This books deals with the existing components of a general theory of classifications, which are mathematical in their presentation, and the reasons why they do not form a complete theory yet. The general approach is to provide a logical description of the classification structures and the algebraic properties of such structures.

The book starts with a discussion on formal concept analysis and ontologies, and explains the properties of the former and why this approach may lead to a combinatorial explosion. The authors then explore the consensual view that hierarchical tree structures are arguably the most important classification structure, and present the known results in the algebra of (finite) trees.

This first part is followed by a discussion on why trees are not the only structures needed for a classification, pointing to the need of structures that deal with overlapping classes. The study of these structures consists of the theory of generalized classifications, encompassing quasi-hierarchies, weak hierarchies, pseudo-hierarchies, and pre-pseudo-hierarchies. The algebra and topology of such structures is much less studied and understood than that of trees, and here the initial statement that, for the moment, a general classification theory “is but a dream” becomes more clear.

Other topics the authors cover include metaclassifications, a very important theme to manage a world in which the production of data is experiencing an exponential growth; an axiomatic theory of classification, a topic quite in accordance with the authors’ view that classification theory is a part of both mathematical and philosophical logic; and alternative theories that deal with higher infinite, inaccessible cardinals, and infinite classification. The presentation of this last topic is detached from the rest of the book, which is restricted to the study of finite trees and finite generalized hierarchies. As such, this last chapter does not follow the same methods and approaches as the previous ones, so it does not blend very well with the rest of the book.

The book does provide a global mathematical view of classifications--a view that has been absent from the literature of logic--describing existing results and pointing to the gaps that persist in the general theory. However, the purely mathematical approach may exclude from its reading audience researchers interested in a more human-centered view of classification, with psychological biases and implications on classification-based human decisions.