[FOM] Antonino Drago on Leibniz

[Gabriel Stolzenberg]

  In "progress of philosophy in mathematics" (13 March 2007)
Antonino Drago writes:

> For instance, why an apparently philosophical commitment of
> mathematics to the philosophical notion of an infinitesimal
> (see Leibniz) was extraordinarely progressive for Western
> mathematics?

   At the end of this message, I've appended one by the historian
of mathematics (and research mathematician), Harold Edwards, that
tells a rather different story.

   My layman's view is (1) there is no philosopical notion of an
infinitesmal and (2) the mathematical notion of which you seem to
be speaking may have participated in "extraordinary progress" in
the calculus, differential equations, etc. but there is no evidence
that it did much more than come along for the ride.  To say more,
one would have to compare a development with infinitesmals with one
without it.  And this would be difficult, if not impossible, to do.
Has anyone tried to do it?

> It is not a satisfying answer to use the after knowledge of
> non-standard analysis for evaluating positively this introduction
> of philosophy within mathematics. Because the question comes again
> as a question about present time mathematics.

   I don't think it "comes again."  Contrary to the standard line,
non-standard analysis hasn't made much of a contribution to
mathematics.  (A good example of this is Abraham Robinson's initial
application.  When a colleague of mine heard about it, he proved it
in about 20 minutes.)  It's interesting and I know a few people who
may have made some use of it--though, again, it's a little hard to
tell without having a proper control.

   Finally, speaking of nonstandard analysis, as I recall, Abraham
Robinson makes the same point that Edwards says Lacroix made--that
Leibniz considered his language of infinitesmals only as a shorthand.

   If anyone would like me to elaborate on any of this, please ask.

   Gabriel Stolzenberg

   P.S.  Here is the message from Harold Edwards.
__________________________________________________________________
Date: Fri, 18 Sep 1998
From: Harold M Edwards <hme1 at SCIRES.ACF.NYU.EDU>

Dear Gabriel,

     I'm afraid I have no information for you.  All I know about
Leibniz is what I read in the newspapers.  Lacroix says something on
the subject in his preface to his calculus that I like to quote (my
translation from the French):

     (Quote from Lacroix's book, "Traite du calcul differentiel
      et du calcul integral," 1797.)

     "Leibniz seems to have believed that those who were able to
     use the differential calculus would easily grasp its spirit,
     by comparing it with the method of the Ancients, because he
     neglected to enter into any detail whatsoever in this regard,
     and his silence was imitated by Bernoulli and L'Hopital; but
     when he was attacked on this subject, he showed by his responses
     that he had reflected maturely on it.  On all occasions, he
     compares his method with that of Archimedes, and makes clear
     that his is nothing but a sort of abbreviation of the other,
     more appropriate to research but in the end it amounts to the
     same thing; because instead of supposing the differential
     actually to be infinitely small, it suffices merely to conceive
     that one can always make them so small that the error one commits
     by omitting them in the calculation is less than any given
     magnitude; and to aid the imagination of his readers he provided
     some concrete examples [here there are page references].

     This method of reasoning, which would seem to be beyond
     reproach, was seen by Fontenelle as a confession on Leibniz's
     part of the inadequacy of his principles, from which would
     follow the collapse of the entire edifice he had erected on
     infinities.  The complaints Fontenelle makes about it in the
     preface to his geometry, and which have been repeated in many
     works, offer an example of the ease with which errors pass
     from book to book, and shows how few people take the trouble
     to form an opinion independent of that of others."

                 (End of quote from Lacroix's book.)

    Not having studied Leibniz "to form an opinion independent of that
of others" I can't say the Lacroix is right, but it has the ring of
truth.  I suspect Leibniz had indeed reflected maturely on the matter
and understood it well, but had difficulty communicating it to others
who had not done the same reflecting.
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