The Principles of Mathematics (1903)

§ 135

Having failed to define wholes by logical priority, we shall not, I think, find it possible to define them at all. The relation of whole and part is, it would seem, an indefinable and ultimate relation, or rather, it is several relations, often confounded, of which one at least is indefinable. The relation of a part to a whole must be differently discussed according to the nature both of the whole and of the parts. Let us begin with the simplest case, and proceed gradually to those that are more elaborate.(§ 135 ¶ 1)

  1. Whenever we have any collection of many terms, in the sense explained in the preceding chapter, there the terms, provided there is some non-quadratic propositional function which they all satisfy, together form a whole. In the preceding chapter we regarded the class as formed by all the terms, but usage seems to show no reason why the class should not equally be regarded as the whole composed of all the terms in those cases where there is such a whole. The first is the class as many, the second the class as one. Each of the terms then has to the whole a certain indefinable relation[93], which is one meaning of the relation of whole and part. The whole is, in this case, a whole of a particular kind, which I shall call an aggregate: it differs from wholes of other kinds by the fact that it is definite as soon as its constituents are known.(§ 135 ¶ 2)

  2. But the above relation holds only between the aggregate and the single terms of the collection composing the aggregate: the relations to our aggregate of aggregates containing some but not all the terms of our aggregate, is a different relation, though also one which would be commonly called a relation of part to whole. For example, the relation of the Greek nation to the human race is different from that of Socrates to the human race; and the relation of the whole of the primes to the whole of the numbers is different from that of 2 to the whole of the numbers. This most vital distinction is due to Peano[94]. The relation of a subordinate aggregate to one in which it is contained can be defined, as was explained in Part I, by means of implication and the first kind of relation of part to whole. If u, v be two aggregates, and for every value of x x is a u implies x is a v, then, provided the converse implication does not hold, u is a proper part (in the second sense) of v. This sense of whole and part, therefore, is derivative and definable.(§ 135 ¶ 3)

  3. But there is another kind of whole, which may be called a unity. Such a whole is always a proposition, though it need not be an asserted proposition. For example, A differs from B, or A's difference from B, is a complex of which the parts are A and B and difference; but this sense of whole and part is different from the previous senses, since A differs from B is not an aggregate, and has no parts at all in the first two senses of parts. It is parts in this third sense that are chiefly considered by philosophers, while the first two senses are those usually relevant in symbolic logic and mathematics. This third sense of part is the sense which corresponds to analysis: it appears to be indefinable, like the first sense—i.e., I know no way of defining it. It must be held that the three senses are always to be kept distinct: i.e., if A is a part of B in one sense, while B is a part of C in another, it must not be inferred (in general) that A is a part of C in any of the three senses. But we may make a fourth general sense, in which anything which is part in any sense, or part in one sense of part in another, is to be called a part. This sense, however, has seldom, if ever, any utility in actual discussion.(§ 135 ¶ 4)

§ 135 n. 1. Which may, if we choose, be taken as Peano's ∈. The objection to this meaning for ∈ is that not every propositional function defines a whole of the kind required. The whole differs from the class as many by being of the same type as its terms.

§ 135 n. 2. Cf. e.g. F. 1901, § 1, Prop. 4. 4, note (p. 12).