NU (XIV) METAPHYSICS
[Nu is devoted to some ad hoc arguments: (1) about contraries, in
chapters i-ii, (2) more rebuttals of the various theories of number
in chapter iii, (3) about the Good and the Beautiful, in chapter
iv, and (4) about numbers again in chapters v and vi. In several
places discussions of number and of contraries are mixed, as in
Iota (although Iota seems the clearer). The topical transitions do
not always match chapter limits. All this leaves an impression of
a rather confused end of our text, a veritable rag-bag of tag ends.
(For those interested, there are questions raised by scholars about
the relation of Mu and Nu: see Jaeger, Entstehungsgeschichte, pp.
41 ff., Aristotle, 2nd ed., p. 184, note 2, and Aristotelis
Metaphysica (Oxford Classics ed.), p. 292; also Ross, II, pp. 461-
462, 470. But the text may speak for itself well enough.)
[Contraries had been an important topic of traditional Greek
speculation, as may be seen in the Presocratics. Aristotle makes
this clear in his first sentence. Chapters i and ii are tightly
knit. Chapter i deals mainly with the contraries, the one and the
many (or equivalents of the many). Chapter ii deals with the
contraries, being and not-being. Both pairs were traditional
primary pairs of contraries, very closely associated]
Chapter i
Contraries, the one and the many, etc.
1. so much for that ousia [supra-sensible, para tas aisthEtas,
1086a25-1087a25], but everyone makes the principles contrary, as
[shown] in Physics [A, v; line 188a19 is nearly identical] and
likewise in the Metaphysics [Iota, iii ff.], peri men oun tEs
ousias tautEs eirEsthO tosauta, pantes de poiousi tas archas
enantias, hOsper en tois phusikois, kai peri tas akinEtous
ousias homoiOs, 1087a29-a31
2. if there cannot be something prior to the principle of all [the
first principle], another principle [as subject] cannot be that
principle, as if someone were to call white qua white a
principle, yet it is white as a predicate of something else.
That [something] would be prior. But all things generated from
contraries are like from a subject. This applies most of all to
contraries. All contraries are predicates, and none is separate,
whereas nothing seems contrary to ousia, as reason testifies
[the old problem: there is no contrary to being. The contrary of
being is not! We will shortly meet this head on in the next
chapter]. So none of the contraries is the leading principle of
all, but something else [is], a31-b4
3. some make the other of the contraries to be matter, hoi de to
heteron tOn enantiOn hulEn poiousin, b4-b5 [he means the right-
hand column of a sustoiecheia, and not the Pythagorean
sustoicheia that Aristotle gives in A, v, 986a, but a set like
Plato uses in the Parmenides. This included (among others pairs)
being not-being
one many
equal unequal
limit unlimited]
a. some [Plato] make the unequal contrary to the one, hoi men
tOi heni to anison, as this is the nature of the many, hOs
touto tEn tou plEthous ousan phusin, b5-b6
b. others [Speusippus] make the many contrary to the one, b6
c. (because, according to the former, numbers are generated from
the unequal Dyad, the Great and the Small; according to the
latter, from the many; following the the ousia of the one in
both cases, gennOntai gar hoi arithmoi tois men ek tEs tou
anisou duados, tou megalou kai mikrou, tOi de ek tou
plEthous, hupo tEs tou henos de ousas amphoin), b7-b9
d. and because, saying that the elements are the unequal and the
one, but the unequal is from the Great and the Small Dyad, he
[Plato] speaks as though the unequal and the Great and the
Small were one [i.e. the same], and he doesn't distinguish
them in definition, or in number, b9-b12
e. moreover the principles which they call elements they don't
explain well, alla mEn kai tas archas has stoicheia kalousin
ou kalOs apodidoasin, b12-b13
f. [Plato] naming the Great and the Small, following the one,
makes these elements of numbers three, two matter [the Great
and the Small] and one [the One] the Form, b13-b16
g. others speak of the many and the few, to polu kai oligon,
because the Great and the Small are more appropriate in
nature to magnitude, b16-b17
h. others use the more general terms, excess and defect, to
huperechon kai to huperechomenon, b17-b18
i. none of these differ with respect to the consequences, but
only with respect to their verbal difficulties, which they
defend on account of the difficult language. Except that
according to this argument [that] excess and defect are
principles, but not the Great and the Small, number should be
prior to the two, because both [i.e. excess and defect, and
number] are more general. But some say so; some do not, b18-
b26
4. some [raise] the antithesis, the other and the one [Plato, in
the dialogue, the Parmenides]; others, the many and the one
[much the same]. If beings are from contraries, as they would
have it, the many is contrary to the one, the unequal to the
equal, the other to the same; most of all those who oppose the
one to the many hold such opinions, but not adequately. They
overlook the few as a contrary to the many, b26-b33
5. [Aristotle's objections to all this]
a. the one is a measure, and in all things something else is
[its] subject. Examples. And the measure is indivisible, in
form and to the sense, so there is no absolute one as ousia.
The one is a measure of the many as a measured number (thus
reasonably one is not a number, because measures are not a
measure, but the one is a measure and the origin [cf. Iota,
chapters i and ii], b33-1088a8
b. the same thing should always be the measure of any set
(pasi). Examples. a8-a14
6. [objections to Plato's doctrine]
a. those who make the unequal one, the Infinite Dyad out of
great and small, are very far off the current opinion and the
possible, hoi de to anison hOs hen ti, tEn duada de aoriston
poiountes megalou kai mikrou, porrO lian tOn dokountOn kai
dunatOn legousin. These are properties and attributes rather
than subjects of numbers and magnitudes, the many and the few
of number, the great and the small of magnitude, like even
and odd, smooth and rough, straight and curved, a15-a21
b. and in addition to that error the great and small and such
must be relatives. Relations are the least of all some nature
and ousia. They are posterior to quality and quantity, an
attribute of quantitiy, as said, but not matter, a21-a25
c. since the relative is something different in general and in
its parts and kinds. There is nothing, neither great or
small, many or few, or generally relative, which is not
something else [than what it is relative to]. A sign that it
is not ousia or some being [in its own right] is that there
is no coming to be or ceasing to be, no change in quantity or
quality or place, etc., of it alone. Without changing it can
become more or less or equal, when the other changes. The
matter of each thing or ousia must be its potentiality, but
the relative is neither potentially or actually ousia. It is
absurd, impossible to make what is not ousia an element of,
and prior to ousia. All the [other] categories are posterior
[to ousia], a25-b4
d. furthermore, elements are not predicates. The many and few,
etc., are separate from and predicated of number. If there is
some quantity which is always few, like the two (if it were
many, the one would be few), there would be a simple [i.e.
absolute, permanent] many. The ten would be many if there is
not more [to compare it to], or ten thousand. How do you get
number from the few and the many this way? Either both or
neither needs be predicable [of it], but now [they would have
it] only one or the other is predicated, b4-b13
EFL, 10/11/97
NU (XIV) METAPHYSICS
Chapter ii
Contraries, cont.: being and not-being
[one and many, being and not-being, we saw, were closely related
pairs in the minds of ancient thinkers. Plato's Parmenides, for
example, is about both pairs, along with other pairs too]
1. is it possible that eternals [he means numbers] are composed of
elements? Elements are matter and potentialities. These can not
-be. What can not-be is not eternal. No eternal is composed of
elements [this is directed at Plato's numbers], 1088b14-b28
2. there are some [Xenocrates?] who make the element beyond the one
the Indefinite Dyad, rejecting the unequal [Plato's - Chapter i,
1087b5-b6] with good reason on account of the impossibilities
that occur. They are relieved of such difficulties as follow
upon making the unequal and the relative an element, but not of
the other difficulties attendant upon making Ideal or
mathematical numbers out of those out of [the Indefinite Dyad,
the Great and the Small, etc.], b28-b35
3. many are the reasons for the digression about these causes, but
most of all the old problem: it seemed to them that all beings
will be one, absolute being, if someone will not resolve and
meet the saying of Parmenides [DK VII, line 1], "It shall never
be proved, that not-being is." We must show that not-being
exists. Like when it means being something else [as Plato
pointed out in the Sophist, 241D, 253-259], polla men oun ta
aitia tEs epi tautas tas aitias ektropEs, malista de to aporEsai
archaikOs. edoxe gar autois pant' esesthai hen ta onta, auto to
on, ei mE tis lusei kai homose badieitai tOi Parmenidou logOi
"ou gar mEpote touto damEi, einai mE eonta," all anagkE einai to
mE on deixai hoti estin. houtO gar, ek tou ontos kai allou
tinos, ta onta esesthai, ei polla estin, 1088b35-1089a6
4. first of all, if there are many kinds of being (ousia, quality,
quantity and the rest of the categories), what kind of unity are
all beings, if not-being will not be? Will it be the ousiai or
the attributes and other such, or all [of them]? And will this
and thus and so much be one, and whatever else mean one thing?
But that is absurd. Moreover it is impossible that one nature is
the cause of the what, the what kind, the how much and the
where, kaitoi prOton men, ei to on pollachOs (to men gar ousian
semainei, to d' hoti poion, to d' hoti poson, kai tas allas dE
katEgorias) poion oun ta onta panta hen, ei mE to mE on estai;
poteron hai ousiai, E ta pathE kai ta alla dE homoiOs, e panta,
kai estai hen to tode kai to toionde kai to tosonde kai ta alla
hosa hen ti semainei; all' atopon, mallon de adunaton, to mian
phusin tina genomenEn aitian einai tou tou ontos to men tode
einai to de toionde to de tosonde to de pou, a7-a15
5. again, from what sort of not-being and being [come] beings? Not-
being is many things, as much as being. Not being this means not
being a man; not being such, not being straight; not being so
much, not being three feet long. From what kind of being and
not-being are the many beings? epeita ek poiou mE ontos kai
ontos ta onta; pollachOs gar to mE on, epeidE kai to on, kai to
men mE anthrOpon <einai> sEmainei to mE einai todi, to de mE
euthu to mE einai toionde, to de mE tripEchu to mE einai
tosondi. Ek poiou oun ontos kai mE ontos polla ta onta; a15-a19
6. he [Plato or some pupil] means not-being is the false and such,
from which and being come many beings, so he said one must make
a false assumption, like surveyors do in their scale drawings,
to podiaian einai tEn mE podiaian, but this is impossible to
sustain because the premiss is not part of their argument, and
no being comes from not-being or is done away with in this
manner [Sophist, 237A, 240D, but it seems like some other source
and pupil may possibly be referred to], a20-a26
7. but since not-being is meant in as many ways as there are
categories, in the same ways not-being and the potential are
called false, from which there is coming into being. From the
not-man potentially, a man; and from the not-white potentially,
white; and in the same way one whether one or many [Aristotle's
notion of potentiality is his means of solving the old problem
of not-being and being], a26-a31
8. the inquiry, how are things many, seems to be respecting ousia,
because what are produced are numbers and lines and bodies. But
it is absurd to ask how substantial being is many, but not how
qualities or quantities, phainetai de hE zEtEsis pOs polla to on
to kata tas ousias legomenon, arithmoi gar kai mEkE kai somata
ta gennOmena estin. atopon dE to hopOs men polla to on to ti
esti zEtEsai, pOs de E poia E posa, mE, a31-a35
9. the Infinite Dyad and the Great and the Small are not the cause
that there are two white things, or many colors or tastes or
shapes, because these would be numbers and units. If they looked
at these [things, colors, etc.], they would have seen the cause
in them. These, and the like, are cause, ou gar dE hE duas hE
aoristos aitia oude to mega kai to mikron duo leuka E polla
einai chrOmata E chumous E schEmata, arithmoi gar an kai tauta
Esan kai monades. alla mEn ei ge taut' epElthon, eidon an to
aition kai to en ekeinois, to gar auto kai to analogon aition,
a35-b4
10. this error is the reason for [Plato et al.] seeking the
opposite of being and the one (from which with these are beings)
to establish the relative and unequal. This not the contrary or
the contradiction [of being and one], but a single nature like
a thing or a quality. They need to look into this: how the
relative is many, not one. They ask why there are many units
beside the first one, but not why many unequals beside the
unequal], b4-b10
11. they use and speak of large and small, many and few, from which
the numbers; long and short, wide and narrow, high and low, from
which lines, planes and solids; and the many forms of relative:
what is the cause of the many in these? b11-b15
12. it is necessary, as we say, to suppose the potential in each
[of these] (this [Plato] showed, that the particular is not the
absolute being, and that it is relative [to dunamei of course
was Aristotle's invention. What he means here is that Plato more
or less foresaw or anticipated it with his notion of relative,
pros ti], like he spoke of quality, which cannot be either the
one or being nor a denial of them, but is some one particular
being), and to look for the many in all categories, as stated
[a34], b15-b24
13. in the other categories there is another difficulty, how they
are many (since they are not separate, the substrate becomes
many qualities and quantities, and yet there must be some
inseparable matter in each), b24-b28
14. but in the case of particular things there is some reason they
are many, if they are not both a thing and some [Ideal] nature.
The problem really is: how are there actually many [things] and
not one? But if a particular and quantity are not the same, it
is not said how and why beings are many, but how their quantity
is many. All number means some quantity; and the unit, if not a
measure, the quantitatively indivisible. If quantity and the
thing are different, we are not told what the thing is from, or
how there are many. If they are the same, the speaker suffers
contradictions, b28-1090a2
[The text here turns back to a discussion of number, which
continues in the next chapter, and will be taken up with it.
[The reference to Parmenides is important, indicative of the hold
he had on the imagination of those thinkers of that time. But they
only half-way saw what Parmenides was up to (probably Parmenides
himself only half-way understood what he was doing). They had no
clear idea of the mental nature of Parmenides' One. Indeed, Plato's
effort at clarification in The Parmenides, has only succeeded in
mystifying readers to this day. Henri Bergson was right: "The human
intellect feels at home among inanimate objects . . . our concepts
have been formed on the model of solids, L'intelligence humaine se
sent chez elle tant qu'on la laisse parmi les objets inertes . . .
nos concepts ont ete formees a l'image des solides," L'Evolution
creatrice, i]
EFL, 10/18/97
NU (XIV) METAPHYSICS
Chapter iii, Whence numbers?
[This is really wonderful, this beating around the bush to figure
out what numbers are. To appreciate it, however, we have to make
the imaginative leap of putting ourselves back into the state of
mind of those early investigators. They were starting from scratch.
We have to admire their persistence, and Aristotle's penetrating
common sense]
1. whence our belief that numbers exist? To the proponent of Ideas
[Plato] they provide a cause of things, if each of the numbers
is some Idea, and the Idea is the cause of being in others
somehow. So be it. But one who doesn't agree [Speusippus]
because he sees difficulties with Ideas, and they do not make
numbers, makes mathematical numbers [only]. Why must we believe
there are such, and what use? For they are the cause of nothing,
but they are by themselves and of their own nature. All these
theories of arithmeticals are established in connection with
sensibles, as has been said , 1090a2-a15
2. those [Plato et al.] who say there are Ideas, and these are
numbers, and each Idea is a unity over many things [e.g. hen epi
pollOn], try to tell why they exist somehow, but since this is
neither necessary or possible, number is not to be explained in
such a way, 1090a16-a20. The Pythagoreans, seeing many aspects
of number inhere in sensible bodies, make beings numbers that
are not separate. Beings are made of them. Why? Because there
are attributes of number in music, in the heavens and in many
other things, a20-a25. Those who say there are mathematicals
only [Speusippus] can say no such thing according to their
assumptions, but it was said there will be no sciences of
[sensibles], a25-a28
3. we say there are mathematicals, as we said previously [M, ii,
iii], and they are not separate, because the attributes of
separate things cannot exist in bodies, a28-a30
4. the Pythagoreans are not to be blamed for such [notions] making
physical bodies out of numbers, weighty out of weightless. They
seem to be talking about some other world and other bodies, and
not about the sensible, a30-a35.
5. Those who make [mathematicals] separate [Platonists again] do
so, because there are no axioms about sensibles, but axioms are
true and gladden the mind. They assume that they exist and are
separate, like mathematical magnitudes. Clearly contrary
reasoning will proclaim the contrary, and one must solve the
problem just raised, why the attributes of things that are not
in sensibles, are themselves in sensibles, a35-1090b5
6. there are some [Pythagoreans] who think that because points,
lines and planes are limits and ends of lines, planes and
solids, they must be substances. Let us look at this argument,
if it is not very weak. These ends are not ousiai but rather
limits. That the limit of motion is some thing or ousia, is
absurd. Furthermore, if they are, they will all be of such
sensibles (the argument was about those [lines, planes and
solids]). Why will they be separate? b5-b13
7. and one may insist, if we are not indulgent about all number
and mathematicals, that they contribute nothing to each other,
prior to posterior. (If there is not number, nevertheless there
will be magnitudes for those who say there are only
mathematicals [Speusippus; see Lambda, x, 1075b37-1076a1; Zeta,
ii, 1028b21-b24], and if these do not exist, [there will be] the
mind and the sensible bodies. Nature is not an incoherent jumble
of phenomena, like a bad play), b13-b20
8. this does not apply to those who assume Ideas. They make
magnitude out of matter and number [Xenocrates]; the line, of
two; the plane, of three; the solid, of four or other numbers.
It makes no difference, but will these be Ideas, or what is
their character? and what do they contribute to beings? Nothing.
Neither the mathematicals nor the Ideas. Moreover, no theory of
theirs stands up, if one doesn't try to fiddle with the
mathematicals and make up one's own notions. It is not difficult
to take any hypothesis and spin it out at length and on and on.
In their desire for mathematicals in addition to Ideas these
people fail utterly, b20-b32
9. the first to create two kinds of numbers, eidetic and
mathematical, didn't say and can't say how or from what the
mathematicals [are created]. They put them between the Ideal and
the sensible. But if [made] out of the Great and the Small, it
will be same as the former, the Ideas (does he [Plato] make
magnitudes from any other Great and Small?). And if he mentions
something else, he raises more elements, b32-1091a2
10. if one thing is the principle of each, it will be common to
them [both], and it must be asked how the one is many, and [how]
at the same time number cannot be generated other than from the
one and Indefinite Dyad. All these things are absurd and
conflict with each other and with good sense, and the tall story
of Simonides resembles them, or the yarn slaves will spin when
they have nothing worthwhile to say. The Great and the Small
seem to protest being dragged in. In no way can number be
generated except by repeating the one, a2-a12
11. it is absurd to create eternal beings, and what is more it is
impossible. As for the Pythagoreans no one should be in doubt
whether they do not or do create numbers, because they clearly
state how when the one is put together out of planes or surfaces
or seeds or whatever, straightway the nearest of the unlimited
is drawn into and penetrated by limit. But because they are
constructing a world-system, and want to talk physically, they
appropriately examine the physical side, and should be excused
from our present investigation, because we are looking for the
principles of unchanging things, and so for the genesis of
numbers of this sort, a12-a22
12. [first seven lines of chapter iv conclude the matters of iii]
they [Plato et al.] say there is no generation of odd number, as
clearly there is of even. Some construct the first even of the
unequal, the Great and the Small equalized [see M, vii, 1081a17-
a29]. The unequal must be prior to the equalizing, but if they
were always equals, the unequals were not prior. So it is clear
that they do not bring up the generation of numbers for the sake
of their theory, hOste phaneron hoti ou tou theOrEsai heneken
poiousi tEn genEsin tOn arithmOn, a23-a29 [Indeed, in Plato's
account of number, the one and the Infinite Dyad, and the Great
and the Small, he sounds more like a poet than a philosopher]
EFL, 10/18/97
NU (XIV) METAPHYSICS
Chapter iv
The Good and the Beautiful
1. there is a problem, and a warning to whoever disregards it: what
is the relation of the elements and principles to the good and
the beautiful, echei d' aporian kai euporEsanti epitimEsin pOs
echei pros to agathon kai to kalon ta stoicheia kai hai archai.
The problem is: is the Good one of these [elements, principles],
as we would like to say, or not, [but] later in origin? aporian
men tautEn, poteron esti ti ekeinOn hoion boulometha legein auto
to agathon kai to ariston, E ou, all' husterogenE, 1091a29-a33
2. the theologoi seem to have agreed with some moderns in saying
"No", but the good and the beautiful became manifest after the
nature of things had evolved. (They do this, wary of a true
difficulty that befalls those who say the one is a principle.
The difficulty is not in conceding goodness to the principle as
a quality in it, but making one a principle and element, and
[deriving] number from the one, touto de poiousin eulaboumenoi
alEthinEn duschereian hE sumbainei tois legousin, hOsper enioi,
to hen archEn. esti d' hE duschereia ou dia to tEi archEi to eu
apodidonai hOs huparchon, alla dia to to hen archEn kai archEn
hOs stoicheion kai ton arithmon ek tou henos), a33-b3
3. the old poets [Hesiod et al.] likewise said the first things
like Night and Heaven and Chaos do not rule, but Zeus. But they
said such things on account of the changing ruling principles of
things, while contemporaries who did not explain everything
mythically, like Phercydes and some others, regarded the first
production best, and the Magi and the last of the wise men like
Empedocles and Anaxagoras, who made Love an element and Mind a
principle. Of those who spoke of changeless ousiai, some said
that the Good is the One [Plato]. They thought the One is its
ousia, b3-b15
4. so the question is, which is right? It would be surprising, if
in the first and eternal and most self-sufficient [archE] the
first were not good. It is not due to some other [principle]
that it is eternal and good and self-sufficient. So it would
seem truly reasonable to say so. But that the one is this [Good]
or an element and an element of number, is impossible, to mentoi
tautEn einai to hen, E ei mE touto, stoicheion ge kai
stioicheion arithmOn, adunaton. To avoid such difficulties, some
[Speusippus], agreeing that the one is a first principle and
element, have said: of mathematical number [only], b15-b25
5. All units become good, and there is a great abundance of them.
And if Ideas are numbers, they become good. Moreover let
anything one pleases be Ideas. If only of goods, they will not
be ousiai. If of ousiai, every animal and vegetable will be
good. This is absurd, and the opposite element [in the double
list of sustoicheia], the many, the unequal, and the Great and
the Small, will be bad which is why [Speusippus] declined to
join the good with the one as a necessary being, since genesis
is from opposites, and [declined to say] that the nature of many
is bad. Others [Plato, Xenocrates] called the unequal bad, b25-
b35
6. all beings partake of the bad except one, the absolute One, and
numbers more so than magnitudes. The bad is the place for the
good [to go to work], participating in and striving for its
destruction, because contrary is destructive of contrary. And as
we said that matter is the potentiality to be each thing, like
potential fire of actual fire, the bad will be potentially good,
b35-1092a5
7. [recap] there are all these [theories]: some [like the
Physicists] make elements all the principles; others [like
Empedocles, et al.], contraries; others [Plato, Xenocrates, et
al.], the one; others [Speusippus], numbers, separate and
eidetic, tauta dE panta sumbainei, to men hoti archEn pasan
stoicheion poiousi, to d' hoti tanantia archas, to d' hoti to
hen archEn, to d' hoti tous arithmous tas prOtas ousias kai
chOrista kai eidE, a5-a8
[What have mathematics and number to do with the good and the
beautiful? Aristoxenus, a pupil of Aristotle, in his Elementa
Harmonica (ed. Meursius, 1616 and Macran, Oxford, 1902), tells us
of a report by Aristotle of a lecture that Plato gave in the
Academy "On the Good" (this is accessible in Konrad Gaiser's
Platons Ungeschriebene Lehre, Stuttgart, 1963, page 452, Nr. 7,
whence I cite). "Everyone attended expecting to hear something
about human goods, like wealth, health, power and prosperity. But
when the talk turned out to be about mathematics and numbers and
geometry and astronomy, and the limit, and that one is the good, I
think it appeared quite the contrary to what they expected, and
some despised it, while others belittled it." The association of
the One and the Good was as long as the Platonic tradition, as
exemplified eight hundred years later by the Neoplatonist, Proclus,
writing, "Goodness, then is unification, and unification goodness;
the Good is one, and the One is primal Good," estin ara kai hE
agathotEs henOsis, kai hE henOsis agathotEs, kai to agathon hen,
kai to hen prOtOs agathon, Elements of Theology, ed. Dodds, Oxford,
1963, Prop. 13, p. 16-17. It was of course a religious association,
and Aristotle was not one to have part in that]
EFL, 11/1/97
NU (XIV) METAPHYSICS
Chapter v
Numbers
1. if not putting and putting the good among the first principles
impossible, it is clear that they have not been accounted for
correctly, nor [have the] first things, ei oun kai to mE
tithenai to agathon en tais archais kai to tithenai houtOs
adunaton, dElon hoti hai archai ouk orthOs apodidontai oude hai
prOtai ousiai, 1092a9-a11
2. [against Speusippus]
a. one [Speusippus] is mistaken if he compares the principles of
the whole to that [principle] of animals and plants, ouk
orthOs d' hupolambanei oud' ei tis pareikazei tas tou holou
archas tEi tOn zOiOn kai phutOn, because as the more finished
comes from the indefinite and unfinished, so he says it is
with first things: the one itself is not some being. [But he
is mistaken] because here on earth the first principles are
complete: man produces man. The sperm is not first, a11-a17
[See Lambda, vii, 1072b30]
b. [furthermore] it is absurd to confuse place with mathematical
solids (space belongs to particular things which may be
segregated, but mathematicals are nowhere), and to say that
it is somewhere, but not explain what it is, a17-a21
c. and those who say that beings are from elements, and numbers
are the first beings [Speusippus, see Nu, iv, 1091b20-b26]
should distinguish how they come one from the other, and
tell us in what way number is from the principles. (1) By
mixture? But everything is not mixed, and what becomes mixed
is different, and the one will neither be separate nor
another nature, as they wish. (2) By combination, like a
syllable? But that involves position, and one thinks of the
one and the many as separate. Number will be unity and
multiplicity, or the one and the unequal. Is a number
composed of indwelling constituents, or not? Only things that
come into being are composed of constituents. (3) From seed?
It can't come from the indivisible. (4) From a contrary that
does not remain? But there is something else [a substrate]
that remains. Since one makes one contrary to the many;
another, to the unequal, treating the one as equal, so number
would be from contraries. Why then do other products or
partakers of contraries perish, but not number? They don't
say, a21-b8
3. [against the Pythagoreans]
a. they don't say whether numbers are the causes of beings [1]
as limits, horoi (like the points around a magnitude, as
Eurytus assigned a number to man or horse by putting pebbles
around the figure, like they used to do with a triangle or a
rectangle) or [2] because their harmony is a numerical
proportion? How are numbers attributes? It is clear that
numbers are not ousia, or the causes of form, because ousia
is the ratio, and number is matter, hoti de ouch hoi arithmoi
ousia oude tEs morphEs aitioi, dElon, ho gar logos hE ousia,
ho d' arithmos hulE, b8-b18
b. number is the ousia of flesh and bone thus: that they are
three parts fire and two parts earth [Empedocles, DK96].
Number is always [number] of something elemental or
consisting of units, but ousia is the proportion of one such
to another, according to their mixture. This is not number
but a ratio of the mixture of the corporal or whatever parts,
b18-b23
c. number is none of the four causes of things, oute oun tOi
poiEsai aitios ho arithmos, oute holOs ho arithmos oute ho
monadikos, oute hulE oute logos kai eidos tOn pragmatOn. alla
mEn oude hOs to hou heneka, b23-b25
EFL, 11/8/97
NU (XIV) METAPHYSICS
Chapter vi
Pythagorean numbers, cont.
d. one might ask what the function of numbers is in mixture,
whether simple ratios or unusual [irrational?] ones. A three
three mixture of honey and water is no healthier, but an
unmeasured watered mixture would be better than a neat one
measured according to numbers. Mixtures are ratios of
numbers, not the numbers [themselves], like three to two, not
three twos. It is the same sort of thing in multiplications,
where there is a least common factor, 1092b26-1093a1
e. [shared numbers]
(1) if all [sorts of] things share numbers; many would be
the same. Examples. But the same numbers must change
[their meaning or function] when shared, a1-a9.
(2) different things can fall under the same number, and
would have the same form, a9-a13
(3) but why are [numbers] causes? Examples of various
applications of the number, seven, a13-a19
(4) examples of application of a number to concords and to
double consonants [for the technicalities of this
discussion, see Ross, II, 498], a20-a26
(5) applications of the numbers, seventeen and twenty four,
a26-b6
f. the praised natures of number and mathematicals, and their
contraries, that some people [Pythagoreans still] speak of
and make causes of nature, seem to vanish away upon
inspection (in none of the [four] ways described is any of
them a cause, kat' oudena gar tropon tOn diOrismenOn peri tas
archas ouden autOn aition), b7-b11
g. as they put it, it is clear there is good [in number], and in
the sustoicheia the odd, the straight, the square and the
powers of some numbers reside in the good [left-hand column
of contraries]. And the seasons, and number. So they look
like they belong because they are attributes. They all
conform to one another, like straight in a line, smooth in a
surface, odd in number, white in color, b11-b21
h. ideal numbers are not causes of harmonies and such (because
even equal they differ) so that doesn't make them Ideas.
Ideal numbers are unique, but there are many same
mathematical numbers. The distress about the origin of ideal
numbers and their inseparability from sensibles seems
evidence that [numbers] are not principles, b21-b29
[Like Mu, Nu devotes alot of attention to Ideas, number, and the
various theories thereof, theories of the Pythagoreans, Platonists,
Speusippus, Xenocrates, and perhaps of others. There is redundancy
here, yet the viewpoint seems slightly altered, as exemplified by
the discussion of opposites. There are many possible explanations
for this, but perhaps we had better just accept it at its face
value, crediting Aristotle's effort to cut through them all, to
attain some rational, not mystical, not mythical view of number and
Ideas, especially number here.
[Numbers, according to Aristotle, are not separate in the then
received sense, similar to the sense in which Plato's Ideas were
understood (ekei). They are in things here (entautha). But they are
not in things in the manner that the old Pythagorean tradition
seems to have held. Between this Scylla and Charybdis Aristotle is
looking for another way. He has not found it yet. At least he has
not found a way to express it clearly. Nor would it have been
received, if he had. The natures and roles of mental products; the
difference between conception and perception: all this was not
clearly understood. The role of the mind in relation to brain and
body is still difficult of understanding for us, so many centuries
later. Should we be surprised if it was difficult for them?]
OUTLINE OF NU
Chapter i, Contraries, the one and the many
ii, Contraries, being and not-being
iii, Whence numbers?
iv, The One and the Good
v, Numbers
vi, Pythagorean numbers