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We call classification the operation consisting of distributing objects in classes or groups which are, in general, less numerous than them. The relations between these classes may be (or not) partially or totally ordered. So there exist many kinds of classification schemes. Formally speaking, a classification may be a lattice, a semilattice, a chain, a hypergraph, a matroid, a tree, etc. Our purpose is to find the underlying mathematical structure of all these classifications. We explain how we can represent them in a unique way, constituting what has been called before a ”metaclassification” and which is, in fact, a Chu space. Thus, category theory can describe in terms of morphisms different operations on classifications that, according to Barwise and Seligman, we report in the following. Finally, we show that such a ”metaclassification” is the fundamental brick from which we can get some information about the mathematical continuum.

We want now to take a glance at the logical form a theory of classifications (whatever these may be) could get in general. We must remark, first, that a logical approach to classifications (especially from an intensional viewpoint) never won unanimous support among the ancient philosophers and logicians (as proven by the numerous discussions between Aristotle and Plato, Ramus and Pascal, Jevons and Joseph).

We have seen that classic representations of classifications give hierarchical structures, which are the chains of partitions of a partition lattice. But as we have seen also, such structures may change with time, either because of the internal modifications in the description of objects’ properties (intrinsic instability), or because of the external increasing or decreasing of the number of objects to be classified (extrinsic instability). For these two reasons, it is clear that it would be pleasant to get an algebra of all the possible transformations of classifications, and particularly explicit rules to explain how we can extend them, without breaking their initial structures.

We have presented in the previous chapter, a theory able to generate the infinite set of classifications as the continuum, each classification being in a one-to-one correspondence with a real number. But, as we know, there are, in the mathematics of the infinite, since the works of Cantor, Suslin and others, a lot of possible views of the continuum. Moreover, since the undecidability results of Cohen, there exist also a lot of possible set theories. So, it may be useful to ask some questions about what happens concerning the existence of classifications in those alternative theories, in particular, when they admit higher forms of the infinite. Though the risk is obvious, there, to end up at some undecidability results, several arguments speak for such an extension.

As we have seen (see Chap. 5), classifications cannot be reduced to the classic representations of Barbut and Monjardet (Ordre et classification, algebre et combinatoire, vol. 1, Librairie Hachette, Paris, 1970) or of Lerman (La classification automatique, Paris, 1970; Classification automatique et analyse ordinale des donnees, Dunod, Paris, 1981). Note, however, that Lerman (La classification automatique, Paris, 1970, 73) has proposed to extend partitions search’s algorithms to covering problems. A more recent reference on the subject is (Alpert and Kahng in Integr. VLSI J. 19(1–2):1–81, 1995. It is obvious that these representations are not sufficient for describing k-classifications proposed by Jardine and Sibson (Math. Biosci. 2:465–482, 1968). But as soon as we admit overlapping classes, classifications are no more chains of partitions. They are made of covers. So we define first this notion (Sect. 6.3), count the number of covers on a set (Sect. 6.4), study the set of all covers on a set (Sect. 6.5) and the lattice of all minimal covers (Sect. 6.6). It appears that, when the generalized classifications still keep a hierarchic form, they are in fact chains of covers (Sect. 6.7). Now, as we shall see, if those covers are not complete ones, classifications become chains of parts of covers. And as a part of a cover is just the same a part of a partition, i.e. a class, such classifications are just chains of classes (Sect. 6.8). As the lattice of minimal covers does not contain all possible generalized classifications, and as the set of all covers over a set is not a lattice, we are necessarily led to introduce topological notions (Sect. 6.9), because topological relations are the only relations that subsist on this kind of classifications. We can then apply to classifications some basic theorems of topology: first, it is a useful way to get information on them (Sect. 6.10); but, more generally, it is possible to develop a topological representation of information (Haouas et al. in J. Appl. Sci. 8(20):3743–3747, 2008). Finally, those systems of classes happen to be a kind of hypergraphs, so, it appears that the language of relational structure (Sect. 6.11) is the more convenient to describe those structures.

As most of the classifications man has constructed during a long period going from Plato and Aristotle to Linnaeus (and even to the XIXth century) were hierarchical classifications, we devote this complete chapter to the exposition of mathematical models of hierarchies. After some historical views on the subject (Sect. 3.2), we introduce (Sect. 3.3) the basic notions of partition, partition lattice, and chain of partitions, this last one being the exact mathematical model of a hierarchical classification, classically represented by a tree diagram. In Sect. 3.4, we give the structure of the set of all hierarchical classifications on a finite set (which allows us to know exhaustively all the possible hierarchies we can make). Then, in Sect. 3.5, we present the exact correspondence between tree diagram, hierarchical structure, and the distance we can define on it, which is an ultrametric. These ideal models and their algebraic representations (Sect. 3.6) are those that the taxonomists want generally to obtain, but the appearance of the real world, in general, is quite far from such nice orders. So, we explain (Sect. 3.7) how we can replace the empirical quasi-chaotic data with due mathematical taxonomies. Though, in many cases, we are able to do that and can easily adapt our numerical models to empirical data, some problems arise in this operation (Sect. 3.8): either because of the existence of mathematical limits within the models (intrinsic instability) or because of the presence of changes in human perception of the world in the course of time (extrinsic instability). However (Sect. 3.9), we list finally some possible answers to these important questions.

The ArgumentDagognet's work shows that making algorithmic compressions seems to be one of the major targets of scientific progress. This effort has been so successful that until recently one might have thought everything could be algorithmically compressed. Indeed, this statement, which might be seen as a scientific translation of the Hegelian thesis in its strong form (“the real is rational and the rational is real”), admits to some objective limits in computer science. Though a lot of algorithms are successful, there exist today, and perhaps forever, logical and physical limits that cannot allow us to cherish the dream of a “theory of everything.” Moreover, a complete mastery of complexity does not seem possible — because some domains of reality are too complicated to be computable, because the human brain is too limited, because computers cannot do that much better than the human brain, and because, ultimately, there are some kinds of things it would make no sense to compress. This paper shows that Dagognet's work came to recognize what a glance at the history of algorithmics has made evident.

Depuis la fin du XIXe et le début du XXe siècle, la loi d’attraction universelle de Newton a été transposée en géographie, constituant ce qu’on a pu appeler le « modèle gravitaire ». Dès la fin des années 1960, les géographes utilisent, par ailleurs, des espaces non-euclidiens. Or on sait que les espaces de Riemann, depuis la théorie de la relativité générale d’Albert Einstein (1915), constituent le cadre formel le plus approprié pour comprendre la gravitation, assimilée à une courbure de l’espace-temps. Il restait à mettre en relation les phénomènes géographiques avec ce cadre physico-mathématique, en examinant les difficultés épistémologiques soulevées par cette transposition. Ce travail, qui trouve son inspiration dans un texte de R. Brunet, rappelle d’abord l’origine et la nature du modèle gravitaire ainsi que les problèmes posés par la généralisation de la formule de Newton. Après avoir rappelé quelques définitions concernant les notions mathématiques de distance, courbure et espace, il présente ensuite de façon succincte les fondements de la théorie einsteinienne, restreinte et générale, et développe, pour finir, les fondements d’une théorie de la relativité géographique. Inspired by Newton's law of gravitation, gravity models appeared in geography more than one century ago. For about forty years, non-Euclidean spaces have been of current use in geography. We know also that Riemannian spaces are the formal apparatus of General Relativity. In Einstein's theory (1915), gravitation is nothing but space-time curvature. Throughout this paper, we shall try to extend gravity models by substituting Newton's law by a relativistic model based on a Riemannian space. We also try to answer the different kinds of problems raised by our theory: for instance, discontinuity of space, non symmetric matrices of distances, non-constant curvature, locally non-Euclidean Riemannian spaces, etc.

Parrochia Daniel. Le progrès des instruments scientifiques aux XVIIe et XVIIIe siècles. In: Littératures classiques, n°43, automne 2001. Le temps au XVIIe siècle. pp. 181-191.

In this paper, I am dealing with some epistemological aspects of what Christopher Langton (1989) and some other scientists have been calling recently «artificial life», whose history, in fact, is far older. I want to take a view on the origin, further developments and latest issues of these models, and try to point out the major philosophical and epistemological problems arising with them.

Depuis le developpement accelere d'Internet, dans le courant des annees 1980, la pensee contemporaine a du mal a saisir les aspects multiples de ce nouvel objet, qui, comme tous les instruments de communication qui sont aussi des instruments de puissance induit de multiples effets allant de la seduction fascinante a la crainte respectueuse, en passant par nombre de delires. D'ou les limites, selon nous, d'un certain type de discours qui, sautant par-dessus une veritable etude de philosophie des techniques, oscillent entre l'envolee lyrique et les propos de Cassandre. Dans le texte qui suit, nous nous appuierons sur la realite (actuelle) d'Internet pour depasser ces opinions en vue de saisir ulterieurement quelques-unes des transformations dans lesquelles nous sommes pris avec le phenomene de l'interconnexion des reseaux.

Notre ambition, dans cet article, est tout a fait modeste. Il s’agit de presenter, comme notre titre l’indique, quelques aspects historiques de la notion de reseau. Cette entreprise n’est pas la premiere du genre. Nous avons, naguere, en un temps ou la notion de reseau etait moins en vogue qu’aujourd’hui, contribue a mettre cette histoire en lumiere. Depuis, Pierre Musso a, a plusieurs reprises, developpe une critique tres utile des representations auxquelles elle a donne lieu. Toutefois, comme Hegel l’a montre, des representations de concepts ne sont pas des concepts, c’est pourquoi la mythologie, l’art, la religion, ne sont pas des moyens adequats pour l’expression de la pensee. De plus, nous entendons nous limiter, pour l’essentiel, a l’histoire du noyau rationnel de cette notion, sans pretendre evidemment, dans la limite de l’espace qui nous est imparti, a l’exhaustivite. Soucieux de montrer de facon claire comment la notion de reseau s’est peu a peu degage du halo d’images et d’obstacles epistemologiques qui l’entouraient, nous irons de ses aspects prehistoriques (le reseau comme filet) a ses usages effectifs dans la theorie des graphes et des reseaux de transport en montrant comment, de l’organisation de la matiere a celle du territoire puis des moyens de communication, cette notion s’est averee, peu a peu indispensable pour decrire les societes contemporaines ou, selon une these celebre de Mac Luhan, les echanges et la re-production deviennent aussi importants, sinon plus, que l’appareil productif lui-meme.

En quoi la mathematique et l'informatique peuvent-elles concerner les philosophes et les penseurs de la politique ? Naguere, la reponse eut consiste a dire que les structures de l'autorite et du pouvoir trouvaient leur representation formelle la plus simple dans des structures d'ordre typique, largement repandues dans ces disciplines, comme, par exemple, la structure d'arbre1. Certes, on savait qu'il existait differentes formes d'arbres formels et que tout arbre n'a pas une racine. Il restait que l'arbre apparaissait clairement comme la structure de commandement par excellence, d'autant que l'informatique apprenait encore qu'on pouvait ordonner la structure de ses nœuds2. L'arbre etait d'ailleurs devenu, au fil du developpement de l'informatique theorique et de l'algorithmique, non seulement une structure de controle, mais une structure de decision.

Maugeri Salvatore. Parrochia Daniel, Philosophie des reseaux.. In: Revue francaise de sociologie, 1996, 37-1. pp. 132-136.

Resume Le but de cet article est ?on;etudier quelques aspects des rapports entre Huygens et Spinoza. Apres un court examen de ľoptique des deux auteurs, nous montrons que cette science a servi de paradigme a la metaphysique du philosophe. Nous considerons aussi bien la methode (correction des aberrations) que le contenu (la Nature, comme la lumiere, est un milieu enveloppant et expressif). Mais il y a ?on;autres paradigmes: la mecanique pendulaire, par exemple, dont la contradiction avec ľoptique peut etre attenuee, grâce a la geometrie de Huygens (theorie des developpees); ou encore le debut de la theorie des probabiles, au moyen de laquelle nous presentons une nouvelle interpretation des fins de ľElhique. Summary The purpose of this article is to study some aspects of the relations between Huygens and Spinoza. After a short examination of the two authors' optics, we show that this science has been used as a “paradigm” for philosopher's metaphysics. We take account of the method (correction of aberrations) as well as the matter (Nature, like light, is an involving and expressive medium). But there are other paradigms: pendulum mechanics, for example, those contradiction with optics may be reduced, thanks to Huygens' geometry (theory of developed curves); or probability theory's beginning, with which we present a new explanation of the aims of Ethics. Zusammenfassung In diesem Artikel werden einige Aspekte des Verhaltnisses zwischen Huygens und Spinoza untersucht. Nach einer Untersuchung der Optik der beiden Autoren wird gezeigt, dass diese Wis-senschaft als Paradigma fur die Metaphysik des Philosophen gedient hat. Wir ziehen sowohl die Methode (Korrektur der Abweichungen) als auch den Inhalt (die Natur, wie das Licht, ist ein umfangendes und expressives Medium) in Betracht. Es gibt aber auch andere Paradigmen: die Mechanik des Pendels, z. B., deren Widerspruch mit der Optik dank der Geometrie Huygens' abgeschwacht werden kann; oder auch der Beginn der Wahrscheinlichkeitstheorie, mit deren Hil-fe wir eine neue Interpretation der Zwecke in der Ethik vorlegen.

International audience; Based on field studies in the Northern California and on available literature, the goal is to bring what is generically called "nanotechnology" back to the convergence of the three main fields of research and innovation of the last fifty years: biotechnology, nanosciences and ICT. This convergence is a long-term process requiring specific inquiries in human and social sciences, quite different from usual (largely) misleading debates.; L'objectif est de ramener ce qu'on nomme génériquement " nanotechnologies " au processus de convergence des trois grands champs de recherche et d'innovation des cinquante dernières années : les biotechnologies, la nanoscience des matériaux et les technologies numériques. Cette convergence est un projection de long terme qui virtuellement reconfigure les relations entre la recherche, l'innovation, les institutions politiques et le développement politique et social. Cette convergence est un type de gouvernance de la société par la science et la technologie. L'objectif est de montrer qu'un processus d'une telle ampleur requiert une réponse adaptée des sciences humaines et sociales, en particulier la formation d'une épistémologie politique.

International audience

International audience