数学 1854

对思维规律的研究

乔治·布尔

逻辑是一种代数——把推理化成服从定律的符号。

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In depth · the introduction

乔治·布尔有一个既古怪又强大的念头：逻辑的规则，可以像代数里的算式一样，被写下来、被计算出来。

把这个想法拆开看

我们通常把逻辑——「所有 A 都是 B」「这个并且那个」「要么……要么……」——当作关于词语和缜密思考的事。布尔看出，它可以被做成数学。让一个字母代表一个类别（比如 x =「白色的东西」），再让类别的组合，遵循看上去就像代数的规则：「x 并且 y」成了一种乘法，x 乘以 y；「非 x」成了 1 减去 x；万物构成的全域是 1，而空无一物是 0。

让这一切成立的窍门是：他的字母永远只取两个值，0 或 1——关或开，假或真。在白色的东西里再挑出白色的东西，挑来挑去还是那些白色的东西，所以 x 乘以 x 等于 x。这条不起眼的事实——在寻常算术里除了 0 和 1 之外都不成立——正是一整套可以拿来计算的、完整而精确的逻辑的种子。

它从哪里来

布尔是一个几乎全靠自学的英国人，鞋匠的儿子，从未拿过自己的大学学位，却当上了爱尔兰科克市皇后学院的数学教授。他在 1847 年一本薄薄的小册子里勾画了这个想法；1854 年，他在这本书里把它和盘托出，而它那宏大的书名——《对思维规律的研究》——宣告着他的信念：他找到了推理本身背后的数学。他的朋友奥古斯都·德摩根，当时也正追逐着相关的想法。布尔英年早逝，1864 年去世，年仅 49 岁，从未看见他的代数会变成什么。

它为何重要

两千年来，逻辑一直是亚里士多德的——缜密，却是言辞的，也难以推广。布尔把它变成了可以用符号写下、用计算求解的东西，让逻辑向数学的全部力量敞开。这一步——把推理当作计算——是思想史上最深刻的步伐之一，它为后来的一切埋下了伏笔：从数理逻辑，到计算机的设计。

电灯开关

想象两个开关，接到同一盏灯上。每个开关，关时记作 0，开时记作 1。把它们一前一后串成一排，只有两个都开，灯才亮——这就是布尔的「与」，x 乘以 y。把它们并排接上，任一个开，灯就亮——这就是他的「或」。一个会把信号反过来的开关，就是「非」。布尔在 1854 年算出这套代数时，心里并没有任何机器；一个世纪后，它竟恰好描述了计算机内部的电路是怎么「思考」的。拨动下面的开关，看着逻辑发生。

它落在哪里

布尔站在本馆「推理」这条故事线的一处枢纽上。在他身后，是欧几里得——证明了真理可以靠纯粹的逻辑步骤推导出来；在他前方，是哥德尔——探问形式系统的极限，以及克劳德·香农——他的工作，把布尔的二值代数径直带进了数字时代。当你用「与」和「或」做检索，当任何一台计算机把两个数相加，你用的，正是布尔在这本书里写下的代数。

The original document

Original source text

George Boole · An Investigation of the Laws of Thought · London: Walton and Maberly, 1854

Chapter I — Nature and design of this work

The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind.

The universe of discourse: 0 and 1

Boole lets a literal symbol such as x stand for a class — say, all white things — and writes xy for the things that are at once x and y. He fixes the bounds of the discussion with two limiting symbols: 0 for Nothing, and 1 for the Universe — the class “comprehending every conceivable class of objects whether actually existing or not.” Every other class is some selection within that 1.

The fundamental law of thought

Because a class selected twice over is the same class — the white things among the white things are simply the white things — the symbols obey x·x = x, which Boole writes x² = x. In ordinary algebra only 0 and 1 satisfy that equation; that is the hinge of the whole book.

Let us conceive, then, of an Algebra in which the symbols x, y, z, &c. admit indifferently of the values 0 and 1, and of these values alone.

That axiom of Metaphysicians which is termed the principle of contradiction, and which affirms that it is impossible for anything to possess a quality, and in the same time not to possess it, is a consequence of the fundamental law of thought, whose expression is x² = x.

[ … ]

The second half of the treatise carries the same symbolic method into the theory of probabilities, treating a probability as a value between 0 and 1 and deriving the probability of compound events from the logical relations among them.

Queen's College, Cork · 1854