It remains to discuss the notion of the. This notion has been symbolically emphasized by Peano, with very great advantage to his calculus; but here it is to be discussed philosophically. The use of identity and the theory of definition are dependent upon this notion, which has thus the very highest philosophical importance.(§ 63 ¶ 1)

The word the, in the singular, is correctly employed only in relation to a class-concept of which there is only one instance. We speak of the King, the Prime Minister, and so on (understanding at the present time); and in such cases there is a method of denoting one single definite term by means of a concept, which is not given us by any of our other five words. It is owing to this notion that mathematics can give definitions of terms which are not concepts--a possibility which illustrates the difference between mathematical and philosophical definition. Every term is the only instance of some class-concept, and thus every term, theoretically, is capable of definition, provided we have not adopted a system in which the said term is one of our indefinables. It is a curious paradox, puzzling to the symbolic mind, that definitions, theoretically, are nothing but statements of symbolic abbreviations, irrelevant to the reasoning and inserted only for practical convenience, while yet, in the development of a subject, they always require a very large amount of thought, and often embody some of the greatest achievements of analysis. This fact seems to be explained by the theory of denoting. An object may be present to the mind, without our knowing any concept of which the said object is the instance; and the discovery of such a concept is not a mere improvement in notation. The reason why this appears to be the case is that, as soon as the definition is found, it becomes wholly unnecessary to the reasoning to remember the actual object defined, since only concepts are relevant to our deductions. In themoment of discovery, the definition is seen to be true, because the object to be defined was already in our thoughts; but as part of our reasoning it is not true, but merely symbolic, since what the reasoning requires is not that it should deal with that object, but merely that it should deal with the object denoted by the definition.(§ 63 ¶ 2)

In most actual definitions of mathematics, what is defined as a class of entities, and the notion of the does not then explicitly appear. But even in this case, what is really defined is the class satisfying certain conditions; for a class, as we shall see in the next chapter, is always a term or conjunction of terms and never a concept. Thus the notion of the is always relevant in definitions; and we may observe generally that the adequacy of concepts to deal with things is wholly dependent upon the unambiguous denoting of a single term which this notion gives.(§ 63 ¶ 3)