Milinda’s Questions

The details of why I came to prepare the following aren’t worth going into. Suffice it to say that it was motivated by extreme boredom at work a couple of weeks ago — while I was scoring a math test, in fact. It’s my own version, slightly rewritten and edited, of the “following a rule” passage from Wittgenstein’s Philosophical Investigations. (This part is most of sections 185 through 187, plus a little bit from 198 and 201.) Editing a text like this to make it a bit easier for readers (I hope) actually forced me to pay closer attention to exactly what it’s saying. So enjoy.

Judged by the usual criteria, the student has mastered the series of integers. Next we teach him to write down other series of numbers and get him to the point of writing down series of numbers of the form

when we give him an instruction of the form “add n”; so at the instruction “add 1” he writes down the series of integers. Let us suppose we have done exercises and given him tests up to 1000.

Now we get the student to continue a series (say add 2) beyond 1000 – and he writes 1000, 1004, 1008, 1012.

We say to him: “Look what you’ve done!” He doesn’t understand. We say: “You were supposed to add two: look how you began the series!” He answers: “Yes, isn’t it right? I thought that was how I was supposed to do it.” Or suppose he pointed to the series and said: “But I went on in the same way.” It would now be no use to say: “But can’t you see….?” – and repeat the old examples and explanations. – In such a case we might say, perhaps: It comes natural to this person to understand our instructions with our explanations as we should understand the instruction “Add 2 up to 1000, 4 up to 2000, 6 up to 3000 and so on.”

Such a case would present similarities with one in which a person naturally reacted to the gesture of pointing with the hand by looking in the direction of the line from finger-tip to wrist, not from wrist to finger-tip.

“What you are saying, then, comes to this: a new insight – intuition – is needed at every step to carry out the instruction ‘add n’ correctly.”

To carry it out correctly! How is it decided what is the right step to take at any particular stage?

“The right step is the one that accords with the instruction – as it was meant.”

So when you gave the instruction “add 2” you meant that he was to write 1002 after 1000 – and did you also mean that he should write 1868 after 1866, and 100036 after 100034, and so on – an infinite number of such directions?

“No: what I meant was, that he should write the next number but one after every number that he wrote; and from this all those directions follow in turn.”

But that is just what is in question: what at any stage we are to call “being in accord” with that instruction (and the mean-ing you then put into the instruction – whatever that may have consisted in.)

“But I already knew, at the time I gave the instruction, that he ought to write 1002 after 1000.”

Certainly: and you can also say you meant it then: only you should not let yourself be misled by the words “know” and “mean”. For you don’t want to say that you thought of the step from 1000 to 1002 at that time – and even if you did think of this step, still you did not think of the other ones. When you said “I already knew at the time…” that meant something like “If I had then been asked what number should be written after 1000, I would have replied ‘1002’.” This assumption is rather of the same kind as: “If he had fallen in the water then, I would have jumped in after him.”

“But how can a rule show me what to do at this point? Whatever I do is, on some interpretation, in accord with the rule.”

That is not what we ought to say, but rather: any interpretation still hangs in the air along with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning.

“Then can whatever I do be brought into accord with the rule?”

Let me ask this: what has the expression of a rule – say a sign-post – have to do with my actions? What sort of connection is there here? Well perhaps this one: I have been trained to react to this sign in a particular way, and now I do so react to it. But a person goes by a signpost only insofar as there exists a regular use of signposts, a custom.

This is our paradox: no course of action can be determined by a rule, because every course of action can be made out to accord with the rule. The answer is: if everything can be made out to accord with the rule, then it can also be made out to conflict with it. And so there is neither accord nor conflict here.