Logic is transcendental.

Logic is not a body of doctrine, but a mirror-image of the world. Logic is transcendental. Mathematics is a logical method. The propositions of mathematics are equations, and therefore pseudo-propositions. A proposition of mathematics does not express a thought.

Indeed in real life a mathematical proposition is never what one needs. Rather, one uses mathematical propositions only to make 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.)

The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics. If two expressions are joined by the sign of equality, 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.

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)'.

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. 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.

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. An equation merely marks the point of view from which I consider the two expressions, it marks their equivalence in meaning.

Intuition is needed to solve mathematical, but language itself provides the necessary intuition. The process of calculating serves to bring about that intuition. Calculation is not an experiment.