Pt3.Ch6.Sb2 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘The Calculus’. summary

by Corry Shores
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[Note: All boldface and underlining is my own. It is intended for skimming purposes. Bracketed comments are also my own explanations or interpretations.]

 

Henry Somers-Hall

 

Hegel, Deleuze, and the Critique of Representation.

Dialectics of Negation and Difference

 

Part 3: Beyond Representation

Chapter 6: Hegel and Deleuze on Ontology and the Calculus

Subdivision 2: The Calculus

Brief Summary:

We will later compare Deleuze’s and Hegel’s interpretation of early calculus. Calculus determines differential relations between quantities that vary with respect to one another. We can think of it as the ratio between an infinitely small amount of one term’s variation to that of another term.

Summary

Calculus deals with “quantities whose relations to one another vary.” (162) Somers-Hall deals mostly with speed, the relation of time to distance. Early developments in the study of rates of change come from Oresme. To find velocity, draw a graph with time as one axis and distance traveled as the other. To determine the speed we look at change in distance over change in time, which also gives us a diagonal between those time-space coordinates on the graph. This only gives us average velocity. If we wanted to know the velocity at any moment, it would have to be at constant speed. But for motion with inconstant velocity, we cannot use this method. Instead we need to draw a tangent to the curve to determine the velocity at that moment. But a point cannot have a direction. Leibniz’s solution is to begin with the diagonal between the point in question and some other arbitrarily selected point (which gives the average velocity within the bounds of that time frame), and then bring the arbitrary point infinite close to the one in question. This slides the diagonal to indicate the velocity at that point. In fact, we can do this merely working with the graph’s functions in their numerical form. (163d)

For Hegel, calculus’ method is expressed in the formula [see §608 of this entry]

or

The second one is the gradient function for a curve, change in x over change in y. Now consider Dr. Siddique’s animation below, which will show us how we find the tangent. Notice first the triangular form.

(Thanks Dr. Siddique / faculty.uncfsu.edu)

Notice first the triangular form. On the ends of the bottom horizontal leg of the triangle, we have two x values, at the left corner and at the right corner. In Hegel’s formula, the change in x (or the length of the triangle’s horizontal leg) is called i. In our equation, we find the y values by means of the function y = f(x). The  y value on the lower left is f(x) and the y value at the upper right is f(x + i). We will now modify it to get the second equation.

We do this by substituting the infinitesimal for i. We will arrive at this equation more simply by assuming that our function is y = x². We will find that dy/dx is 2x. So recall the first formula.

Notice in the animation how the lines decrease. The decrease to the infinitely small. So they still have a value, but not a finite one. When the infinitely small amount of x , called dx , and the infinitely small amount of y , called dy , enter into a ratio relation, they produce the finite value, telling us the tangent at the point along the curve/function. So our P value is dx/dy.

The second value on the top, f x , is x² we said.
The function squares the value of the top right formulation.

We said that the i value is the change in x, but we are substituting for it the infinitesimal amount of x, so:

Let’s now multiply the top left value by itself.

That gives us

The two x² ’s cancel.

The dx in the denomoneator and one in the numerator cancel.

Now as we said, dx with respect to dy has a finite value. However, dx with respect to x has no finite value. So it can be removed.

This is the same as.

Altogether we have:

The procedure that we have just carried out is called differentiation and has given us the result that the gradient of any point on the curve y = x2 is always equal to 2x, or two times the point's position on the x axis. (165)

Now recall Hegel’s first formulation.

Now consider if we substitute the value two in for n. We get

dx² = 2x^2-1 dx

which reduces to

dx = 2x

We see that the two formulas are expressing the same operation. We can also reverse this procedure to obtain the primitive function of the derivative, and this procedure is called integration. There are paradoxes with the use of the infinitesimal, and a set theoretical solution regards the dy/dx as a whole, called the differential. We then use the concept of the limit of an infinitely diminishing series. So we are not using the last term in that series but rather the limit to which the series is headed.

By defining the derivative to be the limit of the ratio, rather than the values of dy and dx, questions about what happens when or whether the ratio actually reaches this limit are put out of play. This is the method of Weierstrass, which allowed Russell to move away from an antinomic interpretation of mathematics and therefore also from his early Hegelianism. (165)

 

 

 

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

 

Images from that text and

Geometrical Derivative animation:
http://faculty.uncfsu.edu/msiddiqu/Maple_Animations.htm
http://faculty.uncfsu.edu/msiddiqu/images/images/Gif_Folder/Definition%20of%20Derivative18.gif
Thanks Dr. Siddique of Fayetteville State University