Abstract
In principle, a general theory of classifications should have to say, first, what is a classification, or even what it must be. So, in a textbook or in a monograph, we should have to begin with some rigorous definition of the notion of “classification”. From this mathematical standpoint, a good method would be to take some weak structure, for instance a “system of classes” or a “hypergraph” in the sense of Berge (Graphes et hypergraphes, 1970), and, by adding restrictive properties, to construct richer structures (for instance: covers, partitions, hierarchy of partitions…), for getting at the end a more precise view of the notion available in some special fields of knowledge. However, using such a method would imply that we already know for what we are searching, i.e. the means of unifying the whole domain of classifications.
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Notes
- 1.
Some similarities between forms (Gestalten) and classifications have already been mentioned in an ancient paper of Grelling and Oppenheim (see [210], 92–96), where forms are investigated, following Carnap, through the help of the notions of classifier (a concept which determines a classification), of state-classifier (a concept which assign certain values to the “positions” in a “domain of positions”), of connection, division, articulation and transposition. The forms themselves, defined as invariants of transpositions (like melodies, that can be played in different tones), are, in fact, equivalence classes of correspondences.
- 2.
Maybe it is an imaginary one, which has never actually existed, but it does not matter.
- 3.
We can observe that this class is not a simple class. It is, obviously, the class C of all classes included in the classification. So, we could believe it is the classification itself. But it is not, since there are other classes outside which are defined by other predicates.
- 4.
The partition of the ten main classes thus gives successively 100 divisions and 1000 sections.
- 5.
As we have seen, they share this property with the classification of H. Bliss.
- 6.
We shall explicitly introduce relational structures in Chap. 6.
- 7.
A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group.
- 8.
Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups.
- 9.
Let us say, briefly, that a measure-preserving transformation T:X→X, on a measure-preserving dynamical system (X,B,μ,T), where X is a set, B a sigma-algebra over X, μ a probability measure, is a transformation which is measure-preserving, i.e. which preserves the measure μ so that each A∈B satisfies μ(T −1(A))=μ(A).
- 10.
In the case of infinite classifications, this requirement, of course, must be weakened: we may only want the (infinite) cardinal of the classification to be less than or equal to the (infinite) cardinal of the set of objects to be classified.
- 11.
The sense of it will have to become clearer.
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Parrochia, D., Neuville, P. (2013). Philosophical Problems. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_1
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