of the thing... not as if we really understood any more, what principle or virtue that is, which moveth a stone downwards, than we know who moveth it upwards, when it is separated from the projicient, or who moveth the moon round, except only the name, which more particularly and properly we have assigned to all motion of descent, namely gravity."4 In his discussion on the tides he severely criticizes Kepler for explaining the moon's influence on the tides in terms that sound like the occult qualities of the scholastics, judging it better for people "to 'to pronounce that wise, ingenious, and modest sentence, I know it not,'" rather than to "suffer to escape from their mouths and pens all manner of extravagances."49 Galileo was by no means consistent in this positivism. In some cases he allowed his own speculations to run rampant. He did not hesitate to explain the spots on the sun as black smoke given off by the ethereal pabulum which the sun is continually devouring in constant supply in order to continue spreading light and heat; nor to account for the miracle of Joshua 50 by supposing, with Kepler, that the planetary revolutions on their axes were caused by the sun's revolution on his, hence a temporary cessation of the latter might explain the stoppage of the former. It is difficult to tell, however, whether such a remark was meant for more than religious consumption. Yet that this positivistic trend in his thought was something vital is amply proved by the fact that at times even the fundamental questions of the creation of the universe and its first cause he is tempted to relegate to the realm of the unknown, at least until, on the basis of the positive achievements of mechanics, we find it possible to proceed to their solution. Profound considerations of this sort belong to a higher science than ours. We must be satisfied to belong to that class of less worthy workmen who procure from the quarry the marble out of which, later, the gifted 48 Two Great Systems, p. 210, ff. sculptor produces those masterpieces which lay hidden in this rough and shapeless exterior."51 It is difficult indeed to leave Galileo without pausing a moment to reflect on the simply stupendous achievements of the man. The space at our disposal forbids such supererogatory disquisitions, but just consider that the history of thought must turn to this single individual as the one who, by experimental disproof, overthrew a hoary science, who confirmed by sensible facts a new theory of the universe that hitherto had rested on a priori grounds alone, who laid the foundations of the most stupendous intellectual conquest of modern times, the mathematical science of physical nature; and then, as if these accomplishments were not enough, we must turn to him likewise as the philosopher who sufficiently perceived the larger implications of his postulates and methods to present in outline a new metaphysic-a mathematical interpretation of the universe-to furnish the final justification for the onward march of mechanical knowledge. Teleology as an ultimate principle of explanation he set aside, depriving of their foundation those convictions about man's determinative relation to nature which rested upon it. The natural world was portrayed as a vast, self-contained mathematical machine, consisting of motions of matter in space and time, and man with his purposes, feelings, and secondary qualities was shoved apart as an unimportant spectator and semi-real effect of the great mathematical drama outside. In view of these manifold and radical performances Galileo must be regarded as one of the massive intellects of all time. In every single respect of importance he broke the ground or otherwise prepared the way for the only two minds in this advancing current of thought comparable to his ownDescartes and Sir Isaac Newton. 51 Two New Systems, p. 194. CHAPTER IV DESCARTES DESCARTES' importance in this mathematical movement was twofold; he worked out a comprehensive hypothesis in detail of the mathematical structure and operations of the material universe, with clearer consciousness of the important implications of the new method than had been shown by his predecessors; and he attempted both to justify and atone for the reading of man and his interests out of nature by his famous metaphysical dualism. While still in his 'teens, Descartes became absorbed in mathematical study, gradually forsaking every other interest for it, and at the age of twenty-one was in command of all that was then known on the subject. During the next year or two we find him performing simple experiments in mechanics, hydrostatics, and optics, in the attempt to extend mathematical knowledge in these fields. He appears to have followed the more prominent achievements of Kepler and Galileo, though without being seriously affected by any of the details of their scientific philosophy. On the night of November 10th, 1619, he had a remarkable experience which confirmed the trend of his previous thinking and gave the inspiration and the guiding principle for his whole life-work1. The experience can be compared only to the ecstatic illumination of the mystic; in it the Angel of Truth appeared to him and seemed 1 An admirable account of this event in the light of the available sources, with critical comments on the views of other Cartesian authorities, is given in Milhaud, Descartes sadanı, Paris, 1922, p. 47, ff. 96 to justify, through added supernatural insight, the conviction which had already been deepening in his mind, that mathematics was the sole key needed to unlock the secrets of nature. The vision was so vivid and compelling that Descartes in later years could refer to that precise date as the occasion of the great revelation that marked the decisive point in his career. (A) Mathematics as the Key to Knowledge The first intensive studies into which he plunged after this unique experience were in the field of geometry, where he was rewarded within a very few months by the signal invention of a new and most fruitful mathematical tool, analytical geometry. This great discovery not only confirmed his vision and spurred him on to further efforts in the same direction, but it was highly important for his physics generally. The existence and successful use of analytical geometry as a tool of mathematical exploitation presupposes an exact one-to-one correspondence between the realm of numbers, i.e., arithmetic and algebra, and the realm of geometry, i.e., space. That they had been related was, of course, a common possession of all mathematical science; that their relation was of this explicit and absolute correspondence was an intuition of Descartes. He perceived that the very nature of space or extension was such that its relations, however complicated, must always be expressible in algebraic formulae, and, conversely, that numerical truths (within certain powers) can be fully represented spatially. As one not unnatural result of this notable invention, the hope deepened in Descartes' mind that the whole realm of physics might be reducible to geometrical qualities alone. Whatever else the world of nature may be, it is obviously a geometrical world, its objects are extended and figured magnitudes in motion. If we can get rid of all other qualities, or reduce them to H these, it is clear that mathematics must be the sole and adequate key to unlock the truths of nature. And it was not a violent leap from the wish to the thought. During the following ten years, besides his numerous travels, Descartes was engaged in further mathematical studies, which were written down toward the end of this period, and he was also working out a series of specific rules for the application of his all-consuming idea. In these rules we find the conviction expressed that all the sciences form an organic unity, that all must be studied together and by a method that applies to all3. This method must be that of mathematics, for all that we know in any science is the order and measurement revealed in its phenomena; now mathematics is just that universal science that deals with order and measurement generally. That is why arithmetic and geometry are the sciences in which sure and indubitable knowledge is possible. They "deal with an object so pure and uncomplicated that they need make no assumptions at all that experience renders uncertain, but wholly consist in the rational deduction of consequences." This does not mean that the objects of mathematics are imaginary entities without existence in the physical world. Whoever denies that objects of pure mathematics exist, must deny that anything geometrical exists, and can hardly maintain that our geometrical ideas have been abstracted from existing things. Of course, there are no substances which have length without breadth or breadth without thickness, because geometrical figures are not substances but boundaries of them. In order for our geometrical ideas to have been abstracted from the world of physical objects, granted that this is a tenable hypothesis, that world would have to be a geometrical world-one fundamental characteristic of it is extension in space. It may turn The Philosophical Works of Descartes, Haldane and Ross translation, Cambridge. 1911. Vol. I, p. 1, ff., 9. Vol. I, p. 306. Vol. I, p. 13. • Vol. I, p. 4, ff. • Vol. II, p. 227. |