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Empirical Clustering and Classic Hierarchies

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Towards a General Theory of Classifications

Part of the book series: Studies in Universal Logic ((SUL))

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Abstract

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.

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Notes

  1. 1.

    Finding the number of partitions of a set is different from finding the number of partitions of a number n, but it is connected to it (see [5, 307]) and our Sect. 3.4.2.

  2. 2.

    Prof. Stéphanie Ruphy, a former student of D. Parrochia in Toulouse, has shown that stellar kinds will not please the essentialist monism, because of the existence of continuous parameters, transitory properties, and several kinds to which a given star may belong, so that stellar kinds cannot be said “natural kinds”. See [432], 1109–1120: “Not only does the stellar world not come prepackaged with a unique set of objective, privileged (in an essentialist sense) divisions, but also it does not come prepackaged with objective divisions, tout court” (1118). But, on the opposite, stellar kinds will not please the pluralist embracing promiscuous realism. There exist in fact objective properties connected to the main aims of scientific research in Astrophysics. “Astrophysicists want to know how stars form, evolve, and disappear. Their theoretical understanding of the behavior of gaseous spheres tells them that parameters such as temperature, density, or mass loss are determinant parameters in stellar evolutionary processes, whereas proper motion or distance from the earth are not; hence, we have their choice of the former, and not the latter, as taxonomic parameters. In short, kind membership is conferred by structural properties central for explaining a large variety of stellar behaviors (and those properties translate into spectral features that are directly observable)” (1111).

  3. 3.

    On the contrary, in the case of a similarity coefficient s, the objects e and e′ would have to be as much more similar as s(e,e′) gets a bigger value. On the definition of a dissimilarity coefficient, see Sect. 3.5.1.

  4. 4.

    We resume here the presentation of Boorman and Olivier [58]. For another presentation, see B. Leclerc [299].

  5. 5.

    The same type of generalization has been presented in the form of fuzzy equivalences in the case of certain similarities and ultrasimilarities proposed by some other authors (see, for instance, [433, 434]).

  6. 6.

    In a strict sense, a similarity relation S is a reflexive, symmetric, but not transitive relation.

  7. 7.

    In fact, this assumption of empirical clustering may be questioned. In particular, Russell (see [436]) has shown that Keynes’ principle of limited variety (Treatise on probability, Chap. XXII, 258) was not valid, so that classification constraints cannot be explained by empirical observations but are to be founded on mathematical regularities.

  8. 8.

    The question raised with the Russell’s paradox leads, beyond the problematic answer Russell himself has proposed with the theory of types, to the beginning of theories of universes with classes and sets. After the classical solution of Zermelo and Fraenkel (the ZF axiomatic), will come non-classical theories like those of Finsler, Kelley-Morse, Quine, Von Neumann, etc. Some interesting attempts of the French mathematician Claude Frasnay about new definitions of classes and sets must also be mentioned here (see [180]).

  9. 9.

    Plato (Protagoras 331de), already said that “it is not fair to describe things as like which have some point alike, however small, or as unlike that have some point unlike”. Nelson Goodman goes further in the beginning of a well-known paper: “Similarity, I submit, is insidious. And if the association here with invidious comparison is itself invidious, so much the better. Similarity, ever ready to solve philosophical problems and overcome obstacles, is a pretender, an impostor, a quack. It has, indeed, its place and its uses, but is more often found where it does not belong, professing powers it does not possess.” (See [200], 437.) Of course, resemblance alone is not enough for representation, it may be superfluous in the case of descriptions of replicas of inscriptions or events, it does not explain metaphors and does not account for our predictive, or more generally, our inductive practice. If defined between particulars, it does not suffice to determine qualities and can hardly be measured in terms of possession of common characteristics: the “seven strictures” constitute a relentless criticism. Stressed again in a more recent book (see [201]), Goodman’s opinion received many comments and surely made a deep impression in philosophers or psychologists of the end of the XXth century. However, in the beginning of the 2000s, Hahn and Ramscar try to couch categorization in terms of more sophisticated and precise notions of similarity (see [222], and the review of Bradley C. Love [321]). Moreover, if we can easily share some of the critics of Goodman, we support the idea—already present, as we have seen, in Russellian rejection of the doctrine of natural kinds ([436], 461) or in Quine’s criticism of perceptual similarity in favor of a more intellectual way of conceptualizing category membership (see his paper on “natural kinds” in [406] and also the comments of [463])—that similarity must be founded on “good” mathematical structures.

  10. 10.

    In fact it is well known, since the middle of the 1980s, that such a problem is, in the case or hierarchical clustering, NP-hard (see [284]), and, in the case of overlapping clustering, NP-complete (see [285]).

  11. 11.

    A “key” is the operator that gives access to some set of documents, according to a particular aspect of the requirement. As the aspects of the requirement may be totally ordered, so that some of them which are, for instance, more general or more basic ones might be processed before others, one can define also a total order over the set of keys.

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Authors and Affiliations

  1. Department of Philosophy, Université Jean Moulin – Lyon III, Lyon, France

    Daniel Parrochia

  2. Ecole Nationale Supérieure des Sciences de l’Information et de la Bibliothèque, Villeurbanne, France

    Pierre Neuville

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  1. Daniel Parrochia
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Parrochia, D., Neuville, P. (2013). Empirical Clustering and Classic Hierarchies. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_3

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