Was Deleuze's understanding of the infinitesimal calculus primitive?

Question

According to the SEP:

Deleuze was one of the targets of the polemic in Sokal and Bricmont 1999. As much of their chapter on Deleuze consists of exasperated exclamations of incomprehension, it is hard to say what it is that Sokal and Bricmont think they have accomplished. One thing is clear though: Deleuze was perfectly aware of the finitist revolution in the history of the differential calculus, despite Sokal and Bricmont's intimations otherwise.

What is meant by the finitist revolution? The only finitist position I know of is certain limits placed on a theory of sets by generally abandoning the axiom of infinity altogether.

He writes in Difference and Repetition, “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...”

This seems to me to be fairly obvious from any reading of the physical & mathematical literature. That is Cauchys invention of limits placed Newtons calculus on a rigourous basis for mathematicians (leading onto the invention of analysis, topology and many other things) but at the price of exorcising these intuitive or barbaric methods. In fact these methods are still used in Physics where they were first introduced, and one might suppose that first, there are other axiomatic forms that brings out the intuitive character; and secondly that notation itself may inspire different interpretations. In fact, both of these have been achieved in some way - the first being synthetic geometry and the second being the theory of forms.

It seems obvious here that Deleuze's treatment of early forms of the differential calculus is not meant as an intervention into the history of mathematics, or an attempt at a philosophy of mathematics, but as an investigation seeking to form a properly philosophical concept of difference by means of extracting certain forms of thought from what he clearly labels as antiquated mathematical methods. (For positive views of Deleuze's use of mathematics as provocations for the formation of his philosophical concepts, see the essays in Duffy 2006.)

Here, the author of the article writes 'clearly labels as antiquated mathematical methods'. But in the fact Deleuze writes 'old so-called barbaric or pre-scientific'. That is he recognises them as forming part of the pre-history of science, but gives them the respect they are due as originating ideas.

Is this the usual understanding of Deleuze's remarks on mathematics? For example do De Landa in Virtual Mathematics make similar remarks or do they punt in a different direction?

Answer 1

finitist revolution

I believe that refers to Cauchy/Weierstrass, epsilons and deltas and so on replacing the pre-rigorous (pre-surreal, e.g. Newton/Berkeley debate) notion of a fluxion. Which you allude to in your response to the dx part.

dx

From the passage you quoted I would tend to side more with what you said about the suggestiveness of the notation. Or that's the interpretation I would suppose first ... but don't really know what Deleuze meant. Remember Newton used ẋ and series, not dx.

extracting

I did a very cursory reading of some freely available Deleuze text and put a few thoughts here (definitely non-scholarly, just speculation/discussion).

Answer 2

He was here not interested in mathematics for the sake of mathematics, but for a sort of extraction of the problematics underpinning a mathematical idea. He would not consider measures or functions as tied to particular problems to be objects of great interest in philosophical development. It is the formalizaion of a problem that is of concern --that which sets the stage for an answer, a work, a line of inquiry, even entire schools of thought etc--. The problems one sets out to answer all presuppose a problematic. That's why someone like Deleuze would focus on things expressed in Leibniz that mathematicians would not see any value in. He's not a mathematician, but a philosopher.

From an idea standpoint the employment of the discussion around Leibniz's use of dx & dy is closer to what Leibniz himself did in his development of differential calculus and the idea contagion of the concept of infinitesimals with the development of monads. With Deleuze's particular case he was addressing Kant's framework of making Ideas subject to external constraints. The thought looked at here was that of something being undetermined, determinable and effectively determined (dx,dy)/(dx/dy), (values of dx/dy). Or put differently, understanding the idea of differentials under 1) a principle of determinability; 2) a principle of reciprical determination and 3) a principle of complete determination. A 'differential' for Leibniz under these principles is very different than the Kantian approach that precedes by via representation (or sub-representation).

I believe the term "finitist" only makes sense in the context of this history of mathematical discussion around infinitesimals which is not discussed so much in the book they cite above, Difference & Repetition, but in the later book on Leibniz. The SEP author is correct that the history of mathematician's treatment of Leibniz's ideas isn't in any sense lost on him, it just had no relevance at all to the discussion on the history of philosophical ideas being had in the book.