Lecture 9 — Refutation of Anaxagoras’ opinion about an infinite number of principles

The theory of Anaxagoras that the principles are infinite in multitude was probably due to his acceptance of the common opinion of the physicists that nothing comes into being from non-being.

For this is the reason why they use the phrase “all things were together,” and the coming into being of such and such a kind of thing is reduced to change of quality, while some spoke of combination and separation.

Moreover, the fact that the contraries proceed from each other led them to the conclusion.

The one, they reasoned, must have already existed in the other, for since everything that comes into being must arise either from what is or from what is not, and it is impossible for it to arise from what is not (on this point all the physicists agree),

they thought that the truth of the alternative necessarily followed, namely, that things come into being out of existent things—that is, out of things already present but imperceptible to our senses because of the smallness of their bulk.

So they assert that everything has been mixed in everything, because they saw everything arising out of everything.

But things, as they say, appear different from one another and receive different names according to the nature of the particles that are numerically predominant among the innumerable constituents of the mixture.

For nothing, they say, is purely and entirely white or black or sweet, bone or flesh, but the nature of a thing is held to be that of which it contains the most.

Now the infinite as infinite is unknowable, such that what is infinite in multitude or size is unknowable in quantity and what is infinite in variety of kind is unknowable in quality. But the principles in question are infinite both in multitude and in kind. Therefore it is impossible to know things that are composed of them, for it is when we know from what and from how many it is composed that we suppose we know a complex.

Further, if the parts of a whole may be of any size in the direction either of greatness or of smallness (by “parts,” I mean components that are actually present and into which a whole can be divided), it is necessary that the whole thing itself may be of any size. Clearly, therefore, since it is impossible for an animal or plant to be indefinitely big or small, neither can its parts be such, or the whole will be the same. But flesh, bone, and the like are the parts of animals, and the fruits are the parts of plants. Hence it is obvious that neither flesh, bone, nor any such thing can be of indefinite size in the direction either of the greater or of the less.

Again, according to the theory, all such things are already present in one another and do not come into being, but are constituents that are separated out, and a thing receives its designation from that which abounds in it. Further, anything may come out of anything—water by separation from flesh and flesh from water.

Hence, since every finite body is exhausted by the repeated abstraction of a finite body, it seems obviously to follow that everything cannot subsist in everything else.

For let flesh be extracted from water, and again, let more flesh be produced from the remainder by repeating the process of separation; then, even though the quantity separated out will continually decrease, still it will not fall below a certain magnitude.

If, therefore, the process comes to an end, everything will not be in everything else (for there will be no flesh in the remaining water); if, on the other hand, it does not, and further extraction is always possible, there will be an infinite multitude of finite equal particles in a finite quantity—which is impossible.

Another proof may be added: since every body must diminish in size when something is taken from it, and flesh is quantitatively definite in respect both of greatness and smallness, it is clear that no body can be separated out from the minimum quantity of flesh, for the flesh left would be less than the minimum of flesh.

Finally, in each of these infinite bodies, there would be already present infinite flesh and blood and brain—having a distinct existence, however, from one another, and no less real than the infinite bodies, and each infinite. This is contrary to reason.

The statement that complete separation never will take place is correct enough, for passions are indeed inseparable. If, then, colors and states had entered into the mixture, and if separation took place, there would be a “white” or a “healthy” that was nothing but white or healthy, that is, was not the predicate of a subject.

So, the intellect which seeks these impossibilities is absurd, if he truly wishes to separate such things and it is impossible to do so, both in respect of quantity and of quality—of quantity, because there is no minimum magnitude; and of quality, because passions are inseparable.

Nor is Anaxagoras right about the generation of things of the same species. It is true there is a sense in which clay is divided into pieces of clay, but there is another in which it is not. Water and air are and are generated from each other, but not in the way in which bricks come “from” a house and again a house “from” bricks.

And it is better to assume a smaller and finite number of principles, as Empedocles does.

58. Having set forth the opinions of the natural philosophers concerning the principles, he here pursues one of these opinions, namely, that of Anaxagoras. For this opinion seemed to assign a common cause for all the species of motion.

The discussion is divided into two parts.

In the first part, he sets forth Anaxagoras’ argument;

in the second part, he raises objections against it, at now the infinite (187b7; [64]).

Concerning the first part, he makes three points.

First, he sets forth those things that Anaxagoras supposed and from which he argued.

Second, at the one, they reasoned (186a32; [62]), he sets forth the order of his argument.

Third, at but things, as they say (187b2; [63]), he sets forth Anaxagoras’ response to a certain tacit objection.

59. Anaxagoras assumed two things from which he argued. The first of these is a point that is assumed by all of the natural philosophers, namely, that nothing comes to be from nothing. And Aristotle says that, because of this, Anaxagoras seemed to have held the opinion that the principles are infinite. For he accepted as true the common opinion of all philosophers of nature, namely, that what simply is not in no way comes to be. For they assumed this as a principle and then developed their different opinions.

60. Lest they should be forced to hold that something new comes to be that previously was in no way at all, some held that all things from the beginning existed together, either in some one confused state, as Anaxagoras and Empedocles held, or in some material principle, such as water, fire, and air, or some intermediate between these.

And, in accordance with this, they posited two modes of production.

Those who held that all things preexisted together as in one material principle said that coming to be is nothing other than being altered. For they said that all things come to be from that one material principle through its condensation and rarefaction.

Others, however, who held that all things preexisted together in some one confused state and mixture of many, said that the coming to be of things is only a joining together and a separation.

All of these philosophers were deceived because they did not know how to distinguish between potency and act. For being in potency is, as it were, a mean between pure non-being and being in act. Therefore, those things that come to be naturally do not come to be from non-being simply, but from being in potency, and not, indeed, from being in act, as they thought. Hence things that come to be did not necessarily preexist in act, as they said, but only in potency.

61. Next, at moreover, the fact that (187a31), he mentions the second thing that Anaxagoras assumed. Anaxagoras said that contraries come to be from each other. For we see the cold come to be from the hot, and vice versa. And from this he concluded that, since nothing comes to be from nothing, one of the contraries preexists in the other. And this is true, of course, in respect to potency. For the cold is in the hot in potency, but not in act, as Anaxagoras thought. For he was not aware of being in potency, which is a mean between pure non-being and being in act.

62. Next, where he says, the one, they reasoned (187a32), he sets forth the deductive order of the argument. Anaxagoras proceeded as follows. If something comes into being, it is necessary that it should come to be either from being or from non-being.

But he excluded one of these alternatives—namely, that something should come to be from non-being. He did this because of the common opinion of the philosophers mentioned above.

From this, he concluded that the remaining alternative was correct, namely, that a thing comes to be from being. For example, if air comes to be from water, then air preexisted. For it cannot be said that air comes to be from water unless air preexisted in water.

Hence he wished to say that everything that comes to be from something preexisted in that from which it comes to be.

But, because this seemed to be contrary to what appears to the senses (for it is not apparent to the senses that that which is generated from something preexists in it), he forestalled this objection by holding that that which comes to be from something preexists in it as certain most minute parts that are not sensible to us because of their smallness. For example, if air comes to be from water, certain minute parts of air are in the water, but not in that quantity in which it is generated. And so he said that, by the gathering together of these parts of air by themselves, and by their separation from the parts of water, air comes to be.

Having accepted, therefore, that everything that comes to be from something preexists in it, he further assumed that everything comes to be from everything. Hence he concluded that everything would be mixed in everything else as minute, non-sensible parts. And, because an infinite variety of things can come to be from another, he said that there are infinite, minute parts in each thing.

63. Next, at but things, as they say (187b2), he heads off a certain unspoken objection. It is possible for someone to object as follows. If infinite parts of everything are in everything, it would follow that things neither differ from each other nor appear to differ from each other. Therefore, as if he were answering this objection, Anaxagoras says that things appear to differ from each other and are diversely named because of that which is dominant in them, even though there is an infinite number of minute parts contained in any mixture. And so nothing is purely and totally white or black or bone. Rather, that which abounds in each thing seems to be the nature of that thing.

64. Next, at now the infinite (187b7), Aristotle refutes the above mentioned position.

Concerning this, he makes two points.

First, he disproves the position absolutely.

Second, at and it is better (188a17; [74]), he compares it to the opinion of Empedocles.

Concerning the first part, he makes two points.

First, he sets forth arguments to disprove the opinion of Anaxagoras.

Second, at the statement that (188a5; [72]), he disagrees with Anaxagoras’ way of understanding his own position.

Concerning the first part, he gives five arguments.

The first of these is as follows. Every infinite thing is unknown insofar as it is infinite. He explains why he says as infinite. If it is infinite in respect to multitude or magnitude, it will be unknown with respect to quantity. If, however, it is infinite with respect to species (for example, if it is composed of an infinite variety of species), then it will be unknown according to quality. And the reason for this is that what is known by the intellect is grasped by the intellect with respect to all that belongs to that thing. But this cannot happen with regard to something infinite. If, therefore, the principles of a thing are infinite, they must be unknown either with respect to quantity or with respect to species. But, if the principles are unknown, those things that are from the principles must be unknown. He proves this as follows. We think that we know any composite when we know from what and from how many it is composed, that is, when we know both the species and the quantity of the principles. It follows, therefore, from first to last that, if the principles of natural things are infinite, then natural things are unknown either with respect to quantity or with respect to species.

65. At further, if the parts (187b13), he gives the second argument, which is as follows. If the parts of a whole do not have a determinate quantity, either great or small, but can be any size, either great or small, it is necessary that the whole not have a determinate greatness or smallness. Rather, the whole could have any size. This is so because the quantity of the whole comes from the parts. (But this must be understood of the parts existing in act in the whole, as flesh and nerve and bone exist in an animal. Hence he says, by “parts,” I mean components that are actually present and into which a whole can be divided. And by this, he excludes the parts of a continuous whole that are in the whole in potency.)

But it is impossible that an animal or a plant or some such thing be related indeterminately to any size, whether great or small. For there is some quantity so large that no animal exceeds it in size. So also there is some quantity so small that no animal is found to be smaller. And the same must be said of plants. Therefore, by denying the consequent, it follows that the parts are not of indeterminate quantity. For what is true of the whole is true of the parts. But flesh and bone and things of this sort are parts of an animal, and fruits are parts of plants. Therefore, it is impossible that flesh and bone and such things should have an indeterminate quantity, either greater or smaller. Therefore, it is not possible that there should be certain parts of flesh or bone that are non-sensible because of smallness.

66. It seems, however, that what is said here is contrary to the statement that a continuum is divisible to infinity. For if the continuous is divisible to infinity, and flesh is, indeed, a kind of continuum, it seems that flesh is divisible to infinity. Therefore, some part of flesh, according to a division to infinity, goes beyond every determinate smallness. But it must be pointed out that, although a body is divisible to infinity when considered mathematically, the natural body is not divisible to infinity. For in a mathematical body, nothing but quantity is considered. And in this there is nothing repugnant to division to infinity. But, in a natural body, the form also is considered, which form requires a determinate quantity, and also other accidents. Hence it is not possible for quantity to be found in the species of flesh except as determined within some termini.

67. He gives the third argument at again, according to the theory (187b21).

Concerning this, he makes two points.

First, he sets forth certain things that are the basis of the argument.

Second, at for let flesh (187b27; [68]), he sets forth the deductive order of the argument.

Concerning the first part, he proposes three things.

The first is that, according to the position of Anaxagoras, as was said above, all things are together. And, from this, Aristotle wishes to reduce Anaxagoras’ argument to absurdity. For Anaxagoras said, as was pointed out, that all such things—that is, all things that are of like parts, such as flesh and bone and the like—are in each other and do not come to be from nothing, but are separated from that in which they preexist. And each thing is named from that which abounds in it, that is, from the largest number of parts existing in the thing.

The second point is that everything comes to be from everything, as water comes to be by separation from flesh and flesh likewise comes to be from water.

The third point is that every finite body is exhausted by a finite body. If, from some finite body, however large, a finite body, however small, is taken away, the smaller can be taken away from the larger until eventually the greater whole is consumed by the smaller through division.

And, from these three points, Aristotle concludes what he primarily intended, namely, that each thing is not in each thing. And this is contrary to the first of these three points. For this is how the reductio ad absurdam works: the conclusion contradicts some premise.

68. Next, at for let flesh (187b27), he develops his argument and assumes what was concluded in the preceding argument. He says that, if flesh is removed from water (since flesh is generated from water), and if again another separation of flesh is made from the remaining water, then, although there will always remain a smaller quantity of flesh in the water, still the size of that flesh will not fall below a certain magnitude; that is, there happens to be a certain small measure of flesh than which there will not be any smaller flesh, as is clear from the argument given above.

Therefore, having established that there is some smallest particle of flesh than which there is no smaller, he proceeds as follows. If flesh is separated from water, and again other flesh, either the process of separation will stop or it will not. If it stops, then there is no flesh in the remaining water, and everything will not be in everything. If it does not stop, then some part of flesh will always remain in the water.

Thus, in the second separation, the remaining flesh is smaller than in the first, and in the third, it is smaller than in the second. And, since we cannot proceed to infinity in smallness of parts, as was said, then the smallest parts of flesh are equal and infinite in number in some finite body of water. Otherwise, separation could not proceed to infinity.

It follows, therefore, that, if the separation does not stop, but flesh is always removed from water to infinity, then, in some finite magnitude—namely, water—there are certain things that are finite in respect to quantity and equal to each other, but infinite in respect to number—namely, the infinite, smallest parts of flesh. But this is impossible and contrary to what was said above, namely, that every finite body is reduced by some finite body. Therefore, the first point—namely, that everything is in everything, as Anaxagoras held—is also impossible.

69. We must note that it is not without reason that the Philosopher used the term equal in stating the last absurdity to which this position leads. For if the definition of quantity is considered, it is not absurd that an infinity of unequal parts be in a finite body. For if a continuum is divided according to the same proportion, it will be possible to proceed to infinity, such as if we take a third of a whole, and then a third of the third, and so on. In this case, however, the parts were not taken as equal in quantity. But, if the division is made according to equal parts, we will not be able to proceed to infinity even if we consider only the account of quantity that is found in a mathematical body.

70. He gives his fourth argument at another proof may be added (187b35). The argument is as follows. Every body becomes a smaller one when something is taken from it, because every whole is greater than its parts. Since, then, the quantity of flesh is determinately great or small, as is clear from what was said above, there must be some smallest bit of flesh.

Therefore, nothing can be separated from this, because the remaining flesh would be smaller than this smallest piece of flesh. Therefore, it is impossible that everything comes to be from everything by separation.

71. At finally, in each (188a2), he gives his fifth argument, which is as follows. If infinite parts of each thing are in each thing and everything is in everything, it follows that infinite parts of flesh and infinite parts of blood and brain are in an infinite number of bodies. And, regardless of how much is separated, the same amount would always remain. Therefore, it would follow that the infinite is in the infinite infinitely. But this is unthinkable.

72. Next, at the statement that (188a5), he disproves the position of Anaxagoras according to Anaxagoras’ own understanding of it.

He does this in two ways.

First, he shows that Anaxagoras did not understand his own position.

Second, at nor is Anaxagoras (188a13; [73]), he shows that Anaxagoras did not have sufficient evidence for holding this position.

Therefore, he says first that, although Anaxagoras has in a certain respect spoken the truth, he himself did not understand what he said when he held that the process of separation would never end. For accidents can never be separated from substance, yet he held that there was a mixture not only of bodies, but also of accidents. When something becomes white, he said that this happened by an abstraction of white from the previously existing mixture.

If, then, colors and other accidents of this sort are mixed together, as he said, and if someone says, on this supposition, that all things that are mixed can be separated, it would follow that there would be white and healthy but no subject of which these are predicated and in which they are. But this is impossible. Therefore, the truth is that, if accidents are in the mixture, it is impossible that all mixed things can be separated.

Another problem results from this. Anaxagoras held that all things were mixed from the very beginning, but intellect began to separate them. Now, any intellect that attempts to do what cannot be done is not worthy of the name “intellect.”

Hence, that intellect will be inconsistent, intending the impossible, if he truly wishes, that is, wishes to separate things completely. For this is impossible both from the point of view of quantity—because there is no smallest magnitude, as Anaxagoras said, since something can be subtracted from any small quantity—and from the point of view of quality, because accidents are not separable from their subjects.

73. Next, at nor is Anaxagoras (188a13), he disproves this position by reason of the fact that Anaxagoras did not have sufficient evidence. Since Anaxagoras saw that a thing is made large by the coming together of many small parts that are similar, as a stream is made from many brooks, he believed this to be the case for all things.

And thus Aristotle says that Anaxagoras did not correctly understand the generation of things of the same species; that is, he did not understand that a thing is always generated by things that are similar in respect to species.

For some things are both generated from and are resolved into things like unto themselves, as clay is divided into pieces of clay, but in other instances, this is not so. For some things are generated from that which is dissimilar.

And, in these instances, there is not merely one mode of production. For some things are made by alteration from that which is unlike, as the sides of a house are made from clay and not from sides; but other things are made by composition, as the house is not made of houses, but of sides.

It is in this way that air and water come to be from each other, that is, as from the unlike.

Another reading here is as the sides are from the house. And thus he sets forth a twofold way in which things come to be from the unlike: through composition, as the house is made of sides, and by resolution, as the sides come to be from the house.

74. Next, at and it is better (188a17), he disproves the position of Anaxagoras by comparing it with the opinion of Empedocles. He says that it is better to make the principles smaller in number and finite, as Empedocles does, than to make them many and infinite, as does Anaxagoras.