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

Negation and Negativity

“Hegel is willing to say some extraordinary things about the concept he sometimes calls ‘negation’, sometimes ‘negativity’. What he has been taken to mean has been the source of most of the criticism of Hegel: that he confused logical negation with actual opposition, as in the oppositions of forces or magnitudes in general in the world; that he thought everything in the world contradicted itself, and so believed that pairs of contradictory judgments could both be affirmed; or simply that what he said about negation and contradiction cannot be coherently understood. And any commentator must face the fact that he invoked the notion of negation in many different contexts in many different ways. So the first task is to have in view that variety of contexts before we can understand what they all might have in common (if anything)” (Pippin, Hegel’s Realm of Shadows, p. 139).

Some time ago I touched on the most elementary parts of this thorny issue in Contradiction vs Polarity. My basic sense at this point is that Hegel does indeed have very worthwhile things to say on this very distinctive theme of his, but that the standard connotations of his core vocabulary for talking about it — negation, negativity, contradiction — are so impoverished relative to what he really means in his extensive and varied metaphorical uses of it that the vocabulary does not communicate well.

Pippin identifies five somewhat overlapping contexts that ought to be taken into account in a serious interpretation of Hegelian negation: the nature of thinking; the talk about freedom; the nature of intelligibility; the notion of speculative truth; and the talk about contradiction.

In the context of the nature of thinking, he says “In the simplest sense, we are talking about the logical structure of apperceptive intentional knowledge, as well as the ontological status of agency. What it means to claim that the intelligibility of any content of empirical knowledge is not… wholly ‘positive’… is best understood by contrast. If it were not so and were wholly positive, subjectivity would be something like a mere complex registering and responding device (of the same ontological status as a thermometer) ” (p. 141).

“As we have seen, one is not simply wholly absorbed in the presence of the world to one, not wholly and merely reactive to the stimulation of sensibility, and that ‘not’ is the beginning (but certainly not the end) of all the logical issues of negation that emerge in Hegel’s philosophy, at both the phenomenological and the logical level. In making such a judgment I ‘negate’ the mere immediacy or givenness of perceptual content, negate it as immediate and putatively given, and take up, am always taking up, a position of sorts about what is there, what is the case” (p. 142).

The suggestion is that all “taking” of things to be thus-and-such for Hegel implicitly involves a negation of immediacy.

“What thinking is is such a ‘negation’ of one’s immediate ‘positive’ state. (One can say: this negation of mere immediacy is ‘taking a stand’, rather than being put into a state.) Any thinking could be a seeming-to-be-the-case, not what is the case, and that possibility is constitutive of the act’s being a judging in the first place…. The constitutive feature attended to in a Hegelian philosophical logic is the fact that judgments are potentially responsive to reasons and revisions just qua judgments” (ibid).

Pippin thus cites responsiveness to reasons as another non-obvious instance of negation for Hegel.

“And in being an agent, I am not simply causally responsive to inclinations and desires; there is no ‘fullness of positive being’ here either. I interrupt or negate positive being (what I feel inclined to do, experience as wanting to do) by deliberating and resolving what to do” (p. 143).

Deliberation is another non-obvious instance.

“As noted before, the closest first and general approximation of what he means is Aristotelian: subjectivity (thinking and acting according to norms) is the distinct being-at-work… of the biological life form that is the human, reason-responsive substance; this in the same sense in which Aristotle says, if the eye were body, seeing would be its form, its distinctive being-at work” (ibid).

Subjectivity in general is another.

“Instead of thinking of the fundamental act of understanding as a synthesis of independent, originally unrelated elements, either by subsuming an individual under a concept, or by including one concept under another, we should understand ourselves, both in experience and in logical reflection, beginning with ‘wholes’, never with experiential or logical subsentential simples [individual words] or atoms” (pp. 143-144).

Hegel presents this last point as a sort of inversion of Kant, and it does speak to some of Kant’s language. But I would argue that the true starting point of Kantian synthesis is not experiential or logical atoms, but rather the sensible manifold in intuition, which is only potentially differentiable. This brings Kant closer to Hegel. True, the manifold has such a loose unity that arguably it might have trouble qualifying as a “whole”, but even less does it consist in already predifferentiated “atoms”. In Kant I think we have no basis for identifying putative atoms in experience until “after” the figurative synthesis of imagination has done its work. (It is the preconscious, in a non-Kantian sense “spontaneous” figurative synthesis that for Kant creates an inevitable gap between being and thinking, and that for Hegel gives us the fixed representations relied upon by common sense.)

In the context of freedom, Pippin says that “In the same way that judging, insofar as it is genuine, holds open the possibility of its negation or disconfirmation, just by being judging, not by virtue of any second, reflective act…, a deliberation about action, if it is to be a deliberation, is open to the force of reasons the agent has already accepted by deliberating at all, a possibility criterial for his acting at all…. [Hegel] is insisting on the logical or categorical requirements of the normative, and in that sense (the sense in which freedom is normatively constrained judgment and rational action), the negative (here only the possibility of not doing what I am powerfully inclined to do) is ‘that by which a person is free’. (And he does not mean any uncaused causality, but that which counts to the subject as a reason” (p. 147).

Normativity in general thus counts as depending on a kind of negation.

“In terms of the structure of the Logic, what Hegel will want to argue is that we cannot adequately explain freedom if we consider just a determinate property that some beings happen to have…, and we cannot explain it either as a kind of essential ideal, manifest in but never adequately expressed in its appearances, in concrete individual actions…. We need the logic of the Concept, in which concepts are said to ‘give themselves their own content’ and be ‘self-determining’ in a way indebted to [the] Kantian claim on self-legislation…. Any philosophical determination of actuality must be understood as ‘self-legislating’ in the broad sense [that] reason relies only on itself in determining such a normative structure. These are not empirical questions. There is no flash of ‘essence intuition’ (Wesenschau) giving access to a world of abstract immaterial objects, essences” (p. 148).

Self-legislation too involves some subtle kind of negation.

In the context of the nature of intelligibility, Pippin says “Every determination of every sort of content in the Logic is a negation of some insufficient determinacy that must be able to be conceived positively” (p. 153). This is another way of characterizing the basis of the notorious “movement” in Hegel’s logic.

The determination of content for sure involves negation.

In the context of speculative truth, he mentions that “[E]valuative judgments, like ‘this action is good’, or our familiar ‘this is a bad house’, will be paradigmatic examples of judgment in the logic of the Concept. They do not qualitatively specify a thing by distinguishing it from other things; they do not identify the appearances that show the ‘essence’ of the thing; they understand the content ‘in terms of its concept'” (p. 154).

Evaluation of anything against its concept is another subtle variety.

In the context of the notorious “contradiction”, Pippin says “So in a general sense, one has to say that a thing… ‘includes’ its contrary, or more precisely its relation to its contrary, in order to be, and be known to be, what it is. Neither of these ‘moments’ of negation involves contradiction in the Aristotelian sense because ‘is’ and ‘is not’, while said of the same thing and at the same time, are not meant in the same sense. But Hegel wants for various reasons to call such an analyzed state a ‘contradiction’, and there is some ground in the use of the term for saying that…. In the simplest sense, personifying the process, what someone intends to say, means to say, can be ‘contradicted’ by what is actually said, what he finds he has to or can say” (p. 156).

Evaluation of any outcome against intent is another.

Pippin notes that Hegel says retrospectively at the end of the Logic that the abstract treatment typical of the logic of being and the logic of essence is due to “mere opinion” or lack of awareness of what it is actually treating. “It cannot say what it means to say” (p. 157). This is yet another way of characterizing the Logic‘s “movement”.

Logical “movement” is another.

“Hegel’s speculative notion of contradiction is not predicate or sentence negation…. [H]e means to focus attention on concept negation…. Now concepts, understood as rules, have content by being understood to have the content that they have. To understand a rule is to know how to use it, and in using it, to know one is following it…. So understanding… is not… ‘grasping a content’ but understanding what the rule instructs us to think. That is, the concept is always already a moment of discursive activity, a thinking through of its implications” (pp. 158-159).

“[T]he considerations discussed so far should not be understood to be matters of formal clarity…. As Hegel insisted, we are not studying how we think about (or talk about) matters (or even how we ‘must’ think). The question is a question about ‘any possible intelligibility’, and so about being in its intelligibility, … not ‘our ways’ of rendering intelligible” (p. 160).

All the above varieties of “negation” involve normativity in one way or another. Here Pippin again emphasizes the universality of criteria of intelligibility that Hegel counterposes to the Kantian gap between thinking and being.

“The forms of intelligibility are the forms of what could be true, although they do not settle the question of what, in particular, is true” (p. 175).

“[O]ur first orientation in trying to understand Hegelian negation should be not the logical operation of predicate or sentence negation, but real opposition…. [Michael] Wolff also thinks that the controversy in the eighteenth century about positive and negative magnitudes, especially as it surfaced in Kant, as well as the emerging clarity about negative numbers, played a far larger role in the development of Hegel’s thinking about negation and contradiction than did a reflection on the logic of the formal operator” (pp. 177-178).

Real opposition is an instance of contrariety, which is actually much more relevant to Hegel than formal logical contradiction.

“Hegel’s category of becoming, so important at the beginning of the logic of being, owes much to Hegel’s defense of Newton and the latter’s doctrine of the becoming equal of magnitudes” (pp. 178-179).

“Critics like Crusius were aghast at the idea of forces having positive and negative values. But Kant understood that such ‘values’ (and here again an important precedent for Hegel) had those values in relation to each other, not absolutely, that they were relative values, arbitrarily reversible even” (pp. 179-180).

Negative magnitudes are another instance of contrariety.

“This is not the sense of contradiction throughout the Logic, but it gives us enough background… to appreciate that Hegel is neither a lunatic for saying that ‘everything is contradictory’, nor a mystical Heraclitean” (p. 180).

Pippin devotes nearly half of this chapter to critical remarks about Brandom’s interpretation of Hegelian negation as material incompatibility, which involves a more nuanced form of contrariety. He says Brandom’s reading very well captures the meaning of negation in Hegel’s logic of being, but is inadequate for what Hegel goes on to do in the logic of essence and the logic of the concept. For example, material incompatibility alone is insufficient to explain things like the “self-legislation” of the concept or the idea of the good, but Hegel at least claims that these have something to do with the concept of negation.

Brandom’s interpretation of Hegelian negation seemed to me incomparably clearer than Hegel himself, so until now I have adopted it enthusiastically as a charitable rendering of what Hegel ought to have said to better express his meaning. This is the first of many counter-arguments to Brandom I have seen that really seems to me to at least raise a serious question, but for now I will forego another lengthy tangent.

Next in this series: Deeper Truth