I am working on 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 part 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, whose 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 questions which are difficult today.

The translation will take months or years. I summarise it below.

Section V of Disputation 40 is arranged as a classic scholastic disputation. It begins with 10 objections to his view. It is followed by a middle section outlining and arguing for his view. Finally, there are replies to the 10 objections. The objections themselves 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. God could remove all points from a 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.

(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' (uniformiter difformiter) 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 these authorities 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? Suarez' argument at this point is very hard to follow - he distinguishes between 'extrinsic' and 'intrinsic' termination, for example. I think this is the same as the modern mathematical distinction between an open and a closed interval, but cannot be sure.

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