CHAPTER I

UNIVERSAL MATHEMATICS VERSUS UNIVERSAL PSYCHOLOGY

Had Lord Bacon known that in the century following the publication of the Advancement of Learning no school of philosophers would acknowledge him as master, he would not have been seriously disheartened at the prospect. Splendid as was the ambition of the scholar who chose all knowledge for his province, that ambition did not include the founding of a school. In truth, to his mind such an accomplishment seemed so slight, and the distinction it won so petty, that he was content to leave it to ingenious but narrow-minded men. What he wished to found was not a school of philosophy, but philosophy itself—or science, if you please, for in his day the two terms were still synonymous. But had he known that by far the most important movement of thought during the next three generations was to be in direct and conscious opposition to his most cherished principles-in England as a reaction against his influence, on the continent in contemptuous disregard of him-only a sublime faith in their truth could have saved him from utter discouragement. Writing in 1739, the young David Hume comments upon the fact, that the whole period of the pre-Socratic philosophy in Greece was “nearly equal to that betwixt my Lord Bacon and some late philosophers of England, who have begun to put the science of man on a new footing." Yet the institution of a body of experimental "sciences of man" was the part of Bacon's program that was nearest his heart, and that he himself did most to forward.

The phenomenon is certainly a striking one. Bacon had taught that deduction, as a scientific method, was useful only for purposes of instruction, and even so was better fitted to produce a showy than a real and thorough knowledge; and that for the discovery and establishment of truth induction and ex

periment were all-important. The great rationalists of the seventeenth century-Hobbes, Descartes, Spinoza, and Leibniz, for example however great their differences in detail, were agreed upon the general point, that deduction is the sole ultimately satisfactory mode of proof; that experimental methods are wholly subordinate devices, which may, indeed, be indispensable in the course of a complex investigation, but which the completed theory must in every case cast aside. Bacon had taught that science must begin with particulars, rising by successive inductions to more and more general laws, and arriving at its supreme explanatory principles only at the last stage of its endeavors. According to the rationalists, that whole ascent is a mere preliminary to the task of science; science itself begins with secure first principles, and its problem is the explanation of the more particular laws of nature as necessary consequences of the first principles. Finally, whereas the rationalists one and all regarded precise definition and the consistent use of terms as prime necessities for scientific discussion, and counted upon these as most potent aids to the discovery of truth, the great chancellor held that the establishment of definitions belongs not to the beginnings of science but to its consummation, and that in the meantime the effort at verbal consistency is only too apt to issue in self-deception.

It would be beyond our present purpose to attempt a complete explanation of this phenomenon the temporary unsuccess of Bacon's polemic. It has been customary to attribute it in great measure to personal defects in him; especially to a lack of plodding thoroughness, that made his brilliant suggestions mere suggestions, and left his programs of scientific advancement unsupported by actual solid contributions to knowledge. Two other causes were probably more important. The first of these was the silent influence of Aristotle. It is true that in the seventeenth century it was not the fashion to refer to Aristotle except for the purpose of emphasizing one's disagreement with him and one's contempt for his authority. But "he who flees is not yet free"; and never did the perennial vigor of the ancient rationalism

show itself more clearly than in the control which it exerted over the development of rationalism in the seventeenth century. A good part of Hobbes's Computation, or Logic is scarcely more than a simplified restatement of the leading principles of Aristotle's methodology, in terms of the already traditional English nominalism, and not improbably profited by some study of the Greek original. In the case of the continental rationalists, the dependence is generally more indirect—through the continued prevalence of conceptions inherited from scholasticism—but not less evident. The opening paragraphs of Descartes's Discourse on Method afford a singular illustration of this. He is inclined to think that the intellectual differences between men cannot have to do with their reason, because that faculty is the distinguishing characteristic of the human species, which completes its definition, and consequently must be present equally in all members of the species. From this one fossil vestige, well-nigh the whole skeleton of the classical logic might be safely reconstructed.

The other influence to which we referred was that of the mathematical sciences, and especially of the geometry of Euclid. It is difficult for us today to realize what the possession of this work meant to the thinkers of the later renaissance. To these pioneers of modern science it was the very image of all that they hoped to do, and, more than that, an unquestionable guarantee of the competence of the human mind to solve the riddles of the universe. While physics and physiology were still the sport of vain and conflicting theories, here, at least, was a science. With all of the unfounded pretensions and lamentable failures of the Greeks, so much they had accomplished. This was their great bequest to the modern world. Accordingly we can understand that in the seventeenth century the hope of a science meant the hope of a new geometry. Whatever modern methods, experimental or analytical, might be employed in its construction, the finished product was to be of the one uniform type.

How was this type understood? In the most natural and

perfectly obvious fashion. At its basis were conceived to be a certain number of indemonstrable but self-evident propositions, involving a certain number of indefinable but self-explanatory terms. Resting upon this basis were series of propositions of ever narrowing generality and increasing complexity. The truth of the later propositions was supposed to result from, and to be guaranteed by, that of the earlier propositions, without giving to these any reciprocal support. It seems to have been popularly supposed that the order in which the propositions followed upon one another was quite fixed, or admitted, at any rate, of no radical alteration; and, although, of course, mathematicians were well aware that this was not the case, they were nevertheless inclined to think that one order alone could represent with perfect clearness the exact interrelations of the concepts involved, and that all others were therefore open to ultimate logical criticism. The discovery of this ideal order was therefore regarded as a very great desideratum.

The influence of mathematical conceptions upon philosophy was due in part to the fact that two of the great rationalists, Descartes and Leibniz, were among the founders of modern mathematical science, and that many lesser members of the school were competent mathematicians. With Descartes, indeed, whose system was the point of departure for the whole movement, the philosophy was the result of a deliberate attempt at an extension of the mathematics. Inspired by his success in developing the great discovery of his early manhood—the application of algebraic analysis to the solution of geometrical problems-he thought to apply a similar analysis to the fundamental problems of all departments of science. Had Descartes lived a little earlier, Bacon would surely have cited his system as the superlative instance in all history, of the Idol of the Cave. After telling us how Aristotle, when he had discovered and classified the various forms of demonstration, was thenceforth driven to interpret all the phenomena of nature and society in terms of this new logic; and after taking his fling at his countryman Gilbert, who had

pondered for years over a magnet, until he saw magnetism everywhere and in everything; he would have capped the climax with the inventor of analytical geometry and author of the Discourse on Method.

But rationalists who were by no means distinguished as mathematicians were scarcely, if at all, less under the influence of mathematical conceptions. The most obvious example is Spinoza, composing his Ethics in "geometrical order," and illustrating the invariability of natural causation by the necessity with which the idea of a triangle implies that the sum of its angles is two right angles. Hobbes, too, who was so far from competence in geometry that he is remembered in its history only as the most fatuous of circle-squarers, must nevertheless be said to have owed the first flush of his enthusiasm for science, as well as his first clear conceptions of scientific method, to a copy of Euclid's Elements. On the other hand, Leibniz, the greatest mathematician of the whole group, was not least a slave to mathematical notions, though in various directions he strained these notions to their breaking-point. His writings are, indeed, remarkable for their constant use of principles which in their manifest implications far transcend the rationalistic standpoint. More than any other modern philosopher-except perhaps Bacon-he was a man of the world with the most far-reaching social and political interests. Yet his logical theory remained mathematical to the core, though the uses to which he endeavored to put it were strikingly, nay absurdly, concrete. Thus, for example, he was not above enforcing a practical social optimism by a reference to the law of the parallelogram of forces. That this is the best of possible worlds might be seen in the fact, that every change that takes place in the world comes about with the least possible expenditure of energy; so that, considering the state of affairs at each moment of the world's history, as much as possible is always happening!

We have mentioned several dogmas upon which all the great rationalists are found to be united. The essential point is prob