Wittgenstein gives an example. I teach someone the series "Add two," that runs two, four, six, 8 , and he writes it to my satisfaction up to 1000, but beyond 1000 he begins writing 1004, 1008, 1012 . On what grounds can we say he is following th e rule incorrectly? When I said, "Add two," I meant that he should write "1002" after "1000," but I surely did not have those two numbers specifically in mind when I gave the rule. I simply assumed that that is what I would do at that stage.
We want to say that even if "'1002' follows '1000'" is not directly on my mind, something in what I say or mean determines all the steps in advance. The algebraic formula for a series determines every step in advance insofar as people have been trained in such a way that everyone will normally write down the same series given that algebraic formula. There is nothing in the formula itself (nothing in what I say or mean) that determines the steps. Without training, the formula is meaningless.
The idea that a formula determines every subsequent step is like the idea that a machine at rest contains in it the possibilities of its movement. The possibility of its movement is not an observation of past experience or a prediction of future movement: it would seem to be something in the present state of the machine. But there is nothing in the machine that we could call its "possibility of movement." this is just an expression that tells us the kinds of movement we anticipate in the machine.
If we cannot prove that following "1000" with "1004" is incorrect, we might conclude that any interpretation of a rule can be correct, and that at every step, a new interpretation is needed. Wittgenstein counters this conclusion by suggesting that followi ng a rule does not usually consist of interpretation at all. If I follow a sign-post, I am not interpreting the sign-post; I am according myself with the custom or institution of sign-post-following that is a common practice in my community. There cannot be a society in which there was only one rule that was obeyed only once, because rules can only exist as public practices.
Following a rule correctly is not guided by guessing the rule-giver's intention, hearing an inner voice, or finding some logical justification. When I teach a rule, I am teaching a certain practice, and when I follow a rule I am obeying that practice. The practice need not rely on any further justification. This absence of justification does not mean that I am free to interpret and then follow a rule however I choose. The question of interpretation or choice does not occur to me when I obey a rule.
Though there is no ultimate justification for following rules as we do, we do not dispute how to follow a signpost or follow the order "Add 2." These practices of rule following are forms of life that are prior to questions of justification and interp retation. If we could not agree on how to follow these rules, there would be little point in disputing them because the level of our misunderstanding would be so deep as to render any meaningful communication impossible.
The example of the student who incorrectly adds two is meant to prove that the student's only understanding of the concept "add two" comes from our having written out the first five or ten terms in the series and then saying, "now go on like this." This t eaching applies equally well to the series we would understand as "Add two" and to the series we would understand as "Add two until 1000, and then add four after that." Indeed, if we have only written down the series up to twenty, then it would seem that there is an infinite number of interpretations the student could draw from our teaching.
If the student does not interpret the rule as we intended it, how can we say the student has done it wrong? We can say "I would have written '1002' after '1000,'" but there are an infinite number of these conditional claims, and they cannot all have been on our mind when we explained the series. What we need is some sort of super-fact that grounds all these different conditionals.
The difficulty that comes up again and again is that any further rule, explanation, or justification that we provide is equally open to various interpretations, and so cannot determine each step in the series any better than the initial order, "Add 2." W e might provide an interpretation of the initial order, but then we will also need to provide an interpretation of the interpretation, an interpretation of that interpretation, and so on.
We are looking for an absolute standard for correctness. Wittgenstein tells us that no such standard exists. Nothing in the rule tells us what is right or wrong. Because every step in following a rule can be interpreted in countless different ways, every step we take requires a new act of interpretation, a new choice on our part in how we will follow the rule. Wittgenstein answers the question of how we can know what is the correct interpretation of the rule by saying there is no correct interpretation, a nd if we all obey the rule "add two" in the same way, that is simply a matter of convention.
Another reading of this passage is that Wittgenstein is not simply telling us that there is no standard for correctness, but showing us that the very notion of an ultimate ground of correctness is incoherent and misleading. The example of a student adding four after 1000 is odd because we generally do not think of writing out a series of even numbers as requiring knowledge of what is the correct interpretation. There is not only no standard for correctness, but also, there is no act of interpretation, whe n writing out a series.
We can connect this reading to Wittgenstein's discussion of skepticism. The trouble with skeptical doubts such as "how do I know this is a hand?" or "how do I know the world existed five minutes ago?" is that if these sorts of things are open to doubt, it is impossible to have any fruitful discourse on pretty much anything. How am I supposed to discuss history, or even current events, with someone who genuinely doubts that the world has existed for more than five minutes?
Not everything is open to doubt, or else we would not be able to understand one another. Similarly, not everything is open to interpretation. If we lived in a society where writing out the series "add two," was open to interpretation, or where the rule wa s constantly interpreted in different ways without there being any means of agreement on how to interpret it, such rules would lose their point entirely. In dismissing the idea that there is any absolute standard for correctness in interpreting a rule, Wi ttgenstein is not telling us that rules are not as solid or definite as we may think. On the contrary, he is telling us that if rules were the kinds of things that constantly required interpretation and justification, they would not be of any use to us at all.