Let us now compare language-games of which we should say
25.
that they are played with a limited set of numerals with language-games of which we should say that they are played with the endless series of numerals.

23).   Like 2) A orders B to bring him a number of building stones. The numerals are the signs “1”, “2”, etc. … “9”, each written on a card. A has a set of these cards and gives B the order by shewing him one of the set & calling out one of the words, “slab”, “column”, etc.
24).   Like 23), only there is no set of indexed cards. The series of numerals 1 … 9 is learned by heart. The numerals are called out in the orders, & the child learns them by word of mouth.
25).   An abacus is used. A sets the abacus, gives it to B, B goes with it to where the slabs lie, etc..
26).   B is to count the slabs in a heap. He does it with an abacus, the abacus has twenty beads. There are never more than 20 plates in a heap. B sets the abacus for the heap in question & shews A the abacus thus set.
27).   Like 26). The abacus has 20 small beads & one large one. If the heap contains more than 20 plates, the large bead is moved. (So the large bead in some way corresponds to the word “many”).
28).   Like 26). If the heap contains n plates, n being more than 20 but less than 40, B moves n-20 beads, shews A the abacus thus set, & claps his hand once.
29).   A & B use the numerals of the decimal system (written or spoken) up to 20. The child learning this language learns these
26.
numerals by heart, etc., as in 2).
30).   A certain tribe has a language of the kind 2). The numerals used are those of our decimal system. No one numeral used can be observed to play the predominant role of the last numeral in some of the above games (27), 28)). (One is tempted to continue this sentence by saying, “although there is of course a highest numeral actually used”). The children of the tribe learn the numerals in this way: They are taught the signs from 1 to 20 as in 2) and to count rows of beads of no more than 20 on being ordered, “Count these”. When in counting the pupil arrives at the numeral 20, one makes a gesture suggestive of “Go on”, upon which the child says (in most cases at any rate) “21”. Analogously, the children are made to count to 22 & to higher numbers, no particular number playing in these exercises the predominant role of a last one. The last stage of the training is that the child is ordered to count a group of objects, well above 20, without the suggestive gesture being used to help the child over the numeral 20. If a child does not respond to the suggestive gesture, it is separated from the others and treated as a lunatic.
31).   Another tribe. Its language is like that in 30). The highest numeral observed in use is 159. In the life of this tribe the numeral 159 plays a peculiar role. Supposing I said, “They treat this number as their highest”, – – but what does this mean? Could we answer: “They just say that it is the highest”? ‒ ‒ They say certain words, but how do we know what they mean by them? A criterion for what they mean would be the occasions
27.
on which the word we are inclined to translate into our word “highest” is used, the role, we might say, which we observe this word to play in the life of the tribe. In fact we could easily imagine the numeral 159 to be used on such occasions, in connection with such gestures and forms of behaviour as would make us say that this numeral plays the role of an unsurmountable limit, even if the tribe had no word corresponding to our “highest”, and the criteria for numeral 159 being the highest numeral did not consist of anything that was said about the numeral. 32).   A tribe has two systems of counting. People learned to count with the alphabet from A to Z and also with the decimal system as in 30). If a man is to count objects with the first system, he is ordered to count “in the closed way”, in the second case, “in the open way”; & the tribe uses the words “closed” & “open” also for a closed and open door.