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

MICHAEL DUMMETT
a method. Normally one would say that the sieve of Eratosthenes
was such a method; but with a large number one would not
-could not-use the sieve, but would resort to some more power-
ful criterion. It will be said that this is a mere practical, not a
theoretical, matter, due to the comparative shortness of our
lives. But if some fanatic devoted his life to computing, by means
of the sieve, the primality of some very large number proved
to be prime by more powerful means, and arrived at the con-
clusion that it was composite, we should not abandon our proof
but say that there must be some error in his computations. This
shows that we are taking the "advanced" test, and not the sieve,
as the criterion for primality here: we use the theorem as the
standard whereby we judge the computation, and not conversely.
The computation is of no use to us because it is not surveyable.
A mathematical proof, of which computations are a special case,
is a proof in virtue of our using it to serve a certain purpose;
namely, we put the conclusion or result in the archives, that is,
treat it as unassailable and use it as a standard whereby to judge
other results. Now something cannot serve this purpose, and
hence is not a mathematical proof, unless we are able to exclude
the possibility of a mistake's having occurred in it. We must be
able to "take in" a proof, and this means that we must be certain
of being able to reproduce the same proof. We cannot in general
guarantee that we shall be able to repeat an experiment and get
the same result as before. Admittedly, if we get a different
result, we shall look for a relevant difference in the conditions
of the experiment; but we did not have in advance a clear
conception of just what was to count as a relevant difference.
(It is not quite clear whether in saying that we must be able
to reproduce a proof Wittgenstein means that one must be able
to copy from the written proof before one and be certain that
one has copied without error, or that one must be able to read
the proof and understand it so that one could write it down
without referring to the original written proof, so that the
possibility of a misprint becomes more or less irrelevant. It does
not seem to affect the argument which interpretation is adopted.)
Thus the computation, for a very large number proved prime
by other means, of its primality by means of Eratosthenes' sieve
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