バートランド・ラッセル『数理哲学序説』第1章「自然数列」_ 冒頭

* 出典:バートランド・ラッセル(),平野智治(訳)『数理哲学序説』(岩波書店,1954年8月刊。276+7pp. 岩波文庫 青33-649-1)
* 原著: Introduction to Mathematical Philosophy, 1919)

Chap. 1: The Series of Natural Numbers, n.1
 この相違を別の仕方でのべてみる。数学でもっとも明白で容易なことがらは、論理的に最初にくるものではなく、論理的演繹からみるとなかほどにくるものである。いちばん容易に見える物体は、非常に近かったり遠かったりするものでも、非常に小さかったり大きかったりするものでもない。これと同様にいちばんわかりやすい概念は、きわめ複雑なものやきわめて単純なもの(「単純」を論理的意味で使って)ではない。われわれの視力を拡大するために望遠鏡と顕微鏡という2種類の道具を使うように、われわれの論理能力を拡大するためには2種類の道具を必要とする。その1つはより高等な数学へと導くものであり、他の1つは数学で承認されていると思いがちなことがらの論理的基礎へと戻ってゆくものである。通常の数学的諸概念を分析する後方への旅の後で、新しい前進方向をとることによって、新鮮な洞察力、新たな能力、さらに、新しい数学的対象全体に到達する手段がえられることになるであろう。本書の目的は、初等的な扱い方ではほとんど不可能な、わかりにくい部分は論じないで、技術的でなくまた簡単に数理哲学を説明することである。十分なとり扱いは『数学原理』(Principia Mathematica, 3 vols., 1910-1913)にあるので、本書の扱い方はただ入門のためのものとする。
MATHEMATICS is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers ; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics. But it should be understood that the distinction is one, not in the subject matter, but in the state of mind of the investigator. Early Greek geometers, passing from the empirical rules of Egyptian land-surveying to the general propositions by which those rules were found to be justifiable, and thence to Euclid's axioms and postulates, were engaged in mathematical philosophy, according to the above definition; but when once the axioms and postulates had been reached, their deductive employment, as we find it in Euclid, belonged to mathematics in the ordinary sense. The distinction between mathematics and mathematical philosophy is one which depends upon the interest inspiring the research, and upon the stage which the research has reached; not upon the propositions with which the research is concerned.
We may state the same distinction in another way. The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle. Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very great, so the easiest conceptions to grasp are those that are neither very complex nor very simple (using "simple" in a logical sense). And as we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward to the higher mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics. We shall find that by analysing our ordinary mathematical notions we acquire fresh insight, new powers, and the means of reaching whole new mathematical subjects by adopting fresh lines of advance after our backward journey. It is the purpose of this book to explain mathematical philosophy simply and untechnically, without enlarging upon those portions which are so doubtful or difficult that an elementary treatment is scarcely possible. A full treatment will be found in Principia Mathematica; the treatment in the present volume intended merely as an introduction.