> So my question is, has this
[error in Wantzel's proof]
been noticed before? And since
> Wantzel's proof has an error, to whom do we credit the
> first correct proof?
About the first question, I don't know, but my answer to
the second, on which I expatiate below, is "Wantzel".
> The problem in Wantzel's paper also raises some broader
> questions: what constitutes sufficient proof for a result in
> any historical period, and what are the different ways in
> which a proof of one period may be considered insufficient
> by a later generation of mathematicians? Was the error in
> Wantzel's paper not noticed by all those who give him
> credit, or was it knowingly tolerated and accepted as a
> correct proof?
>
> As usual, any comments or feedback will be welcome.
I've thought long and often about such questions, and have a
fairly firm opinion about them, which is, at least, a considered one.
Let me remark first that this sort of thing is the norm rather
than the exception. It would be hard to come up with a proof from
this period or earlier which would nowadays pass muster. So if
we were to reject all such invalid proofs, we'd end up saying
that there was nothing proved before about 1850 (and if we took a
very hard line, we could make this date much later).
Here are my proposed two tests:
1) Is the proposed "bug" something that could have been noticed
by a contemporary?
and
2) If so, and had the contemporary confronted the author with it,
is it likely that the author would have found it trivial to
patch it up?
Test 1) guards against retrospective application of changes in
standards. It would clearly be unfair, for instance, to void a
proof because it was not expressed in a formal language invented
at a later time.
Your bug certainly passes this test - it was as clearly a bug in 1837
as it is today.
Test 2) is to prevent us from being too strict about the legalities.
We obviously mustn't, for instance, dismiss a proof just because it has a
typographical error in it, and in my view, the same principle should and
must be extended to minor errors of the author, provided they don't
really affect the viability of the proof.
Of course there's room for argument about just how far to go here,
because we don't want to "pass" a proof that has a really significant
bug in it. My test is admittedly not an operational one, since it
depends on our feeling for what a long-dead author might do; but it
can work, precisely because it only asks about the plausibility of
a trivial patch.
^^^^^^^
In this case such a trivial patch is definitely plausible. I don't
say certain, because plausibility is all that's required to give
Wantzel the benefit of the doubt. My guess is that he'd agree with
you that this is a bug (so it passes test 1), but quite likely come
back with a cure in 5 minutes (so it fails test 2).
You might be inclined to be more legalistic, but if so, please
consider what your attitude would be if you found such a bug in
some work of a living colleague, who 5 minutes later telephoned
you with a valid correction. Don't be harsher with Wantzel!
John Conway