## 5 Jan 2013

### Pt3.Ch6.Sb3 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘Hegel and the Calculus’. summary

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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 3: Hegel and the Calculus

Very Brief Summary:

For Hegel, the dy/dx of Newton’s fluxions expresses the genuine infinite.

Brief Summary:

Hegel does not base his theory of true infinity on Leibniz’s infinitesimal calculus but rather on Newton’s fluxions. Here dy/dx is a relation expressing a moment of a curve’s variation, that is a moment of variation in the relation between dy and dx. To understand the spurious infinite, Hegel has us consider 2/7 = 0.285 7 142 . . . The right side is trying to express the left side, but this means it ought always to add another decimal determination. This is not the true infinite. For that we need two terms that express the relationship itself. The left side expresses the infinity of the right side wholly in its simple ratio formulation. However, the terms themselves can be substituted with others, so they in themselves are not expressing the relationship. However, the dy/dx of Newton’s fluxions is a finite whole. But it expresses a relation, and the parts of that relation have vanished, so they are not finite values. Yet, dy and dx have no value outside their differential relation. So dy/dx does not express independent values for each term but rather just the value of their relation. So here a finite value contains within it something not-finite, and the not-finite contents, as differentially related, contain within this relation the finite value it expresses. So this is like Hegel’s true infinite where the finite and infinite contain one another and are sublated.

Summary

Previously we saw how we obtain the differential relation between infinitely diminished values that vary with respect to one another. This can find the gradient of a curve at a point.

Now we turn to Hegel’s treatment of the calculus. So the gradient can be expressed

To find the gradient at a point, Leibniz used an infinitesimal amount of change in x and y . (166)

As this difference was infinitely small, it could be discounted for the purposes of calculation, but, as it retained a magnitude relative to dx, it could be used to form a ratio, dy/dx, which had a determinate value. (166)

For Hegel, Leibniz method must be rejected, because the fact that it neglects the infinitesimal values it works with makes it not have the rigor needed for a mathematical proof. [see § 583 of this entry]. Hegel instead turns to Newton’s method.

Slightly before the time of Leibniz’s invention of the infinitesimal calculus method, Newton invented the method of fluxions. He considers the line to be composed of points that are in continual motion. So a curve contains the movement of a point at a certain velocity. The gradient at a point would be the instantaneous velocity of that point. His quantities, then, are not infinitely small, but rather are vanishing at the limit point.

Under this interpretation, the differentials, called fluxions in Newton's system, do not have to be seen as being too small to affect the result of the calculation, but can actually be seen to vanish at the limit point, when dt=O: "Quantities, and the ratios of quantities, which may in any finite time converge continually to equality, and before the end of time approach nearer to each other than by any given difference, become ultimately equal. If you deny it, suppose them to be ultimately unequal, and let D be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is contrary to the supposition." (167)

Newton uses the idea of an ultimate ratio to argue that at the limit case, where the numerator and denominator are reduced to 0/0 (where the lines become a point), we can | still interpret the point as having a determinate ratio: "And, in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of quantities, not before they vanish, nor after, but with which they vanish." (167-168)

Hegel keeps this idea of the ultimate ratio, but does not keep the notions of motion and velocity of the fluxion.

[Note, previously we were dealing with Hegel’s qualitative infinity; now with his quantitative infinity] We need to distinguish true and spurious infinite. Both come into being together through the equation. (168) In the qualitative infinite, both the finite and the infinite are both negated and yet preserved in the whole. “This overcame the spurious infinite, which was merely the perpetual repetition of the finite”. (168) [Hegel’s true infinite will have two terms that are together in the same ratio.] So consider 2/7 = 0.285 7 142 . . . The value on the left represents an ‘ought-to-be’, there ought to be a repetitive continuation of the decimal determinations. This is because we make the decimal form equal to the fractional form, which means the decimal form needs to live up to the completion of the fractional form. But the form on the left gives us the relation itself, not its quantitative determination, and so it is an approximation to the qualitative infinite.

what we have on the left of the equation is no longer a quantitative notion but has passed on into a qualitative determinateness, as the ratio is not tied to any specific value, but instead to their relation. This ratio cannot be captured by the expansion of the purely quantitative series of numbers on the right, no matter how closely it may approach the fraction. The fractional representation, as including that which is unobtainable from the quantitative determination (the moment of difference), has therefore become our first approximation to the qualitative mad1ematical infinite. (169)

However, we can substitute other values, like 4/14 and 6/21. So“while the unity of the different terms is essential for the fraction itself, it is not essential for the terms that make up the fraction.” (169) [What we are looking for is a relation between terms that vary with respect to each other, and not vary together proportionally.]

the fraction itself, it is not essential for the terms that make up the fraction. Similarly, Hegel notes that moving to an algebraic description of the fraction does not overcome this limitation, as an algebraic formulation such as y/x = a can equally be written in a form that does not contain a ratio, such as y = ax, the equation of a straight line. Variability, therefore is shown not to be the defining characteristic of the qualitative mathematical infinite, as even in the case of the algebraic variable, we still have a symbol standing in for an arbitrary quantum, that is, we have variability, but variability still conceived of as magnitude. (169)

But recall Newton’s ultimate ratio, dy/dx.  Either term alone has no value. They rather express moments of the variable relation [moments of the variation in the relation].

In Newton's ultimate ratio, as we are dealing with the ratio of values at a point, neither of the terms in the ratio, dy/dx, can have any meaning outside of the ratio itself. "Apart from their relation they are pure nullities, but they are intended to be taken only as moments of the relation, as the determinations of the differential co-efficient dx/dy" (SL, 25). In the differential relation, we therefore have a situation whereby both the ratio itself and the terms can only be understood as a totality. (169)

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