FOM: Leibniz and the actual infinity

[Charles Silver]

On Wed, 23 Sep 1998, Alexander Zenkin wrote:

> On Sun, 13 Sep 1998, Julio Gonzalez Cabillon wrote (in particular):
> 
> >Leibniz wrote a couple of months before his death in which he states without
> euphemisms <  >"I told them that I  [= Leibniz]  did not believe at all in the
> existence of magnitudes actually infinite or actually infinitesimal..."
> 
> Thank you for the direct Leibniz' citation (today, to my regret, I have
> no possibility to attend a library - because of the Russian government's
> "jokes"). 
> 
> If my memory serves (what follows, I cites by my very old outlines), Cantor
> himself writes in his "To Study of Transfinity" [in Russian: "K ucheniju o
> Transfinitnom"]: "… in my monography "Grundlagen einer allgemeinen
> Mannigfaltigkeitslehre" (Leipzig, 1883), I decline … the Leibniz authority who
> turned out very inconsequent in this question [AZ: on the actual infinity]
> 
> I think the fact of Leibniz's inconsequence,
> 
> > Leibniz's opinions/beliefs (?) about the infinite seemed to vary
> according to 'pompa & circunstancia',

	First, I am sorry that the situation in Russia is so bad at the
moment.  I hope things improve soon. 


	Leibniz, according to my understanding of him (which is not that
great) holds many original, fascinating, and provocative views that are
often difficult to render consistent with one another.  Often he provides
tricky arguments that supposedly reconcile these seeming inconsistencies.

	Regarding the "actual infinity," I think it is absolutely clear
that he believed in it.  He says: "I am so much in favor of the actual
infinite, that rather than admit nature abhors it, as one says vulgarly, I
hold that nature exemplifies it everywhere, in order to display better the
perfections of her author." (Phil. I, p. 416)

	However, he also rejects the notion of "infinite number".  He
says: "I do not at all admit any true infinite number, though I concede
that the multitude of things surpasses every finite number, or rather
every number." (Phil., vi, p. 629)

	He does explain how these two seemingly incompatible views are
possible, but I am not competent to do his position justice.  Whatever is
involved in his rejection of infinite number (which looks to me to be
related to his application of his infinitesimal calculus to problems in
physics), he has to maintain the real existence of at least two actual
infinities:  (1) the number of possible worlds is infinite, from which God
chooses the best one to actualize, and (2) the number of distinct
individuals (monads) in any given possible world is also infinite.  These
two actual infinities provide the bedrock of his entire metaphysical world
view, and I don't think there is any textual evidence indicating that he
ever changed his mind about either of them.

	(I am tempted to add that both (1) and (2) above reflect how
things look "from God's view."  From some other point of
view--ours?--there is *another* way to analyze the number of things in our
world, and this analysis does *not* yield an infinite number of them.  I
am also tempted to add that I think it is Leibniz's infinitesimal calculus
that supposedly resolves this incompatibility.  But, I'll just stick with
saying that I'm "tempted to add" these remarks, rather than actually
maintaining their truth, since I don't understand Leibniz all that well
here.  I would appreciate hearing others' opinions on these points, but I
think messages should be sent to me directly, as the topic seems far
afield of f.o.m.) 
 

Charlie Silver
Smith College