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.