The conclusion is plain. If logic is to have room in its household for both truth and error, if it is to avoid the old predicament of knowledge that is trifling or miraculous, tautologous or false, if it is to have no fear of the challenge of other sciences or of practical life, it must be content to take for its subject-matter the operations of intelligence conceived as real acts on the same metaphysical plane and in strictest continuity with other acts. Such a logic will not fear the challenge of science, for it is precisely this continuity that makes possible experimentation, which is the fundamental characteristic of scientific procedure. Science without experiment is indeed a strange apparition. It is a λόγος with no λέγειν, a science with no scire; and this spells dogmatism. How necessary such continuity is to experimentation is apparent when we recall that there is no limit to the range of operations of every sort which scientific experiment calls into play; and that unless there be thoroughgoing continuity between the logical demand of the experiment and all the materials and devices employed in the process of the experiment, the operations of the latter in the experiment will be either miraculous or ruinous.
Finally, if this continuity of the operations of intelligence with other operations be essential to science, its relation to "practical" life is ipso facto established. For science is "practical" life aware of its problems and aware of the part that experimental—i.e., creative—intelligence plays in the solution of those problems.
INTELLIGENCE AND MATHEMATICS
HAROLD CHAPMAN BROWN
Herbart is said to have given the deathblow to faculty psychology. Man no longer appears endowed with volition, passion, desire, and reason; and logic, deprived of its hereditary right to elucidate the operations of inherent intelligence, has the new problem of investigating forms of intelligence in the making. This is no inconsequential task. "If man originally possesses only capacities which after a given amount of education will produce ideas and judgments" (Thorndike, Educational Psychology, Vol. I, p. 198), and if these ideas and judgments are to be substituted for a mythical intelligence it follows that tracing their development and observing their functioning renders clearer our conception of their nature and value and brings us nearer that exact knowledge of what we are talking about in which the philosopher at least aspires to equal the scientist, however much he may fall below his ideal.
For contemporary thought concerning the mathematical sciences this altered point of view generates peculiarly pressing problems. Mathematicians have weighed the old logic and found it wanting. They have builded themselves a new logic more adequate to their ends. But they have not whole-heartedly recognized the change that has come about in psychology; hence they have retained the faculty of intelligence knit into certain indefinables such as implication, relation, class, term, and the like, and have transported the faculty from the human soul to a mysterious realm of subsistence whence it radiates its ghostly light upon the realm of existence below. But while they reproach the old logic, often bitterly, their new logic merely furnishes a more adequate show-case in which already attained knowledge may be arranged to set off its charms for the observer in the same way that specimens in a museum are displayed before an admiring world. This statement is not a sweeping condemnation, however, for such a setting forth is not useless. It resembles the classificatory stage of science which, although not itself in the highest sense creative, often leads to higher stages by bringing under observation relations and facts that might otherwise have escaped notice. And in the realm of pure mathematics, the new logic has undoubtedly contributed in this manner to such discoveries. Danger appears when the logician attains Cartesian intoxication with the beauty of logico-mathematical form and tries to infer from the form itself the real nature of the formed material. The realm of subsistence too often has armed Indefinables with metaphysical myths whose attack is valiant when the doors of reflection are opened. It may be possible, however, to arrive at an understanding of mathematics without entering the kingdom of these warriors.
It is the essence of science to make prediction possible. The value of prediction lies in the fact that through this function man can control his environment, or, at worst, fortify himself to meet its vagaries. To attain such predictions, however, the world need not be grasped in its full concreteness. Hence arise processes of abstraction. While all other symptoms remain unnoticed, the temperature and pulse may mark a disease, or a barometer-reading the weather. The physicist may work only in terms of quantity in a world which is equally truly qualitative. All that is necessary is to select the elements which are most effective for prediction and control. Such selection gives the principle that dominates all abstractions. Progress is movement from the less abstract to the more abstract, but it is progress only because the more abstract is as genuinely an aspect of the concrete starting-point as anything is. Moreover, the outcome of progress of this sort cannot be definitely foreseen at the beginnings. The simple activities of primitive men have to be spontaneously performed before their value becomes evident. Only afterwards can they be cultivated for the sake of their value, and then only can the self-conscious cultivation of a science begin. The process remains full not only of perplexities, but of surprises; men's activities lead to goals far other than those which appear at the start. These goals, however, never deny the method by which the start is made. Developed intelligence is nothing but skill in using a set of concepts generated in this manner. In this sense the histories of all human endeavors run parallel.
Where the empirical bases of a science are continually in the foreground, as in physics or chemistry, the foregoing formulation of procedure is intelligible and acceptable to most men. Mathematics seem, however, to stand peculiarly apart. Many, with Descartes, have delighted in them "on account of the certitude and evidence of their reasonings" and recognized their contribution to the advancement of mechanical arts. But since the days of Kant even this value has become a problem, and many a young philosophic student has the question laid before him as to why it is that mathematics, "a purely conceptual science," can tell us anything about the character of a world which is, apparently at least, free from the idiosyncrasies of individual mind. It may be that mathematics began in empirical practice, such philosophers admit, but they add that, somehow, in its later career, it has escaped its lowly origin. Now it moves in the higher circles of postulated relations and arbitrarily defined entities to which its humble progenitors and relatives are denied the entrée. Parvenus, however, usually bear with them the mark of history, and in the case of this one, at least, we may hope that the history will be sufficient to drag it from the affectations of its newly acquired set and reinstate it in its proper place in the workaday world. For the sake of this hope, we shall take the risk of being tedious by citing certain striking moments of mathematical progress; and then we shall try to interpret its genuine status in the world of working truths.