By Evert W. Beth (Eds.)
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Extra info for A Study in the Philosophy of Science
In particular, I wish to submit a few comments on an observation which he makes in his explanation of the term nolov : quality is, "in general, that which is inherent in the essence, apart from quantity,. 55) The usual interpretation of this text can be summed up as follows: (i) When we abstract from the specific qualities of the things, and retain only their mere individuality, then we obtain a mathematical number, that is, a multitude of unities, which are only numerically, but not specifically, distinct; (ij) When we abstract from 19 THE THEORY OF ABSTRACTION the multitude of the individual representatives of a species, then we grasp the essence of the species which, as a unity of all specific qualities, is present in each of its specimens.
In order to justify this step, Plato goes on to show that all things can indeed be deduced from these two principles. f-) All terms are divided into absolutes (for instance: horse) and relatives; relatives again are divided into opposites (for instance: healthy-unhealthy, equal-unequal), and relatives proper (for instance: large-small, left-right). In a pair of opposites, each term is generated by the destruction of the other one, whereas in relatives proper the terms are always generated and destroyed together.
6. PLATO'S THEORY OF IDEAL NUMBERS Plato himself attempted to solve the difficulties raised by the theories of separation and inherence by establishing an entirely new theory which is known as the Theory of Ideal Numbers. I shall try to reconstruct the historical background of this theory before giving a summary of the theory itself as it has been handed down to us. I have shown that a primitive atomistic theory of the continuum had been upset by Zeno's paradoxes. After the discovery of incommensurable magnitudes, it became clear that no repair of the atomistic theory was possible.
A Study in the Philosophy of Science by Evert W. Beth (Eds.)