ABSTRACT
The logic of the differential Having located the logic of the differential from the differential point of view of the infinitesimal calculus in the geometrical example of Spinoza’s Letter XII, with the assistance of Leibniz, the subsequent developments that this logic undergoes will now be examined in relation to the work of some of the key figures in the history of this branch of the infinitesimal calculus. These figures are implicated in an alternative lineage in the history of mathematics by means of which the differential point of view of the infinitesimal calculus is aligned with the differential calculus of contemporary mathematics, a lineage which effectively bypasses the methods of the differential calculus which Hegel uses in the Science of Logic to support the development of the dialectical logic. The logic of the differential from the differential point of view of the infinitesimal calculus is then implicated in Deleuze’s project of constructing a philosophy of difference by means of Spinoza’s discussion of the physics of bodies in the second part of the Ethics. According to Deleuze, the most simple bodies of Spinoza’s Ethics correspond directly to the infinitely small differentials of the geometrical example in Letter XII. The logic of the differential from the differential point of view of the infinitesimal calculus is thereby implicated in Deleuze’s interpretation of Spinoza’s theory of relations in the Ethics, and therefore in the development of Deleuze’s project of constructing a philosophy of difference. The manner by means of which the figures in the history of the differential point of view of the infinitesimal calculus are implicated in an alternative lineage in the history of mathematics will now be examined. Ironically, one of the mathematicians who develops the differential point of view of the infinitesimal calculus is Karl Weierstrass, who considers the differential relation to be logically prior to the function in the process of determination associated with the infinitesimal calculus, that is, rather than determining the differential relation from a given function, the kinds of mathematical problems that Weierstrass dealt with involved investigating how to generate a function from a given differential relation. Weierstrass develops a
recovers this theory in order to restore the Leibnizian perspective of the differential as the genetic force of the differential relation to the differential point of view of the infinitesimal calculus, by means of the infinitesimal axioms of non-standard analysis. Hegel, on the contrary, locates the genetic force of the calculus in the dialectical logic. According to Deleuze’s reading of the infinitesimal calculus from the differential point of view, a function does not precede the differential relation, but is rather determined by the differential relation. The differential relation is used to determine the overall shape of the curve of a function primarily by determining the number and distribution of its distinctive points, which are points of articulation where the nature of the curve changes or the function alters its behaviour. For example, when the differential relation is zero, the gradient of the tangent at that point is horizontal, indicating that the curve peaks or dips, determining therefore a maximum or minimum at that point. These distinctive points are known as stationary or turning points. The differential relation characterizes or qualifies not only the distinctive points which it determines, but also the nature of the regular points in the immediate neighbourhood of these points, that is, the shape of the branches of the curve between each distinctive point. Where the differential relation gives the value of the gradient at the distinctive point, the value of the derivative of the differential relation, that is, the second derivative, indicates the rate at which the gradient is changing at that point, which allows a more accurate approximation of the nature of the function in the neighbourhood of that point. The value of the third derivative indicates the rate at which the second derivative is changing at that point. In fact, the more successive derivatives that can be evaluated at the distinctive point, the more accurate will be the approximation of the function in the immediate neighbourhood of that point. This method of approximation using successive derivatives is formalized in the calculus by a Taylor series or power series expansion. A power series expansion can be written as a polynomial, the coefficients of each of its terms being the successive derivatives evaluated at the distinctive point. The sum of such a series represents the expanded function provided that any remainder approaches zero as the number of terms becomes infinite; the polynomial then becomes an infinite series which converges with the function in the neighbourhood of the distinctive point.1 This criterion of convergence repeats Cauchy’s earlier exclusion of divergent series from the calculus. A power series operates at each distinctive point by successively determining the specific qualitative nature of the function at that point. The power series determines not only the specific qualitative nature of the function at the distinctive point in question, but also the specific qualitative nature of all of the regular points in the neighbourhood of that distinctive point, such that the specific qualitative nature of a function in the neighbourhood of a distinctive
function by generating a continuous branch of a curve in the neighbourhood of a distinctive point. To the extent that all of the regular points are continuous across all of the different branches generated by the power series of the other distinctive points, the entire complex curve or the whole analytic function is generated. 1 So, according to Deleuze’s reading of the infinitesimal calculus, the differential relation is generated by differentials and the power series are generated in a process involving the repeated differentiation of the differential relation. It is due to these processes that a function is generated to begin with. The mathematical elements of this interpretation are most clearly developed by Weierstrassian analysis, according to ‘the theorem on the approximation of analytic functions’. An analytic function, being secondary to the differential relation, is differentiable, and therefore continuous, at each point of its domain. According to Weierstrass, for any continuous function on a given interval, or domain, there exists a power series expansion which uniformly converges to this function on the given domain. Given that a power series approximates a function in such a restricted domain, the task is then to determine other power series expansions that approximate the same function in other domains. An analytic function is differentiable at each point of its domain, and is essentially defined for Weierstrass from the neighbourhood of a
1 Given a function, f(x), having derivatives of all orders, the Taylor series of the
function is given by (k)f (a) k!k=0
∞∑ k(x-a) where f (k)(a) is the ‘kth’ derivative of ‘f’ at ‘a’. A function is equal to its Taylor series
if and only if its error term Rn can be made arbitrarily small, where
is given by
converges at x = 0, or for all real ‘x’, or for all ‘x’ with –R < x < R for some positive real ‘R’. The interval (–R, R) is called the circle of convergence, or neighbourhood of the distinctive point. This series should be thought of as a function in ‘x’ for all ‘x’ in the circle of convergence. Where defined, this function has derivatives of all orders. See also H.J.
