Pt3.Ch6.Sb4 Somers-Hall’s Hegel, Deleuze, and the Critique of Representation. ‘Berkeley and the Foundations of 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 4: Berkeley and the Foundations of the Calculus

Very Brief Summary:

Hegel’s dy/dx expresses the sublation of being and nothingness.

Brief Summary:

Hegel notes that the fluxions in dy/dx are vanishing, but are still determinate. That means they are still being and not yet nothing. But this is being at the brink of being nothing, so the dy/dx expresses a sublation of being and nothingness.

Summary

 

Previously we saw how for Hegel, the dy/dx of Newton’s fluxions expresses the genuine infinite.

One problem with Newton’s fluxions is “whether such a thing as an ultimate ratio can actually be said to exist, since it also appears implicitly to rely on the dual properties of the ratio not yet having, but at the same time already having a determinate quantity.” (170) [Berkeley notes that these calculus methods first assume values, dy and dx. On the basis of these values, we determine other values, so the 2x in the example of y = x².  So if we delete the dx in the calculation, we need also delete the values we obtained by means of them.] Berkeley criticizes the foundations of the calculus:

"If with a view to demonstrating any proposition a certain point is supposed, by virtue of which certain other points are attained; and such supposed point be itself afterwards destroyed or rejected by a contrary supposition; in that case, all other points, attained thereby and consequent thereupon, must also be destroyed and rejected, so as from thence forward to be no more supposed or applied in tl1e demonstration. This is so plain as to need no proof." Thus, Berkeley attacks Newton's notion of the ultimate ratio for both appearing to be unequal to zero (as the terms forming the ratio can be divided by one another), but also equal to zero (in order for the ratio to be applied to an instant). Going on to question the fluxions that make up the elements of both Newton's ultimate ratio and Hegel's mathematical infinite, Berkeley asks: "What are these fluxions! The velocities of evanescent increments! And what are these same evanescent increments! They are neither finite quantities, nor quantities infinitely small nor yet nothing. May we not call them the ghosts of departed quantities!" (170)

Hegel notes that the vanishing values are not nothing but a determinate state. So there is not an indeterminate state between being and nothingness in their vanishing. [Hegel wants us to think of that moment of vanishing being the relation between when there is and is not values.]

Thus, the true foundation of the calculus, according to Hegel, is to be found in the results obtained in the dialectic of being and nothing that opens the Science of Logic. The differential calculus is therefore seen as being grounded in the fundamental dialectical moment of transition, within which the two moments of the fluxion, being and nothing, are to be taken as immanently related. We thus have a ratio of two terms, both of which are on the brink of vanishing, but which, when related to one another, give a determinate value. (171)

[So because the fluxions are vanishing but are determinate, they are the dialectical sublation of being and nothingness.]

When these fluxions are incorporated into the ultimate ratio itself, we have a structure that is isomorphic with the structure of both contradiction and the infinite: "The truth is rather d1at mat which has being solely in the ratio is not a quantum; the nature of quantum is such that it is supposed to have a completely indifferent existence apart from its ratio, and its difference from another quantum is not supposed to concern its own determination; on the other hand, the qualitative is what it is only in its distinction from an other. The said infinite magnitudes, therefore, are not merely comparable, but they exist only as moments of comparison" (SL, 254-55). We thus have the unity of moments that can only exist in their difference through this unity. What this analysis has attempted to show is how, for Hegel, the differential calculus both requires a move to a dialectical understanding of mathematics and also, in its dialectical development, | comes to represent the structure of the system as a whole as it incorporates the movement from being to nothing. (171-172)

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