The aims and justification of taxonomy are discussed, followed by an examination of the foundations of ordination and classification. The Adansonian or phenetic philosophy is critically examined and it is concluded that its claims of objectivity and precision are ill-founded, since subjective or arbitrary choices and definitions are necessary concerning acceptable or relevant attributes, homologies and correspondences, measures and commensurabilities of attributes, and measures of similarity. Phylogeny, represented topologically as a temporal branching sequence, is held to be the nearest approach to a firm basis of reference in nature for biological classification. The charge that phylogenetic reconstruction involves viciously circular reasoning is discussed and rejected, though some positive feedback is admitted. The theory and application of numerical taxonomy are discussed in general and the potential value of numerical phyletics is stressed. The possibility of using DNA base matching as a solid foundation is briefly examined. It is shown that, while phenetic classifications are infinitely variable, topological phyletic reconstructions do not themselves supply the kind of taxonomy that is usually demanded, since they do not adequately express significant evolutionary changes in patterns of organisation. No optimal classification can be defined, but improvement is possible up to a point of inherent instability. While the underlying facts and processes can be scientifically studied as part of systematics, classification itself remains largely a disclipined art, which is not convertible to an exact science by any form of arbitrary quantification. The necessity for compromise and continued synthesis is stressed. The suggested replacement of the Linnaean hierarchy by “numericlature” is mentioned but held to be premature and to have serious disadvantages. Mathematical concepts are discussed wherever relevant to the foundations of the subject.