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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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[Your author is not a mathematician, he's just an admirer of Edwards & Penney's book. Please consult the text to be certain about anything in the summary below.]

[Other Entries in the Edwards & Penney's Calculus Series]

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.

Dot Products in Strang's Linear Algebra, Class 1

Presentation of Gilbert Strang's lecture, by Corry Shores
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Dot Products in Strang's Linear Algebra, Class 1

The fundamental problem of linear algebra is to solve a system of linear equations.

An example with an equal number of equations as unknowns (n equations, n unknowns):

The coefficient matrix is a rectangular array of numbers. In this case it is two rows and two columns. Strang lists the coefficients as they are arranged above, except without the variables.

Strang now displays the unknowns. They are in a column. He calls this a 'vector.'

And there are two numbers for the right-hand side vector.

He calls the matrix of coefficients, 'A.' He calls the vector of unknowns X. The right-hand side he calls b. So linear equations are AX = b

Strang now gives the "row picture" for these equations.

And he will place all the points that will satisfy the equations. He begins with the first equation, and he wonders where it crosses the horizontal x line. When x is 0, y is 0 too.

When we substitute 1 in for x, we get 2 for y.

We are dealing with a linear equation. So we may draw a line connecting the two points. This line satisfies all possible solutions for the equation.

Now we will display the second equation. If we say that y is 0, then we get x is -3.

If we take x to be -1, then y is 1.

We then draw the line for the second equation. The two lines intersect at "the all important point."

It is at x = 1 and y = 2.

These values solve both equations.

Strang will now give the "column" picture.

Recall the original equations.

He will now just pull-out the values from the x column.

And then he does the same for the other two columns.

This now gives us three vectors. It wants us to combine the first and the second vectors in just the right amounts so to get the third vector. The combination of the first two columns is called the linear combination of columns.

This is the algebra representation of the equations. Now he will draw the geometry version. The vectors have two components, so he draws two axis.

The first column is 'over two, down one.'

The second column is 'minus one, up two.'

Now he will take a combination that will produce 0,3. We say x is one of those, and y is two of these.

For the first part of his linear combination, he will draw a line from the one that goes in the direction of line two.

But we see that we need two for that vector, so he draws another such line added to the first.

[Now, what we want is a mixture of their directionalities, proportioned according to the solution to the equations. So we begin with the first value, starting at 0, and we follow it according to its proportional value. It is one times: over two, down one. The other one is weighted more heavily, twice as much. So we modulate the first one according to its direction and weight. It is two times: back one, up two. Then we obtain a synthetic direction that is the combination of the other two, given their values and the weights modifying those values.]

So we now look at how the equation was solved originally to give us the 1 for the first column and the 2 for the second.

We start with top numbers. We see that 1(2) + 2(-1) = 0. And for the bottom values, it is 1(-1) + 2(2) = 3.

Strang will now do an equation with three variables.

He displays the matrix shorthand.

He will make a picture that shows all the points that solve the equations. We are in three dimensions.

And he will solve x, y, and z, one at a time. He begins with the second equation.

He notes that the origin is not one of the points that satisfies it. Now if x is 1, then y and b can be zero, for the second equation.

And z can be 1, and x and y can be 0.

And x can be 0, z is 0, then y is minus a half.

Now he wants all the points to satisfy the equation. If there would only be two variables, the solution would be a line. But since there are three variables, the solution is a plane.

This plane above solves the second equation. And this is the plane for the first equation.

The two planes intersect at a line. Now we show the third plane. All three meet at a point.

That was the row picture. We will now look at the column picture.

First he draws the first vector.

Then the other two.

We have to now find the correct values for the variables so to produce the right-hand column. We see that the z column is the same as the right-hand column.

So we want one z and none of the other columns.

Hence the answer is the same as the third column, b.

Now, the matrix form of his equation/system is some matrix A times some vector x equals some right-hand side b.

This is multiplication, A times x; matrix times vector. How do we multiply them? Strang creates an example matrix and vector.

There are two ways to multiply them. Strang begins with his favorite: it's a column at a time. We see that our vector is this:

And our matrix is this:

So we take 1 of the matrix's first column:

And 2 of the matrix's second column.

We can notate the additions this way: one of the first column plus two of the second column.

We then add across the rows after factoring-in the weights. So the top row would be 2 plus 10 equals 12. The bottom row would be 1 plus 6 equals 7.

Now he will show the way find the dot product. In this way, we do it a row at a time. 2 times 1 plus 5 times 2 gives the 12. Then 1 times 1 plus 3 times 2 equals 7. [Below is a video of Strang explaining dot products]

What we find is that Ax is a combination of the columns of A.

From MIT OpenCourseWare. Creative Commons Licence. All blackboard images and video from Prof. Gilbert Strang's 18.06 Linear Algebra , Video Lecture 1.