Deleuze, Peirce, and the Habits of Things
Much of the discussion that compares the work of Deleuze and Peirce has focused upon Peirce’s theory of signs. This is understandable given the emphasis Deleuze himself places on Peirce’s typology of firstness, secondness, and thirdness in his Cinema books, as well as the importance of a theory of signs in his Proust and Signs. In the following chapter I will explore a subtler but equally significant interplay between the work of Deleuze and Peirce by showing how they both come to use a concept of habit in order to account for the emergence of individuated, determinate identities; moreover, this account of the emergence of individuated identities by way of habits (or passive syntheses, as Deleuze will argue) entails the necessary affirmation of the actual infinite, or the possibility of supertasks.1
The importance of passive synthesis as set forth in the second chapter of Difference and Repetition has been routinely discussed. What is of particular significance for our purposes is that with the notion of passive synthesis, Deleuze is able to argue for a process whereby an identity comes to be individuated in a way that does not presuppose an identity that predetermines the process—it presupposes, instead, an actual infinite or indeterminate chaos. Something comes to be the identifiable, individuated thing that it is not because it actualizes a predetermining essence but rather because the active, indeterminate, and infinite processes become contracted into a habit, and it is only then that it takes on the formal, identifiable features by which we come to identify this something as the determinate individual it is. Although this theme has been widely discussed among Deleuze scholars, it is a lesser-known theme among Peirce scholars. But the importance of habit, it will be argued, is, in precisely the Deleuzian sense just sketched, equally important for Peirce in accounting for the individuation of things. That habit is integral to Peirce’s theory of belief is well established, but its ontological and metaphysical significance, and its implications for understanding pragmatism, have not received the attention they deserve.
In the first section I will introduce the problem of supertasks and discuss the traditional response to this problem, which has been, simply put, to reject the possibility of such tasks, or, as this is more commonly known in the literature, to reject the actual infinite. The second section will turn to Kant’s First Antinomy—namely, the problem of believing whether or not the world has a beginning in time. Kant argues that neither option is tenable, and this is primarily because a belief in such a world would involve a super-task. To avoid the antinomy, Kant rejects the very notion of a world in itself and as a consequence any belief in the world. This leads us to the third section where I will argue that Peirce’s metaphysics sets out to restore a belief in the world, and belief in an infinite world. The fourth and final section shows how a key concern of Deleuze’s project was precisely to restore belief in the world, a belief that involves the supertask of affirming the actually infinite, which occurs, as mentioned above, in the processes of individuation, or in the passive synthesis and contraction of habits that are inseparable from determinate realities. It is at this point where the metaphysical implications of pragmatism come to fruition in Peirce—implications that are then taken up by Deleuze, and especially by Deleuze and Guattari in What Is Philosophy? where an instance of the affirmation of the infinite occurs with the philosophical task of creating concepts. I will call this task “infinite pragmatics.”
Let us begin with an example. I put an apple on a table, wait half a minute, and then remove the apple. I wait a quarter of a minute and put the apple back on the table, removing it after 1/8th of a minute, putting it back on after 1/16th of a minute, and so on ad infinitum. Let us assume for the sake of the argument that at the end of one minute I have completed an infinite sequence of placing and removing the apple. At the end of the minute, is the apple on the table or not?2 This question seems unanswerable and has led many to assume that such a task, often called a supertask, is impossible. Zeno, however, in his well-known paradox of Achilles and the tortoise, sought to show that if such supertasks are indeed impossible, then even the most mundane of tasks becomes impossible as well, despite all appearances to the contrary. As Aristotle recounts Zeno’s paradox, the conclusion one is led to is that “the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold the lead” (Aristotle 1984, 239b15). If the tortoise has a ten-meter lead on Achilles, and even if Achilles runs ten times faster than the tortoise, he must first reach the point where the tortoise was, and since there is an infinite series of such points, Achilles will never catch the tortoise because he must first reach an infinite number of points. But clearly Achilles will catch the tortoise. Given enough information, a simple mathematical calculation will enable us to determine at what point the two will be tied, after which Achilles will take the lead. So has Achilles performed a supertask?
The standard response to this question, beginning with Aristotle and continuing on through Peirce and beyond, is to say that of course Achilles did not actually have to do the impossible and reach an infinite number of points in the process of catching up to the tortoise. For Aristotle, what Zeno fails to recognize is that there is an important difference between the actual distance covered between any two points in a given finite amount of time and the potential for this distance or time to be subdivided to infinity. On Aristotle’s view, the actual is finite, not infinite; and the infinite is only the potential to continually divide the actual ad infinitum but without actually ever reaching the infinite. Peirce will likewise agree with Aristotle that Achilles does not actually reach an infinite number of points in catching up with the tortoise, but this is for a significantly different reason. Peirce counters Zeno’s paradox as follows:
All the arguments of Zeno depend on supposing that a continuum has ultimate parts. But a continuum is precisely that, every part of which has parts, in the same sense. Hence he makes out his contradictions only by making a self-contradictory supposition. In ordinary and mathematical language, we allow ourselves to speak of such parts—points—and whenever we are led into contradiction thereby, we have simply to express ourselves more accurately to resolve the difficulty.3
In short, we never reach points, for they are simply tropes, manners of speaking, and what we might in everyday or mathematical language speak of as a point is itself composed of parts—parts that are in turn composed of parts, and so on ad infinitum.4 Zeno’s mistake was thus twofold. First, Zeno failed to see that the continuum is irreducible to points, with points being merely abstractions from the continuum, and yet it was precisely the points reached along the way to catching the tortoise that did the heavy lifting in Zeno’s formulation of the paradox. His second mistake was to be confused by language itself. In both mathematical and ordinary language we will speak of points or parts, but in doing so Peirce claims we overlook the reality that is the continuum. If we are to avoid the contradictions that give rise to paradoxes such as Zeno’s, then for Peirce it is necessary to align our everyday and mathematical language with the ontological reality of the continuum.
Peirce is willing to draw upon the resources of pre-Kantian metaphysics, albeit while fully aware of the Kantian critical project, and as a result Peirce is much less adverse to affirm a metaphysics of the infinite than most of the philosophers who follow in his wake. Deleuze will continue in this Peircean direction as he develops his own metaphysical position, which I will call “infinite pragmatics.” For present purposes, and in order to connect Peirce to Deleuze’s understanding of philosophy as the creation of concepts, I want first to highlight the problem of the infinite that is associated with grasping the extension of a concept. To state the problem differently: does grasping a concept and its corresponding extension entail performing a supertask, and if so, does this invalidate the act itself? In what way does understanding the concept “lemur,” “banyan tree,” or even Descartes’s concept of the “cogito” entail performing a supertask? One can follow Hume from his Treatise. There Hume argues that an abstract idea is neither an idea that is abstracted from all the qualities and quantities of the particular such that it becomes the idea of nothing in particular, nor is it an abstract idea that synthesizes all the qualities and quantities of the particular that fall under it, for this would indeed imply “an infinite capacity in the mind,” (Hume 1978, 18), or a supertask that would be impossible. Hume’s solution is to reduce an abstract idea to a custom or habit that is revived each time we see a particular of a given type. Kant, on the other hand, rejects Hume’s solution, although he also rejects the notion that an idea or concept entails an infinite capacity or supertask. Kant thus moves to transcendental idealism in order to restore the viability of concepts without affirming the actual infinite. For Peirce and Deleuze, however, a concept involves a supertask, and far from invalidating the reality or possibility of such concepts, it is the actual infinite itself that is the condition for the possibility of concepts. To begin to see how this works, let us turn to Kant.
The impossibility of completing a supertask is at the heart of Kant’s approach to the First Antinomy of pure reason. In addressing the question of whether the world does or does not have a beginning in time, Kant argues that if one assumes it does not, then “an eternity has elapsed, and there has passed away in the world an infinite series of successive states of things,” but since an infinite series “can never be completed through successive synthesis [it] follows that it is impossible for an infinite world series to have passed away” (Kant 1965, A428/B456). Because a supertask is impossible, the world must have a beginning in time. If, however, we assume the world has a beginning in time, then the world proceeds from a “time in which the world was not, i.e. an empty time.” An empty time, however, lacks any determinate content and thus it lacks any “distinguishing condition of existence rather than non-existence.” There is thus no way to think of a world that begins in time, for this would involve a thought without content, which for Kant is another impossible task (although not a supertask in this case, for rather than an infinite regress of content to be synthesized we have no content, and hence no thought).
Kant’s solution to the antinomy is straightforward—he rejects the very notion that the world exists in itself as either a finite or infinite totality. One cannot have a thought or belief in this world, for it does not, in the end, exist, and to claim otherwise is to suffer from an illusion—what Kant calls a transcendental illusion. Kant is clear on this point:
If we regard the two propositions, that the world is infinite in magnitude and that it is finite in magnitude, as contradictory opposites, we are assuming that the world, the complete series of appearances, is a thing in itself that remains even if I suspend the infinite regress in the series of its appearances. If, however, I reject this assumption, or rather this accompanying transcendental illusion, and deny that the world is a thing in itself, the contradictory opposition of the two assertions is converted into a merely dialectical opposition. Since the world does not exist in itself, independent of the regressive series of my representations, it exists in itself neither as an infinite whole nor as a finite whole. It exists only in the empirical regress of the series of appearances, and is not to be met with as something in itself.
(Kant 1965, A504–5/B532–3)
Kant’s argument, in essence, largely repeats Aristotle’s response to Zeno’s paradox. As we investigate the conditions of empirical phenomena, we launch upon an infinite regress in that “however far we may have advanced in the ascending series [of conditions], we must always enquire for a still higher member of the series, which may or may not become known to us through experience” (Kant 1965, A518/B546). The question, then, is whether the world is what we get at the end of an infinite series—at the end of a supertask—or whether we must remain content with an empirical regress of forever advancing through an unending series of conditions. In rejecting the possibility of supertasks, Kant rejects the existence of a world in itself and thus for Kant we do not have a regress to infinity but rather a “regress in the series of appearances, as a determination of the magnitude of the world, [that] proceeds in indefinitum” (Kant 1965, A521/B549). The regress is therefore neither an infinite regress nor a finite regress but is instead an indeterminate process of determining, in accordance with a rule, “how experience, in conformity with its object, is to be obtained and further extended” (Kant 1965, A521/B549).
We can now return to Peirce. In his 1868 essay, “Questions Concerning Certain Faculties Claimed for Man,” one of a series of articles written for the Journal of Speculative Philosophy (also known as the “cognition series”), Peirce examines a number of incapacities that philosophers had traditionally taken to be capacities. I will focus on the first such capacity, our ability to distinguish between an intuition that is a primitive, unquestioned given that is not determined by any previous thoughts or cognitions, such as past experiences, education, habituation, etc., and a cognition that always is, according to Peirce, determined by previous cognitions. An intuition, on this view, serves as the premise upon which a chain of thoughts and cognitions can be founded, and many would like to believe they can accurately distinguish between premises (i.e., intuitions) and the arguments that are grounded upon them. For Peirce, however, this is not a capacity we have. Peirce offers the example of eleventh-century theologian, Berengarius, to make his point. Berengarius had the audacity to suggest that “the authoritativeness of any particular authority must rest upon reason” (Peirce 2:194). Berengarius’s contemporaries thought such a suggestion was absurd and impious. The “credibility of authority,” Peirce points out, “was regarded by men of that time simply as an ultimate premise, as a cognition not determined by a previous cognition of the same object, or, in our terms, as an intuition” (194–195). The lesson Peirce draws from this example is that what we take to be intuitive today—namely, the data of sense intuition, what Peirce will call “internal authority”—may tomorrow come to be seen as cognitions rather than intuitions. Peirce thus asks, rhetorically: “Now, what if our internal authority should meet the same fate, in the history of opinions, as that external authority has met?” (195).
The next question for Peirce, and with this Peirce’s concerns dovetail with Kant’s, is whether it is even possible for there to be an intuition at all or whether all cognitions are determined by other cognitions, and so on ad infinitum. The short answer for Peirce is that it is not possible—all cognitions are determined by previous cognitions. To support this claim, Peirce relies upon the principle of sufficient reason. First, Peirce argues that it is problematic to argue “[f]or something entirely out of consciousness which may be supposed to determine it [consciousness], [but] can, as such, only be known and only adduced in the determinate cognition in question” (209). In other words, in the tradition of Berkeley, to think the condition that is outside all thought and cognition is to think this condition, and hence we have not made the case for a cognition that is not determined by another cognition. If we persist, however, and argue for a condition that is “absolutely external,” as Peirce puts it, meaning beyond any thought, then such a condition becomes inexplicable, for in order for an explanation to be successful it would entail being recognized and understood as such—in short, it will involve cognition. To rely upon the inexplicable as a means to explain cognition is thus contradictory. Peirce’s conclusion, therefore, is that all cognitions are determined by previous cognitions.
At this point a clear contrast emerges between Peirce and Kant. For Kant the supertask of synthesizing an infinite regress of representations—or in Peirce’s terminology, points—is impossible, and any conclusions that rely upon such a task are to be rejected and chalked up as being illusory. For Peirce, on the contrary, the real itself is a supertask, or the continuum as processual unfolding that is the condition for the possibility of the successive points and representations that are derivative abstractions conditioned by, rather than conditions for, the continuous reality. Let me refer to two brief examples from Peirce where this point becomes clear. In the first, which is again from the 1868 essay, Peirce compares the successive chain of thoughts to an inverted triangle. The waterline on this triangle corresponds to a cognition, and as we move the triangle up and down in the water we have further cognitions, each one being determined by the movement from the previous one. The triangle, however, does not contact the water at a point, for every point, to recall our earlier discussion, can be further divided, or in this case each line has a smaller segment below it. For Peirce, then, we either have cognitions—waterlines on the triangle—or the triangle is out of the water and we have no cognitions at all. Reality, if we extend Peirce’s understanding of the analogy, is simply the infinite continuum of the triangle.
The second example comes from Peirce’s later essay, “Synechism, Fallibilism, and Evolution,” and it does not rely upon an analogy. In this essay Peirce rejects the exclusive disjunction between existence and nonexistence, and argues that “all things are continuous, [and that] the universe must be undergoing continuous growth from non-existence to existence. There is no difficulty in conceiving existence as a matter of degree” (Peirce 1955, 358). There is thus for Peirce no determinate fixed points of existence—reality itself is simply the process of becoming more and more existent. Reality itself is a supertask, a process that always already presupposes the infinite continuum, but just not an infinite continuum of extensive points, which Peirce argues, as we have seen, is contradictory. Even the laws of nature, Peirce claims, “are results of evolution; that underlying all other laws is the only tendency which can grow by its own virtue, the tendency of all things to take habits” (1955, 359). As will be discussed in more detail later, it is this continual process and tendency of settling into habits that is the principle of sufficient reason for all that appears, for all determinate, particular phenomena. This tendency, moreover, as with Deleuze’s understanding of passive synthesis, is not predetermined by an already identified form or essence. As a result of this move, Peirce is able, unlike Kant, to believe in the world, and for Peirce this is an infinite world or infinite continuum. It is only when we come to think of the world in terms that reduce it to the discrete that we then encounter problems, such as reducing it to the synthesis of discrete representations for Kant or the discrete points Achilles reaches on the way to catching the tortoise.
Peirce affirms the infinite world, but we can see that he does so at the expense of the discrete and the singular. By contrast, in his effort to develop what we could see as Peircean pragmatism, Deleuze will affirm an infinite world teeming with singularities, but singularities that are not extensive and discrete. It is to this that we now turn.
As we turn to Deleuze, the clearest way to summarize where we have been and where we are going is as follows: in the wake of various paradoxes associated with the infinite, Aristotle and Kant reject the actually infinite and take refuge in the potentially infinite (Kant’s in indefinitum); Peirce affirms the infinite world as a continuum of “growth from non-existence to existence,” but this is a world where the discrete and particular become derivative abstractions; and Deleuze, finally, affirms the infinite world as actually infinite, and as teeming with what he will call pre-individual singularities. It is this radical affirmation of the infinite that accounts for why the infinite looms so large in the definitions of two of Deleuze and Guattari’s key concepts from What Is Philosophy?—namely, the concept of a philosophical concept and the concept of chaos. Given that Deleuze and Guattari’s answer to the question, “What is philosophy?” is that it creates concepts—“philosophy is the art of forming, inventing, and fabricating concepts” (Deleuze and Guattari 1994, 2)—the subsequent question is, “What is a concept?” Deleuze and Guattari are forthright with their answer: “The concept is defined by the inseparability of a finite number of heterogeneous components traversed by a point of absolute survey at infinite speed” (Deleuze and Guattari 1994, 21, emphasis mine).
This definition of a concept draws significantly from the concept of passive synthesis that Deleuze develops in Difference and Repetition. Passive syntheses, as Deleuze understands them, are generalizations of Hume’s famous thesis, which is the thesis that “repetition changes nothing in the object repeated, but does change something in the mind which contemplates it” (Deleuze 1994, 70). In the repetition of AB, for instance, there is nothing in A itself, as Hume famously argued, that connects it to B; it is only by virtue of a change in the mind, a developed habit and expectation, whereby one is led to expect B on the appearance of A. This is an example of the “contemplation,” the change in the mind, that enacts a passive synthesis that is irreducible to the elements themselves; or, as Deleuze argues, it is a relation that is external to the terms while not being separate or separable from these terms. Deleuze will then generalize upon this notion and argue, for example, that “[e]very organism, in its receptive and perceptual elements, but also in its viscera, is a sum of contractions, of retentions and expectations” (1994, 73). By absolute survey, Deleuze and Guattari draw from Raymond Ruyer’s work, but they do so with a Humean notion of passive synthesis clearly at stake, for just as Ruyer argued that the relationship of the perceiver to their visual field is one of “absolute survey,” meaning they are immediately present to and inseparable from all aspects of the field, so too for the “contemplation” that enacts the passive synthesis—it is inseparable from and yet connected to all the elements of the synthesis.5 This is the sense then that the contemplation is a “point of absolute survey at infinite speed”—there is not a finite movement from one element to another, but an infinite movement (i.e., a supertask) that draws all the elements together without being reducible to these elements.
The infinite will return later in What Is Philosophy? when they define chaos: “chaos is characterized less by the absence of determinations than by the infinite speed with which they take shape and vanish” (Deleuze and Guattari 1994, 42). In short, we can say that the infinite is the condition of possibility and impossibility for determinate individuation. The contemplations of passive synthesis perform the supertask of a contemplation “at infinite speed” that allows for the possibility of an individuated and determinate entity (an organism, for example, as we saw above). At the same time, such individuation must forestall and stave off the infinite as that which undermines the possibility of connections being drawn together and hence the possibility of passive synthesis. It is with this understanding of the infinite that Deleuze and Guattari will set out to differentiate philosophy from science, including the philosophy that aligns itself with science. They argue that “the problem of philosophy is to acquire a consistency without losing the infinite,” whereas “the problem of science… [is] to provide chaos with reference points, on condition of renouncing infinite movements and speeds and carrying out a limitation of speed first of all. Light, or the relative horizon, is primary in science” (1994, 42). To accommodate Deleuze and Guattari’s use of the notion of infinite speeds and movements, it is critical to reorient one’s philosophical perspective away from the Kantian critical tradition and extend Peirce’s project in order to embrace the actual infinite. One way to state this reorientation is to characterize it as the effort to restore belief in the world, a belief Kant sought, as we saw, to render impossible.
This effort to maintain belief in the world, with all that this entails, is a central issue in Deleuze and Guattari’s What Is Philosophy? As they put it: “It may be that believing in this world, in this life, becomes our most difficult task, or the task of a mode of existence still to be discovered on our plane of immanence today” (1994, 75). In attempting this task, moreover, one inevitably confronts the problem of supertasks, for in creating a philosophical concept, one creates something with infinite speeds and thus, in contrast to science, a philosopher’s task is to affirm the actual infinite rather than to restrict the infinite and place it within its proper limits (e.g., speed of light). To do this, however, Deleuze and Guattari seek to find a middle path between affirming infinities that are determinate—whether this be the world as a determinate, infinite totality or constituted of actually infinite and extensive parts—and they likewise want to avoid the Peircean conclusion that all is continuum, that every determinate, extensive entity is further divisible into parts and so on ad infinitum (the “gunky” view of matter in contemporary analytic metaphysics [see fn. 4]). The reason for steering clear of the continuum is to affirm the reality of differences, and the extensive determinations such differences make possible. In the case of Peirce, difference is ultimately subsumed by the identity of the continuum, an identity that surfaces in Peirce’s philosophy as the Truth and opinion fated to be agreed to by all; that is, the opinion that correctly represents the infinite continuum.6 For Deleuze and Guattari, by contrast, the world is between the atomism of regressive external differences and a single continuous reality (we will discuss the implications of this for a theory of truth later). On just this point, Deleuze and Guattari are in agreement with the argument of José Benardete, who, in his long overlooked book, Infinity: An Essay on Metaphysics, had already put forth an argument that sought to chart a similar path between atomism and continuum. The central thesis of this book is to make a case for affirming the actual infinite, and among the many arguments Benardete makes, he will also propose a middle path between atomism and the continuum and argue that “there [is] perhaps some tertium quid that would enable us to eschew both the minim [basic elements] and the continuum [atomless gunk] at once” (Benardete 1964, 202). In the case of Deleuze and Guattari this tertium quid is the differential.
To avoid the Kantian rejection of the belief in the world, Deleuze turns to the pre-Kantian tradition, most notably Spinoza and Leibniz. From Leibniz Deleuze adopts the concept of differential relations as infinitesimals. In the differential relation dy/dx, for example, when y and x become infinitely small we end up with dy = 0 and dx = 0 (or dy/dx = 0/0 as it was commonly written in the seventeenth and eighteenth centuries). We thus end up doing away with the terms but not the differential relation, for the relation subsists as the infinite, intensive supertask even when the terms have been eliminated, or, as Deleuze will frequently put it, the relation is external to the terms. The differential is precisely Deleuze’s version of Benardete’s tertium quid, for it is not extensive and hence a minima—the extensive terms have vanished—nor are they absorbed into the continuum for the determinable relation subsists as an irreducible relation, an intensive quantity that cannot be further reduced into parts, and parts of parts, etc.7
Deleuze’s understanding of the differential is also critical to his reading of Kant’s First Antinomy. Kant’s solution, Deleuze points out, is that it is made possible by discovering “within representation an element irreducible to either infinity or finitude”—this is the regress of representations—which is in turn related to “the pure thought of another element which differs in kind from representation (noumena)” (Deleuze 1994, 178). The problem with Kant’s move, Deleuze argues, is that this “pure thought,” the noumena, to the extent that it “remains undetermined—or is not determined as differential,” continues to remain tied to the framework of external conditioning and representation in that the noumena is external to and conditions the possibility of the regressive series of representations. In other words, as with his critique of Aristotle, who subjects specific difference to “the identity of an undetermined concept (genus)” (Deleuze 1994, 32), so too does Kant subject the differences between the representations of the regressive series to the identity of the undetermined noumena. For Deleuze, however, his effort to develop a metaphysics of difference leads him to account for identity in terms of difference rather than difference in terms of identity, and it is here, again, where the differential emerges as a key conceptual tool for Deleuze.8
The differential, and differential relations, will also become integral to Deleuze’s concept of multiplicity since differential relations are not isolated relations but presuppose other relations, and so on ad infinitum. For example, in the differential relation dy/dx, as the determinate values for x and y become infinitely small we end up with a differential relation (or an intensive difference or quantity) that is external to the terms and tends toward a third term that is its limit, a term that does have a finite value, z let us say—thus, dy/dx = z.9 Every determinate, finite term, therefore, can be understood as the limit of a differential relation—this is the sense in which Deleuze understands the differential relation to be constitutive of identity rather than dependent upon identity, and this is how Deleuze sets out to develop his metaphysics of difference. Thus even the determinate terms of our initial relation, dy/dx, will involve their own constitutive differential relation, a differential relation with its own determinate terms and its own series of constitutive differential relations, and so on ad infinitum. For Deleuze this is precisely the principle of sufficient reason for all phenomena, meaning all determinate, extensive phenomena:
Every phenomenon is composite because not only are the two series which bound it heterogeneous but each is itself composed of heterogeneous terms, subtended by heterogeneous series which form so many sub-phenomena. The expression “difference of intensity” is a tautology. Intensity is the form of difference in so far as this is the reason of the sensible. Every intensity is differential, by itself a difference. Every intensity E-E’, where E itself refers to an e—e’, and e to ε—ε’ etc.… We call this state of infinitely doubled difference which resonates to infinity disparity. Disparity—in other words, difference or intensity (difference of intensity)—is the sufficient reason of all phenomena, the condition of that which appears. (Deleuze 1994, 222)
To rephrase for the sake of clarity and to bring us to the concept of a multiplicity, each differential relation is the constitutive condition for “every phenomenon,” meaning every determinate, extensive phenomenon (“that which appears”). Each phenomenon presupposes an infinite series as its sufficient reason, and each phenomenon is itself in an infinite series of differential relations with other phenomena, and this for precisely the reason that an infinitely doubled series—disparity—is the sufficient reason of all phenomena. If a given phenomenon were to be incapable of entering into relations with other phenomena, then we would have an end to the series—the series would end with this phenomenon and hence be a finite series, a conclusion Deleuze rejects. Every phenomenon is thus, Deleuze argues, echoing Leibniz’s theory of monads, related to every other phenomenon, and infinitely so.10
From the claim that every differential relation is in relation with other relations, and so on ad infinitum, we come to Deleuze’s concept of multiplicity. As Deleuze argued, and for reasons noted above, “every phenomenon is composite.” More to the point, every phenomenon is a composite substance of differential relations, a composite of differential relations that presupposes difference or intensity (disparity) as its sufficient reason. It should also be clear by now that we do not have one substance or even a plurality of substances. What is key to the differential relations that compose substance is that they are intensive differences and are thus not to be confused with extensive differences—the differences between terms, for example—and thus a differential relation is not to be identified with a specifiable difference, with a determinate cut that separates one thing from another. It is for precisely this reason that Deleuze will refer to intensive differences as pre-individual singularities, for they are the constitutive differences, as we saw, of determinate identities but are not to be confused with these identities. As a result, the differential relations that compose substance are irreducible to basic entities, although they are fully differentiated in that each differential relation or singularity is irreducible to other singularities—the differential relation dy/dx, as discussed earlier, maintains its irreducible relation even after the terms have disappeared. We thus avoid atomism in that the differential relations are not an extensive and finite minim; they are, rather, intensive differences that presuppose as their principle of sufficient reason an infinite series of differential relations. At the same time the differential relations that compose substance do not compose a single or unitary substance, for this too relies upon an extensive mode of individuation—it calls upon the numerical identity of the determinate, already individuated entity or whole when the substance of differential relations is the very constitutive condition for the possibility of such countably determinate entities. It is, finally, multiplicity that is the substance that is composed of a series of differential relations:
“Multiplicity,” which replaces the one no less than the multiple, is the true substantive, substance itself.… Everything is a multiplicity in so far as it incarnates an Idea. Even the many is a multiplicity; even the one is a multiplicity. Everywhere the differences between multiplicities and the differences within multiplicities replace schematic and crude oppositions. Instead of the enormous opposition between the one and the many, there is only the variety of multiplicity—in other words, difference. (Deleuze 1994, 182)
At this point we can begin to bring together a number of the concepts we have been discussing and see precisely the manner in which we take Deleuze and Peirce to be affirming the actual infinite, or doing infinite pragmatics. We can begin with Peirce’s own understanding of pragmatism, or what he called “pragmaticism.” In “How to Make Our Ideas Clear,” for example, Peirce argues, “Thought in action has for its only possible motive the attainment of thought at rest; and whatever does not refer to belief is no part of thought itself.” In other words, the process of thought or conceiving itself is a matter of attaining belief, and the “essence of belief,” Peirce adds, “is the establishment of habit,” or “the establishment in our nature of a rule of action” (Peirce 3:266). From here Peirce concludes that “the whole function of thought is to produce habits of action.” It is this pragmatic rule or law of mind that will come to characterize Peirce’s own understanding of pragmatism.11
By 1905 Peirce had begun to generalize the establishment of habits by way of the beliefs that serve as a rule for our actions into a metaphysical view of the nature of reality itself, including the laws of nature. In his Reasoning and the Logic of Things lectures, for example (delivered in February and March 1898), Peirce turns again to the principle of sufficient reason and demands an explanation for the laws of nature rather than accepting them as brute facts. Peirce is straightforward in his insistence that the “explanation of the laws of nature must be of such a nature that it shall explain why these quantities should have the particular values they have” (Peirce 1992, 240). Why, for example, does light move at “over 300,000,000 centimeters per second”? Peirce’s explanation is that “the laws of nature are still in process of evolution from a state of things in the infinitely distant past in which there were no laws” (Peirce 1992, 240). In other words, and as we have discussed, any determinate thing, action, or belief—in short, any determinate identity whatsoever—is, given this Peircean metaphysics, ultimately the result of an infinite process (i.e., from the “infinitely distant past”). This infinitely distant past, however, is not a determinate place from which the processes begin; to the contrary, it is indeterminate precisely for the reason that we have chaos, or a lack of habits and consistency, that allows for the possibility of identifying a determinate place at all.
How does this evolutionary process proceed? As Peirce himself recognizes, “this evolution must proceed according to some principle,” some principle that tends toward generalization and in accordance with a rule—in other words, the acquiring of habits. This is just the principle Peirce calls upon: “Now the generalizing tendency is the great law of mind, the law of association, the law of habit taking,” and hence Peirce concludes that he was “led to the hypothesis that the laws of the universe have been formed under a universal tendency of all things toward generalization and habit-taking” (Peirce 1992, 241). In his final lecture of The Reason and the Logic of Things, titled “The Logic of Continuity,” Peirce provides a more detailed account of how this works, an account that shows the importance for Peirce of the infinite continuity presupposed by each and every thing, including the laws of nature—in short, Peirce’s infinite pragmatics.
To detail Peirce’s entire arguments regarding continuity would take us too far afield, but we can see their relevance to our current discussion by focusing on an example Peirce offers during his lecture—an example that connects quite well with Deleuze’s infinite pragmatics as well. As an example of continuity, Peirce offers the clean blackboard. The blackboard, Peirce claims, “is a continuum of two dimensions, while that which it stands for is a continuum of some indefinite multitude of dimensions” (Peirce 1992, 261). The blackboard thus “stands for” the infinite continuum that is reality itself, or for the chaos and continuity upon which the generalizing, habit-taking tendency of reality relies. To this blackboard Peirce draws a chalk line, and thereby introduces a discontinuity, which he claims “is one of those brute acts by which alone the original vagueness could have made a step toward definiteness” (261–262). The line itself, however, is its own continuity, and the discontinuity that emerges does so as the limit of two continuities. Peirce is clear on this point: “the only line that is there is the line which forms the limit between the black surface and the white surface. Thus discontinuity can only be produced upon that blackboard by the reaction between two continuous surfaces into which it is separated, the white surface and the black surface.” Stated in Deleuze’s terms, and as discussed above, the discontinuous, extensive difference between a black and white surface will be the result of a differential, dy/dx, where dy and dx are continuous, infinitesimals that become vanishingly small as they converge upon zero, or dy/dx = 0/0 (Deleuze and Guattari will refer to this as degree zero in A Thousand Plateaus). Once a mark on the blackboard gets to the point where it “will stay for a little while,” then, Peirce concludes, we have “some beginning of a habit [that] has been established by virtue of which the accident acquires some incipient staying quality, some tendency toward consistency” (1992, 262). In short, for Peirce each determinate identity, each thing, is itself the result of a habit-taking tendency, and a tendency that develops from an infinite continuity. Some of our actions, therefore, or the things of reality itself, exemplify the habits already taken on by the generalizing tendency or “law of mind.” As Peirce makes clear, however, since even the laws of nature are themselves in evolution, not everything is determined by law.12 As a result, and on this point Deleuze could not agree more—while some actions may be ordinary actions that simply follow well-established habits and “laws of nature,” others will be singular and extraordinary and begin the process of establishing new habits, a process that is simultaneously, for reasons we have seen, a supertask.13 Here, ultimately, is where Deleuze follows Peirce in pursuing an infinite pragmatics.
In confronting the problem of supertasks, the most common strategy, we saw, was simply to reject the possibility of the actually infinite. Achilles does not actually reach an infinite number of points in catching the tortoise; one does not actually complete the supertask of infinitely placing and removing the apple from the table. With Peirce, however, the continuum that is presupposed by the habit-taking tendency of reality is indeed actually infinite, and we only encounter the paradoxes when we attempt to understand the infinite continuum by way of irreducible points and particulars, when they too consist of parts, and parts with parts, and so on ad infinitum. For Peirce, however, the affirmation of the actual infinite is itself placed within relation to the limit that is the Truth or the opinion all are fated to settle upon at the end of the road of inquiry. Deleuze, by contrast, affirms the actual infinite with no holds barred, and this is an infinite that is not held in abeyance to any identity, whether this be Peirce’s True opinion, Hegel’s Spirit, or any other ultimate identity, and it is irreducible to the extensive properties and qualities that make the scientific propositions possible, and hence the “true” correspondence relation between the propositions and properties and qualities referred to by them. In contrast to the infinite convergent series where one can be lulled into the belief that the series does indeed converge at the limit, the divergent infinite series of placing and removing the apple does not converge on the apple being either on the table or not, nor does it converge upon an ultimate stability and Truth as with Peirce, but it instead remains undecidable, or metastable (to use Simondon’s term). It is precisely the metastability and chaos of reality, or the differential as disparity as we saw, that is the actually infinite, according to Deleuze, and it is this that is the principle of sufficient reason for all that appears. Deleuze’s Peircean pragmatism does indeed follow in the tradition of Peirce’s metaphysics by placing the infinite at the heart of reality, but rather than affirm the identity of the infinite cosmos as Peirce does, Deleuze affirms the difference and disparity of the infinite chaosmos.
1 For discussion of supertasks in the literature, see Thomson (1954). This essay is in response to Max Black’s famous essay, “Achilles and the Tortoise,” (Black 1950).
2 I borrow this example from José Benardete (1964).
3 Charles Sanders Peirce, “Validity of the Laws of Logic,” in Writings of Charles S. Peirce: A Chronological Edition, vol. 2, ed. Edward C. Moore et al. (Bloomington, IN: Indiana University Press, 1982–present), 256 [hereafter referred to be referred to as Peirce, followed by volume and page number, e.g., Peirce 2:256].
4 Contemporary analytic metaphysicians will use the term “gunk” when they refer to the (Peircean) position that asserts everything has proper parts and there are no basic elements. See, among many examples, John Hawthorne and Brian Weatherson (2004) and Zimmerman (1996).
5 In What Is Philosophy?, Deleuze and Guattari cite Ruyer (1952), chaps. 9–11. For an excellent discussion of Ruyer, see Mary Beth Mader (2012).
6 For Peirce’s discussion of Truth as the opinion we are all fated to hold, see Peirce 1955, 288: “That is to say, I hold that truth’s independence of individual opinions is due (so far as there is any ‘truth’) to its being the predestined result to which sufficient inquiry would ultimately lead.”
7 Daniel Smith has highlighted this aspect of Deleuze’s work as well. In his essay, “The Conditions of the New” (in Smith 2012), he argues, in the context of arguing that for Spinoza “simple bodies are actually infinite… [that] [t]he formula of the actually infinite, however, is neither finite nor indefinite. On the one hand, it says that there are indeed ultimate or final terms that can no longer be divided—thus it is against the indefinite; but on the other hand, it says that these ultimate terms go to infinity—thus they are not atoms but rather terms that are ‘infinitely small,’ or as Newton would say ‘vanishing terms’” (Smith 2012, 249).
We can contrast Deleuze’s project as understood here with Bergson. Bergson’s work was extremely influential upon Deleuze’s own understanding of difference. Deleuze, like Bergson, understands fundamental difference, difference in itself, as intensive rather than extensive. Bergson also provided Deleuze with the notion of a multiplicity that is irreducible to the one or the multiple. Despite these similarities, Bergson’s approach resonates too strongly with Peirce’s distinction between the continuum as real (for Bergson this is the durée) and the points that are extensive abstractions (for Bergson this is quantity). With the notion of differential, and by drawing upon the pre-Kantian metaphysical tradition, Deleuze argues for a notion of intensive quantity that sets his project apart from Bergson’s.
8 Deleuze announces this task of prioritizing difference in the Preface to the English Edition of Difference and Repetition, where he claims, “All that I have done since [the initial publication of DR] is connected to this book, including what I wrote with Guattari,” and this was to develop a concept of difference and avoid the fate whereby the “majority of philosophers had subordinated difference to identity or to the Same, to the Similar, to the Opposed or to the Analogous…” (Deleuze 1994, xv).
9 To borrow an example from Smith (2012, 246), “z” may be the trigonometric tangent.
10 See Leibniz, Monadology §56: “Now this connexion or adaptation of all created things to each and of each to all, means that each simple substance has relations which express all the others, and, consequently, that it is a perpetual living mirror of the universe.”
11 In a 1905, for example, Peirce argues that his earlier maxim regarding pragmaticism (referring to “How to Make Our Ideas Clear” and “Fixation of Belief”) was to “consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have” (Peirce 1955, 290).
12 See ibid, 240: “But if the laws of nature are still in process of evolution from a state of things in the infinitely distant past in which there were no laws, it must be that events are not even now absolutely regulated by law.”
13 The importance of ordinary versus singular (or distinctive) points is an important theme in Deleuze’s philosophy. See Deleuze (1994, 24, 47, 181).
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Zimmerman, Dean W. 1996. “Could Extended Objects Be Made out of Simple Parts? An Argument for ‘Atomless Gunk,’” Philosophy and Phenomenological Research, 56: 1–29.