THE HARRIED MATHEMATICS INSTRUCTOR
IT IS initially bewildering why, in the course of his discussion of knowing how to go on in the Philosophical Investigations, Wittgenstein introduces the following example:
185. . . . Now we get the pupil to continue a series (say +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 meant to add two: look how you began the series!' -- He answers: 'Yes, isn't it right? I thought that was how I was meant to do it.' -- Or suppose he pointed to the series and said; 'But I went on 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 order with our explanations as we should understand the order: '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.
There need perhaps not be much argument what Wittgenstein is asking us to suppose here. We are to suppose, not merely that the student makes this mistake, but that he takes everything we say by way of showing him that it is a mistake, as confirmation that he has proceeded correctly. Like a rational person, he stays to talk about it, and has something to say in response to any point we make; and what he says is said with apparent earnestness and because it seems right to him; but whether he is rational in the further sense that what he says, although wrong, is nevertheless such that (a) we can understand someone thinking it right, and (b) given it, he had proceeded correctly, is an important question on which Witt-