Bacon’s Pope, 1955, in Deleuze’s Francis Bacon commentary

 

by Corry Shores

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artimage.org.uk

]

 

[The following is quotation. My commentary is in brackets (except for painting numbers in the quotation). Proofreading is incomplete, so please forgive my typos.]

 

 

Francis Bacon

 

Pope, 1955

 

(Thanks source: artimage.org.uk)

 

Painting 57 of Deleuze’s
Francis Bacon: Logique de la sensation. Tome II - Peintures
Painting [19] of the English translation
and Painting [57] of the Seuil 2002 French

[Note: The date given for this painting in the French editions is 1954]

 

 

From the text:

Plus encore le sourire goguenard, presque intenable, insupportable, du Pape de 1954 ou de l’homme assis sur le lit : on sent qu’il doit survivre à l’effacement du corps. Les yeux et la bouche sont si bien pris sur les lignes horizontales du tableau que le visage se dissipe, au profit des coordonnées spatiales où seul subsiste le sourire insistant. [57, 59]

(Deleuze 1981a: 23c; 2002: 34a)

 

Furthermore, there is the scoffing, almost untenable, and insupportable smile of the 1954 Pope [19] or of the man sitting on the bed [11]: one senses that the smile will survive the effacement of the body. The eyes and the mouth are so completely caught up in the horizontal lines of the painting that the face is dissipated, in favor of the spatial coordinates in which only the insistent smile remains.

(Deleuze 2003a: 26a; 2003b: 28c; 2005: 21a)

 

Commentary:

[Deleuze discovers forces acting in and on the Figures in Bacon’s paintings. By means of these forces, the Figure’s body is trying to escape from itself through some opening in the body, like the mouth for example. As it is trying to escape itself, the body in a sense is dissipating or becoming effaced. The forces acting on and in the body are highly contortive. It is hard to imagine the Figure actually enjoying it or finding it pleasing in the sort of relaxed way that invites a smile. So it is odd to see the smiles on Bacon’s figures.

(askyfilledwithshootingstars.com)

(In this case, the smile seems to be scoffing, and it is “untemable” and “unsupportable” perhaps because such a smile could not endure for long under the tense conditions in the rest of the body.) So there is a disjunction of sorts between the smile and the body. The smile displays what is beginning to escape the body, because the contortive body behind the smile is what is beginning to dissipate. We are to recall the Cheshire Cat whose smile remains after the disappearance of its body.]

 

 

 

 

From the text:

 

L’hystérique, c’est á la fois celui qui impose sa présence, mais aussi celui pour qui les choses et les êtres sont présents, trop présents, et qui donne á toute chose et communique á tout être cet excès de présence. [...] Bacon peut dire avec humour que le sourire hystérique qu’il peint sur le portrait de 1953, sur la tête humaine de 1953, sur le pape de 1955, vient du « modèle » qui était « très nerveux, presque hystérique ». Mais c’est tout le tableau qui est hystérisé.

(Deleuze 1981a: 36c; 2002: 52c)

 

The hysteric is at the same time someone who imposes his or her presence, but also someone for whom things and beings are present, too present, and who attributes to every thing and communicates to every being this excessive presence. [...] Bacon explains rather testily that the hysterical smile he painted on the || 1953 portrait [11] , on the human head of 1953 [13] , and on the 1955 Pope [19] came from a “model” who was  very neurotic and almost hysterical.” But in fact it is the whole painting that is hystericized.

(Deleuze 2003a: 44b; 2003b: 50||51; 2005: 36c)

 

Commentary:

[When an image represents a situation, we might say that the image provides itself in its presence, and the represented element is given indirectly through the representation. Processing images representationally is something more or less cognitive and cerebral. Bacon’s paintings avoid representation while still providing intense sensations, and thus we are not distracted by the cognitive elements and instead are left with the raw presences of the image and its sensations. And we might also note that when we experience an overwhelming sensory experience, it attests both to the presence of those sensations as well as to the presence of whatever may be causing them, including the “forces” that are at work in imposing themselves upon our sensory apparatus. And given how overwhelming the sensations are, it attests to the excessiveness of this presence that has such a disruptive effect on our body’s workings. The Figures in Bacon’s paintings can be said to both themselves experience and express this excessive presence as well as causing us to have an overwhelming sensory experience, and thus they communicate that excessive presence to us. We might also find that in such experiences of sensory overload, our bodies begin to undergo confusions of many physical sorts, like certain components of the sensation being at odds with others. We may for example want to look away from the image in disgust while at the same time being drawn to it with rapt and eager curiosity. While under this mode of experience, we could perhaps notice certain traits of hysteria, coming especially from the lack of control of all the influences, tensions, sensations, and so on wrestling throughout our body. Deleuze’s point in these passages is that the hysteria of these Figures is something found not just in the smiles but throughout their bodies and in fact throughout the rest of the painting. For, the other elements in the painting are involved in the play of forces that give rise the Figure’s hystericized body and smile.]

 

 

Texts:

 

Deleuze, Gilles. 1981a. Francis Bacon: Logique de la sensation. Tome I. Paris: Éditions de la différence.

Deleuze, Gilles. 1981b. Francis Bacon: Logique de la sensation. Tome II - Peintures. Paris: Éditions de la différence.

 

Deleuze, Gilles. 2002. Francis Bacon: Logique de la sensation. Paris: Éditions du seuil.

 

 

Deleuze, Gilles. 2003a. Francis Bacon: The Logic of Sensation [with translator’s introduction (Smith’s “Deleuze on Bacon: Three Conceptual Trajectories in The Logic of Sensation”) and author’s introduction to the English edition]. Translated by Daniel W. Smith. Minneapolis: University of Minneapolis.

 

 

Deleuze, Gilles. 2003b. Francis Bacon: The Logic of Sensation [with translator’s preface, preface to the fourth edition by Alain Badiou and Barbara Cassin, author’s foreword, and author’s preface to the English edition]. Translated by Daniel W. Smith. Minneapolis: University of Minneapolis.

 

 

Deleuze, Gilles. 2005. Francis Bacon: The Logic of Sensation [with translator’s preface, preface to the fourth edition by Alain Badiou and Barbara Cassin, author’s foreword, and author’s preface to the English edition]. Translated by Daniel W. Smith. London/ New York: Continuum.

 

 

 

Images obtained gratefully from:

https://artimage.org.uk/13180/francis-bacon/pope--1955

 

.

Bacon’s Study after Velazquez’s Portrait of Pope Innocent X, 1953, in Deleuze’s Francis Bacon commentary

 

by Corry Shores

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

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[Francis Bacon (painter), entry directory]

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[Entry collecting the images of the paintings in this text.]

[I am profoundly grateful to the source of this image:

artinvest2000.com

Credits given at the end.]

 

[The following is quotation. My commentary is in brackets. Proofreading is incomplete, so please excuse my typos.]

 

 

Francis Bacon

 

Study after Velazquez’s Portrait of Pope Innocent X, 1953

 

(Thanks artinvest2000.com)

 

 

Painting 54 of Deleuze’s
Francis Bacon: Logique de la sensation. Tome II - Peintures
Painting [16] of the English translation
and Painting [54] of the Seuil 2002 French

 

 

From the text:

Tout le corps s’échappe par la bouche qui crie. Par la bouche ronde du pape [54] ou de la nourrice [55] le corps s’échappe comme par une artère.

(Deleuze 1981a: 23c; 2002: 33c)

 

The entire body escapes through the screaming mouth. The body escapes through the round mouth of the Pope or the nurse, as if through an artery [16, 24].

(Deleuze 2003a: 25c; 2003b: 28a; 2005:20d)

 

Commentary:

[In Bacon’s paintings, forces apply constricting pressure upon the Figures while at the same time there are forces internal to the Figure that are pushing outward, perhaps as a result of the constriction, like when squeezing a water balloon. On account of this play of forces, the Figure’s body tries to escape itself; and given the chance, it will dissipate through that escaping action. One exit for the escape is through the screaming mouth.

]

 

 

From the text:

À ce point extrême de la dissipation cosmique, dans un cosmos fermé mais illimité, c’est bien évident que la Figure ne peut plus être isolée, prise dans une limite, piste ou parallélépipède : on se trouve devant d’autres coordonnées. Déjà la Figure du pape [54] qui crie se tient derrière les lames épaisses, presque les lattes d’un rideau de sombre transparence : tout le haut du corps s’estompe, et ne subsiste que comme une marque sur un suaire rayé, tandis que le bas du corps reste encore hors du rideau qui s’évase. D’où l’effet d’éloignement progressif comme si le corps était tiré en arrière par la moitié supérieure. Et sur une assez longue période, le procédé est fréquent chez Bacon. Les mêmes lames verticales de rideau entourent et raient partiellement l’abominable sourire de « Étude pour un portrait [59] », tandis que la tête et le corps semblent aspirés || vers le fond, vers les lattes horizontales de la persienne.

(Deleuze 1981a: 24b; 2002: 34||35)

 

At this extreme point of cosmic dissipation, in a closed but unlimited cosmos, it is clear that the Figure can no longer be isolated or put inside a limit, a ring or parallelepiped: we are faced with different coordinates. The Figure of the screaming Pope [16] is already hidden behind the thick folds (which are almost laths) of a dark, transparent curtain: the top of the body is indistinct, persisting only as if it were a mark on a striped shroud, while the bottom of the body still remains outside the curtain, which is opening out. This produces the effect of a progressive elongation, as if the body were being pulled backwards by its upper half. For a rather long period of time, this technique appeared frequently in Bacon’s works. The same vertical curtain strips surround and partially line the abominable smile of Study for a Portrait [11], while the head and the body seem to sink into the background, into the horizontal slats of the blind.

(Deleuze 2003a: 26d; 2003b: 29b; 2005: 21c)

 

Commentary:

[We often see in Bacon’s paintings the Figure’s being confined within the limits of a geometrical structure, like a circular ring or platform or a rectilinear parallelepiped. But in this painting, which is during Bacon’s malerisch period, there are vertical smear stripes that seeming pull the Figure downward and sideward out of the bottom of its confining structure.

Bacon: Study after Velazquez’s Portrait of Pope Innocent X, 1953

(Thanks artinvest2000.com)

]

 

From the text:

Il faut considérer le cas spécial du cri. Pourquoi Bacon peut-il voir dans le cri l’un des plus hauts objets de la peinture [54] [55] ? « Peindre le cri… »

(Deleuze 1981a: 41a; 2002: 60b)

 

We must consider the special case of the scream. Why does Bacon think of the scream as one of the highest objects of painting? “Paint the scream . . .” [16, 24].

(Deleuze 2003a: 51a; 2003b: 60a; 2005: 42d)

 

Commentary:

[Recall from the first commentary the notion of the forces acting upon and within the Figure. To paint the scream is to paint those invisible forces.]

 

 

 

Texts:

 

Deleuze, Gilles. 1981a. Francis Bacon: Logique de la sensation. Tome I. Paris: Éditions de la différence.

Deleuze, Gilles. 1981b. Francis Bacon: Logique de la sensation. Tome II - Peintures. Paris: Éditions de la différence.

 

Deleuze, Gilles. 2002. Francis Bacon: Logique de la sensation. Paris: Éditions du seuil.

 

 

Deleuze, Gilles. 2003a. Francis Bacon: The Logic of Sensation [with translator’s introduction (Smith’s “Deleuze on Bacon: Three Conceptual Trajectories in The Logic of Sensation”) and author’s introduction to the English edition]. Translated by Daniel W. Smith. Minneapolis: University of Minneapolis.

 

 

Deleuze, Gilles. 2003b. Francis Bacon: The Logic of Sensation [with translator’s preface, preface to the fourth edition by Alain Badiou and Barbara Cassin, author’s foreword, and author’s preface to the English edition]. Translated by Daniel W. Smith. Minneapolis: University of Minneapolis.

 

 

Deleuze, Gilles. 2005. Francis Bacon: The Logic of Sensation [with translator’s preface, preface to the fourth edition by Alain Badiou and Barbara Cassin, author’s foreword, and author’s preface to the English edition]. Translated by Daniel W. Smith. London/ New York: Continuum.

 

 

 

Images obtained gratefully from:

http://www.artinvest2000.com/bacon_study-innocent-x.htm

 

.

Somers-Hall, (1.7), Deleuze’s Difference and Repetition, ‘1.7 The Eternal Return (40–2/50–2)’, 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.]

Summary of

Henry Somers-Hall

Deleuze’s Difference and Repetition:
An Edinburgh Philosophical Guide

Part 1
A Guide to the Text

 

Chapter 1. Difference in Itself

1.7 The Eternal Return (40–2/50–2)

Very brief summary:

By means of the Nietzschean idea of the eternal return, we may have a univocal understanding of being that would see difference as intensive. All being is variations in power between competing forces that express their will and potency at their fullest each moment. Such a purely affirmative view of the world would mean that we find every moment of our lives as perfect and as full as they can be, and thus we would want to relive these moments infinitely throughout eternity.

Brief summary:

In Nietzsche’s eternal return, we re-experience every moment of our lives exactly as we have, but infinitely more times throughout eternity. If we view the world as made of delimitable subjects that may make moral choices as to how much power to exert over others, then we would be inclined to think that certain circumstances in our lives could have been better somehow. We therefore would not want to affirm the eternal returnability of every aspect of our lives and self-experience. However, if we saw that fundamentally there are no such substantial divisions in our world but rather just forces competing and always expressing their will and power to their fullest in any instance, given their struggles with other forces, then we would see that every moment of our lives is as perfect as can possibly be. And for that reason, we would want to affirm the eternal returnability of our entire lives. By means of this concept of eternal return, then, we may have a univocal understanding of being, since all being is variations in differential power relations, while also maintaining an intensive view of differences in being.

 

Summary

[SH will draw from Nietzsche’s formulation of the eternal return in The Gay Science, §341.]

Nietzsche formulates the eternal return as follows [the following is Nietzsche quotation]:

What if some day or night a demon were to steal after you into your loneliest loneliness and say to you: ‘This life as you now live it and have lived it, you will have to live once more and innumerable times more’ . . . Would you not throw yourself down and gnash your teeth and curse the demon who spoke thus? Or have you once experienced a tremendous moment when you would have answered him: ‘You are a god and never have I heard anything more divine.’ (Nietzsche 2001: §341)
(SH 42)

SH identifies two questions regarding the eternal return that are relevant at this place in Difference and Repetition (DR). 1) What is eternally returning? and 2) what is the eternal return a test for? SH begins with the first question. We previously saw Deleuze’s two ways to understand how the world is made-up: a) sedentary distribution “understands the world as a collection of things with properties” (SH 42). But Deleuze is not interested in this sort of understanding of the world as based on identities, so we will not regard what is returning as being a subject as a center of identity. Rather, b) “Instead, what returns is the nomadic distribution” (43). [What SH says at this point I do not entirely grasp, but that is probably because he is giving a preview of the elaboration to come. Somehow what returns is “the intensive, nomadic distribution”. This has something to do with the ground for modes being a “a pre-judicative field of becoming” (43). Whatever does return is something prior to identity. But how it is that the distribution returns is not entirely clear to me, especially since we cannot understand the distribution itself as an identifiable thing (as it is in the sentence) that returns.]

For Deleuze, taking up the eternal return, the ground (or, as we shall see, unground) for modes is going to be a pre-judicative field of becoming: it is the intensive, nomadic distribution which returns. ‘Only the extreme, the excessive, returns; that which passes into something else and becomes identical . . . Eternal return or returning expresses the common being of all these metamorphoses . . . of all the realised degrees of power’ (DR 41/51). The priority of difference does not, therefore preclude the existence of identities, but asserts that what returns is not these identities themselves, but something prior to identity, which Deleuze characterises as difference.
(SH 43a.b)

The ‘test’ of the eternal return is whether or not we can only see the world as a sedentary distribution or if we can instead also see it as a nomadic distribution. [I do not understand clearly what this has to do with the ‘what we can do’ idea, but perhaps the idea is if we can see that nothing about our lives is ever lacking, since each moment and its conditions are an expression of power in its fullness. And I suppose if we see each moment as perfectly full and lacking in no way, that we would want to affirm it and desire to re-experience it for eternity.]

The eternal return appears as a test – whether we can bear the heaviest burden of the demon’s truth. What is this a test for? The lamb and the bird of prey both see the world in terms of different distributions; the former according to the sedentary distribution, the latter according to the nomadic distribution. In this case, deciding between them is straightforward, but it may be difficult to see whether something is governed by a sedentary or nomadic distribution. The eternal return allows us to differentiate those two classes. Only that which is pure affirmation, or which is not separated from what it can do, can truly will the repetition of everything that makes it what it is. Those who cannot affirm this do not have their ground in the affirmative field of differences, but are instead, like the lamb, grounded in the sedentary distribution. The fact that they make a distinction between what can be done and what is done (they posit agency), means that they as agents are not the same as their actions. For the lamb, therefore, positing its own return is not identical with positing the return of everything which is. The eternal return therefore allows us to differentiate ‘the superior form of everything that “is” ’ (DR 41/51) from those beings that are really not (as the sedentary distribution is not a well-founded way of understanding the world). In doing so, it allows us to characterise that set of entities which genuinely are, and are not merely secondary effects, just as the lamb’s attitude is a secondary effect of the bird of prey’s.
(SH 43)

 

 

Citations from:

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

Or if otherwise noted:

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

 

Nietzsche, Friedrich (2001), The Gay Science, trans. Josefine Nauckhoff, Cambridge: Cambridge University Press.

 

 

Somers-Hall, (1.6), Deleuze’s Difference and Repetition, ‘1.6 Nietzsche (36–7/45–7, 40–2/50–2, 52–5/63–7)’, summary

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Deleuze Entry Directory]
[Henry Somers-Hall, Entry Directory]
[Henry Somers-Hall’s Deleuze’s Difference and Repetition, Entry Directory]

 

[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.]

Summary of

Henry Somers-Hall

Deleuze’s Difference and Repetition:
An Edinburgh Philosophical Guide

Part 1
A Guide to the Text

 

Chapter 1. Difference in Itself

1.6 Nietzsche (36–7/45–7, 40–2/50–2, 52–5/63–7)

Very brief summary:

A Nietzschean univocal view of reality would say that there are not primarily independent beings choosing to exercise power morally or immorally. Rather, there are nothing more than competing forces in flux, all exercising their power purely affirmatively, since each force is exerted to the fullest it can be. This is a univocal understanding, since all being is understood as power. And, it is based on difference and affirmation.

Brief summary:

If we understand ‘being’ as power rather than as substantiality, then we can grasp it as univocal and affirmative. Deleuze builds from Nietzsche’s notion of subjectivity being a fabrication created by the weak in order to externalize blame for their weakness. In reality there are just competing forces expressing themselves at their fullest and finding relative value in their competitions. This is a nomadic understanding of what makes up the world. It sees the world as made of pure difference, that is, exclusively of differential relations between competing forces. And it is affirmative, since these powers are understood as being as great as they can be and never arbitrarily self-limiting. They are expressions of a pure affirmative will.  A sedentary understanding would instead section off regions in this field of differential power relations and say we have substances with different moral values, depending on how they seemingly choose to dominate others. This sees one thing as being defined by the limits that separate it from other things, thus it is based on negation rather than affirmation. Also, it can only come after the more basic differential field of change and becoming.

 

Summary

[We previously examined Spinoza’s univocal understanding of being. There is one substance (being/God), but it has an infinity of intensive modal variations/determinations and an infinity of essences. SH will now look at a critical problem Deleuze finds in this, which is that it still in a sense is like Aristotle’s genus-species predicative structure and thus may have its problems of representation. For Aristotle, the genus is a predicate to the species. For example, “Man is (a type of) animal).” Also, the genus is like a substance and the species are like determinations of it. So, an animal can be more determinately a man, a dog, etc. Spinoza’s system still places modes as determinations of being, and in that sense takes this form of (predicative) judgment and thus of representational thinking. I will venture a guess at how this might look. For example maybe we could say something like: ‘this modal variation in the attribute of extension, specifically, myself, is a modal variation of being’. In that way, being still functions like a genus to which we are a predicated determination.]

Despite the fact that Spinoza represents an advance over the work of Scotus, Deleuze claims that for Spinoza, ‘substance must be said itself of the modes and only of the modes. Such a condition can be satisfied only at the price of a more general categorical reversal according to which being is said of becoming, identity of that which is different, the one of the multiple, etc.’ (DR 40/50). Deleuze’s point is that the relation of modes to being is still structured like the terms of a judgement. The modes are said of being, in the same way that we might say of a man that he is rational, and so we still understand being as if it were a subject, even if we know that in reality it is singular rather than one, and thus different in kind from the object of a judgement.
(SH 38)

[While Spinoza (and similarly Scotus) provided an intensive understanding of difference, he still in a sense places being (God/substance) as the highest term in the hierarchy. So we turn now to Nietzsche.] “In order to overcome this limitation, we need somehow to replace our account of | being as the highest term in our hierarchy with difference, whilst retaining the insights given by the intensive understanding of difference; it is Nietzsche, Deleuze claims, who provides the means to do this” (SH 38-39). Although SH will return to Nietzsche later, he will in the following locate two parts of Nietzsche's writings that Deleuze bases his argument on: 1) section 13 of Essay I of Genealogy of Morals, “where Deleuze sees Nietzsche as opposing the subject–property view of reality,” and 2) aphorism 341 of the Gay Science “where Nietzsche presents the eternal return.” For Deleuze, Nietzsche’s eternal return is a univocal principle (SH 39).

In section 13 of Essay I of Genealogy of Morals, Nietzsche contrasts “two basic attitudes towards the world, that of the lamb and the bird of prey” (SH 39). The lamb is weak, and it views its predator birds as evil, since they gain dominance and advantage over the lambs. Whatever is opposite to the birds, that is, whatever is more like the lambs, would then be good in the eyes of the lambs. Their values then are not based on some intrinsic value to the birds’ traits and behaviors, but rather based on whatever is superior in power and more threatening to them. Consider instead good and evil from the perspective of the birds. The predator birds do not in this relation see any threats or superior powers. So there is no evil here. And in fact, the lambs are delicious to them. So lambs are not evil but rather good. [I am not sure if this next point is inferable from the prior ones we just made, but the next idea we move to in the Nietzsche quotation is that there are no subjects or actors. This is a mistake we make by using language (and its conceptual structures, like subject-predicate) to understand how events transpire.  For example, we see lightning and its flash. We conceptualize it (in certain Western languages) in a formulation like, “The lightning flashed.” This grammatical structure leads us to conceptualize the lighting as distinct from its flash, as if there is an actor that may choose one action or another. In reality there are just competitions of power operating according to quantities of force (drive, will, action). Perhaps in the case of lighting, we do not have a subject, lighting. Rather we have a competition of positive and negative charges, and as a result of that struggle of force quantities, there is the event of lightning-flash. But I am not sure if I am picturing this properly, since in this illustration, we seem to have two distinct actors, namely, the competing forces or ‘charges’. At any rate, the relevance here is that in reality there is an amoral situation. The powerful force prevails over the less powerful. But both are reciprocally determined in their mutual conflict, so more basically there is just force-differentiation. Language misleads us to misunderstand the situation. We think that there is a dominating party acting immorally upon a dominated party. We hold a supposed subject responsible for the immoral act. But really as we said there is no blame to be given and no subject who could receive blame anyway. Another idea here is that the more powerful side of the conflicts does not really have a choice to be less powerful. This may be because again there is no such actor who might have that option, or perhaps it is because the weak cannot help but be weak and similarly the strong cannot help but to be strong.]

Nietzsche presents a contrast between two basic attitudes towards the world, that of the lamb and the bird of prey [the following up to citation is quotation of Nietzsche Genealogy of Morals]:

There is nothing strange about the fact that lambs bear a grudge towards large birds of prey: but that is no reason to blame the large birds of prey for carrying off the little lambs. And if the lambs say to each other, ‘These birds of prey are evil; and whoever is least like a bird of prey and most like its opposite, a lamb, – is good, isn’t he?’, then there is no reason to raise objections to this setting-up of an ideal beyond the fact that the birds of prey will view it somewhat derisively, and will perhaps say: ‘We don’t bear any grudge at all towards these good lambs, in fact we love them, nothing is tastier than a tender lamb’ . . . A quantum of force is just such a quantum of drive, will, action, in fact it is nothing but this driving, willing and acting, and only the seduction of language (and the fundamental errors of reason petrified within it), which construes and misconstrues all actions as conditional upon an agency, a ‘subject’, can make it appear otherwise . . . no wonder, then, if the entrenched, secretly smouldering emotions of revenge and hatred put this belief to their own use and, in fact, do not defend any belief more passionately than that the strong are free to be weak, and the birds of prey are free to be lambs: – in this way, they gain the right to make the birds of prey responsible for being birds of prey. (Nietzsche 2006a: §13)

Nietzsche is here presenting an argument which combines moral and ontological aspects. The natural state of affairs is that of the bird of prey, who exercises his strength, and sees itself as good. The lamb, however, sees the bird of prey as evil, and therefore sees itself as good. The symmetry between these two positions is misleading, however, and each rests on fundamentally different ways of seeing the world. For the bird of prey, its action is simply an expression of its strength, or, in more Nietzschean terms, we might say that the bird of prey itself is an | expression of strength: ‘It is just as absurd to ask strength not to express itself as strength . . . as it is to ask weakness to express itself as strength’ (Nietzsche 2006a: §13).
(SH 39-40)

[The next paragraph continues ideas made in our prior bracketed commentary regarding grammar and subjectivity.]

The lamb’s reaction is a moral reaction, and one that is made possible by an illusion fostered by grammar: it posits a subject who is responsible for exercising its strength. Nietzsche gives the further example of lightning. When we say that ‘lightning strikes’, we are forced by the structure of language to posit a distinction between a subject (‘lightning’) and an act (‘striking’). Now we might recognise in this case that in fact there is nothing other to the lightning than its striking itself – there is no hidden subject behind the act – but language opens up a way of thinking of the world in terms of agents and actions. Once the lamb understands the bird of prey as an agent acting, he can posit the (illusory) possibility of the agent withholding this action. Thus, the bird of prey, once it is seen as a subject, becomes culpable for what it does.
(SH 40)

SH will now use this distinction [I think, between a subject-structured understanding of events and a force-conflict understanding] to explain Deleuze’s other distinction between sedentary and nomadic distributions. A distribution is a way “of thinking about what something is essentially, or more generally, what kinds of things the world is composed of” (SH 40). [Perhaps it is considered a ‘distribution’ because it is concerned with how compositional parts are arranged.] Our attention in the following will be working for a while on sedentary distributions, regarding which Deleuze writes, “‘A distribution of this type proceeds by fixed and proportional determinations which may be assimilated to “properties” or limited territories within representation’ (DR 36/45)” (SH 40). We are here reminded of Aristotle’s system of division which differentiates things on the basis of clear defining limits that define what is special and proper to each thing. In the above bird/lamb illustration, the lamb would be working with sedentary distributions, since it regards itself and the birds as different actors with distinct traits that have very different moral values (40).

We also note that in this Aristotelian sedentary conception, limits serve to define what makes one thing what it is and what makes something else not that thing but rather something different entirely. This notion of delimiting then makes us conceptualize notions spatially, as if one could occupy one part of conceptual space and have a limit, outside of which it does not belong but in where other concepts do [think for example of Venn diagrams]. It also involves a concept of negation [since we understand each concept in terms of how it is not the others, and vice versa]. [This conception also would have us judge the degree to which an actual instance fits with its proper concept. Perhaps for example we might characterize something as more typical or exemplary of its class.] “Finally, it provides ‘a hierarchy which measures beings according to their limits, and according to their proximity or distance from a principle’ (DR 36/46); in other words, according to how closely a being conforms with its essence or is a degenerate instance of it” (SH 41). This means that the sedentary distribution regards the world on the basis of opposition and negation.

A sedentary distribution therefore is a way of ordering the world that is hierarchical, and proceeds by the delimitation of the world according to oppositional determinations. The notion of difference is grounded in negation and operates according to a spatial metaphor.
(41)

The Aristotelian system regarded difference as being a matter of ‘this and not that’. Yet we saw how for Scotus and Spinoza we may regard difference not as difference in kind but rather as intensive qualitative variation, that is, having more or less a degree. We are not in these cases working with “a spatial conception of organisation” (41). Thus “Deleuze introduces the univocal conception of being in order to explain those features of the world which escaped something like an Aristotelian conception of the world” (41). [I do not completely grasp the next quotation about Oedipus. Perhaps it is similar to the idea we examined before that for Scotus, there is just an intensive difference between God’s infinite being and our own finite being, but still we make a difference-in-kind distinction between the two. Or we might also think that the difference between ice and water is quantitatively (and intensively) a small change, but qualitatively it is a complete difference in kind.]

Deleuze introduces the univocal conception of being in order to explain those features of the world which escaped something like an Aristotelian conception of the world. The nomadic distribution is intimately connected to this univocal conception: ‘Oedipus’ chorus cries: “which demon has leapt further than the highest leap?” The leap here bears witness to the unsettling difficulties that nomadic distributions introduce into the sedentary structures of representation’ (DR 37/46).
(SH 41)

[The next part is a bit confusing for me. My impression is that it is a reference to Deleuze’s Spinozistic notions of ‘what a body can do’, limit, and perfection. In this talk here Deleuze discusses the difference between spatialized limit and the limit of power or action. There is more discussion of ‘what can a body do’ here, and also in this and this talk. And see again especially Deleuze’s commentary on Spinoza’s correspondence with Blyenbergh regarding the perfection of essence and modal expression. It seems in the quoted Deleuze passage that instead of limit being understood spatially, we are to understand it more as a limit of action or power, a limit of what one body can do. One main idea in the Blyenbergh correspondence is that everything is doing everything that it can do in any given moment. If it seems to be weak, that is because it finds itself in a certain relation of forces, including the competing forces within it and also including the competing forces between itself and other external competing forces, in which it is expressing the fullest power it is able to have in that moment. Deleuze gives the example of meditating in the dark, and someone turns on the light suddenly. The person loses power, since the light dazzles them and makes them lose meditative concentration. The person is still functioning at their optimum even when blinded, but just under these circumstances the optimum is relatively lower than before. If however the person were looking for their glasses in the dark, and someone gradually turns up the light so they can better see them, then their power is increased relatively given that situation. These fluctuations in power are not however fluctuations in perfection. All is equally perfect in a sense, since everything is operating at its maximum (and also, from the perspective of substance itself, nothing is being subtracted from it even though its modal parts are varying in power in relation to one another). It is perhaps for this reason that Deleuze says in the quote to follow that “the smallest becomes equivalent to the largest”, in that all are equally affirmative of the power they have in any given situation. The point SH seems to be making here is that we do not think of the world as being made up of things that choose whether or not to exercise their power but rather we have competing forces in a world of fluctuating changes. SH’s further points become slightly more difficult for me to grasp. The next point is that for Deleuze, negation results from affirmation. To get to this point, perhaps we first recognize that the substance-property based ontology which fixes subjects who perform actions and express properties is based on negation, since it depends on definitional limits. But if we follow Deleuze’s Nietzschean-Spinozism, we would say that more primarily there are competitions of forces, whose limits are not spatially substantial but are rather intensive degrees of moreness or lessness. This is purely affirmative, since everything expresses its power in complete fullness, given the affective circumstances of their situation. Then secondly, on account of feelings of powerlessness, we externalize blame to subjectivities we fabricate in this indeterminate field of forces. (I still cannot make my account consistent regarding how it is that  we can have entities which place blame on other entities they fabricate but supposedly there never were any entities to begin with). So first the world in its basis is affirmative (in that there is will to express power in fullness each moment) and composed of pure difference (in the form of differential relations between competing forces). Only secondly and on a less fundamental level do negations arise in the form of fabricated or conceptualized limits between supposed agents wrongly understood to be making moral choices of whether or not to artificially limit their own power.]

If a sedentary distribution is fundamentally tied to an understanding of the world in terms of subjects and properties, how are we to understand this notion of a nomadic distribution? The key point is Deleuze’s claim that everything goes to the limit of what it can do. He elaborates on this as follows [the following up to citation is quotation of Deleuze]:

Here limit [peras] no longer refers to what maintains a thing under a law, nor what delimits or separates it from other things. On the contrary, it refers to that on the basis of which it is deployed and deploys all of its power; hubris ceases to | be simply condemnable and the smallest becomes equivalent to the largest once it is not separated from what it can do. (DR 37/46)

When we separate the bird of prey from its action, or lightning from its striking, we institute the two moments of an ontology of judgement: the subject and the property. This moment of separation of something from what it can do is what gives us the Aristotelian idea of a world of fixed things. If something is not separated from what it can do, then instead of an ontology of being, we have an ontology of forces, or becoming. There are not static points from which movement originates, but rather just movement itself. We can tie together a number of results at this stage. Just as Scotus shows that analogy can only operate within a prior univocal framework, Nietzsche shows that the point of view of the lamb is derivative of that of the bird of prey. Deleuze similarly argues that ‘negation results from affirmation: this means that negation arises in the wake of affirmation or beside it, but only as the shadow of the more profound genetic element – of that power or “will” which engenders the affirmation and the difference in affirmation’ (DR 55/67). Difference is therefore primary in this scheme. This leads us to the last aspect of Deleuze’s discussion of univocity: how are we to conceive of a univocal conception of becoming?
(SH 41-42)

 

 

 

Citations from:

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

Or if otherwise noted:

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

Nietzsche, Friedrich (2006a), On the Genealogy of Morality, trans. Carol Diethe, Cambridge: Cambridge University Press.

 

 

 

Pt3.Ch7.Sb2 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘Force and the Understanding.’ summary

by Corry Shores
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Henry Somers-Hall

 

Hegel, Deleuze, and the Critique of Representation.

Dialectics of Negation and Difference

 

Part 3: Beyond Representation

Chapter 7: Force, Difference, and Opposition

Subdivision 2: Force and the Understanding

Very Very Brief Summary:

Hegel could critique Deleuze’s actual and virtual by showing that the forces involved in each are dialectically inherent to one another, thus collapsing the ontological distinction on which Deleuze bases his transcendental empiricism.

Very Brief Summary:

A Hegelian critique of Deleuze’s ontology would show how Deleuze’s virtual and actual dialectically transform into one another. This might be done on the basis of Hegel’s philosophy of force, which would say that Deleuze’s sensible actual is the flux of differences, which come about from a multiplying force, and the supersensible virtual is the persistence of that flux, and comes about through a unifying force. So we would say there is some force that makes Deleuze’s virtual always the supersensible conditions of appearing, and another force that makes the stable virtual continually actualize differently, but, these two forces are dialectically inherent to one another, thus placing the actual and virtual on the same ontological plane.

Brief Summary:

In Deleuze’s transcendental philosophy, the things that appear and the conditions of their appearing are on two different ontological levels, namely, the virtual and the actual. We want to see if a Hegelian can show that objects have such a doubled aspect that converge on a single ontological plane, that virtual and actual transform into one another. Hegel could do so with his concept of force. There is a constant flux of different appearings. There is force involved in the sense that force causes there to always be difference, yet there is also force in the sense that there is this ‘always be’ of the flux, its persistence or stability in that it is always this flux. This ‘always being differences’ is the condition for there being determinate differences, but this ‘always being difference’ is as well merely the process of the flux, which is on the same ontological plane as the appearances themselves. Nonetheless, it is supersensible, unlike the determinate appearings. But also for Hegel, the supersensible and the sensible are dilectically inherent to one another, as the process of appearance is inherent to the determinate appearences, and the determiante appearances are inherent to the process of the flux. Deleuze however does not dialectally mix the supersensible and the sensible, the virtual and the actual; there are objects and underlying them is a play of forces. So a Hegelian critique might involve us dialectically mixing the sensible and the supersensible in Deleuze’s philosophy.  To explain the nature of force, we might follow the way Hegel turns to the concept of law, which would say that there is always a flux of differences. But confirming the necessity of this will involve tautology. Instead we will turn to the inverted order of law.

Summary

We saw how for Hegel, essence is the process of appearing. Deleuze has a transcendental philosophy [which means he thinks the grounds for intelligible and sensible appearing are of a different ontological order than what appears].

Such a result would clearly present difficulties for a transcendental philosophy such as Deleuze's, where the object is conceived of as doubled, provided we could show that the Deleuzian concept of transcendental field can be captured within this structure of the dialectic. Hegel's dialectic of essence would allow the 'doubled' aspect of the object to be reincorporated into a single ontological plane, thus collapsing the distinction between virtual and actual. (188)

Hegel also take this approach when discussing Schelling, who tries to overcome the subject object duality in Kant’s system. Hegel says he fails because he does not use an immanent development from the speculative Idea but rather follows external reflection (189). Schelling uses formal deduction, which means there can be nothing in the conclusion that was not in the premises, so he begins by assuming the resolution of the dualism, which is also what he concludes. Hegel’s method is better because “movement of the concepts themselves to show the real status of the subjective and objective” (189)

the subjective signifies the transformation of itself into the objective, and that the objective signifies its not remaining such, but making itself subjective" (LHP, 3, 526). [189]

The Hegelian looking at Deleuze’s philosophy would need to show the virtual and actual transform into one another. In Hegel’s section ‘Force and Understanding’ in the Phenomenology he explains the dialectic between “a field of differences (appearance) and a field of identity (the tautological law) .” (189) We will explain Hegel’s claim

that "the supersensible is the sensuous and perceived posited as it is in truth; but the truth of the sensuous and perceived is to be appearance. The supersensible is therefore appearance qua appearance" (PS, 89). [189]

Consciousness cannot resolve the dialectic of the one and the many at the level of pure perception. But also an understanding using the universal is also inadequate. Consciousness thinks of the object as a bundle of properties. Because the properties are determinate, they are all separate from each other, but this makes the object itself dissolve into a plurality of properties. If the object is something more than its properties, then it itself has no properties, making it indeterminate. The understanding solves this with the notion of force, which reconciles unity and difference. Self-identical essence is posited by “focusing on the moment of transition between the simple unity and the many.” (190) In the transition of evanescent calculus values, there was a problematic, contradictory, ambiguous status of the fluxion being both a determinate quantity and yet also equal to zero. There is not this problem with force, because there are “two moments or aspects of force governing the thing”. For the object, “ ‘on the one side, a universal medium of many subsistent 'matters,' and on the other side, a One reflected into itself, in which their independence is extinguished’ (PS, 81)”. (190) Force is distributed across two levels: “force proper, which preserves the unity of force, and force as expressed manifoldly in appearance.”  (190) Manifoldly expressed force is “the dispersal of independent 'matters' in their immediate being,” and this is the expression of force. Force proper is what the independent matters have disappeared into; it is “ ‘Force which has been driven back into itself from its expression" (PS, 81). [Not sure what all this means. It seems force expresses itself in a multiplicity, but this multiplicity returns into itself, so there is both identity and difference, unity and multiplicity.] “Force therefore allows the understanding to reconcile the two aspects of the object, difference and identity, without having to bring in the notion of transition.” (190)

Thus we have “two moments: force as the One, and force as its expression”. (190) Force proper is the One. Its essence is to express itself. It needs some other to impress or force itself upon. This means that force’s expression is called forth or solicited by something else. Then, as soon as force expresses itself, it becomes expressed force, which is the many, and it is separate from its essence, which is One-ness. So after force expresses itself, it becomes “ ‘a medium of unfolded “matters” ‘ (PS, 83)” (191). So force proper goes outside itself as something other. But then to go back to itself, it needs something else to solicit its self-reflection. What makes force return to itself is again force [but this time a repulsive force somehow.]

In order to allow force to return to itself, it requires some other to solicit its self-reflection. As Hyppolite notes, this other will in turn be force itself. As he writes, "to every attraction corresponds a repulsion; otherwise, the matter of the whole universe would coagulate at one point." When we think back over the nature of force, this result should seem inevitable, to the extent that the concept of force was brought in precisely to allow two seemingly contrary ideas, identity and difference, to coexist with one another. From the start, it was therefore implicit in our understanding of force that it could only be force that was solicited by force, or in other words, that it is the nature of force itself that necessitates its expression. (191)

So [somehow] force begins unified, it is broken into two independent forces, but these forces are dependent on each other. Since “force can only be in relation to the other, ultimately, it lacks all determinacy.” (191) The two moments of force then collapse into an undifferentiated unity.

Force has now become, as Hyppolite puts it, "the thought of the phenomenal world which, as an interplay of forces, is now no more than the incessant exchange | of determinations, a perpetual instability whose unity and consistency lie only in thought." (191-192)

[The understanding only sees the vanishing.]

For the understanding, therefore, the play of forces is conceived of as a nonbeing, a surface show (Schein), the totality of which is appearance. Appearance, as a nonbeing of being, a being that vanishes, brings with it the corresponding notion of the being of being, "the inner being of things qua inner" (PS, 86), which is mediated by immediate appearing. (192)

But now we have the problem of a beyond. But the beyond is simply a void. Hegel thinks “consciousness has two choices in relating to this beyond.” (192)

[1] Leave the inner world and consider it “an inaccessible beyond of consciousness.” (192) The problem is that consciousness has already characterized the beyond as the truth of appearance. Thus it we would perceive it as true while knowing to not be true.

[2] Populate the void of the beyond with determinations. These determinations must have their root in appearance.

The reason for this is clear, in that if consciousness relates not to the beyond, but to appearances, then the only content it is able to apply to the beyond is that found in appearance itself. If consciousness were able to apply the determinations of the beyond to the beyond, then it would be clear that the beyond was not in fact a beyond, as it would be accessible to consciousness. (193)

Appearance is the filling of the supersensible.

Hegel's point is that within the flux of appearance, in which nothing is stable, we become aware of at least one permanence: the permanence of the flux itself. "The truth of the sensuous and the perceived is to be appearance" (PS, 89), which for Hegel is the structure of appearance itself. (193)

So what is appearing is in flux, but what it is appearing in, the structure of appearance, is stable. “the truth of the flux of appearance being the process of its appearing.” (193) [The supersensible perhaps is the stable structure of appearance.] So the supersensible is appearance qua appearance but not mere sensibility. Deleuze does not equate the supersensible with appearing [the virtual with the actual] So

The dialectic, which begins with the play of forces and ends with the equation of the supersensible with appearance, if it could be shown to apply to Deleuze's philosophy, would clearly offer strong grounds to reject his dual ontology of virtual and actual. This is particularly clear in relation to Deleuze's early work, Nietzsche and Philosophy, where the world of objects is conceived of as the expression of an underlying field of forces, necessarily inaccessible to consciousness. (193)

Hegel uses the concept of law to overcome certain shortcomings. So the truth of appearance is its own ceaseless movement. What is in the flux is always different, so there is a universal difference of the flux. This is the truth of the flux, and it is the ‘simple element in the play of Force’, so it is the law of force. The beyond has an element of universality. Law takes this universality, and law itself subsists in the sensible flux. But law combines two conflicting tendencies: [1] the law must capture all the difference in the play of forces and [2] the laws must express the unity of force in appearance.

Either we maintain the diversity of laws, in which case, we end up with a diversity which cannot capture the unity of appearance, which is their purpose, or else, we abstract from these general laws in order to generate a unified theory, but in this case, the specificity of the law is lost, and we return to a beyond as, at the limit, an abstract universal. (194)

 

We have another problem, we have no way to establish the necessity of the laws of force. When trying to do so, the understanding uses tautology. [I cannot explain this, here is the passage].

In order to introduce the necessity that Hegel argues is lacking from the notion of law, the understanding is forced to resort to explanation by tautology. This is achieved by creating an artificial distinction between two different moments, followed by the cancellation of this difference. In the case of the law itself, this is achieved through taking a phenomenon as if it were structured from indifferent elements. Thus, for instance, "motion is not itself thought of as something simple, or as a pure essence, but as already divided; time and space are in themselves its independent parts or essences, or, distance and velocity are modes of being or ways of thinking, either of which can well be without the other" (PS, 94). This structure of the law is then repeated in that which is to explain it: "The single occurrence of lightning, e.g., is apprehended as a universal, and this universal is enunciated as the law of electricity; the explanation then condenses the law into Force as the essence of the law. This Force then, is so constituted that when it is expressed, opposite electricities appear, which disappear again into one another; that is, Force is constituted exactly the same as law; there is said to be no difference whatever between them" (PS, 94-95). The movement of the law is therefore paralleled by a movement of force. It turns out that explanation in terms of law simply moves from the same to the same. The implication of this is that what is posited as a ground for appearance is in fact no ground at all but is merely the same as | appearance itself. (194-195)

So we cannot explain the unity of the world appearance using law. Instead, we will see, the understanding posits an inversion of the order of law. (195)

 

 

Somers-Hall, Henry (2012) Hegel, Deleuze, and the Critique of Representation. Dialectics of Negation and Difference. Albany: SUNY.

The DeComposition of Our Body's Functions. Review and philosophical application of Compound Functions in Edwards & Penney's Calculus (Section 1.4)

presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

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[I your author am not a mathematician; I am merely an admirer of Edwards & Penney's wonderful calculus book. Please consult the text or other references to be certain about anything in the summary below. I mean this emphatically.]

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The DeComposition of Our Body's Functions

Review and philosophical application of
Edwards & Penney's Calculus
Compound Functions
in
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

What Do Compound Functions Got to Do With You?

Consider if we are trying to lose weight. So we go on a diet. Over the course of a day, we begin to feel our appetite increase. So this would be one function: as time increases, our appetite increases. But then, as our appetite increases, there is a decrease in our willpower to hold strictly to the diet. So our decrease in willpower is like a function compounded on the first function. Our lives are filled with many such highly complex functional relations.

Brief Summary:

Functions can be compounded one-upon-another as if the output of one machine instantly becomes the input of another.

Points Relative to Deleuze
[Under Ongoing Revision]

We might find a composition of functions in Deleuze's account of sensation, building from his portrayal of Spinoza's affection. Deleuze's example is that we are looking for our glasses in a dark room. Someone enters and turns on the light. This causes us to make a transition from the way we are affected by the dark to the way we are affected by the light. Because we are searching for our glasses, this affection increases our power to act. Consider instead if we are meditating in the dark room. Then when someone turns on the light, this dazzles us and ruins our concentration, and we lose the power to act.

Affection for Spinoza is a matter of our body's composition. What gives our body its power is the differential relations between the speeds of its simple bodies. But its simple bodies are like calculus differentials, infinitely small parts that only obtain a finite value when put into a differential relation with other simple bodies. Consider if we take poison. This will disrupt the ways the simple bodies in our body differentially relate to one another. This is why we die. Our bodies no longer express the range of differential variations that define us. Instead, they eventually express for example dirt.

Now, Deleuze also places a value on our internal decomposition when it comes to the sensations that affect us. The discord of our faculties is what makes sensations more intense [see here and here.] Also consider Spinoza's Ethics Part IV, Proposition 39, Scholium. He writes:

It sometimes happens, that a man undergoes such changes, that I should hardly call him the same. As I have heard tell of a certain Spanish poet, who had been seized with sickness, and though he recovered therefrom yet remained so oblivious of his past life, that he would not believe the plays and tragedies he had written to be his own : indeed, he might have been taken for a grown-up child, if he had also forgotten his native tongue. If this instance seems incredible, what shall we say of infants? A man of ripe age deems their nature so unlike his own, that he can only be persuaded that he too has been an infant by the analogy of other men. (Elwes translation)

From what Spinoza says about the decomposition of the body, it would seem that to change from child to man, or from poet to someone different, would be a decomposition, and thus a loss of power. But could it be that Deleuze takes the opposite view? To change from child to man is not an expression of our body's frailty. It rather shows our body's power to become something new. Imagine we get a sickness like the Spanish poet, and we no longer recognize our past selves. This would seem to be a loss of power. But that would only be so when we take the perspective of the self whom our body no longer expresses. But our bodies are always as perfect as they can be, given the conditions affecting them at any given moment. So to become someone else is not really a deprivation. It could be a testament to the power of our bodies to become new things, to express new essences. The question for Deleuze might not so much be "What can a body do?" as much as it might be "What can a body become?"

So here would be the composition of functions in this case. A certain affection might for Deleuze increase the intensity of sensation. But that means a decrease in the organization of our faculties and our body's compositional relations. Yet this again would be an increase in our body's power to become new things, to express new relations and essences. So it ultimately is an increase in our body's expressive power.

Edwards & Penney's Calculus
Compound Functions
in
Section 1: Functions, Graphs, and Models
Subsection 4: Transcendental Functions

Recall the function machine.

We can think of functions like these 'black box' machines. The machine takes-in a number from one set of values, and it gives-out a value from another set. So consider the squaring function:

The machine takes-in a 2, for example, and then gives-out a 4.

We can also compound our functions. So we can take the result of one function, and then put it into another 'function machine', if you will.

So let's begin with the function g(x). We put in x, and it gives-out u. We then have another function f(u), which is the same as f(g(x)), and we read it as, 'f of g of x'. So we plug the u into the f function, and it gives out the value for f(u). We can then call f(u), or f(g(x)) a new name. We can call it h(x). So,

h(x) = f(g(x))

This requires that all the x values in g's domain correspond (by means of g) to values in f's domain. We call the g function here the 'inner function', and the f function would then be the 'outer function' in this case.

Edwards & Penney now give an example that will illustrate something about notation for composition of functions.

We will first find f(g(x)). First note the g(x). If our number for x is larger than 1, then we will have a negative number. But when we apply it to the f function, we will have the square root of a negative number. So instead we qualify it:

But let's instead consider if we first take f(x), and then after that apply it to the g function. To avoid having the square root of a negative number, we must first begin with a positive. Then after that, there are no restrictions for the g function. So

Edwards and Penney would like then to distinguish two notations:

f(g(x))

and

f ○ g

Our example above shows that the composition of functions f and g is not the same as g and f.

f ○ g ≠ g ○ f

So the notation f(g(x)) might remind us of multiplication notation. In the case of multiplication, it does not matter what order the functions are multiplied. But as we see in the composition of functions, the order does matter sometimes.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp. 36-37.

Spinoza. Ethics. Transl. Elwes. available online at:
http://ebooks.adelaide.edu.au/s/spinoza/benedict/ethics/index.html