Fermat and Publication

Here is Fermat's famous "marginal note" in his copy of Diophantus, 
describing what became known as his Last Theorem:

 "It is impossible to divide a cube into two cubes, a fourth 
   power into two fourth powers, and in general any power except 
   the square into two powers with the same exponents,...I have 
   discovered a truly wonderful proof of this, but the margin is 
   too narrow to hold it."

This note appeared in Fermat's edited works, published by Fermat's 
son Samuel in 1670.  It's worth mentioning that this manner of 
announcing results (stating the result, and that he had a proof of 
it, but that he lacked the time or the paper or whatever to actually 
spell out the proof) was frequently used by the senior Fermat.  As
a result, Euler and others had to reconstruct the proofs for most of
Fermat's claimed theorems more than a century later.

It is sometimes said that Fermat's marginal comments were merely
intended as private notes to himself, and that he never imagined they
would be published, but this is not really correct.  It's true that 
the senior Fermat never got around to publishing his comments on 
Diophantus, but it's misleading to call it a private notebook.  It 
was the style of his day to write (and publish) commentaries and 
observations on ancient works.  Typically the original text was 
printed with wide margins in which the modern commentator would 
expound on the subject and, if possible, show that he was even smarter 
than the revered ancient author.  The impulse to "outdo the ancients" 
was a major psychological factor driving the scientific revolution 
in the West during the 16th and 17th centuries.

With this in mind, it's clear that Fermat's notes were in the form 
of a commentary to be published.  Obviously no one would write a note 
to himself to inform himself that he had discovered a truly wonderful 
proof, but that unfortunately he didn't have room to tell himself what 
the proof was.  This is somewhat reminiscent of Galios's comments that 
he scribbled in the margins of one of his papers the night before his 
death: 

   "There are a few things left to be completed in 
    this proof.  I have not the time. (Author's note.)"  

As Tony Rothman observed, "It is unfortunate that Galois tarnished
some of the romance by including the parenthetical 'author's note'".
The point is that it's usually fairly easy to tell the difference
between something that is really a personal note someone has written 
to himself and a note intended to be read by others.  Clearly Fermat's 
commentary on Diophantus was written with the idea of it being read by 
others.  

Of course, this is not to suggest that Fermat ever made a decision 
to actually publish the commentary.  In fact, for various reasons, 
Fermat almost never decided to publish anything.  (This had both 
advantages and disadvantages for his reputation; it denied him some 
of the credit for differential calculus, but it also preserved 
"plausible deniability" for his various mis-statements.)  Needless 
to say, if he had ever published his Observations on Diophantus he 
would have omitted the above-quoted comment, having long since 
realized that his wonderful proof didn't work.  So, in a sense, it 
was fortunate for the development of number theory that he never 
edited his papers.  

There's actually a subversive alternative theory, which is that 
Fermat Sr. never wrote that particular comment at all, and it was 
in fact an original contribution of Samuel's.  This actually makes 
some sense, given the fact that the senior Fermat never once mentioned 
the fully general problem in any conversation or letter, even as a 
challenge question to other mathematicians  - and he LOVED to stump 
other mathematicians.  Indeed his life's passion seems to have been
number theory, and he was always striving (usually in vain) to cite
examples of propositions, theorems, and conjectures from this field 
of research that would impress people with the depth and subtlety of 
the subject.  It seems exceedingly strange that with such a striking 
proposition - one that is supported by lots of numerical evidence, 
and for which he had once believed he had a truly marvelous proof -
that he would never have returned to the question again, even in 
passing.  Especially since he returned many times to the question 
for cubes and 4th powers.  It is really quite incongruous.  Even
in its technical substance it was out of character because, as Weil 
notes, the "Last Theorem" was the ONLY time that Fermat ever mentioned 
a problem that involves a curve of genus greater than 1.