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# Wittgenstein on rule following

Sat 26 Jun, 2010 03:59 pm
The rule following passages in the investigations have to be one of the more interesting aspects of Wittgenstein's later philosophy. Certainly, his thoughts on logic, mathematics, necessity, private language, and scepticism are related to his position on rules.

Section 185 of the Investigations seems to be a good place to centre the conversation:
"Let us return to our example (143). 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.
"

Wittgenstein clearly thinks that there is nothing in the correct segment of the series (the bit up to 1000) that makes 1002 the correct continuation of the +2 series, nor in the verbal/written expression of the rule, "Add two", and section 186 makes it clear that there is no fact discoverable by introspection about the pupil's or the learner's mental content that makes 1002 the correct continuation. So, what, if anything, makes it the case that the correct continuation of the series + 2 is „998, 1000, 1002, 1004, 1006 . . .‟ and not „998, 1000, 1004, 1008, . . .‟?
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jack phil

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Sun 27 Jun, 2010 05:44 pm
It is sometimes referred to as Wittgenstein's paradox, i.e. that there is no number sequence that could not be made out to conform to a rule. I think the quote you provided was an example of how someone might do honest mathematical work and the answers are not predetermined. He seemed to have been trying to make something as simple as counting a creative endeavor, but then, maybe not.

Consider how children learn to count. I heard a story just the other day about how one child counted 'out of order', as the person related it to me. I thought about suggesting to the mother that the 5 year old read Wittgenstein.
mickalos

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Mon 28 Jun, 2010 10:34 am
@jack phil,
jack phil wrote:

It is sometimes referred to as Wittgenstein's paradox, i.e. that there is no number sequence that could not be made out to conform to a rule. I think the quote you provided was an example of how someone might do honest mathematical work and the answers are not predetermined. He seemed to have been trying to make something as simple as counting a creative endeavor, but then, maybe not.

Consider how children learn to count. I heard a story just the other day about how one child counted 'out of order', as the person related it to me. I thought about suggesting to the mother that the 5 year old read Wittgenstein.

I don't think Wittgenstein wants to make calculating anything other than it is. Wittgenstein's writings are wholly descriptive, his aim, as I see it (not just on rule following), is to dissolve problems that have been created by the poor descriptions of his philosophical forbears and contemporaries. Rule following is a case in point. If we see following a rule as supplying an interpretation, then, yes, anything goes. Yet, clearly anything does not go; 1004 is an incorrect continuation of the series. Thus, construing rule following as being entirely about supplying an interpretation is wrong. As Wittgenstein puts it, "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."

Wittgenstein's answer is not to swallow the sceptical paradox, but to say, "It can be seen that there is a misunderstanding here from the mere fact that in the course of our argument we give one interpretation after another; as if each one contented us at least for a moment, until we thought of yet another standing behind it. What this shews is that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" and "going against it" in actual cases." To follow a rule is not to give the right interpretation, but to master a technique, to be indoctrinated into the practices of the rule following community and act in accordance with them. 1002 is the correct continuation because that is how we calculate.

He does take calculating to be a creative endeavour in a sense, I suppose; for, in Remarks on the Foundations of Mathematics he calls mathematicians inventors rather than discoverers, but this is only in the sense that they have it in their power to create new rules and practices that we all must go by. In no sense can a child simply continue a series in any way he pleases. Something that is implied by this, though, is a conventionalism and relativism about necessity: 12x12=144 because we all agree that it does. Certainly, we might imagine that our practices were different. We might even imagine a Martian coming along, looking at our maths textbooks, and saying "Try looking at it like this ...", going on to construct a proof of 12x12=143. There is nothing to stop us accepting the proof, and accommodating it in our practices.
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