THE CAMBRIDGE COMPANION TO WITTGENSTEIN

            We take any of the three alternatives as a criterion of going according to rule, then “ it would be possible,” wittgenstein hold, “ that 4 + 1 sometimes made 5 and sometimes something else. It would be think able, that is to say, that an experimental investigation would shew whether 4 + 1 always makes 5. “ empiricism in mathematics lead to anarchism every bit as much as psychologism does. We could no more talk about adding than frege’s psychological logicians could discuss the properties of the moon. Denying the special and peculiar role of mathematical proposition makes mathematics arbitrary and everchanging, subject to the shifting winds of feeling, intuition, and practical consequence. [ if I bet on a winning horse because I miscalculated the odds, does that mean I didn’t  really make a mistake in mathematics?]

            Wittgenstein continues with his own account of the criterion of going according to the mathematical rule :

            If it is not supposed to be an empirical proposition that the rule [+ 1] leads from 4 to 5, then this, the result, must be taken as the criterion for one’s having gone by the rule.

            Thus the truth of the proposition that 4 + 1 makes 5, is so to speak, overdetermined. Overdetermined by this, that the result of the operation is defined to be the criterion that this operation has been  carried out.

            The proposition rests on one too many feet to be an empirical proposition. It will be used as a determination of the concept “ applying the operation +1 to 4.” For we now have a new way of judging whether someone has followed the rule [RFM, VI-16]

            In the next section we will deepen our sketch of widgestaen’s later view by contrasting it with a famous misinterpretation