Now suppose that I am looking at a bright red patch. I may say ‘this is my present percept’; I may also say my present percept exists’; but I must also say ‘this exists,’ because the word ‘exists’ is only significant when applied to a description as opposed to a name. This disposes of existence as one of the things that the mind is aware in objects.
I come now to understanding of numbers. Here there are two very different things to be considered: on the one hand, the propositions or 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 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 with only symbols.
The History of Western Philosophy (1972 ed.), 155.
