On Quantity - "Categories" Ch. 6 - Aristotle

Transitioning from Chapter V on Substances to Chapter VI on Quantities, here's an interesting philosophical point: If continuous quantities have to do with boundaries being connected, then really all of space and matter is one continuous quantity. But this doesn't make sense because we talk about the "many" all the time, not just one thing. How do we do this? Well, quantity (the many) very specifically is defined as a "part," meaning it is something only in relation to a greater "whole". Substances therefore deal with wholes and quantities (and all accidents) with parts. The substance is the boundary between continuous quantities that makes it a “1” thing as to its identity, and thus generates many parts (quantity) in relation to that 1 whole (substance). 

On Quantities Proper

4>20. Of quantities some are discrete, others continuous; and some are composed of parts which have position in relation to one another, others are not composed of parts which have position. 4°22. Discrete are number and language; continuous are lines, surfaces, bodies, and also, besides these, time and place. For the parts of a number have no common boundary at which they join together. For example, if five is a part of ten the two fives do not join together at any common boundary but are separate; nor do the three and the seven join together at any common boundary. 

There are two fundamental types of quantities, discreet and continuous. Aristotle defines continuous quantities as: "composed of parts which have position in relation to one another...". Aristotle defines discrete quantities as: "not composed of parts which have position in relation to one another." He then lists examples which fall into each category. Discrete include numbers and spoken language, and continuous include lines, surfaces, bodies, time, and place. 

Numbers are discrete because they are not physical things, and so they are mental quantities which do not need to be physically connected to one another to have meaning. 

Nor could you ever in the case of a number find a common boundary of its parts, but they are always separate. Hence number is one of the discrete quantities. Similarly, language also is one of the discrete quantities (that language is a quantity is evident, since it is measured by long and short syllables; I mean here language that is spoken). For its parts do not join together at any common boundary. For there is no common boundary at which the syllables join together, but each is separate in itself. A line, on the other hand, is a continuous quantity. For it is possible to find a common boundary at which its parts join together, a point. And for a surface, a line; for the parts of a plane join together at some common boundary. Similarly in the case of a body one could find a common boundary —a line or a surface—at which the parts of the body join together. Time also and place are of this kind. For present time joins on to both past time and future time. Place, again, is one of the continuous quantities. For the parts of a body occupy some place, and they join together at a common boundary. So the parts of the place occupied by the various parts of the body, themselves join together at the same boundary at which the parts of the body do. Thus place also is a continuous quantity, since its parts join together at one common boundary.  

Language is also discrete because Aristotle is talking about about spoken language which is made up of individual words in a sentence. These words have no physical boundaries with one another, as they are abstract, nor do their sounds since there are pauses between them, and they can also be reordered with the same meaning. 

5215. Further, some quantities are composed of parts which have position in relation to one another, others are not composed of parts which have position. For example, the parts ofa line have position in relation to one another; each of them is situated somewhere, and you could distinguish them and say where each is situated in the plane and which one of the other parts it joins on to. Similarly, the parts of a plane have some position; here again one could say where each is situated and which join on to one another. So, too, with the parts of a solid and the parts of a place. With a number, on the other hand, one could not observe that the parts have some position in relation to one another or are situated somewhere, nor see which of the parts join on to one another. Nor with the parts of a time either; for none of the parts of a time endures, and how could what is not enduring have any position? Rather might you say that they have a certain order in that one part of a time is before and another after. Similarly with a number also, in that one is counted before two and two before three; in this way they may have a certain order, but you would certainly not find position. And language similarly. For none of its parts endures, once it has been uttered it can no longer be recaptured; and so its parts cannot have position, seeing that none of them endures. Some quantities then are composed of parts which have position, others are not composed of parts which have position.  

Paryonymous Ways of Using Quantity

5238. Only these we have mentioned are called quantities strictly, all the others derivatively; for it is to these we look when we call the others quantities. For example, we speak of a large amount of white because the surface is large, and an action or a change is called long because the tame is Jong. For it is not in its own right that each of these others is called a quantity. For example, if one is to say how long an action is, one will determine this by the time, saying that it isa-year-long or something of that sort; and in saying how much white one will determine it by the surface—whatever the size of the surface one will say that the white too is that size. ‘Thus only those we mentioned are called quantities strictly and in their own right, while nothing else is so in its own right but, if at all, derivatively. 

5>r1. Next, a quantity has no contrary. In the case of definite quantities it is obvious that there is no contrary; there is, for example, no contrary to four-foot or fivefoot or to a surface or anything like that. But might someone say that many is contrary to few or large to small? None of these, however, is a quantity; they are relatives. For nothing is called large or small just in itself, but by reference to something else. For example, a mountain is called small yet a grain of millet large—because one is larger than other things of its kind while the other is smaller than other things of its kind. Thus the reference is to something else, since if a thing were called small or large in itself the mountain would never be called small yet the grain of millet large. Again, we say that there are many people in the village but few in Athens—though there are many times more here than there; and that there are many in the house but few in the theatre—though there are many more here than there. Further, ‘four-foot’, ‘five-foot’, and the like all signify a quantity, but ‘large’ or ‘small’ does not signify a quantity but rather a relative, since the large and the small are looked at in relation to something else. So it is clear that these are relatives.  

Relatives Are Not Quantities Properly Speaking 

5630. Moreover, whether one counts them as quantities or does not, they have no contrary. For how could there be any contrary to what cannot be grasped just in itself but only by reference to something else? Further, if large and small are to be contraries it will turn out that the same thing admits contraries at the same time, and that things are their own contraries. For the same thing turns out to be at the same time both large and small—since in relation to this thing it is small but in relation to another this same thing is large; so the same thing turns out to be both large and small at the same time and thus to admit contraries at the same time. But nothing seems to admit contraries at the same time. In the case of a substance, for example, while it seems to be able to receive contraries, yet it is certainly not at the same time ill and well nor is it at the same time pale and dark; nor does anything else admit contraries at the same time. It turns out also that things are their own contraries. For if large is contrary to small, and the same thing is at the same time large and small, a thing would be its own contrary. But it 1s impossible for a thing to be its own contrary. Large, therefore, is not contrary to small, nor many to few. So that even if someone says that these belong not to relatives but to quantity, it will still have no contrary. 

Forming the Principle of Non Contradiction (Substances Can Admit Relatives)

611. But it is most of all with regard to place that there seems to be contrariety of a quantity. For people regard up as contrary to down—meaning by ‘down’ the region towards the centre—because the centre is at the greatest distance from the limits of the world. And they probably derive from these their definition of the other contraries also; for they define as contraries those things in the same genus which are most distant from one another. 

Paronymous "Contraries of Quantity" - Space

641g. A quantity does not seem to admit of a more and a less. Four-foot for example: one thing is not more fourfoot than another. Or take number: we do not speak of a three as more three than a five, nor of one three as more three than another three. Nor yet is one time called more a ume than another. Nor is there a single one, among those we listed, as to which a more and a less is spoken of. Hence a quantity does not admit of a more and a less. 

Paronymous "Contraries of Quantity" - Equal and not equal

6226. Most distinctive of a quantity is its being called both equal and unequal. For each of the quantities we spoke of is called both equal and unequal. For example, a body is called both equal and unequal, and a number is called both equal and unequal, and so is a time; so also with the others we spoke of, each is called both equal and unequal. But anything else-—-whatever is not a quantity-—is certainly not, it would seem, called equal and unequal. For example, a condition is certainly not called equal and unequal, but, rather, similar; and white is certainly not equal and unequal, but similar. Thus most distinctive of a quantity would be its being called both equal and unequal.