You ask me to pass to the consideration of knowledge of a new kind, knowledge of mathematical truth. “Here at least,” you say, “severe reasoning dominates supreme, and Imagination has no place.” “Two and one make three,” “The angles at the base of an isosceles triangle are equal:” “surely we may assume that Imagination has nothing to do with these propositions. They must be decided by pure Reason.” Never was assumption more grotesque. Excuse me; but by what other adjective can I characterize the statement that the Imagination has “nothing to do with” propositions for the very terms of which we are indebted to the Imagination? I maintain without fear of contradiction that the knowledge of these propositions requires an effort of the Imagination so severe that the very young and the completely untrained cannot attain to it. For, in the first place, what do you mean by “one,” “two,” and “three”? I have never had any experience of such things; nor have you; nor can you. “Two” oranges, “two” apples, and the like, we have had experience of, and can realize; but to think of “one” or “two” by themselves (“one” or “two” with “anythings”, or with “nothings” after them), “one” or “two” as “abstract ideas”—this really is a most difficult or rather (I am inclined to say) an impossible task. When I say “one” and “two,” I think I see before me dimly “one” or “two” dots or small strokes, and I perceive that two and one of these dots or strokes make up three dots or strokes. When I speak of “twenty” and “thirty,” I do not see any images of these existences; and when I say that “twenty” and “thirty” make “fifty,” I do not realize the process of addition at all visibly; I merely repeat the statement on the authority of previous observations and reasonings mostly made by others and not by myself. But so far as I approximate to the realization of an abstract number, I do it by a kind of negative imagination. And in any case we can hardly deny that all arithmetical propositions, since they employ terms that denote mere imaginary ideas, must be regarded as based on the imagination.