# The Principles of Mathematics (1903)

## § 75

We may ask, as suggested by the above discussion, what is to be said of the objects denoted by a man, every man, some man, and any man. Are these objects one or many or neither? Grammar treats them all as one. But to this view, the natural objection is, which one? Certainly not Socrates, nor Plato, nor any other particular person. can we conclude that no one is denoted? As well might we conclude that every one is denoted, which in fact is true of the concept of every man. I think one is denoted in every case, but in an impartial distributive manner. Any number is neither 1 nor 2 nor any other particular number, whence it is easy to conclude that any number is not any one number, a proposition at first sight contradictory, but really resulting from an ambiguity in any, and more correctly expressed by any number is not some one number. There are, however, puzzles in this subject which I do not yet know how to solve.(§ 75 ¶ 1)

A logical difficulty remains in regard to the nature of the whole composed of all the terms of a class. Two propositions appear self-evident: (1) Two wholes composed of different terms must be different; (2) A whole composed of one term only is that one term. It follows that the whole composed of a class considered as one term is that class considered as one term, and is therefore identical with the whole composed of the terms of the class; but this result contradicts the first of our supposed self-evident principles. The answer in this case, however, is not difficult. The first of our principles is only universally true when all the terms composing our two wholes are simple. A given whole is capable, if it has more than two parts, of being analyzed in a plurality of ways; and the resulting constituents, so long as analysis is not pushed as far as possible, will be different for different ways of analyzing. This proves that different sets of constituents may constitute the same whole, and thus disposes of our difficulty.(§ 75 ¶ 2)