[Sidenote: The Meaning of Abstractness in Truth.]
§ 52. But though the abstractness involved in scientific description is open and deliberate, we must come to a more precise understanding of it, if we are to draw any conclusion as to what it involves. In his "Principles of Human Knowledge," the English philosopher Bishop Berkeley raises the question as to the universal validity of mathematical demonstrations. If we prove from the image or figure of an isosceles right triangle that the sum of its angles is equal to two right angles, how can we know that this proposition holds of all triangles?
"To which I answer, that, though the idea I have in view whilst I make the demonstration be, for instance, that of an isosceles rectangular triangle whose sides are of a determinate length, I may nevertheless be certain it extends to all other rectilinear triangles, of what sort or bigness soever. And that because neither the right angle, nor the equality, nor determinate length of the sides are at all concerned in the demonstration. It is true the diagram I have in view includes all these particulars; but then there is not the least mention made of them in the proof of the proposition."[140:6]
Of the total conditions present in the concrete picture of a triangle, one may in one's calculations neglect as many as one sees fit, and work with the remainder. Then, if one has clearly distinguished the conditions used, one may confidently assert that whatever has been found true of them holds regardless of the neglected conditions. These may be missing or replaced by others, provided the selected or (for any given investigation) essential conditions are not affected. That which is true once is true always, provided time is not one of its conditions; that which is true in one place is true everywhere, provided location is not one of its conditions. But, given any concrete situation, the more numerous the conditions one ignores in one's calculations, the less adequate are one's calculations to that situation. The number of its inhabitants, and any mathematical operation made with that number, is true, but only very abstractly true of a nation. A similar though less radical abstractness appertains to natural science. Simple qualities of sound or color, and distinctions of beauty or moral worth, together with many other ingredients of actual experience attributed therein to the objects of nature, are ignored in the mechanical scheme. There is a substitution of certain mechanical arrangements in the case of the first group of properties, the simple qualities of sense, so that they may be assimilated to the general scheme of events, and their occurrence predicted. But their intrinsic qualitative character is not reckoned with, even in psychology, where the physiological method finally replaces them with brain states. Over and above these neglected properties of things there remain the purposive activities of thought. It is equally preposterous to deny them and to describe them in mechanical terms. It is plain, then, that natural science calculates upon the basis of only a fraction of the conditions that present themselves in actual experience. Its conclusions, therefore, though true so far as they go, and they may be abstractly true of everything, are completely true of nothing.