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The SEP explains in its introduction to Deleuze that

"Deleuze conceived of philosophy as the production of concepts, and he characterized himself as a “pure metaphysician.” In his magnum opus Difference and Repetition, he tries to develop a metaphysics adequate to contemporary mathematics and science—a metaphysics in which the concept of multiplicity replaces that of substance, event replaces essence and virtuality replaces possibility"

What does it mean that Deleuze "develop[s]... a metaphysics in which the concept of multiplicity replaces that of substance"? To my understanding, substance is a philosophical concept dating from antiquity. In what ways might it be metaphysically problematic? What does "multiplicity" mean here?

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3 Answers

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Deleuze picks up the concept of multiplicity from Riemann and Bergson and develops it in many of his works and in a variety of ways. In general, he rejects the One-Many dialectic and proposes multiplicity instead:

[M]ultiplicity must not designate a combination of the many and the one, but rather an organisation belonging to the many as such, which has no need whatsoever of unity in order to form a system. The one and the many are concepts of the understanding which make up the overly loose mesh of a distorted dialectic which proceeds by opposition. (Deleuze 1994, p. 182)

Substance theory from Aristotle to Spinoza operates freely with the One-Many dyad (e.g. monism reduces the variety of things in the world to the unity of one). Deleuze opposes this and that is why it can be said that his metaphysics replaces substance with multiplicity:

'Multiplicity', which replaces the one no less than the multiple, is the true substantive, substance itself. ... Even the many is a multiplicity; even the one is a multiplicity. ... Everywhere the differences between multiplicities and the differences within multiplicities replace schematic and crude oppositions. ... Instead of the enormous opposition between the one and the many, there is only the variety of multiplicity - in other words, difference. (Deleuze 1994, p. 182)

In Anti-Oedipus the concept of multiplicity is directly related to desiring-production (which, in turn, is directly related to the notions of desiring machines and flows):

It is only the category of multiplicity, used as a substantive and going beyond both the One and the many, beyond the predicative relation of the One and the many, that can account for desiring-production: desiring-production is pure multiplicity, that is to say, an affirmation that is irreducible to any sort of unity. (Deleuze 1983, p. 42)

References:

Deleuze, G. (1983). Anti-Oedipus : capitalism and schizophrenia. Minneapolis: University of Minnesota Press.

Deleuze, G. (1994). Difference and repetition. New York: Columbia University Press.

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+1 for a great answer, this is especially relevant to my current reading, thank you ;) –  Dr Sister Jan 29 '13 at 23:25
    
although I've accepted your answer, I only understood most of it now. Its only the last section that remains problematic for me now. –  Mozibur Ullah Dec 13 '13 at 7:58
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paraphrasing the entry for multiplicity in The Deleuze Dictionary, ed. Adrian Parr:

Although Multiplicity is used throughout Deleuzes work in many different ways and contexts there are some essential traits: It is a complex that does not refer to a prior unity - either a fragmented whole, or manifold expressions of a single concept. He insists it should be understood substantively and not adjectivally.

As Tsapkou mentions this concept is borrowed from Riemann & Bergson. From Riemann, he takes the idea that a situation is an ensemble without becoming a whole. More profoundly from Bergson, multiplicity has two modes - extensive numerical & intensive continuous. The first characterises space, the second time. The first can be divided without changing its nature; in the second, division implies change of nature.

D&G also link the concept of the virtual with that of the multiplicity - the virtual multiplicity. This is real without being neccesarily embodied in the world. (D&G frequently quote Prousts adage regarding memory when discussing the virtual 'Real without being actual, ideal without being abstract'). Rather than expressing abstract alternative possibilities, they form something like the real openess to change that inheres in every particular situation.

The virtual and abstract effect changes in each other. Existence is then actual multiplicities - states of affairs - and virtual multiplicities - particular intensive movements of change.

The concept of multiplicity makes no reference to a transcendent realm, that contains the laws of existence. The virtual counterparts of our actual multiplicities make possible continued change even at the points of world of actuality seems rigid and oppresive.

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"multiplicity", substantive, "in-itself". well articulated in terms of 'different-in-itself' (and not 'different-from') and 'repetition of difference', or repetition within the flux of continous change occuring as difference, i.e. multiplicity. think echoes, traces, vibrations, ripples, etc. the segmented body of the tapeworm with each seperate fragment differentiating into another body of an organism itself.

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