On Folds

Explicit discussion of the fold arises extensively in Deleuze's book on Leibniz, and in a very interesting comment on subjectivity in the book on Foucault.  There are also references to Leibniz more generally in several other places, including Difference and Repetition.  Why is this notion of the fold so important? I think there are three reasons, best displayed in the work on Leibniz:

1.  If we want to understand natural phenomena, we have to have some notion of curved lines (see notes on Leibniz) .  Curved lines provide one definition of folds.  Curved lines are everywhere rather than the straight lines dealt with in classic Greek geometry and in positivist science and mathematics.  We see curved lines in the shapes of trees and leaves, rivers, and, more abstractly, in the distribution of qualities of various kinds.  While the old approach tried to simplify curved lines by seeing them or sequences of straight lines, Leibniz and others argue the reverse -that straight lines are really highly compressed curve minds.  Leibniz was to celebrate the curved line in his mathematical attempts to develop the calculus.  Whereas we can estimate the slope of the curved line by calculating the slope of its tangent, we get into some infinite regress very rapidly, since there are an infinite number of tangents that we can draw on each point of a curved line.  If I understand it correctly (which is by no means guaranteed), Leibniz's  major contribution was to develop an abstract notion of the relationship between verticals and horizontals (rise, or changes on the Y axis, over run, changes on the X axis) that we used to calculate the gradient of straight lines.  As Deleuze argues, this led to some new and incredible arithmetical conclusions, for example suggesting that the ratio can persist even where the values (judged by normal arithmetical students) are actually zero for both run and rise.

What seem to be straight lines are really highly compressed curved lines, including those straight lines that appear as plateaus.  These are particularly linear areas of curved lines.  They seem to be self sufficient and 'objective', but they are connected by curved lines under the surface, as it were.  This explains how we might move from one plateaus to another by shifting towards these subterranean curved lines, as argued in Thousand Plateaus. One you get the hang of this, you see folds everywhere,and Deleuze lists a few common ones: fans, complex double folds in rock formations (lovely ones on the metamorphosed shale at Boscastle Harbour, Cornwall) , elaborate folds in painting and sculpture, Greek folds like this one:

greek folds in

It follows that if natural phenomena are really organized by equations expressing curved lines, or folds, that our understanding of them will also be affected.  This is argued most spectacularly in the discussion on perception.  The perceiving subject occupies a position on a curved line.  This is clearly indicated in the diagram below.  It is the curved line that provides us with what Deleuze's was to call 'percepts'.

Leibniz also describes the upper floors of the monad in terms of the projection of sensory data onto folds in the soul or in consciousness ( the diagram below is from Deleuze's book on the fold p.4) .  In other words, consciousness also operates by a process of folding and unfolding, connecting things, and also explicating particular folds such as concepts. The implications are unfolded in subsequent reflection: this is analysis and explanation in the 17th century, quite unlike anything empirical like sociologists do.


A particularly interesting case arises when a concave curve provides us with a particular enclosure of reality.  This is what Deleuze uses to explain Foucault's conception of subjectivity, as in the diagram below.  Here, the concave curve or fold is turned into something like a hem, by the actions of social institutions which prioritize particular forms of closure.

We also have a diagram illustrating the use of the fold to explain the problem of subjectivity in Foucault (see Deleuze's book on Foucault),and maybe in Deleuze too. Here is the diagram:

foucault on

Note that the fold in the middle is closed off at the top. It is like a hem ( and Deleuze says Mrs Deleuze though of this).What is insied is expereinced as subjective, personal, ourr individualthoughts and expereinces,but the whole thing is really an enclosed chunk of hte outsiode world allalong. It is constrained by social strat ( things like class andpiwer sysrtems) oneither side. It ispossivle to adopt a stategic approach to the outrside world through what Foucualt calls strategies,garabbing elementsor atoms of the outside worlk ( biutnot,tgypoically ists structure or ontology). The heavy baclk shading that close ofthe hem represnets the role of social institutions,including language and culture,  which make llife manageable for us by shuttingout a lot of the complexity of hte world

2.  The fold seems to have informed a particular kind of the baroque aesthetic.  Because they could, and, I think, because they wanted to demonstrate kind of capitalist excess, baroque architects and fashion designers developed an elaborate a form of folded cloth or folded stucco to decorate the exteriors of their bodies or buildings.  Deleuze just notes this aesthetic, and perhaps traces it through to the emergence of some modern aesthetic forms, notes the importance of unfolding as a non deterministic account of embryology and evolution, and admires modern conceptions such as the Mandelbrot set.

folded clothes

3.  I think I can also detect a theological importance of the fold, combined with the 17th century notion of sufficient reason.  This notions said that everything that exists must have a reason, and, in philosophical terms, this meant that it must be explicable in terms of some underlying concept (it is not at all like empirical notions of reason which explain existing phenomena in terms of various empirical laws which produce regularities).  If I have understood this in Leibniz, who was a Christian, the purpose was to show that God could be found literally in everything, and that god was not fragmented in any sense, even though empirical objects and things appeared to be quite separate and self sufficient.  The answer was that God took a shape like a fold, producing empirical objects at particular points on the curve, may be particularly densely folded points, but persisting throughout the other regions of the curve.  If I have understood this correctly (and there are the same reservations as above), this was also an argument found in Spinoza, although related to Substance rather than God.

I think Deleuze uses this argument to show how it is that virtual reality actualizes itself in every aspect of empirical reality, although there might be different stages of actualization, ending in full realization in the book on Leibniz, or offering different species of differenTiation and differenCiation in Difference and Repetition, as virtual reality differenTiates itself first, through mathematical variation, and then empirical reality differenCiates itself through various empirical processes such as evolution or a move towards autonomy and complexity.