著者

Yamanaka Hideya 桃山学院大学文学部

出版者

桃山学院大学総合研究所

雑誌

国際文化論集 = INTERCULTURAL STUDIES (ISSN:09170219)

巻号頁・発行日

no.24, pp.3-26, 2001-07-10

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In his presentation of the secondary grade curriculum for the young, the Athenian stranger refers to the matter, which it is disgraceful not to know: the problems concerning "the essential nature of the commensurable and the incommensurable." (820c) The reference appears immediately after the Athenian stranger's introduction to three branches of learning: arithmetic, geometry, and astrology (817e), whose detailed study is necessary for only 'a few' (818a). Referring to the instruction of the essential nature of the commensurable and the incommensurable, what matter did Plato exactly have in his mind? G. R. Morrow said that the introduction of the problem of incommensurable magnitudes to the curriculum was "an innovation to which Plato attaches peculiar importance" and also that although the belief that "all magnitudes are somehow commensurable with one another is natural to us," seeing that this belief is in fact false would be to rise above one's human nature." Although the saying seems to hit the mark, Morrow did not fully explain why Plato's introduction to the learning of incommensurable magnitudes was "an innovation" which had a "peculiar importance" and how the subject-matter of the incommensurable could "emancipate the student from his instinctive sense-bound beliefs and thus, ・・・・・・raise him above his human nature." (pp.346-7) However, it is worth listening to his other words which bear on the significance of mathematical thinking in the context of legislation: "the law, as we learn in the Philebus, results from the application of the Limit (περα〓) to an indefinitely varying qualitative continuum (απειρου), and the Limit is conceived of as analogous to mathematical order." The words could be settled more aptly in the context of the incommensurable. I insist that Plato here in the dialogue Laws, referring to the peculiar importance of the incommensurable, gives an inkling of the fact that the scientific knowledge of the incommensurable has an isomorphic property with that faculty that discerns "the identical element which pervades all the four virtues" (965c). That is to say, Plato identifies the method by which the nature of the incommensurable magnitudes can be explained as a significant model in order to "hold very tight and not to let go"(965d) until we can adequately explain the essential nature of the object existing as a unity. In other words, the detailed study of the incommensurable is of great advantage to 'a few' (818a) who are expected to be members of the Nocturnal Council and who "must not only be able to pay regard to the many, but must be able also to press towards the one (pros to hen) so as to discern it and, on discerning it, to survey and organize all the rest with a single eye to it" (965b), since the fruit of this offers them a convenient paradigm in medium of which they may master an isomorphic way in order to see the invisible one. Plato's reference to the problem of the incommensurable in the Laws is at first sight negative. After his labor to explain the necessity of a subject concerning the incommensurable, the Athenian stranger says that it is to be laid down provisionally only, 'like pledges capable of redemption, apart from the rest of our constitution, in case they fail to satisfy either us who enact them or you for whom they are enacted.' (820e). However, the utterance is paradoxical in the same way as the introduction of the Nocturnal Council in the closing books of the Laws appears paradoxically to be an appendix, whereas in reality the whole constitution of Magnesia is rooted in the Council. Both facts correspond to one another and put Plato's metaphysical thinking out of our sight. Focusing the implications of philosophical dimensions of the incommensurable in the Laws, I will try to bring them into the relevance to Plato's metaphysical thought in the other dialogues and the Seventh Letter VII, where the deepest tenet on the incommensurable or the cognate ideas make their appearance vividly.