"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 he 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."
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Bertrand Russell A History of Western Philosophy (1945), Book One, Part II, Chapter XVIII, Knowledge and Perception in Plato, p. 155
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