Daniel Parrochia

Université Lyon III - Jean Moulin
Lyon, France

Classification problems are one of the basic topics of scientific research : in mathematics and physics, as in natural sciences in general, in social sciences and, of course, in the domain of library and information sciences, taxonomies are very useful to organize an exponentially increase of knowledge and to perform information retrieval. But, from a strictly mathematical viewpoint, classes are also concrete sets that need a general theory, whose foundation might be different from that of usual sets.

The main purpose of this tutorial is not to provide a complete exposition of a perfect mathematical theory of classifications, that is, a general theory which would be available to any kind of them : hierarchical or not hierarchical, ordinary or fuzzy, overlapping or not overlapping, finite or infinite, and so on, founding all possible divisions of the real world. For the moment, such a theory is but a dream.

Our aim is essentially to expose the «state of art» of this moving field. We shall speak of some advances made in the last century, discuss a few tricky problems that remain to be solved, and, above all, show the very ways open for those who do not wish to stay any longer on the wrong track.

The three parts of the tutorial are the following ones:

1. History of classifications and epistemological problems.
After a brief introduction, we shall go on with a historical overview of the whole domain, recalling the long history of classifications, from the Greek (Plato and Aristotle) to Ranganathan (a well-known indian 20th libarian) and his faceted classification, through the advances made in the 18th century (Kant’s logic of natural classifications) and in the 19th century, the century of big library classifications (DDC and UDC especially).

2. Exposition of some formal models and search for a unified language.
In this second session, we shall introduce the mathematics of finite classifications, which are based, since G. Birkhoff, on the main concepts of order theory (partitions, chains of partitions, semilattice of chains...). We shall study, before all, the well-known domain of hierarchical classifications, that has been studied by the french school (Barbut, Monjardet, Benzecri, Lerman), but also have a look at some kinds of overlapping taxonomies (in the sense of Brucker and Barthelémy). In the end, we shall examine the problems raised by the idea of an algebra of classifications and investigate its possible models (reverse polish notation (Lukasiewicz), new forms of parenthesized products (Wedderburn-Etherington), or new kinds of k-algebras (dendriform algebras for trees, for example).

3. Towards a general theory of classifications.
Starding with the idea of a general theory of classification, as it has been developed in the XIXth century (A. Comte, A.P. de Candolle) and in the middle of the XXth (Léo Apostel), we shall recall the different types of classifications that exist now. Then we shall explain how we can represent them by means of a unique figure (ellipses of the plane) which leads to a kind of «classification of classifications» (or «metaclassification»). Here is the end of our own approach. But a general theory has to take into account the other points of view on classifications, even if it means criticizing them. One of them developed in Germany under the influence of K. Wille, through the notion of «conceptual analysis». This approach starts with the Galois lattice, that contains all the information we can get about objects and properties (which is sometimes too much information to be suitably handled and end in a useful classification). Other approaches suppose some extensions – more or less convincing – of the concept of «partition». It is the case, for example, of Barry Smith’s approach, who introduces the concept of «granular partition». We shall also speak briefly of the approach of Category theory, which developed in an interesting paper of R.S. Pierce in the 1970s. In the end, we shall say a few words about infinite classifications in relation with set theoretical problems.

Bibliography

Apostel L., «Le problème formel des classifications empiriques», La Classification dans les Sciences, Bruxelles, Duculot, 1963.

Barbut, M., Monjardet, B., Ordre et Classification, 2 tomes, Paris, Hachette, 1970.

Beth, E., Foundations of mathematics, Amsterdam North-Holland, 1965.

Brucker, F., Barthelémy, J.-P., Eléments de classifications, Paris, Hermès-Lavoisier, 2007.

Gondran, M., «La structure algébrique des classifications hiérarchiques», Annales de l'INSEE, 1976, 22-23, p. 181-189.

Jech, Th., Set Theory, Berlin-Heidelberg-New York, Springer Verlag, 2003.

Lerman, I. C., La classification automatique, Paris, 1970;

Neuville, P., «La difficulté de classer», Bulletin des Bibliothèques de France, t. 39, n°1, 1994.

Parrochia, D., «Trames, classifications», définitions, Math. Inf. Sci. hum., 29e année, no 116 (1991), pp. 29-46.

Parrochia, D., «Classifications, histoire et problèmes formels», Bulletin de la société francophone de classification, 1998 (non paginé : 20 p.)

Parrochia, D., Neuville P., Towards a general theory of classifications, Birkhäuser, Basel, 2012.

Neuville P., Parrochia D., «Epistémologie des classifications» in L. Obadia, G. Carret (ed.), Représenter, classer, nommer, Intercommunications, EME, Fernelmont, 2007, pp. 25-40.

Shelah, S., «Taxonomy of universal and other classes», Proceedings of the International Congress of Mathematics, Berkeley, Californie, 1986, pp. 154-162.