Page:Mind (New Series) Volume 15.djvu/518

 IV. SYMBOLIC REASONING (VIII.). 1 BY HUGH MACCOLL. 1. THE main subject of this article will be paradoxes. We meet with them everywhere in logic, in mathematics, and in science generally. They nearly always spring from the ambiguities and obscurities more or less inherent in all languages the symbolic languages of logic and mathematics not excepted. The same words or symbols suggest different concepts to different minds, and even to the same mind at different times. Take the word infinite or infinity, which mathematicians usually represent by the symbol oo. Works on modern geometry often speak of a series of straight lines meeting at " the point at infinity," when, as a matter of fact, their points of intersection at infinity, instead of being one, may be many, or may even be non-existent. And they also speak of a series of points being all in li the line at infinity," when there may be no real line containing all the said points, either at infinity or elsewhere. Similarly algebraists some- times speak of the infinity oo as if it were one definite huge number or ratio which differed from a million or a billion in only one respect, that of being much larger ; whereas there are numberless infinities, each of which differs from a million or a billion not only in being much larger, but also in another important quality of which I shall speak presently. The truth is that the real ' infinity ' of mathematics denotes not a single individual ratio but a whole class, and that the symbol oo, when it represents a reality (which it does not always), sometimes stands for an infinite ratio oo 1} at another time or in another place for a different infinite ratio oo 2 , at another time or place for an infinite ratio oo 3 , and so on. Thus, when we meet such a statement as oo 2 oo = ix , which seems to assert the absurdity that infinity is equal to its double and also to its half, we must understand it to mean oo ! = 2oo 2 = 00 3, a perfectly self-consistent statement which only asserts that the infinity oo x is double the infinity oo 2 and 1 For VII. see MIND, July, 1905.