Dummett - Wittgenstein's Philosophy of Mathematics.pdf - Ted Sider

MICHAEL D UMMETT
seven girls there are twelve children. If he accepts these argu-
ments it will be quite true that he will have adopted a new
criterion for saying that there are twelve children present, and
again a new criterion for saying, "I must have miscounted."
Before, he would say, "I miscounted," only when he noticed
that he had, for example, counted one of the children twice over;
now he will say, "I miscounted," when he has not observed
anything of this kind, simply on the ground that he got the result
that there were five boys, seven girls, and thirteen children.
But we wish to say that even before we met this person and
taught him the principles of addition, it would have been true
that if he had counted five boys, seven girls, and thirteen
children, he would have been wrong even according to the
criteria he himself then acknowledged. That is, he must have
made a mistake in counting; and if he made a mistake, then
there must have been something that he did which, if he had
noticed it, he himself would then have allowed as showing that
he had miscounted.
If we say that if he counted five boys, seven girls, and thirteen
children, then there must have been something which, if he had
noticed it, he would have regarded as a criterion for having
miscounted, then the effect of introducing him to the concept of
addition is not to be simply described as persuading him to adopt
a new criterion for having miscounted; rather, he has been
induced to recognize getting additively discordant results as a
symptom of the presence of something he already accepted as a
criterion for having miscounted. That is, learning about addition
leads him to say, "I miscounted," in circumstances where he
would not before have said it; but if, before he had learned,
he had said, "I miscounted," in those circumstances, he would
have been right by the criteria he then possessed. Hence the
necessity for his having miscounted when he gets additively
discordant results does not, as it were, get its whole being from
his now recognizing such results as a criterion for having mis-
counted.
If on the other hand we say that it is possible to count five
boys, seven girls, and thirteen children without there being
anything other than the fact of getting these results such that,
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