Brown Book: Difference between revisions

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

 

{{parBB|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?

{{BBB TS reference|Ts-310,24}} The unbounded game is opened with the question, “How high shall we go?” If the players look up the rules of this game in a book of rules, they will find the phrase “& so on” or “& so on ad inf.” at the end of certain series of rules. So the difference between the two hands a) & b) lies in the tools we use, though admittedly not in the cards they are played with. But this difference seems trivial and not the essential difference between the games. We feel that there must be a big & essential difference somewhere. But if you look closely at what happens when the hands are played, you find that you can only detect a number of differences in details, each of which would seem inessential. The acts, e.g., of dealing & playing the cards may in both cases be identical. In the course of playing the hand a), the players may have considered making up more cards, & again discarded the idea. But what was it like to consider this? It could be some such process as saying to themselves or aloud, “I wonder whether I should make up another card”. Again, no such consideration may have entered the minds of the players. It is possible that the whole difference in the events of a hand of the bounded, and a hand of the unbounded game lay in what was said before the game started, e.g., “Let's play the bounded game”.

Let us now compare language-games of which we should say

 

{{BBB TS reference|Ts-310,25}} 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.

 

 

{{parBB|29}} A & B use the numerals of the decimal system (written or spoken) up to 20. The child learning this language learns these

 

{{BBB TS reference|Ts-310,26}} numerals by heart, etc., as in 2).

{{parBB|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

 

{{BBB TS reference|Ts-310,27}} 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. {{parBB|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.

 

{{BBB TS reference|Ts-310,28}} that we could also have construed the preceding games as beginnings of a system. In 29) the systematic aspect of the numerals used is even more conspicuous than in 28). One might say that there was no limitation imposed by the tools of this game, if it were not for the remark that the numerals up to 20 are learnt by heart. This suggests the idea that the child is not taught to “understand” the system which we see in the decimal notation. Of the tribe in 30) we should certainly say that they are trained to construct numerals indefinitely, that the arithmetic of their language is not a finite one, that their series of numbers has no end. (It is just in such a case when numerals are constructed “indefinitely” that we say that people have the infinite series of numbers.) 31) might shew you what a vast variety of cases can be imagined in which we should be inclined to say that the arithmetic of the tribe deals with a finite series of numbers, even in spite of the fact that the way in which the children are trained in the use of numerals suggests no upper limit. In 32) the terms “closed” & “open” (which could by a slight variation of the example be replaced by “limited” and “unlimited”) are introduced into the language of the tribe itself. Introduced in that simple and clearly circumscribed game, there is of course nothing mysterious about the use of the word “open”. But this word corresponds to our “infinite”, & the games we play with the latter differ from 31) only by being vastly more complicated. In other words, our use of the word “infinite” is just as straight forward as that of “open” in 31 || 32?), and our idea that its meaning is

{{BBB TS reference|Ts-310,29}} “transcendent” rests on a misunderstanding.)

We might say roughly that the unlimited cases are characterized by this: that they are not played with a definite supply of numerals, but instead with a system for constructing numerals (indefinitely). When we say that someone has been supplied with a system for constructing numerals, we generally think of either of three things: a) of giving him a training similar to that described in 30), which, experience teaches us, will make him pass tests of the kind mentioned there; b) of creating a disposition in the same man's mind, or brain, to react in that way; c) of supplying him with a general rule for the construction of numerals.