[FOM] Suarez on the Continuum

[Dean Buckner]

I am just finishing off the first draft of a translation (from the
Latin) of Section V of the 40th Metaphysical Disputation of Francisco
Suarez, originally published in 1597.  I don't believe this section has
ever been translated into English.  There, Suarez argues against the
nominalist view that points, lines &c have no real existence, and gives
his arguments for the view that there are both 'terminating
indivisibles' and 'continuing indivisiables' that have actual existence
in the continuum.

Suarez is important for a number of reasons.  1.His work was highly
influential at the time, and was a standard textbook in Germany until
the 18th century.  It is at least a guide to what would have been
received wisdom for educated Europeans of the early modern period.  He
is known to have influenced Leibniz. (Baumann, in his Lehre von Raum,
Zeit und Mathematik in der neuern Philosophie (Berlin, 1868-9),
frequently cited by Frege, devotes the opening section of the first
volume (1-67) to the exposition of mathematically relevant terms and
concepts in the Disputationes. Baumann gives his reason as the
importance of the work of Suarez as a generally used textbook during the
early modern period).  2.He supplies many references to works on the
continuum, many of which are not referenced in modern histories of the
subject.  This suggests there is more to be understood about the history
of thought about the continuum up to the early modern period.  3.Suarez
was a leading thinker of the Jesuit order, and his advocacy of an actual
infinity casts some doubt on the conventional wisdom about the
development of mathematical ideas in that period.  The conventional
wisdom, I take it, is that Catholic theology was opposed to the concept
of actual indivisibilia.  (My view is the opposite: those who opposed
actual infinity were nominalist and generally anti-Catholic philosophers
such as Hobbes, Locke, Hume &c).  4.Suarez' work on the continuum is
interesting in itself, and raises some questions which are still
difficult today.

As it is unlikely I will publish this translation I summarise it below,
for FOM readers who may be interested.


--------- Suarez on the continuum ----

Section V of Disputation 40 is arranged as a classic scholastic
disputation.  First, there are 10 arguments *against* the house view.
This is followed by a middle section outlining and arguing for the house
view.  Finally, there are replies to the 10 arguments.

The 10 arguments are as follows: 1.There are no terminating points,
because, in whatever way two lines are joined, they are joined
continuously.  There is nothing 'in between', as it were.  2.There no
reason to suppose such points, for what effect does it have if they are
removed?  An added indivisible does not make a line larger, and
consequently does not make it smaller when removed.  3.There are no
'continuing points' because there are no terminating points, and a
continuing point is simply that which terminates the two parts of the
line that it divides.  4.If a continuing point is necessary to unite the
parts, why can't the parts be joined directly?  If some third thing is
necessary to join them, why not a fourth thing and so on ad infinitum?
5.To suppose the existence of indivisible points in the continuum
implies the existence of an actual infinity.  6.(An argument I recognise
from Ockham) that God could remove all points from the line.  This leads
to two impossibilities: that there would remain a continuous thing
divided in every part, and that there would be a multitude of points
everywhere discrete.  7.In a cylinder of finite length there could exist
a line of infinite length.  This involves Buridan's argument about the
'linea gyrativa', a spiral whose first turn is 1/2 of the cylinder, the
second a 1/4, the third 1/8 and so on.  8.There is no physical subject
in which a point could be given.  9.A point cannot connect the parts of
a line, for it either touches the parts in some indivisible thing, in
which case the whole line consists of points, or in some divisible
thing, which is impossible, for an indivisible cannot touch an
indivisible. 10.According to Aristotle, a surface is only potentially in
the middle of a body, not actually.

Suarez then argues the case for indivisibilia.  Some of the arguments he
gives are (A) that a perfectly spherical object must touch a perfectly
flat surface in a single point.  If it touched in some extended part,
then either the sphere would have a flat bit, or the surface would have
parts which are equidistant from the centre of the sphere, which
contradict the definition of 'sphere' and 'surface'.  (B) Similarly, a
perfect cylinder lying on a flat surface, must touch the surface by a
line.  (C) An argument that depends on the idea of 'uniformly varying'
properties, which originated with Oresmus and was taught by the Merton
school (I think).  Suarez argues that light grows less bright
continuously in proportion to its distance from the sun.  There must
therefore be a surface of the same determinate degree of brightness,
such as where the light meets the sea.  This surface must actually exist
even if the sea exists were replaced by air, because that degree of
brightness actually exists.  (D) A similar argument about a fire
continuously burning through flax, forming a surface that moves as it
burns.  (E) This view is not contradictory to Aristotle or Aquinas,
because, when they say that these indivisibilia are in the continuum
'potentially', this is meant to exclude real division, but not real
existence.

Finally, Suarez replies to the original 10 arguments one by one.  Here,
there are some interesting parts where he admits difficulties.  For
example the continuum is 'full up', as it were, yet how can it be that
we can draw the first point on the line, but not the second, and we can
draw the last, but not the next to last.  Thus 'hanc infinitatem longe
diversae rationis ab infinitate quantitatis discretae' the continuum is
hugely different from a multitude of discrete quantitity.  The infinity
of the continuum is actual only 'secundum quid' - actual in a 'qualified
way' (about which qualification S. is vague).    He also mentions the
case where a surface consists of different colours (black paint on
white, say) and where the two colours form a boundary.  What colour are
the points on the boundary?  They can't be both colours at the same
time, if the colours are contrary.  But if they are one colour, why one
colour rather than the other?  S. argument at this point is very hard to
follow - he distinguishes between 'extrinsic' and 'intrinsic'
termination, for example.