At 12:41 AM 11/02/1999 +0300, Alexander Zenkin wrote:
| Can anybody tell me where does Aristotle state his famous thesis
| "Infinitum Actu Non Datur", and how does he substantiate it ? (Please,
| with references and short citations since, unfortunately, I myself have
| not a possibility to attend a library now).
Apropos of the Aristotle's famous dictum on infinity, I believe most
historians and philosophers will basically agree with Struik when he
states that
"The scholastic writers of the Middle Ages, especially St. Thomas
Aquinas, accepted Aristotle's _infinitum actu non datur_ [= there
is no actual(ly) infinite], but considered every continuum as
potentially divisible ad infinitum. Thus there was no smallest
line. A point, therefore, was not a part of a line, because it
was indivisible: _ex indivisibilibus non potest constare aliquod
continuum_ [= a continuum cannot consist of indivisibles]. A point
could generate a line by motion. Such speculations had their
influence on the inventors of the infinitesimal calculus in the
seventeenth century and on the philosophers of the transfinite in
the nineteenth; Cavalieri, Tacquet, Bolzano, and Cantor knew the
scholastic authors and pondered over the meaning of their ideas."
Is this the case?... Comments are welcome.
Aristotle's _infinitum actu non datur_ is taken from his "Physics",
Book III (especially part 5)
Since Professor Zenkin specifically asked for references with quotations
I have taken myself the liberty of posting entire parts 4-8, translated
into English by R.P. Hardie and R.K. Gaye. Fascinating reading, I'm afraid.
Hope to hearing from you,
Julio Gonzalez Cabillon
-------------------------
Book III
Part 4
The science of nature is concerned with spatial magnitudes and
motion and time, and each of these at least is necessarily infinite
or finite, even if some things dealt with by the science are not,
e.g. a quality or a point -- it is not necessary perhaps that such
things should be put under either head. Hence it is incumbent on
the person who specializes in physics to discuss the infinite and
to inquire whether there is such a thing or not, and, if there is,
what it is.
The appropriateness to the science of this problem is clearly
indicated. All who have touched on this kind of science in a way
worth considering have formulated views about the infinite, and
indeed, to a man, make it a principle of things.
(1) Some, as the Pythagoreans and Plato, make the infinite a
principle in the sense of self -- subsistent substance, and not as a
mere attribute of some other thing. Only the Pythagoreans place
the infinite among the objects of sense (they do not regard number
as separable from these), and assert that what is outside the
heaven is infinite. Plato, on the other hand, holds that there
is no body outside (the Forms are not outside because they are
nowhere), yet that the infinite is present not only in the objects
of sense but in the Forms also.
Further, the Pythagoreans identify the infinite with the even. For
this, they say, when it is cut off and shut in by the odd, provides
things with the element of infinity. An indication of this is what
happens with numbers. If the gnomons are placed round the one, and
without the one, in the one construction the figure that results is
always different, in the other it is always the same. But Plato has
two infinites, the Great and the Small.
The physicists, on the other hand, all of them, always regard the
infinite as an attribute of a substance which is different from it
and belongs to the class of the so-called elements -- water or air
or what is intermediate between them. Those who make them limited
in number never make them infinite in amount. But those who make
the elements infinite in number, as Anaxagoras and Democritus do,
say that the infinite is continuous by contact -- compounded of the
homogeneous parts according to the one, of the seed-mass of the
atomic shapes according to the other.
Further, Anaxagoras held that any part is a mixture in the same
way as the All, on the ground of the observed fact that anything
comes out of anything. For it is probably for this reason that he
maintains that once upon a time all things were together. (This
flesh and this bone were together, and so of any thing: therefore
all things: and at the same time too.) For there is a beginning of
separation, not only for each thing, but for all. Each thing that
comes to be comes from a similar body, and there is a coming to be
of all things, though not, it is true, at the same time. Hence
there must also be an origin of coming to be. One such source
there is which he calls Mind, and Mind begins its work of thinking
from some starting-point. So necessarily all things must have been
together at a certain time, and must have begun to be moved at a
certain time.
Democritus, for his part, asserts the contrary, namely that no
element arises from another element. Nevertheless for him the
common body is a source of all things, differing from part to part
in size and in shape.
It is clear then from these considerations that the inquiry concerns
the physicist. Nor is it without reason that they all make it a
principle or source. We cannot say that the infinite has no effect,
and the only effectiveness which we can ascribe to it is that of
a principle. Everything is either a source or derived from a source.
But there cannot be a source of the infinite or limitless, for that
would be a limit of it. Further, as it is a beginning, it is both
uncreatable and indestructible. For there must be a point at which
what has come to be reaches completion, and also a termination of
all passing away. That is why, as we say, there is no principle of
this, but it is this which is held to be the principle of other
things, and to encompass all and to steer all, as those assert who
do not recognize, alongside the infinite, other causes, such as
Mind or Friendship. Further they identify it with the Divine, for
it is 'deathless and imperishable' as Anaximander says, with the
majority of the physicists.
Belief in the existence of the infinite comes mainly from five
considerations:
(1) From the nature of time -- for it is infinite.
(2) From the division of magnitudes -- for the mathematicians also
use the notion of the infinite.
(3) If coming to be and passing away do not give out, it is only
because that from which things come to be is infinite.
(4) Because the limited always finds its limit in something, so
that there must be no limit, if everything is always limited
by something different from itself.
(5) Most of all, a reason which is peculiarly appropriate and
presents the difficulty that is felt by everybody -- not only
number but also mathematical magnitudes and what is outside
the heaven are supposed to be infinite because they never
give out in our thought.
The last fact (that what is outside is infinite) leads people to
suppose that body also is infinite, and that there is an infinite
number of worlds. Why should there be body in one part of the void
rather than in another? Grant only that mass is anywhere and it
follows that it must be everywhere. Also, if void and place are
infinite, there must be infinite body too, for in the case of
eternal things what may be must be. But the problem of the infinite
is difficult: many contradictions result whether we suppose it to
exist or not to exist. If it exists, we have still to ask how it
exists; as a substance or as the essential attribute of some
entity? Or in neither way, yet none the less is there something
which is infinite or some things which are infinitely many?
The problem, however, which specially belongs to the physicist is
to investigate whether there is a sensible magnitude which is
infinite.
We must begin by distinguishing the various senses in which the
term 'infinite' is used.
(1) What is incapable of being gone through, because it is not in
its nature to be gone through (the sense in which the voice is
'invisible').
(2) What admits of being gone through, the process however having
no termination, or what scarcely admits of being gone through.
(3) What naturally admits of being gone through, but is not
actually gone through or does not actually reach an end.
Further, everything that is infinite may be so in respect of
addition or division or both.
Part 5
Now it is impossible that the infinite should be a thing which
is itself infinite, separable from sensible objects. If the
infinite is neither a magnitude nor an aggregate, but is itself
a substance and not an attribute, it will be indivisible; for
the divisible must be either a magnitude or an aggregate. But
if indivisible, then not infinite, except in the sense (1) in
which the voice is 'invisible'. But this is not the sense in
which it is used by those who say that the infinite exists, nor
that in which we are investigating it, namely as (2) 'that which
cannot be gone through'. But if the infinite exists as an
attribute, it would not be, qua infinite an element in substances,
any more than the invisible would be an element of speech,
though the voice is invisible.
Further, how can the infinite be itself any thing, unless both
number and magnitude, of which it is an essential attribute,
exist in that way? If they are not substances, a fortiori the
infinite is not.
It is plain, too, that the infinite cannot be an actual thing
and a substance and principle. For any part of it that is taken
will be infinite, if it has parts: for 'to be infinite' and
'the infinite' are the same, if it is a substance and not
predicated of a subject. Hence it will be either indivisible or
divisible into infinites. But the same thing cannot be many
infinites. (Yet just as part of air is air, so a part of the
infinite would be infinite, if it is supposed to be a substance
and principle.) Therefore the infinite must be without parts
and indivisible. But this cannot be true of what is infinite in
full completion: for it must be a definite quantity.
Suppose then that infinity belongs to substance as an attribute.
But, if so, it cannot, as we have said, be described as a
principle, but rather that of which it is an attribute -- the air
or the even number.
Thus the view of those who speak after the manner of the
Pythagoreans is absurd. With the same breath they treat the
infinite as substance, and divide it into parts.
This discussion, however, involves the more general question
whether the infinite can be present in mathematical objects
and things which are intelligible and do not have extension,
as well as among sensible objects. Our inquiry (as physicists)
is limited to its special subject-matter, the objects of sense,
and we have to ask whether there is or is not among them a body
which is infinite in the direction of increase.
We may begin with a dialectical argument and show as follows
that there is no such thing. If 'bounded by a surface' is the
definition of body there cannot be an infinite body either
intelligible or sensible. Nor can number taken in abstraction
be infinite, for number or that which has number is numerable.
If then the numerable can be numbered, it would also be possible
to go through the infinite.
If, on the other hand, we investigate the question more in
accordance with principles appropriate to physics, we are led
as follows to the same result.
The infinite body must be either (1) compound, or (2) simple;
yet neither alternative is possible.
(1) Compound the infinite body will not be, if the elements are
finite in number. For they must be more than one, and the contraries
must always balance, and no one of them can be infinite. If one of
the bodies falls in any degree short of the other in potency --
suppose fire is finite in amount while air is infinite and a given
quantity of fire exceeds in power the same amount of air in any ratio
provided it is numerically definite -- the infinite body will obviously
prevail over and annihilate the finite body. On the other hand, it is
impossible that each should be infinite. 'Body' is what has extension
in all directions and the infinite is what is boundlessly extended,
so that the infinite body would be extended in all directions ad
infinitum.
Nor (2) can the infinite body be one and simple, whether it is,
as some hold, a thing over and above the elements (from which
they generate the elements) or is not thus qualified.
(a) We must consider the former alternative; for there are some
people who make this the infinite, and not air or water, in order
that the other elements may not be annihilated by the element
which is infinite. They have contrariety with each other -- air is
cold, water moist, fire hot; if one were infinite, the others by
now would have ceased to be. As it is, they say, the infinite is
different from them and is their source.
It is impossible, however, that there should be such a body; not
because it is infinite on that point a general proof can be given
which applies equally to all, air, water, or anything else -- but
simply because there is, as a matter of fact, no such sensible
body, alongside the so-called elements. Everything can be resolved
into the elements of which it is composed. Hence the body in
question would have been present in our world here, alongside air
and fire and earth and water: but nothing of the kind is observed.
(b) Nor can fire or any other of the elements be infinite. For
generally, and apart from the question of how any of them could be
infinite, the All, even if it were limited, cannot either be or
become one of them, as Heraclitus says that at some time all things
become fire. (The same argument applies also to the one which the
physicists suppose to exist alongside the elements: for everything
changes from contrary to contrary, e.g. from hot to cold).
The preceding consideration of the various cases serves to show us
whether it is or is not possible that there should be an infinite
sensible body. The following arguments give a general demonstration
that it is not possible.
It is the nature of every kind of sensible body to be somewhere,
and there is a place appropriate to each, the same for the part
and for the whole, e.g. for the whole earth and for a single clod,
and for fire and for a spark.
Suppose (a) that the infinite sensible body is homogeneous. Then
each part will be either immovable or always being carried along.
Yet neither is possible. For why downwards rather than upwards or
in any other direction? I mean, e.g, if you take a clod, where will
it be moved or where will it be at rest? For ex hypothesi the place
of the body akin to it is infinite. Will it occupy the whole place,
then? And how? What then will be the nature of its rest and of its
movement, or where will they be? It will either be at home
everywhere -- then it will not be moved; or it will be moved
everywhere -- then it will not come to rest.
But if (b) the All has dissimilar parts, the proper places of the
parts will be dissimilar also, and the body of the All will have no
unity except that of contact. Then, further, the parts will be
either finite or infinite in variety of kind. (i) Finite they cannot
be, for if the All is to be infinite, some of them would have to be
infinite, while the others were not, e.g. fire or water will be
infinite. But, as we have seen before, such an element would destroy
what is contrary to it. (This indeed is the reason why none of the
physicists made fire or earth the one infinite body, but either
water or air or what is intermediate between them, because the
abode of each of the two was plainly determinate, while the others
have an ambiguous place between up and down.)
But (ii) if the parts are infinite in number and simple, their
proper places too will be infinite in number, and the same will be
true of the elements themselves. If that is impossible, and the
places are finite, the whole too must be finite; for the place and
the body cannot but fit each other. Neither is the whole place
larger than what can be filled by the body (and then the body would
no longer be infinite), nor is the body larger than the place; for
either there would be an empty space or a body whose nature it is
to be nowhere.
Anaxagoras gives an absurd account of why the infinite is at rest.
He says that the infinite itself is the cause of its being fixed.
This because it is in itself, since nothing else contains it -- on
the assumption that wherever anything is, it is there by its own
nature. But this is not true: a thing could be somewhere by
compulsion, and not where it is its nature to be.
Even if it is true as true can be that the whole is not moved (for
what is fixed by itself and is in itself must be immovable), yet
we must explain why it is not its nature to be moved. It is not
enough just to make this statement and then decamp. Anything else
might be in a state of rest, but there is no reason why it should
not be its nature to be moved. The earth is not carried along, and
would not be carried along if it were infinite, provided it is held
together by the centre. But it would not be because there was no
other region in which it could be carried along that it would
remain at the centre, but because this is its nature. Yet in this
case also we may say that it fixes itself. If then in the case of
the earth, supposed to be infinite, it is at rest, not because it
is infinite, but because it has weight and what is heavy rests at
the centre and the earth is at the centre, similarly the infinite
also would rest in itself, not because it is infinite and fixes
itself, but owing to some other cause.
Another difficulty emerges at the same time. Any part of the infinite
body ought to remain at rest. Just as the infinite remains at rest
in itself because it fixes itself, so too any part of it you may
take will remain in itself. The appropriate places of the whole
and of the part are alike, e.g. of the whole earth and of a clod
the appropriate place is the lower region; of fire as a whole and
of a spark, the upper region. If, therefore, to be in itself is the
place of the infinite, that also will be appropriate to the part.
Therefore it will remain in itself.
In general, the view that there is an infinite body is plainly
incompatible with the doctrine that there is necessarily a proper
place for each kind of body, if every sensible body has either
weight or lightness, and if a body has a natural locomotion towards
the centre if it is heavy, and upwards if it is light. This would
need to be true of the infinite also. But neither character can
belong to it: it cannot be either as a whole, nor can it be half
the one and half the other. For how should you divide it? or how
can the infinite have the one part up and the other down, or an
extremity and a centre?
Further, every sensible body is in place, and the kinds or
differences of place are up-down, before-behind, right-left; and
these distinctions hold not only in relation to us and by arbitrary
agreement, but also in the whole itself. But in the infinite body
they cannot exist.
In general, if it is impossible that there should be an infinite
place, and if every body is in place, there cannot be an infinite
body.
Surely what is in a special place is in place, and what is in place
is in a special place. Just, then, as the infinite cannot be quantity
-- that would imply that it has a particular quantity, e.g. two or three
cubits; quantity just means these -- so a thing's being in place means
that it is somewhere, and that is either up or down or in some other
of the six differences of position: but each of these is a limit.
!!!!
It is plain from these arguments that
THERE IS NO BODY WHICH IS ACTUALLY INFINITE.
!!!
Part 6
But on the other hand to suppose that the infinite does not exist
in any way leads obviously to many impossible consequences: there
will be a beginning and an end of time, a magnitude will not be
divisible into magnitudes, number will not be infinite. If, then,
in view of the above considerations, neither alternative seems
possible, an arbiter must be called in; and clearly there is a
sense in which the infinite exists and another in which it does not.
We must keep in mind that the word 'is' means either what potentially
is or what fully is. Further, a thing is infinite either by addition
or by division.
Now, as we have seen, magnitude is not actually infinite. But by
division it is infinite. (There is no difficulty in refuting the
theory of indivisible lines.) The alternative then remains that the
infinite has a potential existence.
But the phrase 'potential existence' is ambiguous. When we speak of
the potential existence of a statue we mean that there will be an
actual statue. It is not so with the infinite. There will not be an
actual infinite. The word 'is' has many senses, and we say that the
infinite 'is' in the sense in which we say 'it is day' or 'it is the
games', because one thing after another is always coming into
existence. For of these things too the distinction between potential
and actual existence holds. We say that there are Olympic games,
both in the sense that they may occur and that they are actually
occurring.
The infinite exhibits itself in different ways -- in time, in the
generations of man, and in the division of magnitudes. For generally
the infinite has this mode of existence: one thing is always being
taken after another, and each thing that is taken is always finite,
but always different. Again, 'being' has more than one sense, so that
we must not regard the infinite as a 'this', such as a man or a horse,
but must suppose it to exist in the sense in which we speak of the
day or the games as existing things whose being has not come to them
like that of a substance, but consists in a process of coming to be or
passing away; definite if you like at each stage, yet always different.
But when this takes place in spatial magnitudes, what is taken persists,
while in the succession of time and of men it takes place by the
passing away of these in such a way that the source of supply never
gives out.
In a way the infinite by addition is the same thing as the infinite
by division. In a finite magnitude, the infinite by addition comes
about in a way inverse to that of the other. For in proportion as we
see division going on, in the same proportion we see addition being
made to what is already marked off. For if we take a determinate part
of a finite magnitude and add another part determined by the same
ratio (not taking in the same amount of the original whole), and so
on, we shall not traverse the given magnitude. But if we increase
the ratio of the part, so as always to take in the same amount, we
shall traverse the magnitude, for every finite magnitude is exhausted
by means of any determinate quantity however small.
The infinite, then, exists in no other way, but in this way it does
exist, potentially and by reduction. It exists fully in the sense in
which we say 'it is day' or 'it is the games'; and potentially as
matter exists, not independently as what is finite does.
By addition then, also, there is potentially an infinite, namely,
what we have described as being in a sense the same as the infinite
in respect of division. For it will always be possible to take
something ah extra. Yet the sum of the parts taken will not exceed
every determinate magnitude, just as in the direction of division
every determinate magnitude is surpassed in smallness and there will
be a smaller part.
But in respect of addition there cannot be an infinite which even
potentially exceeds every assignable magnitude, unless it has the
attribute of being actually infinite, as the physicists hold to be
true of the body which is outside the world, whose essential nature
is air or something of the kind. But if there cannot be in this way
a sensible body which is infinite in the full sense, evidently there
can no more be a body which is potentially infinite in respect of
addition, except as the inverse of the infinite by division, as we
have said. It is for this reason that Plato also made the infinites
two in number, because it is supposed to be possible to exceed all
limits and to proceed ad infinitum in the direction both of increase
and of reduction. Yet though he makes the infinites two, he does not
use them. For in the numbers the infinite in the direction of
reduction is not present, as the monad is the smallest; nor is the
infinite in the direction of increase, for the parts number only up
to the decad.
The infinite turns out to be the contrary of what it is said to be.
It is not what has nothing outside it that is infinite, but what
always has something outside it. This is indicated by the fact that
rings also that have no bezel are described as 'endless', because it
is always possible to take a part which is outside a given part. The
description depends on a certain similarity, but it is not true in
the full sense of the word. This condition alone is not sufficient:
it is necessary also that the next part which is taken should never
be the same. In the circle, the latter condition is not satisfied:
it is only the adjacent part from which the new part is different.
Our definition then is as follows: A quantity is infinite if it is
such that we can always take a part outside what has been already
taken. On the other hand, what has nothing outside it is complete
and whole. For thus we define the whole -- that from which nothing is
wanting, as a whole man or a whole box. What is true of each
particular is true of the whole as such -- the whole is that of which
nothing is outside. On the other hand that from which something is
absent and outside, however small that may be, is not 'all'. 'Whole'
and 'complete' are either quite identical or closely akin. Nothing
is complete (teleion) which has no end (telos); and the end is a limit.
Hence Parmenides must be thought to have spoken better than Melissus.
The latter says that the whole is infinite, but the former describes
it as limited, 'equally balanced from the middle'. For to connect the
infinite with the all and the whole is not like joining two pieces of
string; for it is from this they get the dignity they ascribe to the
infinite -- its containing all things and holding the all in itself
-- from its having a certain similarity to the whole. It is in fact
the matter of the completeness which belongs to size, and what is
potentially a whole, though not in the full sense. It is divisible
both in the direction of reduction and of the inverse addition. It is
a whole and limited; not, however, in virtue of its own nature, but in
virtue of what is other than it. It does not contain, but, in so far
as it is infinite, is contained. Consequently, also, it is unknowable,
qua infinite; for the matter has no form. (Hence it is plain that the
infinite stands in the relation of part rather than of whole. For the
matter is part of the whole, as the bronze is of the bronze statue.)
If it contains in the case of sensible things, in the case of
intelligible things the great and the small ought to contain them.
But it is absurd and impossible to suppose that the unknowable and
indeterminate should contain and determine.
Part 7
It is reasonable that there should not be held to be an infinite in
respect of addition such as to surpass every magnitude, but that there
should be thought to be such an infinite in the direction of division.
For the matter and the infinite are contained inside what contains them,
while it is the form which contains. It is natural too to suppose that
in number there is a limit in the direction of the minimum, and that in
the other direction every assigned number is surpassed. In magnitude,
on the contrary, every assigned magnitude is surpassed in the direction
of smallness, while in the other direction there is no infinite
magnitude. The reason is that what is one is indivisible whatever it
may be, e.g. a man is one man, not many. Number on the other hand is
a plurality of 'ones' and a certain quantity of them.
Hence number must stop at the indivisible: for 'two' and 'three' are
merely derivative terms, and so with each of the other numbers. But in
the direction of largeness it is always possible to think of a larger
number: for the number of times a magnitude can be bisected is infinite.
Hence this infinite is potential, never actual: the number of parts
that can be taken always surpasses any assigned number. But this number
is not separable from the process of bisection, and its infinity is not
a permanent actuality but consists in a process of coming to be, like
time and the number of time.
With magnitudes the contrary holds. What is continuous is divided ad
infinitum, but there is no infinite in the direction of increase. For
the size which it can potentially be, it can also actually be. Hence
since no sensible magnitude is infinite, it is impossible to exceed
every assigned magnitude; for if it were possible there would be
something bigger than the heavens.
The infinite is not the same in magnitude and movement and time, in
the sense of a single nature, but its secondary sense depends on its
primary sense, i.e. movement is called infinite in virtue of the
magnitude covered by the movement (or alteration or growth), and time
because of the movement. (I use these terms for the moment. Later I
shall explain what each of them means, and also why every magnitude
is divisible into magnitudes.)
Our account does not rob the mathematicians of their science, by
disproving the actual existence of the infinite in the direction of
increase, in the sense of the untraversable. In point of fact they
do not need the infinite and do not use it. They postulate only that
the finite straight line may be produced as far as they wish. It is
possible to have divided in the same ratio as the largest quantity
another magnitude of any size you like. Hence, for the purposes of
proof, it will make no difference to them to have such an infinite
instead, while its existence will be in the sphere of real magnitudes.
In the fourfold scheme of causes, it is plain that the infinite is
a cause in the sense of matter, and that its essence is privation,
the subject as such being what is continuous and sensible. All the
other thinkers, too, evidently treat the infinite as matter -- that is
why it is inconsistent in them to make it what contains, and not what
is contained.
Part 8
It remains to dispose of the arguments which are supposed to support
the view that the infinite exists not only potentially but as a
separate thing. Some have no cogency; others can be met by fresh
objections that are valid.
(1) In order that coming to be should not fail, it is not necessary
that there should be a sensible body which is actually infinite. The
passing away of one thing may be the coming to be of another, the All
being limited.
(2) There is a difference between touching and being limited. The
former is relative to something and is the touching of something (for
everything that touches touches something), and further is an attribute
of some one of the things which are limited. On the other hand, what
is limited is not limited in relation to anything. Again, contact is
not necessarily possible between any two things taken at random.
(3) To rely on mere thinking is absurd, for then the excess or defect
is not in the thing but in the thought. One might think that one of
us is bigger than he is and magnify him ad infinitum. But it does not
follow that he is bigger than the size we are, just because some one
thinks he is, but only because he is the size he is. The thought is
an accident.
(a) Time indeed and movement are infinite, and also thinking, in the
sense that each part that is taken passes in succession out of existence.
(b) Magnitude is not infinite either in the way of reduction or of
magnification in thought.
This concludes my account of the way in which the infinite exists, and
of the way in which it does not exist, and of what it is.