Overview
M (XIII) AND N (XIV) METAPHYSICS
Overview
[Books M and N are two appendices, so to speak, ad hominem and ad
rem rebuttals of current doctrines about Ideas, mathematicals,
numbers. They bear on what has been said, but they tend to be more
polemical. They are supplemental to Aristotle's main presentations
in the earlier Books. Here is a summary outline:
Book Mu (XIII)
I. Introductory: The opinions of others about immovables,
mathematicals and Ideas, chapter i, 1076a8-a32
II. Mathematicals, chaps. ii, iii, a32-1078b9
III. Ideas, chaps. iv, v, b9-1080a11
IV. Numbers, Plato, chap. vi, vii, a12-1083a20
V. Speusippus, Xenocrates, Pythagoreans, chap. viii, a20-b23
VI. More arguments: to mega kai to mikron, limitless & limited,
the one and multitude, geometricals, chaps. viii, ix,
b23-1086a21.
VIII. Platonic Ideas, chaps. ix, x, a21-1087a25
Book Nu (XIV)
I. Opposite first Principles: to mega kai to mikron, to hen
kai ho aoristos duas, etc., chap. i, ii, a26-1089a31
II. Multiplicity, chap. ii, a31-1090a2
III. Numbers, chaps. ii, iii, a2-1091a29
IV. The Good and the Beautiful, chap. iv, a29-1092a8
V. the One: rebuttal of an argument of Speusippus', v, a9-a21
VI. Numbers: their generation and use, v, vi, a21-1093b29
[The context is somewhat different here in M and N than in Alpha.
His attention is mainly on his nearer contemporaries in and around
the Academy. When he treats here of Plato, it is often a different
Plato than the one we are familiar with. It is the Plato of the so-
called "Unwritten Doctrine," lectures in the Academy, especially a
notorious one "On the Good," about which we would like to know more
than we do. It is from such that the doctrines of to mega kai to
mikron (the Great and the Small) and hE aoristos duas (the
Indefinite Dyad) came. Speusippus was Plato's nephew, head of the
Academy after Plato's death. Xenocrates was another colleague who
followed Speusippus in that position. They all had different
theories, contradicting one another. Aristotle makes a point of
just this fact near the end of Mu (186a14-a16), as evidence that
their assumptions and principles are false. It is these theories
that are presented here. These two Books are more important for
capturing this scene than for any important substantive
contribution to metaphysics. None of these people had the genius of
Plato and Aristotle, or caught on to what those two were really
after. No surprise, since even Plato and Aristotle were not
entirely clear about it either. Yet Plato and Aristotle were
intellectually way ahead of their contemporaries like Speusippus
and Xenocrates and late Pythagoreans of their time]
Mu (XIII) Chapter i, Introductory
1. we have said what the ousia of sensibles things is, in the
investigation about matter in the Physics, and later about
actuality. [Our] investigation is whether or not there is
something beside sensible ousia, [something] motionless and
eternal, and if so, what it is, . . . peri men oun tEs tOn
aisthEtOn ousias erEtai tis estin, en men tEi methodOi tEi tOn
phusikOn peri tEs hulEs, husteron de peri tEs kat' energeian.
epei d' hE skepsis esti poteron esti tis para tas aisthEtas
ousias akinEtos kai aidios E ouk esti, kai ei esti tis esti, .
. . 1076a8-a12 [The first reference is pretty clearly to the
Physics, but the second ("later about actuality," husteron de
peri tEs kat' energeian) is perhaps to Zeta - Theta above]
2. the opinions of others, ta para tOn allOn legomena: there are
two notions, duo d' eisi doxai peri toutOn, a12-a17
a. that mathematicals are ousiai, a17-a19 [mathematicals for
Aristotle are numbers, points, lines, planes and solids as
such, (hoion arithmous kai grammas kai ta suggenE toutois,
1076a18), but here in the early chapters of Book Mu he has in
mind chiefly the latter, the points, lines, planes and
solids. Numbers he takes up later, in chapter vi. See also
Physics, B, 194a7-a11]
b. that Ideas [are ousiai], a19
c. [the lineup of opinions about these:] (1) some [Plato]
posited two kinds of ousia, Ideas and numbers, tas te ideas
kai tous mathematikous arithmous [numbers were some sort of
intermediate entity, metaxu], (2) some [Xenocrates], that
both these are one nature, mian phusin amphoterOn, (3) others
[Speusippus], that there are mathematical ousiai only, tas
mathEmatikas monon ousias einai phasi, a19-a22 [cf. Zeta, ii,
1028b18-b27, Ross, Intro., I, lxxi-lxxvi (4) unmentioned just
here are the Pythagoreans, probably late Pythagoreans. They
will receive notice in the ensuing discussion. It is this
division of opinion, and of course his own matter of fact
approach, that gives rise to the complications of M and N,
and more or less dictates their structure]
3. plan of inquiry to follow, a22-a32
a. mathematicals [chaps. ii-iii], a22-a26
b. Ideas [chaps. iv-v], a26-a29
c. are numbers and Ideas ousiai and archai?
[chaps. vi-x], a29-a32
[The last six lines, a32-a37, introduce the close inquiry into
mathematicals, and will be read with the next chapter]
EFL, 7/19/97
MU (XIII) METAPHYSICS
Chapter ii
Mathematicals: in sensibles or separate?
[By mathematicals, mathEmatika, he means number, points, lines,
planes and solids, but here he treats mostly of the geometricals.
Numbers are treated separately later. I usually translate stigmE as
"point", and monas as "unit." The difference is made plain by
Aristotle in Delta, vi, 1016b24-b31: points and units are one and
indivisible, points (stigmai) having position, units (monades)
having no position]
1. if there are mathematicals, they must be in sensibles, some say,
[the Pythagorean tradition there], or separate from sensibles
[the Platonic tradition], or if neither, either they do not
exist, or they exist in some other fashion, so our question will
be not about [their] existence but about the mode, anagkE d',
eiper esti ta mathEmatika, E en tois aisthEtois einai hathaper
legousi tines, E kechOrismena tOn aisthEtOn, E ei mEdeterOs, E
ouk eisin E allon tropon eisin, 1076a32-a37 [See in this another
capsule example of Aristotle's, indeed the Greek penchant for
dialectical statement]
2. they cannot be in sensibles, a38-a39
a. as stated in our problems [B, ii, 998a9-a11, a13-a14], two
solids cannot exist in the same place, a39-b1
b. other potentialities and natures would co-exist in sensibles
too, not separately, kai hoti tou autou logou tas allas
dunameis kai phuseis en tois aisthEtois einai kai mEdemian
kekOrismenEn, b1-b3
c. no body could be divided, if a point cannot be divided [as is
indeed the case], ei tEn stigmEn dielein adunaton, because it
would be divided at planes, these at lines, and these at
points, b3-b8
d. what difference, whether such natures [mathematicals] are
sensibles, or are not but are present in them [somehow], ti
oun diapherei E tautas einai toiautas phuseis, E autas men
mE, einai d' en autais toiautas phuseis; The result is the
same, because when sensibles are divided, they are divided,
or they are not sensibles, to auto gar sumbEsetai,
diairoumenOn gar tOn aisthEtOn diairethEsontai, E oude hai
aisthEtai, b8-b11
3. and such natures cannot be separate, alla mEn oude kechOrismenas
g' einai phuseis toiautas dunaton, b11-b12
a. if there are other solids separate and prior to sensibles,
there will be other planes, lines and points separate from
solids, and lines and points separate from planes, and points
separate from lines [somewhat confused by Aristotle; see
Ross' note, II, 412-13], an absurd accumulation, b12-b36
b. the same reasoning for numbers: beside points there will be
other units, and beside individual sensibles, mental, so
there will be a class of mathematical numbers, ho d' autos
logos kai peri tOn arithmOn. par hekastas gar tas stigmas
heterai esontai monades, kai par' hekasta <ta> onta ta
aisthEta, eita ta noEta, hOst' estai genE tOn mathEmatikOn
arithmOn, b36-b39
c. the questions we raised in our problems [B,ii, 997b12-998a6],
how are we to solve them? For subjects of astronomy and
geometry, sounds and sights, and living beings, there will be
entities beside the sensible and individual, estai para ta
aisthEta kai ta kath' hekasta, b39-1077a9
d. and there will be another ousia intermediate between Ideas
and intermediate mathematicals [B, ii, 998a6-a13]. If these
are impossible, those are impossible, a9-a14
e. in general this is the opposite of the truth and of what we
are accustomed to. Because separate mathematicals would have
to be prior to sensible magnitudes, but in truth they follow.
The incomplete magnitude is prior in becoming, but posterior
in being, like the inanimate to the animate, a14-a20
(f. and how and when will mathematical magnitudes be unified?
Things in this world are unified by the mind or part of it or
by something else, but what unifies these divisible
quantities? a20-a24) [Ross call this, rightly it seems, a
digression. It interrupts the flow of thought from e. to g.]
g. lines, planes and the third dimension (to bathos) come
into being in that order. If what is posterior in becoming is
prior in being, body is prior and more complete and whole
than line and plane. How would lines and planes be animate?
The priority is with our perceptions, a24-a31
b. and ousia is body, eti to men soma ousia tis. How could
ousiai be lines? hai de grammai pOs ousiai; neither as form
or matter. They are logically prior, but not everything
logically prior is prior in ousia, tOi men oun logOi estO
protera, all ou panta hosa tOi logOi protera kai tEi ousiai
protera, a31-b4
c. properties [like points, lines, planes] cannot exist
separately. They may be prior logically but not in ousia, ei
gar mE esti ta pathE para tas ousias, hoion kinoumenon ti E
leukon, tou leukou anthrOpou to leukon proteron kata ton
logon all' ou kata tEn ousian. ou gar endechetai einai
kechOrismenon all' aei hama to sunolOi estin (sunolon de legO
ton anthrOpon ton leukon), . . . Hoti men oun oute ousiai
mallon tOn somatOn eisin oute protera tOi einai tOn aisthEtOn
alla tOi logOi monon, oute kechOrismena pou einai dunaton,
eirEtai hikanOs. epei d' oud' en tois aisthEtois enedecheto
auta einai, phaneron hoti E holOs ouk estin E tropon tina
esti kai dia touto ouch haplOs estin, pollachOs gar to einai
legomen, b4-b17
[So geometricals (points, lines, planes, and solids) cannot be IN
sensibles, nor can they exist separately, as the main heads here
and the arguments under them show. But what other mode (tis allos
tropos) is there? What does it mean to be "in" sensibles? Does it
mean to have a substantial or material existence within another
sensible object, like a heart within a body, a room within a house,
or such? a "doubling up" of ousia? Clearly points, lines and planes
are not in a sensible object in this sense. But there is some sense
in which they are. What is it?
[Equally clearly, they are not separate as separate entities, a
sort of existential reduplication of the object in which they
inhere. Yet in some sense they are separate. What is that?
[If perchance I have mistaken some detail here or there, or if the
text were corrupt, it would be of little moment. The larger
structure of the chapter (as is true of the whole) is more certain,
and counts for more in the final reckoning]
EFL, 7/26/97
MU (XIII) METAPHYSICS
Chapter iii
Mathematicals: what kind of existence?
[This chapter continues chapter ii, asking, what "allos tropos,"
what other mode of existence mathematicals might have]
1. [protasis] just as generalities in mathematics are not about
separate [things] other than their magnitude and numbers, but
about these [magnitude and number] and not what has bulk or is
divisible, [and] it is clear that there can be arguments and
proofs about sensible magnitudes, not qua sensible but qua such
[abstractions], hOsper gar kai ta katholou en tois mathEmasin ou
peri kechOrismenOn esti para ta megethE kai tous arithmous alla
peri toutOn men, ouch hE de toiauta hoia echein megethos E einai
diaireta, dElon hoti endechetai kai peri tOn aisthEtOn megethOn
einai kai logous kai apodeixeis, mE hE de aisthEta all hE
toiadi, 1077b17-b22
2. [protasis, cont.] and just as there are many accounts [of
movables] qua movable only, regardless of what it is that moves
or their attributes, and it is not necessary it is something
separate from sensibles or that there is some distinct nature in
these, and so there will be rules and knowledge about moving
things not qua moving things but as mere bodies and planes and
lines, divisible and indivisible, having and not having
position, hOsper gar kai hE kinoumena monon polloi logoi eisi,
chOris tou ti hekaston esti tOn toioutOn kai tOn sumbebEkotOn
autois, kai ouch anagkE dia tauta E kechorismenon ti einai
kinoumenon tOn aisthEtOn E en toutois tina phusin einai
aphOrismenEn, houtO kai epi tOn kinoumenOn esontai logoi kai
epistEmai, ouch hE kinoumena de all' hE somata monon, kai palin
hE epipeda monon kai hE mEkE monon, kai hE diaireta kai hE
adiaireta echonta de thesin kai hE adaireta monon, b22-b30
3. [apodosis] so, to tell the truth, there are not only separate
[things] and non-separate (like moving things), but there are
mathematicals, such as they say, hOst' epei haplOs legein
alEthes mE monon ta chOrista einai alla kai ta mE chOrista
(hoion kinoumena einai), kai ta mathEmatika hoti estin haplOs
alEthes eipein, kai toiauta ge hoia legousin [e.g. Plato. These
are a third kind, metaxu], b31-b34
4. and to speak truly, just as other sciences are not about an
attribute, . . . but about the [subject] of each . . . so with
geometry: if what they are about happens to be sensible, it is
not qua sensible, [and] mathematical sciences will not be of
sensibles or of something separate from them. Male and female
have different attributes, yet they are not something different
from animal: so it is with lines and planes, kai hOsper kai tas
allas epistEmas haplOs alEthes eipein toutou einai, ouchi tou
sumbebEkotos . . . all' ekeinou hou estin hekastE . . . houtO
kai tEn geOmetrian. ouk ei sumbebEken aisthEta einai hOn esti,
mE esti de hE aisthEta, ou tOn aisthEtOn esontai hai
mathEmatikai epistEmai, ou mentoi oude para tauta allOn
kechOrismenOn . . . , b34-1078a9 [Jaeger in his app. crit. notes
the difficulty with the redundant negative, ouk . . . ou, in
lines a3-a4 there, and excises the ouk. Agree. So apparently
does Tredennick. It makes sense]
5. and insofar as [mathematical sciences] are about the prior in
reasoning, and simpler, to this extent they are more precise,
and more so without magnitude than with; most of all without
motion; or if with motion, most of all with the first and
uniform motion, a9-a13
6. the same with the theory of music and with optics and mechanics:
one investigates these as such separate from their accidents.
Thus one investigates best, when one separates what is not
separate, like arithmetic and geometry do [cf. E, i, 1026a9; De
anima, Gamma, vii, 431b15 ff.], a14-a23
7. man is indivisible, but a geometer does not regard him so, but
as a solid, all' hE stereon. The geometers are right, and they
discuss things that exist, orthOs hoi geOmetroi legousi, kai
peri ontOn dialegontai, kai onta estin, because being is of two
kinds, complete and material, ditton gar to on, to men
entelecheiai, to d' hulikOs, a23-a31
8. the good and the beautiful: they err who say that the
mathematical sciences have nothing to do with them, a31-b6
[Much circumlocution here, but the gist is simple and amplifies
chapter ii: mathematicals are not the things they are in, nor are
they separate from them, but they are in things in their own
peculiar way. For Plato, mathematicals were ta metaxu, in between.
In between what? Ideas and things (copies), of course. But
Aristotle has replace Plato's Ideas with form, so that will no
longer do. Instead they are between sensibles and non-sensibles,
being neither of those. What is this limbo? It is nevertheless
where all abstractions are to be found: in the mind. But neither
Plato nor Aristotle make this unambiguously clear yet. Everything
is still too corporeal. Even Ideas were, considering the way people
in general hypostasized them. They took the Phaedo too literally.
They still do. Did Plato mean them to do so? He speaks quite
differently in the Parmenides. The Phaedo is fine poetry and
perhaps some good history, but it is not good ontology!
EFL. 8/2/97
MU (XIII) METAPHYSICS
Chapter iv, Ideas
1. so much for mathematicals, that they exist and what they are.
Next let us look at the doctrine of Ideas, without reference to
their relation to numbers, but as their first proponents brought
them up in the beginning, peri de tOn ideOn prOton autEn tEn
kata tEn idean doxan episkepteon, mEthen sunaptontas pros tEn
tOn arithmOn phusin, all' hOs hupelabon ex archEs hoi prOtoi tas
ideas phEsantes einai [aside: isn't Aristotle suggesting there
that Plato's doctrines about number came later in his career
than his inquiries about Ideas?], 1078b7-b12
2. origin of [Plato's] doctrine of Ideas:
a. a response to Heracliteans' doctrine of change: knowledge
requires something stable, sunebE d' hE peri tOn eidOn doxa
tois eipousi dia to peisthEnai peri tEs alEtheias tois
Herakleitoiois logois hOs pantOn tOn aisthEtOn aei hreontOn,
hOst' eiper epistEmE tinos estai kai phronEsis, heteras dein
tinas phuseis einai para tas aisthEtas menousas. ou gar einai
tOn hreontOn epistEmEn, b12-b17
b. Socrates' treating the ethical virtues and seeking general
definitions (among the physicists Democritus had touched on
[these] a little, and the Pythagoreans in a few cases gave
names to numbers, such as the fit [7], the just [4 or 5] and
marriage [3 or 6], but Socrates sought definitions reasonably
and tried to argue [sullogizesthai]. The beginning of
argument is definition. There was at that time no dialectical
skill that enabled [anyone] to study the contraries other
than to name them [as in lists like the sustoicheia], and
[consider] if there was one science of contraries. One may
justly give Socrates credit for two [innovations]: logic and
general definitions. These are the beginning of knowledge),
b17-b30 [compare A, vi, 987a29-b9; see also Ross, I, xxxiii,
ff.]
c. Socrates did not make the general definitions separate.
These [Plato and his followers] did, and called them Ideas,
so it occurred to them almost immediately that there are
Ideas of all universals, all' ho men SOkratEs ta katholou ou
chOrista epoiei oude tous horismous. hoi d' echOrisan, kai ta
toiauta tOn ontOn ideas prosEgoreusan, hOste sunebainein
autois schedon tOi autOi logOi pantOn ideas einai tOn
katholou legomenOn, b30-b34
3. objections
a. it is like someone thinking he couldn't count a few things,
so he added to them, because Ideas are additional entities,
and the causes of things. And they are a collective, kai
paraplEsion hOsper an ei tis arithmEsai boulomenos elattonOn
men ontOn oioito mE dunasthai, pleiO de poiEsas arithmoiE.
pleiO gar esti tOn kath' hekasta aisthEtOn hOs eipein ta
eidE, peri hOn zEtountas tas aitias ek toutOn ekei proElthon.
kath' hekaston te gar homOnumon ti esti kai para tas ousias,
tOn te allOn hen estin epi pollOn, kai epi toisde kai epi
tois aidiois, b34-1074a4
b. in no case are the ways in which they demonstrate Ideas
evident: some the arguments are not compelling; in others
there are ideas for things they don't want them for [an echo
of Parmenides' (i.e. Plato's) question in the Parmenides,
130C-D, Ideas of hair, mud, dirt]. There will be Ideas of all
things of which there is knowledge: collectives, negations,
perishables, phantoms, a4-a11
c. and in the strictest sense of the word some create Ideas of
relatives, [most notably, the Same and the Other in Plato's
Sophist, 254E ff.] which they say are not a class by itself;
while others speak of the "third man" [infinite regress, as
in Plato's Parmenides, 132A]. The arguments about Ideas do
away with the very things that the proponents of Forms wish
[to preserve]. The dyad is not primary, but number is; and
the relative [is prior to] number; and to the absolute; their
opinions about Ideas contradict their principles, a11-a19
d. and according to their notion of Ideas, there will be Forms
not only of ousiai but of many other things (because thought
is not only of ousiai but also of many other things like the
notion of the one; knowledge is not just of ousiai. There are
many such things). But according what they teach about them,
if Forms are participated in, they must be Ideas of ousiai,
a19-a26
e. [these things] do not participate [in Ideas] accidentally but
only as not predicated of a subject [i.e., as subjects
themselves] (e.g., if something participates in doubleness,
and in eternalness, this [latter] is accidental, because
doubleness is eternal), so Ideas will be ousia, a26-a31
f. the same words mean ousia in this world and the intelligible
world, or what does a collective name, hen epi pollOn, mean?
And if the form of Ideas and their participants is the same,
they will be some thing in common. Why, in the case of
perishable dyads and the Idea of the dyad, will the two be
one, or in the case of the absolute and the particular
[dyad]? But if they are not the same form, they would be
same in name only, just as if one called a man and a wood-
carving, "Kallias," a31-b3
g. if we allow other parts to join the common definitions of
Ideas, like adding "plane figure," etc. to the Idea of a
circle, the essence is enlarged. And it must be inquired
whether this is not altogether in vain. To what is it being
added? The middle or the surface or the whole [of the
circle]? All the things in the ousia are Ideas, and inherent
natures like their class, b3-b11
[These two chapters bear comparison with Alpha, vi and ix]
EFL, 8/9/97
MU (XIII) METAPHYSICS
Chapter v, Ideas, cont.
[This chapter continues Aristotle's objections to Ideas in iv.
These two chapters bear comparison with Alpha, vi and ix]
h. what do forms contribute to the eternal sensibles or to the
created and perishable sensibles [cf. Lambda, i, 1069a31 for
this subdivision of sensible (movable) ousia]? They are no
cause of motion or change. They don't contribute to knowledge
of other things, because they are not the essence of these,
or they would be in them. Nor to the existence, not being in
the participants. This is the same as saying that the white
is mixed in a white thing, but this argument, used by
Anaxagoras and Eudoxus, is very easily refuted (it is easy to
collect objections to such an opinion), PantOn de malista
diaporEseien an tis ti pote sumballontai ta eidE E tois
aidiois tOn aisthEtOn E tois gignomenois kai phtheiromenois.
oute gar kinEseOs estin oute metabolEs oudemias aitia autois.
alla mEn oute pros tEn epistEmEn outhen BoEthei tEn tOn allOn
(oude gar ousia ekeina toutOn. en toutois gar an En), out'
eis to einai, mE enuparchonta ge tois metchousin . . . ,
1079b12-b23
i. in no fashion are other ordinary things [composed] of Forms.
To call them paradigms and say other things participate in
them is empty talk and poetic metaphor. What is accomplished
with Ideas? Anything can come into being and exist, and not
be a copy. Socrates was what he was, without an eternal
paradigm, alla mEn oude ek tOn eidOn esti talla kat' outhena
tropon tOn eiOthotOn legesthai. to de legein paradeigmata
einai kai metechein autOn ta alla kenologein esti kai
metaphoras legein poiEtikas. ti gar esti to ergazomenon pros
tas ideas apoblepon; endechetai te kai einai kai gignesthai
otioun kai mE eikazomenon, . . . , b23-b30
j. there will be more paradigms and Ideas of the same thing,
like animal, two-footed and man itself, for a man, b31-b33
k. and Ideas will not just be paradigms of sensibles, but of
themselves, as genus of species, and the same will be both
paradigm and copy, b33-b35
l. it seems impossible for ousia or what has ousia to be
separate, so how can the Ideas be the ousia of things, since
they are separate? This is how Plato describes them in the
Phaedo , where the Forms are the cause of being and becoming,
eti doxeien an adunaton chOris einai tEn ousian kai hou hE
ousia, hOste pOs an hai ideai ousiai tOn pragmatOn ousai
chOris eien; en de tOi PhaidOne touton legetai ton tropon,
hOs kai tou einai kai tou gignesthai aitia ta eidE estin,
b35-1080a3
m. nothing comes into being from Forms, if there is no mover,
yet many other things do come, like artefacts, to which they
do not ascribe Forms. So it is clear that those things of
which they say there are Ideas can be and come into being for
such causes as [I have] named, but not because of Forms. But
there are many things that we could add both in this manner
and with more logical and more precise arguments, similar to
what we have seen, kaitoi tOn eidOn ontOn homOs ou gignetai
an mE Ei to kinEson, kai polla gignetai hetera, hoion oikia
kai daktulios, hOn ou phasin einai eidE . . . , a3-a11
[The cat is out of the bag in line 1080a2. With his mention of the
Phaedo, Aristotle gives away the source of his view of Plato's
Ideas, and the source of the view of most of us ever since. The
Phaedo has always been made the test case of Plato's so-called Form
Theory, but it is only one of many sources of Plato's doctrine of
Ideas. To base our case entirely on the Phaedo (and perhaps some
remarks in Symposium, 211, and Phaedrus, 247) may be a narrow view
and a gross error. Those of you who were with us for the discussion
of the Parmenides will recall my partiallity for that dialogue as
a basis for approach to Plato's Ideas, and my impression that Plato
was asking more questions than he was answering there, that it is
we, not Plato, who have created the "doctrine of Ideas,"
[Why would such an error have been committed? Because it is a
natural propensity of mankind to think in dialectical terms:
heaven and earth, good versus bad, cops and robbers, and so forth,
Plato v. Aristotle. Why are "Western" movies so popular, if not for
their clear-cut dialectical statements? This propensity has taken
over Plato and Aristotle, and Plato's Ideas and Aristotle's form,
and shaped them to our own image. We can do better.
[But what about Aristotle? What did he do? Of course he saw the
impossibility of Plato's reputed doctrine. His forms bring the
Ideas back to this world, entautha, from that other world, ekei. He
objects all these ways we have just seen here to the transcendent
Ideas, but he does not yet have a clear and explicit view of the
role of the mind, although he seems to have a suspicion of it. He
does not self-consciously say what he is doing. He just DOES it.
[Finally: does it matter what Plato really thought? Isn't it the
contrast that really counts? To the first question: yes, it does
matter, and to the second: no, it is not what really counts! Why?
Because it prevents us from seeing the evolution going on there,
the slow groping for an understanding of the life of the mind.
Since we are still haggling over just that question, it may pay us
to see how old a problem it is, and indeed how unanswerable it is
with our either-or method]
EFL, 8/16/97
MU (XIII) METAPHYSICS
Chapter vi
Numbers
[From this chapter to the end of the Metaphysics, Aristotle's
attention is devoted mostly to number. He returns once to Ideas in
a remarkable passage in the last chapter of Mu, but otherwise he is
concerned with various theories of number, theories of the
Pythagoreans, Plato, Speusippus, Xenocrates, and perhaps some
others not so clearly identifiable. Much of what we know about
these people is learned right here, or in Book A. There are a few
other testimonies, but they are not abundant, and they are much
later. There are almost no primary sources. This makes it difficult
sometimes to understand what is said here. There are helps, such as
Kirk and Raven on The Presocratics, Konrad Gaiser on the "Unwritten
Doctrine," Lang on Speusippus, Heinze on Xenocrates, and C. J. De
Vogel generally in her Greek Philosophy, vols. I and II, all of
which do a fine job of gathering and organizing the other evidence,
and arranging it topically]
1. having discussed these matters [geometricals and Ideas], it is
well to look again at the properties of numbers according to
those who say they are separate ousiai and the first causes of
beings, epei de diOristai peri toutOn, kalOs exei palin
theOrEsai ta peri tous arithmous sumbainonta tois legousin
ousias autous einai chOristas kai tOn ontOn aitias prOtas,
1080a12-a14 [this would rule out the Pythagoreans, but he does
refer to them occasionally, as you will see]
2. if number is some nature, and its essence is nothing else but
itself, as some say, it follows that, anagkE d', eiper estin ho
arithmos phusis tis kai mE allE tis estin autou hE ousia alla
tout' auto, hOsper phasi tines, a15-a16:
3. number is ordered, each different in form, Etoi einai to men
prOton ti autou to d' echomenon, heteron on tOi eidei hekaston,
a17-a18, and
a. either it consists at once of units and is uncombinable [or
"not addible"] any one with any other, kai touto E epi tOn
monadOn euthus huparchei kai estin asumblEtos hopoiaoun monas
hopoiaioun monadi, [a number is an ideal entity, like the
Indefinite Dyad, the Tetrad, etc., Xenocrates], a18-a20
b. or they are all immediately combinable, as they say of
mathematical number (no unit differs from another), E euthus
ephexEs pasai kai sumblEtai hopoaioun hopoiaisoun, hoion
legousin einai ton mathEmatikon arithmon (en gar tOi
mathEmatikOi ouden diapherei oudemia monas hetera heteras)
[i.e. NOT Ideal number, Speusippus], a20-a23
c. or some are combinable; some, not, E tas men sumblEtas, tas
de mE, the units that make them up are combinable, but the
Ideal Two and the Ideal Three are not [Plato]. The
mathematician counts with the former, a23-a35
d. or there are all three of the above, a35-a37
[As Ross, II, 426 points out, "This sentence is irregular in
structure," but the important and fundamental distinction in
this complex passage, a17-a37, is two-fold: (1) addible or
comparable, sumblEtoi, and (2) non-addible or incomparable,
asumblEtoi, numbers. These are respectively (1) mathematical,
and (2) ideal]
4. numbers are either separate from things, or are not separate but
in sensibles (albeit not like we considered at first [chapter
ii, 1076a38-b11] as sensibles being made of numbers [the
Pythagorean doctrine]), or neither or both, a37-b4
5. these are the only alternatives. Those who say the One is the
principle and ousia and element of everything, and that number
comes from the one and the other, speak in one of these modes,
except those who say that all numbers are uncombinable. And this
is reasonable. There can be no other mode than these, b4-b11
a. some [Plato] say numbers are both: and those having priority
and posteriority [are] Ideas; those that are mathematical
[are] beside Ideas and sensibles [intermediates, ta metaxu].
Both are separate from sensibles, b11-b14
b. some [Speusippus] say there is only mathematical number,
the first of beings, separate from sensibles, b14-b16
c. the Pythagoreans say there is only the mathematical, but not
separate, but the sensible ousiai are made of it. The whole
heaven is constructed of numbers, except not of units,
although they suppose units have bulk. How the first one,
having bulk, is made, they seem to be puzzled, b16-b21
d. someone else [?], that one of the Ideas is the first one,
b21-b22
e. some [Xenocrates et al.] think the ideal and the mathematical
are the same, b22-b23
6. the same with lines and planes and solids, b23-b24
a. some [Plato] say the mathematicals and the Ideal are
different, b24-b25
b. speaking differently, some [Speusippus] speak mathematically
of mathematicals: numbers are not Ideas, and there are no
Ideas, b25-b28
c. some [Xenocrates] speak of mathematicals, but not
mathematically. They don't divide all magnitude into
magnitudes, or [say] that two is [made up of] whatever units.
Every number is a unit, they suppose, monadikous de tous
arithmous einai pantes titheasi, b28-b31
d. except the Pythagoreans who say that the one is the element
and principle of beings. These say that [units] have
magnitude, as we said above [4.c.], b31-b33
7. So much can be said about them, and it is clear that these are
all the modes [of regarding numbers]. All are impossible, some
more than others, b33-b36
[A final word is in order about the two terms introduced in this
chapter, sumblEtoi (addible, combinable) and asumblEtoi (inaddible,
not combinable). These are new terms in our text, previously unused
in connection with number, and they are central to this chapter. It
is my opinion that Aristotle begins with Plato in mind, and these
two terms correspond the two kinds of number discussed by Plato:
(1) the sumblEtoi, addible, are the mathematical numbers that Plato
considered metaxu, between Ideas and copies. They are probably what
we would consider the common material of arithmetic, the numbers
that we manipulate in calculation, 1 + 3 = 4, and so forth. (2) the
others, the asumblEtoi, inaddible, are the One and the Indefinite
Dyad, etc. These are ideal numbers, and as such they are unique.
They are Ideas. [See C. J. DeVogel, Greek Philosophy, vol. I,
paragraph 362d and note 3, also paragraph 367, in agreement.
[If this is right, it would seem possible to sum up Aristotle's
account with the following possibiities, leaving aside nameless
adherents of these and other variations:
(1) numbers are things and things are numbers. The old
Pythagorean doctrine.
(2) there are ideal AND mathematical numbers. Plato.
(3) there are only mathematical numbers. Speusippus.
(4) there are only ideal numbers. Xenocrates.
[It is these terms then that provide the context for Aristotle's
discussion here. This chapter is preparation for a long and
detailed critique in chapters vii, viii, ix, and the first half of
chapter x. All these possibilities are rejected]
EFL, 8/23/97
MU (XIII) METAPHYSICS
Chapter vii
Numbers, cont.: dialectical arguments
[Now Aristotle begins a more detailed critique of the theories of
number introduced in the last chapter. He returns to the
combinable/uncombinable classification (mathematical/Ideal), but
notice that he now subdivides the uncombinable in order to rule out
the mathematical uncombinable, and then show that Ideal are absurd]
1. first it should be investigated whether units are combinable,
[2. below] or not [3. and 4. below], and if not, in which of the
ways we have differentiated, PrOton men oun skepteon ei
sumblEtoi hai monades E asumblEtoi, kai ei asumblEtoi, poterOs
hOnper dieilOmen. For there is [3.] non-combinable [non-addable,
asumblEtai], whatsoever to whatsoever number, and [4.] non-
combinable, as ideal numbers, esti men gar hopoianoun einai
hopoiaioun monadi asumblEton, esti de tas en autEi tEi duadi
pros tas en autEi tEi triadi, kai houtOs dE asumblEtous einai
tas en hekastOi tOi prOtOi arithmOi pros allElas, 1080b37-1081a5
2. combinable units are mathematical, 1081a5-a17
a. if the units are all combinable, and alike and not different,
number becomes mathematical and only one, and they cannot be
Ideas, ei men oun pasai sumblEtai kai adiaphorai hai monades,
ho mathematikos gignetai arithmos kai heis monos, kai tas
ideas ouk endechetai einai tous arithmous, (how will Man or
Animal or any such be Ideas? There is one Idea of each, like
one of Man and another of Animal. But like [numbers] are
indifferent and infinitely many, so this particular Three is
no more the Idea of Man than any other), a5-a12
b. if Ideas are not numbers, they cannot exist (from what
principles will they be? Number is from the One and the
Infinite Dyad, the principles and elements of number. [Ideas]
cannot rank before or after numbers), a12-a17
3. non-combinable, whatever to whatever, 1081a17-b35
a. if units are non-combinable, asumblEtai, any with any, there
can be no mathematical number (since mathematical number is
not different, and operations with it fit together), nor
number of forms, a17-a21
b. there will be no first two, out of the One and the Indefinite
Dyad, and after that the rest of the numbers, as they are
successively called, two, three, four - because units come
into being simultaneously in the first two, as the author of
the theory held, equal in strength, or otherwise. If one unit
is prior to the other, it will be prior to the resulting two,
because where there is order [in the units] there will be
order in the result, a21-a29 [so they are combinable]
c. and if there is an Ideal One first, and after that a second
one first of others, and again a third, second after the
second, and third after the first, then the units from which
they count will be prior to the numbers, proterai an eien hai
monades E hoi arithmoi ex hOn legontai, like in the two there
will be a third unit before the three, etc., a29-a35
d. no one speaks of uncombinable units in this manner, but it
follows their principles logically and is impossible in
truth. Because it is reasonable that there are prior and
posterior units, if there is a first unit and first one,
[i.e. the ideal before the mathematical] likewise with two,
and so forth . . . There cannot be two units before two,
followed by a first two, a35-1081b10
e. it is clear that if all the units are not combinable, there
can be no Ideal two or three or any other numbers. Whether
units are same or they differ, numbers must be counted by
addition [of units]. Numbers cannot be created from the
[Indfinite] Dyad and the one, adunaton tEn genesin einai tOn
arithmOn hOs gennOsin ek tEs duados kai tou henos. Each
successive number is part of its follower. Or four would be
the sum of the first two and the Infinite Dyad, two twos
beside the Ideal two. [Further examples] b10-b29
f. all this is absurd and fictitious, panta gar taut' atopa
esti kai plasmatOdE, and must result if the One and the
Infinite Dyad are elements, anagkE d', epeiper estai to hen
kai hE aoristos duas stoicheia. Such are the necessary
consequences, if the units differ one from another, ei men
oun diaphoroi hai monades opoiaioun hopoiaisoun, tauta kai
toiauth' hetera sumbainei ex anagkEs, b29-b35
4. [if they are incombinable as ideal numbers], b35-1082b19
a. if units in different numbers differ, but units in the same
number do not, there is no less difficulty, b35-b38
b. because in the ideal ten there are ten units, but it is made
of these and also of two fives. Since the ideal ten is not
just any chance ten, nor is it made of any chance fives or
any chance units, the units in this ten differ. If they do
not, the fives will not differ. If the fives differ, so do
the units. But if they [the units] differ, will there or
won't there be other fives, but only these two? If there
won't, that is absurd. If there will, what kind of ten is
made of them? There is no other kind. Similarly with four: a
definite doubling of the Indefinite Dyad makes two twos [i.e.
two kinds of two], 1081a1-a15 [This should remind you of the
song, "New Math," written and sung by Tom Lehrer, the MIT
mathematician, about thirty five years ago]
c. how can there be a two, some nature beside two units, or
three beside three units? Either one shares in the other, or
one is a species of the other, a15-a20
d. there are unity of contact, of mixture, of position, none of
which can be in the units making a two or a three. As two men
do not make one other than the two, so with units. They don't
differ because they are indivisible. Points are indivisible,
but two of them is nothing beside the two, a20-a26
e. nor should this escape us: there are prior and posterior
twos, and likewise other numbers. Let twos be simultaneous in
a four: they will have priority in an eight considered as two
fours, so if the first two is an Idea, so will the latter
twos [be]. The same with units, and there will be Ideas
composed of Ideas, a26-b1
f. generally, to make the units different in any way is absurd
and fictitious (by this I mean a forced hypothesis). We don't
see them differ quantitatively or qualitatively. Number must
be equal or unequal, numbers made of units most of all. If
they are not more or less, they are equal. Equal and not
different we suppose to be same among numbers. If not, the
same twos in the Ideal Decad will not be equal. What cause
will be to call them the same? b1-b11
g. if every unit and another unit make two, the unit from the
Ideal Two and the two from the Ideal Three will be from
different [numbers]. Are they prior to or posterior to the
three? They must be prior. The units must be simultaneous
with the three, and simultaneous with the two, b11-b16
h. We suppose that one and one are two, whether equivalent
things or not, like the good and the bad. Those who preach
Ideal numbers don't even let units [do this], b16-b19
i. if the Ideal three is not more than the Ideal two, that is a
surprise. If it is more, it is clear that there is [a number]
equal to two in it, so this is not different than two. But
that cannot be, if there is first and second number, b19-b23
j. numbers will not be Ideas. They are right who think numbers
will be different, if they are Ideas, as was said earlier.
Because the Form is unique. But units are not different, and
two and three will not be different. Therefore they must say
that counting, one, two, we are not adding to the existing
number (because there is no generation from the Indefinite
Dyad, nor can it be an Idea. One Idea would be in another,
and all Forms parts of one). Thus they state their hypothesis
correctly, but they are altogether wrong, because they will
spoil their argument if they make a question out of whether,
when counting, one, two, three, we are adding steps or taking
portions. We are doing both, and it is laughable to bring up
this old dilemma, b23-b37
[Why such prolonged attention to these arguments? Aristotle sees,
as you and I see, that we can get along without Plato's Ideas, but
we can't get along without numbers. We must count. But numbers have
something in common with Ideas. They are not things, but something
different from the things they count. In some way they are
"separate." How? The nature of this separation must be explored.
Are numbers countable, computable, sumblEtoi? Or are they some
other kind of separate, asumblEtoi? (We dismiss the thought that
they are the things themselves, as the Pythagoreans would have us
believe.) Since they must be countable, they cannot be Ideas
(paragraph 2.); if they were non-countable, they would be Ideas,
and useless and absurd (3. and 4.)]
EFL, 8/30/87
MU (XIII) METAPHYSICS
Chapter viii, to 1083b23
Numbers, cont.: ad hominem arguments
Plato, Speusippus, Xenocrates and the Pythagoreans
1. [Plato's inconsistencies], 1083a1-a20
a. first of all it is well to distinguish, what is the
difference of number and unit, if there is any, pantOn de
prOton kalOs echei diorisasthai tis arithmou diaphora, kai
monados, ei estin. They must differ either in quantity or
quality, although it seems neither of these can be the case,
anagkE d' E kata to poson E kata to poion diapherein, toutOn
d' oudeteron phainetai endechesthai huparchein. But number
[differs] in quantity, all' hE arithmos, kata to poson. If
units differed in quantity, a number would also differ from
a number which is equal in the number of units, ei de dE kai
hai monades tOi posOi diepheron, kan arithmos arithmou
diapheren ho isos tOi plEthei tOn monadOn. Whether the former
[number] are more or less, do the latter [units] increase or
the opposite? All that is absurd, eti poteron hai prOtai
meizous E ellatous, kai hai husteron epididoasin E
tounantion; panta gar tauta aloga, 1083a1-a8 [this is a
criticism of Plato's treatment of the One, to hen, as an
Idea, and something other than the number, one, as a unit, hE
monas. Such was Plato's wont in the Parmenides and in the
notorious lecture "On the Good." Aristotle's position vis-a-
vis these matters was made clear in Book Iota, i-ii]
b. but they cannot differ in quality, alla mEn oude kata to
poion diapherein endechetai. Because they can have no
quality, outhen gar autais hoion te huparchein pathos.
Because they say quality belongs to numbers later than
quantity, husteron gar kai tois arithmois phasin huparchein
to poion tou posou. And it wouldn't arise from the one or the
two, eti out' an apo tou henos tout' autais genoito out' an
apo tEs duados, because the one is not a quality, and [two]
is a certain quantity, to men gar ou poion hE de posopoion.
Nature herself is the cause of multiplicity, a8-a14
c. if it is otherwise, they should have said so right in the
beginning, ei d' ara echei pOs allOs, lekteon en archE
malista touto, and distinguished the differences of units,
kai dioristeon peri monados diaphoras . . . but if Ideas are
numbers, it is clear that all the units cannot be combined or
uncombined with each other any way, eiper eisin arithmoi hai
Ideai, oute sumblEtos tas monadas hapasas endechetai einai,
phaneron, oute asumblEtous allElais oudeteron tOn tropOn (cf.
chapter vi), a14-a20
2. while others speak about numbers, it is not well done, alla mEn
oud' hOs heteroi tines legousi peri tOn arithmOn legetai kalOs,
a20-a21
a. [Speusippus] there are those who think there are no Ideas
either themselves or as numbers, eisi d' houtoi hosoi ideas
men ouk oiontai einai oute haplOs oute hOs arithmous tinas
ousas, but there are mathematicals, and numbers are the first
of beings, and their origin is the One itself, mathEmatika
einai kai tous arithmous prOtous tOn ontOn, kai archEn autOn
einai auto to hen. What they say is absurd, that the one is
the first of [all] ones, but not the two of twos, or three of
threes. The same reasoning applies to all. If this is the way
it is with numbers, and one assumes there is only
mathematical number, the one cannot be a principle. If the
one is a principle, it must be rather as Plato said, and
there is a first two, and three, and numbers are not
combinable with each other, kai ou sumblEtous einai tous
arithmous pros allElous. We showed the impossibility of that
[in chapter vii]. It must be one way or the other, so if it
is neither, there can be no separate number, alla mEn anagkE
ge E houtOs E ekeinOs exein, hOst' ei mEdeterOs, ouk an
endechoito einai ton arithmon chOriston, a21-1083b1
b. [Xenocrates] the third way of putting it is the worst, that
Ideal and the mathematical are the same number, phaneron d'
ek toutOn kai hoti cheirista legetai ho tritos tropos, to
einai ton auton arithmon ton tOn eidOn kai ton mathEmatikon,
because this must combine two errors in one opinion. There
cannot be mathematical number in this sense, oute gar
mathEmatikon arithmon endechetai touton einai ton tropon, b1-
b8
c. [Pythagoreans] the way the Pythagoreans put it has less
difficulties than aforementioned, but it has its own. Not
making number separate does away with many impossibilities,
but to compose bodies of numbers, and call this number
mathematical, is impossible, ho de tOn PuthagoreiOn tropos
tEi men elattous echei duschereias tOn proteron eirEmenOn,
tEi de idias heteras. to men gar mE chOriston poiein ton
arithmon aphaireitai polla tOn adunatOn, to de ta sOmata ex
arithmOn einai sugkeimena, kai ton arithmon touton einai
mathEmatikon, adunaton estin. Because it is not true to say
there are indivisible magnitudes, oute gar atoma megethE
legein alEthes, and if this is what they really mean, units
do not have magnitude, ei th' hoti malista touton echei ton
topon, ouch hai ge monades megethos echousin. How can
magnitude be composed of indivisibles? megethos de ex
adiairetOn sugkeisthai pOs dunaton; Arithmetical number
consists of units, alla mEn ho g' arithmEtikos arithmos
monadikos estin. Those people say beings are number. Their
theories attribute it to bodies that numbers are made of
them, b8-b19
d. [summary] if number must be some Ideal being in one of these
ways, and none of them is possible, it is obvious that there
is no such nature as they create who separate it, ei toinun
anagkE men, eiper estin arithmos tOn ontOn ti kath' hauto,
toutOn einai tina tOn eirEmenOn tropOn, outhena de toutOn
endechetai, phaneron hOs ouk estin arithmou tis toiautE
phusis hoian kataskeuazousin hoi choriston poiountes auton,
b19-b23
[In sum, Aristotle considers that all these actual theories are
inadequate in these several ways. It is his service to take the
measure of the confusion about these matters on the part of his
contemporaries. Number is not separate, the way Ideas are separate,
but it is not in things the way the Pythagoreans would have it.
Where and what is it?]
EFL, 9/6/97
MU (XIII) METAPHYSICS
Chapter viii, 1083b23 to end
Plato's ideal numbers
1. the Great and the Small, to mega kai to mikron, 1083b23-b36
[another of Plato's Ideas, opposite of the One. Recall Book A, vi,
987b18-b22, where the Great and the Small is called the material
cause of all beings, and the One is called the ousia, epei d' aitia
ta eidE tois allois, takeinOn stoicheia pantOn OiEthE tOn ontOn
einai stoicheia. hOs men oun hulEn to mega kai to mikron einai
archas, hOs d'ousian to hen. ex ekeinOn gar kata methexin tou henos
einai tous arithmous (or ta eidE einai). Elsewhere the Great and
the Small is the Indefinite Dyad (aoristos duas) and the Ideal Two]
a. is each unit [of the two] from the Great and the Small
equally, or one from the Small and the other from the Great?
If the latter, neither is [composed] of all the elements, nor
is the same (because they are opposite in nature), eti
poteron hekastE monas ek tou megalou kai mikrou isasthentOn
estin, E hE men ek tou mikrou hE d' ek tou megalou; ei men dE
houtOs, oute ek pantOn tOn stoicheiOn hekaston oute
adiaphoroi hai monades . . . , 1083b23-b28
b. how [is each] in the Ideal Three? Because there is an extra
one. Perhaps for this reason they put the Ideal One in the
middle of the odd number, b28-b30
c. if each of the units is from both equally, how will two,
being one nature, come from the great and the small? or why
will it [the two] differ from the unit? b30-b32
d. and the unit is prior to the two (if it is removed, the two
is gone) so there would have to be an Idea of an Idea, prior
to the Idea, generated previously. From what? Because it is
the Indefinite Dyad that makes two [they say], b32-b36
2. number must be either unlimited or limited, because they make it
separate [i.e. Ideal], so it must be one or the other of these,
eti anagkE Etoi apeiron ton arithmon E peperasmenon, chOriston
gar poiousi ton arithmon, hOste ouch hoion te mE ouchi toutOn
thateron huparchein, b36-1084a1
a. clearly it cannot be unlimited
(1) the unlimited is neither odd nor even, but the production
of numbers is always odd or even. Thus dropping one, an
odd number becomes even; or one doubled becomes even; and
even is the other of odd, hOdi men tou henos eis ton
artion piptontos perittos, hOdi de tEs men duados
empiptousEs ho aph' henos diplasiazomenos, hOdi de tOn
perittOn ho allos artios, a2-a7
(2) furthermore, if every Idea is of something, and numbers
are Ideas, the unlimited will be an Idea of something, of
sensibles or of something else. That cannot be, according
to their thesis or according to reason. But that is how
they propose their Ideas, a7-a10
b. but if limited [there are also difficulties]
(1) to what extent? mechri posou; this must be stated, and
not just what but why. Numbers reach ten, as some [Plato] say
[recognizing their base ten]. Then their forms "run dry,"
epileipsei [repeat]. If three is the Idea of Man, what will
the Idea of Horse be? Because each number up to ten is an
Idea, and it must be one of these numbers, but there are more
Ideas than there are numbers up to ten, a10-a17. (2) At the
same time, if three is the Idea of Man, so will other threes
be, and there will be unlimited ideas of Man and men, a18-
a21. (3) and if man is two and horse is four, man will be
part of horse, a21-a25. (4) it is strange for the numbers up
to ten to be Ideas, but eleven and higher, not, a25-a27. (5)
or there are numbers of which there are not Forms. Why? This
means Forms are not causes, a27-a29. (6) and it is strange
that number up to ten is more being and Form than the Form of
Ten, and there is no generation of them as of one, while
there is, of the decad. But they try to show number complete
up to ten, a29-a32. (7) And they generate derivatives, like
void, analog, odd, etc., all within the decad. They assign
some to principles, some to numbers. The odd is one; if it
were three, how would five be odd? [They do] the same with
magnitude: line is the first [limit], indivisible; then two
and so on up to the decad, a32-b2
3. [Further difficulties with Ideal numbers]
a. if number is separate [an Idea] one might wonder whether the
one or the three or the two is prior, eti ei esti chOristos
ho arithmos, aporEseien an tis poteron proteron to hen E hE
trias kai hE duas, 1084b2-b4
b. if number is composite, the one [is prior]; if universal and
form, the number [is prior, the two or three, etc.]. Because
each of the units is part of the number as matter; the number
is the form, hEi men dE sunthetos ho arithmos, to hen, hEi de
to katholou proteron kai to eidos, ho arithmos, hekaste gar
tOn monadOn morion tou arithmou hOs hulE, ho d' hOs eidos,
b4-b6
(1) like the right angle is prior to the acute in definition
[tOi logOi] but the acute is prior as a divisible part of
the right angle. The acute angle is the element and the
unit, prior as matter; as related to form and definition
and ousia the right angle [is prior], and the whole is
made out of matter and form. Both are nearer to the form
and the definition, but [the whole] comes into being
later, b7-b13
c. how is the one a principle? by being indivisible, they say.
But the universal and the particular and the element are
indivisible, all' adiaireton kai to katholou kai to epi
merous kai to stoicheion. But in another way, some in
definition, some in time, alla tropon allon, to men kata
logon to de kata chronon. And which way is one a principle?
As said [b. above], as the right angle [is principle] of the
acute [tOi logOi], and the acute seems prior [hOs hulE], and
each is one, hOsper gar erEtai, kai hE orthE tEs oxeias kai
autE ekeinEs dokei protera einai, kai hekatera mia. They make
the principle the one both ways, b13-b19
d. that is impossible, because one is form and ousia, and one is
part and matter. Truly [it is] potentially one (if a number
is not just a collection of units, as they say), actually
not; each is a unit, esti de adunaton, to men gar hOs eidos
kai hE ousia to d' hOs meros kai hOs hulE. esti gar pOs hen
hekateron - tEi men alEtheiai dunamei (ei ge ho arithmos hen
ti kai mE hOs sOros all' heteros ex heterOn monadOn, hOsper
phasin), entelecheia d' ou, esti monas hekatera, b19-b23
e. the cause of [their] error is that at the same time they seek
[the one] in their mathematics, and in their reasonings about
universals, so from the former they assume the one as a point
and an origin (because the unit is a point without position,
although some others compose it of smallest beings, and these
people say the unit is the matter of numbers, prior to the
dyad, and again the two being posterior to the one is a
whole), but [in the latter mode] they seek the universal, and
speak of the predicated one as if [it is] a part [i.e. as
Form is a part of ousia], but they can't have it both ways,
b23-b32
f. but if the Ideal One is abstract, atheton, and two is
divisible, while the unit is not, the unit is more like the
Ideal One, and The Ideal One will be more like the unit than
the Dyad. So each unit will be prior to the two. But they say
not; they produce the two first [before its constituent
units], b32-1085a1
g. if the Ideal Two and the Ideal Three are one, they both make
two. What is the Ideal Two made of? a1-a2
[Plato had a fondness for "likely stories" (Timaeus, 29D) which his
pupils and later followers often overlooked. Speusippus, Xenocrates
and many others since have tended to take these with utmost
seriousness. Hence the imagery of not just the Timaeus and of many
of his myths, but of Ideal Number, of the Great and the Small, the
Indefinite Dyad, and all that. It was Aristotle's task to combat
that with every argument he could think of]
EFL, 9/13/97
MU (XIII) METAPHYSICS
Chapter ix, to 1086a21
Ideal numbers and magnitudes
[Werner Jaeger's proposal, early in this century, that the text at
1086a21 returns to an early version of the Metaphysics, has more or
less stood the test of time. There is unity of substance however in
this chapter.
1. if there is no contact among numbers, but there is succession,
nothing between their units, one might wonder, whether there is
succession to the Ideal one, or not, and whether the two is the
next successor, or either of the two units [Ross'note, II, 454,
explains], 1085a3-a7
2. difficulties with lines and planes and bodies, a7-a9
a. some [Plato] produce these from the Forms of the Great and
the Small, ek tOn eidOn tou megalou kai tou mikrou, like the
line from the Long and the Short, the plane from the Wide and
the Narrow, and bodies, tous ogkous, from the High and the
Low. These are species of the Great and the Small, a9-a12
b. Others [Xenocrates] do differently, make the principle of
these singular [as befits Ideas], tEn de kata to hen archEn
alloi allOs tithesi tOn toioutOn, although in these there
seem to be countless impossibilities and fictions and
contradictions of good sense. Because they would be separated
from each other, if the principles did not accompany each
other, like the broad and narrow and long and short (yet if
this [is the case], the line will be a plane, and the plane
a cube, and how will angles and figures and such be
distinguished?) The same applies to number, because
properties of magnitudes are numerical, although magnitude is
not [made] from them, as line is not from straight and
curved, or solids from smooth and rough, a13-a23
c. like all these is the problem about Forms as genera, when one
deals with universals, [e.g.] whether the Idea of the animal
is in the animal or is separate from the animal itself,
poteron to zOion auto en tOi zOiOi E heteron autou zOiou. If
it is not separate, there is no problem. But if one and
numbers are separate, as they say, it is not easy to resolve,
if it is not easy to speak the impossible. When someone
thinks of the unit in two and in number generally, does he
think of an Idea, or something else? a23-a31
d. some [above] produce magnitude out of such matter [the Great
and the Small]; but others [Speusippus], from the point (the
point seems to them to be not one, but like the one) and
another material like the many, to plEthos, but not the many.
There is no less of a problem about these. If the material is
one, line and plane and solid will be the same because they
will be made of the same. If the materials are many and
different for the line and the plane and the solid, either
they follow one another, or not. The result will be the same:
either the plane will not include a line or it will be a
line, a31-b4
3. [one and multitude, hen kai plEthos], 1085b4-b34
a. how number can be from the one and the many [Speusippus], is
not shown. They have the same difficulties as those as those
who say it is from the One and the Indefinite Dyad [Plato].
One group produces number from the universal and not from a
particular multitude; the other, from a particular first
multitude (the first multitude is two). So there is no
difference in what they say, but the same problems ensue:
mixture, arrangement, genesis, etc., b4-b12
b. one would do better to ask, if each unit is one, what it is
of, because every unit is not an absolute One. [According to
them] it must be from the Ideal One and the Many or part of
it. To say the unit is a many is impossible, since it [i.e.,
the unit] is indivisible. [To say] it is [made] of parts, has
many other difficulties, because each of the parts must be
indivisible (or they are a many and divisible) and the one
and many are not elements (each unit is not made of many and
one), b12-b21
c. they do nothing else but make another number, because number
is a multitude of indivisibles, b21-b22
d. we should ask them whether number is unlimited or limited.
Because there is, as it seems, limited multitude, from which
and from the one come limited units. The Ideal Many and the
unlimited many are different. What kind of Many is an element
along with the One? b22-b27
e. likewise we should ask about the point and the element out of
which they make magnitudes, because this is not just a single
point. Where does each of the other points come from? Not
from some division and an Ideal point. Nor can indivisible
parts come from the division, as [they can] from the
multitude made of units. Number is composed of indivisibles,
but not magnitude, b27-b34
4. [conclusion] all these and other such [impossibilities] make it
clear that it is impossible for there to be separate numbers and
magnitudes, and the disagreement of the explanations about
number is a sign of their untruth, b34-1086a2
a. some [Speusippus] creating mathematicals only beside the
sensibles, perceiving the difficulty and fiction in Ideas,
distance themselves from Ideal number, and create the
mathematical, hoi men gar ta methEmatika monon poiountes para
ta aisthEta, horOntes tEn peri ta eidE duschereian kai
plasin, apestEsan apo tou eidEtikou arithmou kai ton
mathEmatikon epoiEsan, a2-a5
b. others [Xenocrates] wanting to create the Forms along with
numbers, and not perceiving how there will be mathematical
number in addition to the Ideas, if one posits both, make
them the same in theory, although in fact the mathematical is
done away with (they propose their own, not mathematical,
hypotheses), hoi de ta eidE boulomenoi hama kai arithmous
poiein, ouch horOntes de, ei tas archas tis tautas thEsetai,
pOs estai ho mathEmatikos arithmos, para ton eidEtikon, ton
auton eidEtikon kai mathEmatikon epoiEsan arithmon tOi logOi,
epei ergOi ge anEirEtai ho mathEmatikos (idias gar kai ou
mathEmatikas hupotheseis legousin), a5-a11
c. [Plato] the first to propose that there are Forms and
numbers, separated the existence of Forms and mathematicals
with good reason, so all [these hypotheses] are partly right,
if not wholly right. The others admit as much by their
contradiction of each other. The reason is their assumptions
and principles are false. "Ill begun, ill done," as
Epicharmos said, ho de prOtos themenos ta eidE einai kai
arithmous ta eidE kai ta mathEmatika einai eulogOs echOrisen,
a11-a18
d. but enough about numbers. More is hardly needed to pursuade
the [already] persuaded, and will hardly persuade the
unpersuaded, a18-a21
EFL, 9/20/97
MU (XIII) METAPHYSICS
Chapters ix, 1086a21 to end
Platonic Ideas
1. having stated what those who speak only of sensible ousiai had
to say about the first principles and elements, some in the
Physics (others are not a part of our present discussion), and
what they say who say that there are other ousiai beside the
sensible, we will look next into the latter. Since there are
some who say Ideas and numbers are such [beside the sensible],
and their elements are the elements and principles of beings,
let us look at them, and what and how they say it, 1086a21-a29
2. some only posit mathematical numbers [Speusippus], and will be
looked at later [in Nu], a29-a30
3. let us now contemplate the method and the problem of those who
espouse Ideas [Plato]. Because they make the Ideas universal,
and at the same time separate and of particulars, hama gar
katholou te poiousi tas ideas kai palin hOs chOristas kai tOn
kath' hekaston. That this cannot be, has been discussed before
[A, ix, etc.], a31-a35
4. the reason that the proponents of universal ousiai [make them
separate] is that they don't make them in sensibles, aition de
tou sunapsai tauta eis tauton tois legousi tas ousias katholou,
hoti tois aisthEtois ou tas autas epoioun. They hold that
sensible individuals change, and none of them remain, while
universals are outside these, para tauta, and are different.
This, as we said above [A, vi and M, iv], Socrates brought up in
[his search for] definitions, but he did not separate them from
particulars; he correctly thought that they were not separate.
This is clear from the results: there is no knowledge gotten
without the universal, but to separate them is the cause of the
difficulties befalling Ideas, a35-b7
5. if there are to be some ousiai beside changing sensibles, they
must be separate. Having no others, they set aside the said
universals, so that the universals and the particulars were
almost the same natures, hoi d' hOs anagkaion, eiper esontai
tines ousiai para tas aisthEtas kai hreousas, chOristas einai,
allas men ouk eichon tautas de tas katholou legomenas exethesan,
hOste sumbainein schedon tas autas phuseis einai tas katholou
kai tas kath' hekaston. This is one of the difficulties
mentioned above [a32-a34 here; A, ix, and in B, ii, 997b, the
famous "men and horses" passage], b7-b13
[We are far enough along that we may survey this book as a whole,
and see its organisation, revising our tentative outline at the
beginning (Introductory, July 19):
OUTLINE OF MU METAPHYSICS
Chapter i, Introductory: plan of inquiry
Chapter ii, Mathematicals: in sensibles or separate?
Chapter iii, Mathematicals: what kind of existence?
Chapter iv, Ideas: origin of Plato's doctrine & objections
Chapter v, Ideas: objections continued
Chapter vi, Numbers: review of possibilities
Chapter vii, Numbers, cont.: are they combinable or not?
Plato, Speusippus, Xenocrates and the Pythagoreans
Chapter viii, to 1083b23, Numbers, cont.: ad hominem arguments
Plato, Speusippus, Xenocrates and the Pythagoreans
Chapter viii, 1083b23, to end: Plato's ideal numbers
the One, the Great & the Small, the Indefinite Dyad
Chapter ix, to 1086a21, Ideal numbers and magnitudes
Plato, Speusippus, Xenocrates
Chapter ix, 1086a21 to end, Platonic Ideas
Chapter x, Platonic Ideas, cont.
[Xenocrates was not far off the mark when he related numbers and
Ideas. But he had no notion, how? For Aristotle numbers are not
separate. They are in things, although not the way the Pythagoreans
would have it. Then how? What is missing here is the awareness that
there are various meanings of "separate" (pollachOs legetai
chOristos), that there is a way that ideas and numbers can be "in"
things, and yet still separate from them]
EFL, 9/27/97
MU (XIII) METAPHYSICS
Chapter x
Ideas, finale
[Careful inspection of chapter x reveals a precise structure, and
an extraordinary example of the Greek dialectical mind at work.
First there is a detailed statement of a problem. Then there is the
solution of it.
[The statement of the problem, 1086b14-1087a10, takes the shape of
a "Chinese box" of dilemmas, that is a dilemma within a dilemma
within a dilemma within a dilemma!]
1. if there are Ideas, 1086b14, tois legousi,
a. if they are not separate ousiai, b16-b18, ei men gar,
b. if they are separate ousiai, how will one regard their
elements and principles? b19-b20, an de tis,
(1) if their elements are individual, b20-b21, ei men gar,
(2) if their elements are universal, b37, alla mEn eige,
(a) either ousiai are universal, b37, E,
(b) or non-ousia is prior to ousia, a1, E,
2. if there are not Ideas, 1086b15, tois mE legousin, and
1087a7-8, ei de mEthen . . . kai mEthen . . .
[or in the Greek text]
1. ho de kai tois legousi tas ideas echei tina aporian, 1086b14
a. ei men gar tis mE thesei tas ousias einai kechOrismenas, kai
ton tropon touton hOs legetai kath' hekasta tOn ontOn, b16-
b18
b. an de tis thEi tas ousias chOristas, pOs thesei ta stoicheia
kai tas archas autOn; b19-b20
(1) ei men gar kath' hekaston kai mE katholou, b20-b21, with
a long illustrative analogy in parenthesis, b22-b37,
based on the parallel meaning of stoicheia, the alphabet
(2) alla mEn eige katholou hai archai, b37
(a) E kai hai ek toutOn ousiai katholou, b37-1087a1
(b) <E> estai mE ousia proteron ousias, 1087a1-a7
2. kai tois mE legousin, 1086b15, and ei dE mEthen kOluei hOsper
epi tOn tEs phOnEs stoicheiOn [reference to the illustrative
analogy, b22-b37] polla einai ta alpha kai ta beta kai mEthen
einai para ta polla auto alpha kai auto beta, esontai heneka ge
toutou apeiroi hai homoiai sullabai, 1087a7-a10
[All the horns of these dilemmas eventuate in further dilemmas, or,
finally, unsatisfactory results:
1. Either
a. they do away with ousia as we [Platonists]
understand it, anairEsei tEn ousian hOs boulometha
legein, b18-b19, or
b. either
(1) there will be as many beings as elements, and the
elements will be unknowable, tosaut' estai ta onta
hosaper ta stoicheia, kai ouk epistEta ta
stoicheia, b21-b22, or
(2) either
(a) ousia is universal, b37-1087a1, or
(b) non-ousia is prior to ousia, a1-a7
2. or there will be an unlimited number of like
syllables (i.e. by analogy: substances), a7-a10
[The solution is given in dialectical distinctions commencing at
1087a10. First of all universals are distinguished from Platonic
Ideas] The universals are not separate substances [as the Ideas are
supposed to be] and it is these universals that are the object of
knowledge, to de tEn epistEmEn einai katholou pasan, hOste
anagkaion einai kai tas tOn ontOn archas katholou einai kai mE
ousias kekOrismenas, a10-a13. [Aristotle does not distinguish what
kind of separation he intends here, but in the context his meaning
is clear: it is the physical sort of separation that Ideas were
believed to have. If the universals have separability of another
kind, connected as they are with knowledge, none of this is clear
yet. All he says is that] this is so, and not so, echei men malist'
aporian tOn lechthentOn, ou mEn alla esti men hOs alEthes to
legomenon, esti d' hOs ouk alEthes, 1087a13.
[How do we know both universals and particulars, unless there are
two kinds of knowledge? Aristotle makes use here of his theory of
potentiality and actuality:] Knowledge of universals and of the
unlimited is a potentiality. Actuality is definite [knowledge] of
the definite thing, hE gar epistEmE, hOsper kai to epistasthai,
ditton, hOn to men dunamei to de energeiai. hE men oun dunamis hOs
hulE katholou ousa kai aoristos, tou katholou kai aoristou estin,
hE d' energeia hOrismenE kai hOrismenou, tode ti ousa toude tinos,
a15-a18
[Aristotle often equates form with ousia (substance). In this he
was mistaken, as evidenced by his own words: ousia is a sunolon, a
combination of matter and form. In the present context both matter
and form are potentialities; it is the combination that is the
actuality (Z, iii). Now we all know that elsewhere he says
something quite different from this, prefering as he does the
"reality" of form. And we know as well that such statements gave a
toe-hold for the "realist" interpretation of the medieval schools
and of modern neo-scholasticism. But evidence to the contrary is
right here in these last lines of Book Mu (1087a15-a25), hE men gar
epistEmE, hOsper kai to epistasthai, ditton, hOn to men dunamei to
de energeiai. hE men oun dunamis hOs hulE katholou ousa kai
aoristos tou katholou kai aoristou estin, hE d'enargeia hOrismenE
kai hOrismenou, tode ti ousa toude tinos . . . dElon hoti esti men
hOs hE epistEmE katholou, esti d' hOs ou. Knowledge is of two
kinds: (1) of universal as potential and unlimited, and related to
matter somehow as well as form (hOs hulE katholou ousa); (2) actual
knowledge is definite, and of the limited. It is particular, of a
particular. Knowledge is related there to (1) potentiality, matter,
form (the universal) and the unlimited, and (2) to the actuality
and the definite and the particular. This is a sunolon. The double
nature of knowledge is explicit. The same relation of (1) potential
and infinite and mental, and of (2) actual and finite and physical,
is to be found in the Physics, in Aristotle's solution of Zeno's
problems of motion.
[There is more than one way to read Aristotle, and it isn't just
the words that are ambiguous, but the very subject; and not just
metaphysics, but life as a whole is ambiguous!]
EFL, 10/4/97