lacking particular representations, applicable alike to a house and to any other manifestation of the real. The same may be said of the cat and of the rose, since it is evident that cats and roses have appeared on the earth at a definite time and will disappear at another, and that while they endure, they can be looked upon as something fixed and precise, only when we have regard to some particular group of cats and of roses, indeed to one particular cat or rose at a definite moment of its existence (a gray cat or a black cat, a cat or a kitten; a white rose or a red rose, flowering or withered, etc.), elevated into a symbol and representative of the others. There is not, and there cannot be, a rigorous characteristic, which should avail to distinguish the cat from other animals, or the rose from other flowers, or indeed a cat from other cats and a rose from another rose. These and other fictional concepts are, therefore, representative, but not ultrarepresentative; they contain some objects or fragments of reality, they do not contain it all. [Sidenote: _or universals void of representations_] The conceptual fictions of the triangle and of free motion have an analogous but opposite defect. With them, it appears, we emerge from the difficulties of representations. The triangle and free motion are not something which begins and ends in time and of which we are not able to state exactly the character and limits. So long as thought, that is to say, thinkable reality, exists, the concept of the triangle and of free motion will have validity. The triangle is formed by the intersection of three straight lines enclosing a space and forming three angles, the sum of which, though they 'vary from triangle to triangle, is equal to that of two right angles. It is impossible to confuse the triangle with the quadrilateral or the circle. Free motion is a motion, which we think of as taking place without obstacles of any sort. It is impossible to confuse it with a motion to which there is any particular obstacle. So far so good. But if those conceptual fictions let fall the ballast of representations, they ascend to a zone without air, where life is impossible; or, to speak without metaphor, they gain universality by losing reality. There is no geometric triangle in reality because in reality there are no straight lines, nor right angles nor sums of right angles, nor sums of angles equal to that of two right angles. There is no free motion in reality, because every real motion takes place in definite conditions and therefore among obstacles. A thought, which has as its object nothing real, is not thought; and those concepts are not concepts but conceptual fictions. [Sidenote: _Critique of the doctrine which considers them to be erroneous concepts,_]