[FOM] Davis on Torricelli

Dean Buckner Dean.Buckner at btopenworld.com
Sun Nov 2 04:38:19 EST 2003

Tait wrote [FOM: 31 Oct 2003]

> Leibniz also believed (at least sometimes) in the actual
> infinite---which was not a commonly held belief at the time.

I am not an expert on Leibniz, but I was quoting what Mancosu said in the
book mentioned by Davis.  I also benefited from an (excellent) paper that
Richard Arthur has privately circulated.

Tait's example from Swinehead was interesting.  What is the reference for
this please?

> In any case, independently of the issue of how one is to understand the
> distinction between actual and potential infinite , the very
> possibility of these figures (Torricelli's and the one I wish
> Swineshead had constructed)  contradicts Aristotle, which would seem to
> support Martin Davis's statement.

That is true, but I did say it was a simplification, not that it was
incorrect.  Davis statement rather implies everyone that X, until
Torricelli, then they thought Y.

I wrote
> > It was shocking to Hobbes, but only because of
> > his (indefensible) position that the infinite cannot be given in sense
> > experience, a position that arose from his extreme empiricism &
> > antagonism
> > to scholastic (i.e. Aristotelian) philosophy.

> This is a strange observation: it was Aristotle's view and the almost
> unanimous view of the medieval philosophers (after the transmission of
> Aristotle's works) that there is no actual infinite and, in particular,
> that the infinite is not given in sense experience---in the sense that
> there are no infinities in the natural world at any particular time.
> (The qualification is needed because the infinity of days and nights up
> to any given time was a problem for them.) So Hobbes is in agreement
> with them on this.

Hobbes extreme empiricism is undoubtedly the reason he opposed this sense of
the actual infinite.  And you only have to read a little Hobbes to get a
sense of how much he dislikes Aristotle.  As for "almost unanimous" - what
does "almost" mean?  As I said in the posting, the "official" view as by
Aristotle, then Aquinas (can't get any more official than Aquinas), is that
it is potential.  But Ockham (as I said) believed in actual infinity (in the
sense characterised by Shipman earlier)

Gregory of Rimini also held that "every continuum has a plurality of parts,
and not so many finite in number that there are not more (non tot finitas
numero quin plures), and has all its parts actually and at the same time"
Also John of S. Thomas.  Suarez (1548-1617) possibly the "last" scholastic
of the original scholastic tradition, thought that the parts of the
continuum though no actually divided are "actually distinct", existing
"formally" as parts joined by "continuing indivisibles".  Continuing
indivisibles are the points that are as it were within the continuum, and
which join together the parts that can be separated at them.  I'm  not
saying any of that makes sense, just that people had these thoughts, and
that they were not unanimous.

Of his near-contemporary Suarez, Hobbes wrote "When men write whole volumes
of stuff (meaning Suarez first book), are they not mad, or intend to make
others so?" (Leviathan I. viii)


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