Abstract
As of now, we do not know exactly what the structure of the set of classifications over a set is, and a complete theory of classifications is still to arise. Only a few parts of it exist in scientific literature. As we have seen, the more convincing of them are concerned, for the moment, with finite partitions, hierarchies and chains. But the global situation is quite different. The boisterous ocean of partitions, the unruly storm of classes, usually escape the searchers and widely elude, at the present time, the most powerful mathematical tools we have at our disposal. If we try, however, to shed new light on the problem, we must be able to tell the difference between the approach of a general theory of classifications, carried out in Chap. 6, and the one we performed in this chapter, that we have called, using a prefix familiar to mathematicians, “metaclassification”.
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Notes
- 1.
This structure is not so far from the one invented by Apostel (see [8]) for classifying languages, and which was a graph whose points were themselves graphs.
- 2.
Conjectures and even more or less imprecise—if not utopian—plans (see, for example, the Langlands program, sometimes called “Langlands philosophy”) belong to mathematics.
- 3.
Proteomics is the large-scale study of proteins, particularly their structures and functions.
- 4.
The meanings of these terms are as follows: cubic (or isometric): three equal axes, intersecting at right angles; hexagonal: three equal axes, intersecting at 60°, angles in a horizontal plane, and a fourth, longer or shorter, axis, perpendicular to the plane of the other three; tetragonal: two equal, horizontal axes at right angles and one axis longer or shorter than the other two and perpendicular to their plane; orthorhombic: three unequal axes intersecting at right angles; monoclinic: three unequal axes, two intersecting at right angles and the third at an oblique angle to the plane of the other two; rhombohedral (or trigonal): three equal axes intersecting at oblique angles); and triclinic: three unequal axes intersecting at oblique angles.
- 5.
Maybe Pierce should have added “with bijection”, because, otherwise, we would have \(\mathrm{End}(A) = \mathrm{Mor}_{\mathfrak{U}}(A, A)\). This detail would not matter much if it was possible to get Pierce’s results with endomorphisms as well as with automorphisms. But unfortunately, it is not the case, and we cannot write, for example: End(α)=αϕα −1.
- 6.
In particular, Pierce does not say how we can construct some empirical C-category, nor how we can build surjective functors able to solve the classification problem of C-categories.
- 7.
It is the case very often—still now—in life sciences. Take the example of Eucaryotes: it is quite common to observe that classic and cladistic viewpoints lead to different classifications (see [259]). In the case of Procaryotes, the situation is even more confusing, so that some disciples of Hennig (see [237, 238]) like Woese think that this domain is divided into three (and not only two) classes (see the paper of Mayr (see [337]) and the answer of Woese (see [507])).
- 8.
If X and Y are topological spaces, a function f:X⟶Y is a local homeomorphism if, for every point x in X, there exists an open set U containing x, such that f(U) is open in Y and f|U:U⟶f(U) is a homeomorphism.
- 9.
We shall just say that, in the language of Category theory, a presheaf of C on X is a contravariant functor from the category of open subsets of X, with inclusions as morphisms, to C (“contravariant” means that the restriction morphisms F(U)⟶F(V) go in the opposite direction of the inclusion V⊂U). Now a sheaf is a presheaf satisfying an additional axiom (that we do not make precise) which captures the idea of pasting together the structures F(U).
- 10.
We may even get a bit stronger structure. A classification (observed by \(\mathcal{C}\)) is, in fact, what Godement called an “espace étalé”. In the language of Grothendieck, where \(\mathcal{U}\) is a Grothendieck universe, if Ouv(X) is now the category of open sets of X with the canonical topology, Top(X) will be the “topos” of the \(\mathcal{U}\)-sheaves on Ouv(X). This topos is equivalent to the category of the “étalés” topological spaces over X when we associate to some space X′ on X the sheaf U↦Γ(X′/U)=Hom X (U,X′) on Ouv(X) (see [13], 311–312). So the category of classifications (observed by \(\mathcal{C}\)) is, in fact, a Grothendieck topos.
- 11.
It means that w 1 and w 2 are necessarily complex numbers, as a trivial calculus may show.
- 12.
This representation of classifications is a very common one, especially in automatic classification theory, where, as we have said before (see 3.6.1), it brings some help for interpreting the direction and extension of classes in the factorial space (see [257]). Here we generalize this representation, and apply it even in the case when we have not defined an explicit distance or dissimilarity between the objects to be classified.
- 13.
It is clear that the intersection of some classes (i.e. parts of partitions or covers) is the empty set. So an empty class may be defined on the set of all classes. However, we cannot give any sense to the notion of an empty partition (resp. cover) and no more to the notion of an “empty classification”. So there is no topology in the classic sense on the set of partitions or on the set of classifications.
- 14.
We must thank Prof. Luisa Iturrioz for having suggested to us this possible approach, that we cannot develop here.
- 15.
Simplicial theory allows us to define polyedra or complexes. By associating a dual cell to every simplex, one can facilitate the description of them. But the boundary of an n-dimensional simplex is an (n−1)-dimensional chain, and the boundary of a chain is defined by linearity. Within our metaclassification theory, the group of boundaries depends on space, not on simplicial decomposition. For avoiding any misunderstanding, the best is to define ellipsoids according to a minimum, average or maximum number of connections, computed by the means of cell transformations. But it is always possible to put some cell boundaries in correspondence within the simplexes. When some order of connections allows us to speak of a manifold (in this case, an ellipsoid) which is not so different from another manifold (in this case, a cell), using n-tuple open sets of \(\mathbb{R}^{n}\) happens to be far less precise. The method we adopt here allows us to compute numbers of connections by the means of transformations expressed into incidence matrices. Mathematicians will certainly put them in evidence in the future.
- 16.
One may speak, sometimes, of “Poincaré rotation numbers”. In the beginning, the rotation number is an invariant of homeomorphisms of the circle. It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. if fand g are two homeomorphisms of the circle and hof=goh for a continuous map h of the circle into itself (not necessarily homeomorphic) then f and g have the same rotation numbers. According to the fact that the rotation number of f may be rational or irrational, Poincaré and Denjoy proved that one may construct a topological classification of homeomorphisms of the circle. In the case of the torus T 2, isotopic to identity, the general rotation set is convex and is equal to the convex hull of the pointwise and measure rotation sets. We get results similar to those for circle maps. In particular, if the rotation set is a polygon, then all its interior points and vertices are “good”.
References
Apostel, L.: Le problème formel des classifications empiriques. In: La Classification dans les Sciences, pp. 157–230. Duculot, Bruxelles (1963)
Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des topos et cohomologie étale des schémas. Springer, Berlin (1972). (SGA4-1)
Bahnot, G., Alexe, G., Venkataraghavan, B., Levine, A.: A robust metaclassification strategy for cancer detection from MS data. Proteomics 6(2), 592–604 (2006)
Berger, M.: Geometry II. Springer, Berlin (1987)
Beth, E.W.: The Foundations of Mathematics, reviewed edn. Harper, New York (1966)
Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume. Math. Ann. 70(3), 297–336 (1911)
Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung). Die Gruppen mit einem endlichen Fundamentalbereich. Math. Ann. 72(3), 400–412 (1912)
Bourbaki, N.: Théorie des Ensembles. Hermann, Paris (1966)
Bourbaki, N.: Eléments de mathématiques, Topologie générale 1. Hermann, Paris (1971)
Dzega, D., Pietruszkiewicz, W.: Classification and metaclassification in large scale data mining application for estimation of software projects. In: IEEE SMC UK and RI, 8th Conference on Cybernetic Systems, September 9–10, 2009, pp. 35–52
Faure, R., Heurgon, E.: Structures ordonnées et algèbres de Boole. Gauthier-Villars, Paris (1971)
Fisher, R.A.: The use of multiple measurements in taxonomic problems. Ann. Eugenics 7, 179–188 (1936)
Fung, B.Y.M., Ng, V.T.Y.: Meta-classification of multi-type cancer gene expression data. BIOKDD, 31–39 (2004)
Godement, R.: Topologie algébrique et théorie des faisceaux. Hermann, Paris (1960)
Gramain, A.: Formes d’intersection et d’enlacement sur une variété. Mém. Soc. Math. Fr. 48, 11–19 (1976)
Hammon, P.S., de Sa, V.R.: Preprocessing and meta-classification for brain-computer interfaces. IEEE Trans. Biomed. Eng. 54(3), 518–525 (2007)
Hennig, W.: Grundzuge einer Theorie der phylogenetische Systematik. Deutscher Zentralverlag, Berlin (1950)
Hennig, W.: Phylogenetic Systematic. University of Illinois Press, Champaign (1966)
Heyting, A.: Die intuitionistische Grundlegung der Mathematik. Erkenntnis 2, 106–115 (1931)
Jambu, M.: Classification automatique pour l’analyse des données, 2 tomes. Bordas-Dunod, Paris (1978)
Janvier, Ph., Tassy, P., Thomas, H.: Le cladisme. La Recherche 117, 1396–1406 (1980)
Jech, Th.: Set Theory. Springer, Berlin (2003)
Lin, W.-H., Hauptmann, A.: Meta-classification: Combining Multimodal Classifiers. In: Mining Multimedia and Complex Data. LNCS, vol. 2797/2003, pp. 217–231. Springer, Berlin (2003)
Maimon, O., Rokach, L.: Decomposition Methodology for Knowledge Discovery and Data Mining. World Scientific, Singapore (2005)
Markov: Insolubility of the problem of homeomorphy. In: Proc. Intern. Congress of Mathematics, pp. 300–306. Cambridge University Press, Cambridge (1958)
Mayr, E.: Two empires or three? Proc. Natl. Acad. Sci. USA 95(17), 9720–9723 (1998)
Mc Lane, S., Gehring, F.W.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998)
Miyamito, S., Nakayama, K.: A hierarchical representation of citation relationship. IEEE Trans. Syst. Man Cybern. smc-10(12), 899–903 (1980)
Morariu, I.D., Vintan, L.N., Tresp, V.: Meta-classification using SVM classifiers for text documents. Int. J. Math. Comput. Sci. 1(1), 54–59 (2005)
Mormann, Th.: Russell’s many points. In: Hieck, A., Leitgeb, H. (eds.) Reduction, Abstraction, Analysis, pp. 239–258. Ontos Verlag, Frankfurt (2009)
Neuville, P.: Vers des formalismes algébriques pour la théorie des classifications polycopié (1984). Université Lyon 1
Nour, A.: Sémantique algébrique d’un système logique basé sur un ensemble ordonné fini. Math. Log. Q. 45(4), 457–466 (1999)
Parrochia, D.: Mathématiques et existence. Champ Vallon, Seyssel (1991)
Parrochia, D.: Classifications, histoire et problèmes formels. Bull. Soc. Francophone Class., 1998 (not paged: 20 p.)
Pierce, R.S.: Classification problems. Math. Syst. Theory 4(1), 65–80 (1970)
Ng, C.H., Rasiowa, H.: Plain semi-post algebras as a poset-based generalization of post algebras and their representability. Stud. Log. 48, 509–530 (1989)
Russell, B.: An Essay in Foundation of Geometry, Routledge, London (1896), reed. 1996
SanJuan, E.: Algèbres de Heyting avec opérateurs booléens et applications aux systèmes d’information. Thèse de doctorat, Université Claude Bernard, Lyon 1, 22 décembre 2000
Serre, J.-P.: Cours d’Arithmétique. P.U.F., Paris (1980)
Siersdorfer, S., Weikum, G.: Using restrictive classification and meta classification for junk elimination. In: Advances in Information Retrieval, ECIR 2005 (European Conference on IR research No 27, Santiago de Compostela, 21–23 March 2005). LNCS, vol. 2005, pp. 287–299
Smith, B., Kumar, A.: Towards a proteomics metaclassification. In: Proceedings of Fourth IEEE Symposium on Bioinformatics and Bioengineering (BIBE2004), Taiwan, 19–21 May 2004
Wallace, A.H.: Introduction à la topologie algébrique. Gauthier-Villars, Paris (1973)
Woese, C.R.: Default taxonomy: Ernst Mayr’s view of the microbial world. Proc. Natl. Acad. Sci. USA 95, 11043–11046 (1998)
Zassenhaus, H.: Über einen Algorithmus zur Bestimmung der Raumgruppen. Comment. Math. Helv. 21, 117–141 (1948)
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Parrochia, D., Neuville, P. (2013). Metaclassification. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_7
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