[FOM] constructive cauchy
Gabriel Stolzenberg
gstolzen at math.bu.edu
Fri Nov 9 21:08:09 EST 2007
It is said that Cauchy, in his argument in Cours d'Analyse about
a limit of continuous functions being continuous, say, on [0,1],
failed to see that the convergence must be uniform. (Actually, it
is easy to see that, classically, uniform convergence is sufficient
but not necessary.)
Nonstandard analysis readings of this argument go back at least
to John Cleave in around 1970. Here I wish to give a very sketchy
first draft of a constructive reading of it.
* * * * *
In constructive mathematics, a set comes with an equality relation.
A function from one set to another is a rule that preserves equality.
So, if, for example, we take a real number to be a Cauchy sequence of
rationals and define two to be equal if their difference is a null
sequence, then
'f is a real-valued function on [0,1]'
means that
for all x in [0,1], f(x + a) = f(x) if a = 0.
(Note. Here a = 0 is a natural definition of 'a is an infinitesmal.')
Cauchy gives two definitions of 'continuous function,' related
by 'in other words.' But there is an ambiguity. One can be read as
saying that, for all x in [0,1],
f(x + a) -> f(x) as a -> 0,
This is a standard definition of 'a pointwise continuous function
on [0,1].'
Cauchy's second definition of 'continuous function' can be
read as saying that
f(x + a) = f(x) if a = 0,
which, as I noted above, when read constructively, is the standard
definition of 'f is a real-valued function on [0,1].' (If we alter
the variable by an infinitesmal, then the value also changes by an
infinitesmal.)
So, on these readings, the two definitions are different. In the
first, we consider f(x + a) for a range of different values of a as
a -> 0; whereas, in the second, we consider f(x + a) for only a single
real number, a = 0. My guess is that Cauchy conflated the two.
* * * * *
As in classical mathematics, the standard constructive reasoning
about the continuity of the limit assumes uniform convergence. However,
for reading Cauchy, it may be relevant to note that, in constructive
but not classical practice, we expect to find (on the page!) that any
given point-wise convergent sequence comes with a modulus of uniform
convergence. (The classical counterexamples do not work because, in
those cases, the limit function is not constructively defined on all
of [0,1].)
Furthermore, on a meta-level, the claim that "point-wise = uniform"
is a consequence of the meta-theorem that an integer-valued function of
a real variable is locally constant.
Similarly, if, using the second definition, we read Cauchy's argument
as proving merely that a point-wise limit of functions is a function,
then, both in practice and on the meta-level, we can see that all of
these functions, including the limit, come with a modulus of continuity
and that the convergence is uniform.
* * * * *
In reading Cauchy's argument more carefully than I have here, it
may be relevant to note that he allegedly accepted Abel's criticism
of it. If he did, what was his understanding of where and how he had
gone wrong? Finally, I would like to suggest that, in speculating
about these and similar questions, we do well to recall Imre Lakatos'
remark that history is a caricature of its rational reconstructions,
bearing in mind that he was not talking about politics (although it
is true for it too--think of the Boston Tea Party or the storming of
the Bastille) but about science and mathematics.
Gabriel Stolzenberg
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