[FOM] Suarez on Proper Subsets and Actual Infinities

Dean Buckner d3uckner at btinternet.com
Sat Mar 11 08:17:33 EST 2006

```Yet another interesting passage from Francisco Suarez, the 16C Jesuit
philosopher.  The Latin is below.

As mentioned in previous posts, Suarez believes that there are
infinitely many points actually existing in the 'continuum'.  He
discusses the problem that any line drawn from the centre of the small
circle contained in a larger circle, intersects each of the circles at a
point.  Thus to each point in the small circle there corresponds a point
in the larger one, and conversely.  There is  one to one correspondence
(certa proportio) between the points in the circles.

Hence, if there were an actual infinity of points constructed in this
way, there would be as many points in the smaller circle than the larger
one, which is a contradiction.  Against the view of Gregory of Rimini
and Scotus (and discussed later by Galileo), that the notion of
'smaller' and 'larger' do not apply to infinite quantities, Suarez
argues that there are two senses of 'greater than'.

In the first sense, ('in certa proportione' - similar to one-one
correspondence), one infinity is no greater than the other.  In the
second sense, one infinity is greater than another, because it is
related as whole to part, and 'the whole contains whatever is in some
part, and something more'.  This seems to be the 16C version of one set
being a proper subset of the other.

Thus, one infinity can be greater than another in the sense that it
contains it as a proper part, but not in the sense that there is a one
to one correspondence between the indivisibilia that compose it.

---------------
Sed, licet verum sit unum ex his infinitis non posse esse maius alio in
certa aliqua proportione, tamen absolute capere non possum quin plura
sint in toto quam in partibus sigillatim et divisim sumptis, quia totum
continet quidquid est in aliqua parte, et aliquid amplius.

Quapropter, eo modo quo unum infinitum esse potest pars alterius, non
est cur repugnet plura puncta esse in toto quam in parte [...].

[Metaphysical Disputations 40, section 5, para 49].

```