In an earlier post, I made reference to Jacob Klein’s essay
about Husserl’s history of the origin of geometry. Klein’s own work is very
impressive as well (Burt Hopkins has a recent book on both Klein and Husserl [a NDPR review is here),
and reading through Klein's book has helped me to see one reason why Deleuze so freely
and regularly draws from both mathematics and art, though not just any
mathematics or any art. Deleuze was interested in a problematic as opposed to
axiomatic mathematics; and he was interested in a figural as opposed to
figurative art. What the two have in common is a certain form of abstraction.
In Klein’s historical narrative, the emergence of the “modern concept of number’” emerges in stages by way of Diophantus, François Viète, Simon Stevin, Descartes, and John Wallis. The general trajectory of this narrative for Klein is the gradual but persistent move away from numbers being understood as signs subordinate to determinate and countable objects that are potential objects of intuition to becoming abstract symbols without any determinate connection to an intuition or countable entity.
Simon Stevin was a crucial link in this chain for he broke with the traditional view that founded the status of numbers on the basis of a determinate entity – even if only the countable monad as such, abstracted from all other properties and qualities – and argued instead that the zero is the basis for numbers, much as the dimensionless point is for the line. Moving quickly on to Descartes, it was his “great idea,” Klein argues, to identify “the ‘general’ object of his mathesis universalis,” not with a determinate, countable entity, but “with the ‘substance’ of the world, with corporeality as ‘extensio.’” (197) A key aspect of this move for Descartes is that when one tries to represent this ‘extensio’ one would, as Klein summarizes the key passage from Rule 14, “necessarily arrive at contradictions—for in the imagination the ‘idea’ of extension cannot be separated from the ‘idea’ of body, nor the ‘idea’ of number from the ‘idea’ of the thing enumerated…” (199). This anticipates Spinoza’s own claim, in his famous Letter 12, that when we attempt to understand reality by way of numbers we end up with the “grossest absurdities.”
Since we cannot rely on the imagination, Descartes’ solution is the use of “symbolic figural representation,” which for Klein is “not a representation of a determinate number of units of measurement…but the ‘symbol’ of an indeterminate multitude obtained by ‘symbol-generating abstraction’.” (205) The figure enables us to get traction and avoid contradictions precisely by being divorced from determinate bodies and spaces much as numbers were for Stevin (and later Wallis) divorced from countable entities and intuitions.
Deleuze’s understanding of much of twentieth-century painting tracks a Cartesian path. In fact, in his book on the twentieth-century English painter Francis Bacon, Deleuze develops the contrast between the figurative and the figural in painting: ‘the figurative (representation),’ Deleuze argues, ‘implies the relationship of an image to an object that it is supposed to illustrate…’ while the figural, ‘through extraction or isolation,’ attains an autonomy that cannot be reduced to a represented object. (see Logic of Sensation, p. 6) Figurative paintings stay safely within the logic of the model-copy relationship, while figural paintings employ elements (sensations) that elude this logic.
Francis Bacon’s paintings, for example, are figural for they do not begin with an effort to draw what one sees but begin rather with what Bacon referred to as “involuntary free marks.” Through these marks Bacon slowly develops the painting, in a process Bacon refers to as “coagulation.”
Even in cases where Bacon is beginning with an existent object, such as in his self-portraits or in his Study after Velázquez’s Portrait of Pope Innocent X:
When Bacon begins with a determinate object such as another painting or a picture, etc., there remains an element of the figurative that Bacon himself admits cannot be eliminated, but in the process of painting he disrupts the relationship between these givens as a model to be represented by interjecting a diagram, which Deleuze claims is “indeed chaos, a catastrophe, but it is also a germ of order or rhythm. It is a violent chaos in relation to the figurative givens, but it is a germ of rhythm in relation to the new order of the painting.” (83)
This is where the Deleuzo-Cartesianism as I see it enters the scene, for while Bacon’s figural paintings inject chaos into the figurative and thus disrupt the model-copy relationship, it does so without succumbing to the chaos itself. As Deleuze puts it, the methods of nonfigurative artists “must not be given free rein, and the necessary catastrophe must not submerge the whole,” (89) much as the use of the figural for Descartes must avoid the contradictions that follow upon using the imagination to grasp “extension.” Deleuze claims that the work of Jackson Pollock commits such an error. Chaos has submerged the whole:
At the same time, however, a work can be nonfigurative and yet fail to inject the “chaos,” “catastrophe,” and “rhythm” into the work. The result here can be a work that suffers from excessive abstraction (or it exemplifies and empty formalism). Deleuze cites Mondrian as an example of the latter:
Francis Bacon's work is thus able, as Deleuze sees it (and I made a similar point in an earlier post regarding Kafka and David Foster Wallace), to avoid allowing chaos to submerge the whole - or, which was Descartes' concern, of succumbing to the contradictions and absurdities that result when the imagination is used to think extension - and Bacon's paintings avoid being figurative, representational works - or they allow for the emergence of something new much as Descartes' rules and methods were set forth in order to enable the mathematician not simply to repeat what we already know but to extend this knowledge and discover something new and previously unknown.
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