6.2 Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions.
6.2 数学是一种逻辑方法。数学命题是等式，因此都是伪命题。
6.21 A proposition of mathematics does not express a thought.
6.21 数学命题不表达思想。
6.211 Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)
6.211 在现实生活中我们要得到的并非教学命题；或者说，我们应用数学命题只是为了从一些不属于数学的命题推论出另一些同样也不属于数学的命题。
（“我们使用这个词或者这个命题究竟为了什么？”这个问题在哲学中往往导致有价值的领悟。）
6.22 The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.
6.22 逻辑命题在重官式中显示的世界的逻辑，数学在等式中显示出来。
6.23 If two expressions are combined by means of the sign of equality, that means that they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form.
6.23 如果两个表达式用等号连接起来，这就意味着它们可以彼此代换。但是事实是否如此，两个表达式本身必可显示出来。
两个表达式可以彼此代换，这表明它们的逻辑形式的特征。
6.231 It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.
6.231 肯定可以看作双重否定，这是肯定的一个性质。“1＋1＋l＋1”可以看作“（1＋1）＋（l＋1）”，这是‘1＋1＋1＋1”的一个性质。
6.232 Frege says that the two expressions have the same meaning but different senses. But the essential point about an equation is that it is not necessary in order to show that the two expressions connected by the sign of equality have the same meaning, since this can be seen from the two expressions themselves.
6.232 弗雷格说，上述两个表达式有相同的指谓，但是有不同的意义。
但对于等式来说，具有根本意义的一点是：为了显示用等号连接的两个表达式有相同的指谓，等式并非必要，因为这一点从两个表达式本身即可以看出来。
6.2321 And the possibility of proving the propositions of mathematics means simply that their correctness can be perceived without its being necessary that what they express should itself be compared with the facts in order to determine its correctness.
6.2321 而数学命题证明的可能性，不过意味着数学命题的正确性可以直接察知，而无须将它们表达的东西本身同事实比较以确定其正确性。
6.2322 It is impossible to assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different.
6.2322 两个表达式指谓的同一是不能断言的。因为，为了能够断言关于它们指谓的任何东西，我就必须知道它们的指谓，而一旦知道了它们的指谓，也就知道了它们所指的是否相同。
6.2323 An equation merely marks the point of view from which I consider the two expressions: it marks their equivalence in meaning.
6.2323 等式不过标志我考察两个表达式的角度，即它们的指调相等的角度。
6.233 The question whether intuition is needed for the solution of mathematical problems must be given the answer that in this case language itself provides the necessary intuition.
6.233 在解决教学问题中是否需要直觉，这个问题应该这样回答：这里语言已经提供了必须的直觉。
6.2331 The process of calculating serves to bring about that intuition. Calculation is not an experiment.
6.2331 演算过程正好引进了这种直觉。演算并非试验。
6.234 Mathematics is a method of logic.
6.234 数学是一种逻辑的方法。
6.2341 It is the essential characteristic of mathematical method that it employs equations. For it is because of this method that every proposition of mathematics must go without saying.
6.2341 数学方法的本质特征在于它是用等式来工作的。正是由于这种方法，每个数学命题本身必须足以表明自己的成立。
6.24 The method by which mathematics arrives at its equations is the method of substitution. For equations express the substitutability of two expressions and, starting from a number of equations, we advance to new equations by substituting different expressions in accordance with the equations.
6.24 数学用来得到等式的方法是代换法。因为等式表达两个表达式的可代换性；我们从一定数目的等式出发，按照等式的条件，通过代换不同的表达式而推进到新的等式。
6.241 Thus the proof of the proposition 2 t 2 = 4 runs as follows: (/v)n'x = /v x u'x Def., /2 x 2'x = (/2)2'x = (/2)1 + 1'x = /2' /2'x = /1 + 1'/1 + 1'x = (/'/)'(/'/)'x =/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x. 6.3 The exploration of logic means the exploration of everything that is subject to law . And outside logic everything is accidental.
6.241 因此，命题2×2＝4的证明进行如下：
6.2 数学是一种逻辑方法。数学命题是等式，因此都是伪命题。
6.21 A proposition of mathematics does not express a thought.
6.21 数学命题不表达思想。
6.211 Indeed in real life a mathematical proposition is never what we want. Rather, we make use of mathematical propositions only in inferences from propositions that do not belong to mathematics to others that likewise do not belong to mathematics. (In philosophy the question, 'What do we actually use this word or this proposition for?' repeatedly leads to valuable insights.)
6.211 在现实生活中我们要得到的并非教学命题；或者说，我们应用数学命题只是为了从一些不属于数学的命题推论出另一些同样也不属于数学的命题。
（“我们使用这个词或者这个命题究竟为了什么？”这个问题在哲学中往往导致有价值的领悟。）
6.22 The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.
6.22 逻辑命题在重官式中显示的世界的逻辑，数学在等式中显示出来。
6.23 If two expressions are combined by means of the sign of equality, that means that they can be substituted for one another. But it must be manifest in the two expressions themselves whether this is the case or not. When two expressions can be substituted for one another, that characterizes their logical form.
6.23 如果两个表达式用等号连接起来，这就意味着它们可以彼此代换。但是事实是否如此，两个表达式本身必可显示出来。
两个表达式可以彼此代换，这表明它们的逻辑形式的特征。
6.231 It is a property of affirmation that it can be construed as double negation. It is a property of '1 + 1 + 1 + 1' that it can be construed as '(1 + 1) + (1 + 1)'.
6.231 肯定可以看作双重否定，这是肯定的一个性质。“1＋1＋l＋1”可以看作“（1＋1）＋（l＋1）”，这是‘1＋1＋1＋1”的一个性质。
6.232 Frege says that the two expressions have the same meaning but different senses. But the essential point about an equation is that it is not necessary in order to show that the two expressions connected by the sign of equality have the same meaning, since this can be seen from the two expressions themselves.
6.232 弗雷格说，上述两个表达式有相同的指谓，但是有不同的意义。
但对于等式来说，具有根本意义的一点是：为了显示用等号连接的两个表达式有相同的指谓，等式并非必要，因为这一点从两个表达式本身即可以看出来。
6.2321 And the possibility of proving the propositions of mathematics means simply that their correctness can be perceived without its being necessary that what they express should itself be compared with the facts in order to determine its correctness.
6.2321 而数学命题证明的可能性，不过意味着数学命题的正确性可以直接察知，而无须将它们表达的东西本身同事实比较以确定其正确性。
6.2322 It is impossible to assert the identity of meaning of two expressions. For in order to be able to assert anything about their meaning, I must know their meaning, and I cannot know their meaning without knowing whether what they mean is the same or different.
6.2322 两个表达式指谓的同一是不能断言的。因为，为了能够断言关于它们指谓的任何东西，我就必须知道它们的指谓，而一旦知道了它们的指谓，也就知道了它们所指的是否相同。
6.2323 An equation merely marks the point of view from which I consider the two expressions: it marks their equivalence in meaning.
6.2323 等式不过标志我考察两个表达式的角度，即它们的指调相等的角度。
6.233 The question whether intuition is needed for the solution of mathematical problems must be given the answer that in this case language itself provides the necessary intuition.
6.233 在解决教学问题中是否需要直觉，这个问题应该这样回答：这里语言已经提供了必须的直觉。
6.2331 The process of calculating serves to bring about that intuition. Calculation is not an experiment.
6.2331 演算过程正好引进了这种直觉。演算并非试验。
6.234 Mathematics is a method of logic.
6.234 数学是一种逻辑的方法。
6.2341 It is the essential characteristic of mathematical method that it employs equations. For it is because of this method that every proposition of mathematics must go without saying.
6.2341 数学方法的本质特征在于它是用等式来工作的。正是由于这种方法，每个数学命题本身必须足以表明自己的成立。
6.24 The method by which mathematics arrives at its equations is the method of substitution. For equations express the substitutability of two expressions and, starting from a number of equations, we advance to new equations by substituting different expressions in accordance with the equations.
6.24 数学用来得到等式的方法是代换法。因为等式表达两个表达式的可代换性；我们从一定数目的等式出发，按照等式的条件，通过代换不同的表达式而推进到新的等式。
6.241 Thus the proof of the proposition 2 t 2 = 4 runs as follows: (/v)n'x = /v x u'x Def., /2 x 2'x = (/2)2'x = (/2)1 + 1'x = /2' /2'x = /1 + 1'/1 + 1'x = (/'/)'(/'/)'x =/'/'/'/'x = /1 + 1 + 1 + 1'x = /4'x. 6.3 The exploration of logic means the exploration of everything that is subject to law . And outside logic everything is accidental.
6.241 因此，命题2×2＝4的证明进行如下：
