§189

Wittgenstein does not want to deny that, "the steps are determined by the formula..." makes sense. He just asks how the expression is used. His method, I assume, is to get rid of conceptual confusion by drawing attention to uses of the expression which make sense.

Training/education is mentioned here. Would it be correct to say that the right way to continue the series is the way that people with the right sort of education would continue? Presumably not - people educated in mathematics might make mistakes. But there is the connection with mastery of a technique here - gaining the appropriate abilities.

Wittgenstein suggests that we might say that "for these people [trained to take the same step at the same point] the order '+3' completely determines every step from one number to the next". But this doesn't sound right. Surely it would be correct for someone to write '1006' after '1003' even if they hadn't shared in the training of the relevant group. Or is Wittgenstein's point that another group might have different concepts similar to our number concepts?

"'y = x(squared)' determines a number y for a given value of x' is a grammatical truth - so it's unclear what to make of the question 'is y = x(squared) a formula which determines y for a given x?'

What was incorrect in the interlocutor's view? - It is a mistake to think that the formula makes certain steps correct (as if there is some intermediary between formula and application) a given step is correct - in accord with the rule.