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THE PLATONIC THEORY OF THE CONTINUUM T HE proper correlation of the continuous and the discrete remains one of the fundamental problems of philosophy . Various solutions have been proposed, from the early discussions of the Pythagoreans and the Eleatics to the contemporary controversies about the theory of aggregates, the foundations of topology, or the methods of integration. Some of them are strictly mathematical; while others cover the wider field of metaphysics. But none seems to be satisfactory enough to all thinkers. The fullest theories which have come down to us from ancient times are those of Plato and Aristotle. Many writers have extolled the remarkable intuitions of the former at the expense of the doctrines of the latter, as the Platonic views seemed to be more in line with the classical conceptions of the calculus. Such an interpretation, however, does not take into account the recent views about the foundations of mathematics , which involve more Aristotelian than other elements. On the other hand, the Platonic views are not only mathematical , but mainly philosophical: hence they should be criticized in themselves, whatever be their partial connection with successful mathematical methods. Owing to obvious analogies between the classical Greek conceptions and the modern views on the continuous and the discrete, we propose to offer here a short discussion, of the Platonic Theory of the Continuum, as a background to a critical and historical approach to this important question. I. IRRATIONALS AND INDIVISIBLES Having found in mathematics the means of rationalizing the world of knowledge and existence, Plato had to consider the rationalization of mathematics itself as a preliminary requirement . This operation became particularly urgent when it was 179 180 THOMAS GREENWOOD shown that geometry involved many notions which could not be accounted for by the arithmetic of integers. It was necessary to explain how the discontinuous series of the integers can generate the geometrical continuum, and how the integers themselves are obtained. The difficulties of the early Pythagoreans in their treatment of similar problems were due .to their inability to establish a generalized arithmetic ·and to avoid inconsistencies like Zeno's arguments. This faihJ.re became evident when the irrationals were discovered. The parallelism between geometry and arithmetic was broken: the Pythagorean concept of number was no longer adequate. to account for all geometrical forms and, consequently, for the empirical things corresponding to their geometrical patterns. In order to save the rational value of knowledge as illustrated by the truth of mathematics, it was necessary to revise the conception of number itself, to widen it so that it might become possible to define " irrational " numbers as well, and to formulate laws for their addition and multiplication in terms of the arithmetic of integers. In fact, Plato could not consider the irrational quantities as being beyond reason. He even used them in his doctrine of Reminiscence. In a well-known passage of the Meno, Plato tried to explain that teaching is only re-awakening in the mind of the learner the memory of something. A slave is introduced into a room containing objects the mere sight of which will make him aware, when subjected to the dialectical treatment, of universal truths concerning them. Meno is to watch whether the boy is taught by Socrates in any of his answers; whether he answers anything at any point otherwise than by way of reminiscence and really out of his mind, as the reasonable questions of Socrates fall like water on the reed-ground. By putting to the slave a carefully prepared series of questions, Socrates leads him to recognize that double the square of any straight line is not the square on double the line, but the square on the diag()nal of the original square.1 1 Memo, 8ft B-85 B: THE PLATONIC THEORY OF THE CONTINUUM 181 Surely, the so-called irrationals must be rational somehow, if the mind has contemplated their patterns in the world of ideas, and if it " remembers " them when confronted with their actual geometrical illustrations. " See him now; how he remembers in the logica,l order, as he ought to remember." And again," Just now, as in a dream, these opinions have been stirred up within him...

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