What does Deleuze mean by “the world can be regarded as a 'remainder'”?

Question

Deleuze in Difference & Repetition writes in the chapter named as the Assymetrical synthesis of the sensible:

It is therefore true that God makes the world by calculating, but his calculations never work out exactly [juste], and this inexactitude or injustice in the result, this irreducible inequality, forms the condition of the world. The world 'happens' while God calculates; if the calculation were exact, there would be no world. The world can be regarded as a 'remainder', and the real in the world understood in terms of fractional or even incommensurable numbers.

To some extent, this appears as a 'translation' of the Manicheasm (the world is corrupted) and Occasionalism (God intervenes not only at creation but at all times) into contemporary philosophical diction, or at least into (Deleuzian diction or even posthumously invented Pythagorean diction).

But what is meant by the last sentence:

and the real in the world understood in terms of fractional or even incommensurable numbers.

I'd suggest it being as cognisable (fractional as ratio as rational) or incognisable (incommensurable as not a ratio, and thus not rationable).

Is this correct? Or is there a better interpretation?

Answer 1

More of an extended in passing comment than an answer, but I thought some of it might be constructive. Deleuze takes up the theme of "rigorous and inexact" notions at the heart of science and philosophy elsewhere in Negotiations. Even though they're discussing A Thousand Plateaus, the discussion there maybe has some valences in common here, in terms of liberating systems of differential relations from interpretations according to axiomatically "pre-formatted" multiplicities, either quantitative or qualitative.

Another quick thought. If we suppose the world as the result of calculation, perhaps God-or-nature is computational! One of the appendices to Logic of Sense is about simulacra and might be a potentially useful place to go for a little more in this vein (although note that the term more or less disappears from Deleuze's work after D+R).

Answer 2

This is not a direct answer to the question, but more context, and thus it may be useful for others.

To me, Deleuze is either a genius or a madman. His works seem as some form of nonsensical free-verse poetry littered with alliteration and assonance, yet there may be something meaningful within that is encrypted therein, requiring more than intelligence to open but also a keen eye for puzzle solving and lots of patience.

Here is the full quote in context. I added footnotes for the principles he mentions.

Chapter V

Asymmetric Synthesis of the Sensible

Difference is not diversity. Diversity is given, but difference is that by which the given is given, that by which the given is given as diverse. Difference is not phenomenon but the noumenon closest to the phenomenon. It is therefore true that God makes the world by calculating, but his calculations never work out exactly [juste], and this inexactitude or injustice in the result, this irreducible inequality, forms the condition of the world. The world 'happens' while God calculates; if the calculation were exact, there would be no world. The world can be regarded as a 'remainder', and the real in the world understood in terms of fractional or even incommensurable numbers. Every phenomenon refers to an inequality by which it is conditioned. Every diversity and every change refers to a difference which is its sufficient reason. Everything which happens and everything which appears is correlated with orders of differences: differences of level, temperature, pressure, tension, potential, difference of intensity. Carnot's principle^[1] says this in one fashion, Curie's principle^[2] in another. There are locks everywhere. Every phenomenon flashes in a signal-sign system. In so far as a system is constituted or bounded by at least two heterogeneous series, two disparate orders capable of entering into communication, we call it a signal. The phenomenon that flashes across this system, bringing about the communication between disparate series, is a sign. "The emerald hides in its facets a bright-eyed water-sprite . . .": every phenomenon is of the "bright-eyed water-sprite" type, made possible by an emerald. Every phenomenon is composite because not only are the two series which bound it heterogeneous but each is itself composed of heterogeneous terms, subtended by heterogeneous series which form so many sub-phenomena. The expression ‘difference of intensity’ is a tautology. Intensity is the form of difference in so far as this is the reason of the [object]. Every intensity is differential, by itself a difference. Every intensity is E – E’, where E itself refers to an e – e’, and e to ε – ε’ etc.: each intensity is already a coupling (in which each element of the couple refers in turn to couples of elements of another order), thereby revealing the properly qualitative content of quantity. We call this state of infinitely doubled difference which resonates to infinity disparity. Disparity—in other words, difference or intensity (difference of intensity)—is the sufficient reason of all phenomena, the condition of that which appears.

_{[1] Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain, which thus solely depends on the difference between the hot and cold temperature reservoirs.
http://en.wikipedia.org/wiki/Carnot%27s_theorem_%28thermodynamics%29}

_{[2] The Curie symmetry principle (Curie, 1894) is the causality relation between the symmetry of the cause and that of the effect. The principle is composed of three parts: - If certain causes yield the known effects, the symmetry elements of the causes should be contained in the generated effects. - If the known effects manifest certain dissymmetry (absence of symmetry elements), this latter should be contained in the causes which have generated those effects. - The converse to these two previous propositions is not true, at least in practical: i.e., the effects may have higher symmetry than the causes which generate these effects. Curie's principle expressed in other words: a crystal under an external influence will exhibit only those symmetry elements that are common to the crystal without the influence and the influence without the crystal.
http://www.mi.sanu.ac.rs/vismath/visbook/sydchiba/
http://www.mx.iucr.org/iucr-top/comm/cteach/pamphlets/18/node8.html}

Answer 3

Firstly - in DR Deleuze is primarily giving a theory of individuation, using Maimon's critique of Kant as a wedge (that knowing the conditions of possibility is not enough - these ground, as it were, multiple potential realities. We need a complimentary account of the conditions of genesis of the actual).

Deleuze's theory of individuation is based on difference, in this case referred to as the remainder - if there is no remainder, there is no difference, and so no individuation - and so no world/things/people. Total homogeneity can never give rise to anything - emergence requires minute intensities and scarcities in the underlying milieu, between which must exist gradients of energetic intensity (this is what the reference to Carnot and Curie attempts to get at).

This is the thesis, anyway. Almost every sentence in the paragraph refers to earlier ones. it's not really a justificatory passage but a bringing-together or disparate strands. From what I remember most of the justification/argumentation is in ch. 1-3, the rest being elucidation and drawing out.

Re. 'the real' - the real for Deleuze includes both actual and virtual (potential). I expect that the irrational numbers are the numerical equivalent of that which cannot be given to sensation - i.e. the virtual, pre-individual ground of the actual which is Difference and out of which the actual is individualised.

Answer 4

Be more literal IMO. He probably means that reality consists of both the world and God's poor mathematics, by saying any remainder can be expressed as a fraction that is the real.

http://en.wikipedia.org/wiki/Commensurability_%28mathematics%29

Moreover, in saying that the world and the remainder of it is even expressible as a ratio, that all of reality falls under the domain of math.

This follows from basic reading comprehension if you take his word for it.

Looks like you're trying to take a short cut.

The world can be regarded as a 'remainder', the real in the world understood in terms of fractional or even incommensurable numbers.