The Function of Cesurae. Review and philosophical application of Edwards & Penney's Calculus, Algebraic Functions, in Section 1.3

presentation of Edwards & Penney's work, by Corry Shores

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The Function of Cesurae

Review and philosophical application of
Edwards & Penney's Calculus
Algebraic Functions
in
Section 1: Functions, Graphs, and Models
Subsection 3: Polynomials and Algebraic Functions

What do algebraic functions have to do with you?

During our experiences, we might feel that our decisions, or the events affecting us or happening around us, are tending in a certain direction. So we feel the pull or push that lends to how things carry-onward. Yet sometimes we make what feels like an absolutely spontaneous decision. At that moment, it is as though we are not in the process of going some specific direction, but that rather we are in a state of suspension, where anything can happen next. And yet, looking back, we see that the moment of spontaneity followed other moments, and came before those happening after. So there is continuity in how we experienced the successions of moments, and yet there was a moment where there were no tendencies, at least no clear ones or unambiguous ones. Algebraic functions are like this. Later we will see how calculus differentiation can be seen as determining the quantities of these tendencies-toward-change. An algebraic function would serve to illustrate an instance when there is continuity at a place where there is also a break (or a cesura) in there being tendencies. We could relate this as well to what Hölderlin says about Oedipus (See his remarks on Oedipus and Antigone, and Beaufret's commentary on these remarks.) Oedipus learns that he is the wanted accursed criminal, despite at first having every reason to believe otherwise. This brings him down into an absolutely deep internal reflection upon himself, which reveals himself to be different from himself. So there is a cesura or break within his selfhood. As well, there is a break in the rhythmic flow of the drama, where the pace is found to be different before and after that moment. So there is continuity in the flow of events, but a break (cesura) in the rhythmic pace. This is also a moment when time is emptied of its contents. Normally tragic characters follow a divinely dictated fate where the end of their life story 'rhymes' with how it begins, usually with such terrible crimes punished by violent death. Oedipus does not die then. It is as though the gods ceased filling his life with a fated direction. So for him, the tendencies he normally felt pushing and pulling his life's progress were in that moment suspended. There was no content to time. It was the pure and empty form of time, that emptiness that events secondarily fill. So for Oedipus, there was a continuity in the carrying-onward of the events. However, there was a break or cesura in the tendencies of development. We will visualize this with the graph for an algebraic function. Consider the one Edwards & Penney give:

Try thinking of the line as if there is movement from left-to-right. On both sides of the y axis, the line is tending a certain direction. But at the limit at 0, the motion is not tending anywhere. It seems miraculously to have turned-out to go a whole new direction, seemingly unannounced prior to the change. This suspension of tendencies in its motion will illustrate for us the continuity without tendency that we see in spontaneous decisions and in moments when we are driven so deep into our fractured selves that we are in a moment where we are not in a time that is headed somewhere, but rather in an experience of the pure empty formal condition of time.

Brief summary

Algebraic functions are power functions modified by algebraic operations. Their graphs might have corners or cusps. Although the function may be differentiable in all other places, it is not so at a cusp.

Points Relative to Deleuze
[Under Ongoing Revision]

Deleuze discuses 'the dice throw' of Nietzsche. We might consider the 'cusp' of an algebraic function's graph to be like such a point where the tendencies-toward-change are not there determined. Deleuze also makes use of Hölderlin's idea of the cesura when associating the split-self with the empty form of time. Again, algebraic functions can be thought to illustrate the cesura. This will be more evident when later we clarify ways Deleuze's work with calculus can help us understand certain other ideas of his that do not seem at first to have any relation at all to mathematics, like selfhood, freedom, and affection.

Edwards & Penney
Section 1.3
Algebraic Functions

We will discuss algebraic functions. Recall the 'function machine'.

A function is like a machine that takes-in a number from one set of numbers, and gives-out another number from another set of numbers. Consider the squaring function.

When we put the number 2 into the machine, it gives out 4.

One type of function is a power function. These have exponents, so they take the form:
f (x) = x^k (where k is a constant)

Algebraic functions have formulas that we can construct by beginning with a power function, and then applying certain algebraic operations, such as:

Addition
Subtraction
Multiplication by a real number
Multiplication
Division, and/or
Root-taking

Polynomials and rational functions, then, would fit this description. One difference between these and algebraic functions is that algebraic functions may have a limited domain. Consider these examples:

Another interesting difference is this: The graphs for polynomial and rational functions have a smoothness to them, even as they change direction. However, graphs for algebraic functions can have parts that are not smooth; we might call them 'corners' or 'cusps'. Consider for example these:

Edwards and Penny write that we will discuss more what a smooth graph is in chapter 3. Here they give a graph that is like the one on the left above.

Later we will discuss differentiation. The idea is that we will be able to say where a line is tending to go at a particular place (a limit). What is notable here, Edwards & Penney seem to be saying, is that there is a continuous progress even as the line crosses the y axis. All before the limit at 0, and all after it, the line is tending a certain direction. However, at the 0 limit, it is not tending anywhere. Nonetheless, the line (the function) is continuous. So algebraic functions can be continuous even at places where we cannot determine its tendencies (that is, differentiate it, in the calculus sense of the term).

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, pp. 3; 31-31; 137.

Heaven Vacated. Jean Beaufret. Summary of "Hölderlin et Sophocle", pp.16-26

by Corry Shores
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Heaven Vacated

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Summary of
Jean Beaufret
"Hölderlin et Sophocle"
pp.16-26

What's This Commentary Got to Do with You?

In our lives, dramatic unpredictable events happen which change the course of events in our lives, but also change us ourselves in a very profound way. What is that moment like? It is as though the normal flow of time evaporated. We are left without the ability to expect what is to come, or to make sense of what has happened. It is as if time is emptied of all its contents. We are purely in the moment. And the immediate moment is unfilled and undetermined, and unconnected with the past and future which now seem to originate from this middle-point, as though we now will understand the past in relation to this current moment, and as well we now will see all future events following from this moment too. Yet maybe these moments are not so rare. Perhaps it is more of a matter of degree. That is to say, the future and the past are always originating now, although maybe only more-or-less explicitly to our awareness. And this is because the immediate moment is one of unpredictable and undetermined variation, from which we secondarily define the past and future. What would make the immediate situation a state of pure variation would be if it were constituted by differences and paradox. So the more intensely we feel the difference which makes the immediate situation what it is, the less we are living in a pre-drawn line-of time. We instead are always shooting out intensively like the differential tangent on a continuously varying curve. This is time at the heart of time, upon its grounding bedrock. This means that we are not fundamentally ‘in’ time. We need not see ourselves as being determined by a past and as determining a future. We could see ourselves more primarily as being immediate difference on the inside, in contact immediately with the differences in the world. When we more clearly see how we are constituted by difference, we can better grasp the reasons for why we are continuously changing.

Brief Summary

Jean Beaufret discusses certain Kantian influences in Hölderlin’s Remarks on Oedipus and Antigone. Hölderlin identifies a rhythmic point in the stories where the pace shifts. This break he calls the cesura, and it happens when the seer Tiresias predicts a future that is not implied from the story’s beginning. Oedipus for example should not be blamed for his crimes, so his suffering might seem unjust. In a way, the gods abandon time itself, because they cease to make time have an internal consistency; the end will no longer ‘rhyme’ with the beginning. But humans too need not act as though their actions will bring about a correspondence with their past. They need not at all view time as a story-line laid-out in advance. Instead, humans can be at the heart of time, in the a priori conditions of time and space. When the gods abandon us by no longer determining a content to the progress of time, it is like them vacating the heavens, leaving behind them a vast vacuum, a pure emptiness of time and space.

Points Relative to Deleuze
[Under Ongoing Revision]

For Deleuze, our split-self is what explains the basis of time. And also, temporality in its rawest form is ‘rhythm’ for Deleuze, and in an even more basic form, time is void of all contents. Deleuze elaborates these ideas by looking at Hölderlin’s Remarks on Oedipus and Antigone, where he explains the disruptive rhythmic cesura when human-kind falls into an a priori pure and empty form of time. Oedipus Rex, for example, has two main parts, divided according to the pace of dramatic development unfolding in each section. The breaking point, the cesura, which balances the two unequally weighted sections, is the moment that Tiresias declares a past and future unknown to Oedipus. In fact, he declares Oedipus to be a person completely contrary to who Oedipus thought himself to been throughout his past. For all his life, Oedipus tried not to fulfill his prophesy of killing his father and marrying his mother. So in his mind, he absolutely is not such a person who would do these things. As it turns out, he in fact did do these things, but completely without knowing it.

1) So in the first place, we note that at the moment of the prophesy, Oedipus faces himself as a self completely foreign to himself. The next thing we note is that there is a lot of dramatic density and intensity at the beginning of the play. After the prophesy, it is a tortured gradual process of Oedipus self-unraveling. This means, at the point of the prophesy, we are experiencing a pure variation of speed. This makes it like an instantaneous change, which we see for example in physics’ instantaneous velocities. Before and after this moment we might regard their being extents of dramatic unfolding with their own general pace. But at the moment of change, in between these extents, is a quantity that indicates the tendency toward change. If we consider terms from the physics example, we could imagine this quantity as being like the relation between an infinitely small amount of time to an infinitely small amount of dramatic development. What we get is a value telling us the tendency toward changing in dramatic pace. But this tendency does not extend in space or time (it is based on vanishingly small values). It is non-extensive, and in fact, it would be intensive. The prophetic moment is an intensity, a piece of reality that is full of reality, but is not expressed explicitly in the actuality of the extending world birthed from these intensities. So in a sense, it is a pure quantitative variation, a pure difference, a difference without terms.

2) We normally consider rhythm as a regular pattern. But we could also think of it as being a feeling of change. A rhythm too steady would homogenize with itself. Our attention fades when a rhythmic pattern is too regular, like how we struggle to pay attention to every tick of a clock. So rhythm instead can be thought of as having a continuous variation. Rhythm could be the pure feeling of change. This then places the cesura at the heart of the dramatic rhythm of the tragedies.

We will see that for Deleuze, temporality in its rawest expression is a continuous quantitative variation in how things are given to us, that is, how they appear to us. In other words, the raw experience of the motion of time is an experience of rhythm, and at the heart of rhythm is an immediate break in our experience of the world and in our experience of ourselves.

3) The parts before and after the cesura do not ‘rhyme’. In Aeschylus’ tragedies, for example, it could be apparent to the characters that their actions will obviously bring tragic consequences. Oedipus story, however, does not fit this. He is punished even though he should not be held responsible for his crimes. And also, his tragic end is not the expected sort. Usually the characters suffer a painful death at the end of the story. Oedipus, however, goes on living. He wanders until eventually making a peaceful, painless, and glorious transition into the afterlife. So the beginning and the end do not rhyme, in his story. There is not the accordance like in Aeschylus where the initial crime finds its mete end. So at the cesura, it is like a poetic modulation. It is like writing a new poem in the middle, so that the next line does not rhyme, but seems like it should, because it caries-on from where the poem was previously developing. Deleuze regards this as the unbending of the circle of time into a line. We might find that this line is like the tangent line which could illustrate an instantaneous velocity, for example. It is a non-extending line of time. Not time as a linear continuity, like our historical time-lines. But time as a direction and tendency for development in that direction, expressed immediately and not over an extent of time. This would be time unraveling down upon its immediate a priori conditions, it being made of pure differential relations.

4) This pure form of time is empty. Beaufret makes this point very nicely. Normally in Greek tragedy, there is a fated path our lives follow, and it is dictated by the gods. But after Oedipus’ self-punishment, he is left to wander. So there is no longer a determinate content laid-out before him, at least not one that is clear in his moment of realization. And also, he had his own past pulled from underneath him like a rug. In that moment of disorientation, no past is clear to him, because at that moment when his own self becomes unclear, different pasts become equally believable and doubtable to him. So there is a complete suspension of the future and the past. And without future and past present to him, even the present loses its temporal meaning. We would not even say that he is in the present, because 'present' has little meaning without relation to past and future. Instead, we would say he is in pure immediacy. It is a non-temporal experience. Nonetheless, this experience demonstrates what lies at the heart of our experience of time. Hölderlin says that it brings us to the a priori conditions of space and time, their being pure and empty. It is not that time ceases to exist for Oedipus in his moment of realization, but rather that time is emptied of all its actual content. He experiences just the emptiness of time, what secondarily gets filled by actual events of change. Beaufret puts this very nicely. The gods reside in the heavens. When they ceased filling Oedipus’s life-time with fated content, it is like they vanished from the heavens. This leaves just the pure emptiness of the sky. This is like how he faces time not as filled by change or as flowing-by, but as the pure conditions of emptiness which allow secondarily for change to find its temporal relational ‘space’ of occurrence.

Jean Beaufret


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Summary of
"Hölderlin et Sophocle"
pp.16-26

A main theme in Sophocles' Oedipus is the gods’ turning their backs on humans. So it is not atheist in the sense of us not believing in god. Rather, it is godless, in the sense of the gods existing while yet also ceasing to involve themselves in dictating the events of our lives. Oedipus is not struck-down by the gods. Rather, he is condemned to wander the earth. (16) Normally a tragic character dies from his crimes. In Oedipus’ case, he somehow survived. His death is delayed. His destiny then is no longer imposed on him. However, the indeterminacy of his destiny makes it more profound, and also more like our modern sense of the tragic, which is not violent death as in the classic Greek, but more of an abandonment to an indeterminate fate, with no recourse to divine determination. (17)

In a way, this is a shortcoming in the gods ability to interfere in Oedipus’ human life. We are no longer bound into a line of development in our lives that leads to a fate predetermined by the gods. Rather, we live entirely in the moment, but for that reason we forget ourselves. The gods usually represent a justice in time. They normally punish grave crimes with a tragic painful death. But Oedipus does not end this way. So time for him is different now. And there is infidelity both in time and in humanity. Because there is no fated divine justice in Oedipus’s life-story, there is no cyclical return of restoration and atonement. Instead, the beginning and end do not correspond to each other; they do not ‘rhyme’. So time itself has a sort of infidelity with itself. The time-path of Oedipus’ life is now wandering and indeterminate in a way. It constantly verges from its beginning. [It in a sense is a continual process of becoming low-fi, to borrow a concept from Richard Pinhas.] The other infidelity is on Oedipus’ part. He can no longer restore his life-situation to how it began. So he too continually diverges from his origin. He lives now only in the moment, with no faithfulness to how time will progress in a way that produces justice. (18)

So we no longer think of time as having a back-side and a forward-side that could harmonize or ‘rhyme’. We instead live in the moment where we feel not a cycle but something more like an oscillation of a sort, like our ship is rocking in the sea as it sails-on.

By living in the moment, and thereby forgetting themselves, humans give birth to a new memory of themselves that is even more profound. They see themselves more authentically as being apart from the gods. (19)

We might here consider Kant’s distinction between the phenomenon and the neumenon, putting aside what he says about sensible cognitions being intellectual ones as well. Hölderlin reads Kant differently than does Fichte, Hegel, and Schelling. (20)

Recall how Hölderlin writes that the gods are nothing more than time. He also says that in the utmost form of suffering, there exists nothing but the conditions of time and space. These conditions are not like affections, which give us content. There is no content to a priori space and time. Kant calls them pure or empty forms of space and time. Before Oedipus, the gods determined the time spanning-out before a person’s life. The gods in a way are like the content to time. But when they depart from us, as if leaving the sky, do they not leave us with a pure void, an empty heaven, the pure form of space and time in all its vast emptiness and purity?

God is no longer “cette immense horloge de lumière qui mesure ce qu’elle manifeste et manifeste ce qu’elle mesure”. (21)

Before, the Homeric time of the gods prevailed. But with Oedipus, during such extraordinary moments of inner confusion when the gods become faithless to humans, we enter a time and space that is as empty as a desert. (22-23)

We then pose a question: so, there is this empty time which is more like an a priori “condition” for our temporal experiences; how then are we to comprehend it as having an end which does not “rhyme” with its beginning? Beaufret says that we must in the first place remain within the context of the tragedies, and consider their beginnings and ends, rather than jumping to conclusions regarding temporality in general.

At the beginning of his Remarks on Antigone, Hölderlin returns to this idea of the cesura. On the one hand, the cesura is what forces-together both sides of the tragedy, yet as a break, it as will is what places them over-against one another. The cesura is the balancing point, and it leans more either toward the end or the beginning, so to balance the ‘weight’ of the paces of dramatic development. Hence, each tragedy has a different “rhythm”. At this cesura point when the dramatic pace explodes, gods and men are forced together. (23)

We also saw divine prophetic intervention in Aeschylus’s Agamemnon. Cassandre foresees the murders and return of Orestes. But this prophesy does not serve as a cesura. What she says is something expected. [Oedipus’ crimes were unknown to him, so the consequences were for him unforeseeable. Agamemnon, however, when sacrificing his daughter, could know ahead of time that this would later bring grave consequences. As well, Agamemnon was rightfully reluctant to walk-upon the red carpet, which he knew would insult the gods.] In Aeschylus, the story in a way is finished even as soon as it begins; the telos or final end is set from the beginning. (24) Oedipus, on the contrary, begins nobly. His crimes were unintentional and unknowable. His later blind wandering does not at all rhyme with the beginning. It is like a note that does not fit, as if a new melody started playing, or a new poem began from the middle of another one. [A cesura is a break in the sense of a new time beginning immediately. But as the beginning, nothing fills it. Deleuze might have us see things as always this way. Each moment is a new beginning. We are ‘ever begun’ or ‘always initial’, rather than the more common motto, ‘always ever’. This 'always initial' is like the innocence of becoming; we are always initiates, ever having our first experience, which is the experience of the pure difference of immediacy.] The cesura breaks-apart what has happened ‘so far’ ('thus') from what shall happen ‘henceforth’, making a fundamental change in the path of development.

We might trace Hölderlin’s understanding of this sort of a radical break to Kant’s Religion within the Limits of Reason Alone. Here, becoming moral is not just an improvement, but a true revolution within us, a complete change of our heart [Herzenänderung]. In this case of a fundamental internal transformation, the beginning and end of someone’s life will not rhyme. (25) Here then we can see what is probably another Kantian influence on Hölderlin’s thinking. (26)




Beaufret, Jean. "Hölderlin et Sophocle," preface to Hölderlin: Remarques sur Oedipe. Remarques sur Antigone. Transl. François Fédier. Paris: Union générale d'éditions, 1965.


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