### Summary

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 twenty-one, 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 twenty-two, 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 twenty-three to thirty-two, 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 thirty-three, 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.

### Analysis

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.