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)

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.