The principles in question must be either (a) one or (b) more than
one. If (a) one, it must be either (i) motionless, as Parmenides and
Melissus assert, or (ii) in motion, as the physicists hold, some
declaring air to be the first principle, others water. If (b) more
than one, then either (i) a finite or (ii) an infinite plurality. If
(i) finite (but more than one), then either two or three or four or
some other number. If (ii) infinite, then either as Democritus
believed one in kind, but differing in shape or form; or different
in kind and even contrary.

    A similar inquiry is made by those who inquire into the number of
existents: for they inquire whether the ultimate constituents of
existing things are one or many, and if many, whether a finite or an
infinite plurality. So they too are inquiring whether the principle or
element is one or many.

    Now to investigate whether Being is one and motionless is not a
contribution to the science of Nature. For just as the geometer has
nothing more to say to one who denies the principles of his
science-this being a question for a different science or for or common
to all-so a man investigating principles cannot argue with one who
denies their existence. For if Being is just one, and one in the way
mentioned, there is a principle no longer, since a principle must be
the principle of some thing or things.

    To inquire therefore whether Being is one in this sense would be
like arguing against any other position maintained for the sake of
argument (such as the Heraclitean thesis, or such a thesis as that
Being is one man) or like refuting a merely contentious argument-a
description which applies to the arguments both of Melissus and of
Parmenides: their premisses are false and their conclusions do not
follow. Or rather the argument of Melissus is gross and palpable and
offers no difficulty at all: accept one ridiculous proposition and the
rest follows-a simple enough proceeding.

    We physicists, on the other hand, must take for granted that the
things that exist by nature are, either all or some of them, in motion
which is indeed made plain by induction. Moreover, no man of science
is bound to solve every kind of difficulty that may be raised, but
only as many as are drawn falsely from the principles of the
science: it is not our business to refute those that do not arise in
this way: just as it is the duty of the geometer to refute the
squaring of the circle by means of segments, but it is not his duty to
refute Antiphon's proof. At the same time the holders of the theory of
which we are speaking do incidentally raise physical questions, though
Nature is not their subject: so it will perhaps be as well to spend
a few words on them, especially as the inquiry is not without
scientific interest.

    The most pertinent question with which to begin will be this: In
what sense is it asserted that all things are one? For "is" is used in
many senses. Do they mean that all things "are" substance or
quantities or qualities? And, further, are all things one
substance-one man, one horse, or one soul-or quality and that one
and the same-white or hot or something of the kind? These are all very
different doctrines and all impossible to maintain.

    For if both substance and quantity and quality are, then, whether
these exist independently of each other or not, Being will be many.

    If on the other hand it is asserted that all things are quality or
quantity, then, whether substance exists or not, an absurdity results,
if the impossible can properly be called absurd. For none of the
others can exist independently: substance alone is independent: for
everything is predicated of substance as subject. Now Melissus says
that Being is infinite. It is then a quantity. For the infinite is
in the category of quantity, whereas substance or quality or affection
cannot be infinite except through a concomitant attribute, that is, if
at the same time they are also quantities. For to define the
infinite you must use quantity in your formula, but not substance or
quality. If then Being is both substance and quantity, it is two,
not one: if only substance, it is not infinite and has no magnitude;
for to have that it will have to be a quantity.

    Again, "one" itself, no less than "being", is used in many senses,
so we must consider in what sense the word is used when it is said
that the All is one.

    Now we say that (a) the continuous is one or that (b) the
indivisible is one, or (c) things are said to be "one", when their
essence is one and the same, as "liquor" and "drink".

    If (a) their One is one in the sense of continuous, it is many,
for the continuous is divisible ad infinitum.

    There is, indeed, a difficulty about part and whole, perhaps not
relevant to the present argument, yet deserving consideration on its
own account-namely, whether the part and the whole are one or more
than one, and how they can be one or many, and, if they are more
than one, in what sense they are more than one. (Similarly with the
parts of wholes which are not continuous.) Further, if each of the two
parts is indivisibly one with the whole, the difficulty arises that
they will be indivisibly one with each other also.

    But to proceed: If (b) their One is one as indivisible, nothing will
have quantity or quality, and so the one will not be infinite, as
Melissus says-nor, indeed, limited, as Parmenides says, for though the
limit is indivisible, the limited is not.

    But if (c) all things are one in the sense of having the same
definition, like "raiment" and "dress", then it turns out that they
are maintaining the Heraclitean doctrine, for it will be the same
thing "to be good" and "to be bad", and "to be good" and "to be not
good", and so the same thing will be "good" and "not good", and man
and horse; in fact, their view will be, not that all things are one,
but that they are nothing; and that "to be of such-and-such a quality"
is the same as "to be of such-and-such a size".

    Even the more recent of the ancient thinkers were in a pother lest
the same thing should turn out in their hands both one and many. So
some, like Lycophron, were led to omit "is", others to change the mode
of expression and say "the man has been whitened" instead of "is
white", and "walks" instead of "is walking", for fear that if they
added the word "is" they should be making the one to be many-as if
"one" and "being" were always used in one and the same sense. What
"is" may be many either in definition (for example "to be white" is
one thing, "to be musical" another, yet the same thing be both, so the
one is many) or by division, as the whole and its parts. On this
point, indeed, they were already getting into difficulties and
admitted that the one was many-as if there was any difficulty about
the same thing being both one and many, provided that these are not
opposites; for "one" may mean either "potentially one" or "actually
one".