FOM: Leibniz and the actual infinity

[Moshe' Machover]

Further to the postings by Julio Gonzales Cabillon, Alexander Zenkin and
Charles Silver regarding Leibniz on the infinite, let me draw your
attention to the detailed discussion in ch. 10 of A Robinson's Non-Standard
Analysis on Leibniz' view of infinite and infinitesimal *magnitudes*.

Before I summarize Robinson's discussion, let me make two remarks.

1. Robinson had made a special study of Leibniz, in whose philosophy he was
interested already in his student days. (I know this first-hand, as R was
my teacher. But see also Dauben's biography of R.)

2. The issue of the reality or otherwise of infinite/infinitesimal
*magnitudes* is not quite the same (at least it was not the same for
Leibniz) as the issue of the existence of actual infinity. The former
concerns mathemtics; the latter concerns physical reality.

According to Robinson, Leibniz' view on the subject, despite occasional
slight inconsistencies, remained basically unchanged during the last two
decades of his life.

In 1716 Leibniz, writing in French, states quite clearly that he does not
believe at all that there exist magnitudes that are truly (ve'ritablement)
infinite or infinitesimal: in his view they are fictions, but useful
fictions, that can be used to shorten [arguments] and to speak more
generally (pour parler universellement).

But in 1701, also writing in French, he makes his position even clearer. He
says that if one doesn't believe in infinite or infinitesimal lines in
methaphysical rigour and as real things (a` la rigeur metaphysique et comme
des choses re'elles) one can surely still use them *as ideal notions* which
shorten argumentation (qui abre`gent raisonnement) *in a similar way that
one uses imaginary roots in ordinary analysis, as for example the sqrt of
-2.*

>From this it is clear that for Leibniz infinite and infinitesimal
magnitudes were not real in the sense that real numbers are real (whatever
that may be...) but their status is similar to that of imaginary numbers.


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