. The automaton calculates its state and that of its neighbors, applies the predefined rule, then determines its next state. Once all the cells have recalculated their respective states, the process starts again for each time step. In this way, we can demonstrate that, starting from extremely simple operating rules and initial conditions, cellular automata are capable of revealing varied, unpredictable and complex behavioral patterns. From their earliest conceptualizations, by mathematicians Stanislaw Ulam and John von Neumann, cellular automata were perceived as organisms disembodied from the physical world, evolving in parallel by simple calculation and proceeding from a purely logical basis. Since then, experiments with cellular automata have made it possible to simulate and better understand a whole range of complex phenomena, demonstrating in particular their aspects of growth, aggregation, reproduction, competition and evolution. The most exhaustive and accessible study for the general public is the book by British physicist Stephen Wolfram, who has devoted over twenty years to this subject (Stephen Wolfram, A New Kind of Science, Champaign, Wolfram Media, 2002). The Game of Life, invented by John Conway in 1970, is one of the most famous models of cellular automata.<\/p>\n

<\/a>Combinatorial optimization<\/a><\/strong>: an approach derived from complexity theory and theoretical computer science, which brings together a range of techniques for solving highly complex computational problems. Using various algorithms, it seeks to find the optimum solution from a finite number of choices. Combinatorial optimization can be used to model constrained decision problems (logistics, planning, production management, transport, etc.). The traveling salesman problem is a typical example of combinatorial optimization.<\/p>\n

<\/a>Cybernetics<\/a><\/strong>: the science of command and control, founded in the 1940s by mathematician Norbert Wiener. It studies organisms and machines using systemic approaches based on the theories of information, automata, and games. Cybernetics was a crucible of multidisciplinary research that gave birth to cognitive science, and in particular, artificial intelligence. The first wave of cybernetics corresponds to a mechanistic perception of systems, in which, for example, the brain is conceived of as a logical machine, while the second wave, which dates back to the 1960s, is inspired by the logic of living organisms and developed around the notions of self-organization, order and disorder.<\/p>\n

<\/a>Entropy<\/a><\/strong>: measure of the level of disorder in a physical system. According to the second law of thermodynamics, every system tends towards maximum entropy, and therefore towards increasing disorder, which is its final state.<\/p>\n

<\/a>Feynman\u2019s sum<\/a><\/strong>, named after the 1965 Nobel Prize-winning American physicist (1918-1988) who formulated it, is a procedure in quantum theory that allows us to consider all the possible paths a particle can take to get from one point to another.<\/p>\n

<\/a>Fitness landscape<\/a><\/strong>: mathematical metaphor introduced in 1931 by biologist Sewall Wright (1889-1988) to visualize the evolutionary dynamics of one or more organisms. The evolutionary landscape is a map of a space where each point in the space corresponds to a number that defines a fitness level. The surface of this landscape can thus be more or less “rough,” dotted with “adaptive peaks” separated by “valleys.” The highest peaks of this landscape correspond to the best aptitude. Between the peaks, organisms wander, subject to the constraints of contingency and selection. The mathematical modeling of the evolutionary landscape developed by Stuart Kauffman is now used in various disciplines and research activities (theoretical biology, turbulence physics, chemistry, genetic algorithms, combinatorial optimization, etc.) to describe complex problems involving a selection process in a more intuitive way.<\/p>\n

<\/a>Frankfurt School<\/strong><\/a>: current of thought that emerged in 1923 within the Institut f\u00fcr Sozialforschung at Frankfurt University. Founded by Max Horkheimer and Theodor Adorno, this movement proposed a critical theory of capitalism that sought to re-found the social sciences on the basis of the Marxist analytical framework. Fleeing Nazism in the 1930s, the members of the Frankfurt School emigrated to the United States and discovered American sociology. Gradually, they abandoned the Marxist dimension of their analysis and introduced psychoanalysis and other multidisciplinary approaches.  The main representatives of the Frankfurt School include Erich Fromm, Leo L\u00f6wenthal, Friedrich Pollock, Walter Benjamin, J\u00fcrgen Habermas, and Herbert Marcuse.<\/p>\n

<\/a>Gauss, Carl Friedrich (1777-1855)<\/a><\/strong>, a German mathematician, physicist, and astronomer, and Nikolai Ivanovich Lobachevsky (1792-1856), a Russian mathematician, were the discoverers of the first non-Euclidean geometries, geometries of curved surfaces where the parallel postulate is not verifiable.<\/p>\n

<\/a>Growth problems<\/a><\/strong>: refer to phenomena associated with the large-scale formation of various patterns of matter, such as sand dunes, distinctive cracks in dried clay soil, coral reefs, bacterial colonies, frost on a glass pane, snowflakes, etc. Born from processes like evolution, epidemics, freezing, percolation, or crystallization, these diverse physical and chemical growth phenomena are currently being studied through a common mathematical perspective. We employ computer models\u2014such as Diffusion Limited Aggregation (DLA)\u2014that allow us to represent these growth phenomena through simulations, revealing the formation of branching structures with fractal geometry. The fractal growth mode of the DLA type simulates a process of collaging identical particles or clusters of particles to explore the formation of dendrites found in neuronal structures and electric discharges, among other places. The principles of digital growth involve abstracting the physical processes at play in various phenomena to enhance our understanding of the universality of their properties.<\/p>\n

<\/a>Hilbert, David (1862-1943)<\/a><\/strong> was a German mathematician who addressed practically all fields of mathematics, but particularly logic, geometry, and number theory. His speech at the International Congress of Mathematicians in Paris in 1909 became famous: he presented a list of problems to be solved in the 20th century, and indeed, these problems were debated and developed by several generations of mathematicians. Following others (including Leibniz), David Hilbert attempted to unify mathematics with a “formalism program”, but in the 1930s, the work of logicians Kurt G\u00f6del and Alan Turing definitively ended Hilbert’s belief that any formalized problem in mathematical logic could be solved in principle.<\/p>\n

<\/a>Incompleteness theorem<\/a><\/strong>: formulated by Kurt G\u00f6del (1906-1978), an Austrian-American logician and mathematician, in 1931, it states that if a formal system containing elementary arithmetic is assumed to be consistent, meaning that an assertion and its opposite cannot be demonstrated simultaneously, then it is incomplete, which means the assertion cannot be proven true or false.<\/p>\n

<\/a>Koch von, Helge (1870-1924)<\/a><\/strong> was a Swedish mathematician who, in 1904, discovered a continuous, infinite curve with no bounded tangent within a finite area. This mathematical object, known as a “snowflake”, is obtained by successive iterations. Each segment of the curve is self-similar, i.e. the same structure (an equilateral triangle) is reproduced at all scales. The Koch flake is one of the most well-known examples of a fractal object. Koch’s curve, along with that of the Italian mathematician and logician Giuseppe Peano (1858-1932)\u2014which is of infinite length and is constructed by filling in all the points of a square\u2014was put forward at the beginning of the 20th century as an exception, a singularity signaling certain conceptual limits of mathematics. A century later, this type of fractal object is cited as a generic case of the structures and shapes produced by Nature.<\/p>\n

<\/a>Kuhn, Thomas<\/a><\/strong>: American philosopher and historian of science Thomas Kuhn proposed, in his book The Structure of Scientific Revolutions (University of Chicago Press, 1962), an approach to the history of science based on the notion of paradigm. For him, the history of science is not a cumulative process of knowledge, but an evolution punctuated by successive intellectual crises leading to scientific revolutions. Thus, Kuhn sees the succession of crises in the history of science as paradigm shifts, i.e. changes in the explanatory model shared by a community of researchers within a scientific discipline.<\/p>\n

<\/a>Laplace, Pierre-Simon de (1749-1827)<\/a><\/strong>,<\/a> was a French mathematician, physicist, and astronomer, who greatly contributed to the development of science and the modernization of its teaching during the French Revolution. An influential figure during the Empire and the Restoration, he is historically remembered for his famous response to Napoleon Bonaparte. When the Emperor asked him, “What place does God have in your system?” he retorted, “Sire, I have no need of that hypothesis,” thus expressing the triumphant determinism of his time. To further clarify his conception of a deterministic universe, Laplace proposed the idea of a “demon,” a superhuman intellect with unlimited calculating power, capable of reading the past and the future: “An intellect which at a certain moment would know all the forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movement of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.” Laplace’s demon reflects the science of the 19th century, but chaos theory, first demonstrated by the mathematician Henri Poincar\u00e9, and later quantum physics, would challenge this determinism that claimed to deduce perfectly the final conditions from the initial conditions of a system.<\/p>\n

<\/a>Learning curves<\/a><\/strong>:  models describing actual and projected improvements in solving problems such as inventories, estimating future costs, or determining optimum production rates.<\/p>\n

<\/a>Lorenz, Edward<\/a><\/strong> was a meteorologist from MIT (Massachusetts Institute of Technology) and is considered one of the first scientists of the contemporary era to have been interested in the study of chaos and non-linear dynamics. In 1963, he highlighted the chaotic nature of weather conditions and coined the metaphor \u201cbutterfly effect.\u201d While studying a model for a meteorological system on his computer, he happened to notice that by performing the same calculation twice, he obtained radically different results. In both simulations of his weather system, the initial conditions were nearly identical, but Lorenz eventually realized that, for his second calculation, he had accidentally introduced a slight difference in the initial conditions of the systems; this difference, on the order of 1 in 1000, corresponded to neglecting to write the last three digits of a number with six decimal places. This tiny difference in the initial conditions was sufficient to produce divergent final conditions. He concluded that meteorological conditions cannot be predicted long-term due to their \u201csensitive dependence on initial conditions.” Lorenz\u2019s discovery, initially published in the Journal of Atmospheric Sciences, went unnoticed until a certain number of physicists, as well as mathematicians, took an interest in it, providing the mathematical foundation for the study of chaotic phenomena.  <\/p>\n

<\/a>Mandelbrot, Beno\u00eet<\/a><\/strong>, a Polish-born French mathematician, conceptualized fractal geometry in the early 1960s, which proved to be closely related to deterministic chaos theory. Mandelbrot coined the term “fractal” from a Latin root, fractus, meaning “broken”, from frangere: “to break, to shatter”.<\/p>\n

<\/a>Moore\u2019s Law<\/a><\/strong>: an observation describing the development of information technology, formulated in the mid-sixties and named after the founder of the Intel microprocessor company. It predicts exponential growth in processor performance\u2014the computing power of a computer chip doubles approximately every eighteen months for the same cost. However, according to Moore’s Law, at the rate of growth in processor power and the miniaturization of transistors etched on silicon, a fundamental physical limit was reached at the threshold of 2020. From this threshold onwards, transistors are the size of a few silicon atoms, and the puzzling aspects of quantum physics come into play. Because of this physical limitation that component miniaturization inevitably encounters, scientific researchers are working on alternative technologies such as quantum, optical, and DNA computers.<\/p>\n

<\/a>Occam’s razor<\/a><\/strong>: a medieval philosophical principle named after the English theologian William of Occam, or d’Ockham (1290-1349), who formulated it. This principle states that “plurality should only be considered in cases of necessity” (pluralitas non est ponenda sine neccessitate). Scientists often use it as a reminder that, in the presence of several theories describing the same reality, the simplest is preferable. Occam’s razor is referred to by various names, such as the principle of simplicity, the principle of economy or the law of parsimony.<\/p>\n

<\/a>Power law<\/a><\/strong>: this law indicates that a variable has no characteristic size; consequently, events of any size can occur. Power laws are said to be scale-invariant and are most often observed during phase transitions.<\/p>\n

<\/a>Randomness<\/a><\/strong>: Chaitin provides a precise definition of this notion, characterized by two key aspects: a lack of structure or pattern, and the incompressibility of the information needed to generate it. According to this strict definition, a sequence of numbers is considered random if there is no more \u201ceconomical\u201d method to generate it than writing it out in full. Likewise, if a number can be expressed more concisely \u2013 for instance, by compacting its redundancies with a compression technique \u2013 then it is not random, as there is a structure that allows for its generation. From this perspective, pi is not a random number, since it can be described and reproduced using words (pi is the \u201cratio of the circumference of a circle to its diameter\u201d) or through a concise computer program.<\/p>\n

<\/a>Red Queen<\/a><\/strong>: a metaphor proposed in 1973 by American evolutionary biologist Leigh Van Valen in which each organism participates in the evolutionary context of other organisms, with the evolutionary landscape thus moving towards increasing complexity. Any species that stops this perpetual evolutionary race is condemned to extinction, according to the principle that you have to run just to stay in the same place. This metaphorical image refers to Lewis Carroll’s Through the Looking Glass, where the Red Queen pulls Alice by the hand and makes her run endlessly while the landscape remains motionless. The Red Queen explains that \u201cNow, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!\u201d<\/p>\n

<\/a>Regulatory genetic network<\/strong><\/a>: theory developed by Stuart Kauffman that proposes a scheme to explain cell differentiation. The idea is to understand how the different cell types required by an organism (256 distinct cells in the case of humans) can be generated from a single genome. During cell differentiation, the set of activated genes varies depending on whether the set specifies a muscle cell, a heart cell, or a neuron.  Kauffman’s simulation is based on a Boolean automaton network in which gene activity is modeled in a binary fashion (each gene being active or inactive). It shows that the system, in fact, stabilizes a number of attractors corresponding to the different cells defined by the genetic program. Kauffman estimated that for a genetic network of 100,000 genes (an estimate, at the time\u20141993\u2014of the size of the human genome), there would theoretically be 2,100,000 possible cell configurations. This staggering number shows that nature proceeds differently to determine a limited number of distinct cell types. Kauffman has calculated that the number of stabilized attractors or cells increases as a function of the square root of the total number of genes: for a network of 100,000 genes, this means 317 attractors or cells. This result produced by Kauffman’s simplified genetic network is relatively close to the 256 human cell types.<\/p>\n

<\/a>Riemann, Bernhard (1826-1866)<\/a><\/strong> was a German mathematician and one of the first, along with Russian mathematician Nikolai Lobachevsky (1792-1856), to develop non-Euclidean geometries, i.e. curved surfaces where the sum of angles is not equal to 180\u00b0 and Euclid’s fifth postulate (that two parallel lines never intersect) is not verified. This discovery, in the last quarter of the 19th century, of the existence of non-Euclidean geometries led to a conceptual crisis comparable to the discovery of deterministic chaos theory a century later. Riemann’s geometry, which is elliptical whereas Lobachevsky’s is hyperbolic, studies curved spaces with a variable number of dimensions; it has had a decisive influence on theoretical physics, particularly on the theory of general relativity and the description of space-time.<\/p>\n

<\/a>Schr\u00f6dinger equation<\/a><\/strong>:<\/a>  describes the evolution of the probability wave of an electron (or its wave function) and is one of the foundations of quantum theory. This equation is named after the Austrian physicist Erwin Schr\u00f6dinger (1887-1961), who sought to define a mathematical framework to describe the wave-particle duality of the electron.<\/p>\n

<\/a>Self-organized criticality<\/strong><\/a>: concept put forward in the late 1980s by the Danish physicist Per Bak at the Niels-Bohr Institute, notably through his “sandpile model”, to shed light on macroscopic phenomena such as earthquakes, Nile floods, stock market crashes, and so on. Self-organized criticality and the sand pile metaphor are used to study the links between certain quantitative and qualitative behaviors typical of complexity.<\/p>\n

<\/a>Spin networks<\/a><\/strong>: simplified physics models describing the disordered behavior of tiny magnets. They are used in a multidisciplinary manner.<\/p>\n

<\/a>String theory<\/a><\/strong>: new field of theoretical physics in which the fundamental objects are extremely small strings, on the order of Planck length (10-20, or a hundred billion billion times smaller than a hydrogen nucleus), whose mode of vibration is thought to be at the origin of the different particles of matter. Each mode of vibration would correspond to the mass and charge of a subatomic particle. This theory aims to unify quantum theory and general relativity, but in the opinion of all the researchers working on it, it would take several decades to develop the highly complex mathematics that would govern it.<\/p>\n

<\/a>Symmetry<\/a><\/strong>: notion that refers to either symmetrical structures that are more likely to be favored by natural selection due to their stability and permanence, or to invariance properties of an object, system, or theory subjected to a transformation (for example, a sphere presents a rotation symmetry, meaning it remains unchanged when it turns on itself). The notion of symmetry is increasingly being used, sometimes to replace the concept of law, and it is usually coupled with the notion of symmetry breaking (The Big Bang is an example of symmetry breaking, because the primordial explosion corresponds to an absolute singularity). Symmetry tends to describe physical properties of regularity, invariance, conservation, and equivalence, while symmetry breaking refers to natural phenomena of complexity, singularity, criticality, and instability. The categories of symmetry and symmetry breaking allow a scientific theory to understand reality better. When coupled together, they signal the dialectic richness that is comparable to that of categories such as finite\/infinite, discrete\/continuous, local\/global.<\/p>\n

<\/a>Tabula rasa<\/a><\/strong>:  philosophical notion defined, among others, by the English empiricist philosopher John Locke (1632-1704) to designate the mind and the acquisition of experience and knowledge. According to Locke, a child’s mind is a tabula rasa (or blank slate) on which all cognitive experiences are gradually inscribed. This idea was proposed to oppose the notion of innate knowledge defended by other philosophers.<\/p>\n

<\/a>Thermodynamics<\/a><\/strong>: science that studies physical systems subjected to variations in temperature, energy (heat and work) and entropy (a measure of the system’s degree of disorder). The First Principle of Thermodynamics postulates the conservation of energy, while the Second Principle asserts that any system evolves over time towards an increasing disorder corresponding to the system’s final state of equilibrium. Its entropy is then said to be maximal.<\/p>\n

<\/a>Traveling Salesman<\/a><\/strong>: this problem is easy to state but practically impossible to solve. A traveling salesman needs to visit n number of cities. Knowing the distance between the different cities, he must determine the optimal itinerary (in terms of fuel costs)\u2014in other words, the shortest possible distance he must travel to visit all the cities. If we examine each itinerary one by one, the time required to find the solution can quickly become astronomical, as it increases exponentially with the size of the problem (as the number of cities increases). For 100 cities, the factorization would take billions of years! The traveling salesman problem is a case study of \u201ccombinatorial optimization\u201d\u2014a type of calculation that requires the development of satisfactory approximations given the size and complexity of the problem.<\/p>\n

<\/a>Turing effect<\/a><\/strong>: refers to the “imitation game” presented by British logician Alan Turing in his article “Computing Machinery and Intelligence” (Mind, vol. 59, no. 236, 1950). This article, which is in a way the birth certificate of artificial intelligence, describes a device for determining whether machines can think.<\/p>\n

<\/a>Turing Machine<\/strong><\/a>: an abstract, ideal machine, and the origin of the theoretical foundations of computer science. It describes and formalizes the operating principle of the computer and the algorithm, i.e., the sequence of instructions leading to the solution of a computational problem. This imaginary device is named after the British logician Alan Turing (1912-1954), who introduced it in a landmark paper published in 1936, “On Computable Numbers, with an Application to the Entscheidungsproblem” (Proceedings of the London Mathematical Society, series 2, vol. 42, 1936-1937, pp. 230-265), one of the most important mathematical writings on the theory of computability. The Turing machine is a modified “typewriter” with a tape head and an infinite ribbon divided into cells, which is its memory.  The tape head has a finite number of states, can move from one square to another, left or right, and can read, write and erase symbols inscribed on these squares. Turing conceptualized it to demonstrate that an automaton programmed in this way could solve any computable problem. Any problem is said to be computable if a Turing machine can calculate it.<\/p>\n

<\/a>Uncertainty principle<\/a><\/strong>, stated in 1925 by the German physicist Werner Heisenberg (1901-1976), posits that one cannot precisely and simultaneously measure the position and velocity of a subatomic particle. According to this essential principle of quantum theory, due to the scales and fluctuations of the microscopic world, one can only know one of these two physical quantities with exact precision. Position and velocity are mathematically linked in an inversely proportional manner: the greater the precision for one, the greater the imprecision for the other.<\/p>\n

<\/a>Weierstrass, Karl (1815-1897)<\/a><\/strong> was a German mathematician who demonstrated in 1872 that certain functions describing a continuous, non-tangent and non-derivable curve had a property of self-similarity. The curves of Weierstrass, Koch and Peano, and Cantor’s set were, at the time of their appearance, perceived as “pathological cases”, even though they represent the first developments in fractal mathematics.<\/p>\n

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Extrait de R\u00e9da Benkirane, La Complexit\u00e9, vertiges et promesses. Histoires de sciences.<\/em> Nouvelle \u00e9dition, Benguerir, UM6P-Press, 2023.<\/p>\n

\n<\/a>[Algorithme<\/a>, Automate<\/a>, Courbes d\u2019apprentissage<\/a>, Criticalit\u00e9 auto-organis\u00e9e<\/a>, \u00c9cole de Francfort<\/a>, Effet Turing<\/a>, \u00c9quation de Schr\u00f6dinger<\/a>, Entropie<\/a>, Gauss Carl Friedrich<\/a>, Hasard<\/a>, Hilbert David<\/a>, Koch von Helge<\/a>, Kuhn Thomas<\/a>, Laplace de Pierre Simon<\/a>, Lorenz Edward<\/a>, Loi de Moore<\/a>, Loi de puissance<\/a>, Machine de Turing<\/a>, Mandelbrot Beno\u00eet<\/a>, Mouvement brownien<\/a>, Optimisation combinatoire<\/a>, Paysage \u00e9volutif ou adaptatif<\/a>, Principe d\u2019incertitude<\/a>, Probl\u00e8mes de croissance<\/a>, Rasoir d\u2019Occam<\/a>, R\u00e9action chimique de Belouzof-Zhabotinsky<\/a>, Reine rouge<\/a>, R\u00e9seaux d\u2019automates bool\u00e9ens<\/a>, R\u00e9seaux de spins<\/a>, R\u00e9seau g\u00e9n\u00e9tique r\u00e9gulateur<\/a>, Riemann Bernhard<\/a>, Sym\u00e9trie<\/a>, Somme de Feynman<\/a>, Tabula rasa<\/em><\/a>, Th\u00e9orie des cordes<\/a>, Th\u00e9or\u00e8me de l\u2019incompl\u00e9tude<\/a>, Th\u00e9orie algorithmique de l\u2019information<\/a>, Thermodynamique<\/a>, Voyageur de commerce<\/a>, Weierstrass Karl<\/a>]\n


\n<\/strong><\/a>A<\/a><\/strong>lgorithme<\/a> : <\/strong> l\u2019ensemble des r\u00e8gles op\u00e9ratoires dont l\u2019application permet de r\u00e9soudre, au moyen d\u2019un nombre fini d\u2019op\u00e9rations, un probl\u00e8me impliquant du calcul. Le mot vient du nom de l\u2019astronome et math\u00e9maticien perse du ixe si\u00e8cle Muhammad ibn Musa Al- Khawarizmi (env. 780-850, latinis\u00e9 en algorithmus<\/em>), \u00e0 qui l\u2019on doit l\u2019introduction en Europe de l\u2019alg\u00e8bre, des r\u00e8gles de l\u2019arithm\u00e9tique et des tables trigonom\u00e9triques. Al-Khawarizmi, savant de l\u2019\u00e9cole de la Sagesse de Bagdad, est entre autres l\u2019auteur du premier livre consacr\u00e9 \u00e0 l\u2019alg\u00e8bre (Kitab al jabr wal muqabala<\/em>) et d\u2019un trait\u00e9 d\u2019arithm\u00e9tique (Al<\/em> jam\u2019 wal tafriq bil hisab al Hind<\/em>, traduit en latin au xiie si\u00e8cle) qui expose l\u2019ensemble des r\u00e8gles de calcul en provenance d\u2019Inde. C\u2019est essentiellement cet ouvrage, dont un chapitre est consacr\u00e9 au calcul li\u00e9 aux activit\u00e9s marchandes, qui va permettre de faire conna\u00eetre le syst\u00e8me de num\u00e9ration indo-arabe dans des soci\u00e9t\u00e9s qui ignorent encore le calcul \u00e9crit.<\/p>\n

<\/a>Automate cellulaire<\/a><\/strong> :  mod\u00e9lisation de l\u2019interaction d\u2019un grand nombre d\u2019\u00e9l\u00e9ments dont le comportement individuel est simplifi\u00e9 \u00e0 l\u2019extr\u00eame pour en permettre l\u2019\u00e9tude. Un automate cellulaire \u00e9volue dans un diagramme o\u00f9 figure son espace-temps, son \u00e9tat d\u00e9pend \u00e0 chaque instant t de l\u2019\u00e9tat de ses voisins, selon une r\u00e8gle que l\u2019on peut moduler \u00e0 l\u2019envi. L\u2019automate calcule son \u00e9tat et celui de ses voisins, applique la r\u00e8gle pr\u00e9d\u00e9finie, puis d\u00e9termine son \u00e9tat suivant. Une fois que toutes les cellules ont recalcul\u00e9 leurs \u00e9tats respectifs, le processus recommence pour chaque pas de temps. Ainsi peut-on d\u00e9montrer qu\u2019\u00e0 partir de r\u00e8gles de fonctionnement et de conditions initiales extr\u00eamement simples, les automates cellulaires sont susceptibles de r\u00e9v\u00e9ler des motifs comportementaux vari\u00e9s, impr\u00e9visibles et complexes. D\u00e8s les premi\u00e8res conceptualisations, dues aux math\u00e9maticiens Stanislaw Ulam et John von Neumann, les automates cellulaires ont \u00e9t\u00e9 per\u00e7us comme des organismes d\u00e9sincarn\u00e9s du monde physique, \u00e9voluant en parall\u00e8le par simple calcul et proc\u00e9dant d\u2019une base purement logique. Depuis lors, les exp\u00e9riences sur automates cellulaires ont permis de simuler et de mieux comprendre toute une gamme  de ph\u00e9nom\u00e8nes complexes, en manifestant notamment leurs aspects de croissance, d\u2019agr\u00e9gation, de reproduction, de comp\u00e9tition et d\u2019\u00e9volution. L\u2019\u00e9tude la plus exhaustive et la plus accessible pour le grand public est le livre du physicien britannique Stephen Wolfram, qui a consacr\u00e9 plus de vingt ans \u00e0 ce sujet (Stephen Wolfram, A New Kind of Science, <\/em>Champaign, Wolfram Media, 2002). Invent\u00e9 par John Conway en 1970, le Jeu de la vie <\/em>est l\u2019un des mod\u00e8les d\u2019automate cellulaire les plus connus.<\/p>\n

<\/a>Courbes d\u2019apprentissage<\/a><\/strong> (learning curves<\/em>) : mod\u00e8les d\u00e9crivant les am\u00e9liorations r\u00e9elles et projet\u00e9es dans la r\u00e9solution de probl\u00e8mes tels que les inventaires, les estimations de co\u00fbts futurs ou la d\u00e9termination des taux de production optimaux.<\/p>\n

<\/a>Criticalit\u00e9 auto-organis\u00e9e<\/a> : <\/strong>concept \u00e9nonc\u00e9 \u00e0 la fin des ann\u00e9es 1980 par le physicien danois Per Bak, au Niels-Bohr Institute, notamment au travers de son \u00ab mod\u00e8le du tas de sable \u00bb, pour \u00e9clairer les ph\u00e9nom\u00e8nes macroscopiques tels que les tremblements de terre, les crues du Nil, les krachs boursiers, etc. La criticalit\u00e9 auto-organis\u00e9e et la m\u00e9taphore du tas de sable sont utilis\u00e9es pour \u00e9tudier les liens entre certains comportements quantitatifs et qualitatifs typiques de la complexit\u00e9.<\/p>\n

<\/a>\u00c9cole de Francfort<\/a><\/strong> : courant de pens\u00e9e n\u00e9 en 1923 dans le cadre de l\u2019Institut f\u00fcr Sozialforschung <\/em>de l\u2019universit\u00e9 de Francfort. Fond\u00e9 par Max Horkheimer et Theodor Adorno, ce courant propose une th\u00e9orie critique du capitalisme qui cherche \u00e0 refonder les sciences sociales \u00e0 partir du cadre d\u2019analyse marxiste. Fuyant le nazisme dans les ann\u00e9es trente, les membres du groupe de Francfort \u00e9migrent aux \u00c9tats-Unis et d\u00e9couvrent la sociologie am\u00e9ricaine. Petit \u00e0 petit, ils vont abandonner la dimension marxiste de leur analyse et introduire la psychanalyse ainsi que d\u2019autres approches pluridisciplinaires.  Les principaux repr\u00e9sentants de l\u2019\u00e9cole de Francfort sont Erich Fromm, Leo L\u00f6wenthal, Friedrich Pollock, Walter Benjamin, J\u00fcrgen Habermas et Herbert Marcuse.<\/p>\n

<\/a>Effet Turing<\/a> <\/strong>: renvoie au \u00ab jeu de l\u2019imitation \u00bb que le logicien britannique Alan Turing a pr\u00e9sent\u00e9 dans son article Pens\u00e9e et Machine <\/em>(\u00ab Computing Machinery and Intelligence \u00bb, Mind, vol. 59, n\u00b0 236, 1950). Cet article, qui constitue en quelque sorte l\u2019acte de naissance de l\u2019intelligence artificielle, d\u00e9crit un dispositif o\u00f9 il s\u2019agit de d\u00e9terminer si les machines peuvent penser.<\/p>\n

<\/a>\u00c9quation de Schr\u00f6dinger<\/a><\/strong> : d\u00e9crit l\u2019\u00e9volution de l\u2019onde de probabilit\u00e9 d\u2019un \u00e9lectron (ou sa fonction d\u2019onde), elle constitue l\u2019un des fondements de la th\u00e9orie quantique. Cette \u00e9quation porte le nom du physicien autrichien Erwin Schr\u00f6dinger (1887-1961) qui a cherch\u00e9 \u00e0 d\u00e9finir un cadre math\u00e9matique descriptif de la dualit\u00e9 onde\/corpuscule de l\u2019\u00e9lectron.<\/p>\n

<\/a>Cybern\u00e9tique<\/a><\/strong> :  science de la commande et du contr\u00f4le fond\u00e9e dans les ann\u00e9es 1940 par le math\u00e9maticien Norbert Wiener. Elle \u00e9tudie les organismes et les machines en recourant aux approches syst\u00e9miques des th\u00e9ories de l\u2019information, des automates et des jeux. La cybern\u00e9tique a constitu\u00e9 un creuset de recherches multidisciplinaires \u00e0 l\u2019origine des sciences cognitives, et en particulier de l\u2019intelligence artificielle. Le premier courant cybern\u00e9tique correspond \u00e0 une perception m\u00e9caniste des syst\u00e8mes o\u00f9 par exemple, le cerveau est appr\u00e9hend\u00e9 comme une machine logique, tandis que le second courant, qui remonte aux ann\u00e9es 1960, s\u2019inspire des logiques du vivant et s\u2019est d\u00e9velopp\u00e9 autour des notions d\u2019auto-organisation, d\u2019ordre et de d\u00e9sordre.<\/p>\n

<\/a>Entropie<\/a> <\/strong>:  mesure du niveau de d\u00e9sordre au sein d\u2019un syst\u00e8me physique. Selon le second principe de la thermodynamique, tout syst\u00e8me tend vers une entropie maximale et donc vers un d\u00e9sordre croissant qui correspond \u00e0 son \u00e9tat final.<\/p>\n

<\/a>Gauss,<\/a><\/strong> <\/a>Carl Friedrich<\/a> <\/strong>(1777-1855), math\u00e9maticien, physicien et astronome allemand, fut avec Nikola\u00ef Ivanovitch Lobatchevski (1792-1856), math\u00e9maticien russe, un des d\u00e9couvreurs des premi\u00e8res g\u00e9om\u00e9tries non euclidiennes, g\u00e9om\u00e9tries des surfaces courbes o\u00f9 le postulat des parall\u00e8les n\u2019est pas v\u00e9rifiable.<\/p>\n

<\/a>Hasard<\/a> <\/strong>: une d\u00e9finition tr\u00e8s pr\u00e9cise de cette notion est propos\u00e9e par le math\u00e9maticien et informaticien Gregory Chaitin correspondant, d\u2019une part, \u00e0 un d\u00e9faut de structure ou de pattern, et d\u2019autre part, \u00e0 une incompressibilit\u00e9 de l\u2019information n\u00e9cessaire pour le g\u00e9n\u00e9rer. Selon cette d\u00e9finition restrictive, une suite de nombres est dite al\u00e9atoire si, pour l\u2019engendrer, il n\u2019existe pas de moyen plus \u00ab \u00e9conomique \u00bb que de l\u2019\u00e9crire int\u00e9gralement. De m\u00eame, si un nombre peut s\u2019\u00e9crire de fa\u00e7on plus concise \u2013 par exemple en compactant ses redondances par une technique de compression \u2013, alors celui-ci n\u2019est pas al\u00e9atoire \u2013 puisqu\u2019il existe une structure permettant de l\u2019engendrer. De ce point de vue, pi n\u2019est pas un nombre al\u00e9atoire puisqu\u2019on peut le d\u00e9crire et le reproduire par des mots (pi est le \u00ab rapport de la circonf\u00e9rence d\u2019un cercle \u00e0 son diam\u00e8tre \u00bb) ou un programme informatique concis.<\/p>\n

<\/a>Hilbert,<\/a><\/strong> <\/a>David<\/a> <\/strong>(1862-1943), math\u00e9maticien allemand, a abord\u00e9 pratiquement tous les domaines des math\u00e9matiques, mais plus particuli\u00e8rement la logique, la g\u00e9om\u00e9trie et la th\u00e9orie des nombres. Son discours, en 1909, au Congr\u00e8s international des math\u00e9maticiens de Paris est rest\u00e9 c\u00e9l\u00e8bre : il y exposait une liste des probl\u00e8mes \u00e0 r\u00e9soudre pour le XXe si\u00e8cle, et effectivement, ces probl\u00e8mes seront d\u00e9battus, d\u00e9velopp\u00e9s par plusieurs g\u00e9n\u00e9rations de math\u00e9maticiens. David Hilbert, apr\u00e8s d\u2019autres (dont Leibniz), a tent\u00e9 d\u2019unifier les math\u00e9matiques en les armant d\u2019un \u00ab programme formaliste \u00bb mais dans les ann\u00e9es trente, les travaux des logiciens Kurt G\u00f6del et Alan Turing mettront d\u00e9finitivement fin \u00e0 la croyance de Hilbert de pouvoir r\u00e9soudre, par principe, tout probl\u00e8me formalis\u00e9 dans la logique math\u00e9matique. Voir aussi l\u2019entretien avec Gregory Chaitin.<\/p>\n

<\/a>Koch von<\/a><\/strong>, <\/a>Helge<\/a> <\/strong>(1870-1924) est un math\u00e9maticien su\u00e9dois connu pour avoir d\u00e9couvert en 1904 une courbe continue, infinie et sans tangente d\u00e9limit\u00e9e au sein d\u2019une aire finie. Cet objet math\u00e9matique, \u00e9galement appel\u00e9 \u00ab flocon de neige \u00bb, s\u2019obtient par it\u00e9rations successives. Chaque segment de la courbe est autosimilaire, c\u2019est-\u00e0-dire que la m\u00eame structure (un triangle \u00e9quilat\u00e9ral) est reproduite \u00e0 toutes les \u00e9chelles. Le flocon de Koch constitue l\u2019un des exemples les plus classiques d\u2019objet fractal. La courbe de Koch, ainsi que celle du math\u00e9maticien et logicien italien Giuseppe Peano (1858-1932) \u2013 qui est de longueur infinie et se construit par remplissage en passant par tous les points d\u2019un carr\u00e9 \u2013, furent mises en avant au d\u00e9but du XXe si\u00e8cle comme des exceptions, des singularit\u00e9s signalant certaines limites conceptuelles des math\u00e9matiques. Un si\u00e8cle plus tard, ce type d\u2019objet fractal est cit\u00e9 comme cas g\u00e9n\u00e9rique des structures et formes produites par la nature.<\/p>\n

<\/a>Kuhn,<\/a><\/strong> Thomas<\/strong><\/a>, philosophe et historien des sciences am\u00e9ricain, a propos\u00e9, dans son livre La Structure des r\u00e9volutions scientifiques <\/em>(Flammarion, 1976), une grille de lecture de l\u2019histoire des sciences \u00e0 partir de la notion de paradigme. Pour lui, l\u2019histoire des sciences ne rel\u00e8ve pas d\u2019un processus cumulatif des connaissances mais d\u2019une \u00e9volution ponctu\u00e9e de crises intellectuelles successives menant \u00e0 des r\u00e9volutions scientifiques. Ainsi, Kuhn voit dans la succession de crises de l\u2019histoire des sciences des changements de paradigme, c\u2019est-\u00e0-dire des changements du mod\u00e8le explicatif partag\u00e9 par une communaut\u00e9 de chercheurs au sein d\u2019une discipline scientifique.<\/p>\n

<\/a>Laplace de,<\/a><\/strong> Pierre Simon <\/strong><\/a>(1749-1827), math\u00e9maticien, physicien et astronome fran\u00e7ais, a beaucoup contribu\u00e9 au d\u00e9veloppement des sciences et \u00e0 la modernisation de leur enseignement pendant la R\u00e9volution fran\u00e7aise.  Personnage influent durant l\u2019Empire et la Restauration, il laissa \u00e0 l\u2019histoire sa fameuse r\u00e9partie \u00e0 Napol\u00e9on Bonaparte \u2013 lorsque l\u2019Empereur lui demanda : \u00ab Que faites-vous de Dieu dans votre syst\u00e8me ? \u00bb, il r\u00e9torqua : \u00ab Sire, je n\u2019ai pas besoin de cette hypoth\u00e8se \u00bb, exprimant ainsi le d\u00e9terminisme triomphant de son temps. Pour mieux pr\u00e9ciser sa conception d\u2019un univers d\u00e9terministe, Laplace \u00e9mit l\u2019id\u00e9e d\u2019un \u00ab d\u00e9mon \u00bb, esprit surhumain dou\u00e9 d\u2019une puissance de calcul illimit\u00e9e, capable de lire le pass\u00e9 et l\u2019avenir : \u00ab Une intelligence qui, pour un instant donn\u00e9, conna\u00eetrait toutes les forces dont la nature est anim\u00e9e, et la situation respective des \u00eatres qui la composent, si d\u2019ailleurs elle \u00e9tait assez vaste pour  soumettre  ces  donn\u00e9es \u00e0 l\u2019analyse, embrasserait dans la m\u00eame formule les mouvements des plus  grands corps de l\u2019Univers et ceux du plus l\u00e9ger atome : rien ne serait incertain pour     elle, et l\u2019avenir comme le pass\u00e9 serait pr\u00e9sent \u00e0 ses yeux. \u00bb Le d\u00e9mon de Laplace refl\u00e8te   la science du XIXe si\u00e8cle, mais la th\u00e9orie du chaos, mise en \u00e9vidence pour la premi\u00e8re fois par le math\u00e9maticien Henri Poincar\u00e9, puis la physique quantique mettront \u00e0 mal ce d\u00e9terminisme qui, partant de la connaissance des conditions initiales d\u2019un syst\u00e8me, pr\u00e9tendait d\u00e9duire parfaitement les conditions finales.<\/p>\n

<\/a>Lorenz, Edward<\/a><\/strong>, m\u00e9t\u00e9orologue du MIT (Massachusetts Institute of Technology<\/em>), est consid\u00e9r\u00e9 comme l\u2019un des premiers scientifiques de l\u2019\u00e8re contemporaine \u00e0 s\u2019\u00eatre int\u00e9ress\u00e9 \u00e0 l\u2019\u00e9tude du chaos et \u00e0 la dynamique non lin\u00e9aire. C\u2019est en 1963 qu\u2019il mit en \u00e9vidence le caract\u00e8re chaotique des conditions m\u00e9t\u00e9orologiques et inventa la m\u00e9taphore de \u00ab l\u2019effet papillon \u00bb. Alors qu\u2019il cherchait \u00e0 \u00e9tudier, sur son ordinateur, un mod\u00e8le de syst\u00e8me m\u00e9t\u00e9orologique, il observa de fa\u00e7on fortuite qu\u2019en effectuant deux fois le m\u00eame calcul,  il aboutissait \u00e0 des r\u00e9sultats radicalement diff\u00e9rents. Dans les deux simulations de son syst\u00e8me m\u00e9t\u00e9orologique, les conditions initiales \u00e9taient quasiment identiques mais Lorenz se rendit finalement compte que lors du second calcul, il avait incidemment introduit une diff\u00e9rence minime dans les conditions initiales du syst\u00e8me ; cette diff\u00e9rence, de l\u2019ordre de 1 pour 1 000, correspondait \u00e0 n\u00e9gliger d\u2019\u00e9crire les trois derniers chiffres d\u2019un nombre \u00e0 six d\u00e9cimales. Cette modification infime des conditions initiales suffit \u00e0 produire des conditions finales divergentes. Il en tira la conclusion que les conditions m\u00e9t\u00e9orologiques ne peuvent \u00eatre pr\u00e9visibles \u00e0 long terme en raison de \u00ab la d\u00e9pendance sensible aux conditions initiales \u00bb. La d\u00e9couverte de Lorenz, initialement publi\u00e9e dans le Journal of the Atmospheric Sciences<\/em> passa inaper\u00e7ue jusqu\u2019\u00e0 ce qu\u2019un certain nombre de physiciens, mais aussi de math\u00e9maticiens, s\u2019y int\u00e9ressent pour fournir, au d\u00e9but des ann\u00e9es soixante-dix, l\u2019ossature math\u00e9matique qui allait permettre l\u2019\u00e9tude des ph\u00e9nom\u00e8nes chaotiques.<\/p>\n

<\/a>Loi de Moore<\/a><\/strong> : observation d\u00e9crivant l\u2019\u00e9volution des technologies de l\u2019information, formul\u00e9e au milieu des ann\u00e9es soixante et portant le nom du fondateur de l\u2019entreprise de microprocesseurs Intel, elle pr\u00e9dit une croissance exponentielle des performances des processeurs \u2013 la puissance de calcul d\u2019une puce informatique double environ tous les dix-huit mois pour le m\u00eame co\u00fbt. Or, selon cette loi de Moore, au rythme de croissance de la puissance des processeurs et de la miniaturisation des transistors grav\u00e9s sur du silicium, une limite physique fondamentale a \u00e9t\u00e9 atteinte au seuil de 2020. \u00c0 partir de ce seuil, la taille des transistors correspond \u00e0 celle de quelques atomes de silicium et les aspects d\u00e9routants de la physique quantique entrent en jeu. En raison de cette limitation physique que la miniaturisation des composants rencontre in\u00e9luctablement, les chercheurs scientifiques travaillent sur des technologies de substitution tels que les ordinateurs quantiques, optiques et \u00e0 ADN.<\/p>\n

<\/a>Loi de puissance<\/a><\/strong> :  cette loi indique qu\u2019une variable n\u2019a pas de taille caract\u00e9ristique : par cons\u00e9quent, des \u00e9v\u00e9nements de toute taille peuvent survenir. Les lois de puissance sont dites invariantes d\u2019\u00e9chelle et s\u2019observent le plus souvent lors des transitions de phases.<\/p>\n

<\/a>Machine de Turing<\/a><\/strong> <\/a>: machine abstraite et id\u00e9ale, \u00e0 l\u2019origine des fondements th\u00e9oriques de l\u2019informatique. Elle d\u00e9crit et formalise le principe de fonctionnement de l\u2019ordinateur et de l\u2019algorithme, c\u2019est-\u00e0-dire la s\u00e9quence d\u2019instructions aboutissant \u00e0 la r\u00e9solution d\u2019un probl\u00e8me de calcul. Ce dispositif imaginaire porte le nom du logicien britannique Alan Turing (1912-1954), qui l\u2019a introduit dans un article historique paru en 1936, \u00ab On Computable Numbers, with an Application to the Entscheidungsproblem \u00bb (Proceedings of the London Mathematical Society, <\/em>s\u00e9rie 2, vol. 42, 1936-1937, pp. 230-265)    qui constitue l\u2019un des \u00e9crits math\u00e9matiques les plus importants sur la th\u00e9orie de la calculabilit\u00e9. La machine de Turing est une \u00ab machine \u00e0 \u00e9crire \u00bb modifi\u00e9e qui dispose d\u2019une t\u00eate de lecture et d\u2019un ruban infini divis\u00e9 en cases qui est en quelque sorte sa m\u00e9moire.  La t\u00eate de lecture a un nombre fini d\u2019\u00e9tats, elle peut se d\u00e9placer d\u2019une case \u00e0 une autre, \u00e0 gauche comme \u00e0 droite, et peut lire, \u00e9crire et effacer des symboles inscrits sur ces cases. Turing l\u2019a conceptualis\u00e9e pour d\u00e9montrer qu\u2019un automate programm\u00e9 de la sorte pouvait r\u00e9soudre n\u2019importe quel probl\u00e8me calculable. Tout probl\u00e8me est dit calculable s\u2019il peut \u00eatre calcul\u00e9 par une machine de Turing.<\/p>\n

<\/a>Mandelbrot, Beno\u00eet<\/a><\/strong>, math\u00e9maticien fran\u00e7ais d\u2019origine polonaise, a conceptualis\u00e9 au d\u00e9but des ann\u00e9es soixante la g\u00e9om\u00e9trie fractale, qui s\u2019est r\u00e9v\u00e9l\u00e9e \u00e9troitement li\u00e9e \u00e0 la th\u00e9orie du chaos d\u00e9terministe. Mandelbrot est l\u2019inventeur du n\u00e9ologisme \u00ab fractal \u00bb qu\u2019il a cr\u00e9\u00e9 \u00e0 partir d\u2019une racine latine, fractus<\/em>, signifiant \u00ab bris\u00e9 \u00bb, de frangere <\/em>: \u00ab casser, briser \u00bb.<\/p>\n

<\/a>Mouvement brownien<\/a><\/strong> <\/a>: correspond au mouvement erratique de particules microscopiques dans un fluide, il porte le nom du botaniste britannique Robert Brown (1773-1858) qui observa en 1828 que des grains de pollen en suspension dans l\u2019eau \u00e9taient en proie \u00e0 un mouvement d\u00e9sordonn\u00e9. Le mouvement brownien, qui traduit en fait l\u2019agitation thermique des atomes et des mol\u00e9cules, a \u00e9t\u00e9 \u00e9tudi\u00e9 par Albert Einstein en 1905 puis par le physicien fran\u00e7ais Jean Perrin qui le d\u00e9crivit en 1913 dans son livre Les Atomes<\/em>. Il sera math\u00e9matiquement conceptualis\u00e9 en 1923 par le math\u00e9maticien am\u00e9ricain Norbert Wiener afin d\u2019\u00e9tudier d\u2019autres processus physiques al\u00e9atoires.<\/p>\n

<\/a>Optimisation combinatoire<\/a><\/strong>, approche issue de la th\u00e9orie de la complexit\u00e9 et de l\u2019informatique th\u00e9orique, regroupe un ensemble de techniques de r\u00e9solution de probl\u00e8mes de calcul tr\u00e8s ardus. L\u2019optimisation combinatoire cherche \u00e0 trouver au moyen de divers types d\u2019algorithmes la solution optimale parmi un nombre fini de choix, elle permet       de mod\u00e9liser des probl\u00e8mes de d\u00e9cision soumis \u00e0 des contraintes (logistique, planning, gestion de production, transport, etc.). Le probl\u00e8me du voyageur de commerce est un probl\u00e8me typique d\u2019optimisation combinatoire.<\/p>\n

<\/a>Paysage \u00e9volutif ou adaptatif<\/a><\/strong> (traduction de l\u2019expression fitness landscape<\/em>) : m\u00e9taphore math\u00e9matique introduite en 1931 par le biologiste Sewall Wright (1889-1988) pour visualiser la dynamique \u00e9volutive d\u2019un ou plusieurs organismes. Le paysage \u00e9volutif est une cartographie d\u2019un espace o\u00f9 \u00e0 chaque point de l\u2019espace correspond un nombre qui d\u00e9finit un niveau d\u2019aptitude. La surface de ce paysage peut \u00eatre ainsi plus   ou moins \u00ab rugueuse \u00bb, parsem\u00e9e de \u00ab pics adaptatifs \u00bb que s\u00e9parent des \u00ab vall\u00e9es \u00bb. Aux plus hauts sommets de ce paysage correspond la meilleure aptitude. Entre les sommets, les organismes errent, soumis aux contraintes de la contingence et de la s\u00e9lection. La mod\u00e9lisation math\u00e9matique du paysage \u00e9volutif d\u00e9velopp\u00e9e par Stuart Kauffman est maintenant utilis\u00e9e dans diverses disciplines et activit\u00e9s de recherche (biologie th\u00e9orique, physique de la turbulence, chimie, algorithmes g\u00e9n\u00e9tiques, optimalisation combinatoire, etc.) pour d\u00e9crire de mani\u00e8re plus intuitive des probl\u00e8mes complexes impliquant un processus de s\u00e9lection.<\/p>\n

<\/a>Principe d\u2019incertitude<\/a><\/strong>, \u00e9nonc\u00e9 en 1925 par le physicien allemand Werner Heisenberg (1901-1976), postule qu\u2019on ne peut mesurer pr\u00e9cis\u00e9ment et simultan\u00e9ment la position et la vitesse d\u2019une particule subatomique. Selon ce principe essentiel de la th\u00e9orie quantique, du fait des \u00e9chelles et des fluctuations du monde microscopique, on ne peut conna\u00eetre avec exactitude que l\u2019une ou l\u2019autre de ces deux grandeurs physiques. La position et la vitesse sont math\u00e9matiquement li\u00e9es de fa\u00e7on inversement proportionnelle : plus grande est la pr\u00e9cision pour l\u2019une, plus grande est l\u2019impr\u00e9cision pour l\u2019autre.<\/p>\n

<\/a>Probl\u00e8mes de croissance<\/a><\/strong> : ph\u00e9nom\u00e8nes associ\u00e9s \u00e0 la formation macroscopique d\u2019une vari\u00e9t\u00e9 de motifs de la mati\u00e8re tels que la formation des dunes de sable, des fractures caract\u00e9ristiques d\u2019un sol argileux ass\u00e9ch\u00e9, des bancs de corail, des colonies bact\u00e9riennes, du givre sur une vitre, des flocons de neige, etc. Engendr\u00e9s au cours de processus \u00e9volutifs, \u00e9pid\u00e9miques, de gel, de percolation ou de cristallisation, ces divers ph\u00e9nom\u00e8nes physiques et chimiques de croissance sont actuellement \u00e9tudi\u00e9s au travers d\u2019un m\u00eame \u00e9clairage math\u00e9matique. On a ainsi recours \u00e0 des mod\u00e8les informatiques \u2013 comme celui de l\u2019agr\u00e9gation limit\u00e9e par diffusion (DLA, Diffusion Limited Agregation<\/em>) \u2013 qui permettent de repr\u00e9senter ces ph\u00e9nom\u00e8nes de croissance par le biais d\u2019une simulation r\u00e9v\u00e9lant la formation de structures ramifi\u00e9es \u00e0 g\u00e9om\u00e9trie fractale. Le mod\u00e8le de croissance fractale de type DLA reproduit un processus de collage de particules identiques ou d\u2019amas de particules pour \u00e9tudier la formation de dendrites que l\u2019on retrouve, entre autres, dans les structures neuronales ou les d\u00e9charges \u00e9lectriques. Le principe de ces croissances num\u00e9riques consiste \u00e0 abstraire le processus physique \u00e0 l\u2019\u0153uvre dans toutes sortes de ph\u00e9nom\u00e8nes pour mieux comprendre l\u2019universalit\u00e9 de ses propri\u00e9t\u00e9s.<\/p>\n

<\/a>Rasoir d\u2019Occam<\/a><\/strong> : principe philosophique m\u00e9di\u00e9val qui porte le nom du th\u00e9ologien anglais Guillaume d\u2019Occam, ou d\u2019Ockham (1290-1349) qui l\u2019a formul\u00e9. Ce principe dit que \u00ab la pluralit\u00e9 ne doit \u00eatre envisag\u00e9e qu\u2019en cas de n\u00e9cessit\u00e9 \u00bb (pluralitas non est ponenda sine neccessitate<\/em>). Les scientifiques y ont souvent recours pour rappeler qu\u2019en pr\u00e9sence de plusieurs th\u00e9ories d\u00e9crivant une m\u00eame r\u00e9alit\u00e9, la plus simple est pr\u00e9f\u00e9rable. Le rasoir d\u2019Occam est d\u00e9sign\u00e9 sous diverses appellations, telles que le principe de simplicit\u00e9<\/em>, le principe d\u2019\u00e9conomie<\/em> ou la loi de parcimonie<\/em>.<\/p>\n

<\/a>R\u00e9action chimique de Belouzof-Zhabotinsky<\/a><\/strong> <\/a>: un des exemples les plus typiques de structures dissipatives, c\u2019est-\u00e0-dire de structures subissant des variations au cours     du temps. Cette r\u00e9action, qui porte le nom de deux scientifiques russes l\u2019ayant observ\u00e9e au cours des ann\u00e9es 1950 et 1960, fut la premi\u00e8re manifestation d\u2019auto-organisation en chimie. Elle demeura pourtant longtemps ignor\u00e9e par les chimistes du monde entier  car leur discipline traitait essentiellement de r\u00e9actions chimiques dont les produits proc\u00e8dent d\u2019une \u00e9volution stable et monotone. Ce n\u2019est qu\u2019au d\u00e9but des ann\u00e9es 1970 que les scientifiques se mirent \u00e0 s\u2019int\u00e9resser aux motifs spatio-temporels de cette r\u00e9action chimique. Ces motifs ont \u00e9t\u00e9 \u00e9tudi\u00e9s par simulation informatique, notamment en recourant aux automates cellulaires, et on s\u2019est rendu compte qu\u2019ils rappellent \u00e9tonnamment ceux de plusieurs ph\u00e9nom\u00e8nes biologiques, comme les structures dissipatives mises en \u00e9vidence dans un milieu d\u2019amibes (Dictyostelium discoideum<\/em>) ayant la particularit\u00e9 d\u2019\u00eatre \u00e0 la fois unicellulaire et pluricellulaire au cours de leur cycle de vie ou l\u2019activit\u00e9 des cellules cardiaques, qui oscillent de mani\u00e8re ind\u00e9pendante les unes des autres lors de la fibrillation (d\u00e9synchronisation du rythme cardiaque pouvant conduire \u00e0 la mort). Ces divers ph\u00e9nom\u00e8nes physico-chimiques ont en commun non pas les composants mais la dynamique de leurs interactions.<\/p>\n

<\/a>Reine rouge<\/a><\/strong> : m\u00e9taphore propos\u00e9e en 1973 par le biologiste \u00e9volutionniste am\u00e9ricain Leigh Van Valen pour \u00e9clairer la co\u00e9volution complexe o\u00f9 chaque organisme participe au contexte \u00e9volutif d\u2019autres organismes, le paysage \u00e9volutif allant de la sorte vers une complexit\u00e9 croissante. Toute esp\u00e8ce qui cesse cette course \u00e9volutive perp\u00e9tuelle est condamn\u00e9e \u00e0 l\u2019extinction selon le principe qu\u2019il faut courir juste pour se maintenir \u00e0 la m\u00eame place. Cette image m\u00e9taphorique fait r\u00e9f\u00e9rence au livre de Lewis Carroll, Alice au pays des merveilles<\/em>, o\u00f9, \u00ab de l\u2019autre c\u00f4t\u00e9 du miroir \u00bb, la Reine rouge tire Alice par la main et la fait courir sans cesse tandis que le paysage reste immobile. La Reine rouge explique qu\u2019\u00ab ici, il faut courir aussi vite que tu peux pour rester \u00e0 la m\u00eame place. Si tu veux te d\u00e9placer, tu dois courir au moins deux fois plus vite \u00bb.<\/p>\n

<\/a>R\u00e9seaux d\u2019automates bool\u00e9ens <\/a><\/strong>:  ensembles d\u2019automates qui, connect\u00e9s de fa\u00e7on al\u00e9atoire, g\u00e9n\u00e8rent des comportements collectifs particuliers.<\/p>\n

<\/a>R\u00e9seaux de spins<\/a><\/strong> : mod\u00e8les simplifi\u00e9s de la physique d\u00e9crivant le comportement d\u00e9sordonn\u00e9 de minuscules aimants, ils sont utilis\u00e9s de fa\u00e7on multidisciplinaire.<\/p>\n

<\/a>R\u00e9seau g\u00e9n\u00e9tique r\u00e9gulateur<\/a><\/strong> : th\u00e9orie \u00e9labor\u00e9e par le biologiste Stuart Kauffman qui propose un sch\u00e9ma d\u2019explication de la diff\u00e9renciation cellulaire. Il s\u2019agit de comprendre comment les diff\u00e9rents types de cellules n\u00e9cessaires \u00e0 un organisme (256 cellules distinctes dans le cas de l\u2019homme) peuvent \u00eatre g\u00e9n\u00e9r\u00e9s \u00e0 partir d\u2019un m\u00eame g\u00e9nome. En effet, lors de la diff\u00e9renciation cellulaire, l\u2019ensemble des g\u00e8nes activ\u00e9s n\u2019est pas le m\u00eame selon     que cet ensemble sp\u00e9cifie une cellule musculaire, une cellule cardiaque ou un neurone.  La simulation de Kauffman, bas\u00e9e sur un r\u00e9seau d\u2019automates bool\u00e9ens o\u00f9 l\u2019activit\u00e9 des g\u00e8nes est mod\u00e9lis\u00e9e de fa\u00e7on binaire (chaque g\u00e8ne \u00e9tant actif ou inactif), montre que le syst\u00e8me stabilise en fait un certain nombre d\u2019attracteurs correspondant aux diff\u00e9rentes cellules d\u00e9finies par le programme g\u00e9n\u00e9tique. Kauffman avait estim\u00e9 que dans le cas d\u2019un r\u00e9seau g\u00e9n\u00e9tique de 100\u2019000 g\u00e8nes (estimation, \u00e0 l\u2019\u00e9poque \u2013 1993 \u2013 de la taille du g\u00e9nome humain), il y aurait th\u00e9oriquement 2100 000 configurations cellulaires possibles. Ce chiffre inconcevable montre que la nature proc\u00e8de autrement pour d\u00e9terminer un nombre limit\u00e9 de types distincts de cellules. Kauffman a calcul\u00e9 que le nombre d\u2019attracteurs ou de cellules stabilis\u00e9s cro\u00eet en fonction de la racine carr\u00e9e du nombre total de g\u00e8nes : soit, pour un r\u00e9seau de 100 000 g\u00e8nes, 317 attracteurs ou cellules. Ce r\u00e9sultat produit par le r\u00e9seau g\u00e9n\u00e9tique simplifi\u00e9 de Kauffman est relativement proche des 256 cellules humaines.<\/p>\n

<\/a>Riemann,<\/a><\/strong> <\/a>Bernhard<\/a> <\/strong>(1826-1866), math\u00e9maticien allemand, fut l\u2019un des premiers, avec le math\u00e9maticien russe Nikola\u00ef Lobatchevski (1792-1856), \u00e0 d\u00e9velopper des g\u00e9om\u00e9tries non euclidiennes, c\u2019est-\u00e0-dire des surfaces courbes o\u00f9 la somme des angles n\u2019est pas \u00e9gale \u00e0 180\u00b0 et o\u00f9 le cinqui\u00e8me postulat d\u2019Euclide (qui dit que deux droites parall\u00e8les ne se croisent jamais) n\u2019est pas v\u00e9rifi\u00e9. Cette d\u00e9couverte, dans le dernier quart du XIXe si\u00e8cle, de l\u2019existence de g\u00e9om\u00e9tries non euclidiennes a conduit \u00e0 une crise conceptuelle comparable \u00e0 la d\u00e9couverte de la th\u00e9orie du chaos d\u00e9terministe un si\u00e8cle plus tard. La g\u00e9om\u00e9trie de Riemann, elliptique quand celle de Lobatchevski est hyperbolique, \u00e9tudie des espaces courbes avec un nombre variable de dimensions ; elle a eu une influence d\u00e9terminante dans la physique th\u00e9orique et plus particuli\u00e8rement dans la th\u00e9orie de la relativit\u00e9 g\u00e9n\u00e9rale et la description de l\u2019espace-temps.<\/p>\n

<\/a>Sym\u00e9trie<\/a><\/strong> : cette notion renvoie soit aux structures sym\u00e9triques que la s\u00e9lection naturelle aurait favoris\u00e9es pour des raisons de stabilit\u00e9 et de permanence, soit aux propri\u00e9t\u00e9s d\u2019invariance d\u2019un objet, d\u2019un syst\u00e8me ou d\u2019une th\u00e9orie soumis \u00e0 une transformation (par exemple, une sph\u00e8re pr\u00e9sente une sym\u00e9trie de rotation : elle reste inchang\u00e9e lorsqu\u2019elle tourne sur elle-m\u00eame). La notion de sym\u00e9trie est de plus en plus utilis\u00e9e, parfois m\u00eame en lieu et place du concept de loi, et elle est en g\u00e9n\u00e9ral coupl\u00e9e \u00e0 la notion de brisure de sym\u00e9trie<\/em> (ainsi le big bang est-il une brisure de sym\u00e9trie parce que l\u2019explosion primordiale correspond \u00e0 une singularit\u00e9 absolue). La sym\u00e9trie tend \u00e0 d\u00e9signer des propri\u00e9t\u00e9s physiques de r\u00e9gularit\u00e9, d\u2019invariance, de conservation, d\u2019\u00e9quivalence, tandis que la brisure de sym\u00e9trie d\u00e9signerait des ph\u00e9nom\u00e8nes naturels de complexit\u00e9, de singularit\u00e9, de criticalit\u00e9 et d\u2019instabilit\u00e9. Sym\u00e9trie et brisure de sym\u00e9trie sont des cat\u00e9gories qui permettent \u00e0 une th\u00e9orie scientifique d\u2019acc\u00e9der \u00e0 l\u2019intelligibilit\u00e9 du r\u00e9el     et, mises en couple, signalent une richesse dialectique comparable \u00e0 celle des cat\u00e9gories fini\/infini, discret\/continu, local\/global.<\/p>\n

<\/a>Somme de Feynman<\/a><\/strong>, du nom du physicien am\u00e9ricain (1918-1988), prix Nobel 1965, qui l\u2019a formul\u00e9e, elle est un proc\u00e9d\u00e9 de la th\u00e9orie quantique qui permet d\u2019envisager tous les chemins possibles qu\u2019une particule peut emprunter pour aller d\u2019un point \u00e0 un autre.<\/p>\n

<\/a>Tabula rasa<\/a><\/em><\/strong> <\/em>: notion philosophique d\u00e9finie, entre autres, par le philosophe empiriste anglais John Locke (1632-1704) pour d\u00e9signer l\u2019esprit et l\u2019acquisition d\u2019exp\u00e9rience et de savoir. Selon Locke, l\u2019esprit d\u2019un enfant est une tabula rasa <\/em>(ou tableau blanc) sur laquelle s\u2019inscrivent au fur et \u00e0 mesure toutes ses exp\u00e9riences cognitives. Cette id\u00e9e fut avanc\u00e9e pour s\u2019opposer \u00e0 l\u2019id\u00e9e du caract\u00e8re inn\u00e9 de la connaissance d\u00e9fendue par d\u2019autres philosophes.<\/p>\n

<\/a>Th\u00e9orie des cordes<\/a><\/strong> : domaine r\u00e9cent de la physique th\u00e9orique o\u00f9 les objets fondamentaux sont des cordes extr\u00eamement petites, de l\u2019ordre de la longueur de Planck (10\u201320, soit cent milliards de milliards de fois plus petites qu\u2019un noyau d\u2019hydrog\u00e8ne), dont le mode de vibration serait \u00e0 l\u2019origine des diff\u00e9rentes particules de la mati\u00e8re. \u00c0 chaque mode de vibration correspondraient la masse et la charge d\u2019une particule subatomique. Cette th\u00e9orie ambitionne d\u2019unifier la th\u00e9orie quantique et la relativit\u00e9 g\u00e9n\u00e9rale, mais de l\u2019avis de tous les chercheurs qui y travaillent, il faudrait plusieurs dizaines d\u2019ann\u00e9es pour d\u00e9velopper les math\u00e9matiques hautement complexes qui la r\u00e9giraient.<\/p>\n

<\/a>Th\u00e9or\u00e8me de l\u2019incompl\u00e9tude<\/a><\/strong>, formul\u00e9 en 1931 par le logicien et math\u00e9maticien autrichien Kurt G\u00f6del (1906-1978), il \u00e9nonce que si un syst\u00e8me formel contenant l\u2019arithm\u00e9tique \u00e9l\u00e9mentaire est suppos\u00e9 consistant \u2013 c\u2019est-\u00e0-dire qu\u2019une assertion et son oppos\u00e9 ne peuvent \u00eatre d\u00e9montr\u00e9es en m\u00eame temps \u2013, alors il est incomplet \u2013 c\u2019est-\u00e0-dire qu\u2019une assertion ne peut \u00eatre prouv\u00e9e vraie ou fausse.<\/p>\n

<\/a>Th\u00e9orie algorithmique de l\u2019information<\/a><\/strong> : d\u00e9velopp\u00e9e dans les ann\u00e9es soixante, cette th\u00e9orie, dont Gregory Chaitin fut l\u2019un des artisans avec les math\u00e9maticiens Kolmogorov et Solomonoff, d\u00e9termine le degr\u00e9 de complexit\u00e9 d\u2019un objet ou d\u2019un \u00e9nonc\u00e9 math\u00e9matique en mesurant la quantit\u00e9 minimale d\u2019information n\u00e9cessaire pour g\u00e9n\u00e9rer cet objet ou cet \u00e9nonc\u00e9 math\u00e9matique. \u00c0 partir de ce principe g\u00e9n\u00e9ral, cette approche th\u00e9orique s\u2019est int\u00e9ress\u00e9e   \u00e0 la compression d\u2019information et aux probl\u00e8mes de calculs r\u00e9alisables ou non en science informatique. Elle a par la suite \u00e9t\u00e9 g\u00e9n\u00e9ralis\u00e9e \u00e0 la mesure du contenu d\u2019information   des syst\u00e8mes logiques formels, c\u2019est-\u00e0-dire des ensembles d\u2019axiomes et des th\u00e9or\u00e8mes. Dans cette perspective, les th\u00e9ories scientifiques elles-m\u00eames sont consid\u00e9r\u00e9es comme des algorithmes et des compressions d\u2019information permettant de d\u00e9crire la complexit\u00e9 des ph\u00e9nom\u00e8nes naturels. La th\u00e9orie algorithmique de l\u2019information est ainsi devenue, en l\u2019espace de trois d\u00e9cennies, un outil de mesure universel de la complexit\u00e9.<\/p>\n

<\/a>Thermodynamique<\/a><\/strong> : science qui \u00e9tudie les syst\u00e8mes physiques soumis \u00e0 des variations de temp\u00e9rature, d\u2019\u00e9nergie (chaleur et travail) et d\u2019entropie (mesure du degr\u00e9 de d\u00e9sordre du syst\u00e8me). Le premier principe de la thermodynamique postule la conserva- tion de l\u2019\u00e9nergie tandis que le second principe affirme que tout syst\u00e8me \u00e9volue au cours du temps vers un d\u00e9sordre croissant correspondant \u00e0 l\u2019\u00e9tat d\u2019\u00e9quilibre final du syst\u00e8me. On dit alors que son entropie est maximale.<\/p>\n

<\/a>Voyageur de commerce<\/a><\/strong> : le probl\u00e8me est vite formul\u00e9 et pratiquement impossible \u00e0 r\u00e9soudre. Un voyageur de commerce doit visiter n villes ; connaissant les distances entre les diff\u00e9rentes villes, il doit d\u00e9terminer l\u2019itin\u00e9raire optimal (en termes de co\u00fbt de carburant), c\u2019est-\u00e0-dire la distance minimale qu\u2019il lui faut parcourir pour visiter toutes les villes. S\u2019il s\u2019agit d\u2019examiner les itin\u00e9raires un par un, le temps de r\u00e9solution peut devenir rapidement astronomique, car il cro\u00eet exponentiellement avec la taille du probl\u00e8me (au fur et \u00e0 mesure que le nombre     n de villes augmente). Pour100 villes, le calcul de factorisation prendrait des milliards d\u2019ann\u00e9es ! Le probl\u00e8me du voyageur de commerce est un cas d\u2019\u00e9cole du calcul par \u00ab optimisation combinatoire \u00bb o\u00f9 il s\u2019agit de d\u00e9velopper des approximations satisfaisantes par rapport \u00e0 la taille et la complexit\u00e9 des probl\u00e8mes pos\u00e9s.<\/p>\n

<\/a>Weierstrass, Karl<\/a> <\/strong>(1815-1897), math\u00e9maticien allemand, a montr\u00e9 en 1872 que certaines fonctions d\u00e9crivant une courbe continue, sans tangente et non d\u00e9rivable avaient une propri\u00e9t\u00e9 d\u2019auto-similarit\u00e9. Les courbes de Weierstrass, Koch et Peano, l\u2019ensemble de Cantor furent, au moment de leur apparition, per\u00e7us comme des \u00ab cas pathologiques \u00bb alors qu\u2019ils constituent les premiers d\u00e9veloppements des math\u00e9matiques fractales.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Version : English – Fran\u00e7aise Excerpts translated by Nicolas Sperry-Guillou from R\u00e9da Benkirane, La Complexit\u00e9, vertiges et promesses. Histoires de sciences. Nouvelle \u00e9dition, Benguerir, UM6P-Press, 2023. [Algorithm, Algorithmic information theory, Belouzof-Zhabotinsky chemical reaction, Boolean automata networks, Brownian motion, Cellular automaton, Combinatorial optimization, Cybernetics, Entropy, Feynman\u2019s sum, Fitness landscape, Frankfurt School, Gauss\u2026 Lire plus \/ Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1775,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"ngg_post_thumbnail":0,"footnotes":""},"categories":[],"tags":[],"class_list":["post-2644","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/2644","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2644"}],"version-history":[{"count":29,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/2644\/revisions"}],"predecessor-version":[{"id":3435,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/2644\/revisions\/3435"}],"up":[{"embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=\/wp\/v2\/pages\/1775"}],"wp:attachment":[{"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2644"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2644"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cch.um6p.ma\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2644"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}