代数的精神―ライプニッツにおける認識と方法―

Abstract

It is difficult to unify Leibniz's ideas in his various writings into an intergrated whole. It is partly because various parts of his general system (i.e. his metaphysics, logic methodology, etc.) cannot be settled in such an order that one part can be regarded as primordial and the others can be derived from it. Nevertheless his whole thought is by no means an aggregation without unity, but is "la substance qui est douée d'une véritable unité." We can find one clue to relate these parts in his theory of expression (we follow the terminology of L.E. Loemker) which permeats many branches of his thought. For example, his metaphysical theory about actual universe, i.e. his theory of monade, is based on the idea that the essential function of all individual substance, that is, of all monades, is to express the universe and the God. His epistemology also can be regarded, as an application of this theory of expression. Because (according to Leibniz), of all created incividual substances, only our minds (and angels) are reflective in the sense that they know that they express something. In his epistemology, this reflective expression, i.e., the expression to the self is thought. But, for us human beings, to express something is to make a symbolic expression, that is, a certain configuration of elemental signs the syntactical structure of which corresponds to that of the thing to be expressed. Therefore, it is quite natural that Leibniz should first inquire the general nature of expressions and symobls in order to construct scientific method. This inquiry led him to thinking of "doctorina de expressionibus in universum" or "une espèce d'Algèbre générale." He called this doctorine "Ars Combinatoria" or "la Specieuse générale". From this results his universal characteristic that is the general method of producing symbolic expressions. In short, Leibniz's method was to construct in each sphere a symbolic system that can express all objects of that sphere according to the producing process, just like numeric symbols express all the natural numbers (we call this quality of system "arithmetic"), and that can express all the reasonings through simple operation of symbols, i.e., through calculations (we call this quality "algebraic"). Up to his time, only arithmetic and algebra had such symbolic system and it was the reason why they had priviledged certainty over other sciences. But from the standpoint of his universal characteristic, these two sciences are mere applications of a general method to number and quantity and he tried to apply it outside of mathematics (the most famous example is his symbolic logic which was its application to the sphere of concepts). Moreover he thought that his method could be applied to every domain that could be the object of reason.