One, Many, Same, Different

Book Iota (X) of Aristotle’s Metaphysics extends the discussion in book Delta (V) of things said in many ways, going into detail on what makes things one, many, the same, or different. These are extremely important matters for any sound reasoning, though somewhat technical in nature. It also contains Aristotle’s sharp critique of the saying of Protagoras, “Man is the measure of all things”.

“[T]here are four senses in which something is said to be one primarily and in its own right, rather than incidentally” (ch.1, p. 185).

As to the first, “oneness belongs to what is continuous, either simply or, especially, by nature, and not by contact or a binding cord (and of these that is more so one and is more primary of which the motion is more indivisible and simple)” (ibid).

Continuity in a material is the lowest degree of something being one in its own right. That which is materially united by contact or by any artificial means would not be one in its own right.

As to the second, “[oneness] belongs still more to what is whole and has some form and look, especially if something is of that sort by nature and not by force, as those things are that are so by means of glue or bolts or being tied with a cord, but rather has in itself that which is responsible for its being continuous. And something is of this sort if its motion is one and indivisible in place and time; and so it is clear that, if something that has a source of motion that moves it in the primary kind of the primary class of motions (by which I mean a circular type of change of place), this is one magnitude in the primary sense” (ibid).

Being a whole is a higher degree of unity than being materially continuous. The association of circular motion with a strong unity is relevant to the upcoming argument in book Lambda (XII) about the first cause.

“So some things are one in this way, insofar as they are continuous or whole, but others are one because the articulation of them is one, and of this sort are those things of which the thinking is one, and this in turn is of this sort if it is indivisible, and an act of thinking is indivisible if it is of something indivisible in form or in number” (ibid, emphasis added).

As to the third, “a particular thing is one by being indivisible in number” (ibid, emphasis added).

These would include all the independent and non-independent things that were extensively discussed in book Zeta (VII).

He continues, “but that which is one by means of intelligibility and knowledge is indivisible in form, so [fourth] what is responsible for the oneness of independent things would be one in the primary sense” (ibid, emphasis added).

I expect that this last kind will turn out to be the first cause.

“But it is necessary to notice that one must not take the sorts of things that are spoken of as one as being meant in the same way as what it is to be one, or what the articulation of it is” (p. 186).

This is analogous to the distinction between saying something about something in general, and saying what something is, also discussed in book Zeta. He illustrates this below, with the example of fire.

“The same thing would also be the case with ‘element’ and ’cause’, if one had to speak about them, distinguishing the things to which the words are applied, and giving a definition of the words. For there is a sense in which fire is an element… and a sense in which it is not; for being fire is not the same thing as being an element…. And it is that way also with ’cause’ and ‘one’ and all such things, and this is why being one is being indivisible, just exactly what it is to be a this, separate on its own in either place or form or thinking, or to be both whole and indivisible, but especially to be the primary measure of each class of things, and, in the most governing sense, of the class of things with quantity, for it has come from there to apply to other things” (ibid).

Being one in the third sense above (being a particular thing) is now said to be “just exactly what it is to be a this“. This foreshadows an extensive one-to-one mapping he will develop below, between all the ways of the saying of being he elaborates in book Delta (V), and the ways of the saying of oneness.

He goes on to speak at some length about measures, which we would call units of measurement.

The most important point is that “a measure is always the same kind of thing as what it measures, for the measure of magnitudes is a magnitude, and in particular, that of length is a length, of breadth a breadth, of spoken sounds a spoken sound, of weight a weight, and of numerical units a numerical unit” (p. 188, emphasis added).

The distinction he makes here tracks perfectly with the way that different types, dimensions, and variables are handled separately in the operations defined by modern mathematics.

“And we speak of knowledge or sense perception as a measure of things for the same reason, because we recognize something by means of them, although they are measured more than they measure” (ibid).

The simultaneously humorous and serious caveat that “they are measured more than they measure” means that knowledge and perception are constrained by reality. More precisely, they are involved in mutual dependencies with the realities of things that they at once measure and are measured by.

“And Protagoras says a human being is the measure of all things, as if he were saying that a knower or perceiver were the measure, and these because the one has knowledge and the other perception, which we say are the measures of their objects. So while saying nothing, these people appear to be saying something extraordinary” (ibid).

Aristotle, at the end of his discussion of measure (longer and more detailed than included here), refers to Protagoras’ famous saying, commonly quoted as “Man is the measure of all things”. With uncharacteristic sharpness, he calls this “saying nothing”. Why? This seems worthy of a short digression.

Protagoras was a prominent Sophist, who appears in Plato’s dialogue of the same name. He wrote a controversial treatise entitled Truth, which began with the sentence, “Of all things the measure is man: of those that are, that they are; and of those that are not, that they are not”. Plato and Aristotle both took Protagoras to be asserting a kind of individualist relativism. Reportedly, the skeptic Sextus Empiricus also read him this way. Plato took him to deny any objective reality. Aristotle in book Gamma (IV) shows rare outrage at his other reported claim that “it is not possible to contradict”. I didn’t mention him by name in my account, but Aristotle names him there as well.

Aristotle implies it is a bit more plausible to much more specifically say that knowledge and perception are measures of things, but even that only with the caveat that really “they are measured more than they measure”.

But why go on to add with such sharpness that Protagoras is “saying nothing”? I suspect the answer may lie with the other part of Aristotle’s interpretation: “as if he were saying that a knower or perceiver were the measure, and these because the one has knowledge and the other perception”.

I’m inclined to think Aristotle would regard it as outrageous to transfer what may (ambiguously) be said of specific knowledge and perception, to an unqualified saying about one who is said to possess knowledge and perception in general.

Aristotle has just given an account of what a measure is, that it must be of the same kind as the things that it measures. How could one thing (“man”) possibly be the measure of countless things that have no common measure? That does sound like nonsense. Any measure has to be of one kind or another. Modern mathematics agrees that kinds, dimensions, variables can only be collapsed together if very specific criteria are met.

What Aristotle emphasizes here about knowledge and perception — that they have a remarkable involvement with things that also measure them — seems very consistent with his general views. But the way Protagoras implicitly presents “man” as their possessor makes the possessor stand apart from the mutual involvement with things that Aristotle emphasizes, in which knowledge and perception and things measure one another. In splendid isolation from all constraints of real measure, the possessor seems to have arbitrary freedom to claim whatever she wishes.

I intuitively associate Protagoras with the attitude that what’s true for you is true for you, but what’s true for me is true for me, so don’t tell me I’m wrong, and I won’t tell you you’re wrong! Then and now, such sentiments had and have a superficial appeal, because they seem to express a live-and-let-live attitude, which seems to be a good thing. But the way it is expressed, in fact it completely undermines any possibility of meaningful dialogue, which undermines reason itself, which undermines the very thing that makes us human.

Back to the text, Aristotle turns to illustrating the one-to-one mapping I mentioned between sayings about being and sayings about oneness.

“[S]ince not even being itself is an independent thing as though it were some one thing capable of having being apart from the many beings (since it is common to them), other than solely as a thing attributed to them, it is clear that oneness is not a universal either” (ch. 2, p. 189).

For Aristotle, neither Being nor the One is an independent thing in its own right. He will nonetheless argue in book Lambda (XII) that there is a first cause for all things.

“What’s more, what is true about oneness must hold true in a similar way for all things; and being and oneness are meant in equally many ways” (ibid).

“And the same account applies also to the other classes of things, … and [if] in all instances it is the case both that the number is a number of something and that oneness is some particular one thing, and oneness itself is not the thinghood of it, then it must also be the same way with independent things” (p. 190).

“[S]o too in thinghood, one independent thing is oneness itself; and that oneness in a certain way means the same thing as being, is clear from the fact that it follows along equally through the ways being is attributed, and is not any one of them (for instance, it is not what anything is, nor of-what-sort anything is, but stands similarly toward them just as being does), and from the fact that no other thing is predicated in ‘one human being’ over and above what is predicated in ‘human being’ (just as being is not something over and above what and of-what-sort and how-much a thing is), or in ‘being one’ over and above being any particular thing” (ibid).

Saying something is one human being is the same as saying it is a human being. In book Zeta, he analogously said that “good” and “being good” are the same.

Things are often one in one respect, but many in another. The one and the many are “contraries, and not opposed as contradictories or as what are called relative terms” (ch. 3, p. 190). Next he turns to the meanings of same, other, and different.

“Since the same is meant in more than one way, in one way we sometimes speak of what is the same in number, but we say it in another sense if things are one in meaning as well as in number, as you are one with yourself in both form and material, and in another again if the articulation of the primary thinghood of things is one, for instance in the way equal straight lines are the same, … but in these equality is oneness” (p. 191).

In Fregean terms, things are what Aristotle calls one in number when they have the same reference. They are what Aristotle calls one in meaning when they have the same sense. Frege illustrates how they differ by saying that the morning star and the evening star have the same reference, but different sense.

“Things are alike if, not being simply the same, nor without difference in their composite thinghood, they are the same in form, just as a larger square is like a smaller one…. Other things are alike if they have the same form, and have it in them to be more and less, but are neither more nor less than one another. Other things, if they are the same attribute, and one in form, say white, more and less intensely, people say are alike because their form is one. Other things are alike if they have more things the same than different, either simply or superficially, as tin is like silver insofar as it is white, and gold is like fire insofar as it is yellow and fiery-red” (ibid).

“So it is clear also that other and unlike are meant in more than one way” (ibid).

“[B]ut difference is something other than otherness…. [W]hat is different from something is different in some particular respect, so it must necessarily be the same in some respect as that with which it differs” (p. 192).

For Aristotle, calling two things different presupposes that they can be meaningfully compared in the first place. This is not the case for things that are just “other” than one another. Black is different from white; they are both colors. But an apostrophe is just other than an aardvark.

This is followed by a ten-page discussion of technical details of contrariety, which I will mostly skip. Contrariety supports a definitive ruling out of some things based on other things. This has great importance for reasoning.

Incidentally, Aristotle explains why there is no essential difference between human beings with different colorings.

“And since one sort of thing is articulation while another is material, those contrarieties that are in the articulation make a difference in species, but those that are in what is conceived together with the material do not make such a difference. This is the reason that whiteness of a human being, and blackness, do not make such a difference…. For there is a difference between a white human being and a black horse, but that is not insofar as the one is white and the other is black” (ch. 9, pp. 201-202).

Finally, he argues that destructibility and indestructibility inhere in things by necessity. Whether a thing is destructible or not depends strictly on what genus of things its species belongs to.

“[A] destructible thing and an indestructible thing must be different in genus…. Therefore it is necessary that destructibility either be the thinghood or be present in the thinghood of each destructible thing; and the same argument also concerns the indestructible, since both are among things present by necessity” (ch. 10, p. 202).

Next in this series: Toward a First Cause

Reflection and Dialectic

As with dialogue, reflection provides a kind of model for dialectic. Reflection can be understood as an either metaphorical or literal dialogue with ourselves. We “question ourselves”, which is to say we examine and potentially criticize or refine the basis of our own commitments. Further, actual dialogue is always implicitly dialogue among fellow rational beings, all of whom are engaged at least to some extent in their own reflective activity, just by virtue of being rational beings, so dialogue implicitly presupposes reflection.

Pippin quotes Hegel: “But at issue here is neither the reflection of consciousness, nor the more specific reflection of the understanding that has the particular and the universal for its determination, but reflection in general…. For the universal, the principle or the rule and law, to which reflection rises in its process of determination is taken to be the essence of the immediate from which the reflection began…. Therefore, what reflection does to the immediate, and the determinations that derive from it, is not anything external to it but is rather its true being” (Hegel’s Realm of Shadows, pp. 238-239).

And again: “In general, this means nothing but this: Anything which is, is to be considered to exist not as an immediate, but as a posited; there is no stopping at immediate determinate being [Dasein] but a return must rather be made from it back into its ground, and in this reflection it is a sublated being and is in and for itself. What is expressed by the principle of sufficient reason is, therefore, the essentiality of immanent reflection as against mere being” (p. 239).

In the first quote, Hegel is again emphasizing that what he means by reflection is not just looking in a figurative mirror, but rather something more like finding an orientation among (or building a synthesis of) the potentially infinite mutual reflections in a hall of mirrors. Reflection “in general” is a name Hegel gives to reflection with this kind of potentially infinite dimension. (That the infinity here should be called potential is my friendly Aristotelian interpolation.)

In the second quote, he is saying that this kind of reflection — lifted out from the distinction between reflective activity and what it reflects upon — is what he would call the “truth” of everything that appears to be immediately determinate.

The principle of sufficient reason as formulated by Leibniz effectively says that for everything that is in some definite way, there is a reason why it is that way. Hegel is saying more specifically that such “reasons why” emerge immanently from the reflective grounding of what he is in a nonstandard way calling essence. What Leibniz cannot show is how a particular essence or monadic point of view results in certain predicates and not others; despite great sophistication, he is still to some extent using essence and monads as unexplained explainers to avoid what Hegel calls the “problem of indifference”. Hegel on the other hand explicitly makes essence and explanation interdependent.

“[T]he ‘principles’ of identification and differentiation are deeply intertwined, not independent of each other” (p. 240).

“[A] thing’s determinate properties are not, cannot be, a mark of that thing’s unlikeness from other things, just by being those properties…. If one thing is red and another square, we do not thereby know one is unlike the other; they are just two different things. A locomotive has nothing to do with a melody; it is not unlike a melody. We are trying to account not for determinate otherness, as in the logic of being, but for how objects that share properties (are like) could be, even with an extraordinary degree of such likeness, still unlike” (p. 241).

“Some of this anticipates topics in the logic of the Concept. Two trees are alike in being trees but unlike in being two individual trees. The idea will be that just in their likeness, their way of being alike, that they are unlike (different trees), just in the way each distinctly instantiates ‘treeness’ that they are unlike. Such a different ‘way of being a tree’ is not another property but the way the tree-properties are ‘had’ by the individual” (p. 242).

“Hegel is thinking of the way in which the specifying work of ‘unlikeness’ cannot be a matter of individual properties, atomistically conceived, but unlikeness within likeness is best understood as some content, the unlikeness of which is strict, even within such likeness. Some charge can be both positive and negative; some number, 4, can be both +4 and -4; some quantity of money can be an asset and also a debt pending; some force can be attractive and repelling; some distance marched east is canceled by the same distance marched west, and all these are ‘opposed’ only within some common likeness” (ibid).

I find “either-or” language more appropriate to these cases than the “both-and” language above, but the intent is the same. The distinctions in each sub-case are concrete “opposites” applicable to some specific context, and each definable only in reference to the other. In each case, it is possible to abstract an indifferent thing being measured or assessed — “positive-or-negative-quantity” for the one, and “virtue-or-vice” for the other.

“The ‘world’s being contradictory’ means nothing more than that, as he says, virtue cannot be virtue just by being other than, different from, in comparison with, vice, but only by ‘the opposition and combat in it’ against vice” (p. 243, emphasis in original).

Pippin complicates the matter with this example, because “relative” seems to have a different significance in the context of virtue and vice than it does in, say, that of positive and negative numbers. But the intended point is a very abstract one about constitution of meanings that is common to both cases. Whatever the difference between the two “oppositions” (positive/negative, virtue/vice), in each case the two sub-terms are somehow measured or assessed “against” one another.

“Hegel is trying to specify how affirming contrary predicates (‘in opposition’) does not amount to a logical contradiction. That is the point of his discussion, to make this distinction, not to treat such oppositions as if they were logically contradictory and then to affirm them anyway. As [Michael] Wolff puts it, Hegel’s orientation… is not from sentence or predicate negation, but from developments in the understanding of negative numbers and from Kant’s defense of Newton on positive and negative magnitudes. In general, then, mathematical, not logical negation” (pp. 243-244).

This is extremely important. The status of negative numbers was still controversial in Hegel’s time. Kant and Hegel contributed to their acceptance. Hegel struggled to invent new language to distinguish ambiguous cases in his Logic and to say reasonable things about them, but readers (certainly including myself) have found his unique idioms very hard to follow. Most of the ink spilled over “contradiction” in Hegel has been based on fundamental misunderstandings. (See also Negation and Negativity.)

“To use an empirical example, if the question is something like ‘Why did the ball fall to the ground?’ we want to avoid two kinds of answers: ‘because whenever a heavy object is dropped from a height, it falls’; and ‘it is in the nature of heavy things to fall’. Doing so, avoiding these alternatives, will allow us to see that the relation between a ‘ground’ and ‘what it grounds’ must be understood as a dynamic relation, one whereby the determinacy of the ground and that of the grounded cannot be fixed in isolation from each other” (pp. 245-246).

He quotes Hegel: “But the being that appears and essential being stand referred to each other absolutely. Thus concrete existence is, third, essential relation; what appears shows the essential, and the essential is in the appearance. — Relation is the still incomplete union of reflection into otherness and reflection into itself; the complete interpenetrating of the two is actuality” (p. 246).

“The general point [Hegel] keeps making is: a strict separation of the two moments, and an insistence that the nature of an appeal to an essence, or to a causal law, or to someone’s reason for acting cannot be understood as punctuated moments on the billiard-ball model of causation, but involve a kind of unity, the development of a kind of unity, much closer (yet again) to Aristotle on energeia. This essential-being-as-activity, manifesting itself in its appearances, is what should count as ‘actuality’. This has the implication that many existing things have no actuality, are not really ‘anything’. A lump of dirt, a cough, a strand of wire” (ibid).

“The question for Hegel is the question of ‘actuality’, not ‘existence’, or the sensibly apprehensible, just as for Aristotle, the question is the ‘really real’, to ontos on.” (p. 247).

Pippin quotes from the Encyclopedia Logic, “The logical is to be sought in a system of thought-determinations in which the antithesis between subjective and objective (in its usual meaning) disappears. This meaning of thinking and of its determinations is more precisely expressed by the ancients when they say that nous [“intellect”, or thought in a non-psychological sense] governs the world” (p. 248).

Here “governs” is meant in a constitutive sense. The important point is that the “thought-determinations” here are indifferent to the ordinary distinction between a subjective “thinker” and an objective “thought content”.

In this context he speaks of “this dynamical relation, this Ur-relation of all relations” (ibid).

“I have tried to show in another book that the most important, most clarifying implications of this Ur-relation occur in Hegel’s practical philosophy, both in his account of human agency, and in the implications of that account for the practical theory of freedom in his Philosophy of Right and theory of objective spirit in general (ibid).

“[T]he full demonstration of the truth of this Ur-relation lies in what it actually illuminates, in the cogency and credibility of, for example, an account of agency based on it” (p. 249).

It was the outstanding (and very Aristotelian) account of Hegel’s view of agency in Hegel’s Practical Philosophy that first attracted me to Pippin’s work.

In his own idiom, Hegel says “thus the inner is immediately the outer, and it is this determinateness of externality for the reason that it is the inner; conversely, the outer is only an inner because it is only an outer” (quoted, ibid.)

Pippin comments, “He does not mean here anything as obvious as: when I do something, my ex ante intention is fulfilled and so becomes something outer, just as what was done, the bodily movement, counts as an action because it expressed this ex ante intention. The passage does not say that the inner becomes the outer, nor that the outer is the expression of the inner. It says: there is no ex ante intention except as outer. It is the outer. And there is no outer except as what must count as inner, nor that it expresses a separable inner. There is no such separation” (pp. 249-250; see also Hegel on Willing).

This concludes Pippin’s chapter on the logic of essence. Unsurprisingly, we have not uncovered any magic formula that would tell us which appearances manifest the essence in particular cases. Such a thing seems completely impossible to me; we should not expect to be able to find any general formula covering an unspecified collection of particulars. Any judgments involving particulars must in part at least come back to something like Aristotelian deliberation and practical judgment, which yield only particular results.

Nonetheless, in discussing the logic of essence we have ruled out some important classes of misunderstandings, and we have set the stage for the climax of Hegel’s Logic in the “logic of the concept”. The logic of the concept will take as a starting point the non-separation of “inner” and “outer” that has been shown in the logic of essence.

Next in this series: “The” Concept?