1. Like Empedocles, Anaxagoras denied that there could be any coming into being or passing away (17=B17):
    The Greeks are wrong to accept coming to be and perishing, for no thing comes to be, nor does it perish, but they are mixed together from things that are and they are separated apart. And so they would be correct to call coming to be being mixed together, and perishing being separated apart.

  2. And, like Empedocles, he thinks that there is genuine qualitative difference among things. But, unlike Empedocles, he does not limit himself to 4 elements. In their place, he helps himself to an infinity of different stuffs.

  3. For Anaxagoras, not only Earth, Air, Fire, and Water, but also blood, gold, hair, bone, etc., are all elemental, not reducible to more primitive parts.

    Why does he hold this? Cf. Robinson (p. 176):

    According to Empedocles, bone is made up of earth, air, fire, and water, blended in a certain proportion. It should be possible, therefore, to break it down again into these elements. The difficulty is that when this is done the bone ceases to be bone any longer; and if Parmenides is right, this is impossible. If bone is, it cannot cease to be.

  4. Moreover, Anaxagoras seems to have reasoned that if bone is made up of those elements in that proportion, you should be able to generate bone out of something else, that is not bone. But Anaxagoras tried to be a good Parmenidean. As he writes (10=B10):
    For how could hair come from not hair or flesh from not flesh?

  5. How does Anaxagoras propose to get out of this difficulty? Robinson again:
    Anaxagoras sought to evade this difficulty by insisting that bone is homoiomerous, i.e., made up of parts having the same nature as the whole. No matter how far it is broken down, what remains is bone.

    Bone is not made up of other elements. Every part of bone has the same nature as the whole. Every part of bone is bone; every part of gold is gold, etc. This is Anaxagoras’s notorious principle of Homoiomereity (or uniformity, lit. “like-partedness”):

    (H) Every part of any kind of stuff, S, is itself S.

    It is controversial whether Anaxagoras maintained (H), which is not asserted in any fragments. There is evidence for it in the testimonia, however. Cf. Aristotle’s summaries in 25=A43 (Aristotle, De Caelo 302a28-b3) and 26=A46 (Aristotle, On Coming-to-be and Ceasing-to-be 314a18):

    25: Anaxagoras says the opposite of Empedocles about the elements.  . . . For the homoiomerous things (I mean flesh and bone and each of the things of that sort) are elements, but air and fire are mixtures of these and of all the other seeds, for each of them is a conglomeration of invisible [amounts] of all the homoiomerous things.

    26: Anaxagoras . . . says that the elements are unlimited in number.  For he makes the elements homoiomerous things, such as bone and flesh and marrow, and each of the others whose parts have the same name [as the whole].

  6. Thus, Anaxagoras saw the Empedoclean theory as subject to Parmenidean objections: the creation of a compound out of its elemental parts, or the division of a compound into those parts, would be deriving what is from what is not, or what is not from what is. Cf. KRS, p. 370:
    Anaxagoras seems to have felt that Empedocles had not gone far enough. If everything consisted solely of the four elements, then in putting together the four elements in different proportions to form, say, flesh or bone, Empedocles had not, to Anaxagoras’s mind, succeeded in eliminating the coming-into-being of something new.

    So Anaxagoras’s principle (H) is designed to enable him to deny the existence of real generation and destruction.

    1. Possible Empedoclean reply: compounds (of the elements) aren’t real. The realities are the elements. We should not describe a mixing of elements as a coming to be.

    2. Anaxagoras’s rejoinder: this leaves Empedocles with a narrow ontology that flies in the face of common sense. Empedocles would have to deny that such things as bone and blood are real.

  7. But it certainly appears as if bone can come from what is not bone, or flesh from what is not flesh. E.g., a chicken eats corn, and drinks water, and what comes to be is flesh and feathers. How, then, can we avoid having to admit that there are cases of coming into being from what is not? Anaxagoras solved this problem with the second important principle of his physical theory. He held that there is already flesh, and feathers, etc., in the corn. As KRS continue:

    The only way to do that [i.e., to succeed in eliminating the coming-into-being of something new] was to posit in everything the presence ab initio of everything which might emerge from it.

    This might give us a principle like this: if stuff y can come from stuff x, there must be a portion of y already in x.

    But Anaxagoras went further. For he seems to have imagined that there are no limits on what kinds of changes are possible. That is, anything can come from anything. This leads to the principle of Universal Mixture (11=B11, cf. also 6=B6):

    In everything there is a portion of everything ....

    This is most plausibly construed as a principle about kinds of stuff (cf. Barnes, The Presocratics, pp 320-323):

    (UM) For any kinds of stuffs, S, S': in each piece of S there is a portion of S'.

    That is, every piece of gold contains portions of wood, flesh, hair, water, silver, etc. Anaxagoras’s principle (UM) is designed to enable him to allow for the existence of real change without allowing for real generation and destruction.

  8. (UM) is certainly a bizarre principle, however well motivated it may be. Some critics have found it worse than bizarre. They think it flatly contradicts (H). Thus, Cornford:
    Anaxagoras’s theory of matter rests on two propositions which seem flatly to contradict one another. One is the principle of homoiomereity: a natural substance such as a piece of gold consists solely of parts which are like the whole and like one another - every one of them gold and nothing else. The other is: “There is a portion of everything in everything,” understood to mean that a piece of gold . . . so far from containing nothing but gold, contains portions of every other substance in the world.

  9. Is Cornford right? Or can Anaxagoras, without contradiction, maintain that every part of a lump of gold is itself gold and still maintain that every piece of gold contains portions of all the other stuffs (including ones which are not gold)?

  10. Various ways out have been proposed.

    1. Guthrie: “homoiomerous” was Aristotle’s word, not Anaxagoras’s. Anaxagoras’s elements were the natural stuffs that Aristotle called homoiomerous. But Anaxagoras didn’t think they were like-parted. Rather, they contained everything. So Anaxagoras doesn’t hold (H).

    2. Vlastos: what’s present in every kind of stuff is every opposite, not every kind of stuff. So (UM) doesn’t contradict (H).

  11. But are (H) and (UM) really incompatible? If it can be shown that they are compatible after all, we will not have to look for a way out in the manner of Guthrie or Vlastos.

    Let us reconsider the argument for saying that (H) and (UM) are inconsistent. Suppose we have a lump of gold, L. (UM) tells us that it is a universal mixture, and (H) tell us that all of its parts are gold. So we consider some part of it, P. (H) tells us that P is gold and that P, like L, is an instance of universal mixture - it contains every kind of stuff within it. Why not? There does not seem to be any inconsistency yet. L is both gold and contains a mixture within, and so is P. Evidently, the trouble is thought to come when we consider the parts of P. Are they also both gold and instances of universal mixture? The underlying assumption seems to be that one can maintain that the parts of L are both gold and instances of universal mixture only until one reaches some part, P*, which cannot be further broken down into parts. What can we say about P*? (H) says it is gold, and (UM) says it is a mixture. But how can it be a mixture if it has no parts? So (H) and (UM) cannot both be true about P*.

    But notice that this argument implicitly assumes that a process of division of L will ultimately lead us to an indivisible part, P*. And it is P* that we have shown cannot be both homoiomerous (i.e., pure gold) and a universal mixture.

    So our argument to show that (H) and (UM) are inconsistent has implicitly been assuming a third principle: that every process of division comes to an end - that matter is only finitely divisible.

  12. But Anaxagoras did not assume this. Indeed, he explicitly denied it. Cf. 3=B3:
    For of the small there is no smallest, but always a smaller.

    This is Anaxagoras’s principle of Infinite Divisibility. There are no atoms.

    [A question that naturally arises here is the relation of Anaxagoras to Zeno. For more on this issue, see this supplementary note.]

    As applied to the argument at hand, the principle of  Infinite Divisibility means that we will never reach a part, P*, about which we get the contradictory result that it contains no parts but still satisfies (UM).

  13. But this still leaves us with a puzzle: how can L, our lump of gold, have other things (e.g., silver, lead, sugar) in it, when all of its parts are gold? If no part of L is silver, how can there be silver in the gold? Or flesh in the corn?

    To answer this question, we must distinguish between the (physically discriminable) parts of a substance and the portions of stuff which it contains.

    The distinction between parts and portions is one that only a non-atomist can coherently make. The idea is simple. Suppose we mix flour, water, chocolate, and eggs together, and bake them into a cake. There is a portion of flour in the cake, and a portion of eggs, etc. But no matter how finely we divide the cake up, we will not recover the flour or the eggs. As Anaxagoras would say, “every part of the cake is cake.” We are not forced to say whether the ultimate particles we arrive at are cake-particles or flour-particles, for there are no ultimate particles making up the cake.

    So the reason why there can be silver in a lump of gold, L, even though every part of L is gold, is that not every portion is a part. There are portions of silver, etc., in L even though every part of L is gold.

    (H) tells us that every part of S is itself S.

    (Every part of the cake is cake.)

    (UM) tells us that for every pair of kinds of stuff, S and S', there is a portion of S in S'.

    (There is a portion of every kind of stuff in the cake. And there is a portion of every kind of stuff in every part of the cake. But every part of the cake is cake.)

  14. But (UM) still sounds odd. If everything is in everything, why isn’t every kind of stuff the same as every other kind? How can there be different kinds? The answer is in Anaxagoras’s principle of Predominance:
    (P) Each kind of stuff is called after the ingredient of which it contains most.

    Cf. 24=A52 (Aristotle, Physics 187b2-6):

    . . . things appear to differ from each other and are called by different names from one another based on what is most predominant in extent in the mixture of the infinitely many [components]. Nothing is purely or as a whole pale or dark or sweet or flesh or bone, but whatever each contains the most of is thought to be the nature of that thing.

  15. Many commentators have found (P) to be an incoherent principle. [Strang, AGP (1963), rebutted by Kerferd (“Anaxagoras and the concept of matter before Aristotle,” on reserve).]  The problem, simply put, is that the definition of, e.g., gold as that stuff in which gold predominates, is wildly circular. Gold cannot be defined as that stuff which contains gold as its dominant ingredient. For how do we define the gold which is the ingredient? There are two possible responses on Anaxagoras’s behalf.

    1. One is Kerferd’s (pp. 501-2), according to which Anaxagoras was not attempting to give definitions, but to describe change:
      It is true to say that we cannot give an account of substances such as gold by analyzing them into “a predominance of gold” and so on to infinity. In such a case we have failed to give either a satisfactory definition or a satisfactory account of gold because we have included the term gold in our attempts at definition and description. But it is not an objection to any position maintained by Anaxagoras, as he had no reason to attempt a definition or a description of gold in this way. He is concerned with change and not with description or definition.

    2. Another response appeals once again to the distinction between parts and portions. We cannot define gold as the stuff in which gold predominates. What we can do is to say that the nature of any (naturally occurring) stuff (or of any part of it) is that of its predominant (pure) portion. So what makes this hunk of metal gold is the fact that gold is the element of which it has the largest portion. But the question: “What makes the gold portion of this hunk of gold gold?” is meaningless.

  16. Problems with the theory of Nutrition:

    One reason Anaxagoras maintained (UM) was to account for our ability to take in nourishment. We eat wheat, and our flesh increases. When we eat too many chocolate chip cookies, our bodies bulk up with flesh, not with chocolate chip cookies. The idea is that we extract the flesh already present in the food we eat. An ancient scholiast (Aetius, I, 3, 5) describes the theory (portion of A46 not reprinted in RAGP):

    We take in nourishment that is simple and homogeneous, such as bread or water, and by this are nourished hair, veins, arteries, flesh, sinews, bones and all the other parts of the body. Which being so, we must agree that everything that exists is in the nourishment we take in, and that everything derives its growth from things that exist. There must be in that nourishment some parts that are productive of blood, some of sinews, some of bones, and so on - parts which reason alone can apprehend.

    But notice that the theory of nutrition requires that wheat contain not just portions of flesh, but physically removable parts that are flesh. Unless the flesh that’s in the wheat (as a part, or a portion - i.e., in the wheat in some sense of “in”) can be extracted and join the flesh of the body, then one’s flesh will not, according to the theory, bulk up from eating wheat.

    But if the wheat contains removable parts that are flesh, principle (H) seems to collapse: not every part of wheat will be wheat. Some parts will be flesh.

    One might think to save Anaxagoras by appealing to (UM). For any fleshy part of the wheat that is extracted will, by (UM), contain portions of everything, including wheat. So even the fleshy parts of the wheat are still, at least in part, wheat.

    But this will not do. For (P) tells us that only a mixture in which wheat predominates is wheat. If wheat is only a minority ingredient in some fleshy part, then that part is flesh and not wheat. A minority element in a mixture does not contribute to the determination of the nature of that mixture. (H) and (P) together entail that flesh must predominate in all of the homoiomerous parts of flesh.


    (H), (UM), and (P) are logically consistent: they do not entail a contradiction. But our way of showing them to be consistent reveals that they are incompatible with the theory of nutrition that has been attributed to him.

    {(H), (UM), (P), + Anaxagoras’s theory of nutrition} leads to a contradiction.

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