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?