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Now—judged by the usual criteria—the pupil has mastered the series of natural numbers. Next we teach him to write down other series of cardinal numbers and get him to the point of writing down series of the form
0, n, 2n, 3n, etc.
at an order of the form "+n"; so at the order "+1" he writes down the series of natural numbers. —Let us suppose we have done exercises and given him tests up to 1000.
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 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 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.
--Ludwig Wittgenstein, Philosophical Investigations, passage 185.
at an order of the form "+n"; so at the order "+1" he writes down the series of natural numbers. —Let us suppose we have done exercises and given him tests up to 1000.
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 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 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.
--Ludwig Wittgenstein, Philosophical Investigations, passage 185.
