Algebraic Concepts Characterized

Deleuze’s notion of the Differential

“Just as we oppose difference in itself to negativity, so we oppose dx to not-A, the symbol of difference [Differenzphilosophie] to that of contradiction. It is true that contradiction seeks its Idea on the side of the greatest difference, whereas the differential risks falling into the abyss of the infinitely small. This, however, is not the way to formulate the problem: it is a mistake to tie the value of the symbol dx to the existence of infinitesimals; but it is also a mistake to refuse it any ontological or gnoseological value in the name of a refusal of the latter. In fact, there is a treasure buried within the old so-called barbaric or pre-scientific interpretations of the differential calculus, which must be separated from its infinitesimal matrix. A great deal of heart and a great deal of truly philosophical naivety is needed in order to take the symbol dx seriously: for their part, Kant and even Leibniz renounced the idea.  […] The principle of a general dif- ferential philosophy must be the object of a rigorous exposition, and must in no way depend upon the infinitely small. The symbol dx appears as simultaneously undetermined, determinable and determination. Three principles which together form a sufficient reason correspond to these three aspects: a principle of determinability corresponds to the undetermined as such (dx, dy); a principle of reciprocal determination corresponds to the really determinable (dY/dx); a principle of complete determination corresponds to the effectively determined (values of dY/dx). In short, dx is the Idea – the Platonic, Leibnizian or Kantian Idea, the ‘problem’ and its being.”

Gilles Deleuze, Difference and Repetition p. 170/17.1

cf my discussion of it (in German):

part I (not categories, fantastical notions)

part II (ideas as the differentials of thought)

part III (not the depth of metaphysics, metaphysical surface)

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