History of Western Philosophy - Bertrand Russell [93]
I come now to understanding of numbers. Here there are two very different things to be considered: on the one hand, the propositions of arithmetic, and on the other hand, empirical propositions of enumeration. '2 + 2 = 4' is of the former kind; 'I have ten fingers' is of the latter.
I should agree with Plato that arithmetic, and pure mathematics generally, is not derived from perception. Pure mathematics consists of tautologies, analogous to 'men are men', but usually more complicated. To know that a mathematical proposition is correct, we do not have to study the world, but only the meanings of the symbols; and the symbols, when we dispense with definitions (of which the purpose is merely abbreviation), are found to be such words as 'or' and 'not', and 'all' and 'some', which do not, like 'Socrates', denote anything in the actual world. A mathematical equation asserts that two groups of symbols have the same meaning; and so long as we confine ourselves to pure mathematics, this meaning must be one that can be understood without knowing anything about what can be perceived. Mathematical truth, therefore, is, as Plato contends, independent of perception; but it is truth of a very peculiar sort, and is concerned only with symbols.
Propositions of enumeration, such as 'I have ten fingers', are in quite a different category, and are obviously, at least in part, dependent on perception. Clearly the concept 'finger' is abstracted from perception; but how about the concept 'ten'? Here we may seem to have arrived at a true universal or Platonic idea. We cannot say that 'ten' is abstracted from perception, for
any percept which can be viewed as ten of some kind of thing can equally well be viewed otherwise. Suppose I give the name 'digitary' to all the fingers of one hand taken together; then I can say 'I have two digitaries', and this describes the same fact of perception as I formerly described by the help of the number ten. Thus in the statement 'I have ten fingers' perception plays a smaller part, and conception a larger part, than in such a statement as 'this is red'. The matter, however, is only one of degree.
The complete answer, as regards propositions in which the word 'ten' occurs, is that, when these propositions are correctly analysed, they are found to contain no constituent corresponding to the word 'ten'. To explain this in the case of such a large number as ten would be complicated; let us, therefore, substitute 'I have two hands.' This means:
'There is an a such that there is a b such that a and b are not identical and whatever x may be, "x is a hand of mine" is true when, and only when, x is a or x is b.'
Here the word 'two' does not occur. It is true that two letters a and b occur, but we do not need to know that they are two, any more than we need to know that they are black, or white, or whatever colour they may happen to be.
Thus numbers are, in a certain precise sense, formal. The facts which verify various propositions asserting that various collections each have two members, have in common, not a constituent, but a form. In this they differ from propositions about the Statue of Liberty, or the moon, or George Washington. Such propositions refer to a particular portion of space-time; it is this that is in common between all the statements that can be made about the Statue of Liberty. But there is nothing in common among propositions 'there are two so-and-so's' except a common form. The relation of the symbol 'two' to the meaning of a proposition in which it occurs is far more complicated than the relation of the symbol 'red' to the meaning of a proposition in which it occurs. We may say, in a certain sense, that the symbol 'two' means nothing, for, when it occurs in a true statement, there is no corresponding