Stanford University

Speusippus (Stanford Encyclopedia of Philosophy)

Aristotle gives us an account of Plato’s metaphysical views that goes
considerably beyond what we find in Plato’s dialogues. In fact,
Aristotle gives us what he says are Plato’s views and also those of
others. From this we learn what little we can about Speusippus and his
contemporary, Xenocrates.

According to Aristotle, all three of Plato, Speusippus, and Xenocrates
constructed their worlds operating with two principles
(archai): the One and something opposed to it. The latter
appears under different names, and it looks as if the different names
go with different ones among the three: ‘the Indefinite
Dyad’ or ‘the large and (the) small’ seem to be the
expressions favored by Plato, ‘plurality’ seems to be
associated with Speusippus, and ‘the unequal’ with
Xenocrates. And for Plato, the One was identified with the Good. These
philosophers employed these two principles, according to Aristotle, to
generate derivative entities (substances, beings) of other types:
Forms, numbers, and so on. This generation was laid out as if it were
a process in time, but the generation-story was to be understood
instead as one understands a mathematical construction: the
construction takes place in time, but what is constructed is an
eternal structure. And in this eternal structure the principles were
not temporally first, but first in relation to the dependence on them
of the things constructed.

These principles are primarily employed by these three thinkers in
connection with numbers. But there are two ranges of number
operating in Aristotle’s discussions (see Metaphysics I 6,
XIII 8–9). There are what he refers to as formal
numbers, one for each numeral; these are the (Platonic) Forms for
numbers. But since there is only one for each numeral, and a
mathematician has to be able to add two and two to get four, there is
another range of numbers required: the mathematical numbers,
indefinitely many for each numeral n, each of the
indefinitely many a collection of n units. As Aristotle has
it, Plato accepted both ranges, and Speusippus only mathematical
numbers; Speusippus, then, rejects Plato’s Theory of Forms and
specifically his belief in formal numbers.

There are only two passages in which Aristotle mentions Speusippus by
name in dealing with his metaphysical views: Metaphysics VII
2. 1028b21 and XII 7. 1072b31. Neither passage gives us anything we
could call a ‘fragment’: there is no indication that any
actual quoting is taking place. But it is from these meager hints that
reconstruction of Speusippus’ metaphysics must start.

The method followed in the reconstruction of Speusippus’ views is a
matter of chaining: we start from our two anchor texts, and look for
other passages in which the views ascribed to him in them are under
discussion. Those passages will sometimes bring in new views; we then
ascribe those views to Speusippus, and go looking for passages in
which those views seem to be alluded to. No one needs to be
told how tenuous such chaining is: each link is weak, and compounding
probabilities would tell us that a chain of this type is actually
weaker than its weakest link. But chaining in that way is all we can
do. Fortunately, it results in a fairly coherent picture.

In the first of our two anchor texts, Aristotle, discussing the ways
in which people have answered the question ‘what are the
substances?’, ascribes to Plato the view that there are not only
perceptible substances but eternal ones of two types: forms and
mathematical objects. He then says that Speusippus thought there were
even more types of substance; he “started from the One”
and adopted principles (archai) for each of his types of
substance: “one for numbers, another for magnitudes, then for
soul”. So we have at least four layers of beings: numbers,
magnitudes, souls, and perceptible beings; the One is Speusippus’
starting-point, but he has different principles for each level of
being.

In the other of our anchor texts, Aristotle, having just sketched a
proof for the existence of an unmoved mover, which he refers to as a
principle (arche) and as ‘god’, says something
about Speusippus’ views about principles: that he (and the
Pythagoreans) suppose that a principle is not characterizable as
‘beautiful’ or ‘good’, going on the analogy of
plants and animals, where beauty and goodness emerge in the end state
rather than the beginning. So, although we have principles at each
level, those principles are not themselves complete entities at that
level.

Then we begin chaining.

In Metaphysics XIV 4–5, Aristotle discusses various
Platonistic views about the relationships between the first principles
and goodness; XIV 4. 1091a34–36 refers to contemporaries of
Aristotle’s who deny that the good is among the principles but suppose
it comes in later in the development; the second of our two anchor
texts warrants the guess that Speusippus is among them. And then some
other things about the passage in XIV fall into place: Speusippus’
main way of referring to the principle opposed to the One was
‘plurality’, and, more importantly, he denied a theory
about causality that was part of Plato’s Theory of Forms as we find
that Theory in, say, the Phaedo: a theory to the effect that
the possession of a feature F by a perceptible object was due
to its participation in something supremely F that transmits
its being F to the perceptible object; as opposed to this
‘Transmission Theory of Causality’, Speusippus adopted
what we might call a Principle of Alien Causality: the first cause of
Fs is not itself F. This Principle is going to be
difficult to formulate, and the word ‘first’ in this
formulation needs emphasis: Speusippus surely agrees with Aristotle
that the cause of a human being is one or more other human beings, so
presumably the Principle has to be restricted in some way to
first causes or principles in some more robust
sense. Unfortunately, the texts (even the Iamblichean passage cited
below) give us no help here.

Metaphysics XIV 5. 1092a11–17 returns to the analogy of
animals and plants which XII 7 told us Speusippus used; Aristotle
says:

And one does not rightly understand either if one compares the
principles of the whole to that of animals and plants, because from
indefinite, incomplete things the more complete always arise, which is
why he says it holds thus for the first things, so that the one itself
is not even a being.

Sticking by our method of chaining, we add to Speusippus’ package the
claim that the One is not a being (1092a14–15). And this is an
instantiation of the Principle of Alien Causality: the first cause of
beings is not itself a being.

Aristotle goes on to speak of “those who say that the beings are
derived from elements and that of the beings the numbers are the
first” (1092a21–22). And again we may suppose that
Speusippus is among them: he would say that the elements, not being so
derived, are not beings, and that numbers, derived from those
elements, are the first beings. Here we may go back to the first of
our anchor texts (Metaphysics VII 2. 1028b21) for some
confirmation.

There may be additional confirmation in a more remote source. In a
Latin translation of part of Proclus’ commentary on Plato’s
Parmenides (the original Greek for this part of the
commentary has not survived), Proclus tells us that “the one is
prior and a cause of being, therefore it itself according to its
existence is not a being, as supporting being; nor does it partake of
being”, and that Speusippus also subscribes to this view,
although Speusippus states it “as if reporting the sentiments of
the ancients.” Unfortunately, if this does go back to
Speusippus, it may have been contaminated by neoplatonic elements, for
Proclus has Speusippus saying that the ancients thought “the one
better than being and that from which the being derives”, and
this should not be Speusippus, given his rejection of the claim that
the One is good (see above). But Speusippus’ rejection of this claim
would have been so bizarre to a neoplatonist that perhaps Proclus
simply managed to read past it. Perhaps what Proclus (or his source)
read is that the One is ‘above’ being and that from which
being derives, and, given his neoplatonic mindset, unhesitatingly
thought ‘better than’.

It has seemed to many scholars that Speusippus’ denial that the One is
a being gains plausibility from Plato’s Republic: VI
509b2–10 has often been read as placing the Form of the Good
(which, according to the tradition stemming from Aristotle, Plato
identifies with the One) ‘beyond being’ and so making the
top of Plato’s ontological ladder not a being, either. But this is
inconsistent with other passages in the Republic in which the
Form of the Good is plainly said to be a being (e.g., VII 518cd,
526e), and the phrase often translated ‘beyond being’ in
509b can be read simply as meaning ‘on the far edge of
being’. This does not mean that there is nothing pertinent in
Plato; there is, in the later dialogues, especially the
Parmenides (see below). But Republic VI
509b2–10 cannot be cited in support of Speusippus’ view.

If we accept this much, then, by further chaining, it looks as if we
have a quotation or at least paraphrase of Speusippus’ views
concerning the One, plurality, the numbers, and even geometrical
objects in a chapter of the De communi mathematica scientia
(On Mathematical Science in General, hereafter DCMS)
of  Iamblichus (3rd century C.E.). Iamblichus was a student of
Porphyry, who was in turn a student of Plotinus: this is the beginning
of Neoplatonism, and there are many resemblances as well as
differences between Neoplatonism and Speusippus’ views. But a case has
been made (Merlan 1968) for the idea that chapter iv of DCMS
contains paraphrase if not quotation from Speusippus.

Iamblichus is known for something that we should call
‘plagiarism’: he often quotes without attribution. This
cannot have seemed in any way wrong to him: one of his quotations is a
longish stretch from Plato’s Republic, and he is not, of
course, claiming that for his own. Presumably it is an act of homage:
Iamblichus is thinking that he can certainly not do better for what he
wants to say than Plato, so he, with reverence but not attribution,
quotes. The hope is that he is doing this in iv. There we read:

For the mathematical numbers one must posit two things, the first and
highest principles, the One (which indeed one ought not yet even call
a being, because of the fact that it is simple and because of the fact
that it is a principle for the things that are, while the principle is
not yet such as are the things of which it is a principle), and again
another principle, that of plurality, which can in its own right
provide division as well, and for this reason we may, by way of making
a comparison appropriate to its capacity, assert it to be like some
moist and completely pliable matter; from which, the one and the
principle of plurality, there is produced the first kind, of numbers,
from both of these when combined with the help of some persuadable
necessity.

A bit later in the chapter, Iamblichus or his source says:

But it is fit to call the one neither beautiful nor good, because of
the fact that it is above the beautiful and the good; for it is when
nature proceeds farther away from the things in the beginning that,
first, the beautiful appears, and, second, when the elements have an
even longer distance, the good.

Also in DCMS iv, we find:

But the elements from which the numbers are produced do not yet belong
there as either beautiful or good; but from the combination of the one
and the matter that is cause of plurality number exists, and first in
these that which is and beauty appear, while next from the elements of
lines geometrical substance appears, in which in the same way there is
that which is and the beautiful, but in which there is nothing either
ugly or bad; but at the extreme in the fourth and fifth levels which
are combined from the last elements badness comes-to-be, not directly,
but from inadvertence and failing to master something of that which
accords with nature.

The picture here is not Iamblichus’ own; rather, it fits with, and
expands on, what we so far have of Speusippus. There are four levels
of beings, and above them on a level of its own the One, which is not
a being.

And here, in the first quoted passage, we are given two arguments to
the effect that the One is not a being.

The first is an argument from the assumed simplicity of the One. This
corresponds to things we can find in Plato. In what is frequently
referred to as the ‘first hypothesis’ of Plato’s
Parmenides (137c-142a), Parmenides begins (137c) by laying it
down that the One cannot be many; he then argues, first (137cd), that
if the One is, it must be without parts, since otherwise it would be
many. Subsequent argumentation makes this a proscription against all
attempts to predicate anything positive of the One: any such predicate
would pluralize the One, make it consist of parts. Ultimately that
will include even the predicate ‘being’: it is supposed to
follow that the One “in no way is” (141e9–10), that
it is not; that is to say, that it is not a being (e11). It is
difficult to believe that Plato himself would have bought this
argument and its conclusion. He did buy into some of the train of
thought: see Sophist 244b-245e. But both Plato (even in the
Sophist) and Aristotle seem prepared to infer from a
predication of the form ‘S is P’ a
conclusion of the form ‘S is’, whereas in the first
hypothesis of the Parmenides and in DCMS iv we have
an argument that goes from a predication of the form ‘the One is
F’ to the conclusion ‘the One is not’. This
is not the place to attempt to sort out the relationship between the
Platonic passages and Speusippus, but the interesting suggestion has
been made (Graeser 1997, 1999; see also Halfwassen 1993) that Plato is
responding to Speusippus in the Parmenides. At any rate, the
Argument from Simplicity for the One’s not-being goes back to the
Academy.

The second argument for the not-being of the One depends explicitly on
the Principle of Alien Causality: “the principle is not yet such
as are the things of which it is a principle”; hence the
principle of beings cannot itself be a being.

After the One comes the second level, the first beings, numbers, and
then magnitudes or geometrical shapes: here we first encounter beauty
and goodness. And finally the fourth and fifth levels are going to be
those of souls and perceptible objects, and it is here that ugliness
and badness appear.

The level of numbers comes about when the second principle, plurality,
kicks in. One of the more confusing things about this level is that
Speusippus seems to have supposed that among the numbers was the
number 1: in this he was apparently ahead of his time, for the
generally accepted view was that 1 is not a number (for this see
Aristotle, Metaphysics, book X, Euclid, Elements vii
def. 2, proof of prop. 1, et passim). That Speusippus accepted the
number 1 follows from the mathematics of the one extensive fragment of
his work that we have: a quotation preserved in pseudo-Iamblichus,
Theologumena arithmeticae (see below). But if this is so, it
is essential to recall that we are dealing with Aristotle’s
mathematical’ numbers, where there is a
plurality for each number. So there is a plurality of numbers 1 (so
that we can add 1 and 1 to get 2, etc.). Then the One is not the
number 1; the latter comes about when you get a plurality of units,
any of which counts as a 1, any couple of which counts as a 2, and so
on. That the One is not the number 1 is only what we should expect,
given the Principle of Alien Causality: the principle for numbers is
not itself a number.

Aristotle more than once complains of Speusippus’
‘episodic’ universe: apparently the train of causality did
not start with the One and go down through the various levels, but
started anew at each level. This is explicit in Metaphysics
VII 2, cited above: Speusippus had new principles for each level of
beings. We need not suppose that the One is different at each level of
being in Speusippus’ universe; the text in DCMS would have it
that the One is the same principle throughout, but because of
differences in what it refers to as the ‘matter’ (see
quotation above), the One is realized in different ways at each level:
as the number 1, or the unit, in plurality at the level of numbers, as
the point, in what DCMS calls ‘place’ or, we
might translate, ‘locus’, at the level of geometrical
figures. The net effect is that what operates at the beginning of the
arithmetical level is the realization of the One in plurality, the
unit, which, as it were, represents the One at that level, and what
operates at the beginning of the geometrical level is the realization
of the One in locus, the point, which similarly represents the One at
that level.

We may have a fair amount of information about what Speusippus thought
about numbers, preserved in pseudo-Iamblichus’ Theologumena
arithmeticae
(already mentioned). This book is a compilation from
various authors: Anatolius, Nicomachus of Gerasa, possibly Iamblichus
himself, possibly authors unknown. It discusses various properties of
each of the first ten numbers. When it gets to the number 10, the
‘decad’, after a section that looks as if it descends from
Nicomachus, it begins to speak of Speusippus and a book of his which
it says was entitled ‘On Pythagorean Numbers’ and
was compiled from various Pythagorean writings, especially those of
Philolaus (82.10–15: that the book had that title and that
Philolaus or any other Pythagorean was a source for Speusippus are
both problematic claims). After what purports to be a description of
the book (82.15–83.5), it introduces a long quotation from
Speusippus himself concerning the number 10.

The description preceding the quotation begins (82. 20) with a passage
on the first half of Speusippus’ book, telling us that it talked about
numbers of various types and the five so-called ‘Platonic
solids’ (the tetrahedron or pyramid, the cube, the octahedron,
the dodecahedron, and the icosahedron, assigned in the Platonic
tradition starting with Timaeus 54d-56b to the elements of
which the universe was composed and to the universe itself or to the
fifth element, ether). This is relatively unproblematic.

The second half of the book, the author tells us, was about the number
10, the decad. Unfortunately, the description of that half of the book
that precedes the quotation contains some wording which it seems
impossible to ascribe to Speusippus. The author tells us that
Speusippus showed that the number 10 was “a sort of artistic
form for the cosmic accomplishments, obtaining in its own right (and
not because we use it nor as it happens), and lying before the god
that is maker of the all as an all-complete paradigm” (De
Falco/Klein 1975, p. 83.2–5, Waterfield 1988, p. 112). The
trouble is that we have already got Speusippus rejecting the Theory of
Forms, and in particular denying that aspect of the Theory in which
the forms are treated as ‘paradigms’: ideal cases that
transmit their properties to things that participate in them.

As it happens, this passage is too good to be true: it is almost
verbatim what Nicomachus himself wants (see Nicomachus’
Introductio arithmeticae I vi 1, Hoche 1866 p. 12.1–12,
D’Ooge et al. 1926 p. 189 e.g.). So it seems best not to ascribe it to
Speusippus.

The quotation itself is another matter. But it, too, offers a problem:
if Speusippus is telling us what Pythagoreans (perhaps especially
including Philolaus) thought, need he be subscribing to the views
himself? Speusippus and Xenocrates, among others, were part of a
revival of interest in Pythagoreanism within the Academy; it is often
thought that in the course of this revival Pythagorean views became
distorted through the attempt to assimilate them to the views of one
or another member of the Academy: the revival plainly involved
Academicians adopting the views they were ascribing to the
Pythagoreans. And as for our quotation from Speusippus, what text we
have certainly shows no disposition on his part to be critical of the
views he is allegedly reporting. So perhaps the best conclusion to
adopt, albeit very tentatively, is that he endorses these views.

In the quotation Speusippus tells us that the number 10 is
‘perfect’ or ‘complete’. The explanation in
Euclid vii Def. 23 (Heiberg/Stamatis 1970 p. 105; Def. 22 in Heath
1926 p. 278) for the phrase ‘perfect number’ is
‘number that is the sum of its proper divisors’. This
makes the first perfect numbers 6, 28, and 496: 10 is not a perfect
number in this sense. Rather, Speusippus appears to be speaking of 10
as ‘perfect’ or ‘complete’ in a way that fits
with his claim that completeness is not there in the beginning but
only comes on once the universe has proceeded a certain way, in
particular, far enough that we can speak of numbers. What he means can
be illustrated by a single example.

He tells us that a perfect number has to be even, since otherwise it
would contain more odd numbers than even ones: a perfect number must
have equally many odd and even numbers in it. On that basis he
pronounces 10 perfect.

This does not look like a mathematically (or philosophically)
interesting conception of perfect numbers, and the same is true of the
other points Speusippus makes in favor of the perfection of the number
10. But it does incorporate one feature of great interest: on the face
of it, if Speusippus thinks that within the number 10 there are just
as many odd as even numbers, he must be counting 1 as an odd number,
and therefore a number. (The alternative, which actually must finally
be considered seriously, is that he is not counting 2 as a number,
either, but this does not seem likely.) That makes him, as already
noted, an exception in his time and place.

When we try to consider Speusippus’ metaphysical views as they bear on
the other levels of his universe, unfortunately, we have virtually
nothing further to tell us what is going on: nothing that tells us
either what might be the representative of the One or what the
material principle might be at the level of souls, or at the level of
perceptible bodies. A passage in Stobaeus (Eclogues I 49.32,
Wachsmuth & Hense 1884 p. 364.2–7) that apparently descends
from Iamblichus’ De anima tells us that he defined the soul
“by the extended in every direction”, but this continues
to resist interpretation.

Aristotle says, in Posterior Analytics II 13.
97a6–11:

There is no need for one who is defining and dividing to know all the
things that are. And yet some say that it is impossible to know the
differences between something and each other thing while one does not
know each other thing, and without the differences one cannot know
each thing, for a thing is the same as that from which it does not
differ, and it is other than that from which it differs.

The ancient commentators (see esp. Anonymous In An. post.
584.17–585.2, where the ascription is credited to Eudemus) tell
us that this is Speusippus’ position (it has antecedents in Plato,
Philebus 18c and Theaetetus 208c-e).

It is a version of what is sometimes called ‘holism’:
knowing something involves knowing where it is located among
everything else. In this particular context, knowing a thing appears
to be a matter of knowing its definition, and its definition is
something arrived at by the Platonic method of division (there is room
for controversy here: see Falcon 2000). An example suggested by what
Aristotle says elsewhere in this same chapter (96a24-b1, much
simplified) might be an attempt to define the number 3 as follows:

(Dtriad) a triad =df a number, odd, prime

We reach this definition by successive divisions:

a diagram of the form (number (odd (prime (triad) composite) even))

And then the view being ascribed to Speusippus is that knowing the
number 3 is knowing where 3 is on such a grid, along with knowing
where every other number on the grid is.

It is very difficult to imagine sustaining this epistemological
holism. It might seem easier if we confined the view to things like
mathematics. But we already know that Speusippus’ universe extended
beyond the realm of mathematical objects, and it seems quite likely
that another division suggested by Aristotle in that chapter
(96b30–35) would be acceptable (at least in principle) to
Speusippus:

a diagram of the form (animal (tame (footed (two-footed (man) four-footed) winged swimming) wild))

The difficulty of sustaining the idea that knowing man is a
matter of knowing its position on a tree that locates absolutely every
animal, or even every ensouled thing, or (worse yet) absolutely
everything, is enormous. But it looks as if Speusippus would have been
committed to this.

For we find this in Sextus Empiricus, Adversus mathematicos
vii 145–146 (Bury 1935 pp. 80, 81):

… but Speusippus said that, since of things some are
perceptible and others intelligible, the criterion of the intelligible
ones is the scientific account, and that of the perceptible ones is
the scientific perception. Scientific perception he understood to be
that which partakes of the truth in accordance with the account. For
just as the fingers of the flautist or harpist have an artistic
actuality, yet one that is not completed in them in the first
instance, but perfected on the basis of discipline conforming to
reasoning, and as the perception of the musician has an actuality that
can grasp what is in tune and what is out of tune, and this is not
self-grown but comes about on the basis of reasoning, so also the
scientific perception partakes of its scientific practice as naturally
derived from the account, leading to the unerring discrimination of
the subjects.

The use of the word ‘scientific’ should not lead one away
from the main point: it translates a word that means ‘pertaining
to knowledge’, and Speusippus is claiming that there is
knowledge at the level of perceived objects. There is no indication
that he gave up his holism at this point. So he seems to be committed
to defending a rather drastic position.

And he seems to have pursued knowledge at the level of perceived
objects with considerable zeal. We have book-titles such as
Definitions and Likes in which Speusippus apparently
attempted with some zeal to locate various species of plants and
animals on something like a division-tree, although the details are
not extant. What we hear about Speusippus’ efforts is, for
example:

MARSHWORTS: Speusippus in book II of Likes says they grow in
water, their leaf resembling marsh celery. (Athenaeus II 61c; Gulick
1927 pp. 266, 267.)

Speusippus in the Likes calls the melon a
‘gourd’. (Athenaeus II 68e; Gulick 1927 pp. 298, 299.)

Speusippus in book II of Likes says that trumpet-shells,
purple-fish, snails, and clams are similar. … Again, Speusippus
enumerates next in order, in a separate division, clams, scallops,
mussels, pinnas, razorfish, and in another class oysters, limpets.
(Athenaeus III 86c,d; Gulick 1927 pp. 372, 273.)

Speusippus in book II of Likes says that, of the soft-shells,
the crayfish, lobster, mollusc, bear-crab, crab, paguros are
similar. (Athenaeus III 105b; Gulick 1927 pp. 450, 451.)

There are a total of twenty-five such citations in Athenaeus. Most
simply record that Speusippus said one organism was like another (as
with the last two), or simply record differences of terminology that
happened to catch Athenaeus’ eye (as with the second one). The only
one that promises anything more is the first one above, and that
hardly promises much. But it has been supposed (see Tarán 1981)
that it implies that Speusippus differentiated species of animals on
the basis of features that would not have counted as differentiae in
Aristotle’s way of classifying animals: differences of locale. And it
is certainly possible that Speusippus’ holism involved a rejection of
Aristotle’s essence-accident distinction: if we take the text quoted
above from Aristotle as implying that someone who knows marshworts
must know every respect in which they are like or unlike every other
organism in the universe, and every such differentiating feature is as
good as any other in defining them, then we have abandoned the project
of differentiating organisms as it appeared to Aristotle.

This line of thought has suggested to some (see Tarán 1981)
that the criticism of the Method of Division in Aristotle’s Parts
of Animals
I 2–3 constitutes a criticism of Speusippus.
This possibility remains highly controversial.

It appears from Simplicius’ Commentary on Aristotle’s
Categories
(Kalbfleisch 1907 38.19–24) that Speusippus
applied his method of division, however it worked, to language: there
we hear of a division of terms even more elaborate than Aristotle’s in
the opening chapter of the Categories: there we hear of
homonyms, synonyms, and paronyms, but Speusippus differentiated
between tautonyms (uses of the same term), within which there are
homonyms (same term but different definitions) and synonyms (same term
and same definition), and heteronyms (different terms), within which
there are heteronyms proper (different names, different things),
polyonyms (many names, the same thing), and paronyms (as in Aristotle:
different terms and different things, with one term derivative from
another, as ‘courageous’ and ‘courage’).
Whether this classification was not only more elaborate than
Aristotle’s, but also had a different basis (that in Aristotle being a
classification of things according to how they are referred to, that
in Speusippus being a classification of terms themselves), is in
dispute (Barnes 1971, Tarán 1978).

Finally, we know a little bit about Speusippus’ attitude toward the
epistemology of mathematics. Assuming that Aristotle in
Metaphysics XIV 3. 1090a25–29, 35-b1 is discussing
Speusippus (he is talking about “those who say there is only the
mathematical number”), Speusippus held that the truths of
mathematics are not about perceptible things, and that the axioms from
which they follow “fawn on the soul”: that is, perhaps,
suggest themselves. And Proclus tells us this (Commentary on Book
I of Euclid’s Elements
, Friedlein 1873 p. 179.8–22; Morrow
1970 p. 141):

For universally, Speusippus says, of the things for which the
understanding is making a hunt, some it puts forward without having
made an elaborate excursion, and sets them up for the investigation to
come: it has a clearer contact with these, even more than sight has
with visible things; but with others, which it is unable to grasp
straight off, but against which it makes its strides by inference, it
tries to effect their capture along the lines of what follows.

So we begin from some sort of intuition by which the axioms suggest
themselves and proceed from them to the rest of the mathematical
truths.

We also hear from Proclus of a dispute in the Academy between some,
whom we might call ‘constructivists’, who saw mathematics
as something like a human construct, and referred to the truths of
geometry as ‘problems’, demanding geometrical
constructions, and others, whom we might think of as
‘mathematical realists’, who saw mathematics as describing
an eternal, unchanging realm of objects, and referred to the truths of
mathematics as ‘theorems’ or ‘objects of
contemplation’. Proclus says (Friedlein 1873 pp.
77.15–78.6; Morrow 1970 pp. 63–64):

But already among the ancients, some demanded that we call all of the
things that follow from the principles theorems, as did Speusippus,
Amphinomus, and those around them, thinking that for the theoretical
branches of knowledge the appellation ‘theorems’ was more
appropriate than ‘problems’, especially in that they make
their accounts about eternal things. For there is no coming-to-be
among the eternal things, so that the problem could have no place with
them, it requiring a coming-to-be and making of that which was not
before, e.g. constructing an equilateral triangle or describing a
square when a line is given, or placing a line at a given point. So
they say it is better to say that all these things are, but that we
look at their comings-to-be not by way of producing them but by way of
knowing them, treating the things that always are as if they were
coming-to-be.

So Speusippus, as we might have predicted from the foregoing, was
firmly in the realist camp.

Speusippus certainly wrote about ethics: the bibliography in Diogenes
lists (in iv 4) one book each on wealth, pleasure, justice, and
friendship. But we have nothing that can be properly called a
fragment.

We have already seen that Speusippus rejected Plato’s Theory of Forms,
and that he refused to place the good among the metaphysical first
principles. So he was not obviously prone to the objections of
Aristotle against Platonic ethics in Nicomachean Ethics I 6,
the general upshot of which is that Plato’s theory puts the Good out
of human reach.

Speusippus appears to have adopted a rather down-to-earth goal, and in
this he is a lead-in to Hellenistic concerns. Hellenistic ethics is
dominated by the identification of the ideal of human happiness as
‘undisturbedness’ (ataraxia). According to
Clement, Stromata II 22, 133 (Stählin 1939 p.
186.19–23):

Speusippus, Plato’s nephew, says that happiness is the completed state
in things that hold by nature, or possession of goods, for which
condition all men have desire, while the good ones aim at
untroubledness (aochlesia). And the excellences are
productive of happiness.

These formulations ‘anticipate’ Stoic and Epicurean ones:
the emphasis on ‘natural’ conditions is a feature of
Stoicism, and the notion of ‘untroubledness’ is found in
Epicurus (see To Menoeceus, in Diogenes Laertius x 127).

There is an argument implicit in the clause “for which all men
have desire, while the good ones aim at untroubledness” that
fits in perfectly with a debate about the good and happiness that we
know from Aristotle to have been current in the Academy, and in which
Speusippus played a leading role. Nicomachean Ethics X 2
begins (1172b9–10):

Eudoxus, then, thought pleasure to be the good because he saw all
things, both rational and irrational, aiming at it, ….

This sort of ‘Universal Pursuit Argument’ is one that
became very popular in the Hellenistic period; here we have Eudoxus
applying it in favor of the claim that pleasure is the good:
‘hedonism’. And Speusippus strenuously denied hedonism
(see below).

We might, then, see Speusippus responding to the Universal Pursuit
Argument construed as favoring hedonism by saying: we should not care
about what all things in general aim at, but at what human
beings
aim at, in fact, more specifically, the good ones
among them, and what human beings pursue is not pleasure, but the
completed state in things that hold by nature, and the good
ones, more specifically, pursue untroubledness. The word
‘untroubledness’, aochlesia, comes from a verb
ochleein that can just mean ‘to move’, so
untroubledness might well suggest a certain stillness, lack of motion.
This will fit with Speusippus’ views on pleasure and pain.

Before we turn to those, we have a little more with which to flesh out
Speusippus’ idea of the good for man; none of it, regrettably, is
supplied with argument.

Cicero, in various places, ascribes a view to “Aristotle,
Speusippus, Xenocrates, and Polemon” which he himself rejects.
In the Tusculan Disputations he enumerates (V x 29) various
things such as poverty, ingloriousness, loneliness, pain, ill health,
etc., which many people (but not he: his own view is the Stoic one
that virtue all by itself guarantees happiness) take to be bad things,
and then says (V x 30, King 1927 pp. 454/455–456/457):

Therefore I do not easily give in to … those ancients,
Aristotle, Speusippus, Xenocrates, and Polemon, since they count the
things I have enumerated above among the bad things, and these very
people say that the wise man is always happy.

Cicero is, in the last clause, accusing these people of inconsistency:
they think the wise man is always happy, and yet they think there are
things such as wealth and health the lack of which would bring about
unhappiness.

Seneca (Epistulae morales 85.18, Reynolds 1965 i 292, Gummere
1920 pp. 294, 295) says:

Xenocrates and Speusippus think that one can become happy even by
virtue alone, but not that there is one good, that is morality
(honestum).

This can be seen as expressive of the same reservation Cicero had. If
Speusippus and Xenocrates espoused the view that the virtuous man was
always happy but thought that there were non-moral evils such as
poverty and pain, then they thought that there were non-moral good
things as well: the absence of poverty and the absence of pain. So (on
Cicero’s and Seneca’s account) they were not consistent in maintaining
that virtue (which incorporates only moral goods) was sufficient for
happiness.

And it appears that Speusippus and Xenocrates did think that there
were non-moral goods; Plutarch, in De communibus notitiis adversus
Stoicos
13. 1065a (Cherniss 1976 pp. 704, 705) refers to
Speusippus and Xenocrates as “thinking that health is not
indifferent and wealth not useless”.

We are left with two views to ascribe to Speusippus (and to
Xenocrates): that wisdom, or virtue in general, is sufficient for
happiness, and that there are non-moral goods the lack of which
conduces to unhappiness. Cicero’s question as to how these can be
rendered consistent is a good one, and we have no information about
how Speusippus might have answered it.

Let us turn to the one other topic about which we have some purchase
on Speusippus’ ethical views: that of pleasure. Aulus Gellius,
Atticae noctes IX v 4 (Marshall 1968 p. 284, Rolfe 1927 pp.
168, 169):

Speusippus and the entire old Academy say that pleasure and pain are
two evils opposed to each other, but that the good is what is in the
middle between the two.

The inclusion of the rest of the Academy along with Speusippus is
presumably due to the influence of Cicero’s teacher Antiochus’
unifying efforts. At any rate, there is confirmation that this was
Speusippus’ view, from two passages in Aristotle’s Nicomachean
Ethics
: VII 13, which names Speusippus, and X 2. 1173a5–13,
which does not, but repeats the argument of the former passage.

In the first of those chapters (in 1153b1–7) Aristotle mentions
Speusippus as having attacked the claim that pleasure is something
good. He had canvassed arguments to the effect that pleasure is a bad
thing earlier in the chapter, and some of the arguments certainly
sound like Speusippus (1152b12–20):

And so, in favor of the claim that pleasure is not a good at all, it
is argued (1) that every pleasure is a perceptible coming-to-be toward
a natural state, while no coming-to-be is of the same kind as its
ends, e.g. no house-building is of the same kind as a house. (2)
Again, the temperate man avoids pleasures. (3) Again, the intelligent
man pursues what is painless, not what is pleasant. (4) Again,
pleasures are a hindrance to reflecting, and by as much as one enjoys
them, by that much are they more of a hindrance, as with the pleasure
of sex; for no one could think in the course of that. (5) Again, there
is no art of pleasure, although everything good is the work of an art.
(6) Again, children and wild animals pursue pleasures.

Assuming that we correctly understood Speusippus’ response to the
Universal Pursuit Argument, (2), (3), and (6) are consonant with that
response.

But (1) plainly echoes material in Plato’s Philebus (see esp.
53c–55a). And that leads to the interesting suggestion that in
that dialogue Plato is entering the dispute between Eudoxus and
Speusippus. It has been suggested independently of that passage that
Speusippus is lurking behind the ‘harsh thinkers’, the
anti-hedonists, of 44a-47b (Schofield 1971, Dillon 1996).

If that is right, and if Graeser’s suggestion that Plato in the
Parmenides is responding to some of Speusippus’ metaphysical
views, it is plain that the understanding of some of what is going on
in late Plato would be aided by an understanding of what was going on
in Speusippus. The loss of his writings is regrettable indeed.

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