Leibniz's Relational Conception of Number

Leibniz’s Relational Conception of Number Kyle Sereda, University of California, San Diego Abstract In this paper, I address a topic that has been mostly neglected in Leibniz scholarship: Leibniz’s conception of number. I argue that Leibniz thinks of numbers as a certain kind of relation, and that as such, numbers have a privileged place in his metaphysical system as entities that express a certain kind of possibility. Establishing the relational view requires reconciling two seemingly inconsistent deinitions of number in Leibniz’s corpus; establishing where numbers it in Leibniz’s ontology requires confronting a challenge from the well-known nominalist reading of Leibniz most forcefully articulated in Mates (1986). While my main focus is limited to the positive integers, I also argue that Leibniz intends to subsume them under a more general conception of number. Introduction1 I n a variety of texts, Leibniz advances a conception of number that has gone largely unnoticed by scholars. Few accounts of Leibniz’s thought make mention of it; the most prominent general work that treats his conception of number is that of Russell (1937), but it merely mentions Leibniz’s view, making no attempt to explain it. Furthermore, while there is a large literature on particular aspects of Leibniz’s philosophy of mathematics – for example, the nature of ininitesimals, the foundations of the calculus2, and the foundations of the analysis situs3 – this literature barely touches on the topic of Leibniz’s conception of number4. The aim of this paper is to answer a simple question: what are numbers, according to Leibniz? I pursue this question along two distinct dimensions. The irst is deinitional: how does Leibniz deine numbers? Does he deine all numbers in the same way? The second is ontological: what is the ontological status of numbers according to Leibniz? Are they conceived as Platonic abstract objects, or mental entities, or something else entirely? Do all numbers have the same ontological status? In this paper, I limit my attention to the positive integers. Thus, I address the question, irst, how does Leibniz deine the positive integers; and second, what is their ontological status according to him? There is good reason to investigate The Leibniz Review, Vol. 25, 2015 31 KYLE SEREDA these questions in tandem: Leibniz’s answer to the irst entails an answer to the second; and his answer to the second faces a challenge from the prominent reading of Leibniz’s metaphysics proposed by Mates (1986), which characterizes Leibniz as a kind of nominalist. My goal in this paper is to provide a detailed reconstruction of Leibniz’s deinition of number – limited to the positive integers – and to argue that this deinition entails that numbers have a well-deined place in Leibniz’s metaphysics. Speciically, I will argue that Leibniz deines numbers as relations5; and that as relations, numbers are inhabitants of the realm of divine ideas, expressing certain kinds of possibilities and providing the basis for a class of necessary truths. Establishing the irst of these points will require reconciling two sets of texts that appear to deine numbers in different ways. One set of texts leaves the impression that Leibniz conceives of the positive integers as “aggregates of unities”, in line with a deinition of number handed down from Euclid. Another set of texts indicates that Leibniz conceives of all numbers – not only the integers -- as examples of a special type of relation. In sections 1 and 2, I show how these texts can be reconciled, establishing that Leibniz ultimately conceives of the positive integers as relations that provide the basis for the wholeness and size of aggregates of things taken as unities. I then turn to the question of the ontological status of numbers for Leibniz, given that they are relations. In section 3, I argue that as a kind of relation, numbers have a natural place in Leibniz’s ontological framework, expressing possibilities and grounding a class of necessary truths. In the course of this argument, I address the challenge my reconstruction faces from Mates. The nominalist reading of Leibniz’s metaphysics – especially as formulated by Mates – has been the subject of a great deal of critical scrutiny6, and it will not be my goal to criticize this reading in detail. Instead, I only aim to show that Mates’ reading does not represent a serious challenge to my interpretation of Leibniz’s position on the ontological status of numbers. In the paper’s conclusion, I briely explore the broader implications of my interpretation for Leibniz’s general account of number. 1. “Number in General” and Positive Integers as Aggregates Leibniz’s earliest relections on the concept of number are found in the Dissertation on the Art of Combinations, published in 1666. Such an early work must be read with caution, as Leibniz often changes his views over the course of his career. However, in this case, Leibniz advances several core theses on numbers and their The Leibniz Review, Vol. 25, 2015 32 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER relation to magnitude that do not change in their essentials. Concerning number, he writes: “The concept of unity is abstracted from the concept of one being, and the whole itself, abstracted from unities, or the totality, is called number. Quantity is therefore the number of parts” (L 76). A number – and here Leibniz seems to have speciically the positive integers in mind – has in this early text the role of expressing the wholeness of a collection of beings considered as unities. The view that such an expression of the wholeness of a collection is a relation – as I will ultimately argue Leibniz holds – can also be extracted from the Dissertation. Immediately before the text just cited, Leibniz writes that “every relation is either one of union or one of harmony. In union the things between which there is this relation are called parts, and taken together with their union, a whole. This happens whenever we take many things simultaneously as one” (L 76). Leibniz seems to claim here that the basis of the wholeness of a collection is a relation; if the positive integers provide such a basis, then they must be relations – speciically, relations of union. Remarkably, this conception of the positive integers remains largely constant throughout Leibniz’s career, except that he later subsumes it under a more general conception of number that is designed to accommodate all of the positive real numbers7. Leibniz also provides in the Dissertation an early indication of the importance of number within his metaphysical framework and within his general philosophy of mathematics: “[T]he Scholastics falsely believed that number arises only from the division of the continuum and cannot be applied to incorporeal beings. For number is a kind of metaphysical igure, as it were, which arises from the union of any beings whatever; for example, God, an angel, a man, and motion taken together are four” (L 76-77/GP IV 35). Leibniz’s subsequent writings on number build on the basic conception advanced in the Dissertation. In many texts, Leibniz says that the positive integers are a certain kind of aggregate. I have ordered these texts chronologically to the extent possible: (1) Number is a whole composed from unities. (A.VI.4.31; 1677) (2) Number is [that which is] homogeneous to unity. A whole number is that of which the aliquot part is unity, or a sum of unities. A fraction is a sum of aliquot parts of unity. (A.VI.4.421; 1680-84?) (3) Number is [that which is] homogeneous to unity, and so it can be compared with The Leibniz Review, Vol. 25, 2015 33 KYLE SEREDA unity and added to or subtracted from it. And it is either an aggregate of unities, which is called an integer... or an aggregate of aliquot parts of unity, which is called a fraction. (GM VII 31, undated) (4) An integer is a whole collected from unities. (Grosholz and Yakira, 1998, 99; c.1700?) (5) An integer is a whole collected from unities as parts. (Grosholz and Yakira, 1998, 88; c.1700?) In (1), (4), and (5), Leibniz either deines number (in (1)) or the integers speciically as wholes composed of unities, mirroring his remarks in the Dissertation8. The relational aspect of his view is not apparent here, however. In (2) and (3), he uses slightly different language, deining the integers as aggregates of unities. For ease of exposition, I subsequently refer to the deinition of the positive integers in these texts as a deinition of them as aggregates of unities. Leibniz seems to intend no difference between an aggregate of unities and a “whole collected from unities”, as evidenced by the fact that he uses these terms interchangeably through a long stretch of years. Two of these texts (4 and 5) are drawn from Grosholz and Yakira’s study; it is worth pausing to take account of their interpretation of Leibniz’s conception of number, for the sake of eliminating it before I propose my own reading. Grosholz and Yakira outline an interpretation that takes into account only texts (4) and (5). The authors claim that text (5) in particular supports the thesis that Leibniz intends his deinition of number to have an essentially geometric content, such that the “parts” Leibniz mentions are to be understood in the same way as the parts of a line. Their main textual evidence for the geometrical construal of “part” is Leibniz’s deinition of magnitude as a number of parts. They hold that in deining magnitude as such, Leibniz “associates a number with a geometrical entity” (1998, 80), and that he must think that “to understand what a whole number is, one must know not only that it can be composed of concatenated units, but also that it can be represented by line segments”, such that integers “are understood by analogy with relations among line segments” (1998, 81). My reconstruction of Leibniz’s view of number, by contrast, will indicate no detour through geometrical notions, at least for the positive integers. Grosholz and Yakira’s interpretation can be shown to be inconsistent with other The Leibniz Review, Vol. 25, 2015 34 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER views Leibniz holds about mathematics. First, Leibniz holds that arithmetic is conceptually prior to geometry. Accordingly, he cannot be understood as intending his deinitions of number to presuppose any geometrical content. Leibniz writes that “geometry... or the science of extension is.... subordinated to arithmetic, since... there is repetition or multitude in extension...” (AG 251-252/ GM VI 100). For Leibniz, arithmetic is the highest mathematical science, and geometry is subordinate to it. He writes: There is an old saying according to which God created everything according to weight, measure, and number. But there are things which cannot be weighed, those namely which have no force or power. There are also things which have no parts and hence admit of no measure. But there is nothing which is not subordinate to number. Number is thus a basic metaphysical igure, as it were, and arithmetic is a kind of statics of the universe by which the powers of things are discovered. (L 221/GP VII 184) Given this, it would be puzzling if Leibniz intended his deinition of a basic arithmetical entity – the positive integers – to presuppose geometrical content. Furthermore, when Leibniz says that arithmetic is prior to geometry, one speciic claim he means to make is that geometrical magnitudes cannot be understood fully without recourse to number. This is because we cannot perform certain operations that yield understanding of magnitude if we do not irst possess number concepts. Grosholz and Yakira’s claim that Leibniz deines magnitude as a number of parts is true, but this is evidence against their interpretation. In deining magnitude as a number of parts, Leibniz deines magnitude in terms of number, rendering our understanding of magnitude dependent upon our understanding of number. He writes that magnitude is “measured by the number of determinate parts” (L 254/ GM V 179); the “magnitude in a thing is represented by a number of parts” (GM VII 53). He holds that in order to acquire distinct knowledge of the size of a given geometrical magnitude, we must have recourse to number: “Precise distinctions amongst ideas of extension do not depend upon size: for we cannot distinctly recognize sizes without having recourse to whole numbers, or to numbers which are known through whole ones; and so, where distinct knowledge of size is sought, we must leave continuous quantity and have recourse to discrete quantity” (NE 156). Given this, it is implausible to suppose that Leibniz intends to deine number in a way that presupposes the concept of the division of a line into parts – Leibniz holds instead that in order to understand the magnitude of a line, we must understand how to measure it, and we must in turn understand numbers in order to do that. The Leibniz Review, Vol. 25, 2015 35 KYLE SEREDA It is worth elaborating upon this point using some key texts. Leibniz makes it clear in a number of texts that the way we are able to understand the size of any continuous magnitude is by conceiving it as a collection of unities, which for Leibniz is just to say that we are able to understand the size of a continuous magnitude by conceiving the magnitude in terms of an integer that speciies how many of a given unit of measure the magnitude contains. The integers allow us to recast the question “how much?” in terms of the question “how many?”, for any given continuous magnitude. Thus, we must irst understand integer concepts in order to apply them in determining the size of a continuous magnitude. The size of some magnitude cannot be precisely determined unless some other magnitude is taken as a unity and then repeated until the original magnitude is exhausted. Leibniz writes: The quantity of a thing, e.g. of the area ABCD (ig. 7) is expressed by a number, e.g. a multiple of four [quaternarium], when it has been assumed that some other thing, such as a square foot AEFG, is taken for a primary measure or real unity. For ABCD is four square feet. But if some other unity AHIK is assumed, which is a half-foot squared, then the quantity of the area ABCD would be 16. Thus, for the same quantity a different number is produced, according to the unity that is assumed. And consequently quantity is not a deinite number, but the material for a number, or an indeinite number that is made deinite when a certain measure is assumed. (GM VII 30-31) He elaborates in another text: If a foot should be considered as unity, then a thumb will be 1/12, a cubit will be 3/2, and the armspan will be 6. If a thumb should be considered as unity, then a foot will be 12, a cubit will be 18, and the armspan will be 72. And in this manner the length of every straight line can indeed be represented by a whole number, if the measure has been drawn off several times, for example if a foot has been drawn off three times, and nothing is left over, then it will be ruled three feet in length. But if something remains when the measure, e.g. foot, has been drawn off as many times as can be done, for this too the thing to be measured will be able to be obtained by a deinite part of a foot, for example tenths, which are drawn off afresh from this remainder as many times as can be done. (GM VII 36) For Leibniz, we must understand the notion of a collection of unities – that is, the notion of an integer – in order to be able to measure continuous magnitudes, as that measurement consists in the conception of those magnitudes as collections of unities. The Leibniz Review, Vol. 25, 2015 36 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER With Grosholz and Yakira’s claim about the content of Leibniz’s deinition of number eliminated, I turn back to the statements of that deinition. Crucially, in texts (2) and (3), Leibniz makes a further remark after deining the integers as aggregates of unities. He also deines “number in general” as “that which is homogeneous to unity”9, and he subsumes various kinds of number under that deinition. Concerning the positive integers, two questions arise here: irst, is Leibniz consistent in offering both the general deinition of number as “that which is homogeneous to unity” and the speciic deinition of the positive integers as aggregates of unities? Second, if he is consistent, what is the intended relationship between these deinitions? A complete answer to these two questions will require an explanation of the phrase “that which is homogeneous to unity”, but one can discover a partial answer to the irst question by carefully examining the above texts, without yet investigating the precise meaning of that phrase. It appears that Leibniz at least intends the two deinitions to be consistent, and indeed to it together in an unspeciied manner. In texts (2) and (3), Leibniz characterizes the positive integers as aggregates of unities in the very next sentence after characterizing number in general as that which is homogeneous to unity. The structure of (2) and (3) is nearly the same: Leibniz deines “number in general”, and implies that integers and fractions are related to the general deinition. This intent is slightly clearer in (3), where Leibniz links the deinition of number in general with the deinition of the integers (and of fractions) using “estque”, or “and it is…”, where the antecedent of the pronoun is clearly “numerus”. Accordingly, the text reads “number is [that which is] homogeneous to unity... And it [i.e. number] is either an aggregate of unities, which is called an integer, or an aggregate of aliquot parts of unity, which is called a fraction”. These texts do not establish the precise nature of the relationship between the deinitions, but it is clear that Leibniz intends to offer them as a coherent package, such that the deinitions of the speciic kinds of numbers are consistent with the deinition of “number in general”. In other texts, Leibniz is more explicit about his intention to include all the positive real numbers – including irrational numbers – under his general deinition of number as “that which is homogeneous to unity”. Leibniz evidently intends his general deinition of number as that which is homogeneous to unity to subsume the different kinds of positive real number -- integers, fractions, and irrationals -- so that the deinition of “number in general” and the deinitions of the different kinds of numbers are related as genus and species. This is suggested by (2) and (4), but it becomes more transparent in the following texts: The Leibniz Review, Vol. 25, 2015 37 KYLE SEREDA (6) Number is [that which is] homogeneous to unity. And so not only integers are numbers but also fractions and surds. (A.VI.4.873, 1687?) (7) It is manifest that number in general – integer, fraction, rational, surd, ordinal, transcendental – can be deined by a general notion, as it is that which is homogeneous to unity, or that which is related to unity. (GM VII 24, 1714) (8) You may also deine number in general, which comprehends integer, fraction, surd and transcendental. It is evidently nothing other than [that which is] homogeneous to unity. (LCW 173, 1715) What (6) suggests, (7) and (8) explicitly state: Leibniz intends his general deinition of number to be powerful enough to accommodate all positive real numbers as instances. In these texts, he claims that all these numbers fall under the category picked out by “that which is homogeneous to unity”, though it is not yet clear how this is supposed to work. What is clear is that he intends the general deinition to pick out a genus, number, which subsumes speciic kinds of number as species. It is beyond the scope of this paper to resolve the issue of how Leibniz intends his account of number to accommodate fractions and irrational numbers. Instead, I will focus on how exactly Leibniz’s deinition of the positive integers relates to the deinition of “number in general”. How are the positive integers, as aggregates of unities, a species of “that which is homogeneous to unity?” More fundamentally, what does it mean to be “homogeneous to unity?” Leibniz deines homogeneous things as “those which are similar or can be rendered similar by a transformation” (A.VI.4.872; identical or nearly identical language is found at A.VI.4.508 and GM VII 30). The concept of similarity is essential to Leibniz’s deinition of homogeneity; he offers multiple deinitions of the term, and it is worth noting several of them. In the text from which (3) is taken, he deines similar things as those things “in which, considered by themselves, singly, it is not possible to ind that by which they might be distinguished”10. Another text reads “similar things are those which can be distinguished through themselves if they are together” (A.VI.4.155). Yet another reads “similar things are those which can be distinguished only by co-perception” (A.VI.4.508). The common thread in these deinitions is the idea of the indistinguishability of two things solely by the examination of the things as they are in themselves: to The Leibniz Review, Vol. 25, 2015 38 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER distinguish similar things, one needs to perceive them together. Leibniz’s favorite example of this phenomenon – discussed in tandem with his deinition of similarity in many places -- is that of similar geometrical igures: for example, two differently sized triangles with the same ratios between their respective sides. According to Leibniz, the only way to discern the difference in size is to perceive the two triangles simultaneously, or to use a third igure as a measuring device by which to compare them. One cannot distinguish them merely by the examination of each igure as it is in itself. These igures are also, according to the deinitions just examined, homogeneous, because they are similar, fulilling one of the suficient conditions for homogeneity. But two things are also homogeneous when they are able to be rendered similar by a transformation. Now, an aggregate of unities is intuitively not similar to unity. It would seem that an aggregate of unities can be distinguished from a unity conceptually, and two different aggregates of unities can be distinguished from each other, without the aid of simultaneous perception. Leibniz is never entirely clear about how this distinguishing is supposed to work, but he appears to be convinced that different aggregates of unities are not similar to one another, and presumably, this means that an aggregate of unities is also not similar to unity. He contrasts the case of distinct aggregates of unities with the case of distinct lines: “But what of number? One is... not similar to another, for example four is not similar to three... [A]lthough three is not similar to four, but a line of three feet is similar to a line of four feet” (A.VI.4.933-34). Earlier in the same text, Leibniz provides some indication of why aggregates of unities are not similar to each other, in terms of yet another deinition of similarity: “Similar things are those which... cannot be distinguished one by one through truths demonstrable about themselves; or those of which no different demonstrable predicates can be assigned. Thus every parabola is similar to every parabola, and every circle to every circle... Similar things are those of which all the internal predicates are the same...” (A.VI.4.931). In contrast to the truths demonstrable about different parabolas or circles, it seems clear both that two different aggregates of unities have different truths demonstrable about them, and that different truths are demonstrable about unity than are demonstrable about any aggregate of unities. Despite the lack of similarity between unity and an aggregate of unities, the latter is homogeneous to the former because an aggregate of unities can be rendered similar to unity. It can be transformed into something that cannot be distinguished from a unity: something that is only conceivable as a unity, and no longer as The Leibniz Review, Vol. 25, 2015 39 KYLE SEREDA an aggregate. In text (3) above, Leibniz characterizes the homogeneity to unity borne by aggregates of unities as consisting in the fact that such aggregates can be “compared with unity and added to or subtracted from it” (GM VII 31). The requisite transformation, then, is a kind of subtraction: speciically, the successive removal of the constituents of an aggregate of unities until what remains is simply a unity – and so what remains is indistinguishable from unity. For example, an aggregate of unities can be rendered similar to unity by the successive removal of its constituents until there remains something that is no longer an aggregate, but something only conceivable as a unity. Thus, an aggregate of unities falls under the category of “that which is homogeneous to unity”, since it can be rendered similar to unity by a transformation. Accordingly, Leibniz’s deinition of the integers is both consistent with his general deinition of number and related to it in a straightforward way11. I close this section by noting that the discussion of the integers as aggregates of unities provides another way of ruling out the reading of Leibniz’s conception of number in Grosholz and Yakira’s study. Now that Leibniz’s notion of homogeneity is understood, his remark in texts (2) and (5) that unities are the parts of integers can be cast in the proper light. While Grosholz and Yakira read Leibniz’s use of “part” as essentially geometrical, so that his deinition of the integers presupposes geometrical notions, the opposite is true: Leibniz understands the relation of part to whole in a general way that is detached from geometrical considerations. In several places, Leibniz deines “part” in terms of an ontological dependence relation that makes no reference to geometry and does not require that parts be understood in terms of continuous magnitude. For example, he writes that “a part is a homogeneous ingredient” (GM VII 19), and that “parts are homogeneous inexistents” (A.VI.4.932), where an “inexistent” is something that exists in something else. A part, for Leibniz, is something that “is in” something else and is homogeneous to it. Parts, then, are a species of inexistents: A is a part of B if A exists in B and is homogeneous to B. It should be clear from the argument of this section that Leibniz’s notion of homogeneity does not presuppose any geometrical content, and his notion of “being in” [inesse] is similarly general: “We say that an entity is in or is an ingredient of something, if, when we posit the latter, we must also be understood, by this very fact and immediately... to have posited the entity as well” (GM VII 19). If parts are homogeneous inexistents, then Leibniz does not understand “part” geometrically, but rather in a broader way that includes as one particular case the way that segments are parts of a line, and as another particular The Leibniz Review, Vol. 25, 2015 40 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER case the way that unities are parts of integers12. The relation between part and whole, understood in terms of inesse, is explained succinctly by Rutherford: “to say that parts ‘are in’ a whole... is to say that if the latter are supposed to exist, the existence of the former can immediately be asserted; conversely, if the former are supposed not to exist, it can immediately be asserted that the latter do not exist” (1990, 541). Given Leibniz’s understanding of parthood in terms of inesse, line segments and unities both qualify as “parts” of their respective wholes in that they are both homogeneous inexistents of those wholes. They are inexistents in the sense just noted: without them, the wholes would not exist; and once the whole (line, integer) is posited, the parts (segments, unities) are thereby posited. It is worth reiterating that homogeneity is also not a geometrical notion for Leibniz: it is merely the notion of one thing’s being able to be rendered similar to another by a transformation, where the things in question can be non-geometrical entities such as aggregates or geometrical entities such as lines, and the transformations can involve the addition or removal of constituents (in the case of aggregates) or the lengthening or shortening of segments (in the case of lines). Unities are parts of integers in this sense, existing in aggregates and bearing the relation of homogeneity to them, since a unity can be transformed into an aggregate by the addition of other unities. 2. Positive Integers as Relations The previous section clariied the relationship between Leibniz’s deinition of the positive integers and his deinition of number in general. But it is not yet clear on what grounds Leibniz conceives of the positive integers as relations. All that has been established thus far is that the deinition of the positive integers as aggregates of unities is consistent with, and indeed a species of, the deinition of “number in general” as “that which is homogeneous to unity”. The aim of this section is to show that Leibniz should be understood as holding that numbers are indeed relations. Within the limited scope of this paper, Leibniz’s considered view is that the positive integers are just relations. It turns out, however, that his deinition of the integers as aggregates of unities actually implies that the integers are relations, both because aggregates are relational entities, uniied by a relation among their constituents, and because the relevant aggreganda in the case of the integers are merely abstract unities, so that all we have in this case is a relation expressing the possibility of the aggregation of things taken as unities. The Leibniz Review, Vol. 25, 2015 41 KYLE SEREDA In section 1, I noted a text from the Dissertation on the Art of Combinations that seems to provide a version of the thesis that the integers are relations. There, Leibniz says that some relations are relations of union, which unite several things in a whole, and the way he characterizes the integers in the Dissertation seems to place them in the category of relations of union. However, in several places in his mature writings Leibniz more explicitly characterizes numbers as relations: (9) [N]umber or time are only orders or relations pertaining to the possibility and the eternal truths of things. (GP II 268-269, 1704) (10) Numbers... have the nature of relations. And to that extent in some way they can be called beings. (GP II 304, 1706) (11) Place and position, quantity -- such as number, proportion -- are nothing but relations, results from other things. (C 9, undated). Limiting the focus to the integers, this way of characterizing number appears to conlict with the characterization of the positive integers as aggregates of unities: an aggregate of unities, or any other aggregate for that matter, is not, at least intuitively, a relation. It is dificult, in the absence of further evidence, to see how Leibniz’s claim that numbers are relations might be reconciled with his claim that the positive integers are aggregates. Prima facie, a positive integer cannot be both a relation and an aggregate. Fortunately, supplementary evidence is forthcoming, in the form of Leibniz’s view of aggregates. In Leibniz’s metaphysics, something can only be properly understood as an aggregate on the basis of the relatedness of its constituents. An aggregate of unities, for example, is only a whole -- it is only one thing -- because of a certain relation among its constituents. This view is evident in its infancy in the Dissertation. It is straightforward to apply Leibniz’s concept of relations of union to aggregates, since aggregates are composed of several things united in some way: what makes them a whole is a relation of union between their constituents. But Leibniz makes the view explicit in the following late remark from his comments on a book by Aloys Temmick: “Bare relations are not creatable things, and arise in the divine intellect alone... and such things are whatever results from posits, such as the totality of an aggregate”13. In the New Essays, Leibniz makes much the same point: “[The] unity of the idea of an aggregate is a very genuine one; The Leibniz Review, Vol. 25, 2015 42 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER but fundamentally we have to admit that this unity that collections have is merely a respect or relation” (NE 146). Aggregates, for Leibniz, are a kind of relational being: an aggregate is only a uniied whole in virtue of a relation that provides the basis for the collecting together of certain items. If the positive integers are aggregates of unities, then several possibilities arise as to the meaning of that claim, for example: (a) A given number is identical to a particular aggregate of concrete things taken as unities, e.g. {Leibniz, Spinoza, Locke}. (b) A given number is identical to the set of all aggregates of concrete things taken as unities that can be put in a one-to-one correspondence, e.g. all aggregates that can be put in a one-to-one correspondence with {Leibniz, Spinoza, Locke}. (c) A given number is identical to an aggregate of unities taken in abstraction, e.g. {unity, unity, unity}. Option (a) is not a serious contender for a conception of number: it is not plausible to maintain that the number ive is identical with a particular collection of concrete things. Even if it were plausible, there is no textual evidence that Leibniz ever considered this kind of view. Option (b) is more plausible, but there is again no evidence that it is Leibniz’s view. In Leibnizian terms, the number three is not identical with a either a particular aggregate or the collection of all aggregates that can be put in a one-to-one correspondence with {Leibniz, Spinoza, Locke}. Thus, if Leibniz thinks of the integers as aggregates, then he must conceive of them in terms of option (c): as aggregates of unities taken in abstraction. However, (c) is equivalent, in Leibnizian terms, to the claim that the integers are relations. Consider the content of an aggregate of unities taken in abstraction: no particular aggregate is signiied, only a general possibility of – or basis for – aggregation. In an aggregate of concrete things taken as unities, some determinate concrete aggreganda are uniied by some relation. By contrast, in an aggregate of abstract unities, the aggreganda are merely unities taken in abstraction. Unities taken in abstraction are nothing more than placeholders for individual things; an aggregate of unities taken in abstraction, then, signiies nothing more than the possibility of taking individual things together in a certain way. But it is relations, for Leibniz, that provide the basis for, or underlie, the possibility of aggregation – the possibility of taking things together. Thus, an aggregate of unities taken in abstraction can only be understood as a relation. This line of thought is captured in Leibniz’s remark in the New Essays that “It may be that dozen and score are merely relations and exist The Leibniz Review, Vol. 25, 2015 43 KYLE SEREDA only with respect to the understanding. The units are separate and the understanding takes them together, however scattered they may be” (NE 145). The taking together of things in an aggregate is nothing more than the consideration of a certain relation among them. In sum, according to Leibniz, the positive integers are the relations that provide the basis for the wholeness of, and express the size of, aggregates insofar as those aggregates are composed of unities. In section 1, before providing my analysis of Leibniz’s conception of the positive integers, I noted that the positive integers provide the basis for the measurement of continuous magnitudes by allowing us to conceive of those magnitudes in terms of collections of unities. The integers can now be understood more generally as providing the basis for the counting of collections of things, expressing the wholeness of those collections and answering the question “how many?” with respect to them. In the continuous case, some magnitude is taken as a unity and repeated until the original magnitude has been re-cast as an aggregate of unities; in the case of the counting of individual things, the “measure” is just the notion of unity itself. Each thing is taken as a unity, and the resulting number counts the aggregate in terms of the question “how many unities?”. At bottom, then, the positive integers are those relations that express the homogeneity to unity possessed by aggregates of things taken as unities, signifying that such aggregates are composed of unities and can be rendered into unity by successively removing their constituents. This is not to identify a positive integer with a relation as exempliied by any particular aggregate. Positive integers, rather than being identiied with some collection or other, are the relations of homogeneity that any collection may have to unity, expressing the size of the collection in terms of an answer to the question “how many?”. 3. Positive Integers as Divine Ideas Having answered the deinitional question, I turn to the question of the ontological status of the integers. If Leibniz conceives of the integers as relations, then it is plausible to think that he confers the same ontological status on them that he confers on other relations. But this leaves open several options for Leibniz: does he think of the integers, as relations, along the lines of Platonic abstract objects? Does he think of them as some sort of mental entity? Does he even think that they have any reality at all? Are they part of his metaphysical picture of the world? Leibniz is clear throughout his corpus that relations have a particular ontological The Leibniz Review, Vol. 25, 2015 44 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER status: they are divine ideas, contents of God’s mind. Given this, the integers, as relations, should be understood as contents of God’s mind14. Given how Leibniz conceives of God’s mind, these numbers should be understood along Platonistic lines, in the sense that they have timeless reality independent of the created world. The precise content of Leibniz’s theory of relations has been the subject of much scholarship15; for our purposes, a selection of key texts will sufice to make the point that matters here. Following the terminology of Mugnai, relations in abstracto, or relations considered in the abstract, independently of any particular relata, are contents of the divine mind according to Leibniz. For example, Leibniz writes that “although relations are the work of the understanding they are not baseless and unreal. The primordial understanding is the source of things...” (NE 145); and that concerning relations, “one can say that their reality, like that of eternal truths and of possibilities, comes from the Supreme Reason” (NE 227). He also says in the same work that “the reality of relations is dependent on mind... but they do not depend on the human mind, as there is a supreme intelligence that determines all of them from all time” (NE 265). In his commentary on Aloys Temmick, Leibniz writes that “their reality [of relations] does not depend on our understanding -- they inhere without anyone being required to think of them. Their reality comes from the divine understanding...” (Mugnai 1992, 155). Thus the integers, as relations, have their ultimate ground in God’s mind. In turn, God’s mind, for Leibniz, is the realm where possibilities are expressed and the ground of necessary truths is provided. If the integers are divine mental contents, then they express possibilities and form the basis of a class of necessary truths. That they occupy this place in Leibniz’s metaphysics was telegraphed earlier by the excerpt from his letter to De Volder, where Leibniz says that numbers “pertain to the possibility and the eternal truths of things”. But Leibniz is explicit in many texts that the realm of divine ideas is the realm of that which expresses possibility and provides the basis for necessary truths. The following passage represents a typical statement of Leibniz’s conception of the realm of divine ideas: Essences, truths, or objective realities of concepts do not depend either on the existence of subjects or on our thinking, but even if no one thinks about them and no examples of them existed, nevertheless in the region of ideas or truths, as I would say, i.e. objectively, it would remain true that these possibilities or essences actually exist, as do the eternal truths resulting from them... As in the region of eternal truths, or in the realm of ideas that exists objectively, there subsist unity, the circle, power, equality, heat, the rose, and other realities or The Leibniz Review, Vol. 25, 2015 45 KYLE SEREDA forms or perfections, even if no individual beings exist, and these universals were not thought about. (S 185/A.II.1.590) The contents of the divine mind have reality independent of the created world; they express possibilities; and they form the basis of eternal truths: Neither... essences nor the so-called eternal truths pertaining to them are ictitious. Rather, they exist in a certain realm of ideas, so to speak, namely in God himself, the source of every essence and of the existence of the rest... It is necessary that eternal truths have their existence in a certain absolute or metaphysically necessary subject, that is, in God, through whom those things which would otherwise be imaginary are realized. (AG 151-152/GP VII 304305) Thus, the positive integers, as relations, and so as inhabitants of the divine mind, have reality independent of the created world, express a certain sort of possibility, and are the subject of necessary truths. As relations, the positive integers express the possible ways in which aggregates of things taken as unities can be homogeneous to unity, which is to say they express the possible sizes of such aggregates in terms of their composition out of things taken as unities. As inhabitants of the divine mind, the positive integers also provide the basis for a class of necessary truths: the truths of arithmetic. Each positive integer is a relation, and the truths of arithmetic concern relations between numbers; so arithmetical truths, for Leibniz, ultimately concern second-order relations -- in other words, relations between relations. Second-order relations, in this framework, express ways in which possible groups of related things can be with regard to each other, insofar as the given groups are united by particular irst-order relations. So the eternal truth that 2+3=5, for example, expresses a relation between 2 and 3, such that any aggregate related to unity by the number 2 is related to any aggregate related to unity by the number 3 in such a way that their combination yields an aggregate related to unity by the number 5. It seems, then, that Leibniz’s answer to the ontological question is quite clear: though numbers are a kind of mental entity, they are the kind of mental entity that gives them the status of something resembling Platonic abstract objects. They are not merely contents of human minds, but of God’s mind; as such, they exist timelessly and independently of the created world. This reconstruction faces a challenge based on the reading of Leibniz’s metaphysics proposed by Mates. Mates characterizes Leibniz as a nominalist, claiming that Leibniz denies a fundamental reality to anything other than concrete individual substances and their modiications: in other words, Leibniz excludes everything other than these entities from his fundamental The Leibniz Review, Vol. 25, 2015 46 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER picture of the world. According to Mates, There can be little doubt that [Leibniz] was a nominalist... [in] the sense it bears in current Anglo-American philosophical discussion about so-called ontological commitment. According to this, a nominalist, as contrasted with a Platonist, is one who denies that there are abstract entities, asserts that only concrete individuals exist, and in consequence considers that all meaningful statements appearing to be about abstract entities must somehow be rephraseable as statements more clearly concerning concrete individuals only. (1986, 170) On this interpretation, Leibniz rejects all manner of abstract entities, including those entities that are the subject of this paper: Leibniz would agree wholeheartedly with that notorious pronouncement of present-day nominalism: “We do not believe in abstract entities.” He does not believe in numbers, geometric igures, or other mathematical entities, nor does he accept abstractions like heat, light, justice, goodness, beauty, space or time, nor again does he allow any reality to metaphysical paraphernalia such as concepts, propositions, properties, possible objects, and so on. The only entities in his ontology are individuals-cum-accidents, and sometimes he even has his doubts about the accidents. (1986, 173) For Mates, then, Leibniz’s variety of nominalism consists in two theses: irst, the denial that abstract entities exist, and second, the thesis that statements about those entities can be rewritten in a way that refers only to concrete individuals. Here I will only be concerned with the former16. Mates’ evidence in favor of the former claim consists in Leibniz’s oft-repeated remark that anything that is not a concrete individual substance or modiication thereof has whatever being it has solely as a content of the divine mind17. Mates’ interpretative strategy is to argue that such a move is intended to reduce everything that is not either an individual substance or a modiication thereof to certain of God’s dispositions. Mates proposes that “what [Leibniz] intends is not that there are two kinds of existence, namely, in the mind of God and out of the mind of God, but rather that statements purporting to be about these kinds of entities are only compendia loquendi for statements about God’s capacities, intentions, and decrees” (1986, 177). In other words, talk of abstract objects, universals, or relations is just talk about certain of God’s perfect concepts or ideas, which in turn is nothing more than talk of God’s mental dispositions. Crucially, for Mates, such a reduction amounts to an elimination of the reduced entity from Leibniz’s fundamental ontology. If divine ideas are ultimately dispositional, Mates’ reasoning goes, then Leibniz must The Leibniz Review, Vol. 25, 2015 47 KYLE SEREDA intend to eliminate from his fundamental ontology everything that he characterizes as a divine idea. The rest of this section will make two claims: (1) Mates’ proposed reduction lacks a textual basis, and in fact Leibniz holds the exact opposite view of the nature of divine ideas to that required by Mates’ strategy; (2) even if Mates’ proposed reduction had textual support, it still would not be the eliminative kind of reduction that his nominalist reading requires, and the abstract entities identiied with divine ideas would still have timeless reality independent of the created world and would still provide the basis for classes of necessary truths. Mates’ proposed reduction relies on Leibniz’s well-known identiication, for the human mind, of the idea of something with the disposition to think about that thing. For example, in “What is an Idea?”, Leibniz writes: “In my opinion, namely, an idea consists, not in some act, but in the faculty of thinking, and we are said to have an idea of a thing even if we do not think of it, if only, on a given occasion, we can think of it” (L 206/ GP VII 263). Our ideas are not mental acts, but dispositions to perform mental acts. However, it is well-documented that Leibniz does not think God’s ideas are the same kind of thing as our ideas. God’s ideas cannot be identiied with his dispositions to perform mental acts, since he has no dispositions and is in fact purely active. Mondadori puts the point as follows in his review of Mates’ study: [T]he view according to which ‘having an idea at a given time does not require having an actual thought at that time, but only a disposition to think’ cannot apply to divine ideas: for the (ininite) totality of God’s thoughts includes ‘all at once’, as actually thought, everything that can be thought by an ininite understanding... Hence, we need not ascribe to God any (modally non-vacuous) dispositions to think; hence, divine ideas must be something other than dispositions to think; hence, they cannot be explained away by appealing to dispositions; hence, the reductive scheme put forth by Mates cannot be made to work, since it crucially depends on the claim that talk of ideas is in fact talk of dispositions to think. (1990, 622) In short, God does not have any dispositions to think about anything: God is always (timelessly) thinking about everything, “all at once”. Mondadori marshals a variety of texts that clearly establish this point; one that is particularly forceful reads “God expresses everything perfectly, all at once, possible and existent, present and future”18. Therefore, Leibniz cannot plausibly be understood as reducing divine ideas to divine dispositions, and so he cannot be understood as reducing to divine dispositions the abstract entities that he identiies with certain divine ideas. The Leibniz Review, Vol. 25, 2015 48 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER However, even if Mates’ proposal did have textual support, and Leibniz could be read as reducing divine ideas to divine mental dispositions, the reduction would ultimately lend little support to the nominalist reading. If divine ideas were reducible to divine dispositions, then those dispositions would still have all the features that Leibniz ascribes to the realm of divine ideas in the passages analyzed earlier in this section. Those dispositions would have timeless reality independent of the created world and would provide the basis for classes of necessary truths; and so the abstract entities identiied with those dispositions would still have these features. The place of relations in abstracto, and so the place of the integers, in Leibniz’s metaphysical framework remains the same regardless of whether the divine ideas are purely actual or are reduced to dispositions. Furthermore, although the contents of the divine mind are ideas, and so whatever entities he relegates to the divine mind are “mental entities” to that extent, this does not amount to a denial of their reality. This class of entities would exist even if there were no created world of individual substances, and even if no inite, created mind ever thought about them. While it is certainly the case that these entities do not exist in the same way that individual substances exist – as Ishiguro puts it, when we refer to relations, for example, we are not referring to “entities which are the basic constituents of the world in the manner that individual substances are” (1990, 140) – it is a mistake to infer from this that they do not exist at all. Thus, it is evident that when Mates claims Leibniz does not have two senses of “existence” in mind – one sense in God’s mind and another sense out of God’s mind – he deviates considerably from the textual evidence. It seems that Leibniz does have exactly this in mind. In the created world of individual substances, one will not ind any abstract entities; but one will ind them in the divine mind, and one would ind them there even if God had never created the actual world or any other world. Ultimately, then, the integers do have a kind of reality for Leibniz, and the reality they have resembles that which the Platonist confers upon numbers, though Leibniz’s account differs in its details. As I have argued, the integers are a kind of mental entity for Leibniz, but they are not the kind of mental entity that one would naturally think of. Instead, as contents of God’s mind, the integers have a robust ontological status, possessing reality independent of the created world, expressing possibilities for the aggregation of things in the created world, and providing the basis for a class of necessary truths. Conclusion The Leibniz Review, Vol. 25, 2015 49 KYLE SEREDA The argument of this paper has been limited to Leibniz’s views on the deinition and ontological status of the positive integers, and no stand has been taken on what he might think about other kinds of numbers: fractions, irrationals, negative numbers, complex numbers, ininitesimals, ininite numbers. There are good reasons for this. First, while Leibniz signals his intent to subsume fractions and irrationals under his general account of number, it is a signiicant interpretative task to explain how he does this and whether his attempt is successful. Second, there is a stark difference between Leibniz’s views concerning the positive real numbers – which he intends to subsume under the same general account – and his views of other kinds of number. To some of these other kinds of number – for example, ininitesimals – he denies anything other than symbolic reality, as signs facilitating certain calculations19. To others – for example, complex numbers – he assigns a status that is less clear. He writes that “imaginary quantities... signify something impossible”, but that they also “can be expressed by real quantities” (GM VII 73). Though Leibniz does say that “number is thus a basic metaphysical igure, as it were, and arithmetic is a kind of statics of the universe by which the powers of things are discovered” (L 221/GP VII 184), he does not appear to hold that all numbers are equally real, or that they all have the same role in our efforts to understand the universe. Leibniz’s conception of number represents a fascinating but largely unexplored area of his philosophy, and the present work is intended only to provide an initial examination of some of his most basic views on the subject. Received 24 July 2015 Kyle Sereda Department of Philosophy University of California, San Diego 9500 Gilman Drive #0119 La Jolla, CA, United States of America ksereda@ucsd.edu References Bos, H.J.M. (1974). “Differentials, higher-order differentials and the derivative in the Leibnizian calculus”, in Archive for the History of the Exact Sciences 14, pp. 1-90. The Leibniz Review, Vol. 25, 2015 50 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER De Risi, Vincenzo (2007). Geometry and Monadology: Leibniz’s Analysis Situs and Philosophy of Space. Basel: Birkhauser. Di Bella, Stefano (2005). “Leibniz’s Theory of Conditions: A Framework for Ontological Dependence”, in The Leibniz Review 15, pp. 67-93. Goldenbaum, Ursula and Douglas Jesseph, eds. (2008). Ininitesimal Differences: Controversies between Leibniz and his Contemporaries. New York: De Gruyter. Grosholz, Emily and Elhanan Yakira (1998). Leibniz’s Science of the Rational (=Studia Leibnitiana Sonderhefte 26). Stuttgart: Steiner Verlag. Hartz, Glenn and J.A. Cover (1988). “Space and Time in the Leibnizian Metaphysic”, in Nous 22, pp. 493-519. Hill, Jonathan (2008). “Leibniz, Relations, and Rewriting Projects”, in History of Philosophy Quarterly 25, pp. 115-135. Ishiguro, Hide (1990). Leibniz’s Philosophy of Logic and Language. Cambridge: Cambridge University Press. Klein, Joseph (1968). Greek Mathematical Thought and the Origin of Algebra. Cambridge, MA: MIT Press. Knobloch, Eberhard (2002). “Leibniz’s Rigorous Foundation of Ininitesimal Geometry by Means of Riemannian Sums”, in Synthese 133, pp. 59-73. Mancosu, Paolo (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford: Oxford University Press. Mates, Benson (1986). The Philosophy of Leibniz: Metaphysics and Language. Oxford: Oxford University Press. Mondadori, Fabrizio (1990). “Review: The Philosophy of Leibniz by Benson Mates”, in Philosophical Review 99, pp. 613-629. Mugnai, Massimo (1992). Leibniz’s Theory of Relations (=Studia Leibnitiana Supplementa 28). Stuttgart: Steiner Verlag. Mugnai, Massimo (2012). “Leibniz’s Ontology of Relations: A Last Word?” in Oxford Studies in Early Modern Philosophy 6, pp. 171-208. Neal, Katherine (2002). From Discrete to Continuous: The Broadening of Number Concepts in Early Modern England (=Studies in History and Philosophy of Science 16). Dordrecht: Kluwer. Russell, Bertrand (1937). A Critical Exposition of the Philosophy of Leibniz. London: Allen & Unwin. Rutherford, Donald (1990). “Leibniz’s ‘Analysis of Multitude and Phenomena into Unities and Reality’”, in Journal of the History of Philosophy 28, pp. 525-552. The Leibniz Review, Vol. 25, 2015 51 KYLE SEREDA Notes 1 In addition to those provided by the editor, I use the following abbreviations for Leibniz’s works: LCW = Briefwechsel Zwischen Leibniz und Christian Wolff, ed. C.I. Gerhardt (Halle: H.W. Schmidt, 1860); S = The Shorter Leibniz Texts, ed. Lloyd Strickland (New York: Continuum, 2006). 2 E.g. Bos (1974); Ishiguro (1990); the papers collected in Goldenbaum and Jesseph (2008); Mancosu (1996); Knobloch (2002). 3 E.g. De Risi (2007). 4 Exceptions are found in Grosholz and Yakira (1998) and De Risi (2007). The former attempts to answer what I call the “deinitional” question in some detail; but it is based on only two texts, and cannot be correct for reasons that I describe below. The latter is quite cursory and does not make deinitive, detailed interpretative claims about Leibniz’s conception of number along any of the dimensions I outline in this introductory section, and so I will not be concerned with it. 5 Russell (1937, 14) notes this, but only in a cursory way and without attempting to explicate the content of Leibniz’s relational conception of number. 6 E.g. in Mondadori (1990); Hill (2008); and Ishiguro (1990). 7 It is worth stressing that the general conception of number under which Leibniz eventually includes the positive integers – and which I will have occasion to note in a subsequent section – is absent from the Dissertation. I note the view of the positive integers that appears in the Dissertation because this view itself remains unchanged in its conceptual core; it simply becomes a speciic case of the general deinition of number that Leibniz advances in his mature work, under which he intends to subsume all of the positive real numbers. 8 In doing so, Leibniz takes up the deinition of number handed down from Euclid, who deines number in Book VII, Deinition 2 of the Elements as “a multitude composed of units”. However, as is apparent from these texts, Leibniz intends to accommodate non-integral numbers under a more general deinition of number, whereas Euclid excludes non-integral numbers from the class of genuine numbers. The promotion of fractions and even irrational numbers to the status of genuine numbers, in opposition to the narrower Euclidean conception of number, is common in the seventeenth century; it is beyond the scope of this paper to explore Leibniz’s place in this mathematical landscape. For details on the shifting conception of what The Leibniz Review, Vol. 25, 2015 52 LEIBNIZ’ S RELATIONAL CONCEPTION OF NUMBER counts as a number during this period, see Klein (1968) and Neal (2002). 9 Translating “numerus est homogeneum unitatis” as “number is that which is homogeneous to unity” is required because Leibniz uses the neuter nominative singular “homogeneum” as the predicate for “numerus”. “Numerus” is masculine, so “homogeneum” must be acting as a substantive in the neuter nominative singular. 10 Nearly identical language is found at A.VI.4.514. 11 There are some texts that might, at irst glance, undercut the reconstruction proposed in this section. Leibniz writes that a transformation is “a change that takes place in the original situation of the parts, none being added or removed” (A.VI.4.508); also that a transformation is “when from one thing it becomes another thing, no part having been added or removed” (A.VI.4.628). He also writes that numbers “cannot be rendered similar” (A.VI.4.933). The irst two texts seem to suggest that the addition or removal of the constituents of an aggregate would not count as a transformation for Leibniz, and so it would become unclear how an aggregate of unities could be rendered similar to unity by a transformation. But Leibniz clearly thinks that this is indeed the kind of transformation in virtue of which these aggregates are homogeneous to unity: he says this outright in my text (3), quoted earlier and partially reproduced in this paragraph. Aggregates of unities are homogeneous to unity because they can be “added to or subtracted from it”. Furthermore, in one of these texts (A.VI.4.508), Leibniz is clearly restricting his discussion to transformations performed on continuous bodies: he begins by laying out some deinitions (of “homogeneous things”, “similar things”, “equal things”, “congruent things”, along with “transformation”), and then he provides a short discussion of the ways in which bodies might be transformed. This text, at least, has no bearing on the question whether Leibniz thinks transformations can involve the addition or removal of parts, since he says nothing about the relevant cases, such as how aggregates might be transformed, limiting himself entirely to transformations of continuous bodies. It is also worth noting that my text (3) is undated, whereas the two texts in this note are from 1682 and 1685, respectively. Perhaps Leibniz begins his career with a restricted notion of a transformation and later broadens it to include the sorts of transformations required to manipulate aggregates. Finally, the text in which Leibniz says that numbers cannot be rendered similar directly conlicts with the signiicant amount of evidence presented in this paper that Leibniz holds that aggregates of unities are all homogeneous to unity. If aggregates of unities can be rendered similar to unity by removing their constituents, then they can also be rendered similar to each other by the same sort of transformation. The Leibniz Review, Vol. 25, 2015 53 KYLE SEREDA 12 The same point is made in Rutherford (1990). For detailed discussion of Leibniz’s mereology, see also Di Bella (2005). 13 My translation of a text found in Mugnai (1992, 156). 14 As such, the positive integers inhabit the top level of what is widely accepted to be Leibniz’s “three-tiered” ontology, consisting of monads at the fundamental level or “ground loor,” phenomena (such as bodies) in the middle (grounded on monads), and ideal entities at the top. The locus classicus for this interpretation of Leibniz’s ontology (from which I borrow the phrases just quoted) is Hartz and Cover (1988). 15 E.g. in Mugnai (1992) and (2012); Ishiguro (1990); and Mates (1986). 16 Hill (2008) mounts an incisive argument against the latter. 17 The thesis that everything besides individual substances and their individual accidents exists only in God’s mind appears repeatedly in Leibniz’s corpus. Mates cites (GP VII 305/L 488) and (GP VI 614-616/L 647-648) in particular. What is at issue here is the meaning of that thesis: Mates holds that it amounts to an elimination or rejection of whatever is identiied with a divine idea, but as Mondadori has conclusively established, this is not the case. 18 Mondadori’s translation of GP IV 533. 19 This scholarly consensus is expressed in the sources cited on the topic in my introduction. The Leibniz Review, Vol. 25, 2015 54