Notes on
Physics
Delta :
Number
There is
quite a lot of discussion in Physics Delta that
invokes the
notion of
number. Here are some
passages from elsewhere in Aristotle that give some hint as to
what he means by 'number'*.
Metaphysics Delta 13, 1020a8-14:
Quantity [poson, literally "how much" or "how
many"]
is said
to be that which is
divisible into constituents, each of which is by nature one
[or "a one"] and a "this"
[tode ti, a specific indicable thing]. A plurality
[plethos]
is a kind of quantity if it [the
quantity] is numerable [countable; arithmeton]; a
magnitude
is a kind if quantity if it
[the quantity] is measurable. A plurality is said to be that
which is divisible
potentially into parts which are not continuous**; a
magnitude, on the other hand, is
that which is potentially divisible into parts which are
continuous....Of these, a
limited [peperasmenon] plurality is said to be a
number, a
limited length a line, a
limited width a surface, and a limited depth, a
body.
[Note that this is all presented as definitional, or as a report
of what certain things are said to
be. That is, Aristotle is not saying anything about whether he
thinks that the world works in
the way we describe it as working; he is not saying whether he
thinks that these definitions
are adequate or accurate when it comes to accounting for what
exists and how it works. He
is not saying whether he thinks that these definitions may imply
contradictions or
incoherences, either in themselves or as they are used.]
Metaphysics Iota (I)6, 1057a2-6:
Plurality is as if it were [or "such as"; hoion]
a
genus
of***
number; for number is a
plurality measurable by the one. And in some sense [or "in a
way"] the one and
number [or "a number"] are opposed, not as contraries,
but...as some relative things
are; for the one in so far as it is a measure is opposed to
number in so far as number
is measurable.
[Note the "as if it were"/"such as" and the "in some sense"/"in a
way" (pos). This suggests
that Aristotle may think that the relationships are not exactly
as described; or that he thinks
that there might be problems with describing them this way: from
a certain angle they look
this way, and from another angle not so much so.
What is meant by saying that number and the one are in a way
opposed in the manner
of relative things is that they stand at opposite poles of a
relationship. A one is used as a
measure (in fact, a one is that which is used as a measure), and
a number is measurable by
ones.]
Metaphysics Nu (N)1, 1087b33-1088a15:
The one signifies a measure, evidently. And in each case
there is some different
underlying subject [hupokeimenon, thing laid down],
such as
in the musical scale a
quarter-tone; in magnitude a finger or a foot or some other
such thing; and in rhythm
a beat or a syllable....And this is also according to
formula [or "definition" or
"account": logos]; for the one signifies a measure of
some
plurality and the number
signifies a measured plurality [a plurality that has been
measured] and a plurality of
measures. Therefore it is also with good reason that the one
is not a number; for
neither is a measure measures, but a measure is a principle
[or "source", arche], and
so is the one.
*It can be dangerous to take what Aristotle says about
some thing or concept in one of his works, and
apply it to his references to that thing or concept
in another of his works. The reason problems may occur
is that the context in which a particular thing or concept
is referred to, or in which a particular term is
used, may vary from one work to another, or in some cases
from one discussion to another. However, in the
passages I will cite here, the contexts appear to
me to match or to work together. The Metaphysics passages
seem to be oriented toward the question of what a
given thing is said to be in general, or toward a presentation
of all of the possible meanings or senses of
something - some of which are specifically
identified there as applying in physics.
**This point will be developed in Physics Epsilon
(E).
***'A genus of' means roughly ‘a kind that includes'. Here, all
numbers would be pluralities (for the Greeks, as we
will see, 1 did not count as a number, or at least,
it did not count as a number in the way that 2,3,...
did), but not all pluralities would be numbers.
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