Let us observe, to begin with, that the explicit mention of *any*, *some*, etc., need not occur in Mathematics: formal implication will express all that is required. Let us recur to an instance already discussed in connection with denoting, where `a` is a class and `b` a class of classes. We have(§ 87 ¶ 1)

Any

is equivalent to`a`belongs to any`b`

;(§ 87 ¶ 2)

implies that`x`is an`a`

implies`u`is a`b``x`is a`u`Any

is equivalent to`a`belongs to a`b`

implies`x`is an`a`there is a

`b`, say`u`, such that`x`is a`u`^{[63]};(§ 87 ¶ 3)Any

is equivalent to`a`belongs to some`b`there is a

;(§ 87 ¶ 4)`b`, say`u`, such that

implies`x`is an`a``x`is a`u`

and so on for the remaining relations considered in Chapter V. The question arises: How far do these equivalences constitute definitions of *any*, *a*, *some*, and how far are these notions involved in the symbolism itself?(§ 87 ¶ 5)

The variable is, from the formal standpoint, *the* characteristic notion of Mathematics. Moreover it is *the* method of stating general theorems, which always *mean* something different from the intensional propositions to which logicians such as Mr Bradley endeavour to reduce them. That the meaning of an assertion about all men or any man is different from the meaning of an equivalent assertion about the concept *man*, appears to me, I must confess, to be a self-evident truth--as evident as the fact that propositions about John are not about the *name* John. This point, therefore, I shall not argue further. That the variable characterizes Mathematics will be generally admitted, though it is not generally perceived to be present in elementary Arithmetic. Elementary Arithmetic, as taught to children, is characterized by the fact that the *numbers* occurring in it are constants; the answer to any schoolboy's sum is obtainable without propositions concerning *any* number. But the fact that this is the case can only be proved by the help of propositions about *any* number, and thus we are led from schoolboy's Arithmetic to the Arithmetic which uses letters for numbers and proves general theorems. How very different this subject is from childhood's enemy may be seen at once in such works as those of Dedekind^{[64]} and Stolz^{[65]}. Now the difference consists simply in this, that our numbers have now become variables instead of being constants. We now prove theorems concerning `n`, not concerning 3 or 4 or any other particular number. Thus it is absolutely essential to any theory of Mathematics to understand the nature of the variable.(§ 87 ¶ 6)

Originally, no doubt, the variable was conceived dynamically, as something which changed with the lapse of time, or, as is said, as something which successively assumed all values of a certain class. This view cannot be too soon dismissed. If a theorem is proved concerning `n`, it must not be supposed that `n` is a kind of arithmetical Proteus, which is 1 on Sundays and 2 on Mondays, and so on. Nor must it be supposed that `n` simultaneously assumes all its values. If `n` stands for any integer, we cannot say that `n` is 1, nor yet that it is `2`, nor yet that it is any other particular number. In fact, `n` just denotes `any` number, and this is something quite distinct from each and all of the numbers. It is not true that 1 is any number, though it is true that whatever holds of any number holds of 1. The variable, in short, requires the indefinable notion of *any* which was explained in Chapter V.(§ 87 ¶ 7)

§ 87 n. 1. Here there is a

where `c`,`c` is any class, is defined as equivalent to If

↩`p` implies `p`, and

implies `x` is a `c``p` for all values of `x`, then `p` is true.

§ 87 n. 2. Was sind und was sollen die Zahlen? Brunswick, 1893. ↩

§ 87 n. 3. Allgemeine Arithmetik, Leipzig, 1885. ↩

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