We introduce into our language-games the endless series of
numerals.
But how is this done?
Obviously the analogy between this process & that of
introducing a series of twenty numerals is not the same as that between
introducing a series of twenty numerals and introducing a series of ten
numerals.
Suppose that our game was like 2) but played with the endless
series of numerals.
The difference between it & 2) would not be just that more
numerals were used.
That is to say, suppose that as a matter of fact in playing the
game we had actually made use of, say, 155 numerals, the game we play
would not be that which could be described by saying that we played the
game 2), only with 155 instead of 10 numerals.
But what does the difference consist in?
(The difference would seem to be almost
23.
one of the spirit in which
the games are played.)
The difference between games can lie say in the number of the
counters used, in the number of squares of the playing board, or in the
fact that we use squares in one case & hexagons in the other,
& such like.
Now the difference between the finite and infinite game does not seem
to lie in the material tools of the game; for we should be
inclined to say that infinity can't be expressed in them, that
is, that we can only conceive of it in our thoughts & hence that
it is in these thoughts that the finite and infinite game must be
distinguished.
(It is queer though that these thoughts should be capable of being
expressed in signs.)
Let us consider two games.
They are both played with cards carrying numbers, and the highest
number takes the trick.
22). One game is played with a
fixed number of such cards, say 32.
In the other game we are under certain circumstances allowed to
increase the number of cards to as many as we like, by cutting pieces of
paper and writing numbers on them.
We will call the first of these games bounded, the second
unbounded.
Suppose a hand of the second game was played & the number of
cards actually used was 32.
What is the difference in this case between playing a hand
a) of the unbounded game & playing a hand
b) of the bounded game?