"if you think childlike, you'll stay young. If you keep your energy going, and do everything with a little flair, you're gunna stay young. But most people do things without energy, and they atrophy their mind as well as their body. you have to think young, you have to laugh a lot, and you have to have good feelings for everyone in the world, because if you don't, it's going to come inside, your own poison, and it's over" Jerry Lewis "I don’t believe in the irreversibility of situations" Deleuze
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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.
Summary
[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 = Δs/Δt, 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.
(134)
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’.
(137-138)
[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.
(138)
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.
(140)
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.
(141)
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:
DR:
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.
Guéroult, Martial (1929), La Philosophie Transcendantale de Salomon Maïmon, Paris: Librarie Félix Alcan.
L’Hôpital (1969), ‘The Analysis of the Infinitesimally Small’, in D. J. Struik (ed.), A Source Book in Mathematics, 1200–1800, Cambridge, MA: Harvard University Press, 312–15.
Maimon, Salomon (2010), Essay on Transcendental Philosophy, trans. Nick Midgley, Henry Somers-Hall, Alistair Welchman, and Merten Reglitz, London: Continuum Press.
Höené Wronski, Józef Maria (1814), Philosophie de l’Infini, Paris: P. Diderot L’Ainé.
by Corry Shores
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Henry Somers-Hall
Deleuze’s Difference and Repetition.
An Edinburgh Philosophical Guide
Introduction
Intro sect.3
The Structure of the Text
Brief summary:
The basic structure of Difference and Repetition begins with a new understanding of these concepts in order to develop a way to understand the world without representation or judgment.
Summary
Commentators are not in agreement as to what is the structure of Deleuze’s Difference and Repetition or what exactly its line of argumentation is. Somers-Hall (SH) sees it as having the following structure:
[1] Introduction: Deleuze relates the concepts of difference and repetition. Repetition is not understood in the concept of law but rather on a non-conceptual understanding of difference.
[2] Chapter 1: This chapter gives a logical and metaphysical analysis of our relationship to the world. The traditional conception of difference – x is different from y – is inadequate. A judgment attributes predicates to objects. It, as well as representation, only gives a partial description of the world. For Deleuze, judgments really arise from a world of intensity.
[3] Chapter 2: this chapter examines experience from a transcendental viewpoint. For Kant, our perceived world is synthesized by the subject in accordance with the structure of judgment. This is why traditionally we have used logical concepts of judgment to explain the world. Deleuze thinks that Kant’s syntheses are based on temporal syntheses which do not have the structure of judgment. The time syntheses also explain the structures of the self and the categories of judgment, rather than the other way around as it is in Kant. So our normal means of explaining the world are inadequate, because the structures we use to understand the world are merely “effects of a deeper play of intensity”.
[4] Chapter 3: So there is a deeper level of intensity and a misleading level of judgment. Firstly Deleuze shows how traditional structures “occlude” intensity. Secondly he discusses his eight postulates of the ‘dogmatic image of thought.’
[5] Chapter 4: Certain advances in differential calculus help us understand how it is that the world is fundamentally intensive. Differential calculus deals with entities that cannot be represented, that is, be incorporated into judgments.
Thus, while the calculus is a definite conceptual structure, it is a conceptual structure with a determinate reference beyond the conceptual realm. It is this reference which allows us to prevent our thought from collapsing into the belief that everything can be understood in terms of extensity and judgement.
(5)
Other domains can also understand the world without reducing it to judgment, for example, physics, biology, and sociology have in some cases done this.
[6] Chapter 5: Deleuze previously portrayed thinking in terms of his concept of Ideas rather than of judgments. In this chapter Deleuze shows the relationship between Ideas and intensity in order for Ideas to not replace judgment. Ideas need to be more than ways we understand the world.
The above are the central themes. SH will also note the implications Deleuze draws out from these ideas, for example, how the rejection of extensity pushes us toward a perspectival model of the world, how the move away from representation raises the importance of the arts in exploring genesis, and Deleuze also shows the need for an alternative philosophical tradition following Lucretius, Duns Scotus, Spinoza, Feuerbach, Nietzsche, and others. SH will touch on as many of these themes as possible. (6)
The infinitesimal has, in general, been very vaguely defined. It has been regarded as a number or magnitude which, though not zero, is less than any finite number or magnitude. It has been the dx or dy of the Calculus, the time during which a ball thrown vertically upwards is at rest at the highest point of its course, the distance between a point on a line and the next point, etc., etc. But none of these notions are at all precise. The dx and dy, as we saw in the last chapter, are nothing at all: dy/dx is the limit of a fraction whose numerator and denominator are finite, but is not itself a fraction at all. The time during which a ball is at rest at its highest point is a very complex notion, involving the whole philosophic theory of motion; in Part VII we shall find, when this theory has been developed, that there is no such time. The distance between consecutive points presupposes that there are consecutive points—a view which there is every reason to deny. And so with most instances—they afford no precise definition of what is meant by the infinitesimal.
[336]
There is, so far as I know, only one precise definition, which renders the infinitesimal a purely relative notion, correlative to something arbitrarily assumed to be finite. When, instead, we regard what had been taken to be infinitesimal as finite, the correlative notion is what Cantor calls the improper infinite (Uneigentlich-Unendliches). The definition of the relation in question is obtained by denying the axiom of Archimedes, just as the transfinite was obtained by denying mathematical induction. If P, Q be any two numbers, or any two measurable magnitudes, they are said to be finite with respect to each other when, if P be the lesser, there exists a finite integer n such that nP is greater than Q. The existence of such an integer constitutes the axiom of Archimedes and the definition of relative finitude. It will be observed that it presupposes the definition of absolute finitude among numbers—a definition which, as we have seen, depends upon two points, (1) the connection of 1 with the logical notion of simplicity, or of 0 with the logical notion of the null-class; (2) the principle of mathematical induction. The notion of relative finitude is plainly distinct from that of absolute finitude. The latter applies only to numbers, classes and divisibilities, whereas the former applies to any kind of measurable magnitude. Any two numbers, classes, or divisibilities, which are both absolutely finite are also relatively finite; but the converse does not hold. For example, ω and ω.2, an inch and a foot, a day and a year, are relatively finite pairs, though all three consist of terms which are absolutely infinite.
[337]
The definition of the infinitesimal and the improper infinite is then as follows. If P, Q be two numbers, or two measurable magnitudes of the same kind, and if, n being any finite integer whatever, nP is always less than Q, then P is infinitesimal with respect to Q, and Q is infinite with respect to P. With regard to numbers, these relative terms are not required; for if, in the case supposed, P is absolutely finite, then Q is absolutely infinite; while if it were possible for Q to be absolutely finite, P would be absolutely infinitesimal—a case, however, which we shall see reason to regard as impossible. Hence I shall assume in future that P and Q are not numbers, but are magnitudes of a kind of which some, at least, are numerically measurable. It should be observed that, as regards magnitudes, the axiom of Archimedes is the only way of defining, not only the infinitesimal, but the infinite also. Of a magnitude not numerically measurable, there is nothing to be said except that it is greater than some of its kind, and less than others; but from such propositions infinity cannot be obtained. Even if there be a magnitude greater than all others of its kind, there is no reason for regarding it as infinite. Finitude and infinity are essentially numerical notions, and it is only by relation to numbers that these terms can be applied to other entities.
[337]
The next question to be discussed is: What instances of infinitesimals are to be found? Although there are far fewer instances than was formerly | supposed, there are yet some that are important. To begin with, if we have been right in regarding divisibility as a magnitude, it is plain that the divisibility of any whole containing a finite number of simple parts is infinitesimal as compared with one containing an infinite number. The number of parts being taken as the measure, every infinite whole will be greater than n times every finite whole, whatever finite number n may be. This is therefore a perfectly clear instance. But it must not be supposed that the ratio of the divisibilities of two wholes, of which one at least is transfinite, can be measured by the ratio of the cardinal numbers of their simple parts. There are two reasons why this cannot be done. The first is, that two transfinite cardinals do not have any relation strictly analogous to ratio; indeed, the definition of ratio is effected by means of mathematical induction. The relation of two transfinite cardinals α, γ expressed by the equation αβ = γ bears a certain resemblance to integral ratios, and αβ =γδ may be used to define other ratios. But ratios so defined are not very similar to finite ratios. The other reason why infinite divisibilities must not be measured by transfinite numbers is, that the whole must always have more divisibility than the part (provided the remaining part is not relatively infinitesimal), though it may have the same transfinite number. In short, divisibilities, like ordinals, are equal, so long as the wholes are finite, when and only when the cardinal numbers of the wholes are the same; but the notion of magnitude of divisibility is distinct from that of cardinal number, and separates itself visibly as soon as we come to infinite wholes.
[337-338]
Two infinite wholes may be such that one is infinitely less divisible than the other. Consider, for example, the length of a finite straight line and the area of the square upon that straight line; or the length of a finite straight line and the length of the whole straight line of which it forms part (except in finite spaces); or an area and a volume; or the rational numbers and the real numbers; or the collection of points on a finite part of a line obtainable by von Staudt’s quadrilateral construction, and the total collection of points on the said finite part.* All these are magnitudes of one and the same kind, namely divisibilities, and all are infinite divisibilities; but they are of many different orders. The points on a limited portion of a line obtainable by the quadrilateral construction form a collection which is infinitesimal with respect to the said portion; this portion is ordinally infinitesimal† with respect to any bounded area; any bounded area is ordinally infinitesimal with respect to any bounded volume; and any bounded volume (except in finite spaces) is ordinally infinitesimal with respect to all space. In all these cases, the word infinitesimal is used strictly according to the above definition, obtained from the axiom of Archimedes. What makes these various | infinitesimals somewhat unimportant, from a mathematical standpoint, is, that measurement essentially depends upon the axiom of Archimedes, and cannot, in general, be extended by means of transfinite numbers, for the reasons which have just been explained. Hence two divisibilities, of which one is infinitesimal with respect to the other, are regarded usually as different kinds of magnitude; and to regard them as of the same kind gives no advantage save philosophic correctness. All of them, however, are strictly instances of infinitesimals, and the series of them well illustrates the relativity of the term infinitesimal.
[338-339]
An interesting method of comparing certain magnitudes, analogous to the divisibilities of any infinite collections of points, with those of continuous stretches is given by Stolz,* and a very similar but more general method is given by Cantor.† These methods are too mathematical to be fully explained here, but the gist of Stolz’s method may be briefly explained. Let a collection of points x' be contained in some finite interval a to b. Divide the interval into any number n of parts, and divide each of these parts again into any number of parts, and so on; and let the successive divisions be so effected that all parts become in time less than any assigned number δ. At each stage, add together all the parts that contain points of x' . At the mth stage, let the resulting sum be Sm. Then subsequent divisions may diminish this sum, but cannot increase it. Hence as the number of divisions increases, Sm must approach a limit L. If x' is compact throughout the interval, we shall have L = b − a; if any finite derivative of x' vanishes, L = 0. L obviously bears an analogy to a definite integral; but no conditions are required for the existence of L. But L cannot be identified with the divisibility; for some compact series, e.g. that of rationals, are less divisible than others, e.g. the continuum, but give the same value of L.
[339]
The case in which infinitesimals were formerly supposed to be peculiarly evident is that of compact series. In this case, however, it is possible to prove that there can be no infinitesimal segments,‡ provided numerical measurement be possible at all—and if it be not possible, the infinitesimal, as we have seen, is not definable. In the first place, it is evident that the segment contained between two different terms is always infinitely divisible; for since there is a term c between any two a and b, there is another d between a and c, and so on. Thus no terminated segment can contain a finite number of terms. But segments defined by a class of terms may (as we saw in Chapter 34) have no limiting term. In this case, however, provided the segment does not consist of a single term a, it will contain some other term b, and therefore an infinite number of terms. Thus all segments are infinitely divisible. The next | point is to define multiples of segments. Two terminated segments can be added by placing a segment equal to the one at the end of the other to form a new segment; and if the two were equal, the new one is said to be double of each of them. But if the two segments are not terminated, this process cannot be employed. Their sum, in this case, is defined by Professor Peano as the logical sum of all the segments obtained by adding two terminated segments contained respectively in the two segments to be added.* Having defined this sum, we can define any finite multiple of a segment. Hence we can define the class of terms contained in some finite multiple of our segment, i.e. the logical sum of all its finite multiples. If, with respect to all greater segments, our segment obeys the axiom of Archimedes, then this new class will contain all terms that come after the origin of our segment. But if our segment be infinitesimal with respect to any other segment, then the class in question will fail to contain some points of this other segment. In this case, it is shown that all transfinite multiples of our segment are equal to each other. Hence it follows that the class formed by the logical sum of all finite multiples of our segment, which may be called the infinite multiple of our segment, must be a non-terminated segment, for a terminated segment is always increased by being doubled. “Each of these results”, so Professor Peano concludes, “is in contradiction with the usual notion of a segment. And from the fact that the infinitesimal segment cannot be rendered finite by means of any actually infinite multiplication, I conclude, with Cantor, that it cannot be an element in finite magnitudes” (p. 62). But I think an even stronger conclusion is warranted. For we have seen that, in compact series, there is, corresponding to every segment, a segment of segments, and that this is always terminated by its defining segment; further that the numerical measurement of segments of segments is exactly the same as that of simple segments; whence, by applying the above result to segments of segments, we obtain a definite contradiction, since none of them can be unterminated, and an infinitesimal one cannot be terminated.
[339-340]
In the case of the rational or the real numbers, the complete knowledge which we possess concerning them renders the non-existence of infinitesimals demonstrable. A rational number is the ratio of two finite integers, and any such ratio is finite. A real number other than zero is a segment of the series of rationals; hence if x be a real number other than zero, there is a class u, not null, of rationals such that, if y is a u, and z is less than y, z is an x, i.e. belongs to the segment which is x. Hence every real number other than zero is a class containing rationals, and all rationals are finite; consequently every real number is finite. Consequently if it were possible, in any sense, to speak of infinitesimal numbers, it would have to be in some radically new sense.
[340]
I come now to a very difficult question, on which I would gladly say nothing—I mean, the question of the orders of infinity and infinitesimality of functions. On this question the greatest authorities are divided: Du Bois Reymond, Stolz, and many others, maintaining that these form a special class of magnitudes, in which actual infinitesimals occur, while Cantor holds strongly that the whole theory is erroneous. To put the matter as simply as possible, consider a function f(x) whose limit, as x approaches zero, is zero. It may happen that, for some finite real number α, the ratio f(x)/xα has a finite limit as x approaches zero. There can be only one such number, but there may be none. Then α, if there is such a number, may be called the order to which f(x) becomes infinitesimal, or the order of smallness of f(x) as x approaches zero. But for some functions, e.g. 1/log x, there is no such number α. If α be any finite real number, the limit of 1/xα logx, as x approaches zero, is infinite. That is, when x is sufficiently small, 1/xα log x is very large, and may be made larger than any assigned number by making x sufficiently small—and this whatever finite number α may be. Hence, to express the order of smallness of 1/log x, it is necessary to invent a new infinitesimal number, which may be denoted by 1/g. Similarly we shall need infinitely great numbers to express the order of smallness of (say) e−1/x as x approaches zero. And there is no end to the succession of these orders of smallness: that of 1/log (log x), for example, is infinitely smaller than that of 1/log x, and so on. Thus we have a whole hierarchy of magnitudes, of which all in any one class are infinitesimal with respect to all in any higher class, and of which one class only is formed of all the finite real numbers.In this development, Cantor finds a vicious circle; and though the question is difficult, it would seem that Cantor is in the right. He objects (loc. cit.) that such magnitudes cannot be introduced unless we have reason to think that there are such magnitudes. The point is similar to that concerning limits; and Cantor maintains that, in the present case, definite contradictions may be proved concerning the supposed infinitesimals. If there were infinitesimal numbers j, then even for them we should haveLimx = 0 1/ (log x. xj) = 0since xj must ultimately exceed ½. And he shows that even continuous, differentiable and uniformly growing functions may have an entirely ambiguous order of smallness or infinity: that, in fact, for some such functions, this order oscillates between infinite and infinitesimal values, according to the manner in which the limit is approached. Hence we may, I think, conclude that these | infinitesimals are mathematical fictions. And this may be reinforced by the consideration that, if there were infinitesimal numbers, there would be infinitesimal segments of the number-continuum, which we have just seen to be impossible. [341-342]
Thus to sum up what has been said concerning the infinitesimal, we see, to begin with, that it is a relative term, and that, as regards magnitudes other than divisibilities, or divisibilities of wholes which are infinite in the absolute sense, it is not capable of being other than a relative term. But where it has an absolute meaning, there this meaning is indistinguishable from finitude. We saw that the infinitesimal, though completely useless in mathematics, does occur in certain instances—for example, lengths of bounded straight lines are infinitesimal as compared to areas of polygons, and these again as compared to volumes of polyhedra. But such genuine cases of infinitesimals, as we saw, are always regarded by mathematics as magnitudes of another kind, because no numerical comparison is possible, even by means of transfinite numbers, between an area and a length, or a volume and an area. Numerical measurement, in fact, is wholly dependent upon the axiom of Archimedes, and cannot be extended as Cantor has extended numbers. And finally we saw that there are no infinitesimal segments in compact series, and—what is closely connected—that orders of smallness of functions are not to be regarded as genuine infinitesimals. The infinitesimal, therefore—so we may conclude—is a very restricted and mathematically very unimportant conception, of which infinity and continuity are alike independent. [342]
by Corry Shores
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Bertrand Russell
Principles of Mathematics
Part 5: Infinity and Continuity
Ch.39: The Infinitesimal Calculus
Brief Summary
Leibniz’ infinitesimal calculus used a concept of the infinitely small (the infinitesimal quantity). Russell explains how differential and integral calculus now function with the concept of limit rather than the concept of infinitesimal.
Summary
§303
‘Infinitesimal calculus’ refers to differential and integral calculus, however “there is no allusion to, or implication of, the infinitesimal in any part of this branch of mathematics.” [330]
Leibniz was its inventor, but he considered it to be more practically applicably than metaphysically truthful. “He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation”. [330]
But because Leibniz believed in the actual infinitesimal, he was unable to see that calculus rests on the doctrine of limits. Newton’s fluxions are closer to this truer foundation.
When he was thinking of Dynamics, his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead. And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules. At other times, it is true, he definitely rejects infinitesimals as philosophically valid; but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and | not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor’s sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. [330-331]
Leibniz’ error has misled philosophers and mathematicians from his time to Weierstrass.
it is at any rate certain that, in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus. [331]
§304
“The differential coefficient depends essentially upon the notion of a continuous function of a continuous variable”. [331]
In §254 we noted that a function relates the elements of one set to those of another, usually in an order-preserving way. [Also see Edwards and Penney’s account of functions here.]
And in §277 we defined a continuum as a dense series of terms whose values are all definable by means of limits contained with in it. [Russell says we examined continuous variables in this chapter. There is no use of this term, just continuum and continuous series, which might therefore be equivalent.] If the function is one-valued and ordered correlatively with a continuous variable, then the function is continuous. [332d]. But if the function has an order indepent of correlation, it could possible be that the series obtained in its correlation is not continuous. When the correlation does produce a continuous series in some interval, then the function is continuous in that interval. [From the informal and formal definitions, it seems that a function is continuous at a point when the limits to either side of that point have equal value. If we were thinking in infinitesimal terms, if we were to move infinitesimally to the right or left of the point, it would be the same value (except for the infinitesimal, inassignable difference). But it is discontinuous when the side-limits are different. As Mr. Flatcher writes “The graph of a continuous function has no holes, jumps, or gaps. Think of a continuous function as one that you can graph without ever lifting your pencil.” (Mr. Flatcher / Flatchermatics) Consider for example this function.
At x = 1, the limits on either of its sides should be 0. However, as we can see, the y values are much different from zero.
(Image and function from ‘Classification of Discontinuities’, wikipedia)
As you can see, at the limit right before x = 1, the y value is greater than zero, and at the limit right after, the y value is even greater than that. Below we have an animation showing a transition from continuity to discontinuity. The caption reads: “A sequence of continuous functions fn(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform.”
(Animated diagram and above caption from ‘Continuous function’, wikipedia)
]
If the function is one-valued, and is only ordered by correlation with the variable, then, when the variable is continuous, there is no sense in asking whether the function is continuous; for such a series by correlation is always ordinally similar to its | prototype. But when, as where the variable and the field of the function are both classes of numbers, the function has an order independent of correlation, it may or may not happen that the values of the function, in the order obtained by correlation, form a continuous series in the independent order. When they do so in any interval, the function is said to be continuous in that interval. The precise definitions of continuous and discontinuous functions, where both x and f(x) are numerical, are given by Dini as follows. The independent variable x is considered to consist of the real numbers, or of all the real numbers in a certain interval; f(x), in the interval considered, is to be one-valued, even at the end-points of the interval, and is to be also composed of real numbers. We then have the following definitions, the function being defined for the interval between α and β, and ɑ being some real number in this interval.
“We call f(x) continuous for x = ɑ, or in the point ɑ, in which it has the value f(ɑ), if for every positive number σ, different from 0, but as small as we please, there exists a positive number ε, different from 0, such that, for all values of δ which are numerically less than ε, the difference f(ɑ + δ) − f(ɑ) is numerically less than σ. In other words, f(x) is continuous in the point x = ɑ, where it has the value f(ɑ), if the limit of its values to the right and left of a is the same, and equal to f(ɑ).”
“Again, f(x) is discontinuous for x = ɑ, if, for any positive value of σ, there is no corresponding positive value of ε such that, for all values of δ which are numerically less than ε, f(ɑ + δ) − f(ɑ) is always less than σ; in other words, f(x) is discontinuous for x = ɑ, when the values f(a + h) of f(x) to the right of a, and the values f(ɑ − h) of f(x) to the left of ɑ, the one and the other, have no determinate limits, or, if they have such, these are different on the two sides of ɑ; or, if they are the same, they differ from the value f(ɑ), which the function has in the point ɑ.” [331-332]
But the limit of a function is slightly different than the limit in general (of series) that we have discussed so far. Russell defines the limit in this way. [Put in simple terms, it seems that the limit is the value most immediate to the point, were we using infinitesimal terms.]
A function of a perfectly general kind will have no limit as it approaches any given point. In order that it should have a limit as x approaches a from the left, it is necessary and sufficient that, if any number ε be mentioned, any two values of f(x), when x is sufficiently near to a, but | less than a, will differ by less than ε; in popular language, the value of the function does not make any sudden jumps as x approaches a from the left. Under similar circumstances, f(x) will have a limit as it approaches a from the right. But these two limits, even when both exist, need not be equal either to each other or to f(ɑ), the value of the function when x = ɑ. The precise condition for a determinate finite limit may be thus stated:
“In order that the values of y to the right or left of a finite number a (for instance to the right) should have a determinate and finite limit, it is necessary and sufficient that, for every arbitrarily small positive number σ, there should be a positive number ε, such that the difference yɑ + ε − yɑ+ δ between the value yɑ + ε of y for x = a + ε, and the value yɑ + δ, which corresponds to the value a + δ of x, should be numerically less than σ, for every δ which is greater than 0 and less than ε.” It is possible, instead of thus defining the limit of a function, and then discussing whether it exists, to define generally a whole class of limits. In this method, a number z belongs to the class of limits of y for x = ɑ, if, within any interval containing ɑ, however small, y will approach nearer to z than by any given difference. Thus, for example, sin 1/x, as x approaches zero, will take every value from −1 to +1 (both inclusive) in every finite interval containing zero, however small. Thus the interval from −1 to +1 forms, in this case, the class of limits for x = 0. This method has the advantage that the class of limits always exists. It is then easy to define the limit as the only member of the class of limits, in case this class should happen to have only one member. This method seems at once simpler and more general.
[332-333]
§305
Russell will now discuss the derivative or differential coefficient of the function. [To better grasp Russell’s example, we will draw from our summary of one of David Jerrison’s class lectures on the differential. We will find an equivalent formulation to Russell’s so that we can make what he is saying more concrete. So first let’s understand the formulation. Consider a curve with point P.
The line has a different slope (tendency of variation) at each point. We will ask, what is its slope at x0, or point P? We determine the y value on the basis of the function f(x). And since P = (x,y), then P = (x0,(fx0)). Our calculation will involve looking at the change in x to the change in y (or change in f we might say).
The slope is found at the limit as Δx goes to zero.
We see the coordinates given here:
Slope is rise-over-run, or Δy / Δx. So
m = (y2 – y1) / (x2 – x1)
or in our case,
m = (f2 – f1) / (x2 – x1)
Notice in the above diagram that the y values in P and Q are: f(x0) and (f(x0 + Δx). For (x2 – x1) we only need Δx. So if we substitute these values into the slope formula, we have:
(f(x0 + Δx) – f(x0)) / Δx
The derivative we will denote as f’(x0). Thus
We call that formulation “the difference quotient”.
Now let’s use a specific function.
f(x) = 1 / x
We want to find the derivative at x0, and the dotted line is the tangent whose slope we seek. So we need to find Δf / Δx. The formula for Δf was (f(x0 + Δx) – f(x0)). When we plug in our function, we (multiplicatively) invert the x values, that is, put a one over them, hence we obtain [18]:
When we remove the embedded fractions (by dropping the top couched-denominators to the entire bottom denominator), we get:
![[D14.MIT.fill.1.14.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPrBPVrzKrGYqwuuQWweAQAeLnIl5_Zc-O-Mvc5_WMDvZgnV0Mn2ElYSB_DcEQvMEg2VaqFaY7_Uz_C2hjEJDA5RXIiaw_Ci-_a6Z5dYX9vqs-MZIuCq3xj00oTfMsLCn038X0dq5o7vs/s1600/D14.MIT.fill.1.14.jpg)
As you can see, 1 / Δx is common to both parts. When we factor it out, we get:
![[D15.MIT.fill.2.15.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhneOS4lCrPBHvgwH6hLPqjCA-V83_8QFTbWtWly9G-v9DcNybd7crPxiOiyHmYtHivEVs1WfkgT3Vyx6oT1Cz1o__cgoolMB3uqx5hIHOIiosWgNeSXWiTnGYjhM77nlLz1OVGpTs2o8E/s1600/D15.MIT.fill.2.15.jpg)
The subtracted parts need a common denominator for us to simplify them. To give them both a common denominator, we multiply each by the other’s denominate set over itself (thus equaling 1).
![[D16.MIT.fill.3.16.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG5ISFZUc_DVmGFnQ5ieyYRoLa8YbyV7J8eNQH7kp_bugY4cd3gvufiSU4_FElj-w4RFJ4apM2wHbb1wt32KzIsON9m8RSddEDBqCa4sPnaLKXOG2DV-DinW7obcgGxm4puR55lO_FeQE/s1600/D16.MIT.fill.3.16.jpg)
Let’s combine these figures to get:
![[D17.MIT.fill.4.1.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTRrbIEWZr3Nlc3c5gxrIEuR1PLa3_YWXS2FNtoHdxoTLtYfF34ibTUyySW4TaoIMlXrzezL19NoTNVUNDYupxK9dJVe0yrbfStvKXDiQ6XYQdFbOoNP9u5Vu_VtxbGvnSXLPCmx5kmH0/s1600/D17.MIT.fill.4.1.jpg)
We can now subtract the terms:
![[D18.MIT.fill.5.18.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjf4IgVzZMI9qTx-0zCn2NSCegVk5B9YkoBoZUisNVZWejh_XPNv0tjGUVmNcjjHlYThaCKQf5YMPASO2d7RAJ0rbL8NRrQNEYi3LuDw758d3_8po3VVIHYqCy6sLbVwZtqwFxVYMvboCk/s1600/D18.MIT.fill.5.18.jpg)
We then distribute the negative in the right side of the numerator to make x0 – x0 –Δx, thereby leaving –Δx; hence:
![[D18.5.MIT.PDF.first.after.own.fill.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrnqUGWlmGzVYcgavlubHIV30HAPRyre3A-3c_Itna3M0KfHd8EJ5fHEB4y14PrG94oLqY6xwRxL1XA00DZZDtPdWw4cANnMJiXxONwravdGuk54DR02nQCjRvMjmAhUDHoFwhgdjpbso/s1600/D18.5.MIT.PDF.first.after.own.fill.jpg)
Since both sides share Δx inversely, we can cancel them, leaving us with 1/1 on the left side, which can as well be eliminated, and remaining on the left is:
![[D18.7.MIT.PDF.second.after.own.fill.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_-pSxK8YJ_LHDIP8z1hD5cKnNSWSsoTM_RaPOtHvUkvM4JQA8qJVTUGmHSI_dA2Fhw-h2gH_dqvx6jIGUGhvwHWz-s3qea62LVtighR_pUk5JyfynZVjzzl6gEEuA0cRbCy20SdY-zbg/s1600/D18.7.MIT.PDF.second.after.own.fill.jpg)
The last step is to take the limit, as delta tends to zero, and substitute zero for Δx. We can do this now, because before the numerator and denominator gave us number divided by zero, which is undefined. But through algebraic operations, we were able to make the Δx negate-out of the equation without leaving a zero in the denomenator. Thus we now substitute-in the limit, that is, make Δx equal zero, leaving us with:
![[D18.9.MIT.PDF.third.after.own.fill.jpg]](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioMnb9EEqjC0y5gNkk_mfrPY841uXdG0MSZNX6MFgsPz_8eyvc5PTbl9U9XwMT-hUMYzpNpYISKcKHRs7HjK8j-ESScmKZk5xKyk5Us7wRUa70Kiarz9rK0_CFu68M6L9QtCcwqzevC-I/s1600/D18.9.MIT.PDF.third.after.own.fill.jpg)
Let’s put all of this together into one large formulation:
We then compare with our chart. This is negative, and likewise the slope is negative. Also, as xo goes to infinity, so as x moves to the right, it becomes less steep:
As we will see, Russell uses a very similar formulation, except with δ instead of our Δx]
If f(x) be a function which is finite and continuous at the point x, then it may happen that the fraction
{f(x + δ) − f(x)}/δ
has a definite limit as δ approaches to zero. If this does happen, the limit is denoted by f '(x), and is called the derivative or differential of f(x) in the point x. If, that is to say, there be some number z such that, given any number ε however small, if δ be any number less than some number η, but positive, then {f(x ± δ) − f(x)}/ ± δ differs from z by less than ε, then z is the derivative of f(x) in the point x. If the limit in question does not exist, then f(x) has no derivative at the point x. If f(x) be not continuous at this point, the limit does not exist; if f(x) be continuous, the limit may or may not exist.
[334]
§306
[Russell will say that the notion of the infinitesimal was not used in this definition. This probably results from the notion of the limit as standing outside the series, and the values approaching can only ever get closer and closer with no final value. I challenge this view, because it implies the value of the interval between the limit and the series that approaches it is finite. Let’s take this idea that between any two values is a middle value, infinitely. So a series of diminishing values perhaps could be something like 1/1, 1/2, 1/4, 1/8. What is important regarding Cantor’s infinity is that the cardinal value for infinity α0 is not among the natural numbers. It is the limit to which their law of genesis implicitly strives toward but does not precisely attain. But so long as the interval is finite, which Russell insists it must be, then does it fulfill the definition for “given any number ε however small”? It seems here the idea is not “the interval is so small it is infinitely small and thus continuous with zero” but rather “the interval is very small but still finite, yet it is close enough to zero that we can substitute one for the other.” Perhaps this is where the term “arbitrarily” small comes from. Is it strange that this ‘fudging’ sort of operation where we exchange a finite value for zero is considered more precise than when we think of this value as infinitely small, especially since in both cases we assume that there are an infinity of subdivisions? If intervals really are infinitely sudividable, why is it so hard to conceive of the intervals between them as being infinitely small? If there were not infinitely small, then they would be finitely small, and an infinity of them would compose an infinitely large interval. But an infinity of infinitely small values could conceivably compose a finite interval (for we would multiply infinity times one over infinity, equaling a finite unit after the infinities cancel.)] Russell emphasizes that the fact he has defined the derivative using limits and not infinitesimals is philosophically the most important part of his treatment on calculus. [The philosophical implication of this might be that the law of continuity does not hold, and thus change is not a matter paradoxically co-given contrary states. Also, change or motion would be ‘at-at’; infinitesimal intervals suggest ‘between-between’ or ‘at-at plus at-at’. I think Russell insists on this philosophical point for logical reasons. The problem with the infinitesimal calculus and its law of continuity is that it is a dialetheia, a true contradiction, and Russell will not allow exceptions to his rigid logic of perfect self-consistency. Because of the advances in dialetheic logic, it is no longer illogical to say that change is inherently paradoxical. And on account of the invention of non-standard analysis, it is no longer conceptually sloppy to use the notion of the infinitesimal in calculus. So here too is equally my greatest point of emphasis and the purpose for all these mathematical technicalities: despite Russell’s insistence, we can have a dialetheic between-between theory of motion, and it will not have such oddities like we find in Russell’s account, such as an infinity of finite intervals composing a finite interval and not an infinite one, and a moving object always being in no state other than rest, just rest in different places at different times.]
The only point which it is important to notice at present is, that there is no implication of the infinitesimal in this definition. The number δ is always finite, and in the definition of the limit there is nothing to imply the contrary. In fact, {f(x + δ) − f(x)}/δ, regarded as a function of δ, is wholly indeterminate when δ = 0. The limit of a function for a given value of the independent variable is, as we have seen, an entirely different notion from its value for the said value of the independent variable, and the two may or may not be the same number. In the present case, the limit may be definite, but the value for δ = 0 can have no meaning. Thus it is the doctrine of limits that underlies the Calculus, and not any pretended use of the infinitesimal. This is the only point of philosophic importance in the present subject, and it is only to elicit this point that I have dragged the reader through so much mathematics. [383]
[In the above, it seems Russell is absolutely clear that the small interval does not equal 0, but it also does not equal an infinitesimally small value. Rather, it equals a finite value that is as small as you want it to be.]
§307
[In Russell’s description of the definite integral, we divide the interval up into n portions. We find their ‘areas’ or products. Then we want the sum of all such interval areas or products. As we increase n, the sum tends toward a definite limit, which gives us the integral sum. For more on this operation, see David Jerison’s definite integral class or Edwards & Penney’s section on Riemann sums.]
Just as the derivative of a function is the limit of a fraction, so the definite integral is the limit of a sum. The definite integral may be defined as follows: Let f(x) be a function which is one-valued and finite in the interval α to β (both inclusive). Divide this interval into any n portions by means of the | (n − 1) points x1, x2, . . . xn − 1, and denote by δ1, δ2, . . . δn the n intervals x1 − α, x1 − x2, . . . β − xn − 1 . In each of these intervals, δs, take any one of the values, say f(ζs), which f(x) assumes in this interval, and multiply this value by the interval δs. Now form the sum
This sum will always be finite. If now, as n increases, this sum tends to one definite limit, however f(ζs) may be chosen in its interval, and however the intervals be chosen (provided only that all are less than any assigned number for sufficiently great values of n)—then this one limit is called the definite integral of f(x) from α to β . If there is no such limit, f(x) is not integrable from α to β.
[384-385]
§308
Russell now explains that neither the concept of the infinitesimal nor of infinity were used in this account of integral calculus. In fact, it is not even a sum [which might be impossible were the terms infinite in number]. Rather, this value is the limit of a sum [to boundary value to which its summing is tending]. But it never reaches that value. [If it did, then that could only be by means of an infinitesimally small increment between the limit and the next value near it. But so long as it never gets there but instead continually gets nearer without arriving, then it is made of many finite units. Imagine a segment that is built by adding 1/2 + 1/4 + 1/8 + etc., it is tending toward the total 1. See the divided square diagram on this page for an illustration (found near the end, under “geometric series that halves each time.”) We don’t need infinitely many to know it is tending to that limit. So we do not need the concepts of infinitesimal and infinity. But if we think that it does actually reach that limit, then it could only do so with an infinity of divisions with the smallest being infinitesimal.]
As in the case of the derivative, there is only one important remark to make about this definition. The definite integral involves neither the infinite nor the infinitesimal, and is itself not a sum, but only and strictly the limit of a sum. All the terms which occur in the sum whose limit is the definite integral are finite, and the sum itself is finite. If we were to suppose the limit actually attained, it is true, the number of intervals would be infinite, and the magnitude of each would be infinitesimal; but in this case, the sum becomes meaningless. Thus the sum must not be regarded as actually attaining its limit. But this is a respect in which series in general agree. Any series which always ascends or always descends and has no last term cannot reach its limit; other infinite series may have a term equal to their limit, but if so, this is a mere accident. The general rule is, that the limit does not belong to the series which it limits; and in the definition of the derivative and the definite integral we have merely another instance of this fact. The so-called infinitesimal calculus, therefore, has nothing to do with the infinitesimal, and has only indirectly to do with the infinite—its connection with the infinite being, that it involves limits, and only infinite series have limits.
[386]
Sources [unless otherwise notes, all bracket page citations are from]:
Bertrand Russell. Principles of Mathematics. London/New York: Routledge, 2010 [1st published 1903].
Otherwise:
Mr. Flatcher. Continuity and Differentiability.
http://fletchmatics.weebly.com/continuity-and-differentiability.html
Wikipedia. ‘Classification of Discontinuities’.
http://en.wikipedia.org/wiki/Classification_of_discontinuities
Wikipedia. ‘Continuous function’.
http://en.wikipedia.org/wiki/Continuous_function
