Zeller has cleared the ground by eliminating the purely Platonic elements which have crept into later accounts of the system. These are of two kinds. First of all, we have genuine Academic formulae, such as the identification of the Limit and the Unlimited with the One and the Indeterminate Dyad;[[768]] and secondly, there is the Neoplatonic doctrine which represents it as an opposition between God and Matter.[[769]] It is not necessary to repeat Zeller’s arguments here, as no one will any longer attribute these doctrines to the Pythagoreans of the fifth century.
This simplifies the problem very considerably, but it is still extremely difficult. According to Aristotle, the Pythagoreans said Things are numbers, though that does not appear to be the doctrine of the fragments of “Philolaos.” According to them, things have number, which make them knowable, while their real essence is something unknowable.[[770]] That would be intelligible enough, but the formula that things are numbers seems meaningless. We have seen reason for believing that it is due to Pythagoras himself ([§ 52]), though we did not feel able to say very clearly what he meant by it. 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 something 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 Akousmata ([§ 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.
Aristotle on the Numbers[Numbers].
143. In the first place, Aristotle is quite decided in his opinion that Pythagoreanism was intended to 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.”[[771]] 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 the heavens,”[[772]] though “the first principles and causes of which they made use were really adequate to explain realities of a higher order than the sensible.”[[773]]
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.[[774]] He is equally positive that these “things” are sensible things,[[775]] and indeed that they are bodies,[[776]] the bodies of which the world is constructed.[[777]] This construction of the world out of numbers was a real process in time, which the Pythagoreans described in detail.[[778]]
Further, the numbers were intended to be mathematical numbers, though they were not separated from the things of sense.[[779]] 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 certain other substances, such as fire, water, or anything 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.”[[780]] Accordingly the numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things.[[781]] According to the Pythagoreans, things are made of 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.[[782]] Let us consider these statements in detail.
The elements of numbers.
144. Aristotle speaks of certain “elements” (στοιχεῖα) of numbers, which were also the elements of 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. Primarily, 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,[[783]] and the 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 probability 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 division into equal parts.”[[784]] Now it is plain that we 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 a middle term.”[[785]] 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 field is left, without a master and without a number, showing that it is defective and incomplete.”[[786]] 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.”[[787]] 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 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.[[788]] The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens.[[789]] 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 res extensa, without analysing that conception any further.