Apollo, Athene: The Logical Foundations of Metaphysics

The halls rose in a pyramid, becoming even more beautiful as one mounted towards the apex, and representing more beautiful worlds. Finally they reached the highest one which completed the pyramid, and which was the most beautiful of all: for the pyramid had a beginning, but one could not see its end; it had an apex, but no base; it went on increasing to infinity. That is (as the Goddess explained) because amongst an endless number of possible worlds there is the best of all, else would God not have determined to create any; but there is not any one which has not also less perfect worlds below it: that is why the pyramid goes on descending to infinity…

At this moment Theodorus wakes up…

Gottfried Wilhelm Leibniz, Theodicy¹

Prelude

In his 1928 lectures on Leibniz, Heidegger concludes his “dismantling of the Leibnizian logic”² with the following words: “This argument [that logic is the foundation of metaphysics even as thinking is the foundation of philosophizing] is not arbitrary. It has metaphysical roots and so crops up recurrently, especially wherever a radical foundation is supposedly sought. And so Leibniz time and again comes close to founding metaphysics on logic, probably most clearly and emphatically in the treatise […] Primae veritates [‘Primary truths’]. Contemporary logic shows a new distortion of the problem. Not only is metaphysics reduced to logic, but logic is itself reduced to mathematics. Contemporary logic is symbolic, mathematical logic, and thus a logic which follows the mathematical method.”³

Heidegger sees the founding of metaphysics on logic as a problem. The primacy Leibniz uncritically grants to logic, so he claims, has obscured the question of Being, which alone is philosophically decisive. But must this be so?

I argue in this essay an alternative approach to interpreting Leibniz, which, if I am not mistaken, is more radical and truer to Leibniz’s own intent than Heidegger’s “dismantling.” I argue that, contrary to Heidegger, Leibniz has not merely “come close to founding metaphysics on logic,” but has actually achieved this founding through his doctrine of the Universal Characteristic. Moreover, he conceives this founding between logic and metaphysics, as I shall explicate in this essay, to be reciprocal: logic founds metaphysics as it makes metaphysics possible through providing a universal view of the pre-established harmony of all possible monads wherein the being of an actual monad can be discerned, while metaphysics founds logic as it makes logic actual through grounding this harmony in the apodictic self-evidence of a primary monad.

This alternative approach has two advantages. First, it obtains a more balanced view on Leibniz’s philosophy as a whole. Instead of playing one part against the other, it recognizes his philosophy to be a genuine attempt at unifying logic and metaphysics into one universal science. By explicating their reciprocal founding, therefore, this approach allows for the more fruitful inquiry into a broadened notion of science that is irreducible to metaphysics or logic alone. Second, it recognizes that logic is secured in this universal science as mathesis universalis, which is for Leibniz’s intellectual predecessors, e.g. Descartes and Wallis, the perennial project to mathematize every truth. Leibnizian logic thus marks the beginning of that “symbolic, mathematical logic” which we now possess and the possession of which Heidegger is so critical of. Now, as mathematical symbolism is the very foundation of modern science, it is worth asking: What is symbolism? What does it achieve for science? Since our approach does not neglect the place mathesis universalis has in Leibniz’s system, these questions regarding symbolism can receive their due attention.

My essay will thus focus on the problem of mathesis universalis and the foundational role it plays in Leibniz’s system. Besides the prelude and an epilogue, this essay is divided into four parts. Part I introduces the doctrine of the Universal Characteristic with its defining marks. Part II examines the tension between the characteristic as a calculus and as a language, while suggesting the idea of infinity to be key to resolving the tension. Part III ventures to reconstruct, using mathematical language, the equivalence between the characteristic, now renewed with an understanding of infinity, and monadology. Part IV first investigates the significance of this equivalence, i.e. how the characteristic and monadology reciprocally found each other, then explores why mathesis universalis, i.e. a general symbolic mathematics, is crucial to Leibniz’s system. Finally, in the flight of fancy of an epilogue, I shall recount briefly the story Leibniz told at the end of his Theodicy (whence the epigraph of this essay) to discuss the ethical import of his philosophy.

Part I

In his early years, Leibniz prided himself on his discovery of the very characteristic “that some people call the Adamic language, and Jacob Böhme calls the ‘nature language.’”⁴ This characteristic, as Leibniz describes, resembling the art of arithmetic and algebra, reduces thinking to symbolic calculations. With rhetorical optimism, he avers that after his initial outline for it, “a few chosen persons could complete the task in five years; in two years they could set forth those doctrines most often used in daily life, that is, morals and metaphysics in an unshakable calculus.”⁵ But before I explain how his invention might actually work from the version available to us, I should emphasize a tension inherent to it that will orient all my following discussion.

Throughout his description of the characteristic, Leibniz repeatedly characterizes it as both a calculus and a language. Yet, are these two really compatible? To clarify the difficulty we should start with something simpler. Let us start with counting, as Leibniz himself is quick to the analogy between arithmetic and his characteristic. In his Greek Mathematical Thought and the Origin of Algebra, Jacob Klein shows that the Greek concept of ἀριθμός has undergone significant change since the invention of algebra.⁶ For the Greeks, he argues, the act of counting is directed immediately towards a determinate multitude of determinate objects, such as two pebbles, two cats, two “Platonic” pure monads, two “Aristotelean” abstract units, etc. After the advent of algebra, however, this intuitive thinking is dominated by a symbolic one: now when we think of “a number,” it appears to us primarily to be a symbol that signifies an indeterminate magnitude, i.e. the unknown x. Yet this revolution is not exclusive to algebra, for as we now do arithmetic, we count not with pebbles, but with numerical ciphers.⁷ When we count to thirty-three, for instance, we do not hold in our mind thirty-three objects, but we represent the number with an Arabic numeral 33. We have established rules that meticulously dictate to us how counting is to be done symbolically: if we replace the second digit of 33 with 4, it is addition by one; if we replace it with 2, it is subtraction by one; if we replace the first digit with 4, it is addition by ten, for we are using a decimal system. No intuition is required in this process: the calculation is done purely symbolically, i.e. mindlessly, by following a fixed set of rules. This is how machines as simple as abaci and as complex as computers work.

This does not mean, however, that intuitive thinking is extinct. Rather, we might say we have trust in the symbolic calculus precisely because we believe that, if we need, we can more or less reactivate whatever is glossed over in our mindless manipulation of ciphers. Though we might be physically incapable of intuiting all the sides of a chiliagon, for instance, we are convinced it is not impossible a priori.⁸ Moreover, when the count is friendly to our sensibility, we usually do not invoke symbolic thinking. When we see two pebbles, we immediately see them as two.

Is language then similarly transformed through the Universal Characteristic? To be exact, what Leibniz has in mind here is not language in general, but that domain of language relating to “discovery” and “judgement.”⁹ Namely, he is concerned only with the scientific discourse, a discourse having to do primarily with the determination of theoretical truths.

If a similar revolution of symbolism as we witnessed in mathematics is to occur in the realm of science, what would it look like (for Leibniz)? Certainly, the full-fledged form is only present as the Universal Characteristic, of which I have so far delayed a complete articulation. But it is expected that, even as numeral manipulation is to counting, so should Leibniz’s new symbolic scientific discourse be to that antiquated science handed down by tradition, i.e. the scholastic science which determines truths primarily through contemplating the concrete natures and essences of things. Through the characteristic, old categories of science are to be overturned and renewed alongside the changed general practice of science itself the same way the concept of number is transformed through the advent of algebra. Leibniz’s project here, therefore, is nothing less than what he inherits from Bacon and Descartes: the project of a New Organon.¹⁰ The tension inherent to this invention being both a calculus and a language lies precisely in the tension between the new science and the old. The battle waged is all decisive: it is to determine whether science primarily is (should be) an art of rule-based and mindless symbolic manipulation, or it remains (should remain) in its essence rather the theoretical comprehension of the natures of things.

But if science is essentially a determination of the knowledge of Being and all beings, this revolution cannot fail to also have metaphysical implications. Therefore, Heidegger is correct insofar as he sees there is a metaphysical dimension to the Leibnizian logic. However, in his haste to reduce this dimension to that of the question of Being, he fails to engage fully with what is truly revolutionary here. Indeed, for him, both the attempt to found metaphysics on logic and the attempt to establish a genuine mathesis universalis, which two attempts are at the heart of the Leibnizian project for a new science, are rather to be dismissed as “a distortion of the problem.”¹¹

Let us now advance to a presentation of the characteristic and decide for ourselves how it might have distorted the problem. The caveat is that Leibniz himself has never provided an authoritative version of the characteristic. The version to be presented and interpreted, therefore, is a mere conjectural one based on his notes available to us.

In his notes from 1679,¹² Leibniz lays down these principles governing the construction of any judgement:

1. In every categorical proposition there are two terms, the subject and the predicate. To these are added a copula (“is”), affirmation or negation, that is quality, and finally, the sign, that is “all” or “some,” which is the quantity.

2. In every proposition, the predicate is said to be in the subject, that is, the notion of the predicate is contained in the notion of the subject.

3. Similarly, in a negative proposition, by denying that the predicate is in the subject we affirm by the very act that the negation of the predicate or a term contradictory to the predicate is in the subject.

4. We must consider that every composite notion is composed of other notions, sometimes positive, and sometimes negative.

5. We must also consider the fact that all negative terms have the property that when positive terms are related as genus and species, their negations, on the other hand, are inversely related, as species and genus. For example, “body” is genus, and “animal” is species, for “body” is broader than “animal” since “body” contains animals, plants and other things. But, on the other hand, “nonanimal” is broader than “nonbody.”

He then provides a clear sketch of the characteristic satisfying these principles:

Now that we have understood these things, we can lay down the true foundations of our calculus. Indeed, for every positive (negative) notion, we shall construct its positive (negative) characteristic number, that is, the characteristic number furnished with the “+” (or “–”) sign, by multiplying all of the characteristic numbers of those positive (negative) notions from which the positive (negative) notion of that term is composed.

Thus, suppose: animal = |+ 13, – 5|, rational = |+ 8, – 7|

Then for this term: man

The characteristic number is: man = |+ (13 × 8), – (5 × 7)|

that is: man = |+ 104, – 35|

In constructing these numbers, we must be careful of only one thing, that no number is contained both in the positive and negative part, that is, that the positive and negative numbers are not divisible by one and the same number, that is, that they do not have a common divisor. For if we were to have written this:

animal = |+ 13, – 5|, rational = |+ 10, – 7|

man = |+ 130, – 35|

we would have written an absurdity. For the notion which is signified by “+ 5” is the contradictory of the one signified by “– 5.” And so, since 5 is contained in 10, the positive notion of “rational” (for 10 is divisible by 5, that is, 10 is the product of 5 and 2), that is, since 5 is found in “rational,” while on the other hand, 5 is denied in “animal,” that is, “animal” contains the contradictory of 5, it follows that “animal” and “rational” are incompatible, and therefore “man,” composed of them, implies a contradiction…

It is obvious that wretchedness can be brought about in some special case of the wealthy… For, some special case of the wealthy has a notion composed of the notion of the wealthy, the genus as it were, together with the notion of the difference that distinguishes this wealthy person [i.e. one who is wretched] from another who is not wretched. Let that difference be “|+ 15, – 28|.”

Thus

wealthy person = |+ 11, – 9|, wretchedness = |+ 5, – 14|

the difference [that distinguishes some wealthy person from a general wealthy person] = |+ 15, – 28|

some wealthy person [who is wretched] = |+ (11 × 15), – (9 × 28)|

Now 15 × 11 is divisible by 5, and 28 × 9 by 14. And so it is obvious that one can bring it about that the predicate is in a special case of the subject [i.e. not all wealthy persons are wretched, yet some can be].

We can summarize the rules of this characteristic in the following way. First, each notion is to be assigned a pair of numbers, one positive, one negative. Second, each number (if not a prime) in a notion is a product of numbers from other simpler notions, which are genera to this more complex notion. Finally, a notion is self-contradictory if the positive number assigned to it is not prime to its negative number. From these rules, moreover, we can induce that there must be an infinite number of simplest notions to make up all the composite notions, even as prime numbers make up all the composite numbers through multiplication. Indeed, Leibniz calls these notions “the primitive notions.”¹³

However, if these rules are to regulate appropriately how notions are to be calculated as if they were numbers, they must first be premised on the properly determined ontological status of notions as such, the same way chess cannot be played without our first presuming the general being of pieces at some very basic level, e.g. pieces are movable physical objects always ready to be handled, pieces are all distinctly shaped and colored to be recognizable bearers of specific functions, etc. At a certain point, the line between rules and properties of the things ruled becomes murky. In Part IV, we see monads are such things whose orderings are their very existences.

The immediate question is, what properties must notions have in this characteristic so that these rules are applicable to them? In the following, I posit three such fundamental properties that are of special importance: equality, independence and intensity.

First, all notions are equal. By equality I mean the characteristic is essentially indifferent to the process by which a composite notion is generated out of simpler notions. For if the characteristic is indifferent to the genesis of a notion, then all notions are equal by nature, as nature in the old science denotes precisely the generative source of a thing. One no longer asks with the characteristic how or why a thing is what it is. One simply registers that it is such. For instance, a biologist and a theologian might both be interested in determining the precise nature of the emergence of man out of the genus of animal through the addition of a differentia, e.g. reason, and they might disagree with each other as to whether reason is a gift from God or a product of evolution. The characteristic, however, is oblivious to this disagreement. It is only interested in the fact that man is a rational animal, but the how or why man is that is suppressed from it.

A possible objection here is that this suppression of the how or why question is not essential to the characteristic but only due to a lack of data registered. One might argue that once we have definitively established either that reason is a gift from God or that it is a product of evolution or that these two options do not contradict each other, we can register this discovery into the characteristic through modifying the composition of these notions. For instance, if we can compose the notion of “reason” out of the notion of “God-given” and other notions, then we would have the why or how of the notion contained in the notion itself. Yet, this objection precisely misses what is crucial. The theologian, when confronted with this clever proposal of a solution, might feel even more flustered. He might say, it is preposterous to think that the notion of “God-given” can be included in the notion of “reason” as its predicate, or to think that in turn “reason” is included in “man,” because reason, qua the gift of God, is by its nature both a miracle of faith and the essence of man, not one datum among many such data predicated of man. In other words, he might argue that “reason” is a source and is thereby irreducible to a predicate. It is not a faculty among many of the human psyche. Rather, as the gift of God, reason marks the incessant miracle of humanity through the formative act of God, which sustains humanity from falling back to the genus of mere animal. Reason, according to this view, is what transcends and first holds humanity in a continuous act of genesis, but not something simply extant in the self-enclosed notion of humanity. That is, “reason” is not included in “man,” but rather it owns man and holds sway over him as his essence.¹⁴

Therefore, registering the how of a notion inside a notion only delays the problem but does not solve it. One can always proceed to ask further, given the notion of “God-given” is included in “reason,” how does reason emerge out of the undifferentiated genus of all “God-given” things? Is it not again a gift of God’s will that reason first becomes reason? Should we then include the notion of “God-given” in that differentia that forms reason out of the genus of God-given things? Yet how does this differentia as a species in turn emerge out of the same genus?¹⁵ As soon as one is involved in this vicious circle, one always stumbles over the same problem: what is the genesis of a notion out of another, i.e. the genesis of species out of genus? Yet, the characteristic is precisely silent on this fundamental problem of genesis: we know that in Leibniz’s characteristic there is an infinite number of first notions which he calls primitive notions, but these primitive notions are presented merely as neutral and homogeneous material ready for the production of any composite notion through a process of combination analogous to the multiplication of numbers. In other words, notions can be combined into more complex notions through some sort of notional multiplication. What has always haunted metaphysics, however, is precisely the nature of this multiplication.

The problem regarding the notional multiplication has a traditional formulation. In the Sophist, Plato calls it the problem of συμπλοκή εἰδῶν, “the blending of Kinds.”¹⁶ Elsewhere, in the voice of Parmenides, he emphasizes the problem thus: “You see, Socrates, how great the difficulty is if one marks things off as forms, themselves by themselves.”¹⁷ The difficulty here is that if there are primitive notions, each self-subsisting and independent of others, how can they be mixed and form one unity of a composite notion? Conversely, how can one composite, if it is a genuine one, be composed of many? Traditionally, attempts at solving this problem usually follow the pattern of positing some first and highest self-generating principle, such as the generative interplay between the One and the Dyad in many versions of Platonism, the self-activating Divine Nous of Aristotle, the self-differentiating Concept in Hegelian metaphysical logic, etc. What is left of the task is a deduction of the manifold of all beings from the self-generation of this highest principle. For instance, according to Klein, the unwritten doctrine of Plato is a doctrine of eidetic numbers, which implies a deductive progression from one to many in distinct phases.¹⁸ First, the One as the Good is the highest principle. Yet, immanent to the One is the Indeterminate Dyad, which negates the self-repose of the One and thereby posits movement. The first phase of oneness then transitions to its next, where the self-repose of the One is counterposed to the self-movement of the Dyad. The two moments are then captured as Rest and Motion under the new collective unity of Being, thus concluding the second phase of twoness. Then the phase of threeness, fourness, etc. Similarly, Hegel deduces the concept of Nothing from that of Being, then that of Becoming from a holding-together of both earlier concepts, and from Becoming next comes Determinate Being, so on and so forth, through the dialectical movement immanent to the Concept which enriches itself in this movement.¹⁹

This pattern of deductive progression is utterly foreign to the characteristic. Although all composite notions seem to be “generated” out of simpler notions, the problem of genesis or συμπλοκή εἰδῶν as delineated is never raised as such by the characteristic. What suppresses this problem is, first, the general mindlessness of the characteristic qua symbolic calculus. That is, it is uninterested in investigating any particular genesis, for all geneses of particular notions are reduced to the one homogeneous operation of multiplication. Furthermore, it is uninterested in determining what this multiplication entails in general, e.g. if this notional multiplication is really analogous to the mathematical multiplication. One can say the characteristic is devised for two questions only: 1. for any given notion, how many notions are contained in it?; 2. what are those contained notions? In other words, the only questions of interest to the characteristic are how simple is this? and what are those things that can make this? These, to be sure, are precisely the principal questions the new science asks, i.e. the questions apropos of efficiency and efficient cause.

But this is to say the characteristic is not just passively suppressive of the problem of genesis. Instead, it has actively deprived this problem of any meaning by revolutionizing the paradigm for what counts as a problem. Let us consider this point carefully. The problem of genesis has lost meaning first in the realm of primitive notions. No primitive is ever generated by another, since they are all equally simple and uncomposed as antecedents to composites, even as greatness of value of prime numbers, i.e. their relative positions on the number line, is irrelevant to them being equally prime. Moreover, primitives are not generated by some highest principle either, as a primitive is by definition simple, i.e. absolutely independent and conceived only in itself. It is not dependent on the Absolute but is only expressive of it. Instead of products generated each in its turn by the self-negated highest unity (e.g. the One negated by the Dyad generates Being; Being negated by Nothing generates Becoming), primitive notions are infinitely many expressive attributes posited of the Absolute. Here, the problem of genesis is replaced by the problem of expression.²⁰

Once the problem of genesis is cast out of the realm of primitives, it is but one step away from total obliteration. For as all primitives are equally simple, none of them can make a composite alone. Nor can a composite be generated out of its positive and negative content of many primitives, for such a generation implies either that one primitive among many is principally the generative cause or that some primitives contribute together to generating the new unity, i.e. the composite. The former amounts to saying a composite is in its essence generated by one primitive, whose impossibility is already demonstrated. But the latter is also impossible. Since generation or the “blending of kinds” always implies a unity of product continuous with its sources, the composite, if it were generated, should be more robust than a mere collection of separate elements. In other words, the generated, insofar as it fully incorporates the generative manifold, would be a unity of primitives reduced to many self-negating and interpenetrating moments submerged in an equilibrium generating this unity.²¹ This is impossible because it returns to the refuted position where primitives are said to be implicated in dialectical movements of generation. Moreover, as the positive and negative contents express the composite equally, e.g. salt is equally “hard” and “not sweet,” neither is the negative generated out of the positive nor vice versa. Thus the unity of a composite, as dictated by the characteristic, is rather that of a thing expressed in many ways, each expression independent of the other. The composite is not generated out of the many primitives, but merely expressed through them. Here, the problem of genesis is again replaced by the problem of expression. The composite is not a product of immanently negated moments, but as the expressed, it is posited with those primitives expressive of it as elements for its positive or negative content.

Therefore, I say notions are equal because the characteristic is indifferent to how they are generated, not in the sense that the characteristic abstracts from all particular geneses, as if notions are equal because ultimately and abstractly they all share the same origin, source or essence; but notions are equal precisely because they have no source at all! Even as all primitives are immediately posited as expressions of the Absolute, so are composites also immediately posited, each the expressed of some primitives. The Absolute, the primitives and the composites are equally ungenerated, hence equally unnatural, and coeternally express each other: they do not share the ontological distinction between the dependent, generated products and the independent, generative sources. Therefore we understand why Heidegger claims that symbolic mathematical logic such as the characteristic obscures the “fundamental problem,” which for him pertains only to the question of Being. For the question of Being is precisely a question apropos of the ontological distinction between Being as the generative source and beings as the manifold products generated by the all-immersive sway of Being. With the eviction of this ontological distinction, the “fundamental problem” seems lost to the characteristic.

The hint of Spinozism here is unmistakable. One is even inclined to agree with Schelling that Leibniz has contributed nothing to philosophy but a popularized Spinozism.²² Are not primitive notions strikingly similar to Spinoza’s infinitely many attributes of substance? Are not composite notions and modes alike? Of course, attributes are not mixable, nor are modes really composites. Yet, Leibniz and Spinoza share the same project: the establishment of a new science. If Spinoza models his new science on geometry (e.g. the geometrical form of presentation of Ethics and the strict parallelism of qualitatively different attributes), Leibniz establishes his based on algebra. I shall reserve my explication of the full import of Leibniz’s algebraism to Part IV, where I explicitly discuss the relation between mathesis universalis and his philosophical system as a whole.

Aside from equality, notions also have to be independent. This is manifest in composites as they are each assigned a negative characteristic number. Whatever primitive deployed to the negative content of a composite could not without contradiction also be included in the positive. According to the law of excluded middle, there can be no mediation: A and not-A are mutually exclusive. Therefore, no two contradictory composites are ever related, since relation is precisely the middle term that mediates two potentially disagreeing things. They are absolutely independent. It is also noticed that primitives are not independent in this way. As neutral elements that make composites, they in themselves are neither positive nor negative. They are only later, in making composites, assigned to positive or negative in the composite made. That is to say they are not exclusive in their pristine state, when they still cohere in communion within the Absolute without contradicting each other, even as all attributes for Spinoza cohere in the one substance peacefully; since there is no contradiction, there is no interference. Primitives are thus each independent of one another. Though one can determine a “primitive” through negation, i.e. through singling it out as a collection of only one element in exclusion of all others, what is determined would not be a primitive at all, but rather a composite to which the addition of any notion necessarily causes a contradiction, and as such, it is the primitive degenerated into a dead tautology.

It follows that notions are also intensive, by which I mean no notion is self-transcending, i.e. holding something as external which is at the same time immanent to itself. Extensions are all self-transcending, such as time and space, each of which is a synthesis of opposites or a genesis between mutually disagreeing elements, like past and future, far and near. Now, composites are clearly intensive. Even as “–7” involves “7” but excludes “+7,” the negative of a composite involves only its own primitives, since they are neutral to it and do not disagree with it. Those contradictory to it, i.e. positives that involve any primitive this negative involves, are excluded by it, and once excluded, they are not involved in it; while those involved in it, i.e. its primitives, do not contradict it. In either case, this negative does not involve in itself its own opposite. The negative, therefore, is never self-transcending. The same applies to the positive. Hence, though the unity of a composite implies a collection of many, it has no extension. Rather, it is singular in being the expressed: its unity as such is independent of its many contents. Given this, not only are composites independent as they exclude, but they remain independent as they include, i.e. they as collective unities are indifferent to the unities they include, precisely because no unity included potentially disagrees with the overall unity of the collective. Each unity, included or including, is only itself. That is, though the content of a composite might depend on primitives or other composites, its unity per se is neither generated from nor dependent on other unities that are only expressive of it. All composites are intensive qua expressions. Similarly, as all primitives cohere pristinely in the Absolute without contradiction, they are all intensive: they imply no dialectical movement of self-transcendence generated by contradiction.

All these properties are posited with the ungeneratedness of notions. Notions are equal because, qua ungenerated, they do not differ in their nature, insofar as they do not have a nature at all. Notions are independent because, qua ungenerated and self-positing, they are not dependent on a generative source. Finally, notions are intensive because they are not self-transcending, i.e. generative, as extensities are. All notions, primitive or composite, cohere as many independent expressions of their respective expressed. These, so far, are the metaphysical implications of the characteristic, governed by one motif, i.e. the substitution of the problem of genesis with the problem of expression.

Part II

Undeniably, the problem of genesis is still meaningful in our life. If the characteristic is to secure “those doctrines most often used in daily life, that is, morals and metaphysics in an unshakable calculus,” it must take account of the phenomenon of genesis. I would even go out on a limb to suggest that precisely this is the difficulty preventing Leibniz from completing the characteristic. The difficulty, one might say, lies in a genesis of geneses: where do we first encounter, in the characteristic, the mediations and geneses we talk about daily in our natural life?

We are back to the tension between the characteristic as an art of calculation and a linguistic construction. To formulate the general differences, let us first examine an example of this contrast:²³

Dog is wretched.

This sentence, being an unconditional truth statement, has a subject-copula-predicate structure, the subject here being “dog,” the predicate “wretchedness.” The act of judgement is the simple analysis of the notion of “dog,” which is found, through this analysis, to include the notion of “wretchedness.” This does not contravene the rules of the characteristic.

I am beating a dog.

This second sentence, however, is more complex. In broad strokes, one can characterize it to have a subject-verb-object structure. It does not lack a predicate, but the predicate here signifies an event. As this predicate contains two notions that clarify its content, i.e. “beat” and “dog,” it also intimates various relations determining it as the event: it is “I,” the speaker, that does the beating to the dog, not vice versa; the beating is ongoing right now; etc. All these various relations are neatly captured by the conjugation, “am beating.”

In ancient Greek, for instance, verb conjugation involves a rather exhaustive list of categories, each expressing a particular relation determining the event: person, number, tense, mood, voice, case and aspect.²⁴ Person expresses how the event is related to the speaker, e.g. if “I” am doing the act or another. Number expresses how many partake in the event as the subject, e.g. if the subject is singular, dual or plural. Tense expresses the time of the event, e.g. if it is in the past or present. Mood expresses how the speaker subjectively relates to the event, e.g. if he thinks the event is certain to happen or not. Voice expresses how active the subject is in relation to the event, e.g. if the subject is doing the act or suffering it. Case expresses how the event relates to the object, e.g. if the act proceeds “to” or “from” the object, viz. if the event comes “here” or goes “there.” Finally, aspect expresses the viewpoint one takes toward the event, e.g. either one sees the action internally as continually unfolding or sees it externally as a completed whole, viz. imperfective or perfective.²⁵ This distinction between two aspects can be rephrased as a distinction between what Leibniz would call “reality” and “virtuality:”²⁶ the imperfective expresses the event as if it were unfolded into an ongoing, concrete experience internally differentiated into many moments, hence expresses it as real; the perfective, on the contrary, expresses it as if its internal content were folded into the undeveloped, undifferentiated latency of an “atom” indifferent to what is really experienced, hence as virtual.

Put succinctly, person has to do with the self-other relation, number with one-many, tense with past-present-future, mood with certain-obscure, voice with active-passive, case with here-there, and aspect with real-virtual. All these relations are generative as they are between relative opposites. They are thus incompatible with the characteristic. For, first, as all notions are independent, each notion subsists only in itself while its unity knows no other. Moreover, notions are immediate unities, hence unrelated to the many; notions are coeternal and thus not in time; notions all have a finite number of components and are thus fully analyzable, i.e. absolutely certain; notions, though some include others while some are included by others, remain independent and do not interact among themselves, hence neither act upon nor suffer from each other; notions do not move about, nor do they proceed anywhere; finally, all notions are virtual, inhering in pure possibility qua eternal essences in the Absolute.²⁷ Clearly, then, the characteristic, qua calculus par excellence, cannot be a language, as it knows none of the distinctions we find in natural speech.

We now have a sharper focus on the puzzle. In what ways can the characteristic in its present form either give rise to or at least simulate the categories inborn to our natural speech, which revolves around events? Is this impossible, since Leibniz never completes the task? Yet neither does he ever abandon it! In multiple places, Leibniz is generous enough to give us a clue as to what his solution to the puzzle is, and they unanimously point to a key concept: infinity.²⁸

In The Source of Contingent Truths,²⁹ Leibniz expounds precisely the genesis of all geneses in terms of infinity with these opening words: “The source of contingent truths [i.e. events] in an infinite progression, on analogy with the proportion between incommensurable quantities [is the following]…” He continues: “if the analysis proceeds to infinity and never attains completion, then the truth is contingent, one which involves an infinite number of reasons, but in such a way that there is always something that remains, for which we must, again, give some reason.” Finally, he maintains “because this has not been understood, the union of the soul and the body has also been taken to be inexplicable. For, in metaphysical rigor, they do not flow into one another, nor indeed, does God move the one on the occasion of the other and divert it from its own proper course. But following its own laws from the time they were instituted with an admirable but infallible constancy [directio], each agrees with the other as exactly as they would if there were a true influx.”

“In metaphysical rigor” means in the way God knows and creates things. But here the metaphysical is equated with the mathematical, for we say the characteristic, though symbolic, contains metaphysically decisive implications, as God employs the same calculus to create the world, which Leibniz calls “Divine Mathematics;”³⁰ only what is for us symbolic is for God intuitive. That is, as we use numbers to stand in for notions, God combines notions in themselves. This implies the world as it is created does not follow the paradigm of genesis or natural speech. Hence, when Leibniz says “in metaphysical rigor” there is no “flow into” but soul and body act as if “there were a true influx,” he means there is no genesis, viz. “influx,” at the foundation.³¹ There is only its simulation at infinity.

Natural speech revolves around events, each of which depends on infinitely many others as reasons for its happening. I am beating a dog, because I am angry, because the dog was barking, because now I have my hands untied, etc. These reasons relate to the one utterance, I am beating a dog, as its many intimations. It intimates the beating is happening right now because…, happening between me and the dog because…, I am certain of it because…, etc. Each intimation falls under a category that involves the current event with some other event in a genesis, as that other event is its ground of reason, e.g. a past that generates the present, the action of another that generates my passion, etc. All events are geneses. If the characteristic were to simulate events as it proceeds to infinity, then these categories prescribing the various geneses of an event must be derivable from it, despite its being, strictly speaking, only a calculus.

The list of categories based on verb conjugation is less than exhaustive, but each item in it is nonetheless essential: person, number, tense, mood, voice, case and aspect. In Leibniz’s corpus, these categories can roughly be divided under four topics, each focusing on one particular problem of genesis: 1. the genesis of the apperceiving egoic monad “I” and other monads obscurely implicated in the ego’s self-apperception, which corresponds to person and mood; 2. the genesis of the actively unifying form of a discrete monad and its passive, dispersive matter, which corresponds to number and voice; 3. the genesis of the extensity of space-time, which corresponds to tense and case; 4. the genesis of a substantial reality out of many virtual phenomena, which corresponds to aspect. This is not the only possible ordering, as all these categories overlap and each implicates the whole. Yet I deem it to be a very convenient one. In the next part of this essay, I will attempt to derive all the said categories from the characteristic in this order.

Part III uses a mathematical method and vocabulary. It shares its form with Leibniz’s The Source of Contingent Truths, such that it draws an analogy between the theory of proportions and the theory of truths. Yet, the purpose is precisely to suggest that the kinship between mathematics and metaphysics is much more than mere “analogy.” Rather, their equivalence implies their genuine mutual foundation.

Part III

THE MATHEMATICAL SYMBOLISM —> ITS METAPHYSICAL IMPLICATION

ELEMENTS: The infinite series of prime numbers, 2, 3, 5, 7, 11, 13… —> The infinite series of primitive notions, each an absolute neutrality attributive to the divine understanding.

METHOD: 1. To multiply one prime number with a different prime number to obtain their product, e.g. 2 × 3 = 6. —> To combine one primitive notion with a different primitive notion to obtain a neutral content.

2. To pair up two such product numbers, then assign positive to one, negative to the other, e.g. (6, 35) = |+ 6, – 35|. —> To include or exclude one notion or a series of notions from the notion that is being defined, e.g. “man” is “rational” and “not-winged.” In other words, to obtain a composite notion with its positive and negative content.

3. A pair in which the positive number is prime to the negative number, e.g. |+ 6, – 35| = |+ (2 × 3), – (5 × 7)|. —> A composite notion that is self-consistent, since it does not exclude a notion that it at the same time includes, e.g. a “wingless man.”

A pair in which the positive number has a common divisor with the negative number, e.g. |+ 6, – 15| = |+ (2 × 3), – (3 × 5)|. —> A composite notion that is self-contradictory, since it includes a notion that is to be excluded, e.g. a “winged man.”

DEFINITION: A transformation F is defined such that F (X) designates the product of unique prime numbers lesser than but closest to X, e.g. F (2 × 2) = 2, F (2 × 2 × 3 × 5 × 5) = 2 × 3 × 5. —> God does nothing in vain. That is, for any content, God uses unique primitives only, lest there is redundancy; or, all repetitive content is reducible, e.g. “man” added with “man” remains “man.”

AXIOM: The set Φ of all the products of unique prime numbers is self-enclosed with the double transformation of multiplication and F. That is, suppose x, a product of elements from Φ, when transformed by F, gives x’, then x’ is an element of Φ. —> The divine understanding is absolutely self-subsistent, wherein an infinite totality of all possible unique content is posited.

EXPLICATION OF THE AXIOM: An element of Φ, e.g. 30, is a product of 2, 3 and 5, each of which is a unique prime number. 14, a product of 2 and 7, is also an element of Φ. 420, the product of 30 and 14, is a product of 2, 2, 3, 5, 7, hence not in Φ, as one of its factors, 2, appears twice in the series, hence not unique. But F (420) gives 210, which is in Φ. —> In other words, if the content of a composite, positive or negative, as a combination of primitives is further combined with another primitive to make a different content, this new content is nonetheless included and preconceived by the divine understanding. The divine understanding is therefore The Absolute.

SCHOLIUM I: 1. Besides the distinction between composites, i.e. notions that contain both positive and negative content, and primitives, i.e. notions that are neutral elements for content, it can be deduced there must also exist the following distinctions: The One includes all the primitives for its positive content and excludes nothing, so any negation contradicts it; pure nothingness includes no primitive but excludes them all as negative content, so any true content contradicts it; a pure idea includes some primitives but excludes all others, hence is tautological and any notion added to it contradicts it, as the product must include what is already excluded, even as nothing can be added to justice itself without bringing in impurity it denies; so is that which excludes some primitives but includes all others an opposite pure idea, since any notion added to it excludes something already included, even as nothing can be added to injustice itself without taking away some portion of what is purely unjust; an essence has both a finite positive content and a finite negative content, so some notions contradict it, some do not; an existence has both an infinite positive content and an infinite negative content.

2. The determining of a composite (following the METHOD) is like an eliminative process.³² As it appears to us, God determines for each primitive if it is positive or negative in, or irrelevant to, the ongoing series that would make the composite in question. Once a composite is fully determined, its negative content excludes certain notions, as primitives in its negative are excluded from its positive. Every composite excludes, as every composite has its negative. The divine understanding, however, is a latent virtuality, so that each content possibly involved in a composite as either positive or negative is already implicated in it without contradiction, even as every possible product of unique prime numbers is contained in Φ. Therefore, as to make a composite requires first of all to select what is to be excluded, and to thus select is to negate the totalizing latent virtuality inherent in the divine understanding, each composite is premised on a selection through negation. As there are infinitely many ways to select, there are infinitely many possible composites.

3. Existence is an infinite composite: the eliminative process which determines it is indefinite. Each existence thus contains an infinitely unfolding series of positive and negative content. An existence is considered impossible if it is found to exclude what it includes. Adam cannot both sin and not sin. Yet for Adam prior to his sinning, he has the freedom to either sin or not sin. With respect to Adam prior to his sinning, therefore, sinning and not-sinning are only incompossible, but neither alone is impossible. For neither sinning nor not-sinning contradicts the notion of Adam, but as soon as sinning is included, we have the new notion of a sinning Adam, and not-sinning can no longer be included. We should therefore distinguish impossibility from incompossibility.

4. “[Since] something rather than nothing exists, there is a certain urge for existence or (so to speak) a straining toward existence in possible things or in possibility or essence itself; in a word, essence in and of itself strives for existence. Furthermore, it follows from this that all possibles, that is everything that expresses essence or possible reality, strive with equal right for existence in proportion to the amount of essence or reality or the degree of perfection they contain, for perfection is nothing but the amount of essence […] And in this context, time, place, or in a word, the receptivity or capacity of the world can be taken for the cost or the plot of ground on which the most pleasing building possible is to be built, and the variety of shapes [therein] corresponds to the pleasingness of the building and the number and elegance of the rooms.”³³ God, in creating the world, would want to involve maximum content with minimum cost. The maximum here is obviously the totality of all possible essences, as they all have an equal right to strive for existence, i.e. to express. Yet if all possible essences are involved, there must arise contradictions, as some essences are contradictory. These contradictions are alleviated by God through his orderings (APORIA III).³⁴

APORIA I: A finite number π belonging to the set Φ either includes prime number ∂ in its series of factors or not. If ∂ is excluded, then π is indifferent to ∂, since it remains itself through this exclusion and as number π is self-sufficient. If now ∂ is multiplied to it, it will then persist in the newly made number as one of its factors, but not the newly made number itself. Conversely, if ∂ is included, the removal of ∂ from its series of factors will change its value, and the number π will be lost, since the newly made number will not contain π as a factor. —> The notion of Adam is either prior or posterior to the predication of sin. If the notion is posterior, it would be absurd, since Adam is free to either sin or not sin. For, given the notion is posterior, it would be like a species to a genus. The differentia would be the predicate “sinning,” whose removal should remove the species altogether, and it would no longer be Adam whom we talk about after the removal, but someone else as the genus of Adam. As he is Adam only if he sins, he, as Adam, could not have not sinned, and is thus not free to not sin, which is contrary to our premise. Thus the notion has to be prior. Yet, if it is prior, the notion of Adam would be like a genus, and Adam stays Adam whether he is predicated of sin or not, and sin is therefore indifferent to him as Adam. In this case, there is no sufficient reason for him as Adam to choose sinning over not sinning. He would be the donkey that does not know if he is to drink or to eat. This is contrary to the law of sufficient reason, hence absurd.

SOLUTION TO APORIA I: The removal (addition) of ∂ from (to) an infinite series ∆ does not change the product value of ∆, as it stays infinite; the apparent removal (addition) only obscures (clarifies) whether ∂ is included in ∆ or not. —> The positive and negative content of Adam, qua substance, is each an infinitely unfolding series. Each is prior to the predication insofar as it retains the same content as infinite content when a predicate is removed from (added to) it; each is posterior to the predication as the removal (addition) obscures (clarifies) if the predicate is so far included in the unfolded content or excluded. That is, if the predicate “sinning” is removed, Adam stays the same infinite substance “Adam,” only it is now obscured whether he sins; if added, Adam again stays Adam, only it is now clarified that he also sins.

EXPOSITION OF APORIA I: This aporia is solved because finite and necessary essences are carefully distinguished from infinite and contingent existences. The aporia arises as the notion of Adam is initially misconceived to be a composite that includes only a finite number of primitives for its content, i.e. an essence. In that case, the addition or removal of a predicate would either be absolutely essential or utterly immaterial for the notion of Adam, as “Adam” either is a species defined through the predicate that is its differentia and would be something else without it, or it persists as a genus indifferent to the predicate. In other words, even as 2 times 3 gives 6 and 6 is 6 only with 3 while 2 is 2 with or without 3, Adam either is Adam only through his sin, or stays Adam if he sins or not. Yet, since Adam is a contingent existence composite of infinitely many essences, his content is rather an infinity consisting in an infinitely unfolding series of predicates, the apparent inclusion (exclusion) of any particular predicate into (from) which changes not its content qua infinity, but only clarifies as the series is unfolding whether an element is included (excluded). In other words, an existence is never fully comprehended as a completed whole, but rather as an unfolding infinity is only gradually clarified in its parts if it includes a particular predicate or not.

APORIA II: A set that contains all the members of an infinite series ∆ of unique prime numbers has cardinality ℵ₀, while the set of real numbers ℝ has cardinality ℵ₁. —> If we say the positive and negative content of a substance, such as that of Adam, each consists in an infinite series of unique primitives, it is then implied that each is made of discrete elements. However, the content of a substance is infinitely divisible, i.e. continuous. This implies that the content of a substance is not an infinite series comprised of unique primitive notions, hence contrary to our premise.

SOLUTION TO APORIA II: Since a set Ω that contains all the members of an infinite series ∆ has cardinality ℵ₀, its power set, i.e. the set that contains all subsets of Ω, has cardinality ℵ₁. —> A substance indeed consists in primitives, positive and negative, yet it also consists in things made of them, e.g. composites, which are irreducible to these primitives, as they are equally expressive. Hence from bottom up, a substance first consists in infinitely many primitives, then in the things they make, then in higher things made of these things, ad infinitum; from top down, it is divisible into infinitely many substances, which in turn consists in infinitely many other substances, ad infinitum.

EXPOSITION OF APORIA II: This aporia arises due to the premise that a substance is immediately composed only of unique primitives. Yet this is a false premise. On the contrary, primitives are components of a substance that in turn make other components equally expressive of the substance and hence irreducible to primitives. Suppose a set Ω contains all the primitives in a substance. It is infinite but countable. Now, as each composite contains its own set of primitives, the set of all composites and primitives in this substance is the power set of Ω. It is infinite and uncountable. But just this set contains all the content of a substance. A substance, therefore, contains an uncountably infinite set of elements, and is hence continuous.

APORIA III: If an element ∂ is excluded from series ∆, it is not included. —> If a substance excludes a predicate, it does not include it. Otherwise, the law of non-contradiction is violated. For instance, a physical object, once it is clarified to have the shape of a circle, shall never come to have the shape of a square, as it by the inclusion of the notion of circle has excluded the notion of square. Similarly, Adam, once he has sinned, cannot have not sinned. Yet a circular thing can certainly become square and Adam can be redeemed. This seems contrary to the premise.

SOLUTION TO APORIA III: Though ∂ is excluded from ∆, it can be ordered such that ∆ includes ∂ ad infinitum but excludes it in any determinate part. —> Though two things can be contradictory, they are not contradictory insofar as they are two different parts of a substance. Adam, who sinned before, can be redeemed in the future. In a similar sense, if this piece of paper here is a circle, that piece of paper over there can be a square.

EXPOSITION OF APORIA III: This aporia arises because the finite analysis of an essence is conflated with the infinite analysis of an existence. A finite analysis is concerned with a determinate notion, i.e. an essence, that can be formally written as |X, – Y|, X and Y being definite values that symbolically represent the positive and negative content of an essence. If X and Y are not prime to each other, a contradiction is incurred. On the other hand, an infinite analysis is concerned with an indeterminable notion, i.e. an existence. Similar to essences, all existences can be formally written as |∂₁, – ∂₂|, while both ∂₁ and ∂₂ are products of infinitely many unique prime numbers, symbolically representing the positive and negative content of an existence. As both are infinite, they are neither prime nor not prime to each other. Similarly, existence does not conflict with itself qua existence. We then replace the product values ∂₁, ∂₂ with infinite series ∆₁, ∆₂. Suppose the existence is now represented as |∆₁ = (… × 2 × 5 × 11 × …), –∆₂ = – (… × 3 × 7 × 11 × …)|. In this case, ∆₁ and ∆₂ are not prime, as they both include 11. Yet since they are infinite, the two series can both be ordered in infinitely many ways. Depending on the ordering, one might unfold them indefinitely without running into any of their common divisors. Analogously, though an existence might contradict itself globally, its content can be ordered in such ways that locally no contradiction ever occurs. Existences, qua infinite series, can thus always be arranged such that in order to run into a contradiction, one must go through infinitely many other existences, qua its sub-series. Therefore, if a contradiction is in finite essence, one sees it immediately, whereas a contradiction in infinite existence can always be infinitely mediated through ordering.

PROBLEM I: To construct the apperceiving self and its perceived other.

ARGUMENT FOR PROBLEM I: Self is the center of perception.³⁵ Earlier, we see a substance consists in an infinitely unfolding series. Moreover, substances are continuous, for their content has cardinality ℵ₁. As God constructs the series from the bottom up,³⁶ what he sees is the unfolding of infinitely many primitives in the series and all the things they make. That is, as God sees the series a, b, c, d… which includes both positive and negative primitives, he also sees the series of sets, {∅}, {a}, {b}, {a, b}, {c}, {a, c}… As human beings, however, we are creatures situated at the end of creation and only know a substance from the top down. For us, substances are given as infinities and we start perceiving them only in obscurity. Yet, as a substance is unfolded (by which I mean the content of a substance is analyzed for our knowledge into its immediate constituents) and its constituents in turn are unfolded,³⁷ we gain more and more clarity on the substance and thus expand continuously the realm of our perception into the realm of obscurity. As this expansion is continuous, the division between clarity and obscurity is indefinite. The habitual realm of clarity is called self, while that of obscurity other. As the two realms are continuous, so are selfhood and otherness. I perceive myself in another, as I look into a mirror darkly lit. Those who see the self clearly and universally through a reflection into all things, i.e. whose self-knowledge keeps unfolding, are apperceiving egos such as human beings; those whose realm of clarity remains relatively stagnant or habitually suffers setbacks are animals, plants and lifeless things. When this realm of clarity unfolds, one lives even as one perceives. When it enfolds (that is, when the content unfolded for our knowledge falls back to obscurity), one perishes in numbness. As one lives, selfhood is generated out of otherness. As one dies, the genesis is reversed.

PROBLEM II: To construct the unifying form and its manifold matter.

ARGUMENT FOR PROBLEM II: Form is the center for organizing a manifold.³⁸ Since some substances unfold while some are enfolded into them, substances organize into a hierarchy. Each substance is a unity, as all composites are unities. But this unity here is extensive, as the series is now infinite and contains contradictory elements; for God, in creating the world, has created a totality involving every essence, thus involving infinitely many contradictions, though none occurs locally (SCHOLIUM I, 4). These contradictions are distributed continuously, for substances are continuous, even as salt is dissolved in water. As some substances unfold, they involve more contradictions than others, and become thus disharmonious. Conversely, some substances are harmonious. Harmonious substances, as they disagree less with themselves, are more unified. Since as a substance unfolds, it tends to fold into more disharmony, even as one tends to take in more salt if one simply drinks more salt water, enfolded series tend to be more harmonious than unfolded ones. The more a substance enfolds, therefore, the more it tends to be unified. Moreover, as all substances are in one continuum, one substance folds into another. A harmonious substance is thereby a nexus that recapitulates a larger disharmony, as it is a more unified expression into which divergent expressions are folded. One calls this nexus form. If as it unfolds a substance enjoys less harmony, it becomes less expressive, till it eventually disperses into a passive content, i.e. matter, that is to be recapitulated by many recapitulating forms. A thing lives as it recapitulates matter to enrich its own content without dispersing itself; that is, as it perceives more things, it also strives to perceive them in harmony. Therefore, though we see through our eye, i.e. a perceptive organ whose content unfolds in clarity, we live on our heart, i.e. a vital organ that, though itself folded in obscurity, actively provides agreeable matter it enfolds for our perceiving life, as it is a silent machine that purveys us our lifeblood. Through these vital organs, we generate ourselves on matter as our food when we live. As we die, the genesis is reversed.

COROLLARY: Life mediates between perception, i.e. unfolding, and nutrition, i.e. enfolding. As one enfolds, one loses clarity but tends to gain unity. As one unfolds, one gains clarity but tends to lose unity. Each approaches death, as one either falls into numbness, or is dispersed into many yet numb things.³⁹

PROBLEM III: To construct space and time.

ARGUMENT FOR PROBLEM III: Space and time are orderings of existing things.⁴⁰ As countless contradictions are distributed among continuous substances, they are distributed in accord with the orderings ordained by God. As orderings differ, contradictions are differentially distributed, and they might be more concentrated in some parts than others. Suppose two substances unfold the same number of contradictions, while for one they are concentrated in various parts, for the other they are equally distributed throughout the whole substance. Both substances are hence equally harmonious. Yet, as both unfold, the former encounters radically shifting states of harmony, while the latter sees none. The former is then said to be temporally ordered, the latter spatially. For as one views a landscape, one does not take in the whole landscape instantly. Rather, each glance is composed of infinitely many kinestheses that organize the view part by part. It is only because these kinestheses fail to create an internally differentiated rhythm in perception that the perceived is taken to be a space. But if as a thing unfolds it disperses and recollects in distinguishable moments, it is time. As one instant lives in near absolute intensity, then almost immediately perishes as it falls in abrupt disharmony, and recollects into a next instant, it generates a rhythm, without which there is no time. Space is therefore a homogenous time, and time a differential space. Now, as all substances are continuous, so are their contradictions. Moreover, as a substance unfolds, the distribution of contradictions in it also varies continuously. Therefore, as a substance unfolds, not only do time and space generate each itself, they also generate each other, as a sculpture can reveal its rhythm, and music its expanse.

PROBLEM IV: To construct substances and their phenomena.

ARGUMENT FOR PROBLEM IV: Substance is an unfolded series.⁴¹ As the series unfolds, our perception gains in content. Reality, indifferent to harmony, disharmony or their distribution, measures only how much content is gained: |+ 2, – 3| has less content than |+ (2 × 5), – (3 × 7)|, hence is less real, though they are equally self-consistent. To illustrate this more concretely: “I am loving” expresses something different from “I love” or “I am in love,” yet all three equally designate a duration experienced by the “I” located within this duration, as “love” signifies a continual state; what is differently expressed by “I am loving,” therefore, must be distinct from mere experienced time. Now, by saying “I am loving,” I only intend to unfold from the atomized state of “I love” its internal differentiation, marking a readiness to confess each and everything I suffer from love. Such differentiation is indifferent to time or the contradictions time alleviates. The passion of love can be fraught with contradictions or be ever so evanescent without seeming less real,⁴² if its content unfolds enough internal differences before is dispersed. What expresses such reality concretely unfolded is a substance. As everything strains for reality, everything strains to unfold; enfolded, it is abstraction, or phenomenon.

SCHOLIUM II: 1. As a world-series contains all existences and essences, one is formally able to extract from it any series, which usually corresponds to one of the following categories (SCHOLIUM I, 1): The One, pure nothingness, pure ideas, essences and existences. It is easy to see that The One does not involve any negation, while pure nothingness does not involve any true content, hence both are distinct from the substance, which has infinite and contradictory content. Pure ideas are tautologies. Depending on how one extracts it from the substance, one either retains only a finite positive content or a finite negative content, such that one might claim the idea, e.g. justice itself, is either expanded too much or negated too much by the addition of any other notion. Essences are extractions finite in both positive and negative content. Now, existences, though they might be extracted from other existences, always entail immediately an infinity and the whole totality of existences, as they continuously merge into each other and no division between them is ever definite. This extraction is thus a mere reordering of the world-series selected to bring one aspect of it clearer to light instead of another. All extractions, however, are such reorderings, each different depending on what aspect is selected and how much of the substance is to be unfolded through this ordering.

2. Abstractions, or phenomena, are all indicative of the substance to a greater or lesser extent. As a phenomenon loses this indication and approaches finitude, i.e. as it enfolds, it becomes more like an essence. For instance, color is substantial, as it is a phenomenon indicative of light reflected off an object. Yet, it loses substantiality, though never fully, and becomes pure phenomenon when it is held indifferent to other notions, in the way an essence is indifferent to what is added to it. Conversely, such a quasi-essence in becoming more and more determined might either be re-substantiated or idealized. That is, through many negations and affirmations, the phenomenon might gradually unfold into a full substance and be continually enriched in content. This is substantiation. It is also possible that, in determining it, its content unfolds such that it approaches a pure idea. This is idealization. Now, as all extractable series are of the same continuous world-series, they are only abstractions or phenomena in continuous geneses within it, such that essences and existences are mutually generative, for infinitude and finitude here are only relative concepts measuring how unfolded notions are; so too are ideas, The One and pure nothingness also in continuous geneses. Thus we understand whence the problem of genesis originates: it originates from the presupposition of a substance, the various abstractions it implicates, and the mediate relations that obtain between them.

3. But this is not God’s perspective. Unlike us, who take the world as given, he expresses the world from himself, i.e. from the totality of primitives in their pristine state inhering in his divine understanding. Each substance is thus for him fully unfolded in its constituent elements. We, on the other hand, can never gain intuition into these primitives that are true elements of the world, for our knowledge relies on abstractions, which are all mere top-down selections of particular aspects that always indicate the substance from which they originate, hence always insinuating some remainder yet to unfold from the hidden source. So far, therefore, our knowledge is only inadequate, as substances cloud our vision. Deus Absconditus. He eternally torments us with the problem of genesis and an unfolding more interminable than Penelope’s weaving. But if we ever desire to know things adequately, we need to know the primitives. That means we need to leave the realm of substances once and for all. Herein lies the tension between the new science and the old: as the former takes this flight and soars, the latter remains grounded.

Part IV

Let us recapitulate the equivalence established in Part III, so as to ground it more firmly.

As the composition of notions proceeds to infinity, we encounter substances, i.e. monads. Each monad unfolds into an ordered series implicating a totality of other series. As all these series are themselves orderings, each monad is an ordering of orderings, i.e. each is a perspective on the totality of all monads. All monads are thereby implicated in each one monad as they express themselves in it qua infinitely many orderings ordered by it. These expressions form into indefinite categories, such as essence, existence, etc., each more or less an abstraction of the one implicating monad, as each unfolds it incompletely. Yet, as God knows all primitives immediately, all monads are fully unfolded in his knowledge. The totality of all orderings from all perspectives, i.e. the world, is thus intuitive to him as the pre-established harmony.

Leibniz divides knowledge into the following: clear, distinct, adequate, and intuitive.⁴³ What is clear is what one knows is true yet can give no account of, e.g. an artist knows how to draw well, yet cannot explain his talent. What is distinct is what one knows is true and can give the account of, as one knows the notions that compose it, e.g. I know “man” distinctly if I know he is a “rational animal.” What is adequate is what one knows in every detail. Yet one can know something adequately either in a symbolic or an intuitive way. That is, one either knows the totality of all truths through symbols, or knows them in themselves. Though we aspire to the intuitive, only symbolic knowledge is humanly possible. For, as we are situated at the end of creation, the world is given to us as a substance. Though we unfold it indefinitely, we merely attain more substances or abstractions indicating substances, but never primitives, through which alone can truths be fully clarified. To truly approach Divine Knowledge, therefore, we would have to first destroy this world underneath which all primitives are concealed.

But the characteristic achieves just this destruction for us, as the world becomes denatured through its symbolism. A symbol, unlike abstraction, is an indifferent substitute for any content. It is pure possibility self-enclosed, i.e. “anything whatsoever.” In it, the infinite horizon of possibility unfolding in continuum with actuality is condensed into one homogeneous object. A symbol can represent something actual as well as possible. It indicates no genesis, as it posits the totality of all possibilities immediately, even as the mathematical unknown x posits all possible magnitudes immediately. In other words, all symbols are expressive. Unlike an abstraction indicative of its substance or an image indicative of its original, symbols only express themselves as immediate and total. Through them, the problem of expression is first manifest as such.

With symbols, we attain all primitives quickly, for we can simply posit them, even as we posit infinitesimals in differential calculus. That is, instead of indefinitely unfolding the substance in search for primitives, we assume this unfolding already completed, and use symbols to represent the primitives that are sought. Primitives, along with their infinite geneses out of the substance, are thereby condensed into symbolic objects. Thus posited, they cannot indicate any substance, for their relation to it is pure indeterminacy. But if primitives are not generated from substances, neither are substances from primitives. A monad is thus posited immediately with primitives, essences and existences, all being expressions equally ungenerated, each definable only through a set of formal principles and rules.

Leibniz’s characteristic is precisely an attempt to formulate these rules. Through it, we see primitives, essences and existences all cohere as expressions of one structural totality encompassed by a universal ordering. That is, we glimpse the pre-established harmony of all monads. Admittedly, we still cannot intuit this harmony, yet we nonetheless manage to discern its formal structure with the help of symbols. At least formally, therefore, a universal monadology is established.

The problem of genesis is, in view of this universal monadology, reworked into a problem of simulation. Genesis is merely emergent at the end of creation, where the world is given to us in perspective as an infinitely unfoldable substance. As we are restricted to always start unfolding from what is clearer and hence more actual to us, i.e. ourselves, the world qua the totality of monads becomes for us a horizon continuously receding into obscurity. The world in its actuality, i.e. self-evidence, becomes thus grounded in the self-evidence first given to a particular monad. A monadology that carries out in actuality this grounding of all monads in one is a particular monadology.

Now the reciprocal foundation between metaphysics and logic can be fully explicated. As metaphysics is primarily a pursuit to ground all beings in the generative activity of Being, it is particular monadology. As such, it grounds all monads and their possible orderings into one actual and self-evident ordering that is a particular monad, i.e. my absolute ego. Logic, however, insofar as it studies pure possibilities through symbols, is mathesis universalis, or universal monadology. It founds metaphysics as it clarifies the universal structure wherein actualities must cohere with possibilities. Metaphysics without logic is thus not possible, as it would lose the absolute universality it aspires to as a science of all beings, hence degenerate into a mere world-view. Logic without metaphysics, however, is not actual, as it would be empty of self-evidence. We only approximate Divine Knowledge, which is both intuitive and universal, with both monadologies. As metaphysics grounds particular evidences in a system as they unfold, logic secures systemicity as such; only together can they complete the new science. Even as “thoughts without content are empty, intuitions without concepts are blind,”⁴⁴ logic without metaphysics is but a formality, metaphysics without logic, a myth.

We conclude with an articulation of the structure of Leibniz’s new science. As empirical sciences study particular orderings in relation to the one ultimate ordering, metaphysics studies this ultimate ordering itself, while logic studies all orderings in general. Logic is thus the universal symbolic science of possibility, i.e. mathesis universalis, which founds both metaphysics and empirical sciences, introducing into them a persisting mathematical symbolism such as is witnessed in mathematical physics. But logic in turn is metaphysically grounded, insofar as it is itself imbued with a historically generated self-evidence whose genesis needs investigations on its own account.⁴⁵

Epilogue

Near the end of Theodicy, Leibniz tells a very Borgesian story. Sextus Tarquin, visiting the Oracle of Apollo, desired to know his future. Having learned it, he asked Apollo to change his destiny, so that he would not turn out evil. Apollo claimed that he, as merely a god of oracles, was unable to accomplish this. Sextus then left enraged and fulfilled his destiny. Theodorus, overhearing the conversation, travelled to Athens and fell asleep in the Temple of Athene. In his dream, he was led by the goddess to observe a pyramid composed of infinitely many worlds, the pinnacle of which is the actual world, which is also the most brilliant.

We misunderstand the story if we miss what is beautiful is not only the world on top, but the whole ordering of all worlds, i.e. the pyramid itself. Only in knowing the whole do we see the beauty of a part, even as God cannot choose the best world to actualize without seeing all that is possible first. Without a knowledge of the totality of the possible, therefore, no reason can be given to the actual as to why it is actual.

Only mathesis universalis offers man a glimpse into this reason. It is, for instance, only through a determination of all the paths light can travel from point A to point B (which determination is almost impossible without some mathematical symbolism such as algebra) that we manage to determine the one path that is “best,” and can thus justify the actual path, as it is simply this “best.”⁴⁶ But those who do not dream of the totality of all that is possible, even if they have learned all that is actual from the Oracle of Apollo, are altogether blind to what is best. Ethics qua contemplation of the best is impossible for them; rather, they shed their tears under necessity.⁴⁷ With mathesis universalis, we dream of totality, and are hence first liberated from bitterness and resentment against our fate. The Goddess of Wisdom and the Goddess of Freedom, therefore, are one and the same.

Bibliography

Aristotle. Metaphysics. Translated by Joe Sachs. Santa Fe: Green Lion Press, 2019.

Deleuze, Gilles. Expressionism in Philosophy: Spinoza. Translated by Martin Joughin. New York: Zone Books, 1997.

––––. The Fold: Leibniz and the Baroque. Translated by Tom Conley. London: The Athlone Press, 1993.

Hegel, G. W. F. Phenomenology of Spirit. Translated by A. V. Miller. Oxford: Oxford University Press, 1977.

––––. Science of Logic. Translated by A. V. Miller. New York: Humanity Books, 1969.

Heidegger, Martin. The Metaphysical Foundations of Logic. Translated by Michael Heim. Bloomington: Indiana University Press, 1984.

––––. “On the Essence of Truth.” Translated by John Sallis. In Basic Writings, edited by David Farrell Krell, 111–138. New York: HarperCollins Publishers, 1993.

Husserl, Edmund. Formal and Transcendental Logic. Translated by Dorion Cairns. The Hague: Martinus Nijhoff, 1969.

––––. Logical Investigations. Translated by J. N. Findlay. New York: Routledge & Kegan Paul, 1970.

––––. Philosophy of Arithmetic. Translated by Dallas Willard. Dordrecht: Kluwer Academic Publishers, 2003.

Kant, Immanuel. Critique of Pure Reason. Translated by Marcus Weigelt. London: Penguin Group, 2007.

Klein, Jacob. Greek Mathematical Thought and the Origin of Algebra. Translated by Eva Brann. New York: Dover Publications, 1968.

Leibniz, G. W. Philosophical Essays. Translated by Roger Ariew and Daniel Garber. Indianapolis: Hackett Publishing Company, 1989.

––––. Theodicy: Essays on the Goodness of God. Translated by E. M. Huggard. London: Routledge & Kegan Paul, n.d.

Plato. Parmenides. Translated by Mary Louise Gill. Indianapolis: Hackett Publishing Company, 1996.

––––. Sophist. Translated by Seth Benardete. Chicago: The University of Chicago Press, 1986.

Rothstein, Susan. "Aspect." In The Cambridge Handbook of Formal Semantics, edited by Maria Aloni and Paul Dekker, 360–366. Cambridge: Cambridge University Press, 2016.

Schelling, F. W. J. On the History of Modern Philosophy. Translated by Andrew Bowie. Cambridge: Cambridge University Press, 1994.

von Siebenthal, Heinrich. Ancient Greek Grammar for the Study of the New Testament. Oxford: Peter Lang, 2019.

Notes

1. G. W. Leibniz, Theodicy: essays on the goodness of God, trans. E. M. Huggard (London: Routledge & Kegan Paul, n.d.), 372–373.

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2. Martin Heidegger, The Metaphysical Foundations of Logic, trans. Michael Heim (Bloomington: Indiana University Press, 1984), 107.

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3. Ibid., 106.

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4. G. W. Leibniz, “Preface to a Universal Characteristic,” in Philosophical Essays, trans. Roger Ariew & Daniel Garber (Indianapolis: Hackett Publishing Company, 1989), 5.

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5. Ibid., 8.

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6. Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, trans. Eva Brann (New York: Dover Publications, 1968), especially Chapter 9.

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7. For further discussion on the status of arithmetic as symbolic calculus, see Edmund Husserl, Philosophy of Arithmetic, trans. Dallas Willard (Dordrecht: Kluwer Academic Publishers, 2003), especially Chapter 11.

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8. “And so when I think about a chiliagon, that is, a polygon with a thousand equal sides, I don’t always consider the nature of a side, or of equality, or of thousandfoldedness (that is, of the cube of tenfoldedness), but in my mind I use these words (whose sense appears only obscurely and imperfectly to the mind) in place of the ideas I have of these things, since I remember that I know the meaning of those words, and I decide that explanation is not necessary at this time. I usually call such thinking, which is found both in algebra and in arithmetic and, indeed, almost everywhere, blind or symbolic.” See Leibniz, “Meditations on Knowledge, Truth and Ideas,” in Philosophical Essays, 24–25.

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9. Ibid., “Preface to a Universal Characteristic,” 6.

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10. “I agree with you that it is important once and for all to examine all of our assumptions in order to establish something solid… If [Descartes] had followed precisely what I call the thread of meditating, I believe that he would have achieved the first philosophy… I admit that I have not yet been able to read all his writings with all the care I had intended to bring to them, and my friends know that, as it happened, I read almost all the new philosophers before reading him. Bacon and Gassendi were the first to fall into my hands… That is why I am inclined to applaud all those who examine the least truth to its deepest level; for I know that it is important to understand one perfectly, however small and however easy it may seem. This is the way to progress quite far and finally to establish the art of discovery which depends on a knowledge, but a most distinct and perfect knowledge of the easiest things…” Ibid., “Letter to Foucher,” 1–5.

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11. Heidegger’s early approach to interpreting Leibniz is deeply dismissive. The structure of his “dismantling” as laid out in his lectures from 1928 is as follows: 1. Leibniz’s doctrine of logic is reducible to his theory of judgement; 2. his theory of judgement revolves around the problem of “inherence (inesse),” i.e. the inclusion of a predicate in a subject; 3. since “inherence” already entails the genesis between one and many, self and other, etc., the Leibnizian logic is really grounded in his metaphysical doctrine of monadology, and logic in general grounded in metaphysics; 4. as monadology has to inquire into the problem of genesis, or as Heidegger calls it, “transcendence,” Leibniz is really only concerned with the problem of Being, though he himself is not fully aware of it. Through this “dismantling,” Heidegger therefore reduces Leibniz’s philosophy into a footnote to his own project in Being and Time. See Heidegger, The Metaphysical Foundations of Logic.

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12. Leibniz, “Samples of a Numerical Characteristic,” in Philosophical Essays, 10–14.

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13. Ibid., “Meditations on Knowledge, Truth, and Ideas,” 24.

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14. Heidegger is one of the most relevant and eloquent representatives of this position. For a glance into his self-interpretation regarding the question of “essence,” see Martin Heidegger, “On the Essence of Truth,” trans. John Sallis, in Basic Writings, ed. David Farrell Krell (New York: HarperCollins Publishers, 1993), 111–138.

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15. This difficulty is what causes Aristotle to formulate the injunction that there is no highest genus. As he explains it, suppose there were a highest genus, there would be no reason why a first difference should come from this undifferentiated genus to generate the species under this genus. See Aristotle, Metaphysics, trans. Joe Sachs (Santa Fe: Green Lion Press, 2019), 43.

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16. For simplicity, I did not use Benardete’s translation of this term, i.e. “the weaving together of the species.” See Plato, Sophist, trans. Seth Benardete (Chicago: The University of Chicago Press, 1986), 57.

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17. I changed the phrasing slightly to fit the context of this essay, without hurting the general sense expressed in the original. See Ibid., Parmenides, trans. Mary Louise Gill (Indianapolis: Hackett Publishing Company, 1996), 135.

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18. Klein, Greek Mathematical Thought and the Origin of Algebra, 82–99.

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19. G. W. F. Hegel, Science of Logic, trans. A.V. Miller (New York: Humanity Books, 1969), 82–110.

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20. For further discussion on the problem of expression in the context of Spinoza’s philosophy, see Gilles Deleuze, Expressionism in Philosophy: Spinoza, trans. Martin Joughin (New York: Zone Books, 1997).

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21. For a thorough and detailed analysis of a state of dialectical submergence similar to the one described here, see G. W. F. Hegel, Phenomenology of Spirit, trans. A. V. Miller (Oxford: Oxford University Press, 1977), 67–79.

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22. In fact, Schelling’s remarks are even more abrasive, as he calls Leibniz’s philosophy a “stunted Spinozism.” I do not share his opinion, for I believe he, too, overlooks the importance of mathesis universalis and algebra in shaping Leibniz’s philosophy. See F. W. J. Schelling, On the History of Modern Philosophy, trans. Andrew Bowie (Cambridge: Cambridge University Press, 1994), 75-84.

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23. Lest this example should offend reader’s sympathy, let it be said that I am only borrowing it from Leibniz: “a dog runs away from the stick with which he was beaten, because his memory represents to him the pain which the stick caused him.” See Leibniz, “Principles of Nature and Grace, Based on Reason,” in Philosophical Essays, 208.

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24. A brief overview of these categories is given by Siebenthal. Contrary to his schema, I separate tense from aspect, for I believe them to be distinct categories, and shall give my reasoning in both the next footnote and Part III of this essay. I also disregard the nominal forms of verbs, for these have to do mostly only with the content of the verb, without directly expressing relations that determine the event. For his schema, see Heinrich von Siebenthal, Ancient Greek Grammar for the Study of the New Testament (Oxford: Peter Lang, 2019), 88.

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25. This understanding of grammatical aspect is based on an interpretation of the following passage: “[…] grammatical aspect […] indicates what Smith (1991) calls ‘viewpoint aspect,’ the viewpoint from which the event denoted by the predicate is perceived. The major grammatical aspects are perfective and imperfective, which correspond roughly to the contrast between whether the event is perceived from an external viewpoint which is located outside the running time of the event, or from an internal viewpoint located within the running time of the event.” As shall be clear from Part III, I champion a not-time-exclusive interpretation of grammatical aspect as opposed to the more traditional temporal interpretation presented in this passage. I do not see imperfective aspect as primarily a designation of experienced duration, but rather as a designation of the concreteness of this experience. That is, by saying “I am noticing,” I am not suggesting that my notice has a running time, since to notice is instantaneous. Or, by saying “I am loving,” I am not suggesting something the same as “I love” or “I am in love.” Yet all three, due to their lexical aspect, designate a duration that I, located within the moment, experience. What is differently expressed by “I am loving,” therefore, cannot be simply a suggestion of experienced duration. Rather, through the imperfective, I only imply the content of this act is not atomized, but has inner differentiation and complexity. Though often this differentiation in content marks a temporal differentiation, what is crucial is not time per se. In other words, what I argue is, in the phrase “running time” used by this passage, what is essential is not “time,” but “running,” i.e. differentiation. For further detail, see Susan Rothstein, “Aspect,” in The Cambridge Handbook of Formal Semantics, ed. Maria Aloni & Paul Dekker (Cambridge: Cambridge University Press, 2016), 360–366.

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26. Husserl tries to convey a similar sense when he writes “reell.” What is indicated here is that what is “real” is not merely what is “objective,” “out there,” “positively existing,” as some platitudes of crude realism would suggest. Rather, reality primarily means what is concrete and fleshed out as content for our experience. Leibniz therefore says: “a possible thing is something with some essence or reality, that is, something that can distinctly be understood.” What is real is then what is distinctly known to include an infinite amount of essence. Its contrary is virtuality. According to Leibniz, “everything that must happen to a person is already contained virtually in his nature or notion, just as the properties of a circle are contained in its definition.” Moreover, regarding the recollection theory of knowledge, he claims “our soul knows all these things virtually and requires only attention to recognize truths.” What is virtual is then neither less objective nor less necessary, but merely expresses what lies latent and indistinct in existence. See Leibniz “On Freedom and Possibility,” in Philosophical Essays, 21; “Discourse on Metaphysics”, 45, 58. Also see Ibid., 41.

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27. “I respond that neither those essences nor the so-called eternal truths pertaining to them are fictitious. 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 […] So it is necessary that eternal truths have their existence in a certain absolute or metaphysically necessary subject, that is, in God […]” That is, though essences exist formally in God, their content is only a virtual possibility not yet developed into full reality. Despite some essences being more “real” or perfect than others, they all equally fall short of reality as such, i.e. existence. See Ibid., “On the Ultimate Origination of Things,” 151–152.

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28. Essays relevant to this topic are, besides The Source of Contingent Truths (1685–89?), On Contingency (1686?), On Freedom (1689?), A Specimen of Dynamics (1695). See Ibid., 28, 96, 133.

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29. Ibid., “The Source of Contingent Truths,” 98–100.

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30. Ibid., “On the Ultimate Origination of Things,” 151.

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31. Elsewhere, Leibniz also states: “Strictly speaking, one can say that no created substance exerts a metaphysical action or influx on any other thing.” From this he derives the absence of influx between body and soul. See Ibid., “Primary Truths,” 33.

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32. This simile, however, only applies to the process of determination, which is how human beings regard notions. Notions appear to us to be either determinate or indeterminate. But as was said in Part I, all primitives and composites are coeternal qua expressions of God. For God, though notions can be finite or infinite, they are all already equally determinate. There is no process of the determining of a notion in God himself, and creation does not proceed in time. Rather, God sees the world as his eternal self-expression immediately posited, i.e. his word. All finite composites, i.e. essences, that make up existences as their content, therefore, already emanate from him as his expressions. The only thing that he does determine (this is where Leibniz diverges from Spinoza) is how this totality of expressions is to be ordered, i.e. how the world of existences is to be regulated. See Ibid., “On Freedom and Possibility,” 19–21; “Discourse on Metaphysics,” 39–40; “Principles of Nature and Grace, Based on Reason,” 211; “The Monadology,” 218.

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33. Ibid., “On the Ultimate Origination of Things,” 150.

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34. “For having accepted the notion of necessity everyone accepts, namely that those things whose contrary implies a contradiction are precisely those that are necessary, it readily appears from a consideration of the nature of demonstration and analysis that there surely can be, indeed there must be, truths which cannot be reduced by any analysis to identical truths or to the principle of contradiction, truths endowed with an infinite series of reasons, fully known to God alone. And, it readily appears, this is the nature of everything called free and contingent, especially those which involve place and time. This has sufficiently been shown above from the very infinity of the parts of the universe and from the mutual interpenetration and connection of things.” See Ibid., “On Freedom,” 98.

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35. Ibid., “Discourse on Metaphysics,” 65–66; “Principles of Nature and Grace, Based on Reason,” 208–209; “The Monadology,” 216; “Preface to the New Essays,” 296.

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36. Again, we note that we use this idea of “construction” to describe God’s creation only from a human perspective. All the content being constructed, primitives and composites alike, is already expressed by God. Here, the genuinely “creative” act on God’s part, so to speak, is only God’s ordering of these infinitely various coeternal expressions.

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37. For further discussion on the subject of unfolding, see Gilles Deleuze, The Fold: Leibniz and the Baroque, trans. Tom Conley (London: The Athlone Press, 1993).

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38. Ibid., “Primary Truths,” 34; “Discourse on Metaphysics,” 51–52; “From the Letters to Arnauld,” 88–89; “A New System of Nature,” 142; “The Monadology,” 222.

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39. Ibid., “Discourse on Metaphysics,” 65; “From the Letters to Arnauld,” 78, 88; “A New System of Nature,” 140–141; “Principles of Nature and Grace, Based on Reason,” 208–209, 211.

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40. In this essay I use extension or extensity to denominate both what Leibniz calls “extension,” i.e. space, and “duration,” i.e. time. See Ibid., “Note on Foucher’s Objection,” 146–147; “From the Letters to de Volder,” 175, 178, 179, 184; “From the Letters to Des Bosses,” 199.

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41. Leibniz often does not differentiate carefully his terms such as “substance,” “form,” “substantial form,” “substantial unity,” etc., for these concepts do overlap. Yet, at one place he distinguishes substantiality from unity: “I do not say that there is nothing substantial or nothing but appearance in things that do not have a true unity, for I grant that they always have as much reality or substantiality as there is true unity in that which enters into their composition.” At another he says: “Things are either concrete or abstract. Concrete things are either substances or substantiata […] Substantiata are aggregates […]” For Leibniz, “reality” or “substantiality” is synonymous with “perfection,” while “perfection” is nothing but the amount of essence included. Though a certain unity of form is always requisite, a thing can still be called “substantial” if it only contains these unities, despite itself being relatively dispersed. See Ibid., “From the Letters to Arnauld,” 86–89; “From the Letters to Des Bosses,” 200. Also see Ibid., “Comments on Michel Angelo Fardella,” 103; “On the Ultimate Origination of Things,” 150; “From the Letters to de Volder,” 178, 179, 181; “Letter to Queen Sophie Charlotte of Prussia,” 186; “The Monadology,” 220.

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42. “For we must admit that it is impossible that bare extension, containing geometrical notions alone, is capable of action and passion.” Though “passion” here is rather a concept of physics, our psychological passion follows the same principle. See Ibid., “A Specimen of Dynamics,” 130.

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43. Ibid., “Meditations on Knowledge, Truth and Ideas,” 24–25.

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44. Immanuel Kant, Critique of Pure Reason, trans. Marcus Weigelt (London: Penguin Group, 2007), 86.

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45. For further studies on this topic of the genesis of logic through the grounding of subjectivity, see Edmund Husserl, Logical Investigations, trans. J. N. Findlay (New York: Routledge & Kegan Paul, 1970); Formal and Transcendental Logic, trans. Dorion Cairns (The Hague: Martinus Nijhoff, 1969).

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46. Leibniz, Philosophical Essays, “Discourse on Metaphysics,” 54–55.

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47. “The language of the ethical Spirit is law and simple command, and complaint, which is more the shedding of a tear about necessity.” See Hegel, Phenomenology of Spirit, 396.

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