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Xenocrates (Stanford Encyclopedia of Philosophy)

Most of what we can reconstruct about Xenocrates pertains to his
metaphysics. We do this largely by identifying views of his that
appear in Aristotle’s criticisms of the metaphysical views of his
predecessors and contemporaries, and chaining together with these
other texts that can plausibly be taken as dealing with his views. But
there are a few sources other than Aristotle.

One of them is Proclus, who says, commenting on the
Parmenides (Cousin 1864, 888.11–19, 36–38; fr.
30H, 94IP):

But to the ideas both belonged: both to be intelligible and {to be}
unchanging in substance, ‘mounted on a holy pedestal’,
that is, on pure mind, being such as to complete the things that are
in potentiality and being causes that give them their form; whence
{Plato} going up to these principles makes the whole of coming-to-be
dependent on them, just as Xenocrates says, positing that the idea is
a paradigmatic cause of the {things} that are always constituted
according to nature … . Xenocrates, then, wrote down this
definition of the idea as in conformity with the founder, positing it
as a separate and divine cause; …

‘The founder’ is Plato. The phrase ‘mounted on a
holy pedestal’ comes from Plato, Phaedrus 254b7, where
the soul has been likened to a charioteer who sees the Forms of the
beautiful and temperance so mounted. Some of the phrasing is no doubt
neoplatonist rather than Xenocratean, but the formulation, ‘the
idea is a paradigmatic cause’, seems to be, as Proclus says,
Xenocrates’ attempt to capture Plato’s intent: see here Plato,
Parmenides 132d.

There is disagreement over the rest of the formulation Proclus
attributes to Xenocrates: in speaking of ‘the things that are
always constituted according to nature’, did Xenocrates intend
to rule out forms for individuals, which are transitory, and for
artefacts, which are not constituted according to nature? This is the
way Proclus goes on to interpret Xenocrates, and it is hard to see how
to get around that, although attempts have been made (see Cherniss
1944 [1962], 256). But there is indirect confirmation of Proclus’
interpretation, at least where artefacts are concerned, from Clement
of Alexandria, who tells us (in Stromateis II 5) that
Xenocrates claimed that knowledge of the intelligible substance is
theoretical as opposed to practical ‘judgment’; at that
rate, carpenters are not contemplating forms when they make beds and
shuttles, despite what is said by Plato in Republic X 596b
and Cratylus 389a–b, and (if it is by Plato) Letter
342d. But it should be noted that the rejection of forms for
artefacts is in agreement with what Aristotle has to say about Plato
and Platonists in Metaphysics I 9. 991b6–7, XII 3.
1070a13–19, and in the fragmentary remains of On Ideas
in Alexander (see esp. Hayduck 1891, 79.23–24, 80.6). Likewise
the rejection of forms for individuals squares with Aristotle’s attack
on the ‘argument from thinking’ (Metaphysics I 9.
990b14–15 = XIII 4. 1079a10–11, supplemented by Alexander,
Hayduck 1891, 81.25–82.7): if every object of thought is a form,
then there are forms also “for the perishables” (990b14 =
1079b10) or “for the particulars and perishables, such as
Socrates, Plato” (Alexander, Hayduck 1891, 82.2–3).

The version of the Theory of Forms associated with Xenocrates is that
which Aristotle ascribes to the later Plato (see Metaphysics
XIII 4. 1078b10–12 for the qualification ‘later’),
in which the Forms are ‘generated’ and are, in the first
instance, numbers. Xenocrates operated, in parallel with Speusippus
and Plato (as Aristotle reports Plato), with a scheme in which two
principles–the One and something called any or all of ‘the
everflowing’, ‘plurality’ (Aëtius i 3. 23), or
‘the Indefinite Dyad’ (Theophrastus, Metaphysics
vi)–generate these form-numbers, and then, in turn, lines, planes,
solids, and perceptible things.

The talk of generation Xenocrates reinterpreted as a mere pedagogical
device; we hear about this technique from Aristotle, De caelo
I 10. 279b32–280a2, and Simplicius’ commentary ad loc. (Heiberg
1893, 303.33–34) names Xenocrates in this connection, as does
Plutarch (De animae procreatione in Timaeo 3. 1013a–b,
Cherniss 1976, 168–171). Here it is a device for interpreting
the creation story in the Timaeus; that Xenocrates also
applied it to the generation of the formal numbers we learn from
Aristotle, Metaphysics XIV 4. 1091a28–29 and the
commentary on that passage in pseudo-Alexander (Hayduck 1891,

In trying to understand what Aristotle tells us about formal numbers,
it is necessary to bear in mind the fundamental distinction he draws
between formal numbers and mathematical numbers: both are, according
to Aristotle, composed of units, but formal numbers are composed of
very strange units, such that those in one formal number cannot be
combined with those in any other. The units of which mathematical
numbers are composed can be added and subtracted freely. (See here
Metaphysics XIII 6. 1080a15–b4.) And furthermore there
is only one formal number for each of the numbers 2, 3, 4, etc., where
there are indefinitely many instances of each among the mathematical
numbers. (See here Metaphysics I 6. 987b14–18.) The
mathematical numbers are the ones mathematicians work with, e.g. in
performing arithmetical operations, and that is presumably why they
are called ‘mathematical’. There is a corresponding
division between types of geometrical figures, but we hear too little
about this; most of what follows will be concerned with numbers.

The position that there are both formal numbers and mathematical
numbers Aristotle ascribes to Plato. Speusippus rejects the formal
numbers (and the entire theory of forms along with them; see the entry
The position Aristotle ascribes to Xenocrates is a bit more

In Metaphysics VII 2, Aristotle tells us, in
1028b19–21, that Plato accepted three sorts of entities: forms,
mathematicals, and perceptibles; in this context that means formal
numbers, mathematical numbers, and perceptibles. He then, in
b21–24, talks about Speusippus’ views (see the entry on
In both cases he gives us the names. Then, in b24–27 he says

But some say that the forms and the numbers have the same nature,
while the others, lines and planes, come next, {and so on} down to the
substance of the heavens and to the perceptibles.

Asclepius’ commentary on this passage (Hayduck 1888, 379.17–22)
tells us that it is dealing with Xenocrates.

The core of Xenocrates’ view is that “the forms and the numbers
have the same nature:” that is, the formal numbers and the
mathematical numbers have the same nature. A series of half a dozen
passages in the Metaphysics can, in consequence of this
identification, be associated with Xenocrates (see XII 1.
1069a30–b2, XIII 1. 1076a20, 6. 1080b21–30, 8.
1083b1–8, 9. 1086a5–11, XIV 3. 1090b13–1091a5). From
these passages it appears that he is saying that the distinction
between formal and mathematical numbers (as well as the corresponding
distinction among geometrical objects) is unnecessary; he does this by
assimilating mathematical numbers to form-numbers and telling us that
mathematics can be done entirely with formal numbers. In other words,
since he thinks that mathematics can be done with formal numbers, he
feels it acceptable to call formal numbers mathematical numbers.

1086a5–9 makes it sound as if some part of Xenocrates’ case for
his position was based on the consideration that all that can be based
on the two ultimate principles, the One and the Indefinite Dyad, is
the series of formal numbers. Without some further comment, it is hard
to see much of an argument here.

And the resulting position is possibly quite unstable: Aristotle
certainly thinks so. For Plato and Speusippus, the addition of 2 and 3
is a matter of putting together a group of units that is a
mathematical 2 with a disjoint group of units that is a mathematical 3
(that numbers are such collections of units is a view that can still
be found later, perhaps most importantly, given his influence, in
Euclid, Elements VII def. 2). Aristotle, too, understood
addition in this way, although with a completely different take on the
underlying ontology. We do not know how Xenocrates understood
addition: perhaps as a sort of map telling you that if you are on the
unique formal number 2 and you want to add the unique formal number 3
to it, you cannot, strictly speaking, do that, but taking three steps
on in the series will get you to the unique formal number 5, and that
is what ‘2 + 3 = 5’ really means. There is, as far as I
know, no evidence to support this conjecture, but it has the advantage
of explaining Aristotle’s complaint, voiced more than once in the
passages cited (see 1080b28–30, 1083b4–6,
1086a9–11), that Xenocrates actually makes doing mathematics
impossible: he ends up destroying mathematical number, and if the
above guess should be correct about Xenocrates’ handling of addition,
it is readily seen how someone of Aristotle’s persuasion might think
that Xenocrates is not so much explaining addition as explaining it

Aristotle complains in 1080b28–30 that on Xenocrates’ view it is
not so that every two units make up a pair, and also that on his view
not every geometrical magnitude divides into smaller magnitudes. This
has to do with Xenocrates’ acceptance of the idea that there are
indivisible lines; this idea Aristotle ascribes to Plato in
Metaphysics I 9. 992a20–22, and Alexander’s commentary
on that passage adds the name Xenocrates, in a way that suggests that
Xenocrates’ acceptance of indivisible magnitudes was even better known
than Plato’s (Hayduck 1891, 120.6–7; see also Simplicius on
De caelo, Heiberg 1894, 563.21–22 and many other
passages in the commentators in which this ascription occurs: frs.
41–49H, 123–147IP). As Proclus understood Xenocrates’
position, it applied to the Form of the line rather than to
geometrical or physical magnitudes (see Diehl 1904,
245.30–246.4), but this is very much a minority view: Porphyry
is quoted by Simplicius in the latter’s commentary on the
Physics (Diels 1882, 140.9–13) as saying that,
according to Xenocrates, what is:

… is not divisible ad infinitum, but {division} stops
at certain indivisibles {atoma}. But these are not
indivisible as partless and least {magnitudes}, but while they are
cuttable with respect to quantity and matter and have parts, in form
they are indivisible and primary; he supposed that there were certain
primary indivisible lines and primary planes and solids composed out
of them.

This suggests that Xenocrates might have thought he could do with the
notion of a line what Aristotle was prepared to do with
notions such as man. Aristotle is prepared to say that a man
is indivisible, and so a suitable unit for the arithmetician’s
contemplation, in the sense that if you divide a man into two parts
what you get is not two men (see Metaphysics XIII 3.
1078a23–26). Xenocrates may have thought the notion of a line
could be made to work in the same way: beyond a certain point,
divisions will no longer yield lines. It is difficult to think how he
could have made this plausible; once again, one can see why Aristotle
might have regarded Xenocrates’ position as unmathematical.

Xenocrates’ espousal of indivisible magnitudes has led to the
conjecture that the pseudo-Aristotelian treatise On Indivisible
is at least in part an attack on him, and that the
arguments recounted in its first chapter in favor of the claim that
there are indivisible lines, which are rebutted in the sequel, might
come from Xenocrates. Unfortunately, those arguments are quite
obscure, and the text itself is not in very good shape (an admirably
concise summary of the first four of these arguments may be found in
Furley 1967, 105). But some of the arguments owe a lot to Zeno of
Elea: that Xenocrates was influenced by Zeno is only what one would
expect, and is confirmed elsewhere (see esp. the passage from Porphyry
cited in part above, apud Simplicius on the Physics,
Diels 1882, 140.6–18).

In the passage of Metaphysics VII 2 quoted above, after we
get the identification of formal and mathematical numbers, with the
formal numbers actually carrying the weight, there is a brief
description of the rest of the universe: “while the others,
lines and planes, come next, {and so on} down to the substance of the
heavens and to the perceptibles.” It appears that Xenocrates
pictured the universe as unfolding in the sequence: (1) forms =
numbers; (2) lines; (3) planes; (4) solids; (5) solids in motion, i.e.
astronomical bodies; …; (n) ordinary perceptible things. Solid
shapes aren’t mentioned in this sentence, but they were earlier, in
1028b17–18, and they are a standard stage in this sequence.

There is here an implicit contrast between Xenocrates and Speusippus,
whose universe was to Aristotle discontinuous or disjointed:
Xenocrates’ universe is at least a more orderly one (see the entry on
And something like this rather faint praise is echoed in
Theophrastus’ Metaphysics. Theophrastus complains that
Pythagoreans and Platonists fail to give us a full story about the
construction of the universe: they just go so far and stop
(6a15–b6). Then he says (6b6–9):

and none of the others {does any different} except Xenocrates: for he
places all things somehow around the world-order, alike perceptibles
and intelligibles, i.e. mathematicals, and again even the divine

So we have it from Aristotle that Xenocrates’ universe showed
continuity, and from Theophrastus that it covered everything. Of
course, we do not know how.

Exactly what Theophrastus means by ‘the divine things’ is
hard to say. There are two candidates: the objects of astronomical
studies, which would connect with Aristotle’s account, or those of
theological studies, about which Xenocrates also had much to say.
These are not exclusive candidates. A passage in Aëtius (Diels
1879, 304b1–14) tells us that Xenocrates took the ‘unit
and the dyad’ to be gods, the first male and the second female,
and also thought of the heavenly bodies as gods; in addition he
supposed there were sublunary daimones. These latter were
beings intermediary between gods and men, also mentioned in Plato,
Symposium 202d–203a.

We hear more about the gods, daimones, and men from Plutarch,
who tells us (De defectu oraculorum 416c–d, Babbitt
1936, 386–387) that Xenocrates associated them with types of
triangle: gods with equilateral ones, daimones with isosceles
ones, and men with scalene triangles: as isosceles triangles are
intermediate between equilateral ones and scalene ones, so
daimones are intermediate between gods and men. According to
Plutarch (417b, De Iside et Osiride 360d–f: in Babbitt
1936, 390–391 and 58–61, respectively.), Xenocrates’
daimones come in good and bad varieties: they may have had
something to do with the explanation of the existence of evil.

In addition, there are isolated snatches of other views of Xenocrates
that might fall under the heading ‘metaphysics’.

Simplicius, in his commentary on Aristotle’s Categories
(Kalbfleisch 1907, 63.21–24) tells us that Xenocrates objected
to Aristotle’s list of ten categories as too long: he thought all that
was needed was the distinction, visible in Plato, between things that
are ‘by virtue of themselves’ and things that are
‘relative to something’ (see, e.g., Sophist 255c,
and Dancy 1999). The standard examples help clarify this: the terms
man and horse are of the first sort, whereas
large, relative to small, good relative to
bad, etc., are of the latter type.

There was, it appears from a text also preserved by Simplicius (in his
commentary on the Physics, Diels 1882, 247.30–248.20,
from Hermodorus, an early associate of Plato’s), an internal
connection between these ‘old academic categories’ and the
One and the Indefinite Dyad. The One was the heading over the category
of things that are ‘by virtue of themselves’: such things
are standalone entities, one thing. The Indefinite Dyad was
the heading over the category of relatives: such a term refers to an
indefinite continuum pointing in two directions. All this is referred
to Plato, not Xenocrates, but if Xenocrates accepted Plato’s later
theory, or at least some of it, he presumably accepted this as well,
and saw in Aristotle’s proliferation of categories a threat to the
basic two principles he shared with Plato.

A text preserved in Arabic (see Pines 1961) has Alexander of
Aphrodisias criticizing Xenocrates for saying that the (less general)
species is prior to the (more general) genus because the latter, being
an element in the definitions of the former, is a part of them (and
wholes are subsequent to parts).

A long passage in Themistius’ commentary on Aristotle’s De
(Heinze 1899, 11.18–12.33) seems to stem from
Xenocrates’ On Nature (in 11.37–12.1 Themistius says
“It is possible to gather all these {things} from the On
of Xenocrates”). This is a discussion of a story
about the composition of the soul from the formal numbers 1, 2, 3, and
4 (although 1 was not normally considered a number), mentioned in
De anima 408b18–27. The motivation for this account of
the soul, in both Aristotle and Themistius, is the explanation of how
we can know things about the universe: the universe is derivative from
those numbers, and so, if the soul is similarly derivative, the soul
can know things under the principle that like things are known by
like. This cognitive sort of account is contrasted with another
motivic type of account, that takes as the primary thing to be
explained the fact that the soul can initiate motion.

However, it is quite clear that, even if the story about the reduction
of the soul to numbers stems from Xenocrates’ On Nature, the
numerical reduction was supposed by Themistius not to be Xenocrates’,
but (perhaps) Plato’s. Aristotle and Themistius both give separate
mention to the account of the soul that is traditionally ascribed to
Xenocrates: that it is a self-moving number (De anima
408b32–33; Themistius in 12.30–33; the ascription to
Xenocrates is supported by a large number of texts gathered as frs.
60H, 165–187IP: e.g., Alexander of Aphrodisias on Aristotle’s
Topics, Wallies 1891, 162.17). Both Aristotle and Themistius
characterize this account as an attempt to combine the cognitive and
the motivic ways of thinking about the soul; as Themistius puts it

And there were others who wove the two together into the explanation
of the soul, both moving and knowing, such as the one who asserted the
soul {to be} a number that moves itself, pointing by
‘number’ to the capacity for knowing and by ‘moving
itself’ to that for moving.

Themistius does not here tell us that this is Xenocrates’ account, but
he does later on (see esp. 32.19–34, which refers expressly to
Xenocrates’ On Nature book 5).

As already noted, this heading comes under ‘logic’ in
Sextus Empiricus. No one reports anything for Xenocrates about what we
would think of as pure logic; Sextus (Adversus mathematicos
vii 147–149) gives us a scrap about epistemology. Xenocrates is
supposed to have divided the substances or entities into three groups:
perceptible, intelligible, and believable (also referred to as
‘composite’ and ‘mixed’). The intelligible
ones were objects of knowledge, which Xenocrates apparently spoke of
as ‘epistemonic logos’ or ‘knowing account’,
and were ‘located’ outside the heavens. The perceptible
ones were objects of perception, which was capable of attaining truth
about them but nothing that counted as knowledge; they were within the
heavens. The composite ones were the heavenly objects themselves, and
objects of belief, which is sometimes true and sometimes false.

This scheme descends from that in Plato, Republic V ad
., where the objects of knowledge were differentiated from
those of belief, and from Republic VI ad fin., where
that division is portrayed on a divided line. In the latter passage,
Plato seems actually to have four divisions of types of cognition and
their objects, but this is notoriously difficult (see Burnyeat 1987),
and Xenocrates appears to have rethought it. His tripartite division
of objects looks like that in Aristotle, Metaphysics XII

The phrase ‘epistemonic logos’ is one Sextus (145) also
assigns to Speusippus; it also recalls discussions in Aristotle (e.g.
Metaphysics VII 15) and the end of Plato’s
Theaetetus. An ‘epistemonic logos’ is the sort of
account that carries knowledge with it.

The intelligible domain must have included the formal numbers dealt
with above, which was also, as mentioned, the domain of mathematics,
while the special place for the heavens accords with the fact that one
of the items in D.L.’s bibliography is “On Astronomy, 6

This picture seems to square with Aristotle’s exempting Xenocrates
from the charge, leveled against Speusippus, of producing a
discontinuous universe, and with Theophrastus’ comment to the effect
that Xenocrates’ universe encompassed everything.

Here again we encounter Xenocrates the theologian: Sextus tells us
(149) that Xenocrates associated the three fates with his three groups
of substances: Atropos with the intelligible ones, Clotho with the
perceptible ones, and Lachesis with the believable ones. This sounds a
Xenocratean touch: it connects with the interpretation of Plato (see
Republic X 620d–e) and takes mythology very

Here we are very much in the dark: we have only disconnected snippets
to consider.

Aristotle names Xenocrates in the Topics in connection with
two ethical views: at II 6. 112a37–38 he ascribes to him the
view that a happy man is one with a good soul, along with (perhaps)
the claim that one’s soul is one’s daimon, whatever that
means; at VII 1. 152a7–9 he ascribes to him an argument to the
effect that the good life and the happy life are the same, employing
as premises the claims that the good life and the happy life are both
the most choosable (a little later, in 152a26–30, Aristotle
objects to this argument).

Plutarch claims (De communibus notitiis adversus Stoicos
1069e–f) that Xenocrates made happiness turn on living in
accordance with nature; since this may derive from Antiochus of
Ascalon, whose project it was to assimilate the Academy to Stoicism,
it is suspect. Clement (Stromateis II 22) ascribes to him the
view that happiness is the possession of one’s own excellence in the
soul. This view bears a family resemblance to Aristotle’s (NE
I 7. 1098a16–17, 9. 1099b26). The negative emphasis in
Xenocrates’ evaluation of philosophical activity as “stopping
the disturbance of the affairs of life” ([Galen], Historia
8, in Diels 1879 605.7–8) sounds like a step
in the direction of the Hellenistic goal of undisturbedness.

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