logic, or psychology, these definitions would have to be reduced to forms far more precise; but I purposely refrain from an attempt at exact definition, because I wish to remain on ground common to all who have made the matter in hand the subject of their investigation. For my purpose, it is of little consequence whether or not the distinction here indicated between representations and concepts is accurate and clear; nor is it necessary to determine the exact nature of the relations established in conception between the constituents of a concept, or between the various concepts themselves; it is sufficient to know that both in the representation and in the concept we have in some form a complex of attributes which are ultimately, in the case of material objects at least, traceable to sensible experience, and that the elaboration of representations into concepts involves the establishment of some sort of mental relations between their elements, as well as between the several concepts themselves.
At this point, it is important to guard against a confusion which naturally arises from the fact that logicians and psychologists habitually illustrate the evolution of concepts by examples taken from the abstract sciences. There is a very wide distinction between the relation of a concept to the object of thought in mathematics, for instance, and the corresponding relation between a representation, or concept of a material object, to that object itself. In mathematics, as in all the sciences which are conversant with single relations or groups of relations established (and, within the limits of the constitutive laws of the mind, arbitrarily established) by the mind itself, all concepts are exhaustive in the sense that they imply, if they do not explicitly state, all the properties belonging to the object of thought. Not only the constituents of such an object, but also the laws of their interdependence, being determined by the intellect, they may be strictly deduced, each from the other. Thus, a parabola is a line, every point in which is equidistant from a fixed point and a given straight line: that is one
- The truth of the proposition that the system of forms and relations, whose discussion constitutes the science of mathematics, is of purely subjective determination, does not involve the assumption (erroneously attributed to Kant, who, on the very first page of his "Critique of Pure Reason," expressly draws the distinction between the "beginning of all knowledge with experience," and "the derivation of all knowledge from experience"), that the mind is furnished a priori with ready-made ideas or concepts; nor is it affected by the circumstance that these forms and relations are ultimately referable to the facts of sensible experience. Mill's refusal to recognize this has betrayed him into writing the extraordinary fifth chapter of the second book of his "System of Logic," in which he questions—albeit falteringly—the necessary truth of the propositions of geometry. The inevitable outcome of this is seen in the writings of Mr. Buckle, who not only boldly asserts that there are no lines without breadth (he strangely forgets the thickness), but also that the neglect of this breadth by the geometrician vitiates his conclusions. His comfort is that the error, after all, is not very considerable. "Since, however," is his language ("History of Civilization in England," ii., 342, Appletons' edition), "the breadth of the faintest line is so slight as to be incapable of measurement, except by an instrument used under the microscope, it follows that the assumption, that there