著者

稲岡 大志

出版者

日本哲学会

雑誌

哲学 (ISSN:03873358)

巻号頁・発行日

vol.2010, no.61, pp.165-179_L10, 2010 (Released:2011-01-18)

参考文献数

16

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In this paper, we examine Leibniz's critique of Euclidean geometry and show what the point of his critique is. Leibniz thinks that, for the human mind to acquire geometrical ideas which have their origin in God's intellect, the use of symbols is essential. By Leibniz's theory of expressio, there is a structural isomorphism between the symbol system and the world. So we can acquire an eternal truth by using symbols. In Euclidean geometry, diagrams are used as symbols to introduce geometrical objects, but there is a difference in a precise sense between a diagram which is actually described and a geometrical object as an abstract object, so imagination must abstract this microscopic difference somehow. But, to acquire an intended geometrical object from a given diagram, we must in advance capture it by means of some kind of intellectual intuition. However, Leibniz rejects ideas acquired by intuition. So, he must discard diagrams as symbols capable of introducing geometrical objects appropriately.In fact, criticizing Euclidean geometry, Leibniz recognizes the role and importance of symbols in geometry and becomes keenly aware that diagrams are not capable of introducing geometrical objects. In analysis, Leibniz readily permits abstraction by imagination and he is thereby able to solve many mathematical problems. However, in geometry he cannot use diagrams in this way, for we cannot solve even an easy geometrical problem without expressing the geometrical object appropriately. This means that Leibniz realizes that between geometry and other areas of mathematics there is an essential difference in the function of imagination.Traditionally, Leibniz's critique of Euclidean geometry is interpreted as a kind of technical critique. But, the key point of his critique is that using diagrams as means of introducing geometrical objects involves a difficulty which is not solved in the framework of Leibniz's theory of knowledge based upon symbols and the theory of expressio.Finally, we discuss some features of Leibniz's characteristica geometrica [geometrical character], which he developed in order to overcome the defects of Euclidean geometry.