# [FOM] Question on the number line

Dean Buckner d3uckner at btinternet.com
Wed Nov 16 14:31:27 EST 2005

```Peirce:
> It is certainly true, First, that every point of the area
>  is either black or white, Second, that no point is both
>     black and white, Third, that the points of the boundary are
>     no more white than black, and no more black than white....

Thank you for this quote.  Peirce was familiar with scholastic writings
on continuity - perhaps not the piece I am studying, but the same
examples tend to crop up everywhere, so I wonder if there is a
connection here?  Similarly with Leonardo - the boundary between the sea
and the air is also one which Suarez discusses.

> A continuous line can usefully be thought of as an ordered set of
> points for mathematical purposes, but the  points are "abstractions,"
> and should not be  thought of as corresponding to real "parts" of  any

> actual continuous "substance." ...

Suarez also discusses this idea, concerning what he calls "indivisibilia
substantiae" which seem to be what you allude to as real (but
indivisible) parts of a substance.  He says:

"non possint esse in materia indivisibilia substantiae, alioqui saepe
amitteretur aliquid materiae, et aliquid de novo fieret. Nam per
divisionem continui destruitur una superficies, quae continuabat partes
corporis, et resultant duae terminantes; ergo, si superficiebus
quantitativis correspondent substantiales termini proportionales, fit ut
per illam divisionem amittatur etiam communis terminus continuans
materiam, et duo extremi seu terminantes resultent"

Which is a difficult passage, but which I translate as "there could not
be in matter indivisibles of (?) substance, otherwise there would often
be lost some of the material, and some [of it?] would be created anew.
For by dividing a continuous [body], one surface is destroyed, which was
continuing the parts of the body, and there result two terminating
[surfaces]; therefore, if to quantitative surfaces there correspond
substantial boundaries in proportion, it comes about that by the
division is lost also the common boundary continuing the material, and
two extrema or boundaries will result".

This is an imaginary opponent's view, to which Suarez' answer is that,
though this is a difficult question, there really are "substantial
indivisibilia", and that they really are destroyed and created anew,
either by the power of matter, which is not really inert, but "active"
in some way.  He says "dici etiam potest haec indivisibilia materialia
resultare ex vi illius actionis qua Deus conservat materiam, esseque
veluti quamdam concreationem" - it can be said that these material
indivisibles result from the force of the action by which God preserves
the material, and which are as it were a concreation".  But what is a
concreation?

What is interesting, as I noted before, is that here is a Catholic
philosopher, moreover a Jesuit philosopher, and a leading philosopher of
that order entertaining ideas about the actual nature of indivisibilia,
and of the actual infinity of the continuum, that some historians say
they cannot (after the model of Galileo's detractors) have held.

D.

```