Note note 8 — ANALYSIS OF THE PHENOMENA OF THE HUMAN MIND James Mill

8 This conception of a geometrical line, as the abstract, of which a physical line is the corresponding concrete, is scarcely satisfactory. An abstract name is the name of an attribute, or property, of the things of which the concrete name is predicated. It is, no doubt, the name of some part, some one or more, of the sensations composing the concrete group, but not of those sensations simply and in themselves; it is the name of those sensations regarded as belonging to some group. Whiteness, the abstract name, is the name of the colour white, considered as the colour of some physical object. Now I do not see that a geometrical line is conceived as an attribute of a physical object. The attribute of objects which comes nearest to the signification of a geometrical line, is their length: but length does not need any name but its own; and the author does not seem to mean that a geometrical line is the same thing as length. He seems to have fallen into the mistake of confounding an abstract with an ideal. The line which is meant in all the theorems of geometry I take to be as truly concrete as a physical line; it denotes an object, but one purely imaginary; a supposititious object, agreeing in all else with a physical line, but differing from it in having no breadth. The properties of this imaginary line of course agree with those of a physical line, except so far as these depend on, or are affected by, breadth. The lines, surfaces, and figures contemplated by geometry are abstract, only in the improper sense of the term, in which it is applied to whatever results from the mental process called Abstraction. They ought to be called ideal. They are physical lines, surfaces, and figures, idealized, that is, supposed hypothetically to be perfectly what they are only imperfectly, and not to be at all what they are in a very slight, and for most purposes wholly unimportant, degree.—Ed.