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.

6.2322 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.

6.2323 An equation merely marks the point of view from which I consider
the two expressions: it marks their equivalence in meaning.

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.

6.2331 The process of calculating serves to bring about that intuition.
Calculation is not an experiment.

6.234 Mathematics is a method of logic.

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.

6.24 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.

6.241 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.

6.31 The so-called law of induction cannot possibly be a law of logic,
since it is obviously a proposition with sense.---Nor, therefore, can it
be an a priori law.

6.32 The law of causality is not a law but the form of a law.

6.321 'Law of causality'--that is a general name. And just as in
mechanics, for example, there are 'minimum-principles', such as the law
of least action, so too in physics there are causal laws, laws of the
causal form.

6.3211 Indeed people even surmised that there must be a 'law of least
action' before they knew exactly how it went. (Here, as always, what is
certain a priori proves to