1 Aristotle says distinctly (J/e¢. A, 6. 987 Ὁ 25) that ““ἴο set up a dyad instead of the unlimited regarded as one, and to make the unlimited consist of the great and small, is distinctive of Plato.” Zeller seems to make an unnecessary concession with regard to this passage (p. 368, n. 2; Eng. trans. p. 396, n. I).
? Zeller, p. 369 sqq. (Eng. trans. p. 397 sqq.).
3 For the doctrine of ““ Philolaos,” cf. fr. 1=2 Ch. (R. P. 64); and for the unknowable ἐστὼ τῶν πραγμάτων, see fr. 3=4 Ch. (R. P. 67). It has a suspicious resemblance to the later ὕλη, which Aristotle would hardly have failed to note if he had ever seen the passage. He is always on the lookout for anticipations of ὕλη.
The Pythagoreans 331
There is no such doubt as to his school. Aristotle
says they used the formula in a cosmological sense.
The world, according to them, was made of numbers
in the same sense as others had said it was made of
“four roots” or “innumerable seeds.” It will not do
to dismiss this as mysticism. Whatever we may think
of Pythagoras, the Pythagoreans of the fifth century
were scientific men, and they must have meant some-
thing quite definite. We shall, no doubt, have to say
that they used the words Things are numbers in a
somewhat non-natural sense, but there is no difficulty
in such a supposition. We have seen already how the
friends of Aristoxenos reinterpreted the old Axousmata
(§ 44). The Pythagoreans had certainly a great
veneration for the actual words of the Master (αὐτὸς
ἔφα) ; but such veneration is often accompanied by a
singular licence of interpretation. We shall start,
then, from what Aristotle tells us about the numbers.
143. In the first place, Aristotle is quite decided Aristotle on
in his opinion that Pythagoreanism was intended to ἜΝ be a cosmological system like the others. “Though
the Pythagoreans,” he tells us, “made use of less
obvious first principles and elements than the rest,
seeing that they did not derive them from sensible
objects, yet all their discussions and studies had
reference to nature alone. They describe the origin
of the heavens, and they observe the phenomena of its
parts, all that happens to it and all it does.”’ They
apply their first principles entirely to these things,
“ agreeing apparently with the other natural philosophers
in holding that reality was just what could be perceived
by the senses, and is contained within the compass of
1 Arist. 27εἰ. A, 8. 989 Ὁ 29 (R. P. 92 a).
332 Early Greek Philosophy
the heavens,” ’ though “ the first principles and causes
of which they made use were really adequate to
explain realities of a higher order than the sensible.” ὅ
The doctrine is more precisely stated by Aristotle
to be that the elements of numbers are the elements of
things, and that therefore things are numbers? He
is equally positive that these “things” are sensible
things,* and indeed that they are bodies,’ the bodies of
which the world is constructed.° This construction of
the world out of numbers was a real process in time,
which the Pythagoreans described in detail.’
Further, the numbers were intended to be mathe-
matical numbers, though they were not separated from
the things of sense. On the other hand, they were
not mere predicates of something else, but had an
independent reality of their own. ‘“ They did not hold
that the limited and the unlimited and the one were
1 Arist. AZez. A, 8. 990 a 3, ὁμολογοῦντες τοῖς ἄλλοις φυσιολόγοις ὅτι τό γ᾽ ὃν τοῦτ᾽ ἐστὶν ὅσον αἰσθητόν ἐστὶ καὶ περιείληφεν ὁ καλούμενος οὐρανός.
2 Met. tb. 990 a 5, τὰς δ᾽ αἰτίας καὶ τὰς ἀρχάς, ὥσπερ εἴπομεν, ἱκανὰς λέγουσιν ἐπαναβῆναι καὶ ἐπὶ τὰ ἀνωτέρω τῶν ὄντων, καὶ μᾶλλον ἢ τοῖς περὶ φύσεως λόγοις ἁρμοττούσας,
3 Met. A, 5. 986 a 1, τὰ τῶν ἀριθμῶν στοιχεῖα τῶν ὄντων στοιχεῖα πάντων ὑπέλαβον εἷναι; N, 3. ΙΟ90 ἃ 22, εἶναι μὲν ἀριθμοὺς ἐποίησαν τὰ ὄντα, οὐ χωριστοὺς δέ, ἀλλ᾽ ἐξ ἀριθμῶν τὰ ὄντα.
4 Met. Μ, 6. 1080 b 2, ὡς ἐκ τῶν ἀριθμῶν ἐνυπαρχόντων ὄντα τὰ αἰσθητά; 20. 1080 Ὁ 17, ἐκ τούτου (τοῦ μαθηματικοῦ ἀριθμοῦ) τὰς αἰσθητὰς οὐσίας συνεστάναι φασίν.
5 Met. M, 8. 1083 Ὁ 11, τὰ σώματα ἐξ ἀριθμῶν εἶναι συγκείμενα ; 76. b 17, ἐκεῖνοι δὲ τὸν ἀριθμὸν τὰ ὄντα λέγουσιν " τὰ γοῦν θεωρήματα πρόσ- άπτουσι τοῖς σώμασιν ὡς ἐξ ἐκείνων ὄντων τῶν ἀριθμῶν ; N, 3. 1090 a 32, κατὰ μέντοι τὸ ποιεῖν ἐξ ἀριθμῶν τὰ φυσικὰ σώματα, ἐκ μὴ ἐχόντων βάρος μηδὲ κουφότητα ἔχοντα κουφότητα καὶ βάρος.
6 Met. A, 5. 986 a 2, τὸν ὅλον οὐρανὸν ἁρμονίαν εἶναι καὶ ἀριθμέν: 8. 990 a 21, τὸν ἀριθμὸν τοῦτον ἐξ οὗ συνέστηκεν ὁ κόσμος ; M, 6. 1080 Ἶ ᾿18, τὸν γὰρ ὅλον οὐρανὸν κατασκευάζουσιν ἐξ ἀριθμῶν ; de Caelo, Τ', 1. 300 ἃ 15, τοῖς ἐξ ἀριθμῶν συνιστᾶσι τὸν οὐρανόν " ἔνιοι γὰρ τὴν φύσιν ἐξ ἀριθμῶν συνιστᾶσιν, ὥσπερ τῶν Πυθαγορείων τινές.
7 Met. N, 3. 1091 a 18, κοσμοποιοῦσι καὶ φυσικῶς βούλονται spade
8 Met. M, 6. 1080 Ὁ 16; N, 3. 1090 a 20,
The Pythagoreans 333
certain other substances, such as fire, water, or anything x else of that sort ; but that the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything.” ὦ numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things.” According to the Pythagoreans, things are made of
Accordingly the
numbers in the same sense as they were made of fire, air, or water in the theories of their predecessors.
Lastly, Aristotle notes that the point in which the
Pythagoreans agreed with Plato was in giving numbers
an independent reality of their own; while Plato
differed from the Pythagoreans in holding that this
reality was distinguishable from that of sensible things.®
Let us consider these statements in detail.
144. Aristotle speaks of certain “elements” The elemen
(στοιχεῖα) of numbers, which were also the elements of + ae
things. That, of course, is only his own way of
putting the matter; but it is clearly the key to the
problem, if we can discover what it means. Pri-
marily, the “elements of number” are the Odd and
the Even, but that does not seem to help us much.
We find, however, that the Odd and Even were
identified in a somewhat violent way with the Limit
and the Unlimited, which we have seen reason to
regard as the original principles of the Pythagorean
cosmology. Aristotle tells us that it is the Even which
gives things their unlimited character when it is
contained in them and limited by the Odd,* and the
1 Arist. Met. A, 5.987015. θΣ 3 Met. tb. 986015 (R. P. 66).
3 Met, A, 6. 987 Ὁ 27, ὁ μὲν (Πλάτων) τοὺς ἀριθμοὺς παρὰ τὰ αἰσθητά, οἱ δ᾽ (οἱ Πυθαγόρειοι) ἀριθμοὺς εἶναί φασιν αὐτὰ τὰ αἰσθητά.
4 Met. A, 5. 986217 (R. P. 66) ; Phys. T, 4. 203 a 10(R. P. 66 a).
334 Early Greek Philosophy
commentators are at one in understanding this to
mean that the Even is in some way the cause of
infinite divisibility. They get into great difficulties,
however, when they try to show how this can be.
Simplicius has preserved an explanation, in all prob-
ability Alexander’s, to the effect that they called the
even number unlimited “ because every even is divided
into equal parts, and what is divided into equal parts
is unlimited in respect of bipartition ; for division into
equals and halves goes on ad infinitum. But, when
the odd is added, it limits it; for it prevents its
1 Now it is plain that we
division into equal parts.
must not impute to the Pythagoreans the view that |
even numbers can be halved indefinitely. They had
carefully studied the properties of the decad, and
they must have known that the even numbers 6
and 10 do not admit of this. The explanation is
really to be found in a fragment of Aristoxenos,
where we read that “even numbers are those which
are divided into equal parts, while odd numbers are
divided into unequal parts and have ἃ middle
term. This is still further elucidated by a passage
which is quoted in Stobaios and ultimately goes
back to Poseidonios. It runs: “When the odd is
divided into two equal parts, a unit is left over in the
middle ; but when the even is so divided, an empty
1 Simpl. Phys. p. 455, 20 (R. P. 66a). I owe the passages which I have used in illustration of this subject to W. A. Heidel, ‘* Πέρας and ἄπειρον in the Pythagorean Philosophy ” (Arch. xiv. pp. 384 sqq.). The general principle of my interpretation is also the same as his, though I think that, by bringing the passage into connexion with the numerical figures, I have avoided the necessity of regarding the words ἡ γὰρ eis toa καὶ ἡμίση διαίρεσις ἐπ’ ἄπειρον as “‘ an attempted elucidation added by Simplicius.”
2 Aristoxenos, fr. 81, ap. Stob. i. p. 20, 1, ἐκ τῶν ᾿Αριστοξένου Περὶ ἀριθμη- τικῆς . . . τῶν δὲ ἀριθμῶν ἄρτιοι μέν εἰσιν οἱ εἰς toa διαιρούμενοι, περισσοὶ δὲ οἱ εἰς ἄνισα καὶ μέσον ἔχοντες.
The Pythagoreans _ 335
field is left, without a master and without.a number,
showing that it is defective and incomplete.”* Again,
Plutarch says: “In the division of numbers, the even,
when parted in any direction, leaves as it were within
itself . . . a field; but, when the same thing is done
to the odd, there is always a middle left over from the
division.” It is clear that all these passages refer to
the same thing, and that can hardly be anything else
than those arrangements of “terms” in patterns with
which we are already familiar (§ 47). If we think of
these, we shall see in what sense it is true that
bipartition goes on ad infinitum. However high the
number may be, the number of ways in which it can
be equally divided will also increase.
145. In this way, then, the Odd and the Even
were identified with the Limit and the Unlimited, and
it is possible, though by no means certain, that
Pythagoras himself had taken this step. In any case,
there can be no doubt that by his Unlimited he meant
something spatially extended, and we have seen that
he identified it with air, night, or the void, so we are
prepared to find that his followers also thought of the
Unlimited as extended. Aristotle certainly regarded
it so. He argues that, if the Unlimited is itself a
1 [Plut.] αὐ. Stob. i. p. 22, 19, καὶ μὴν els δύο διαιρουμένων ἴσα τοῦ μὲν περισσοῦ μονὰς ἐν μέσῳ περιέστι, τοῦ δὲ ἀρτίου κενὴ λείπεται χώρα καὶ ἀδέσποτος καὶ ἀνάριθμος, ὡς ἂν ἐνδεοῦς καὶ ἀτελοῦς ὄντος.
2 Plut. de EZ apud Delphos, 388 a, ταῖς γὰρ εἰς ἴσα τομαῖς τῶν ἀριθμῶν, ὁ μὲν ἄρτιος πάντῃ διϊστάμενος ὑπολείπει τινὰ δεκτικὴν ἀρχὴν οἷον ἐν ἑαυτῷ καὶ χώραν, ἐν δὲ τῷ περιττῷ ταὐτὸ παθόντι μέσον ἀεὶ περίεστι τῆς νεμήσεως γόνιμον. The words which I have omitted in translating refer to the further identification of Odd and Even with Male and Female. The passages quoted by Heidel might be added to. Cf., for instance, what Nikomachos says (p. 13, 10, Hoche), ἔστι δὲ ἄρτιον μὲν ὃ οἷόν re εἰς δύο toa διαιρεθῆναι μονάδος μέσον μὴ παρεμπιπτούσης, περιττὸν δὲ τὸ μὴ δυνάμενον εἰς δύο ἴσα μερισθῆναι διὰ τὴν προειρημένην τῆς μονάδος μεσιτείαν. He significantly adds that this definition is ἐκ τῆς δημώδους ὑπολήψεως.
The number: spatial.
336 Early Greek Philosophy
reality, and not merely the predicate of some other
reality, then every part of it must be unlimited too,
just as every part of air is air. The same thing is
implied in his statement that the Pythagorean Unlimited
was outside the heavens.” Further than this, it is
hardly safe to go. Philolaos and his followers cannot
have regarded the Unlimited in the old Pythagorean
way as Air; for, as we shall see, they adopted the
theory of Empedokles as to that “element,” and
accounted for it otherwise. On the other hand, they
can hardly have regarded it as an absolute void ; for
that conception was introduced by the Atomists. It is
enough to say that they meant by the Unlimited the ves
extensa, without analysing that conception any further.