14 Aug 2015

Somers-Hall, (4.2), Deleuze’s Difference and Repetition, ‘4.2 Ideas and the Differential Calculus (170–82/217–30)’, summary

by Corry Shores
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[The following is summary. All boldface, underlining, and bracketed commentary are my own. Proofreading is incomplete, so please forgive my typos and other distracting mistakes. Somers-Hall is abbreviated SH and Difference and Repetition as DR.]

Summary of

Henry Somers-Hall

Deleuze’s Difference and Repetition:
An Edinburgh Philosophical Guide

Part 1
A Guide to the Text

Chapter 4. Ideas and the Synthesis of Difference


4.2 Ideas and the Differential Calculus (170–82/217–30)



Brief summary:
For Deleuze, the Idea has three intrinsically related moments, indetermination, determinability, and determination. He sees their relation expressed in a non-orthodox tradition in the history of the calculus which grants the differential its proper metaphysical status. Bordas-Demoulin shows how the differential gives us the the essence of something rather than a description of any of its particular instances. For example, Descartes’ formula for the circumference of a circle, x2 + y2 – R2 = 0, only tells us how we would expect the x and y variables to relate for some given point on the circumference of some one circle or another. But the differential formulation, ydy + xdx = 0 tells us more what it means to be a circumference, since it tells us that the tendency of variation in the curve of the circumference is of such a sort that it will eventually return to any point of origin. Also in such a differential formulation, we have the undetermined, since it is more about circumference in general and not about the determinate relations of particular circles.  For
Maimon, the differentials dx  and dy each by themselves cannot be given a sensible interpretation, however, the differential relation of the two can. In this way Maimon gives the conditions for determinability. Wronski thinks that the differentials are real, but they fall under a different kind of knowledge than finite values. He is also concerned with moments in a function’s variation where the change is drastic and as well where the value of that change can be numerically determined. Thus we have the three moments intrinsically related: the differentials dx and dy are by themselves undetermined, but they obtain their determinability when brought into differential relation, which can then be determined numerically.





[Recall from the last section how for Kant] “determinability and determination are extrinsic determinations of Ideas” (SH 131) [since they are matters of predicating objects with empirical properties]. Deleuze, however, thinks that all three moments of the idea (indetermination, determinability, and being determined) can be intrinsically incorporated, and he does so “by turning to the differential calculus as a model of thinking: ‘Just as we oppose difference in itself to negativity, so we oppose dx, the symbol of difference [Differenzphilosophie] to that of contradiction’ (DR 170/217)” (SH 131). SH will first outline the calculus in general. One way of dealing with calculus is geometrically, by examining the tangent to curves. It is especially helpful to look at how this is applied in finding instantaneous velocities. [Let me here suggest some other entries that expand on SH’s excellent summary to follow. In this Bergson entry, you could skip down to the video clips of MIT Physics professor Walter Lewin’s explanation of instantaneous velocity, from his second class of Physics I, 1999. In this other entry MIT Mathematics professor David Jerison explains the derivative in this geometrical interpretation, from the first class of Single Variable Calculus, 2007. And finally there is this section of Edwards and Penney’s Calculus. I will mostly quote SH, since he summarizes the material excellently and since any elaboration that I might add can be found in those links above or in others I add later. Also be sure to check out Simon Duffy’s many superb works on Deleuze and the calculus, including The Logic of Expression.]

A first approximation is that the calculus is a field of mathematics dealing with the properties of points on curves (Boyer 1959: 6). As Boyer notes, this concern with properties of points on curves is similar to a concern with the properties of a body in motion, such as its velocity at a given moment in time. If we wanted to determine the average velocity of a body in motion, we would determine this by finding a ratio between two quantities, the distance that the body has travelled in the time period (s), and the time period itself (t). We could represent this, for instance, in the following form: average velocity = Δst, that is, the difference in displacement over the period divided by the difference in time (with Δ symbolising difference). This would give us an average velocity in terms of metres per second, or miles per hour. While this might be effective for average velocities, the problem emerges when we want to determine the velocity of the body at a particular moment in time. When we are talking about a particular moment, we are no longer talking about average velocity, but rather now about instantaneous velocity. If a body is moving at constant speed, then the average and instantaneous velocities of the body will coincide, but if a body is accelerating or decelerating, however, then its instantaneous velocity will be constantly changing, and so we cannot determine it based on its average velocity.

[paragraph break]

Leibniz’s solution to this dilemma was to suggest that if we take the average velocity of the body over a time, beginning with the point we are trying to determine the instantaneous velocity for, and slowly decrease the slice of time we are using to divide the distance travelled, the average velocity will approach the instantaneous velocity. That is, the smaller the segment of time over which we determine the average velocity, the closer it will be to the instantaneous velocity at a point. If we extend this idea, and determine the average velocity over an infinitesimally small stretch of time, then, because this stretch of time is for all intents and purposes 0, the average velocity will actually equal the instantaneous velocity.
(SH 132)

Calculus does not merely deal with geometrical curves but also with the primitive functions that describe them. So, “When we apply the calculus to the equation of a curve, we get what is known as the derivative, which is an equation that gives us the gradient of the curve at each point (in the example of the body in motion, the velocity at each point)” (SH 133). Now, the calculus is concerned not with finite differences but rather “with infinitesimal differences, otherwise known as differentials” (133) [As SH notes below, whether or not calculus deals with infinitesimals or instead with negligibly small finite ones is a matter of debate, and the orthodox position is that they are not infinitesimal. For more on a defense of Leibniz’ notion of infinitesimals, see this excellent article by Katz and Sherry.] “In order to represent infinitesimal differences, Leibniz introduces the symbolism dy / dx” (133). [You can see this usage for example in his Cum Prodiisset. Also very helpful is this letter that he writes giving a simple explanation of the differential relation between infinitesimal “vanishing” values.] Recall how Aristotle defined relations in terms of negation [see for example section 1.6. In Aristotle’s system of division, things are differentiated on the basis of clear defining limits that determine what is special and proper to each thing. These limits serve to define what makes one thing what it is and what makes something else not that thing but rather something different entirely, and thus Aristotle’s system makes use of negation. One thing that interests Deleuze about the calculus is how it defines a differential relation without the concept of negation.] “The differential calculus provides the possibility of developing a theory of relations that relies on reciprocal determination of the elements, dy and dx” (133). But Deleuze thinks that although there is something philosophically very important about this early version of the calculus, there are two mistakes we might make in how we understand it, namely, we should neither think that dy signifies the infinitesimal nor should we say that therefore the differential has no “ontological or gnoseological” value. [To see the problem with the differential as an infinitesimal, we note that in the calculations, it is first treated as if it had a value and then later treated as if it were zero. For how this happens and how Leibniz defends the introduction then removal of the infinitesimal values with his laws of continuity and transcendental law of homogeneity, see again Katz and Sherry, especially pp.572-573.]

In order to understand why we might make these two mistakes, we need to look further at what the term, dx, signifies. Now, as we saw, dx represents for Leibniz an infinitesimal distance between two points. When we want to use this to determine instantaneous velocity, however, we encounter a contradiction. To see this, we can turn to the account of the infinitesimal of L’Hôpital, one of the earliest popularisers of the calculus [the following up to citation is L’Hôpital quotation]:

Postulate I. Grant that two quantities, whose difference is an infinitely small quantity, may be taken (or used) indifferently for each other: or (which is the same thing) that a quantity, which is increased or decreased only by an infinitely smaller quantity, may be considered as remaining the same. (L’Hôpital 1969: 314)

This postulate is needed because dx must be seen as having a determinate value in order to form a ratio, dy/dx, but also has to have no magnitude (=0) in order to capture the gradient at a point, rather than across a length of the curve. Clearly, this is a fundamental difficulty, since the consistency of mathematics is threatened by taking a variable simultaneously to have and to lack a magnitude. In this sense, it appears that Deleuze is right in holding it to be a mistake to give the differential a sensible magnitude, even if this were infinitely small, and | modern readings of the calculus concur, presenting an interpretation of the calculus in terms of a concept of limits that does away with the need to give anything beyond a formal meaning to the differential.
(SH 134)

This does not mean that Deleuze takes this orthodox view of the differential that strips it of its infinitesimal meaning. Instead, Deleuze will draw from the 18th and 19th century metaphysical readings of the calculus from Bordas-Demoulin, Maimon and Wronski, who “all held that the contradiction in the mathematical account of the differential did not entail that the differential itself was contradictory, but rather that a proper understanding of it involved a metaphysical interpretation that brought in resources not available within mathematics itself” (134). SH will now look at how Deleuze takes up their ideas “to present an alternative to the Kantian notion of the Idea” (134). SH explains:

Each of these figures takes up a different moment of the world of appearances. Bordas-Demoulin’s account is concerned with quantities. As a follower of Descartes, he takes matter to be continuous, rather than made up of discrete atoms. In this regard, he is interested in the way in which the calculus allows us to provide an account of these continuous magnitudes. Maimon is concerned with qualities, such as the colours of objects. As such, he is interested in how these qualities are reciprocally determined, and how we are to understand the changes in quality of objects. Finally Deleuze’s discussion of Wronski develops an account of potentiality in terms of the calculus, that is, the moments in the development of an object where its nature itself changes.

So we begin first with Bordas-Demoulin, who “asks how we can represent mathematical universals as they are in themselves” (134). [The basic idea here seems to be that an algebraic formula could tell us the structural relations of any particular shape, but they do not tell us what really makes that shape what it is. So Descartes’ algebraic formula for a circle’s circumference,  x2 + y2 – R2 = 0 (or we perhaps might have come across this as: x2 + y2 = r2) tells us only about how specific values would relate were they to be substituted into the variables. However, the differential calculus formulation ydy + xdx = 0 tells how the variables vary in the circle, and thus tells us about circumference in general. (I am not sure, but perhaps the formula is telling us that given the way that the y value and the x value vary in relation to one another, that in combination they will eventually bring the variable value back to where it began).]

He [Bordas-Demoulin] claims that Descartes, for instance, does not represent the concept of circumference in itself, but only this or that particular circumference. Descartes’ procedure is, according to Bordas-Demoulin, to present the algebraic equation for a circle, x2 + y2 – R2 = 0. If we drew the graph of this equation, then for a specific value of R, all of the solutions to the equation would together give us a circle, centred on the point (0, 0) of the Cartesian coordinate system. Why does this Cartesian definition not give us the true definition of a circle? Bordas-Demoulin puts the point as follows [the following up to citation quotes Bordas-Demoulin]:

In x2 + y2 – R2 = 0, I can assign an infinity of indifferent values to x, y, R, but nevertheless I am obliged to always attribute to them one, that is, one determinate value, and by consequence to express a particular circumference, and not | circumference in itself. This is true for equations of all curves, and finally for any variable function, so called because they give a continuous quantity and its symbol. It is the individual curve or function which is represented, and not the universal, which, accordingly, remains without a symbol, and has not been considered mathematically by Descartes. (Bordas-Demoulin 1843: 133)

In relation to particular circles, algebra functions like the Russellian notion of sense, or the Kantian notion of a condition, in that the variables, x, y, R simply stand in for particular values. It gives us an account of what circumference is in general, but this account can only be ‘cashed out’ by choosing specific values to put into the equation. Ultimately, therefore, we simply define the structure of this or that particular circumference, rather than circumference itself. In order to develop an account of what circumference is in itself, we need to remove these references to the particular terms, and this is achieved by using the differential calculus, ‘whose object is to extract the universal in the functions’ (Bordas-Demoulin 1843: 54). When we differentiate a function, we receive another function that no longer gives us the precise values of the function, but instead, the variation of the function. Moreover, because this function is constituted in terms of dy and dx, which cannot be assigned a value (they are strictly 0 in regard to y and x), we no longer have a function that can be understood simply in terms of possible values of variables. For Bordas-Demoulin, therefore, dx does not represent a variable that can be given different particular values, but rather a radical break with understanding structure in actual terms. ‘Applied to x2 + y2 – R2 = 0, [the calculus] gives ydy + xdx = 0, an equation that does not express any particular circumference, but circumference in general, dx, dy being independent of all determinate or finite magnitudes’ (Bordas-Demoulin 1843: 134).
(SH 135)

[I do not grasp the next points so well, so let me quote it first.]

What Deleuze wants to take from this is the idea that the differential is simply inexpressible in terms of quantity, and so is inexpressible in terms of the primitive function. Nevertheless, if we reverse the operation of differentiation by integrating a function, we get the formulae for particular, actual circumferences. The differential is not simply different from the primitive function, but we can also see that it has an intrinsic relationship with it: ‘If in, ydy + xdx = 0, one still encounters the finite magnitudes y, x, this is because in quantity, no more than in substance, can the universal isolate itself completely and form a separate being’ (Bordas-Demoulin 1843: 134).
(SH 135)

[For the first part of the first sentence, perhaps we can note that in the differential understood infinitesimally as Leibniz has it, the values have vanished, although their ratio remains. I would think that differential calculus is still dealing with quantities, so I am not sure about the idea here. Then perhaps for the second part of the first sentence, we might note how the derivative for y = x2 is dy/dx = 2x. Maybe the y = x2 is the primitive function, and we see that it is replaced when we find the derivative. I am just making guesses. For the next point, I am not sure, but I again am just making a guess. Perhaps the idea is that if we find the anti-derivative for ydy + xdx = 0 then we get the formula or primitive function for a circle, but I do not know at all how all this works. I guess the important philosophical idea here is that the differential both is different from the primitive function while also being very intimately related to it.] [For the next idea, I think we first recall how with Plato, when we experience something, like the imperfect equality of two things, we are encountering a lesser version of a better experience we supposedly once had of perfect equality. We might say now in this discussion that when we encounter a circle or at least its primitive function, it both expresses this empirical expression of circularity as well as what it means to be circular, since we can derive the differential formulation from the function.]

We thus have a situation that parallels the account of Plato that Deleuze has given in | the last chapter. An empirical concept, such as that of circumference, carries within it its Idea, the differential, in comparison with which it falls short. Whereas for Plato the Idea was ultimately understood by analogy with empirical objects (the use of analogy in Plato’s theory of memory), the differential allows Bordas-Demoulin to present a difference in kind between the Idea and its instantiations. In emphasising the degree to which the differential is immanent to the primitive function while different in kind from it, Bordas-Demoulin chooses another figure as a model of the metaphysics of the calculus who might be even better suited to Deleuze’s account: ‘According to this metaphysics [of the calculus], one might say, by way of comparison, that the God of Spinoza is the differential of the universe, and the universe, the integral of the God of Spinoza’ (Bordas-Demoulin 1843: 172).
(SH 135-136)

We turn now to Salomon Maimon. Deleuze says he uses the calculus to overcome the way that Kant reduces the transcendental merely to the role of providing the conditions of experience rather than accounting for its genesis (136). Deleuze works mainly with Guéroult’s The Transcendental Philosophy of Salomon Maimon, so SH will follow this commentary as well. Maimon adds to Kant’s project “a Leibnizian genetic account of the production of space, time and intensity” since for him the differential is “the source of a construction […] of the phenomenal world” (136). Recall Kant’s problem of explaining how faculties that are different in kind can relate and cooperate to produce knowledge. Kant is not interested, however, with “the reasons why we possess faculties that differ in the first place” (136). However, “Maimon instead wants to investigate the genetic conditions of phenomena” (136). Now, for Maimon, what is given is whatever the intellect cannot think. Thus, were our faculty of thinking infinite, what is given would disappear. This is similar to how Leibniz regards the given empirical object as being “a confused form of perception of the true nature of things” (137). But for Leibniz, the difference between a finite intellect with confused perception and an infinite intellect that can know everything clearly is a difference of degree and not of kind. For Maimon, however, there is a difference in kind between the two sorts of thinking. A differential as an infinitesimal “cannot be given a sensible interpretation without contradiction” (137). However, the differential relation of two infinitesimals can have a sensible interpretation, as for example “the formula for the gradient of the points on a curve” (137). [Since we can experience a curve’s gradient but not the infinitesimal differential values expressing it at each location] “The differential is thus like the Kantian noumenon, which can be thought, but cannot be presented in intuition” (137). [In the following, I do not understand very much how all this works. Somehow the differentials of objects are the noumena but the objects themselves are the phenomena. Then there is reference to intuition = 0, which I thought in Kant had to do with an intuition with a zero magnitude of intensity (Critique of Pure Reason A165/B208), but I am not sure what it means in this case. Perhaps the idea is that intuition = 0 is like a vanishing decrease or evanescent increase in intensity of an intuition, but I am not sure. But perhaps the idea is that intuition for Maimon is intensive in the calculus sense of instantaneous variation taking different degrees of the intensity of change. And maybe they are also a matter of two variables varying like they do in instantaneous velocities. So maybe it is not that we see red but rather a degree of variation of redness, or a degree of change from something to red or from red to something else. I am not sure. Let me quote:]

Maimon takes this mathematical interpretation of the differential, and gives it a transcendental interpretation, so the differential, dx, becomes a symbol of the noumenal grounds for the synthesis of phenomena [the following up to citation quotes Maimon]:

These differentials of objects are the so-called noumena; but the objects themselves arising from them are the phenomena. With respect to intuition = 0, the differential of any such object is dx = 0, dy = 0 etc.; however, their relations are not = 0, but can rather be given determinately in the intuitions arising from them. (Maimon 2010: 32)
(SH 137)

[The next idea is also hard for me to grasp, but it seems to be extraordinarily fascinating. Perhaps the idea is that all the determinations of something are these differential relations, and an infinite intellect that can sum them all up would understand the object without intuitions. But again how all this works I cannot conceive or imagine. The main idea seems to be that our intuitions supply us with the information that we cannot obtain through our understanding of the all the differentials making up the object. Another main idea seems to be that intuition gives us these differentials, perhaps indirectly, but over a period of time, and thus the object is synthesized gradually. Let me quote it so we do not miss the idea:]

An infinite understanding is able to think these differential relations, and thus to think the object in its totality without intuition. In this sense, as Deleuze notes, for Maimon, ‘the particular rule by which an object arises, or its type of differential, makes it into a particular object; and the relations of different objects arise from the relations of the rules by which they arise or of their differentials’ (Maimon 2010: 33). Since the differential gives us a rule that governs the infinite relations of the object, however, the finite intellect is unable to think it all at once. In this respect, as opposed to thinking the object a priori according to the | rules governing the way it arises, it can only think of it as given, that is, through sensible intuition. Thus, rather than the extrinsic relation between the faculties, Maimon shows how intuition emerges through the finite intellect’s inability to think the relations of differentials all at once. Instead of thinking the object as a completed synthesis, it must be thought as a synthesis in process, as an ‘arising’ or ‘flowing’.

[The next point is also tricky. We now need to think of the imagination being conscious only of representations. I am not sure what the representations are. I will make some guesses. Perhaps they are just the raw intuitive data that imply the differentials. Or perhaps they are conceptual data that is like the calculations of differentials that the infinite intellect performs. Perhaps then the illusion Guéroult is referring to is simply the idea that we mistake the manifold of intuitions as synthesized by the imagination for the thing itself. The next point about problems I am not getting so well, so let me just quote for now:]

Now, as Guéroult makes clear, the fact that we cannot simply think the object means that we become subject to a transcendental illusion [the following up to citation is Guéroult quotation]:

The imagination is thus never conscious of anything other than representations; it therefore has, inevitably, the illusion that all of the objects of consciousness are representations; it is led by this to also consider the original object or the complete synthesis as a representation. (Guéroult 1929: 66)

It is this illusion that leads us to see problems in the same terms as solutions. We can therefore see in Maimon two different modes of thinking. One that operates in terms of intuition, and provides a philosophy of conditioning, and another that provides a genetic model of thought that attempts to trace the genesis of the given back to its differential roots.

So recall again from last section that for Kant the Idea has three moments: indetermination, determinability, and the determined. The second two are extrinsic, since they are matters of empirical determination. Now with this calculus and Maimon material, we may provide an alternative theory of the Idea where all three moments are intrinsically related. [The Idea as differential is undetermined, because, like Kant’s Idea, it cannot be given in intuition. But, it becomes determinable by placing it into the differential relation dy/dx. It is then determined as it obtains specific values for its variables.]

We can now present the alternative theory of the Idea. Rather than seeing it as a relation between three moments, two of which are extrinsic, the differential calculus relates the three moments intrinsically. It is undetermined in that the differential, dx, cannot be given in intuition. When it is put into a relation, such as dy/dx, it becomes determinable, as it specifies the complete range of values the function can take. Finally, it is determined in terms of specific values that the function takes at particular moments (the instantaneous velocity of a particular point in time in our prior example). Whereas the infinite understanding thinks the curve as a whole, we can only think the process of generation of the curve, equivalent to the actual evolution of the object in intuition. As Guéroult puts it, ‘the differential is, then, the noumenon (that which is simply thought by the intellect), the source of phenomena (which appear in intuition)’ (Guéroult 1929: 60).
(SH 138)

We turn now to Wronski. We first take note of Lagrange, who wanted to do away with the problematic concepts of infinitesimals, evanescent quantities, differentials, or limits. He did not want a metaphysical interpretation of the differential, but instead he wanted to formulate the ideas of calculus using algebraic formulations (138-139) [For  more, see this entry from Boyer, and also this entry for more of the technicalities (which I cannot recall well enough at the moment to summarize. They are really quite technical).] Deleuze turns to Wronski to preserve this metaphysical interpretation. [It seems from the following material that Wronski considers finite and infinitesimal values as belonging to two different classes of knowledge. The finite are matters of our cognition. What he says about the infinitesimals I do not grasp as well, but he says they have to do with the generation of these cognitions. How that works is not explained, but maybe it is similar to what we said before about Maimon. The important point seems to be that for Wronski, we cannot, like Lagrange wanted to do, only deal with finite concepts, since certain ones relevant to calculus are generated by infinitesimal ones.]

As with the other thinkers of the calculus discussed in this chapter, Wronski holds that there is a fundamental distinction between the differential and normal quantity [the following up to citation is Wronski quotation]:

It is this important transcendental distinction that is the crux of the metaphysics of Calculus. – In effect, the finite quantities and indefinite quantities, that is to say, infinitesimal quantities, belong to two entirely different, even heterogeneous, classes of knowledge: the finite quantities relate to the objects of our cognition, and infinitesimal quantities relate to the generation of this same cognition, so that each of these classes must have knowledge of proper laws, and it is obviously in the distinction of these laws that the crux of the metaphysics of infinitesimal amounts is found. (Höené Wronski 1814: 35)

Now, while Lagrange believes that he has escaped from the need to introduce infinitesimals by resorting to the (algebraic) indefinite, which can be understood purely in algebraic terms, Wronski’s claim is that the indefinite itself cannot be understood without the infinitesimal. To bring the infinitesimal into the domain of cognition, it has to be presented in an intuition, which can be done purely as an indeterminate quantity. The indeterminate quantity that is at the centre of Lagrange’s method is thus, for Wronski, still reliant on the differential.
(SH 139)

[I am not sure I completely grasp the next notion about two kinds of points, singular and ordinary. Perhaps it has something to do with the points of inflection Deleuze discusses in The Fold, and which we discuss in this entry. There idea there is that there are points in the curve’s “movement” (see the animated diagram at that link) where the curve changes direction, but while in that change, it has a 0/0 slope, since it is in the process of changing from downward to upward in its orientation, and it is sort of in limbo in between the two states of affairs. Also, regarding what SH says about differentiating the equation to get acceleration, we might consult also the same Lewin lecture at around 25.00, where he says, “That is the instantaneous acceleration. And this, you will recognize is the first derivative of velocity versus time which is also the second derivative of position versus time.”]

In claiming that Lagrange’s method still relies on the differential, Wronski does not deny that, precisely because it is derived from it, it is still correct. In fact, Lagrange’s method produces a series of differentials which allow us to distinguish between two kinds of points on the line: singular points and ordinary points. If we remember our initial example of the calculus, relating distance to time gave us the velocity of a body. If we differentiate this equation once more, we will obtain a relationship between velocity and time, which is the acceleration of a body. Points on this curve, such as where it is flat, indicate singular features of the movement of the body, such as in this case the point at which it is travelling at constant motion. In more abstract curves, points where the gradient is 0/0, or is null or infinite, define points where the nature of the curve | changes. Potentiality thus defines the points at which the nature of the relationship between the terms radically changes.
(SH 139-140)

[The next paragraph is very important, but I do not grasp it completely. SH says that this schema remains abstract for the moment, but it will be elaborated with concrete examples to follow. The basic idea seems to be that these alternative calculus ideas allow us to create an account of the Idea which intrinsically relates  its three moments, the indeterminate, the determinable, and the determined. Perhaps the way this happens is as follows. dy and dx have no determinate values in themselves, since they are not finite. So they are the indeterminate. However, when they come into relation, their differential value becomes determinable. For some particular function, we can determine that value, and thus we have the moment of the determined.]

We can tie these three moments together to develop an account of the Idea where its three moments, the indeterminate, the determinable and the determined, are intrinsic to it. As we saw when we looked at Bordas-Demoulin, the differentials themselves, dy and dx, are completely undetermined with respect to representation, and hence to the field of solutions. Nonetheless, when brought into relation with each other, they give us an equation that is determinable. This equation gives us the rates of change of a function at each point in time (or more correctly, for any value of x). Such an equation, as Wronski shows, contains singular points that determine the points on the curve where its nature radically changes. That is, by specifying a value of x, we can determine the rate of change at any point. Specifying a value of x, therefore determines the Idea. We therefore have a particular determined value (intuition), a determinable equation that subsumes it (the concept), and a field of differentials themselves which engenders both the determinable and determination. The differential, as problem, therefore contains the solution intrinsically, rather than simply being interpreted in terms of it. While this account may seem abstract for now, as we shall see in the following four sections, we can develop concrete examples of the Idea that operate according to this schema.

SH writes that “The remainder of Deleuze’s discussion of the differential calculus draws the consequences from this understanding of the calculus as Idea” (140). For example, there is the question in the history of calculus of whether the infinitesimals are real or fictive. Wronski shows that “this question has traditionally been interpreted in terms of whether differentials can be an object of (representational) cognition, or are fictions” (140). However, Wronski also shows that the differentials engender the objects of cognition and that they are on a different order of knowledge. This makes the “first alternative – real or fictive?” collapse [since they are both real, but only not real on the same orders of knowledge] (140). The next alternative, between infinitesimal and finitist interpretations of the calculus, are both inadequate to the differential, since both are representational (140). Deleuze also places emphasis on the differential dx, which is “constitutive of the primitive function. As such, it is concerned with problems, rather than solutions” (140-141) [but I am not sure exactly why dx is more on the side of problems rather than solutions, like the primitive function is]. Thus:

In this sense, Deleuze claims that rather than talking of a metaphysics of the calculus, we should talk of a dialectics of the calculus, dialectic meaning ‘the problem element in so far as this may be distinguished from the properly mathematical element of solutions’ (DR 178/226). The work of the mathematician, Abel, is therefore important to Deleuze, because he developed a method for determining whether a problem has a solution without resorting to actually solving the problem itself.

SH concludes by summarizing the findings:

We have already seen how the three moments of the Idea are intrinsically, rather than extrinsically, connected in the calculus, and Deleuze reiterates and summarises his discussion in the following passage [the following up to citation is Deleuze quotation]:

Following Lautman’s general theses, a problem has three aspects: its difference in kind from solutions; its transcendence in relation to the solutions that it engenders on the basis of its own determinant conditions; and its immanence in the solutions which cover it, the problem being the better resolved the more it is determined. Thus the ideal connections constitutive of the problematic (dialectical) Idea are incarnated in the real relations which are constituted by mathematical theories and carried over into problems in the form of solutions. (DR 178–9/226)

Each of these three moments is present in the calculus as a method of intrinsically relating two structures that are different in kind from one another. The calculus thus provides a model for an account of the genesis of determinate quantity from something different in kind where each of its moments is intrinsically connected with the others.
(SH 141)

Citations from:

Somers-Hall, Henry. Deleuze’s Difference and Repetition. An Edinburgh Philosophical Guide. Edinburgh: Edinburgh University, 2013.

Or if otherwise noted:

Deleuze, Gilles. Difference and Repetition, trans. Paul Patton, New York: Columbia University Press, 1994/London: Continuum, 2004.

Bordas-Demoulin (1843), Le Cartésianisme, ou La Véritable Rénovation des Sciences, vol. 2, Paris: J. Hetzel.

Boyer, Carl (1959), The History of the Calculus and its Conceptual Development, New York: Dover Publications.

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