Modality, Analysis and Metaphysics in Leibniz[1] [2]

D. Palmieri
University of Western Ontario
(C) 2004
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Introduction

Leibniz not only aimed to build a coherent system of substances and governing metaphysical principles; he also cared to establish rules for proper scientific inquiry, especially for physics and philosophy. These always appear to be two conflicting goals for him, especially in his accounts of the distinction between necessary and contingent truths, definition and analysis, and proper scientific explanation in physics. The conflict lies in the fact that on the one hand, his system is metaphysical by stipulation, dealing only with true, immaterial substances, and on the other hand, he seems to prescribe norms for inquiry into the empirical world.

            Like Leibniz’s normative claims for inquiry in physics, the problem with one version of his modal distinction is that it seems completely misplaced in his system; it is an apparently epistemological distinction, as will be explained in the first part of this paper, and yet it supposedly has a place in his ontological system of monads, which is so far from being epistemologically-motivated that it even seems to take monads as subjects of Leibnizian analysis. The other version seems to conflict with the epistemological distinction, capturing a completely different sense of necessary truth. The conflict mirrors the tension between his metaphysics and physics in that it seems to involve a gross equivocation of epistemological and ontological principles. One purpose of this paper is to elucidate the causes of that equivocation by looking at concrete examples of these problems, beginning with that of Leibniz’s modal distinction.

            My main purpose is to show that each of three aspects of Leibniz’s system essentially contributes to the ultimate aim of his project. These aspects are (i) the modal distinction for propositions, (ii) the account of analysis and definition, and (iii) normative claims, based on metaphysical principles, for inquiry in the domain of physics. The aim is normative and epistemological, but in a sense that needs spelling out. This is done in section 2.1 below. By looking at what appear to be only logically or abstractly related fragments of Leibniz’s thought, I intend to answer the crucial question that overexposure to his work tends to obscure. It is the most important question that we might pose in regard to the prima facie bizarre elements of his system of thought: what are these things for?

            Part 1 of this paper articulates Leibniz’s epistemological modal distinction, as well as its conflicting counterpart, and part 2 deals with the seeming incompatibility of metaphysical norms in physics, as well as the relevance of analysis and definition.

1. Necessity, Contingency and Ignorance

One part of Leibniz appreciates that a proposition can be contingent in an absolute sense, adopting that modal status irrespective of whether anyone is capable of grasping it. But he is also “bothered by the thought that [on this absolute account] sin appears to be necessary and inevitable” (AG 111). Sin appears necessary and inevitable because the world is deterministic, and our actions written for us. Or so the doctrine of pre-established harmony dictates: for anyone who has adequate knowledge of all properties true of some substance at any given moment, everything that preceded and follows that snapshot of the world is transparent.

            Of course, only by God is knowledge adequate in this sense. Hence everything foreseeable only by him, given that every individual substance is infinitely complex. Our feeble minds, and the infinite complexity of every substance, assure us that we will never ourselves be able to foresee the entire future, past or present (AG 19-23). But this does not deter critics from charging Leibniz with “fatal necessity” (AG 69). It was said that if everything were foreseeable, there would result a moral necessity taking from us all freedom and responsibility for our actions. Clearly, if every action were pre-written, we could not choose to do otherwise than what was written. To this Leibniz has several replies, one of which yields the allegedly epistemological distinction between necessarily and contingently true propositions.

On the epistemological view, things are indeed necessarily as they are, propositions necessarily true, and our actions necessarily performed, given the initial condition of God’s nature and foreknowledge. Hence there is only a hypothetical necessity in the proposition that Adam would sin, but no real necessity, given that Adam is certainly free. This seems to be at least one motivation for the epistemologically-inclined modal distinction. Whether there are other motivations will be examined in the final section of this essay. For the moment it suffices to note that there appears to be a modal account in Leibniz that requires our recognition of a proposition’s necessity as a precondition for its classification as necessary. Propositions that we cannot fully grasp as necessary are, in an exact sense to be articulated in the next section, contingent. That contingency is what saves our actions from “fatal necessity”, and it is our ignorance that saves the corresponding propositions from necessity.

            That Leibniz also defends an absolute modal distinction is also clear. In an unusual display of sensitivity to the epistemological and ontological aspects of necessity and contingency, Leibniz points out that a proposition can be necessary even in the event that we are unable to grasp its necessity (NE 433). In criticizing Locke’s failure to make this very distinction, Leibniz explicitly acknowledges this absolute sense of necessity as legitimate. Let us call the epistemological modal distinction the e-distinction, and refer to the more absolute, God’s-eye-view version as the g-distinction.

            Given Leibniz’s acknowledgment that ignorance need not harm a proposition’s necessity, and that there is a difference between absolute and epistemological contingency, there is a prima facie intelligibility to the view that he believed the distinction between absolute and epistemological senses of both contingency and necessity to be legitimate. If he does allow these categories, then the possible kinds of propositions seem to be the following, where “N” and “C” indicate their necessity and contingency and subscript “g” and “e” indicate the Godlike and epistemological attributes of the modal attributes N and C: any proposition must be either (N_{g &} C_e) or (N_{g &} N_e) or (C_g & C_e) or (C_g & N_e).

            Although I present these categories and disjunctions in order to better explicate Leibniz’s views, even they are problematic. The reason is that there are times when Leibniz does not seem to allow for the epistemological / absolute distinction at all. In “The Source of Contingent Truths”, he states that “the root of contingency is infinity in reasons” (AG 100n). When there are infinite steps involved in demonstrating the truth of a proposition, it is thereby contingent. The reason for classifying the proposition as contingent seems to be that we are incapable of demonstrating its truth. Surprisingly, however, Leibniz goes as far as to deny here that there is a category of N_g propositions in stating that even though God “comprehends the infinite” to the extent that it can be comprehended, these infinitely complex propositions “are not necessary” (AG 99).

            If there is no g-contingency or g-necessity, there seems to be an inescapable element of subjectivity in Leibniz’s account. This bothers many scholars who have taken up the questions of whether there are contingent Leibnizian propositions at all, and if there are, whether they are only e-contingent or g-contingent. [3] It is supposed to be bothersome that Leibniz has only an e-distinction, because this has the consequence of making propositions necessary or contingent for seemingly arbitrary reasons—our ignorance. The e-distinction is taken to be quite trivial and the g-distinction interesting. It seems to me that there is no ultima facie good reason to think one distinction more trivial than the other, even though the epistemological one seems to conflict with the popular view that propositions are necessary or contingent independently of our taking them to be so. Nevertheless, whether Leibniz can be said to be affirming an epistemological modal distinction is directly relevant to answering our pressing question, what’s it for? “It”, in our first N sections,  refers to the modal distinction, and what we will gain by answering the question is a coherent picture of the elements of Leibniz’s system that makes sense of his system. Eventually, addressing Leibnizian modality will lead us to the answer to “what’s it for” for his notion of substance, definition and analysis, and the metaphysical principles underlying a successful physics.

            1.1 What the e-Distinction Doesn’t Mean

The e-distinction we find in Leibniz is presented in terms of the possibility of demonstrability via complete analysis, described as follows. Where ‘P’ denotes a conjunction of predicates, each of which corresponds to a unitary property of the subject (or object), s, we analyze s (or its corresponding object) by providing a list of its predicates (L 270). Since all propositions are of the subject-predicate form ‘S is P’, this process of listing predicates is applicable to them all (G.ii 42; cf. R 28). Necessity and contingency are thus defined as attributes of propositions, and a true proposition is contingent whenever an analysis can be performed on it; it is necessary otherwise. That the proposition is true implies that it is reducible to an identity statement consisting of the subject on one side, and the conjunction of all its properties on the other. The equivalence expresses sameness of extension as well as sameness of meaning, which may turn out to be the same, in Leibniz, given that his notion of real definition almost perfectly coincides with that of complete analysis, and definition provides the meaning for terms.

            Analysis, like Leibnizian definition, is not a process by which we create subjects, objects or their meanings, conventionally building them from some arbitrary listing of predicates. The subject is given, so to speak, prior to definition or analysis, and unlike with Hobbesian nominal definitions, the aim of both definition and analysis in Leibniz is to examine and provide an account of the true nature of the subject. There is thus a fact of the matter regarding whether the resulting definition correctly and fully captures the thing being defined (L 128), and to capture it requires that we undergo the process of listing all predicates belonging to the definiendum. This process of listing is, at bottom, the process of definition: all propositions being in subject-predicate form, and their content simplistically combinatorial, the aim is to provide the listing of all simple predicates not further analyzable (L 229).

            Leibnizian e-contingency thus appears to be defined in terms of the impossibility (for us) of realizing infinite “repetition[s] of a measure” to produce a demonstration greatly resembling the process of definition (AG 100): we begin with some notion to be defined (like force or man), which is also the subject in the proposition we are attempting to prove. We then list its properties using predicates, each of which represents one of the corresponding properties of the subject or object. The demonstrative process of analysis differs from that of definition in that the aim of analysis is to produce a proof that all the predicates are contained in the subject, and hence to prove the impossibility of negating the resulting proposition. Such a proof is impossible for a finite mind to perform when the subject is a Leibnizian substance, for substances are infinitely complex. Of course, there is no impossibility for God (AG 99).

            From the fact that infinite analysis is required for a demonstration of propositions with substances as their predicates, Leibniz concludes that such propositions fail to be necessary, even though the demonstration can be performed by God (ibid.). This clearly seems to amount to e-contingency. Contingency and necessity thus amount to C_e and N_e: and because “it is impossible to give demonstrations of contingent truths”, there is no necessity, despite God’s abilities (AG 99). This would seem to rule out the category of propositions that are C_eN_g. So necessity appears to be determined by our ability to perform the said demonstrative analysis, and contingency by our lack thereof.

            Another way to put this is that the contingency/necessity distinction is defined in terms of the ways in which we may come to learn of the truth of propositions. At least this is how is was put by Adams (1994). But he goes as far as to claim that Leibnizian inconsistency is also epistemological. He insists that Leibniz most frequently defines necessity in terms of a proposition’s becoming contradictory if negated, and that a contradiction, “p Ù -p”, really means, in Leibniz, “provably(by us), p Ù -p” (A 147). This approach would make sense of the fact that, although Leibniz seems to want an epistemic account of the modal distinction, like the above account from analysis, he often defines the distinction in terms of jointly exhaustive and disjoint classes of propositions whose negation either is or is not a contradiction. Inconsistent propositions are the ones whose negation results in necessary truth, and contingent propositions can be negated without pain of contradiction: “the opposite of necessary truths is impossible: truths of fact are contingent and their opposite is possible” (Monadology 33; cf. Theodicy 170, 174, 189, 280-282, 367; AG 19, 45, 151, 193).

            Necessity then, even for Adams, is tied to p’s finite demonstrability, and contingency to its infinite demonstrability, as has been discussed; but because contradiction is defined as provable contradiction, the modal distinction is said to be “drawn on logical grounds alone”. Both Adams and Rescher (1967) agree on this point (A 147; RE 44 n24). The characterization of the modal distinction in terms of analysis is explained as a mere consequence of the purely logical approach (A 147). But contradiction-by-provability does not absolve Leibniz in any way, make his system more coherent, or explain anything previously thought mysterious. Two arguments will suffice to show this.

            First, if contradiction is defined in terms of provability in Leibniz, and necessity in terms of provable lack of contradiction, then it would not be a contradiction for Leibniz that Adam is not Adam. The reason is that we cannot conduct a proof to the effect that Adam was not Adam, because the proof would require access to an infinite number of predicates. [4] Without all of the subject’s predicates, we do not have a proof or analysis at all. Hence the proposition is not inconsistent, on Adams’ view of Leibnizian inconsistency. The problem with this is that Leibniz does think such proposition inconsistent: if we were to change any fact about Adam, and hence a predicate, the substance in question would no longer be Adam (cf. Corr, “Leibniz to Count Ernst von Hessen-Rheinfels”, April 12, 1686). For consider what Leibniz means by “Adam”: “[I mean] Adam who has certain individual characteristics and is thus distinguished from an infinity of possible persons very similar to him” (ibid.). If this seems strange, we need only recall that Leibniz’s goal is not to use natural language in order to clarify “existing terminology [recepta nomina] correctly” but to find “a suitable notation [propria signa]” for talking about complete, real substances, of which Adam is one (GI, 71). Hence in a Leibnizian proposition, “Adam” means the substance defined by all its characteristics. This certainly makes it inconsistent that Adam is not Adam. And yet requiring a proof for inconsistency makes it consistent to negate this proposition. 

            Secondly, that the proposition is inconsistent can be seen from the fact that “the only, and the highest, criterion of truth” for a proposition is that it must “be an identity or be reducible to an identity” (L 356). Negating an identity statement yields a contradiction if in fact the conjunction of properties on one side perfectly correspond to the properties of the substance being analyzed. Showing that there is no contradiction in the concept of some alleged substance is therefore sufficient for having demonstrated its possibility (and actuality—given Leibniz’s famous principle that if something can exist, it actually does, because this is the most perfect of all possible worlds [Monadology §§54-66]). This is the very method of which Leibniz speaks for “proceeding demonstratively, as in geometry” (L 356). If the lack of contradiction of which he speaks were epistemic, or related in Adams’ sense to provability, then Leibniz would not have concluded that a demonstration of consistency guarantees beyond doubt the possibility of the substance’s existence. But he does draw this strong conclusion: his famous ontological argument for the existence of God is of just this form, and follows an explicit statement of the absolute, non-epistemic nature of definition and demonstration, as well as an explicit statement and rejection of the epistemic view (L 354-7). So it is wrong to attribute to Leibniz the view that contradiction is just provable inconsistency. It is therefore wrong to conclude that changing what Leibniz means by contradiction illuminates his modal e-distinction.

            The interpretation of inconsistency in terms of provability was supposed to show an underlying coherence in Leibniz’s modal accounts. Although the attempt failed, the problem runs deeper than the failure suggests: it seems that attempting to elucidate Leibniz’s approach by describing his modal account as “purely logical”, given the role played by provability, is explanatorily backwards. The reason is that Leibniz defines proof in terms of “it being shown that a proposition necessarily follows” from assumptions  (L 285). A necessary proposition is then defined here as one who’s “contrary implies a contradiction”, which he calls “the true and unique mark of impossibility” (L 285-6). It is doubtful that what Leibniz means to say here is that necessity is determined by the negated proposition’s provable inconsistency (à la Adams), where provability involves being able to prove the proposition that necessarily, there is a provable contradiction involved in the said negation. This would also implicate Leibniz in a vicious regress, making his own definitions quite unintelligible.

            Perhaps the worst consequence of Adams’ view is that if the significance of the principle of contradiction itself is under scrutiny, then so must be all other, derivative Leibnizian logical principles, since the principle of contradiction is supposed to be one of the two “great principles” upon which all reasoning is based (L 347, 1049). However, if it had been less problematic, it might have explained why the two modal distinctions Leibniz offered are not quite inconsistent. Of course, an explanation of Leibniz’s explicit defense of the absolute distinction would be in order, on Adams’ view. I believe there is a simpler way to think about the conflict, and it requires an understanding of the e-distinction’s role.

            1.2 What is the e-Distinction for?

We may wish to excuse the apparent inconsistency of Leibniz’s modal accounts on the basis that he made an effort to categorize propositions using relevant facts about our knowledge. At some point, he did recognize the absolute modal distinction, and, whatever we choose to infer from the apparent inconsistency resulting from its coexistence with its conflicting epistemological counterpart, it may at least be useful to have an epistemological distinction in practice. As I see it, this might be useful for two reasons. One possibility is that the e-distinction may have convinced the more religiously-inclined that free will remained unharmed by pre-established harmony, or determinism. However, the theological motive is not philosophically interesting, and its import ought to be considered nil. Yet I want to briefly show its lack of import because the strategy is too easily invoked as an explanation of Leibniz’s appeal to the e-distinction, as may have been clear from section 1 above.

            The lack of intelligibility or philosophical import of objections regarding free will was also noted by Leibniz himself, at least in practice and at select stages of his intellectual development, and his reactions to Arnauld testify to this. When faced with the complaint that Adam could not have done otherwise than to sin, Leibniz simply replies to the problematic question in the guise of a conventionalist without the slightest interest in defending a correct account of substance, or a correct semantical analysis of substance: “if he had had other circumstances, this would not have been our Adam, but another, because nothing prevents us from saying this would be another. He is, therefore, another” (AG 73). This is almost as problematic, coming from the lips of Leibniz, as the obscure reply, which Leibniz given elsewhere, would have been for another: when asked to elucidate the sense of freedom and contingency he was defending, he would often resort to comments like the following “under the principle of possibility, [the concept of me] includes existences or truths of fact or the decrees of God upon which facts depend” (L 512).

            It would be difficult to take such attempts seriously which address questions about free will. Of course, Leibniz does have ready replies appealing to God’s own free will in order to explain free human action to those who are threatened by his system. Although he might have taken such explanations to be philosophically interesting himself, he also thought “free will” to be an overused and “highly ambiguous” term, and offered no alternative definition (NE 174-6). Furthermore, he even seems to introduce a third kind of modal distinction when presented with complaints about free will, indicating a certain promiscuity with such categories—at least in the context of issues involving free will. [5] We may thus refuse, with a clear conscience, to insist upon further elucidation of the term and the concerns it wrought. And we may assume that Leibniz would have approved.

There is a second reason why it might have been a good strategy to categorize propositions in accordance with facts about knowledge. The e-distinction may have been motivated by the specific epistemological goal of providing a method by which finite minds could distinguish necessary from contingent propositions. And yet attributing to Leibniz this motivation would entail that very large yet finitely contentful propositions should turn out to be contingent, given their unanalyzability (by us). Nowhere did Leibniz allow such an consequence, although it would be an easy consequence to infer if he had in mind such epistemological goals. It thus seems that there is not even a clear epistemological motivation behind the analysis-based modal account.

            Moreover, if the e-distinction is defined in terms of the number of steps required for analysis, and if the method of analysis involved is being prescribed and hence accessible to us, then what is required is at least a way of individuating the predicates or properties involved in describing our subject. We might think that the method of individuation is obvious: Leibnizian propositions are atomic just in case they contain exactly one predicate. The predicate is a description of sorts, depicting some property of the object (or subject) in question, and is itself unanalyzable into simpler predicates. So we can say that the predicate P describes some property or attribute A of object o. When God describes A with the use of P, P is certainly an accurate depiction of A. That is, in his infinite wisdom, the unique and detailed description of A is captured by P in such a way that (Ao ↔ Po). But given our finite wisdom and general inability to transcend the phenomenal point of view, it is at least likely that for many such attributes of o, our P will not yield the correct equivalence with A. The reason for this is that our own perception of things typically reflects an insight only into “phenomena, abstractions, relations” (AG, 89). Hence at least most of the time, P will not accurately reflect the attribute God sees, and in these cases, if (Ao ↔ Po) holds at all, it must be only by accident, so to speak.

            That we do not have access to the attributes seen by God is important because this means we do not always have access to real attributes. But it can be shown that we may never have such access in Leibniz's system: because a given substance mirrors everything, and everything mirrors it in return, any change in a substance marks a change in every other, which explains why “there are no purely extrinsic denominations” (NE 227). This means that any change in a substance, no matter how small, would mark a change not only in itself, but in every other substance. If these changes happen at the level of predicates, then there is no way that any finite mind could ever grasp them, let alone describe them in analysis. Worse yet, even if there were no changes, the mirroring aspect of substances make it impossible to grasp any individual predicate. The reason is that the predicate, as we saw, would have to correspond to a property. Every property, in turn, is infinitely complex—even the ones that are not further analyzable.

            Leibniz acknowledges that our knowledge is hopelessly obscure, though he insists that we do possess representations of everything in our own minds (which themselves are monads). But the point at issue here is that he seems to invoke an epistemologically-driven modal distinction, and for methodologically prescriptive reasons. In doing so, he also seems to prescribe a method that we could never realistically employ with much success. So unfortunately, the methodologically prescriptive view of Leibniz’s strategy fails to explain his eagerness to introduce an epistemological account of modality. Nor is it clear why Leibniz would think that the possibility of completed analysis should guide our hypothesis formation. Should we not accept or reject hypotheses based on what we do know, as opposed to what we could know, given certain circumstances? Any philosopher prescribing such methods, it seems, should keep in mind that such a counterfactual should not dictate which of our inferences are actually warranted.

            Was Leibniz so confused about the relationship between prescriptive methodological claims and the ontological status of propositions and their parts? This would not seem too surprising, given that it is a common error. [6] It did seem that according to Leibniz, propositions are deemed contingent just in case no analysis could be performed on them. But if the e-distinction is in fact epistemological and virtuous for its applicability to actual practice, should not the distinction take into account whether an analysis was performed? Surely Leibniz should not recommend that we actually make judgments about modal attributes of propositions based on what could be done, given that we have not yet put any effort into actually discovering whether it is necessary or contingent. The rule prescribed thus appears irrelevant to the goal of maximizing rational acceptance and rejection of hypotheses. Hence Leibniz's inconsistent modal views do not even seem to have the advantage of normative epistemological relevance.

2. What it is for

We know thus far that Leibniz put forth the methods of analysis and definition, which we identified above as appearing rather useless in their applicability to actual scientific research, and human inquiry proper. Yet if he is not engaged in some kind of normative epistemological activity, it is unclear what kind of project he has undertaken. Moreover, given our failure to find a commendable philosophical motivation for his epistemological modal distinction, a case is made for the view that the e-distinction, because it is based on the possibility of complete analysis, most resembles a red herring or an attempt to calm theological worries, at best. We also know that despite these alleged difficulties, Leibniz did have the goal of establishing rules of inquiry for scientific pursuits both empirical and mathematical, and for philosophy (philosophy had methodological prescriptions to offer even itself). In fact, we saw that he seemed to be recommending the use of real definition and complete analysis. Thirdly, we know that he engaged in scientific debates by attacked the Cartesian notion of extended matter and Newton’s theory of universal gravitation; he even defended the materialist’s mechanistic explanation in physics against Newtonian explanation. As fragments, these three aspects of Leibniz’s thought are fairly clear, though they do not paint a flattering picture of his system as a whole: after all, how are immaterial monads relevant to the planets and their attraction, and how do we go about doing physics by analyzing monads? What remains to be done is to show that something intelligible joins these three aspects of Leibniz’s thought in such a way that they are not fragments, but parts of one Leibnizian story whose intelligibility is increased, not decreased, by each one and their conjunction. The joining thesis that makes Leibniz’s project transparent has two parts: Leibniz had an underlying epistemological goal that does not resemble the naïve normative epistemology at play in our exposition of the modal distinction above; yet this epistemological goal can nevertheless be seen as normative, despite the seeming inapplicability of his metaphysics to physics.

            2.1 Analysis, Definition, Physics and Metaphysics

Leibniz’s “Against Barbaric Physics” [7] attacks Newton’s theory of universal gravitation, comparing it to “a return to chimeras” because the theory was said to appeal to an irreducible force as mysterious as the “little Gods” that “regulate organic bodies” (AG 315). Mechanistic explanation has the general advantage of not resorting to such methods. We may infer from his defense of mechanistic explanation that Leibniz’s repulsion to Newton’s method is not primarily based on an insistence that physics must always appeal to final causes, or immaterial substances—an insistence that is typically thought to be characteristic of Leibniz. Instead, Leibniz is here emphasizing that physics need not offer such explanations in order to be conducting itself in an intelligible and intellectually prudent manner. What we find as a prescription is that physics must follow the process of analysis in attempting to provide explanations that reduce “composite” physical principles “to simpler things” (AG 314). The goal must be to reach

Principles which could be explained and which we could hope to reduce to prior and simpler principles, and, in the end, to first principles. I think that this is praiseworthy as long as composite things are reduced to simpler things (ibid.).

            The problem with Newtonian explanation in terms of universal gravitation is not its lack of resemblance to ideal Leibnizian explanation in terms of final causes and monads, but that some will believe “that the explanation given is so satisfactory that there is nothing left to explain” (ibid.). Leibniz’s fear was that scientific attempts at explanation would halt at universal gravitation. Inquiry would certainly not halt if the process of analysis were undertaken: in performing analyses, theorists supposedly realize when there are things left unexplained, and when obscure concepts, like universal gravity, are left unanalyzed into their simplest parts. A complete inquiry into nature and its material objects requires that we “begin with accurate but universal enumerations of all possible modifications, such as those of weight, elasticity, light or heat” (L 270).

If we combine these analyses with experiments, we shall discover in any substance whatever the cause of its qualities… [T]his can be very effectively achieved through definitions and a philosophic language” (ibid.).

            The role for philosophy was to provide a “philosophic language” for performing analyses and providing definitions (L 270). The methodological goal of “undertaking the most accurate reasoning” in physics, for instance, requires that we “seek the formal and universal causes of qualities which are common to all hypotheses”. These causes, in secular empirical research, need not correspond to the true causes underlying the true qualities of real substances, or monads. That is, research need not reflect absolute reality in order to be productive, as Leibniz argues lucidly in his defense of mechanistic physics (AG 314ff). That is, although he thought mechanistic explanation to be essentially incomplete because it did not appeal to final causes and metaphysical principles, it was nevertheless thought to be the most productive way to proceed, given what was known at the time.

            In general, aside from being epistemologically relevant to practical goals, if methods of inquiry must be prescribed, they should not be inaccessible to those for whom they supposedly help. But although the methods of definition and analysis are infinitely complex for all substances, their relevance and utility are great, relative to Leibniz's more modest methodological goal of arriving at the most fundamental or primitive notions we can grasp. Universal gravitation was certainly not one of those for him, since he took the concept to be unclear and dismissive of important details about how particular bodies act on one another. But his prescription is clear in its application to Newtonian physics: use a method by which we might better discover the properties of those things appearing most frequently in all our best scientific hypotheses. Whether completed analyses are possible for us is not relevant to the normative account. Leibniz is suggesting that we engage in such analyses with the aim of defining concepts until we arrive at the most primitive ones. In fact, this is exactly the method Leibniz used in his attack on the thesis that extension is the primary attribute of matter: “those who have taken extension to primitive have "erred through a defect in analysis" (AG 251). He shows that reduction into simpler, absolute predicates is necessary, which leaves the concept of extension barren (AG 314-9). The reason: it is an unanalyzed complex.

            Our broad questions about Leibniz’s system thus become answerable: how are infinitely complex analyses of immaterial substances relevant to explanation in physics? The answer is that complete explanations make use of the process of analysis, and this is so whether we take substances to be material or immaterial. Either way the process is applicable and relevant, though not one that can be fully completed for true substances (monads). Analysis, in turn, presupposes real definition, and the form of a real definition is presented as having as its definiendum a complete substance with infinite primitive attributes, each of which must be taken into account for the definition to be complete. We must not halt inquiry before we arrive at those primitives, as Leibniz believed Newton did. As has been shown in section 1.1, analysis dictates that we proceed until we believe to have arrived at those primitives, and have therefore produced a proof of the proposition or hypothesis in question. These points suffice to show not only the role of substance, definition and analysis, but also the non-mysterious nature of Leibniz’s normative metaphysical intrusion into physics via his peculiar notion of semantical analysis. Leibniz’s own application of the process of analysis to criticisms of physical concepts and theories shows that being able to complete an analysis or real definition is not required for applicability in practice.

            Surprisingly, these considerations naturally suggest an interpretation of Leibniz's modal views that does not have the consequence of normative epistemological irrelevance that seemed unavoidable at the end of section 1.2. Leibniz’s goals can indeed be seen as being epistemological in nature, but not in the naïve sense suggested by our treatment of his modal distinctions above. Rather, even in his modal accounts, he can be seen, correctly I think, as recommending a method for determining necessity and contingency based on a criterion that is indeed applicable in practice. That criterion is the possibility of analysis not in the sense of our being capable of conducting an analysis, but in a more absolute sense: infinite analysis is, in fact, not a requirement for individual agents with idiosyncratic (or even human) capacities, but a process that determines the possibility of analysis. I will explain.  

            The unintelligibility of a completed analysis involving infinite predication is due not to what we can or cannot know, but what no one can know. Interestingly, Leibniz provides support for this view by insisting that in the process of analysis, even God’s eye cannot see “the end of the resolution, of course, which does not exist” (AG 96). The contingency resulting from infinite analysis is thus a consequence of the mysterious nature of an infinite series as a whole. An infinite plurality “itself does not constitute a number or a single whole” (L 834). Hence the analyses in question could never yield one concrete snapshot of the world, or of the substance it means to analyze. Given this, the corresponding proposition could not be necessary, even for God, or really understood in the sense that an intelligible concept is understood. The lack of understanding, because it is God’s own, can be seen to underlie not an epistemological distinction, properly speaking, but an absolute one. This explains nicely why Leibniz himself insisted that despite God’s knowledge, there may fail to be necessity in a proposition (cf. AG 99). Hence human capacities aside, analysis is a semantical tool ruling out the possibility of the category of necessary propositions involving claims about infinitely complex substances. Hence the e-distinction above may be preserved, but with a modification in our conception of Leibnizian analysis, and perhaps a re-evaluation of what we mean by the claim that it is epistemological.

            We are thus also left with the consequence that Leibniz’s modal distinction is rationalist in spirit, being less arbitrary than our epistemological distinction suggested. It is the nature of infinity that endows propositions with contingency, and not our corresponding inability to grasp an infinite series, for God shares this inability with us, given that such things are not understood by anyone.

            Finally, given that the most accurate reasoning in physics involves seeking causes that may or may not be the true causes that monads are made of, we eliminate the problem of being denied access to the real attributes of things. This cannot be said to be a problem of Leibniz’s system any longer, since we have seen that analysis does not require access to real properties at all. The more profound metaphysical problems may still be present, but Leibnizian semantical analysis via definition and complete analysis remains unharmed by those things. Nor can his metaphysical views be said to conflict methodologically with physical science, even given his prescriptive intrusion and notion of ideal inquiry. The ideal, as Leibniz knows, cannot be reached; and yet this fact is irrelevant to the utility of his semantical process of analysis. The metaphysical components, like monads, need not play any explicit role.