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?