Mathematics is a logical method.
The propositions of mathematics are equations, and therefore pseudo-propositions.
Mathematical propositions express no thoughts.
In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
(In philosophy the question “Why do we really use that word, that proposition?” constantly leads to valuable results.)
The logic of the world which the propositions of logic show in tautologies, mathematics shows in equations.
If two expressions are connected by the sign of equality, this means that they can be substituted for one another. But whether this is the case must show itself in the two expressions themselves.
It characterizes the logical form of two expressions, that they can be substituted for one another.
It is a property of affirmation that it can be conceived as double denial.
It is a property of “1+1+1+1” that it can be conceived as “(1+1)+(1+1)”.
Frege says that these expressions have the same meaning but different senses.
But what is essential about equation is that it is not necessary in order to show that both expressions, which are connected by the sign of equality, have the same meaning: for this can be perceived from the two expressions themselves.
And, that the propositions of mathematics can be proved means nothing else than that their correctness can be seen without our having to compare what they express with the facts as regards correctness.
The identity of the meaning of two expressions cannot be asserted. For in order to be able to assert anything about their meaning, I must know their meaning, and if I know their meaning, I know whether they mean the same or something different.
The equation characterizes only the standpoint from which I consider the two expressions, that is to say the standpoint of their equality of meaning.
To the question whether we need intuition for the solution of mathematical problems it must be answered that language itself here supplies the necessary intuition.
The process of calculation brings about just this intuition.
Calculation is not an experiment.
Mathematics is a method of logic.
The essential of mathematical method is working with equations. On this method depends the fact that every proposition of mathematics must be self-evident.
The method by which mathematics arrives at its equations is the method of substitution.
For equations express the substitutability of two expressions, and we proceed from a number of equations to new equations, replacing expressions by others in accordance with the equations.
Thus the proof of the proposition 2×2=4 runs:
( Ω ν ) μ ′ x =
Ω ν × μ ′ x Def.