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Wittgenstein considers the different games that might teach someone how to read a table. A table, like an ostensive definition, gives us a rule that we can follow. For instance, we learn to correspond words in one column to images in another. In turn, we need another rule to tell us how to read this table. In Section 21, Wittgenstein gives examples of different possible rules that could explain how we are to read a table with two columns and four rows. For instance, we can imagine a rule that instructs us to simply from read left to right, but a different rule might tell us to read in a criss-cross pattern. This rule could then be adjoined to the table and we could imagine having another rule to explain how we are to make sense of this rule. On the other hand, it is not necessary that we have a rule to explain every rule that we follow.
Game two introduces a finite series of numbers, but the introduction of an infinite series must be learned in a different way. In Section 22, Wittgenstein conceives of two similar card games, one played with thirty-two cards, and one in which there is a pencil and a series of blank cards so that you can add as many cards to the deck as you like. This latter "unbounded" game may differ from the "bounded" game in a number of ways—its rulebooks might use the words "and so on," players might ask "how high shall we go?" before play begins—though the unbounded game could also be played with thirty-two cards and be indistinguishable from the bounded game. There need not be any concept of infinity in the minds of the people playing.
From Sections 23 to 32, Wittgenstein introduces a series of different number systems. He says that the difference between finite and infinite systems is that finite systems introduce a definite supply of numerals to count with, while infinite systems provide a system for counting. This system for counting can be taught either by rigorous training, by developing a mental disposition to proceed in a certain way, or by providing a general rule by which a person can construct further numerals.
In Section 33, Wittgenstein introduces a table in which the letters "a" through "d" represent the four compass directions, and an order such as "aacadddd" can tell someone how to move. The table, but not the order, acts as a rule in this case. Following this rule can be a matter of consulting the table at every move, or it can be a matter of knowing how to move without consulting the table at all. We can also imagine a series of letters—say "cada"—that can provide a rule for the repeated application of the same movements.
We could also give someone general training in how to read tables. That person could then look at any table and respond to orders based on that table. Each table can be seen as a rule or as an expression of a rule: there is no discernable difference between the two. In sections forty-two and forty-three, Wittgenstein considers a game consisting of dots and dashes that represent steps and hops. It is not clear to what extent we can say this game is limited or not limited, nor at what point we can say someone playing this game is guided by the rules or not.
Wittgenstein was the first to recognize the philosophical significance of rule-following. Wittgenstein asks original questions about rules: What is a rule? How do we know how to follow a rule? How do we learn to follow rules? Wittgenstein's answer to the first question helps us appreciate his approach. He consciously decides not to give us definition of a rule. The word "rule" is like the words "game" or "comparing" or "recognizing": no one fixed definition applies to all cases of rules. Rather, there are a number of related concepts, all of which we might call "rule." Wittgenstein stresses that he has not differentiated between what he calls a "rule" and what he calls the "expression of a rule." We could call a table a rule, but we could also call it the expression of a rule.
If pressed to define rules (to which Wittgenstein would object), we could say that rules teach us "how to go on." We can contrast them with orders by saying that orders give us a fixed number of things to do, whereas rules give us a general description of how to obey certain kinds of orders. This is one of the distinctions between teaching someone a finite system of numbers, as in game two, and teaching someone an infinite system of numbers. We can teach someone the numbers one to ten simply by counting out the numbers on our fingers and then drilling the student until she knows all the numbers. To teach the student how to count in our infinite number system, however, we need to provide her with some sort of a rule. We need to teach her not just the numbers one to ten, but how to go on counting beyond ten.
However, in attempting to define a rule as something that tells us "how to go on," we are saying what a rule does, not what it is. We have not identified any feature or essential quality of a rule that makes it possible for these rules to help us go on. In effect, as Wittgenstein says, we have not managed to distinguish between the expression of a rule and the rule itself.
Wittgenstein recognizes that we can also have rules to explain other rules. A table provides us with a general rule for following certain instructions, but how do we know how to read the table? We need another rule that tells us to read from left to right. But how, then, do we know how to follow that rule? Most rules are governed by another set of rules. Wittgenstein is not claiming that an infinite series of rules exists telling us how to follow other rules that tell us how to follow other rules. He says that at some unfixed point, we can understand a certain rule without any further help.
For instance, some people might need to read a table in order to follow the order "aacadddd," while some people might simply understand the order without needing to refer to the table. We might object that the people who simply understand that order must in some way be appealing to the rule: they must be looking at a mental picture of the table or be following it unconsciously. Wittgenstein acknowledges that perhaps someone who follows the order without reference to a table once learned the rule by following the table, but he asserts that learning a rule is different from following a rule. There is no reason for us to think that the person must have a mental picture of the table in order to follow the rule without referring to the table. If that were necessary, we could argue that the person must also have a mental picture of the rule that explains how to follow a table, and the rule explaining that rule, and so on.
This argument cautions against the common temptation to think of "mental pictures" as somehow doing work that written tables cannot do. It is just as absurd to think that there is an infinite series of mental pictures telling us how to follow rules as it is to think that we need an infinite series of written rules and tables to tell us how to proceed. At some point, we simply "know how to go on," and appealing to mental pictures will not help us explain how we know.