5.632 The subject does not belong to the world:  rather, it is a limit of the world.

5.633 Where in the world is a metaphysical subject to be found?  You will say that this is exactly like the case of the eye and the visual field.  But really you do not see the eye.  And nothing in the visual field allows you to infer that it is seen by an eye.

5.6331 For the form of the visual field is surely not like this

5.634 This is connected with the fact that no part of our experience is at the same time a priori.  Whatever we see could be other than it is.  Whatever we can describe at all could be other than it is.  There is no a priori order of things.

5.64 Here it can be seen that solipsism, when its implications are followed out strictly, coincides with pure realism.  The self of solipsism shrinks to a point without extension, and there remains the reality co-ordinated with it.

5.641 Thus there really is a sense in which philosophy can talk about the self in a non-psychological way.  What brings the self into philosophy is the fact that ‘the world is my world’.  The philosophical self is not the human being, not the human body, or the human soul, with which psychology deals, but rather the metaphysical subject, the limit of the world—­not a part of it.

6 The general form of a truth-function is [p, E, N(E)].  This is the general form of a proposition.

6.001 What this says is just that every proposition is a result of successive applications to elementary propositions of the operation N(E)

6.002 If we are given the general form according to which propositions are constructed, then with it we are also given the general form according to which one proposition can be generated out of another by means of an operation.

6.01 Therefore the general form of an operation ’(n) is [E, N(E)] ’ (n) ( = [n, E, N(E)]).  This is the most general form of transition from one proposition to another.

6.02 And this is how we arrive at numbers.  I give the following definitions x = 0x Def., ’v’x = v+1’x Def.  So, in accordance with these rules, which deal with signs, we write the series x, ’x, ’’x, ’’’x, ... , in the following way 0’x, 0+1’x, 0+1+1’x, 0+1+1+1’x, ... .  Therefore, instead of ‘[x, E, ’E]’, I write ‘[0’x, v’x, v+1’x]’.  And I give the following definitions 0 + 1 = 1 Def., 0 + 1 + 1 = 2 Def., 0 + 1 + 1 +1 = 3 Def., (and so on).

6.021 A number is the exponent of an operation.

6.022 The concept of number is simply what is common to all numbers, the general form of a number.  The concept of number is the variable number.  And the concept of numerical equality is the general form of all particular cases of numerical equality.

6.03 The general form of an integer is [0, E, E +1].

6.031 The theory of classes is completely superfluous in mathematics.  This is connected with the fact that the generality required in mathematics is not accidental generality.