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ON THE VERY CONCEPT OF HARMONY IN LEIBNIZ.

IT IS WELL KNOWN THAT LEIBNIZ'S NOTION OF HARMONY plays a crucial role in his philosophical system. Leibniz drew on this concept of harmony in motivating, and explaining, numerous areas of his thought: everything from Leibnizian mathematics and metaphysics to ethics and social philosophy, incorporates the notion of harmony as a central descriptive and explanatory concept. While there has been much discussion of some the applications of harmony in Leibniz's system--especially the mind-body harmony, and the so-called universal harmony of simple perceiving substances--little attention has been paid to Leibniz's very concept of harmony. Yet it seems important for understanding Leibniz's holistic philosophy that we grasp this concept, for he employed the term "harmony" when describing so many features of the created world: there is a harmony between geometry and physics,(1) between efficient and final causation,(2) between nature and grace,(3) to mention just a few.

In this paper, I attempt to elucidate Leibniz's notion of harmony through an examination of the relevant texts. In the first section, I present and discuss Leibniz's favorite characterization of harmony. The analysis there reveals that harmony is not merely an ontological notion, but that it also calls for analysis from an epistemological point of view. Specifically, it calls for analysis from the point of view of Leibniz's theory of relations, and his doctrine of distinct versus confused cognition. Thus, in sections two through five, I present and discuss some texts which explicitly link these features of his thought with the notion of harmony. Throughout, I shall attempt, insofar as the texts allow, to formulate necessary and sufficient conditions for a collection of entities to be in a state of harmony. I conclude in the sixth section on a negative note, pointing out what seem to me to be problems of interpretation.

I

Leibniz's writings on the concept of harmony are in short supply, and most are from early in his career.(4) Although the terminology he used to define harmony sometimes varied from writing to writing, the idea seems to have remain fixed in his thinking from early to late. In various writings from the early 1670s, we find Leibniz defining harmony as "a similitude in dissimilar things" (similitudo in dissimilibus), and referring to that which is harmonious as "uniformly difform" (Harmonicum est uniformiter difforme).(5) Yet he most frequently defined harmony as "diversity compensated by identity" (diversitate identitate compensata) or as "unity in variety" (unitas in varietate).(6) His most elaborated discussion of the concept appears years later in the Elementa Verae Pietatis, Sive De Amore Dei Super Omnia (16797):
 Harmony is unity in variety.... Harmony is when many things are reduced to
 some unity. For where there is no variety, there is no harmony. Conversely,
 where variety is without order, without proportion, there is no harmony.
 Hence, it is evident that the greater the variety and the unity in variety,
 this variety is harmonious to a higher degree.(7)


In all these formulations, although the terminology may slightly change, the idea expressed seems to be the same.

The idea seems to be that in order for some collection of distinct entities to harmonize with one another, they must stand in some mutually unifying relationship. The question is what Leibniz means by these entities being reduced to some unity. Little help is available, it seems, in his suggestion that the variety must be with "order" and "proportion." Nevertheless, based on the above passage, we might commit Leibniz to the following:
 (H) For any set S of entities, and some relation R which holds among the
 members of S, S is harmonious if and only if:

 (i) S contains more than one member;

 (ii) R unifies the members of S above a certain standard of order.


Condition (i) is motivated by Leibniz's claim that "where there is no variety, there is no harmony." I take the implication to be that the existence of more than one entity is a necessary condition for the existence of harmony. Condition (ii) is motivated by his claim that "where variety is without order, without proportion, there is no harmony." I take the implication to be that "order" (or "proportion") is a necessary condition for the collection of entities to harmonize.(8) Finally, I take the upshot of the entire passage to be that (i) and (ii) are individually necessary, yet jointly sufficient for the relevant set to be in a state of harmony.(9)

(H), of course, is far from illuminating. But it appears to be all that we are going to get from the passage quoted above. (i) is clear enough; it is (ii) which calls for explanation. What kind of "unifying," or "orderly" relation does Leibniz have in mind here?

Matters get more puzzling when we consider Leibniz's statements about order, the most famous of which is found in Discourse on Metaphysics [sections] 6. The heading of that section claims that "God Does Nothing Which Is Not Orderly and It Is Not Even Possible to Imagine Events That Are Not Regular." The ensuing discussion aims to bolster the latter of the two claims:
 [N]othing completely irregular occur[s] in the world, [and] we would not
 even be able to imagine such a thing. Thus, let us assume, for example,
 that someone jots down a number of points at random on a piece of paper, as
 do those who practice the ridiculous art of geomancy. I maintain that it is
 possible to fred a geometric line whose notion is constant and uniform,
 following a certain rule, such that this line passes through all the points
 in the same order in which the hand jotted them down.... But when a rule is
 extremely complex, what is in conformity with it passes for irregular.(10)


Leibniz made the same point, this time explicitly in the language of order, in A Specimen of Discoveries About Marvelous Secrets (1686):
 For just as no line can be drawn, with however casual a hand, which is not
 geometrical and has a certain constant nature, common to all its points, so
 also no series of things and no way of creating the world can be conceived
 which is so disordered that it does not have its own fixed and determinate
 order and its laws of progression though as in the case of lines, so also
 some series have more power and simplicity than others, and so they provide
 more perfection with less equipment.(11)


These passages illustrate the need for an adequate understanding of the order so central to Leibniz's notion of harmony since they suggest that any set of entities maintain some kind of order.(12) Indeed, if we take Leibniz's claim--that one cannot even imagine or conceive a set of entities failing to exhibit some kind of order--to indicate metaphysical impossibility, then we easily come away with the conclusion that Leibniz thought it necessarily true that any set of entities exhibit some kind of orderly relationship.

But surely Leibniz did not think that any set of entities may be said to exhibit some level of harmony. Whereas in the above quoted passages, Leibniz suggests that order is a somewhat trivial notion in that any set of entities may be said to exhibit some degree of order, in his discussions of harmony, it is clear that he has in mind a certain kind of order, or perhaps, a certain degree of order. For example, we have already seen Leibniz's claim that "where variety is without order ... there is no harmony," which suggests that some collections of entities do lack order, or better, a certain kind of order. Thus, the upshot seems to be that although any collection of entities exhibits order to some degree (Discourse on Metaphysics [sections] 6), only certain collections of entities achieve the requisite level of ("unifying") order, worthy of the label "harmonious" (as suggested in the passage from the Elementa). Our question is, then: what is that level, or kind, of order?

A natural place to begin a search for this answer is with Leibniz's definition of order. He seems to have defined it consistently throughout several writings. Consider the following sample of definitions:
 Order is the relation of several things, through which any one of them can
 be distinguished from any other.(13)

 Order is the connection [le rapport] of a variety of relations [rapports]
 which are born from a multitude of terms or ingredients, of the sort that
 it is possible to distinguish each of these ingredients from all the
 others. (14)

 Order is the relation between many things, in which any one of them is
 distinguished from any other.(15)


The definitions are roughly the same: the essence of an ordered relationship is one where the related elements, in virtue of their mutual relationship, can be distinguished from each other. Now while it is not clear from these definitions alone whether Leibniz here has in mind the merely trivial conception of order from Discourse on Metaphysics [sections] 6, or the order necessary for harmony, there is reason to think that these definitions reflect (or, at least, come close to reflecting) the kind of order necessary for harmony. Consider the following from A Resume of Metaphysics (1697, the same time period as the above quoted definitions of order):
 Distinct cogitability gives order to a thing and beauty to a thinker. For
 order is nothing other than a distinctive relation of several things;
 confusion is when several things are present, but there is no way of
 distinguishing one from another.(16)


In this passage, Leibniz characterizes order as a "distinctive relation." Presumably, the point, based on the previous definitions of order, is that any one of the entities so ordered can be distinguished in thought, in virtue of its relationship, from all the others. On the other hand, I take the final leg of the passage to state the converse: when several entities cannot be so distinguished in thought, the requisite level of order is not present. Further, I take it that Leibniz would agree that although the several entities present cannot be distinguished, that does not imply that they do not admit of some degree of order or some sort of mutual relationship. Not only does this seem true in itself, it is suggested by Leibniz's writing a few lines later that "pain contains something disordered, though only relative to the percipient; for in the absolute sense all things are ordered."(17)

We can reach the conclusion that the above characterizations of order reflect the kind of order necessary for harmony through the following considerations. Leibniz frequently linked the concept of harmony to that which is "cogitable,"(18) or "contemplatible"--"the state where everything has the potential for observation," or a state which "extricates [the mind] from confusion."(19) Now given that order is itself characterized by Leibniz as imposed by distinct cogitability, and this, the above passage suggests, is to be contrasted with confusion, we can see that harmony and order are both associated with the concept of cogitability, or of what is thinkable or potentially observable. It is not too far a leap to infer, it seems, that although any given set of entities will be ordered in some sense, they are not harmoniously ordered unless they exhibit the level of distinct cogitability. The lesson, then, seems to be that the kind of order necessary for harmony is that which gives rise to distinct cogitability, the kind, that is, where one can distinguish the harmonized entities. We shall see further evidence that this is Leibniz's view in the next section.

To summarize our results thus far, we have seen that according to Leibniz, whether or not a set of entities is harmonious depends on two factors: first, there must be more than one entity in the set; second, the entities must be unified by a certain relation and according to a certain standard of order. We have also seen that Leibniz suggests that any collection of entities exhibits some level of order. Thus, the question then becomes: what is the necessary standard of order for there to be harmony? The answer, we have just seen, is that although any collection is ordered in some absolute sense, only those which are found to be in a distinctive relationship with one another, that is, only those which are such that the entities can be distinguished in thought--in virtue of their relationship--from one another, achieve a level of harmony. In other words, only those collections which achieve the level of distinct cogitability are harmonious.

Two points must be stressed for our purposes here. First, although for the most part we have been concerned with the concept of harmony as an ontological notion, Leibniz's discussions of the concept of order clearly suggest that there is an epistemological component to be considered. That is, whether or not a set of entities is harmonious depends on how those entities are capable of being related in thought. This is a clear consequence of his claim that it is distinct cogitablity which gives the requisite level of order to a thing. Second, and more importantly, order is an essentially relational notion. In every definition of order that we have examined, order is explicitly called by Leibniz a relation. But it is well known that Leibniz believed that relations themselves are real only insofar as they exist in thought or only insofar as they exist in a mind. Again, there is an epistemological component to Leibniz's notion of harmony, for relations themselves are not substantially real beings for Leibniz but only objects of thought. Hence, we will do well to look at Leibniz's notion of harmony via a brief look at his epistemology and his views on relations.

II

Given what was established in the last section--that order, and therefore harmony, are essentially relational notions--it is perhaps no surprise that Leibniz's discussion of harmony in the Elementa Verae Pietatis continues the way it does. For right after defining harmony as "unity in variety," Leibniz continues:
 Harmony is the perfection of cogitability, insofar as there are cogitable
 things [quatenus cogitabilia sunt]. Harmony is when many things are reduced
 to some unity. For where there is no variety, there is no harmony.
 Conversely, where variety is without order, without proportion, there is no
 harmony. Hence, it is evident that the greater the variety and the unity in
 variety, this variety is harmonious to a higher degree. Hence, dissonances
 themselves increase pleasantness, if suddenly they are reduced to agreement
 with other dissonances. The same is true of symmetries. Hence, it is
 clearly evident that Harmony is perfect cogitability. For it is said ...
 that that is more perfect in which there is more reality. Thinking however
 is a kind of reality, and so much the greater because the thing thought of
 is in a way multiplied.... Hence, that is a more perfect manner of
 thinking, where one act of thought extends to many things simultaneously:
 so there is more reality in that thought. This is done, moreover, with the
 work of relations [Hoc autemfit opere relationum], for a relation is a kind
 of unity in multiplicity. And species of relations are the bonds and
 reasons [rationes] of things among themselves, the proportions, the
 proportionalities. It is from all of these taken at the same time in a
 given object that harmony results [resultat]. Therefore, the more relations
 there are in a thinkable object (the aggregate of which is harmony), this
 has more reality, or what is the same, there is perfection in the thought.
 Therefore, it follows that Harmony is cogitability, and, of course, to the
 extent that there are cogitable things, perfection.(20)


This passage has the virtue of bringing out the essential relational features of the concept of harmony.(21) We are told that harmony is "the perfection of cogitability"--terms strikingly similar to those used to describe the kind of order necessary for harmony. More importantly, we are given further insight into how harmony comes about, or how it results from a collection of entities.

To begin with, Leibniz tells us that harmony results in a given object when all the "species of relations" (which he also refers to as "proportions") in it are "taken at the same time." Formulating this claim in more familiar terms, we may say that harmony results from a given collection of entities (in this case relations) when they are simultaneously considered by a mind. Moreover, Leibniz tells us that this kind of thinking--the kind "where one act of thought extends to many things simultaneously"--is done with the work of relations. Perhaps what is most revealing here is that Leibniz explicitly calls a relation "a kind of unity in multiplicity," a description which is used throughout his writings, as we have seen, to characterize the very notion of harmony.

A brief look at some of Leibniz's views on relations will prove helpful in sorting through the above passage.

III

Even a cursory look at certain passages on relations from Leibniz helps to shed light on the way he is elaborating his concept of harmony in the passage from the Elementa. First, we must notice that Leibniz was no Platonist when it comes to relations.(22) That is, according to Leibniz, relations are not substantially real entities, but are rather entia rationis, or things which exist only in a mind:
 Relations and orderings are to some extent "beings of reason" [entia
 rationis], although they have their foundations in things; for one can say
 that their reality, like that of eternal truths and of possibilities, comes
 from the Supreme Reason.(23)

 The reality of relations is dependent on mind, as is that of truths; but
 they do not depend on the human mind, as there is a supreme intelligence
 which determines all of them from all time.(24)


The second thing to note about these passages is that Leibniz does not think that just because relations are mind-dependent, that implies that they are completely unreal. Rather, what (nonsubstantial) reality they have is grounded in the divine mind, the mind which sees at once all the relations between things even prior to creating them. In this connection, neither does the mind-dependence of relations imply that they are purely subjective, for the objectively true description of the relations that obtain in the world lies in the divine mind. But more than that, relations are not Durely subjective because, as Leibniz tells us, "they have their foundations in things." They are grounded in objective qualities of things that are perceived by the mind, and hence are only "to some extent" entia rationis.(25) In short, relations are not simply subjective features superadded to things by a mind. "Although relations are the work of the understanding," Leibniz writes, "they are not baseless and unreal."(26) Although they are not themselves substantially concrete things, they are grounded in concrete things. And the divine mind gives relations whatever (nonsubstantial) reality they may possess, and, in doing so, provides the true description of the relations which obtain among things in the world.(27)

Moreover, relations, often described by Leibniz as concogitabilitas--the word Leibniz uses to describe that act of the mind which is a thinking together of several things at once(28)--are also frequently referred to as results:
 Besides substances, or ultimate subjects, ... there are relations, which
 are not produced per se but result when other things are produced, and have
 reality in our intellect--in fact, they are there when nobody is thinking.
 For they get that reality from the divine intellect, without which nothing
 would be true.

 Bare relations are not created things, and they are solely in the divine
 intellect, ... and such are whatever result from posited things.... Hence,
 they are not entities, for every entity which is not God is created;
 rather, they are truths.(29)


The point seems to be that relations are not themselves created entities, but rather result from things which are created. They result, that is, from created things being concogitabilitatis, or from created things being thought about together in a certain way by a mind.(30) Again, the reality of these relations is grounded--not in real mind-independent existence--but in the fact that relations are grounded in objective qualities of things, and so the omniscient divine mind sees all the ways that entities may be considered together. Hence, the reality of relations lies in the divine mind.

We may summarize what we have just covered by saying that Leibniz is committed to the following:
 (R) For any set S of entities, and any relation R which holds among the
 members of S, R is such that:

 (a) It is not a substantially real entity but exists only in a mind;

 (b) It is grounded in objective qualities of the members of S;

 (c) Its reality and truth are grounded in the divine mind;

 (d) It results from the members of S being thought about together (being
 concogitabilitas) in some fashion.


I take it that (a)-(d) are sufficiently justified by the above quoted passages. But if (a)-(d) are true with respect to Leibniz's view of relations, then we may return to the concept of harmony with a fresh perspective.

IV

Recall that for Leibniz, harmony, in addition to being "unity in multiplicity" (or "variety compensated by identity"), is also characterized as "perfect cogitability." In addition, Leibniz claims that
 that is a more perfect manner of thinking, where one act of thought extends
 to many things simultaneously: so there is more reality in that thought.
 This is done, moreover, with the work of relations, for a relation is a
 kind of unity in multiplicity. And species of relations are the bonds and
 reasons [rationes] of things among themselves, the proportions, the
 proportionalites. It is from all of these taken at the same time in a given
 object that harmony results [resultat].(31)


In this passage, we have harmony explicated in nearly the same exact terms that Leibniz uses to explain relations in general. For example, like relations, harmony is said to result from the consideration of several things at once, in this case from other relations themselves (that is, harmony results from concogitabilitatis).(32) (And this is true, it should be noted, even if the several things (relations) are in one given object.) We have already noted the striking claim that a relation is a "kind of unity in multiplicity." But when taken together with the first sentence of the above passage--that "that is a more perfect manner of thinking, where one act of thought extends to many things simultaneously"--we might be reminded of the familiar claim that "the greater the variety and the unity in variety, this variety is harmonious to a higher degree.(33) That is, presumably one act of thought is capable of extending to many entities at the same time, and presumably unifying (in a way to be explained)--through the work of a relation, as Leibniz says--that collection of entities. If so, and if that collection is related in thought according to the relevant standard of order, that collection will be harmonious. In any event, it is difficult to resist the conclusion that harmony is something which, like a relation, exists only in a mind (though grounded in objective qualities of things), and results from the mutual consideration of several things. Accordingly, its truth and reality, like all relations, will be grounded in the divine mind.(34)

We have thus arrived at the following more precise--though still incomplete--account of harmony:
 (H) For any set S of entities, and some relation R which holds among the
 members of S, S is harmonious if and only if:

 (i) S contains more than one member;

 (ii) R is such that:

 (a) It is not a substantially real entity but exists only in a mind;

 (b) It is grounded in objective qualities of the members of S;

 (c) Its reality and truth are grounded in the divine mind;

 (d) It results from the members of S being thought about together
 (being concogitabilitas); (e) It unifies the members of S above a
 certain standard of order.(35)


While we have made some headway, it is imperative that we at last improve upon (e). We are now in a position to do so in a preliminary way.

Recall that Leibniz's standard definition of order is the "relation of several things, through which any one of them can be distinguished from any other."(36) If we apply what we have learned about Leibniz's views on relations, we may say, then, that according to this definition, a set of entities exhibits order--or is unified by a relation of order (37)--when it is gathered in thought in such a way that the members of the set can be distinguished from one another. It is important to notice that order is itself a relation, and is therefore merely a way of considering objective qualities of things. Leibniz tells us that "distinct cogitability gives [dat] order to a thing."(38) The point presumably is that if the relevant thing, or set of entities, is distinctly cogitable, it may be thought about in a way that permits one to distinguish one entity from another. In any event, we may reformulate Leibniz's notion of harmony thus:
 (H) For any set S of entities, and some relation R which holds among the
 members of S, S is harmonious if and only if.

 (i) S contains more than one member;

 (ii) R is such that:

 (a) It is not a substantially real entity but exists only in a mind;

 (b) It is grounded in objective qualities of the members of S;

 (c) Its reality and truth are grounded in the divine mind;

 (d) It results from the members of S being thought about together
 (being concogitabilitas);

 (e) It is a way of considering together the members of S such that any
 one member of S may be distinguished from any other.


V

Doubts will remain, of course, about the adequacy of (H), particularly of component (e). For what could it mean to be cogniscent of a collection of entities yet not be able to distinguish them one from another? Some might seek help in Leibniz's calling order "nothing other than a distinctive relation of several things,"(39) and in his claim that distinct cogitability is what bestows order on a given thing. The idea here is to follow up on Leibniz's use of the term "distinct," a term with many applications in his philosophy. Indeed, at first glance, this seems like a promising strategy, for Leibniz often explicated his concepts of "distinct" and "confused" in terms of that which contains, respectively, distinguishable and nondistinguishable items. Consider the following famous passage from Meditations on Knowledge, Truth and Ideas (1684):
 Cognition [cognitio] is clear ... when it gives me the ability to recognize
 the thing represented, and clear cognition in turn is either confused or
 distinct. It is confused when I cannot separately enumerate the marks which
 are sufficient to discriminate the thing from others, even though the thing
 really has such marks and requisites into which its concept can be
 resolved: thus colors, odors, tastes, and other particular objects of the
 senses we indeed recognize sufficiently clearly and discriminate from each
 other, but by the simple testimony of the senses, and not by marks that can
 be expressed. So we cannot explain to a blind man what red is, nor can we
 explain such things to others, except by bringing them into the presence of
 the thing, and causing them to see, smell, or taste it ... even though it
 is certain that the concepts of these qualities are composite and can be
 resolved, since they surely have their causes.... A distinct concept
 however is the kind of notion assayers have of gold; one, namely, which
 enables them to distinguish gold from all other bodies by sufficient marks
 and observations. We usually have such concepts about objects common to
 many senses, such as number, magnitude, and figure.(40)


Before analyzing this passage, it must be stressed that Leibniz applied the terms "confused" and "distinct" to a number of things, such as "perceptions," "ideas," "concepts," and "representations," to mention a few. In the above passage, Leibniz speaks of "cognition" and "concepts," as being either distinct or confused. Thus, the emphasis in the above passage, we might say, is on conceptual abilities as opposed to sensory abilities. The former seem to be the ones with which we ought to be concerned since order and harmony, as we have seen, are notions inextricably linked with what is cogitable (thinkable) and not, it seems, with what is sensible.(41)

According to the above passage, cognition is confused when "we cannot separately enumerate the marks which are sufficient to discriminate the thing from others." We are offered the examples of sensible qualities: colors, odors, and tastes. Leibniz's point seems to be that when it comes to the concepts of these qualities, we are unable to say what it is that these concepts consist of or what the ingredients of these composite concepts are. Elsewhere, Leibniz puts the point more simply: an idea is distinct, as opposed to confused, "when I have a definition of it."(42) That is, a distinct (nonprimitive) concept is one which can be decomposed into its constituents by means of a definition, unlike the concepts of sensible qualities. But although "the concepts of these sensible qualities can be resolved [into definitions], since they surely have their causes," we apparently lack the conceptual ability to resolve these concepts into simpler component concepts: "to be able to pick out the causes of odors and tastes, for instance, and the content of these qualities, is beyond us."(43) Hence, our concepts of sensible qualities are confused.(44)

Armed with this understanding of confused/distinct cognition, is there a way of understanding Leibniz's view that harmony results from considering a collection of entities such that they may be distinguished from one another or such that they may be distinctly conceived? It is not easy to see how we are to understand this. In some passages, Leibniz suggests that ordering certain collections (and so distinguishing the members of that collection) simply amounts to ascertaining the number of elements in the collection. Consider the following revealing passage from the New Essays:
 This picture whose parts one sees distinctly, without seeing what they
 result in until one looks at them in a certain way, is like the idea of a
 heap of stones, which is truly confused . . . until one has distinctly
 grasped how many stones there are and some other properties of the heap. If
 there were thirty-six stones, say, one would not know just from looking at
 them in a jumble that they could be arranged in a triangle or in a
 square--as in fact they could, because thirty-six is both a square number
 and a triangular one. Similarly, in looking at a thousand-sided figure one
 can only have a confused idea of it until one knows the number of its
 sides, which is the cube of 10. So what matters are not names but the
 distinct properties which the idea must be found to contain when one has
 brought order into its confusion. It is sometimes hard to find the key to
 confusion--the way of viewing the object which shows one its intelligible
 properties.(45)


Leibniz's emphasis in this passage is on "finding the key to confusion"--on bringing order to a previously confused idea. His claim is that the ideas of certain objects, when viewed from a certain perspective, will reveal that object's intelligible properties. The passage is revealing for our purposes. We are engaged in a search for what it could mean to say that harmony is the result of considering a set of entities in such a way that the elements of that set can be distinguished from one another. Leaving the notion of harmony aside for the moment, the suggestion of this passage is that when the collection is considered in a certain way, one can deduce its intelligible properties. Thus, when one shifts from a confused idea of a heap of stones to a more distinct idea of that same heap of thirty-six stones (a definition?), one thereby expands one's awareness of the range of intelligible properties that the heap possesses. Where one previously did not know, for example, that the heap could be arranged in a square, when one shifts to a distinct idea of the heap--that is, when one orders the content of this idea--one sees that this can be done or that the heap possesses this property. Leibniz also suggests in the above passage, as well as the following one, that ordering the relevant idea simply amounts to ascertaining the number of stones:
 If I am confronted with a regular polygon, my eyesight and my imagination
 cannot give me a grasp of the thousand which it involves: I have only a
 confused idea both of the figure and of its number until I distinguish the
 number by counting. But once I have found the number, I know the given
 polygon's nature and properties very well, insofar as they are those of a
 chiliagon.(46)


In this passage, of course, it is not stones, but sides of a polygon which are under discussion. Nevertheless, the point seems to be the same. When it comes to heaps and polygons, once one has grasped the number involved (sides, stones, or whatever), one has presumably brought "order into its confusion" and now knows "the nature and properties" of the relevant entity (heap, polygon).

There is, perhaps, some intuitive sense in which the heap, and the chiliagon, are more readily understood when one ascertains the number involved, for then, as Leibniz suggests, certain properties can be inferred on the basis of this information. It is perhaps also plausible, as Leibniz seems to have thought, that ascertaining the number involved amounts to mentally relating them in thought in some sense. Indeed, Leibniz seems disposed to treat "quantity, such as number ... as merely [a] relation, which results from other things,"(47) and a relation, we know, is merely a way of considering things. But what is not clear yet (to me at least) is why ascertaining the relevant number should be understood as distinguishing the members of the collection one from another. But it is clear that this is what Leibniz is committed to: if ascertaining the number of stones (or whatever) elevates one's idea from confused to distinct by imposing order on it, and if both order and distinctness are understood in terms of the ability to distinguish the elements of the heap (polygon), it follows that ascertaining the number of stones amounts to distinguishing one stone from the other. But surely I can distinguish one of these elements from another--in some minimal sense, at least--even if I do not know how many there are. I can, of course, have the concept of a heap of stones, and the ability to distinguish the stones one from another, though not be aware of how many there are in the heap.

Perhaps the interpretive key here lies in Leibniz's understanding of concept resolution, or, as he also put it, in providing a definition, for both, as we know, are supposed to be the mark of distinctness. Thus, consider the heap: this, we may suppose, may be defined as a pile of thirty-six stones, and its concept, presumably, consists of the component concepts mentioned in this definition (that is, the concept of a pile, of the number thirty-six, of a stone, and so forth).(48) Hence, we have, I suppose, distinguished the constitutive components of the concept of that heap one from another. But note that if this is the way of understanding the imposition of order, and the resulting distinctness, it is not so much a matter of distinguishing the actual parts of the relevant entity (for example, the stones of the heap), but, more exactly, it is a matter of distinguishing the constitutive component concepts which collectively make up the concept of the relevant heap. In this case, then, harmony results from simultaneously considering a set of component concepts in such a way that one can thereby deduce the intelligible properties of the concept of the entity (heap) which those component concepts collectively constitute. In this case, the relevant entities to be distinguished are component concepts, and they may be distinguished by means of concept resolution (decomposition) or by providing a definition. There is evidence that Leibniz accepts something like this proposal. Besides the plethora of passages where Leibniz links the existence of order and harmony with "appropriate" conceptualization, he writes in a paper entitled On Destiny that "we cannot see [the order in the world] so long as we do not enjoy the right point of view," and "it is only with the eyes of the understanding that we can place ourselves in [this] point of view which the eyes of the body do not and cannot occupy."(49) If we suppose that concept resolution is the business of "the eyes of the understanding," then we might conclude that harmony, though grounded in objective features of things, results from this sort of conceptual activity.

The examples in the above passages about heaps and polygons are, I suppose, convenient ones for Leibniz, for recall his remarking in the passage from Meditations on Knowledge, Truth, and Ideas that "we usually have [distinct] concepts about ... number, magnitude, and figure,"(50) the very properties which pertain to these examples. Presumably, this is because knowledge of these properties involves knowledge of necessary, mathematical, truths--truths which most people, I suppose, have access to. On the other hand, his other favorite example of a distinct idea, the one that assayers have of gold is, of course, not the kind of idea that we would regard as commonplace. This leads Leibniz to remark that "since [the assayers'] art is not known to everyone, it is no wonder that men do not all have the same idea of gold," for "as a rule, only experts have sufficiently accurate ideas of a given material."(51) But what about the assayers' distinct concept of gold? That is, it was Leibniz's view that in simultaneously considering a collection of entities (presumably, a collection of component concepts) such that one can distinguish the members of that collection, one thereby orders that collection. This, in turn, we are led to believe, enables one to deduce--as Leibniz does, once armed with a definition, in the above passage--the intelligible properties possessed by that collection of entities. In the above examples, the move from a confused to a distinct idea involved simply ascertaining the relevant number of sides or stones, which presumably enables one to provide a definition or to perform the concept resolution necessary for distinctness. But obviously a distinct idea of gold does not involve simply ascertaining a number. How then are we to understand the assayers' distinct idea of gold?

Consider Leibniz's definition of gold: "gold is a metal which resists cupellation and is insoluble in aqua fortis; that is a distinct idea, for it gives the criteria or definition of gold."(52) But if that is the definition of gold, and thus a mark of a distinct idea of gold, then we should, it seems, based on the previous examples, be able to deduce certain intelligible properties from this concept based on that definition. What intelligible properties may we deduce from this concept in light of this definition, such that we may regard it as distinctly apprehended?

The two cases presented above, one having to do with heaps and polygons and the other with gold, do not seem strictly analogous. In the case of the former, we can perhaps see where concept resolution does the trick: when armed with the relevant definition, one deduces certain mathematical and geometrical properties. In the case of the latter, it is not clear how the stated definition of gold as that "which resists cupellation and is insoluble in aqua fortis" provides one with the means to deduce further properties of gold. One is tempted to say that with respect to gold, unlike the case of the polygon and the heap, Leibniz has put the cart before the horse: he has given us certain "intelligible" (dispositional) properties of gold rather than a definition from which these properties are supposed to be deduced.

On the other hand, there is evidence that Leibniz was aware of the disanalogy that I have just pointed out. Consider the following:
 I once defined "adequate idea" (or "perfect idea") as one which is so
 distinct that all its components are distinct; the idea of number is pretty
 much like that. But even if an idea is distinct, and does contain the
 definition or criteria of the object, it can still be "inadequate" or
 "imperfect"--namely, if these criteria or components are not all distinctly
 known as well. For example, gold is a metal which resists cupellation and
 is insoluble in aqua fortis; that is a distinct idea, for it gives the
 criteria or definition of "gold." But it is not a perfect idea since we
 know too little about the nature of cupellation and about how aqua fortis
 operates. The result of having only an imperfect idea of something is that
 the same subject admits of several mutually independent definitions: we
 shall sometimes be unable to derive one from another, or see in advance
 that they must belong to a single subject, and then mere experience teaches
 us that they do belong to it together. Thus, "gold" can be further defined
 as the heaviest body we have, or the most malleable, and other definitions
 could also be constructed; but only when men have penetrated more deeply
 into the nature of things will they be able to see why the capacity to be
 separated out by the above two assaying procedures is something that
 belongs to the heaviest metal. Whereas in geometry, where we do have
 perfect ideas, it is another matter.(53)


Thus, the disanalogy noted above stems from the fact that when it comes to our distinct concepts of geometrical figures, such as the one of the chiliagon, the components of these concepts are also distinct, and that enables one to deduce--on the basis of mathematical inference alone--certain properties of the entity. I suppose this is because when it comes to certain mathematical concepts, the resolution can be carried, so to speak, all the way down, so that any properties of the relevant concept may be deduced through pure conceptual analysis. In a word, our concepts of these are "perfect."

However, the assayers' distinct idea of gold, though definable, can be resolved only into concepts which are themselves confused. The result of this, we are told, is that "gold" admits of several mutually independent definitions, none of which can be derived from any other. That is, none can be derived "in advance"--which means, I take it, a priori--from any of the others. Nonetheless, "mere experience teaches us" that these mutually independent definitions do belong together. Thus, given the right experiences, we can infer one from another a posteriori Hence, contrary to our initial suggestion, Leibniz seems to have held that from the definition, "a metal which resists cupellation and is insoluble in aqua fortis," one may come to infer a posteriori certain other intelligible properties about the defined entity, for example, that it is the heaviest metal and the most malleable.

The upshot then is that Leibniz seems to have accepted DC as the appropriate characterization of a distinctly cogitable entity:
 DC: An entity E is distinctly cogitable if and only if the concept C of E
 may be resolved into component concepts, which allows one to infer, either
 a priori or a posteriori, a range of properties of E.


And if Leibniz accepted DC, then his claim that harmony results from simultaneously considering a collection of entities (relations--see below) such that they may be distinguished from one another--such that, that is, the collection may be distinctly conceived--must amount to the claim that one can enumerate the members of the collection in a way that permits one to infer a range of properties of the collection itself. Perhaps then harmony amounts to the following, where (e) in (H) has been replaced, and (iii) has been added, and where it is now formulated in terms of the concept of the collection of harmonized entities:
 (H) For any set S of entities, the concept C of S, and some relation R
 which holds among the members of S, S is harmonious if and only if:

 (i) S contains more than one member,

 (ii) R is such that:

 (a) It is not a substantially real entity but exists only in a mind;

 (b) It is grounded in objective qualities of the members of S;

 (c) Its reality and truth are grounded in the divine mind;

 (d) It results from the members of S being considered together (being
 concogitabilitas);

 (e) It is a way of considering together the members of S such that one
 can resolve the concept C of S into the concepts of its members;

 (iii) One can infer, as a result of (e), a range of properties of S.


(e) and (iii) are motivated, of course, by Leibniz's views on what ordering and distinctness amount to. But the careful reader will recall that in the passage from the Elementa, Leibniz wrote that it is from relations among things "taken together at the same time that harmony results."(54) Thus, we are led to believe that from resolving the concept C of S into the concepts of its members in a way that permits one to deduce a range of properties, one can thereby grasp the relations that hold among the members of the set, relations grounded in objective qualities of those entities, and can therefore grasp the resulting harmony or see that the set is a harmonious collection.(55)

Before proceeding two things should be noted about this formulation. The first thing to note about (H) (particularly (e) is that it requires that the concept of a heap, a polygon, or whatever, just is the concept of a set of entities. It requires, therefore, that the term "set" carry no extra ontological baggage: once you have resolved the set into its members, there is no further resolution to carry out, for the set just is the sum of its members.(56) Second, it seems to me that the phrase "as a result of (e)" in (iii) is crucial: for surely a range of properties can be inferred even from concepts that are not resolvable. One can infer from one's concept of blue, for example, however confused and irresolvable it may be, that, when mixed with yellow, it will produce green.(57) The point is that the relevant properties of the entity must apparently be ones that are deducible from component concepts which constitute the concept of the entity in question. One would like to know, of course, precisely which kinds of properties are relevant here, such that the object achieves the requisite level of intelligibility.(58)`

Although we shall soon note problems with (H),(59) this formulation seems to make sense of a letter Leibniz wrote to Christian Wolff near the end of his life. In the letter, Leibniz repeatedly linked the notion of harmony with the concept of that which can be distinctly observed. He writes, for example, that "the more potential for observation in a thing, the more general properties, the more harmony it contains."(60) In this passage, the fact that a range of properties may be inferred from a certain collection--a collection constituting a "thing"--is explicitly linked with the concept of harmony. In that same letter, Leibniz equated the harmony of things with "the state where everything has the potential for observation, that is, the state of agreement or identity in variety; you can even say that it is the degree of contemplatibility."(61) Years earlier, as we have seen above in the passage from the Elementa, Leibniz explicitly stated that "Harmony is the perfection of cogitability."(62) It seems reasonable to infer, accordingly, that in these passages, Leibniz's idea is that harmony results from considering a set of entities in such a way that one may deduce a range of properties from the set. Insofar as the set may be conceived so as to yield a range of general properties, including dispositional properties,(63) it does, presumably, count as a cogitable, and, hence, harmonious set.

VI

It is surely not my claim that Leibniz's account of harmony, particularly in its epistemological components, is without philosophical problems.(64) In concluding this paper, I note two prima facie challenges, though there are undoubtedly more.

As we have seen, there are numerous passages where Leibniz speaks of a thing or an object being harmonious.(65) The idea here seems to be that if the concept of the relevant thing is resolved into its component concepts in such a way that a range of properties may be inferred from it, the object exhibits a certain level of harmony. But, more often, Leibniz claims that harmony is exhibited by a collection of entities which do not, in any obvious sense, constitute a thing or an object. For example, Leibniz envisioned the totality of efficient causes as harmonizing with the totality of final causes. What is the relevant set of entities in this case? Presumably, it is the set consisting of all efficient and final causes. But there is no obvious sense (or so I say) in which these add up to a thing, or an object. Thus, it is difficult to see how concept resolution is supposed to aid us here, for there is no thing, the concept of which is to be resolved such that we may see the level of distinct cogitability. (Similar remarks apply to the other harmonies noted at the beginning of this paper.) Perhaps the idea is that, nonetheless, once we grasp how these two kinds of causation work, we can come to infer certain intelligible properties about the whole system of causation, understood as involving a harmony between "realms," or "kingdoms."(66) In any event, this difficulty awaits further analysis.

A second related issue concerns Leibniz's repeated characterization of harmony as involving a unity of a collection of entities. Why should we understand the order imposed upon a collection of entities as imposing a kind of unity? This question is all the more perplexing given the observation that certain collections of entities do not obviously constitute one object or thing. There is some suggestion of an answer in Leibniz's claim that a relation is a kind of unity in multiplicity in that, through the work of a relation, many objects are gathered in thought at the same time.(67) This is further suggested by his claim that "the unity that collections have is merely a respect or relation,"(68) and he often treats order as a relation of "concurrence, [which] involves some connection, such as that of cause and effect, whole and parts, position and order, etc."(69) It seems, then, that Leibniz thought that order, because it is a relation, can impose a mental unity on certain collections of entities. What is less clear is how to view the totality of, say, efficient and final causes, as a unified collection, particularly one worthy of being called harmonious. This too, however, awaits further analysis.(70)

(1) See, for example G. W. Leibniz, Hauptschriften zur Grundlegung der Philosophie, ed. Ernst Cassirer (Hamburg: Felix Meiner, 1966), 2:556. An English translation is in G. W. Leibniz, Leibniz: Selections, trans, and ed. Phillip Weiner (New York: Scribners, 1951), 185. In what follows, if an English translation is available, I cite it after the original language source. If no English translation is cited, then none is available, and the translation is my own.

(2) See, for example, G. W. Leibniz, Die Philosophischen Schriften von Gottfried Wilhelm Leibniz, ed. C. I. Gerhardt (Berlin: Weidman, 1875-1890; reprint, Hildesheim: Olms, 1961), 6:599; G. W. Leibniz, Philosophical Writings, trans, and ed. Mary Morris and G. H. R. Parkinson (London: Dent, 1973), 196.

(3) See, for example, Leibniz, Die Philosophischen Schriften, 6:622; G. W. Leibniz, Philosophical Essays, trans, and ed. Roger Ariew and Daniel Garber (Indianapolis: Hackett, 1989), 224.

(4) More specifically, it seems to me that when it comes to discussions of the very concept of harmony, as opposed to various remarks concerning its applications, most of the relevant texts occur in the 1670s. Thereafter, the only extended discussion I am aware of is not until 1714 in a letter written to Christian Wolff. See G. W. Leibniz, Briefweschel zwischen Leibniz und Christian Wolff, ed. C. I. Gerhardt (Hildesheim: Olms, 1963), 166; Leibniz, Philosophical Essays, 230. In the latter part of this paper, however, I will suggest reasons for thinking that there is significant continuity in his thought on this topic.

(5) The former quotation is from G. W. Leibniz, Sdmtliche Schriften und Briefe, ed. German Academy of Sciences (Darmstadt and Berlin: Berlin Academy, 1923-), 2d ser., 1:98; the latter is from 6th ser., 1:484. It is worth mentioning that I have not found any other passages where Leibniz used these last two expressions to define harmony. However, this does not seem too important, since the idea expressed in this terminology appears to be the same idea expressed using other, more frequently used, terminology.

(6) See Leibniz, Sdmtliche Schriften und Briefe, 6th ser., 1:475, 477, 479, 484, and 2d ser., 1:283. According to 6th ser., 2:283, Harmonia ... est unitate plurimorum, which translates more exactly as "harmony is the unity of pluralities." Again, the idea appears to be the same. See also, G. W. Leibniz, Testes Inedits, ed. Gaston Grua (Paris: Presses Universitaires de France, 1948), 12-13.

(7) Leibniz, Textes Inedits, 12.

(8) In what follows, I temporarily leave aside Leibniz's utilization of the notion of "proportion" and focus instead on "order," a notion which he used more frequently in this context. We shall see, however, that the two notions were closely related for Leibniz for both were used to designate certain intelligible relations between entities.

(9) Another implication of the passage, evident from Leibniz's claim that "the greater the variety and the unity in variety, this variety is harmonious to a higher degree," is that harmony admits of degrees: two sets of entities may each be harmonious, yet one may be more harmonious. I leave aside this complication for now and return to it after pursuing a more satisfactory formulation of harmony.

(10) Leibniz, Die Philosophischen Schriften, 4:431; Leibniz, Philosophical Essays, 39.

(11) Leibniz, Die Philosophischen Schriften, 7:312; Leibniz, Philosophical Writings, 78-9.

(12) Some commentators may see in passages such as the two just quoted, the theological claim that God--due to his infinite wisdom--could not create a world which lacked order because infinitely wise beings always act with order (or according to rules, laws, principles). That is, according to this reading, disorderly worlds may be metaphysically possible, but they are not ones God could create (and so are not really possible worlds, I suppose). But if this is Leibniz's point it is difficult to see why he did not just come out and say it. Moreover, both passages make the claim that we cannot even "imagine" or "conceive" of an unorderly arrangement because there is always a mutually ordering relationship to be found. There is no mention--or even suggestion-that any point being made here is relative to divine wisdom. The point seems purely one of logic. On this, see Gregory Brown's "Compossibility, Harmony, and Perfection in Leibniz," The Philosophical Review 96 (1987): 173-203.

(13) G. W. Leibniz, Die Leibniz-Handschriften der K6niglichen Offentlichen Bibliothek zu Hannover, ed. Eduard Bodemann (Hildesheim: Olms, 1966), 124.

(14) Leibniz, Die Leibniz-Handschriften, 70.

(15) Leibniz, Opuscules et Fragments Inedits de Leibniz, ed. Louis Couterat (Paris: Felix Alcan, 1903), 476. The first of these definitions is dated by watermark from the early 1690s (compare Robert Adams, Leibniz: Determinist, Theist, Idealist [Oxford: Oxford University Press, 1994], 235). I have been unable to obtain a dating of the second. The third is from a table of definitions dated from 1702-4. Notice that the sense of the third is slightly different from the other two. The first two have it that an ordered relation is one in which the elements can be distinguished, whereas the third says that the elements are distinguished [discriminatur]. I doubt that much hinges on this, but it is unfortunate since the third again threatens triviality: for if an ordered relation is one in which, say, a finite mind can distinguish the elements (as in the first two definitions), that would presumably indicate a certain level of order--perhaps the one we are looking for. But this is not suggested by the third definition. At any rate, I have found no indication that Leibniz changed his mind on the issue, and, in fact, as we shall see, there is evidence that Leibniz believed the first two definitions to be the more accurate. For this reason I shall lean primarily on them in what follows. (Perhaps the third is a slip, in that it omits any form of the verb possum?)

(16) Leibniz, Die Philosophischen Schriften, 7:290; Leibniz, Philosophical Writings, 146.

(17) Ibid.

(18) Leibniz, Textes Inedits, 13.

(19) Leibniz, Leibniz und Christian Wolff, 170-1; Leibniz, Philosophical Essays, 233-4.

(20) Leibniz, Textes Inedits, 12-13.

(21) It also has the virtue of linking the notion of harmony with that of perfection. The question of how exactly these two concepts are related in Leibniz's thought is a vexing question which I steer clear of here. For discussion of various aspects of the issue, see Gregory Brown, "Compossibility, Harmony, Perfection"; chapter 2 of Donald Rutherford, Leibniz and the Rational Order of Nature (Cambridge: Cambridge University Press, 1995); David Blumenfeld, "Perfection and Happiness in the Best Possible World," in The Cambridge Companion to Leibniz, ed. Nicholas Jolley (Cambridge: Cambridge University Press, 1994), 382-410.

(22) See Leibniz, Textes Inedits, 547, where Leibniz admits to being a nominalis, "at least provisionally." I cannot see that anything in the passage explains this qualifier. He adds "that it suffices to posit substances as real things and to assert truths about these." See also, "Preface to An Edition of Nizolius," in Leibniz, Die Philosophischen Schriften, 4:138; translation in G. W. Leibniz, Philosophical Papers and Letters, trans, and ed. Leroy Loemker (Dordrecht: Reidel, 1969), 121.

(23) G. W. Leibniz, New Essays on Human Understanding, trans, and ed. Peter Remnant and Jonathan Bennett (Cambridge: Cambridge University Press, 1981), 227. The pagination of this work follows that of the original language text in Leibniz, Samtliche Schriften und Briefe, 6th ser., vol. 6. Thus, I will only cite the former henceforth.

(24) Leibniz, New Essays, 265.

(25) Sometimes Leibniz makes this point via his famous dictum that "there are no purely extrinsic denominations, which have absolutely no foundation in the thing itself denominated"; Leibniz, Opuscules et Fragments, 520; Leibniz, Philosophical Essays, 32.

(26) Leibniz, New Essays, 146.

(27) Note also the following passage from the New Essays: "I believe that qualities are just modifications of substances, and that the understanding adds relations.... However, although relations are the work of the understanding they are not baseless and unreal. The primordial understanding is the source of things; and the very reality of all things other than simple substances consists only in their being a foundation for the perceptions or phenomena of simple substances"; Leibniz, New Essays, 146.

(28) See, for example, the passages gathered by Benson Mates, The Philosophy of Leibniz: Metaphysics and Language (Oxford: Oxford University Press, 1986), 224-5. Concogitabilitas is perhaps best translated as "a thinking together."

(29) Quoted from Mates, The Philosophy of Leibniz, 224. The original Latin can be found in Massimo Mugnai's Leibniz's Theory of Relations (Stuttgart: Franz Steiner Verlag, 1992), 155. See also Leibniz, Textes Inedits, 547; Leibniz, Opuscules et Fragments, 9; and Leibniz, Philosophical Papers and Letters, 525.

(30) Some commentators doubt whether or not Leibniz can consistently wed the view that (a) things are not related by anything real beyond the relata and their properties with the view that (b) the reality of relations is dependent on the mind. See, for example, chapter 12 of Mates, The Philosophy of Leibniz: Metaphysics and Language; Jan Cover, "Relations and Reduction in Leibniz," Pacific Philosophical Quarterly 70 (1989): 185-211; Jan Cover, "Review of Massimo Mugnai's Leibniz's Theory of Relations," The Leibniz Society Review 5 (1995): 1-10. Given that relations result as soon as the relevant relata are posited, Cover expresses the doubt clearly: "Where then arises the place for any contribution by the intellect--for any obvious sense in which relations are 'dependent upon the mind'?" As Cover sees it, if his and Mates's account "is correct, the deep ontological facts serving as truthmakers for relational claims include nothing whatsoever about the role of the mind"; Cover, "Review of Massimo Mugnai's Leibniz's Theory of Relations," 4 and 5. For an attempt to answer this challenge, see Massimo Mugnai's "Reply to Cover," The Leibniz Society Review 5 (1995): 11-14.

(31) Leibniz, Textes Inedits, 13.

(32) In this sense, then, harmony is perhaps most appropriately viewed as a second-order relation, for it seems to supervene on other relations. More on this later.

(33) Leibniz, Textes Inedits, 13.

(34) See also Leibniz, Die Philosophischen Schriften, 2:438; Leibniz, Philosophical Essays, 199, where Leibniz writes: "God views not only the individual monads and the modifications of each monad, but also their relations, and in this consists the reality of relations and truths." Presumably, this includes harmonious relations too. For discussions of the role of the divine mind in grounding the reality of substances, their relations, and their phenomena, see Adams, Leibniz: Determinist, Theist, Idealist, 258 and following; Gregory Brown, "God's Phenomena and the Pre-established Harmony," Studia Leibnitiana 19 (1987): 200-14.

(35) Notice that (d) differs from the previous (d) used in (R) to characterize a Leibnizian relation in that I have dropped the phrase "in some fashion." I take it that when it comes to characterizing harmony, (e) provides the manner in which the members of S must be considered together, namely, in such fashion that they achieve the relevant standard of order. Of course, we still need an explanation of the latter.

(36) Leibniz, Die Leibniz-Handschriften, 70 and 124.

(37) While it is not clear now why "order" should be understood as imposing a certain unity on the set, I shall eventually speculate on this matter.

(38) Leibniz, Die Philosophischen Schriften, 7:290; Leibniz, Philosophical Writings, 146.as "perfect cogitability." In addition, Leibniz claims that

(39) Ibid.

(40) Leibniz, Die Philosophischen Schriften, 4:422-3; Leibniz, Philosophical Papers and Letters, 291-2. See also Leibniz, New Essays, 254-5.

(41) Leibniz, Die Philosophischen Schriften, 3:247; Leibniz, Philosophical Essays, 287. See also Leibniz, New Essays, 255-6. In what follows, I shall use the terms "idea" and "concept" interchangeably. However, it must be stressed that strictly speaking these are not equivalent for Leibniz. Ideas, according to him, are more like dispositions, so that one can be said to have ideas even when one is not presently conscious of them. Concepts on the other hand are essentially content bearing entities. Thus, when one actualizes an idea, it becomes a "concept" or "notion." See Leibniz, Die Philosophischen Schriften, 4:452; Leibniz, Philosophical Essays, 59. Provided we bear this in mind, no harm should result from using these terms interchangeably for both pertain to the realm of conceptualization as opposed to sensory presentings.

(42) Leibniz writes: "Whether one says ideas, or whether one says notions, whether one says distinct ideas, or whether one says definitions (at least, when the idea is not absolutely primitive), it is all the same thing"; Leibniz, Die Philosophischen Schriften, 3:248. The bit in parentheses signals a qualification. Leibniz also held that some ideas are primitive, where that is understood as an idea which is not composite or one which does not contain other ideas. Primitive ideas, then, do not yield to definitions because they do not yield to decomposition or resolution.

(43) Leibniz, New Essays, 256.

(44) Margaret Wilson criticizes Leibniz here for running together two senses of "confused," one of which pertains to sense perception (which is confused because it contains an infinity of petites perceptions which cannot be filtered out), the other pertaining to ideas and notions. According to this objection, Leibniz holds that "the necessary confusedness of our perceptions of colors, odors, etc., is ... taken to show that we only (can only?) have confused ideas of these qualities"; Margaret Wilson, "Confused Ideas," Rice University Studies 63 (1977): 129. But this reading of Leibniz strikes me as mistaken, at least with respect to some ideas. Leibniz makes it clear in the New Essays that with respect to things which appear confused to the senses, we can, at the same time, have distinct ideas of those things. Thus, for example, in the New Essays, he accuses Locke of the same view which Wilson attributes to him: "This example shows that the idea is being confounded with the image. If I am confronted with a regular polygon, my eyesight and my imagination cannot give me a grasp of the thousand which it involves: I have only a confused idea both of the figure and of its number until I distinguish the number by counting. But once I have found the number, I know the given polygon's nature and properties very well, insofar as they are those of a chiliagon. The upshot is that I have this [distinct] idea of a chiliagon, even though I cannot have the image of one: one's senses and imagination would have to be sharper and more practiced if they were to enable one to distinguish such a figure from a polygon which had one side less"; Leibniz, New Essays, 261. Thus, Leibniz seems to have held that one could possess a distinct idea of something (in this case, a thousand-sided figure) but nonetheless lack the sensory repertoire for distinct sensory perception of it. So it does not seem that he could argue from the confusion of a sensory perception of something to the confusion of the ideas of those same things.

(45) Leibniz, New Essays, 257-8.

(46) Leibniz, New Essays, 261.

(47) Leibniz, Opuscules et Fragments, 9; Leibniz, Philosophical Writings, 133.

(48) Note that this definition is good only if we are talking about that very heap in Leibniz's example and not any heap in general.

(49) Leibniz, Hauptschriften, 2:131; Leibniz, Selections, 572.

(50) Leibniz, Die Philosophischen Schriften, 4:422; Leibniz, Philosophical Papers and Letters, 291.

(51) Leibniz, New Essays, 338. This same observation has been made, and developed into a "sociolinguistic hypothesis," in our own time, of course, by Hilary Putnam in his "The Meaning of `Meaning'," in Mind, Language, and Reality: Philosophical Papers, vol. 2 (Cambridge: Cambridge University Press, 1975). Putnam writes: "The features that are generally thought to be present in connection with a general name--necessary and sufficient conditions for membership in the extension (`criteria'), etc.--are all present in the linguistic community considered as a collective body; but that collective body divides the `labor' of knowing and employing these various parts of the `meaning' of `gold'" (p. 228). In Leibnizian terms, while many have a "clear" idea of gold, only some in the community (namely, assayers) have a "distinct" idea of gold.

(52) Leibniz, New Essays, 267.

(53) Leibniz, New Essays, 267.

(54) Leibniz, Textes Inedits, 13. It thus appears that harmony is what we might call a second-order relation, resulting as it does from other relations.

(55) I earlier promised to say something about Leibniz's use of the term "proportion" in his claim from the Elementa that "where variety is without order, without proportion, there is no harmony"; Leibniz, Textes Inedits, 12. In a table of definitions from 1702-4, Leibniz defined a proportion as "the relation of two magnitudes, one of which can be determined with the help of the other given one"; Leibniz, Opuscules et Fragments, 476. Thus, like order, proportion is itself a relation among entities which enables one to distinguish the entities related one from another. Note also that things may be related to one another proportionally with respect to a number of objective qualities of those things, for example, size, shape, color, and so forth. Thus, there is every reason to believe that Leibniz's use of the term "proportion" in describing the relations between a collection of harmonious entities is not merely metaphorical, as might be supposed. Rather, like his notion of order, it is a term he apparently uses to designate intelligible relations among things.

(56) I make this point because it is controversial, of course, whether or not the set of, say, frogs, is something more than the sum of all frogs, namely, the frogs and the set. From what I can see, however, Leibniz's views on concept resolution commit him to the position that a given composite concept just is the sum of its component concepts and nothing more.

(57) Indeed, this is Leibniz's own example. See Leibniz, New Essays, 403.

(58) Russell Wahl pointed out to me that Leibniz's grounding the reality of relations, and thus the reality of harmony and distinct cogitability, in the divine mind seems to present a problem. God, qua perfectly omniscient being, can see all the connections in all the possible worlds, for there is no nondistinctly cogitable world for him. Thus, it would seem, then, that every possible world is harmonious. One might seek to defend Leibniz here by claiming that nonetheless, some worlds are more harmonious than others. But it is not clear to me that this defense will work, for again, there cannot be some worlds more distinctly cogitable than others with respect to God, for they are all equally intuitively intelligible to an omniscient being. I suppose one might reply that nonetheless some worlds are more harmonious with respect to finite minds. But then again Leibniz's remarks suggest that the actual world is the most harmonious simpliciter--without reference to any mind. In any event, it is an issue worth investigating, though I haven't the space to discuss it here. My thanks to Wahl for bringing this issue to my attention.

(59) One issue that one would like to see clarified has to do with the difference between order and harmony. What exactly is the difference? I know of no texts where Leibniz addressed this question head-on. We know that harmony presupposes order such that a harmonious collection must exhibit order. And order, as we know, is a relation which holds among entities such that they may be distinctly conceived. But order, it would then seem, presupposes harmony too, for Leibniz suggests that wherever a collection of entities exhibits distinct cogitabilty, it exhibits a certain degree of harmony. But it cannot be (for Leibniz's sake, at least) that the two concepts are identical, for Leibniz often defined harmony, as we know, in terms of order. Perhaps, then, although the two concepts are not to be identified, they are nonetheless coextensive, such that wherever harmony is exhibited, order is exhibited and vice versa. One suggestion is this: order is a first-order relation between entities endowing them with a certain degree of distinct cogitability, while harmony is a second-order relation which supervenes upon relations of order. This would make sense of Leibniz's claim that it is more exactly from a collection of relations (of order?) that harmony results (see Leibniz, Textes Inedits, 13). This is admittedly speculative, as I have found the texts to be largely indeterminate on the exact difference between these two notions.

(60) Leibniz, Leibniz und Christian Wolff, 170; Leibniz, Philosophical Essays, 233. Thus, it seems that one set of entities, as noted earlier, can be more harmonious than another. The greater the range of intelligible properties deducible from a set of entities (which, I suppose, is a function of the plurality of entities--that is, the more concepts there are to resolve--and the relations they bear to one another), the more hamonious is the collection.

(61) Leibniz, Leibniz und Christian Wolff, 170; Leibniz, Philosophical Essays, 233.

(62) Leibniz, Textes Inedits, 12.

(63) Here I have in mind, for example, the dispositional property of the heap to be arranged in a square, and the dispositional property of gold to resist solubility in aqua fortis. These examples of Leibniz's suggest that properties concerning how certain entities would react in certain situations are to be counted among the intelligible properties of that entity.

(64) Wilson, "Confused Ideas," articulates some interesting difficulties associated with Leibniz's account of distinct cogitability, an account which obviously infects his notion of harmony.

(65) Leibniz, Leibniz und Christian Wolff, 170, Leibniz, Philosophical Essays, 233; Leibniz, Die Philosophischen Schriften, 7:290; Leibniz, Philosophical Writings, 146; and Leibniz, Textes Inedits, 13.

(66) See Leibniz, Die Philosophischen Schriften, 6: 622; Leibniz, Philosophical Essays, 224.(67) Leibniz, Textes Inedits, 13.

(67) Leibniz, New Essays, 146.

(69) Leibniz, New Essays, 142.

(70) An earlier version of this paper was read at the 1999 Intermountain Seminar in Early Modern Philosophy, University of Colorado, Denver, April 25, 1999. I would like to thank the participants for many helpful comments and suggestions. Thanks are also due to Mark Kulstad and Gregory Brown for their comments on earlier drafts.
LAURENCE CARLIN
University of Wisconsin Oshkosh


Correspondence to: Department of Philosophy, University of Wisconsin Oshkosh, 800 Algoma Boulevard, Oshkosh, WI 54901.
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Author:CARLIN, LAURENCE
Publication:The Review of Metaphysics
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Date:Sep 1, 2000
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