[FOM] reply to S.S. Kutateladze (19 Mar)

[Gabriel Stolzenberg]

   In his March 19 reply to my "Antonino Drago on Leibniz,"
S.S. Kutateladze wrote,

> The history of infinitesimals is much older and deeper than the
> oversimplified progressivist's views.

   But my message was mainly about Leibniz and infinitesmals, not
infinitesmals in general.  I did say that there is no philosophical
concept of an infinitesmal.  But I shouldn't have because I'm in
no position to make such a blanket statement.  I should instead
have invited Drago, or anyone else, to teach us such a concept, if
he knows one.

> You may look at

> http://arxiv.org/abs/math.HO/0701068
> and
> http://arxiv.org/abs/math.GM/0608298

> for a few details.

   I read the first article quickly and glanced at the second.  I
didn't find them to be of much help but no doubt others will see
them differently.  (I find it hard to read answers to questions that
I'm not asking.)

   After reading the first article, I incline towards the view that
the term "over-simplified" is incorrect for Lacroix's statement.

   My main problem with the article is that I didn't see any concrete
mathematics.  Thus, Euler is praised---he's my hero too--but I did not
see an analysis of any of the particular mathematics he did that is
relevant for this discussion.  (Did I read too quickly?)

   One more point.  In the second article, on monads, the author
writes,

   "In any case, the view of the monad of a standard real number
    as the collection of all infinitely close points is generally
    adopted in the contemporary infinitesimal analysis resurrected
    under the name of nonstandard analysis in the works by Abraham
    Robinson in 1961."

   For what it is worth, in constructive mathematics, real numbers
may be taken to be collections of rational intervals, every pair of
which intersect and among which are ones of arbitrarily small length,
where any two of them are defined to be equal if their union is a
real number, i.e., if they differ by a real number equal to 0, i.e.,
if they differ by an infinitesmal.

   In this case, "infinitely close" means "equal as real numbers" and
the monad of a real number, as described above, is its equivalence
class, the set of all real numbers that are infinitely close (equal)
to it.

   A curiosity question.  Why is this a better candidate for the
monad of a real number than, say, the union of all real numbers
equal to it, which is itself a real number equal to it?  I ask in
the hope of learning more about what a monad is.


   Gabriel Stolzenberg

   P.S.  As for the use of the term, "progressivist," I am not going
to try to guess what it means.