Why is Leibniz
so important to Deleuze?
I came to reading The
Fold rather late, and had noted lots
of references to Leibniz, especially in Logic of Sense,
and Difference and
Repetition. There were
discussions in particular of notions such as
incompossibility instead of contradiction, but,
more generally, some admiration for the
calculus. Other discussions were Leibnizian
even though I did not realize it at the
time, especially concepts like the singularity,
and the stuff about how multiplicities offer a
series of ordinary and singular points stretching
to the neighbourhood of the next
multiplicity. I think I can now see why
Leibniz in general, and the working out of the
calculus in particular, are so important.
Let's take the calculus first. I'm not
mathematician enough to understand the precise
arguments involved in Leibniz's work, but I can
see some general points. In the first place,
the calculus helps us to analyze the shape of, and
the areas under, curved lines, whereas classical
mathematics had operated with straight
lines. Not only is the curved line an
interesting mathematical problem, but analyzing
curved lines helps us grasp real problems, in
nature, which is riddled with curved lines.
Leibniz proceeds, according to the notes I read on the
calculus, to consider infinitesimal differences
and the series of them, and then projects this
notion forward, as it were, to explain the slope
of the lines of tangent, even very very small or
infinitesimal ones. We can see this in terms
of the practical procedures of rendering the
curved line as a series of straight line tangents,
but to get an accurate description, we need an
infinite number of tangents. What Leibniz
did was to manage this problem by saying that we
can describe infinitesimal differences as ratios,
without needing to actually put in real numbers or
concrete measures, and this is dead handy if we
are discussing infinity, and also very handy if we
are discussing the forces that produce curves,
which might not necessarily work on the same
numerical base. We also need to talk in
abstract terms like lines segments rather than
lengths. The particular ratio that
interested Leibniz was the one between rates of
change in the Y and X variables. Those
variables can be seen as offering two sides of a
right triangle with the tangent as the third side,
and if we know the values of X and Y, we can
calculate the value of the tangent. Moving
away from concrete values to ratios, we can talk
about the abstracted changes in X and Y and their
ratio—giving us the famous differential equation
of DY/DX, where D stands for the rate of
change. This is a constant ratio, at least
between singular points on a curve, where the
slope is constant. At singular points, the
curve slope changes.
Although this particular approach needed
modification, and we had to ignore some issues for
the time being, including intrusive philosophical
notions of the infinitely small, we made progress
in explaining the slope of the curve, and, in
turn, this helped us calculate the area under a
curve, or quadrature, and we can do this even for
curves that extend even to the infinitely
large. Well done, Leibniz. But in the
process, he had developed mathematics away from
concrete values, away even from normal algebra, to
the study of relations, something that was to lead
to the notion of mathematical functions.
Deleuze often cites the example which makes this
point, saying that even when the actual values of
X and Y are zero, the relation between DX and DY
persists [if I have to followed this, it is
because the relation is derived from actual
triangles drawn with positive values, which are
not equal, whereas if X and Y are reduced to zero,
they would be made equal, using conventional
rules. So stuff conventional rules, let's
give calculus new rules].
This exercise in the calculus is part of a far
more general approach to develop a mathematical
understanding of philosophy and of real problems,
again by developing a particularly abstract grasp
of them. We can see complex real objects,
for example, as a series of curves. We can
explain continuity in a new way, as regular
movement from one singularity to another.
This is clearly going to assist Deleuze in his own
project to account for chaosmos in terms of
mathematical concepts like multiplicity,
singularity, and force, an underlying process that
produces fixed points.
Equally important is the connection between
mathematical processes and reasoning like this,
and the more general 17th century notion of
sufficient reason. One of the main
implications here, it seems to me, is that if we
invoke the principle of sufficient reason, we
proceed by rational analysis to unpack the
predicates of a particular concept as
subject. Predicates can be events.
Mathematics seems to proceed in this way, to
nonmathematicians like me, by taking a concept
such as the function, and unpacking more and more
implications from it, developing implications in
the form of a whole series of equations, then
exploring the implications, and so on.
Philosophical analysis in general was to proceed
in this way, via a number of ingenious discussions
in Leibniz to extend the principle of sufficient
reason in order to make it consistent with the
other philosophies and other concepts like
identity.
It strikes me that the sort of philosophy embraced
by Deleuze follows exactly this pattern.
When he talks about what the concept is, he means
it in the same sense as Leibniz, a subject
with predicates, something that can be inferred by
working back from its predicates, although this
does not necessarily give us the entire concept,
and often means that we have only 'fuzzy'
definitions. And when he talks about how
forces turn into realities, via actualizations, he
means something like the concept unpacking itself,
producing a chain going the other way, so to
speak, offering more and more limited definitions,
and more and more empirical accounts of the
relations between them. [I can also see some
links with Spinoza here]. This also explains
his disdain for empirical analysis, of the kind
you might find in sociology, which operates on the
surface, to change the metaphor a bit, too far
down the chain towards the empirical end. As
we get closer to actualities, we can define them
empirically and also use relations between them
such as causes: but there are deeper definitions
and deeper forces and relations further back up
the chain, and it is those that philosophy needs
to investigate [I am probably using terms
developed by DeLanda here].
There are some ore similarities. The Leibnizian
method of vicediction, for example, where one examines concrete cases,
including the accidental ones, in order to trace
the operation of the virtual. This is the sort of
procedure developed in
Logic of Sense ,where it is
described as 'counteractualization', going from
the actual to the multiplicity, and in Difference and
Repetition. Deleuze's
admiration for it is repeated in a lecture
he gave on the 'method of dramatization', where he
said it 'consists in
traversing the Idea as a multiplicity', which
produces all cases, the accidental and the
apparently essential alike.
(Facebook 23/01/2017) In
differential calculus, D(x) means let
x be any infinitesimal difference in
x. Let D(y) be any infinitesimal
difference in y. An infinitesimal
difference in either x or y, while
infinitesimal in itself is significant
in the relationship of x to y formulated.
The relationship of x is altered by
any even infinitesimal difference in
x or y [the butterfly effect]. If x
is a straight line and y is a
straight line, any infinitesimal
difference in either one of them
will cause them to intersect. What
this looks like in a formula is
D(x)/D(y)=change in x/change in y,
and when x and y are given values
(actual, determined), the
differential (virtual, determinable)
can be calculated. What gets
calculated is then actual, however
the trajectory of the the calculus,
the curve between variables has
infinite points along the curve,
calculated between 0 and infinity
which are clear limits of reality
expressed mathmatically.
For example, Velocity = distance
divided by time, meaning that we can
use this relationship in reality
(velocity in this case) to see how
differences (variable in
relationship to each other) apply
(distance and time in this case).
Why this matters is because it is an
Actual expression of how Virtual
Ideas operate for Deleuze—they vary
in relationship to one another as
any differences/variables vary in
the real world. From start to
finish, the faster I go and the
further I go in my car, the greater
my velocity (miles per hour).
These values vary from start to
finish, and from distance a, to
distance b, to distance c, etc. At
the same time there would be time a,
time b, time c, etc., so we could
calculate the velocity at any point
a, b, c, etc. by plugging the values
in to the formula: velocity =
distance divided by time, =d/t. So
how many points a, b, c are there
from start to finish? Actually,
there are an infinite number of
points which we could calculate.
These infinite number of points
are actually calculated by
differential calculus, a method to
cover all the possible points. Well
how can that be done? In order to
calculate target point x on the
start to finish line, Leibniz who
had not yet invented differential
calculus) chose to start with the
average velocity of the car over a
time (t) and slowly decrease the
slice of time we use to divide the
distance (d) travelled so that d/t
would approximate (infinitely
approach) the target point x. This
had the effect of making t differ
from d in incrementally smaller time
periods until arriving at the target
point x. Keep this difference, this
derivative, this differential
calculus in mind as Deleuze’s
concept of how Ideas function in
Virtuality. He is merely using
mathematics here as a model of the
Virtual.
We have a limited faculty of
thought, so all we can see, all our
brains can calculate are the given,
the mere appearance of things, like
looking through a darkened glass.
Instead of thinking that we fully
synthesize an object of perception,
we need to realize that our thought
is a synthesis in process, an
‘arising’ or ‘flowing.’ Therefore we
become subject to the transcendental
illusion, or representations of
objects as if they were complete and
original because we dont
automatically do differential
calculus of all variables at all
times (and not given to intuition).
The differential calculus
relates the three moments of the
Idea intrinsically. It is:
1) Undetermined in that the
differential cannot be given in
intuition
2) Determinable when it is put
into a relation dy/dx
3) Determined when given
specific values
The calculus provides symbols of
difference (individual values) which
although propositional, points
beyond itself to the problem itself.
Thus the calculus accounts for how
undetermined elements can become
determinate through reciprocal
relations, not only in mathematics,
but in the world at large
The differential calculus “has a
wider universal sense in which it
designates the composite universal
whole that includes Problems or
dialectical Ideas, the Scientific
expression of problems and the
Establishment of fields of
solution.” (p. 181)
See also my thoughts on
folds
Back to Deleuze
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