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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.