Brief notes
on: Deleuze, G. (2007?) On Leibniz
Dave Harris
NB online so no page numbers for
quotes
It is useful to look at a life of Leibniz,
by Prenant, so that we can get an idea of
writing styles in use at the time, even though
Leibniz exaggerated them and we also find his work
develops not only in books, but in letters and
memoirs.
We can see Leibniz as a great example of what a
philosopher does, how a philosopher creates
concepts, just as artists and musicians
create. It follows that concepts have to be
created. Science also creates thought, but
not concepts exactly: art is closer; both are a
matter of working with flowers, dividing
organizing and connecting them 'around certain
singularities'. Philosophers do not even
have to be intellectuals, as long as they can
extract singularities from flows of thought, from
inner monologues. Even nonphilosophers have
to use concepts invented by named philosophers,
who sign them. Some of these signed concepts
seem appropriate to us in particular
circumstances, and we need not become a complete
disciple. Concepts are ways of living as
much as thoughts, hence what counts is the
'creative activity .. not… a reflexive
dimension'.
But at the same time, concepts can appear as far
away, as arising from 'the scream'.
Philosophers don't sing but scream—they need
concepts, and we need to trace the concept back to
the scream. It is not just a matter of
having passions. Leibniz is a rationalist, a
philosopher of order, including organizing the
city. However, at the same time, 'he yields
to the most insane concept creation that we have
ever witnessed in philosophy'. It all stems
initially from the notion of universal
reason. Some approach the problem calmly and
rationally, as does Descartes, working on the
basis of an entire tradition, and still coming up
with a new concept as in the slogan 'I think
therefore I am'.
Leibniz creates a whole batteries of new concepts,
partly as a result of the power of the German
language: he is mathematician, physicist, jurist,
and politician. He is not always admirable,
for example, when he conspicuously failed to
defend Spinoza—'Leibniz is abominable'. He
also had a sense of humour. He built a
philosophical system with several levels which
'symbolise each other'. These levels are
used as he argues with people at different levels
of insight themselves, which makes the texts look
incoherent and contradictory: we always need to
evaluate the level at which he is working.
He is a 'very difficult philosopher' with strange
or as he calls them 'funny' thoughts, almost like
science fiction. He was very interested in
games, and suggested establishing an academy for
games, which would straddle a range of disciplines
as a source of 'universal exposition'. His
work can be summarized as consisting of a number
of principle propositions.
The first one is the principle of identity,
which can be rendered as saying that a subject
produces an attribute or predicate which is
logically reciprocal—a triangle is a
triangle. In some cases, this helps us
uncover even more attributes, as when we say a
triangle has three angles [true by definition] and
it also follows that a triangle has three
sides. This is a logical necessity, but not
the result of reciprocity: it is inclusion or
inherence. So we have two kinds of
propositions, reciprocal and those demonstrating
inherence or inclusion. Other way of putting
this argument is to say that 'every analytical
proposition is true'[must be, because it is either
tautologous/reciprocal, or logical/inherent,
inclusive]. In analytical propositions, the
predicate is identical with the subject or
included in the subject, and to get to predicates,
or we have to do is interrogate or analyze the
subject. So far so normal, and we haven't
really got on to proper philosophy yet.
The new thought arises from thinking out
reciprocity, and this had to be thought through,
because it met a need. We consider the
inverse of the statement of identity above
and argue that 'every analytical proposition is
true'. This is not tautologous or obviously
logical, or need not be. What it means is
that every true propositions must be analytical
whether we want this to be the case or not, [we
are implying that we will be able to analyze true
propositions because a true statement must contain
within it, necessarily, certain predicates which
will emerge, whether or not we welcome
them]. There must be attribution and
predication already contained in the true
statement. If we think in terms of concepts,
the predicate of the concept is either reciprocal
with its subject, or inherent in it.[I think we
have a slight of hand here  two notions of
truth,one empirical and one logical, as logical
positivists might say. The 'true' in the first
case every true statement is analytical 
is a logical truth. The 'true' in the second case
is really an empirical statement  'every
analytical proposition is true' is really a
claim that analytic propositions point to some
truth about reality, that reality is organized
like that so as to produce analytical statements,
which means it must have predicates {or events}
contained in subjects and all that. Then there is
a confusing weasel since we are denying that it is
real subjects only notions of subjects, which
makes it a logical truth again  but then we let
real subjects slide in. I don't think any of these
C17th 2level explanations work, not Leibniz nor
Spinoza  Delanda has rescued Deleuze by saying
you need some modern physics to explain changes in
state].
We have to work this out, however, because not all
judgments about reality seem to be matters of
attribution or inherence. Consider
statements such as there is a box of matches on
the table: we have a spatial determination, but no
predication or attribution. There are also
relational judgments, where X is larger than Y—we
can be reciprocal here and say that Y is therefore
smaller than X, but again we don't exactly have
predicates. So is the judgement of
attribution universal, or just one form of
judgment? There are three kinds of judgment
cited here already —relational, spatial, and
attribution, and then again maybe many more like
'judgement of existence' like trying to work out
the qualities of god, but adding 'if he
exists'—his existence is not just an attribute.
Leibniz must therefore show how all propositions
'can be linked to the judgement of attribution'
nevertheless, and things like relations,
localizations and existents should all be seen as
the equivalence of attributes. This is going
to be 'an infinite task', and if pursued, it is
going to produce a strange world.
We also have to redefine our terms a bit, because
it's the case that not every analytical
proposition is true [for example the famous
attributes of fictional people or animals like
'the present King of France'?]. On the other
hand, Leibniz insists that 'every true
propositions is necessarily analytical'.
This leads us to the principal of sufficient
reason. Everything must have a
reason. The principle means that 'whatever
happens to a subject... everything that is said
[truthfully] of a subject must be contained in the
notion of the subject'. What is the
notion? It is something also produced by
reason—reason is precisely the notion itself
insofar as it contains all that happens to the
corresponding subject' [classic circularity here,
it seems to me, where sufficient reason requires a
'notion' of the subject, and, happily, the notion
is itself a product of reason?]. The notion
is a characteristic signed Leibnizian concept
So reason involves a notion of a subject, which
must contain 'everything said with truth about
this subject'. This notion clearly underpins
the reciprocal identity principle which sparked
off the whole philosophical investigation.
It is going to produce a 'bizarre world' where
'everything that you say with truth about a
subject must be contained in the notion of the
subject'. It is also going to be a lot of
work to demonstrate this.
As Leibniz thinks about the implications, he is
going to develop some 'truly hallucinatory
concepts' in his metaphysics. It arises from
connection with the original scream [likened to
delirium here]. We can understand this
'conceptual madness', however. First,
though, it looks like Kant offered the notion of
synthetic judgment to accompany analytic
judgments, and this can be seen as opposing
Leibniz—but this is not disagreement, merely a new
concept: philosophers no more contradict each
other than painters do when they develop different
styles.
To take Leibniz's actual examples, it is true that
Caesar crossed the Rubicon ('we have strong
reasons to assume it's true'), and we can also
makes statements such as 'Adam sinned'.
These are propositions about events
('eventual'). The argument is that the
predicate, 'crossed the Rubicon' must be contained
in the notion of the subject Caesar, if it is to
be true. This notion contains everything
that happens, as long as we can make truthful
statements. This is the 'concept of
inherence. Everything that is said with
truth about something is inherent in the notion of
this something'.
This leads to something endless, however.
Unlike Aristotle, Leibniz does not know where to
stop, even when a chasm opens. The problem is that
the notion of the subject contains everything that
has happened to that subject, and this would
involve ultimately 'the totality of the
world'. We can see this by considering
causality: the things that caused Caesar to cross
the Rubicon can be extended to infinity, as all
causes can be, because they themselves have
causes. It also follows from this that
sufficient reason is something more than cause:
causes can only ever be necessary but not
sufficient. Sufficient reason however 'is the
notion of the thing… Expresses the relation
of the thing with its own notion, whereas cause
expresses the relations of the thing with
something else'. Effects similarly stretch
off themselves to infinity.
As cause links with cause, the whole of the world
becomes 'encompassed in the notion of a particular
subject'. This gives to philosophy 'a
transhistorical characteristic'. It also
invokes another term—not inherence or inclusion,
but 'expression', so 'the notion of the subject
expresses the totality of the world'. That
means that each of us do as well. This
raises the possibility of 'the single subject, a
universal subject', with individuals just as
appearances of the subject. Leibniz had to
repudiate this view, however, because this would
simply take over [reduce] the argument about
notions and sufficient reasons [classic
philosophical argument!] . In this he offers
'the first great reconciliation of the concept and
the individual'. This came as new, because
concepts were thought as something general,
applying to several things: concepts referred to
generalities, individuals to singularities, and
individuals could not be reduced to concepts:
individual characteristics, such as possessing a
proper name, cannot be generalized to produce
concepts [there is no concept of Daveness,
although we might be able to make empirical
generalizations about them].
Leibniz always worked with the subject as an
individual, the singularity, and we can generalize
this into 'the perpetual formula in his works:
substance (no difference between substance and
subject for him) is individual'[apparently,
Leibniz weaseled subsequently to say that even the
notion of God did not prevent substance being
irreducibly individual].
There is an implication for individual freedom,
since it follows that Caesar [as a notion anyway 
and here we have pretty well slipped in a
statement about the empirical Caesar? Are notions
separate from actual empirical subjects or not?]
can only unroll, explicate, or unfold 'something
that was encompassed for all times in the notion
of Caesar'. The action of crossing the
Rubicon is eternal [reminds me of Ahab's speech
about how the conflict with the whale was eternal
and predestined]. Leibniz was forced to
address the issue of freedom, while reconciling it
with a Christian regime [discussed below].
If everyone expresses the totality of the world,
how can we distinguish one subject/substance from
another? What is different is the point of
view. Again this leads to a series of
philosophical concepts, although they take the
form, as so often, at first of 'banal formulae',
at which the philosopher 'winks'. The theory
actually develops following his work in projective
geometry, and as part of the general experiments
about perspective [in the Baroque]. The
point of view is not just produced by the subject,
but rather the other way about. Nietzsche
was able to pick this up later, and also argue
that his 'philosophy is a perspectivism'.
It is not just that everything is relative to the
subject, this is banal and meaningless.
Instead, 'the subject is constituted by the point
of view', not the other way about. Henry
James renews the argument in his own
perspectivism, and makes the subjects in his
novels vary according to particular points of
view. Indeed, points of view become the
'sufficient reason of subjects', explaining
them. It follows that there are as many
universes as there are substances, or, in
Christian terms, as many completely different
representations of god.
The points of view defines individual essence,
otherwise they would be coterminous with living
individuals. Points or view were explored
before, for example in portraiture, [and in
experiments with anamorphosis—apparently, a
portrait of Mussolini also depicted his son in law
and the King, depending on points of view].
Leibniz explored it through mathematics and
projective geometry [via perpendiculars to
tangents converging in concave curves etc see the book on Leibniz ]
. Here, the point of view is expressed by
subjects, but only 'in an obscure and confused
way', principally because individuals only operate
with 'the form of minute perception…
Infinitely tiny perceptions… Unconscious
perceptions'.
Deleuze says there is a clear parallel here with
Leibniz and the development of the notion of
infinity in differential calculus, and promises to
explain more fully later. Here, we can see
these minute perceptions as the differentials of
consciousness. It means we have our own tiny
forms of consciousness, without being conscious of
them! Leibniz uses the term apperception
to refer to conscious perception. So the
point of view that I express consciously is but a
tiny portion of the world. It is like having
something in focus in your own little zone, while
everything around it remains fuzzy, confused, in
uproar. The zones can vary in size.
They are never the same [and they are not just a
matter of subjective knowledge or understanding,
but objectively determined?].
In an example [which Deleuze likes], Leibniz says
that we can be conscious of the sound of a wave of
the sea, we apperceive it as something distinct,
but this depends in turn on tiny unconscious
perceptions of the sounds of each drop of water as
a background. We clearly identify 'one
partial result from this infinity of drops' and it
is this that makes up our own little world'. In
this way, an apperception, a conscious synthesis
of unconscious perceptions helps us make our own
little worlds in the form of 'clear and distinct
expression(s)'. Individual and worlds may
not communicate, they may partially communicate
through analogies, or they may overlap [as in a
Venn diagram]. What guides the particular
expressions that we develop is our relation to our
body (to be explained below), so that I can never
express what Caesar expressed. This has got
implications for the conception of the city as
well [a term for society?] .
It is hard to accept that nothing exists outside
points of view, that the world is 'uniquely the
common expressed of all individual
substances'. Leibniz looks like an idealist
here. Since points of view do not always
overlap entirely, the world exists only as 'the
complication of the concept of expression'.
There are problems because the principle of
identity also enables us to argue that some things
must be contradictory or impossible, where A is
not A. This is a relief, because we can at
least demonstrate that some events or concepts are
logically contradictory, such as a square
circle. However, the notion of individuated
little worlds is not necessarily
contradictory. Leibniz also needs to save
the notion of individual freedom as well, to allow
the possibility, for example, that Caesar could
not have crossed the Rubicon: this would not
exactly be a logical contradiction so it needs to
be explained. So there is a difference
between truth produced by the principle of
identity with its logical impossibility of
contradictions, and truths produced by the
principle of sufficient reason, where
contradictory outcomes are at least possible:
'Adam the nonsinner is possible'.
Here Leibniz has to distinguish between 'truths
of essence and those called truths of existence'
[nearly a distinction between logical and
empirical truth then?]. Where do empirical
possibilities come from, if everything is bound up
in the subject? The answer lies in the
concept of incompossibility—what actually
takes place at the level of existence has to be
compossible with the rest of the world
unfolds—'Adam the nonsinner belongs to another
world… [And]… This world was not
chosen. It is incompossible with the
existing world. It is only compossible with
the other possible worlds that have not passed
into existence'.
Why did this world pass into existence? We
get back to a theory of games. God conceives
an infinity of possible worlds, but designs them
so that they are not compossible. He then
chooses 'the best of possible worlds', [the bit
that's parodied by Voltaire's Dr. Pangloss] and
this happens to involve Adam as a sinner.
Compossibility limits the sphere of logical
possibility. For something to exist it must
be logically possible but also compossible in the
real world.
There is a connection with the monad, where
'individual notions have neither doors or
windows'. This implies there is no exterior,
that 'the world that individual notions expresses
is interior, ...included in individual
notions'. Nevertheless, individual notions
share a common world, since each one expresses not
only its own little world, but one that must be
'compossible with what the others express'.
This does not arise as a result of communication
between subjects, but refers to 'pre established
harmony'. This programmed harmony implies
'the idea of the spiritual automaton', and
automata were becoming popular in the 17th
century.
As a background to the general principles there is
a corresponding general problem, relating back to
what it is to do philosophy. We know it is
creating concepts, and Leibniz is a good example
of the 'exuberant creation of unusual
concepts'. These concepts can be seen as
signatures. We can now explore further 'the
notions of Substance, World, and Continuity'.
The first thing to remember is the different
senses of the term inclusion. Predicates
must be included in the notion of the subject if
propositions are true, so if Adam sinned, the sin
is included in the notion of Adam. This is
Leibniz's 'philosophy of predication'. One
problem is, as we saw, but this leads to the idea
that if a single action is included, so must be
all the things that caused it and all the affects
that follow, so that we end with the totality of
the world included in the notion of the subject in
this way, rationalism ends with a kind of
delirium, or madness. It ends in
perspectivism, remembering that this is not the
usual subjectivist kind.
So are all relations and all events just
predicates? If we look at essences first, we
can see that they can be contained in analytical
statements, as in mathematical truths, where the
sum of two integers is obviously contained in the
two integers themselves, indeed, is identical to
them. In a particular piece, Leibniz is able
to show that every number which is divisible by 12
is thereby divisible by six, or 'every duodecimal
number is sextuple'. The way he did this was
to show that the number 12 can be seen as
something divided by two, multiplied by two, then
multiplied by three, whereas numbers divided by
six can also be seen as divided by two multiplied
by three. So the operation 'divide by two
and multiply by three' is included in the
operation 'divide by 12'. In this way, a
finite series of determinate operations can
produce aggregates, as a mathematical version of
showing that predicates emerge from the properties
of the subject, and inclusion. This is also
a way of producing essences, as it were: we
already have an essential statement, where any
number divisible by 12 is divisible by 12 (the
'pure identity'), but we now also know that any
number divisible by 12 is divisible by six.
We have used finite and determinate operations to
produce this result, that one essence is included
in the other. This is how analysis proceeds
in maths in general, using 'a limited number of
determinate operations'.
What about nonessential [non grammatical]
truths? The truth of particular events like
crossing the Rubicon is not an essence in this
way, but a dated event related to existence, as
opposed to the eternal truths of
mathematics. Nevertheless, that predicate
was in the notion of the subject eternally—in all
eternity, Caesar will cross the Rubicon at a
particular date. 'This is a truth of
existence'. [Because his options have
already been limited by the notion of
compossibility]. Nevertheless, there are
still differences between truths of essence and
truths of existence, even though, for both,
predicates are contained in subjects. The
difference lies in that analysis can only be
infinite in truths of existence, because we can
never finish analyzing the totality of the world
that is implied, as we saw. Limited
empirical human beings cannot do this, because it
lies beyond our experience, so 'infinite
analysis… [While technically possible,
is]… Created in the understanding of God'.
However, Leibniz has also helped to pin down the
notion of infinity at both ends of the scale, in
mathematics. Although he denied a direct
link between philosophy and mathematics, Deleuze
detects 'echoes' of the calculus in the
philosophical notion of infinite analysis.
It suggests a way to solve the problem of what
infinite analysis looks like, while using
determinate analysis.
However, there is also another difference between
truths of essence and truths of existence.
The truth of essence cannot be contradicted,
because the terms have an identity. However,
truths of existence can be contradicted—there is a
'contradictory of sinner'. This makes truths
of existence 'contingent truths'. However,
compossibility produces only one option.
Adam not sinning belongs to another world, but so
does Caesar not crossing the Rubicon—are these two
other possible worlds related? Leibniz
addresses this issue by talking about the dream
where Apollodorus visits a goddess's palace which
contains a number of other palaces—an infinite
series of boxes within boxes, all the way down in
a 'labyrinth of continuity'. At each of a number
of infinite levels, different possible worlds
exist, with the same characters doing different
things, sometimes with overlaps. The palace
is a pyramid. God has simply chosen a world close
to the highest level. All the worlds are
struggling to come into existence (as an aspect or
function of their essences), all are possible in
God's understanding, all have a certain 'weight of
reality', but the only one that emerges is the
best one.
This is not the best in moral terms, but in terms
of a theory of games, a result of Leibniz's
interest in probability. As an aside, it is
this originality that has impressed subsequent
thinkers such as Nietzsche, who was drawn to
Spinoza because there was a 'the familiarity
between his very own screams and the concept of
the philosopher'. Apparently, DH Lawrence
had the same reaction to Spinoza or, and Kleist
with Kant. For Leibniz, it was Borges, and
the notion of the forking paths as the world of
compossibilities all displayed in terms of options
and choices.
Compossibility is not just a matter of simple
contradiction, because the different possibilities
are not logical contradictions, but belong to the
truths of existence. There is a connection
with the notion of infinite analysis.
Leibniz explains this in different ways, to suit
different readers, as above. In one text he
says that predicates are contained in subjects,
but either as acts or as virtuality—so Caesar
crossing the Rubicon could be a virtual inclusion
[only a potential, even if it is
actualized]. In another, however, the
infinite analysis of the total world would proceed
through indefinite analysis, moving from one term
to another, all the way down to the infinitely
small [as in the series of infinitesimals—see notes on Bos]. In
this way, infinite analysis is also 'virtual
analysis [in the sense of] an analysis that
goes towards the indefinite'—but this is an
argument for those with little philosophical
background, where virtual is not used particularly
rigorously: in more scholarly texts, it refers
only to truths of essence [the nonphilosophical
version seems to imply that we get to the virtual
by simply adding up examples of the
actual?]. Overall, this means we cannot use
the term virtual to describe inclusion in truths
of existence, but only in truths of essence, as in
the example of dividing by 12.
So infinite analysis [of truths of existence] is
not a matter of approaching the virtual, nor is it
just indefinite analysis, because this implies
that infinity expresses just a limit to what can
be done with the resources available, including
knowledge; and nor can we just plug in God to get
to the infinite. We need something more
rigorous. Reverting to the example of the
progression of infinitesimals [as a series of
decreasing fractions that add up to 1, which is
clearer], the bit that continues, the 'etc.' does
not preexist but is produced by a procedure
[apparently, Kant later could define the
indefinite in terms of a synthesis producing an
aggregate]. In the 17th century,
philosophers tended to have 'an innocent way of
thinking'about the infinite, sometimes seen as a
mixture of philosophy and theology, but, for
Deleuze, that is a post hoc 'stupid'
judgment.
Leibniz's position eventually became one of saying
that the indefinite is virtual in the sense that
the terms that point to it are constituted by
definite procedures. This is unlike infinity
which is 'actual' [must be another sense of actual
here, the actual {!} phrase is 'there is no
infinite except in act'. It might mean that
the infinite is defined only in terms of truths of
existence?]. Even for God, there is no end
in analyzing the predicates contained in the
individual notion. All the terms are given
[empirically?], so this is not a matter of
indefiniteness, where terms have to be produced by
[mathematical] operations.
The infinite analysis means an infinity of
infinitely small given elements, and this can be
seen to be 'actual'[tangible, empirical?].
However we can never in practice reach the end,
even God cannot, because it is an infinite
aggregate. God gets closer than any humans,
though so you can see how things are connected
together, like sinning and Adam, joined by 'an
infinity of other elements actually given', the
entire existing world. So we might pass from
Adam the sinner, to Eve the temptress, to the evil
serpent and so on. We need to clarify what
an infinitely small element is as well, at least
for those who are more knowledgeable: it's really
'an infinitely small relation between two
elements': relations replace the element.
Here, we need to understand differential calculus
[and the way in which specific relations become
abstract functional ones  see notes].
What this means is that the inclusion of the
predicate in the subject, when talking about
existence, is not a matter of identity, not even a
virtual one. Identity governs truths of
essence. The real relation of interest is
the connection between one predicate and another,
the maximum possible continuity.
Continuity becomes the key to understanding truths
of existence, established by infinitely small
relations between elements. Continuity
defines the world, and discontinuity defines
incompossibility. God chooses the maximum
continuity as the best. It might be a
theological puzzle to establish why the sinning
Adam is in the best world, but he is there because
he ensures an important continuity, that would
lead to Christ and redemption.
What we have is a small relation between two
different elements, however, but these tend to
disappear: they are 'evanescent [vanishing,
fleeting] differences', and this is as close as we
can get to logical identity. It is the
infinitesimal analysis that led to the
differential calculus, Leibniz's particular route
compared to Newton [see notes]
. To understand its significance, we need to
discuss differential equations, which are so
important in current physics, although modern
calculus operates without worrying about the
infinite. For Leibniz, the issue was to
examine the relation between different powers with
different magnitudes and quantities, especially
those relating to equations like 'a xsquared +
y'. What makes this equation difficult to
resolve is that the quantities are
noncommensurable [cannot be considered to have the
same base units for measure]. The problem is
deepened with the discussion of incomparable
quantities. The calculus permits you to
'compare quantities raised to different powers'.
We can see that it is possible to move ahead with
'a xsquared +y' by first extracting dx and
dy. Here it is to be defined as a matter of
extracting 'the infinitely small quantity assumed
to be added or subtracted from x or from y' [so
that we can calculate the slope of the line by
looking at the difference between the first and
second values on the dimension concerned.
When we are projecting this operation into
infinity, the differences become infinitely
small?]. The infinitely small can actually
equal zero, so it is clear that the difference can
be the smallest possible conceivable, something
smaller than that which is normally
measured. Nevertheless, even when the
additions or subtractions from x or y is 0, the
relation between dx and dy [dx/dy] is not equal to
zero. This helps you compare two formerly
incomparable quantities affected by different
powers, because it moves us away from quantities
towards relations, [ratios] and this inaugurates a
shift in mathematics towards relations or
functions. Leibniz was to argue that the
differential was already implicit in ordinary
algebra. So when the relation is given
values c and e, and both are zero, we do not end
with 'absolute nothings', but those that 'conserve
the relational difference', so that even when both
are zero, c does not become equal to e. This is a
'great mathematical discovery'. The
differential relation can be determined. [I
think this only follows because we have already
decided that there is the relation or analogy
between the original x and y, and that this is
preserved with subsequent x and y, or in this case
c and e. If this relation holds, then if we
argue that c and e are equal, when they are set at
zero, it would follow that the original x and y
must be as well, and this would be 'totally
absurd'. Again, it seems like a bit of
clever philosophical sleight of hand.
Really, the difference must remain because we have
already defined it as a constant analogy or
ratio?].
We can make similar analogies to argue that 'rest
is an infinitely small movements, or that the
circle is the limit an infinite series of
polygons, the sides of which increased to
infinity', or that, as in the diagram of the
inverted triangles, the single triangle [ Aec] is
'the extreme case of the two similar triangles
opposed to the vertex'. [See the diagram
below, reproduced in the book. Imagine the line ec
being moved progressively to the right until it
runs through point A. Also remember that we are
not measuring line lengths exactly here, but more
changes in length from E to A etc. The
differential compares changes toget a general
formula for slopes  rise over run.]
We could say that this particular triangle,
located at the point [right at A] where the values
of e and c are zero, 'is only there
virtually'. The existence of e and c does
not depend on their numerical quantities, but
because they are in a relation: strictly speaking,
the triangle when e and c are at A ec has not
disappeared, but it has become 'unassignable
'[isn't this just a way of saying that the
infinitely small is found at A, not nothing, even
though the arithmetical value recorded by zero
might be? I think it all depends on Leibniz on
infinitessimals: 0 is surrounded by
infinitessimals so it is never a static point of
nothingness, there are always relations with
infinitessimals before and after it, so 0 is
always in a relation or a movement. Movement never
just stops in an infinite curve. You can say
that the circle is an unassignable polygon in the
same way— because the sides are infinite [and thus
uncountable? Unable to be assigned a
numerical value or quantity?]. This is how rest
can be considered as a special case of movement,
an infinitely small movement.
Thus Leibniz can redefine the virtual as 'the
unassignable yet also determined'. This is
new and rigorous, and Leibniz can demonstrate it
relatively easily using the examples above [and
getting us to agree that the way he expresses the
analogy is consistent— must be one of these
semiimplicit propositions that Deleuze's book on Leibniz
calls 'enthymemes'?]. What this does is no
less than approximate what God does with infinite
analysis, [it approximates the divine
continuity]. It only describes the maximum
of continuity, the extreme case [which is the
closest we get to the end of an infinite
analysis?], which is plainly included in the
normal empirical case [or is it vice versa?] .
This gives us the formula of predication. We
have to use terms like intrinsic case, to refer to
the 'movement the encompasses all movements',
while the extrinsic case is the [result of] that
movement, the circle in relation to the
polygons. We need a concept that
'corresponds to the general intrinsic case and
which still includes the extrinsic case'. This
will explain continuity between the elements, from
the polygon to the circle, [the inclusion of the
normal in the extreme as above].
Eventually this was to lead to Poncelot and modern
projective geometry based on 'a simple axiom of
continuity': take the arc of a circle cut at two
points by a right angle. Make the right
angle recede and there will be a moment where it
leaves the circle no longer touching it.
[Somehow] this raises new possibilities for the
tangent as an extreme case of this movement: when
the points leave the circle they are still there
but both are virtual [hard to follow, but I think
it is the same sort of argument about the relation
persisting even when the concrete values no longer
do].
So we have moved from trying to show how the
truths of existence are not the same things as
truths of essence or mathematical truths. We
have shown this but only by having to introduce a
notion of infinity. When we are exploring
that, we have to resort to mathematical truths
again: 'It's funny, no?'. Leibniz has to
argue back that he has never said that
differential calculus designates reality. It
is a convention, a symbolic system which shows how
you can designate reality. It's a convention
based in mathematics, but it is a '"well founded
fiction"'. Calculus uses concepts that are
not based in arithmetic or algebra: quantities
that equal zero but are not nothing are clearly
nonsensical as arithmetic. It is a useful
fiction though which 'can cause us to think of
existence'. It is a combination of mathematics and
Leibniz's idea of the existent, the 'symbolic of
the existent'. It is not an arithmetical
calculus.
We can now see what 'evanescent difference'
means—'it's when the relation continues when the
terms of the relation have disappeared'.
This relation exists in God's understanding, so
'God was only doing calculus'. God is a
player, but Leibniz describes the game. He
actually has two explanations:
 In one case, the problem
is to precisely fill a surface with regular
figures, like a rectangular surfaced tiled
with circles. This leads to other
problems like dimensions of figures used, and
'with which other figure when you fill in the
empty spaces?'[This is like the practical
demonstration of calculus, where the issue
becomes how to fill in the gap in the
quadrature left between tangent and
arc]. This is the problem of continuity,
and it clearly involves incommensurable
dimensions and incomparable figures. However, rendering this as God
choosing the best of all possible worlds led
to serious criticism, including that by
Voltaire. There are additional
philosophical and political issues
raised. This was not arguing that god
was choosing a world in which suffering was
minimized, however. It is more a matter
of choosing a world that best fits the empty
space that has to be filled in the example
above, even if people suffer. You could
argue that the circle suffers by being only an
extreme case of a polygon, but the circle
'realizes the maximum of continuity'.
That is God's interest even if it brings human
suffering. God also produces other
possible worlds as stylistic variation.
This was not very popular in the 18th century!
 A second kind of game is
the chess game which God also plays.
'The chessboard is a space, the pieces are
notions', so what is the best combination of
moves? it will be 'that which produces a
determinate number of pieces with determinate
values holding or occupying the maximum
space': we place our pieces so that they
command the maximum space. However,
chess and tiling are not helpful in one
respect—'in the conditions of the creation of
the world, there is no a priori
receptacle'. The receptacle itself has
to be realized as the one which 'contains the
greatest quantity of reality or of essence
[not existents] from the point of view of
continuity'.
[Deleuze also answers some questions
arising] For example that it was not an
issue for 17th century people to worry about
whether the calculus was artificial or real.
Leibniz was clear that it was a 'pure
artifice... a symbolic system'.
Everyone saw that it was not possible to reduce it
to conventional mathematical realities as in
arithmetic or geometry. Leibniz never
thought that differential calculus offered a
complete answer to the issue of infinity, and
distinguished, for example between the infinitely
small and the infinitely large, which would
require different calculuses. There are also
qualitative infinities. Leibniz's attempt to
argue for different orders of the calculus was
only stopped by the 'Kantian revolution'.
We find this in history, that problems are defined
in such a way is to produce sufficient
techniques—geometry and algebra were adequate to
classic Greek problems relating to straight lines
and rectilinear surfaces. Problems of curves
had to wait for solutions: the Greeks modelled
curves in terms of 'equations of variable degrees'
[eg by drawing innumerable tangents?].
Solutions get exhausted, and this prepares the
ground for new ones—but solutions are always
connected to specific problems, such as those
dealing with terms raised to different powers,
itself related to the problem of understanding
natural phenomena, which were curvilinear and
determined by different forces and
velocities. Leibniz introduced the notion of
mass times velocity squared, whereas before,
physical problems had been confined to calculating
mass times velocity, and important issue for
modern physics.
[After another intervention]. Differential
calculus raises the issue of rigorous analysis
based on axioms, but this only appeared
later. Leibniz was prepared to operate with
entirely artificial or metaphysical notions,
impure ones that could not be rendered as axioms,
and which were therefore, strictly speaking, not
scientific. So scientific status was finally
achieved, but at the price of sacrificing any
metaphysical notions, including those of infinity
or limit [with what looks like the use of entirely
symbolic not arithmetic terms. Apparently
somebody called Weierstrass develops this, as a
rather 'static and ordinal' system].
Infinity has changed its meaning, and is now
'completely expelled—we can extract dy/dx just
from x and y', [see notes
on Bos] although the point about the
relation persisting even when the values are zero
remains. It still helps us determine further
quantities, whereas the axiomatic version does not
[except tautologically?].
For Leibniz, we can understand infinite analysis
to the extent that the infinitesimal and its
continuities 'are substituted for identity'.
This further implies the infinite analysis is not
grasped by an understanding of identity, but by
'continuity and vanishing differences'. We
have to discuss compossibility, however.
Leibniz invents this term to be 'entirely linked
to the idea of infinite analysis'.
Incompossibility can be defined in three ways:
 It can be a kind of
logical contradiction, rooted at infinity—but
this just might be one of those simpler
explanations that Leibniz offers. He is
more original than that, moving beyond
identity and contradiction to 'contradictory
identity'. Anyway, the notion of the
infinite is not a matter of a series of the
identical, but a matter of continuity.
 The second answer
suggests that we cannot fully grasp ['know
what its roots are'] incompossibility because
our understanding is finite. The notion
introduces a new domain to supplement 'the
possible, the necessary and the real'.
Again, this is a temporary solution for
Deleuze.
 The third answer means we
have to look across the range of material that
Leibniz produces. We have to understand
compossibility as arising from 'the theory of
singularities', which is not systematically
developed in a single location.
Nevertheless it has 'two poles'. It is
'a mathematical  psychological theory'.
The immediate task is to see how this works in
mathematics.
The notion of singularity in mathematics was not
particularly developed by Leibniz. Deleuze's
analysis here raises the old questions for him
about philosophy—does it lead particular
terminology, or is it really open to any one
without special knowledge? If the first, you
have to work with philosophical terms [such as
concept, category and so on just like you do with
mathematics]. The singular has been
classically located in relation to the universal,
or the particular as opposed to the general.
Nevertheless, a judgment of singularity is neither
something particular nor general.
Mathematicians extended conventional logic and
used the expression as something distinct from
regular, something outside the rule. They
also refer to remarkable and ordinary
singularities, but Leibniz just used the singular
as something remarkable or notable, although he
gave the second term a special meaning.
The mathematical extension helped us replace the
old obsession with what was true and false,
because 'in thought, it's not the true and false
that count, it's the singular and the
ordinary'. Kierkegaard was to add 'the
interesting'. We still have to find reasons
to make the singular as something interesting, out
of the ordinary. Mathematics sees the
singular as a matter of 'certain points plotted on
a curve', or, more generally, 'concerning a
figure'. We have to remember that a figure
is 'something determined', and that it will
include singular and regular ordinary
points. So we have a determination for the
singular, and a broader definition of
determination to include the singular and the
ordinary. For example, we can define the
vertices of a square as singular points, and these
are extensions of the size of the square, points
[of inflection] where one line ends and another
begins. The lines are composed of 'an
infinity of [ordinary] points'
What about curves? Are the singular points
found at the extremes? What about points
where curves meet? Leibniz plots such a
graph, on classic Cartesian ordinates, and
considers the segments of the curve [and then a
puzzling bit, where a particular segment is seen
as unique—I can't reconstruct this diagram, nor
find an example on the web, to follow the
argument, but the conclusion is that a segment can
also be unique, or a singularity, and found not
necessarily at the extremes—it can be a minimum or
a maximum, however, or both depending on which are
good. I CAN see this working if we are talking
about the point where 2 arcs cross  this point
is unique. Of course, a line segment can be a
point, you clot Dave! The other points on
the arcs get closer as they approach this
singularity.]
This is why Leibniz was important, in calculating
singularities and maxima and minima, and this is
still important today in symmetry or in some
optical phenomena. We can now develop the
notion of a singularity: ordinary points are below
maximum and above minima and also exist in a
double fashion [they match on two curves, eg they
have the same value on the X axis?]. We can
even see ordinary points as another case, 'a
singularity of another case' [all ordinary points
have the potential to become singular points in
different circumstances? I am beginning to
see a glimmer of light about how this is going to
end in compossibility].
In another example [irritatingly, the transcript
is incomplete] a complex curve has singularities
that can be seen as 'neighbouring points'.
In topology, neighborhoods are where 'something
changes: the curve grows, or it decreases'.
These points of growth or decrease are also
singularities as opposed to the series between two
singularities which features ordinary points,
going from one neighbourhood to another [clear
echoes here of what happens with multiplicities
generating a series of singular and ordinary
points up to the neighbourhood of the next
multiplicity, as in Difference
and Repetition  it wasn't sci fi, it
was Leibniz!]. Mathematical exploration of
singularity has helped us break the implicit tie
that conventional philosophy had to rectilinear
figures! Those would not have helped
generate the new conception of singularity—we
needed to consider complex curves in their own
right.
[He has obviously had to answer a tricky question,
although this is missing. He says he's been
tripped up and will need to think about it].
Poincaré also develops the notion of singularity,
when discussing differential equations. He
identified singular points at crests where two
curves defined by an equation pass through each
other. Then there are knots where an
infinity of curves defined by the equation can
intersect. The third type is foci, around
which curves spiral in. And the fourth type
is a centre producing closed circles of
curves. This is another example of where
mathematics has pushed philosophical notions of
singularity. Something similar can be found
in Leibniz. The domain of singularities
requires techniques like differential calculus: we
can see a singularity as 'the point in the
neighbourhood of which the differential relation
dy/dx changes its sign', as when curves rise and
descend. At the precise point where they
change direction, the differential relation
'becomes equal… to zero or to infinity',
indicating minimum and maximum again. We see
that singularities extend into ordinary points,
that 'the theory of singularities is inseparable
from a theory… of extension'. This is
a definition of continuity, stemming, ironically
enough, from the notion of singularity as
discontinuity.
[Then we leap to philosophical psychology].
It is the same as the connection between ordinary
perceptions and apperception ['perception endowed
with consciousness']. Unconscious
perceptions are necessary, however because they
are make up the elements of the global or the
relative totality, as a necessary parts of a
whole, as a composition. By extension, this
is a fundamental principle 'that there is no
indefinite', and this in turn 'implies the actual
infinite'. The existence of the composite
means there must be some simpler elements, until
we get to the actual infinite [the problem is that
the composite looks indefinite, at least until we
got to proper chemistry?].
We can argue the point using the notion of
causality: what we perceive is an effect, so there
must have been a cause, and we must have perceived
these however dimly, just as we must have
perceived the sound of individual drops of water
in a wave. In fact, for Leibniz, the parts
are the causes, or rather, 'an argument based on
causality and an argument based on parts' must
coexist. We know this in principle, but we
can also know it in experience and this produces
philosophy's 'moment of happiness, [when
everything fits] even if it's personally the
misfortune of the philosopher'. However, it
is also necessary that our experience indicates
that we are not always fully organized in our
consciousness, or that we can become 'invaded by
minute perceptions that do not become... conscious
perceptions'. The example is being given a
blow on the head, becoming dizzy and disoriented
and fainting! Leibniz even thought it might
describe the state of death. We can describe this
is a state of envelopment, perception without
apperception.
Leibniz discussed the unconscious, and some of the
implications extend even to Freud. However,
for Leibniz, the discussion turns on the
soul. It has the faculty of apperception,
and also one of '"appetition" [the conscious or
possibly consciously experienced version of]
appetite, desire'. Appetitions correspond to
minute perceptions [I can see that the two are at
least implicated, they can only develop an
appetite for something once you have perceived the
objects connected with it]. The work
parallels that of an obscure Spanish biologist,
Turro, who saw hunger as the global sensation
produced by lots of 'minute specific hungers',
relating to hunger for proteins, mineral salts or
whatever. We can see global hunger as
'integrating' these minute ones [no innocent
choice of terms, I am sure]. The way the
animals satisfy their hungers is through eating
'minute qualities', grasping not grass but
something green, not proteins, but some other
object. An internal milieu regulates all
these minute perceptions and appetites, in a
'strange communication between consciousness and
the unconscious'. In animals, it is instinct
that connects appetite to perceptions through a
'psychic investment'. Like us most of the
time, animals do not see the mechanism.
Leibniz developed his ideas after having read
Locke on human understanding, with whom he
disagreed. [Apparently he wrote a large book to
condemn Locke but did not publish it after hearing
Locke had died!] Locke talks about
uneasiness as the principle of psychic life,
anxiety, [arousal], which constantly 'swarms'
through appetite and perception. It might
even be God's way of maximizing our perception—or
producing continuity again, 'an indefinite
progress of consciousness'. Actual
unhappiness arises merely from 'unfortunate
encounters', which may or may not arise from the
best continuity.
We can see the links developing between minute
perceptions and infinitesimals and
differentials. 'Following from this, the
Leibnizian unconscious is the set of differentials
of consciousness. It's the infinite totality
of differentials of consciousness'.
Subsequent analyses of the unconscious in
philosophy have followed a similar link to
infinitesimal analysis, developing a whole
'psychomathematical domain'. Both involve
symbols. Freud inherited this tradition, but
did not develop the idea of unconscious
perceptions, except through the notion of
unconscious representations, and saw opposition
between the conscious and unconscious, not
differentials, differences of perception.
We can consider Leibniz's approach as a matter of
unconsciously detecting minute physical vibrations.
The way these are aggregated varies again, because
Leibniz has two formulations. The first one
is discussed as a relation of parts and wholes, or
of composition, where a sense organ totalizes
minute perceptions, eyes totalize minute
vibrations to compose colours[nice physical
vibrations here, of light] . In the second
case, Leibniz talks about derivation, not the same
as composition, and again we return to
infinitesimal calculus and the mathematical notion
of the integral, a special kind of totalization,
not the same as simple addition. Leibniz
talks about augmentation to describe the
transition from minute perceptions to something
which becomes conscious perception.
To proceed further, we have to talk about the
relation between the clear and the distinct, and
although Leibniz uses the term 'distinct'
apparently indistinguishably from terms like
notable and remarkable, there is also a technical
sense. We can see how this describes the
special addition of minute perceptions.
Minute perceptions 'form a series of ordinaries, a
series called regular', but globalization is
different, and not just arithmetical totalization.
Instead we are talking about a singularization,
something that happens in a particular
neighbourhood of a singular or remarkable point,
and the result is conscious perception. We
have to remember that when we're talking about
minute elements, we are talking about
differentials, and not concrete elements or
specific values, but a relation, as in dy/dx. [Bos elaborates on the
difference between integration and addition, and
says Leibniz saw the terms as equivalent at first
until Bernouilh objected]. We also have to
remember that at the singularity, the differential
relation changes its sign [always?].
We can make this look like a prelude to Freud if
we see the issue as the relation between physical
elements and the body indicated by dy and
dx. And perception becomes conscious only
when this 'corresponds to a singularity, that is,
changes its sign'. This in turn depends on
the idea of sufficient closeness, of excitation in
this case [energetic molecules, so to
speak?]. Apparently, Jung displays 'an
entire Leibnizian side', via German Romanticism,
by introducing an notion of the unconscious as
differential, which annoyed Freud and led to the
split. So psychic life is also affected by a
series of ordinary points reaching the
neighbourhood of a singularity, and this is how
psychic life itself is composed of the continuous,
the extension of points from one singularity to
another. This does not just describe 'the
universe of the mathematical symbols, but
also… the universe of perception, of
consciousness, and of the unconscious'.
We now have 'the formula for compossibility'.
You can trace lines from singularities into the
neighbourhood of other singularities, until you
get intersecting structures, a continuity.
The simplest case is the straight line, but there
are other nonstraight lines as well. For
example, you can draw two circles that intersect,
and argue that 'there is continuity when the
values of two ordinary series [along segments of
the circles? ]… coincide'. This would
be a continuity between two more specific kinds of
continuities, just as the square is made up of a
continuity of straight lines and vertices. A
discontinuity arises if 'the series of ordinaries
that derive from singularities diverge'.
Thus a whole world is constituted from continuous
continuites, 'the composition of the continuous':
the best world maximizes continuity. The
discontinuity produces the incompossible, arising
from divergent series of ordinaries produced by
singularities. What makes this difficult to
perceive immediately is that 'God is perverse',
and disperses these continuities, producing
'leaps, ruptures' in our world, such as the gap
between man and animal [apparent only 'to some
among us', however]: in fact, God produced all the
intermediaries between man and animal, but these
are not made visible to us, and might even be
'placed... on other planets of our
world'. He did this to encourage us, to help
us believe that we could dominate nature, to
establish power over nature. And he did at
least ensure that we could perceive something of
the minute differences, so that we can get some
idea of compossibility and incompossibility. This
particular world 'mathematically implicates the
maximum of continuity' and this is what makes it
'the best of all possible worlds'.
A concept is always complex, with 'all sorts of
languages that intersect within it': it is 'always
necessarily polyvocal'. We can make some
progress if we use mathematical apparatus, but
philosophical concepts proper are different for
Leibniz, and contain 'all sorts of different
orders that necessarily symbolize', Including
thoughts and experience. Leibniz made a
great breakthrough by adding mathematical concepts
of the singular to help us see that it was not
necessary to oppose it to the universal, but
rather to the ordinary or regular.
The inspiration might have been mathematical, but
a whole philosophical theory emerges: 'don't pay
too much attention to the matter of true and
false… because what is true and what is
false in your thinking always results from
something that is much deeper'. It is more
important to examine remarkable and ordinary
points. If we think only of singular points,
there is no extension, and if we use only ordinary
points, we get nowhere, and we are not really
thinking. [There is no implication for the
person, or conventional subject here]—'the more
you believe your self to be remarkable (special),
the less you think of remarkable points'.
When we think of singulars, we necessarily become
modest 'because the thinker is the extension onto
the series of ordinaries, and thought itself
explodes in the element of singularity, and the
element of singularity is the concept'.
We have to see how Leibniz both creates new
principles, and deduces implications from them.
We start with the principle of identity
which is used to argue that 'Every principle is a
reason'it is not just the formal statement that A
is A, rather that the thing is what it is, in
other words 'identity consists in manifesting the
proper identity between the thing and what the
thing is'. We might call what the thing is
'the essence of the thing'. So the principle
of identity is 'the rule of essences… the
rule of the possible'. Another way of
putting this is to use medieval terminology and
say that the reason is the ratio of essence, ratio
essendi [an example of why we need
philosophical terminology, and why we must use the
terms rigorously, 'exactly the same as scales on
the piano'. There is also a hint that we
must use these terms for study, and not just as a
kind of examination material on philosophy
courses. We can give two formulations by Leibniz
in each case, a vulgar and a scholarly one.
The scholarly formulation of the principle of
identity is 'every analytical proposition is
true', that is the predicate and subject are
identical in analytical propositions. As we
saw, the truth of an analytical proposition can be
demonstrated 'either by reciprocity or by
inclusion', as in the example of the
triangle. We can argue that reciprocity
involves intuition, while inclusion involves
demonstration. The whole exercise is
designed to answer the old question 'why is there
something rather than nothing?', establishing the
reason for being, which depends on the relation
(ratio) between the essence and the thing.
The principle of sufficient reason, the
reason for existing, the ratio existendi.
Here the question is 'why this rather than
that?'. In vulgar terms everything must have
a reason. In scholarly terms we have to
leave behind the principle of identity, because
that is too formal and abstract, and would permit
us to say what a unicorn is even now they do not
actually exist. Leibniz argues that every
predication, 'the activity of judgment that
attributes something to a subject' has a basis in
the nature of things. Everything said about
a thing is included in the notion of the
thing. That includes the essence said about
the thing, but also 'the entirety of the
affections, of the events that refer or belong to
the thing'. We saw above that this means
that each individual notion expresses the totality
of the world, everything that happens is contained
in the individual notion of the thing.
We are entering the domain of infinite analysis
here, while identities deal with finite
analyses. We can think of the notion of the
reciprocal, and argue that 'the principle of
sufficient reason is the reciprocal of the
principle of identity'. However, there is
been a significant change in that the principle of
sufficient reason now occupies 'the domain of
existences'[unlike the reciprocal action we
performed with the principle of identity to get
tautologies and inclusions]. We have to note
that we can only do this if we invoke an analysis
extending to infinity, so 'the concept of infinite
analysis is an absolutely original notion', and
not one that only takes place in the mind of God,
but one which we can pin down a bit in the form of
a technique of differential analysis and
infinitesimal calculus.
We also need to remember that Leibniz does not
intend here to restore a notion of causality:
sufficient reason is more than causality, and we
need to 'account for reason in causality itself',
and to develop the difference between necessary
and sufficient causes: causes are only necessary
if they are a part of sufficient reason.
Overall, we now have the argument that there is a
concept for everything, and this must include
relations with other things, including causes and
effects. How can we develop this further,
again by thinking about reciprocals (not
necessarily contradictions or opposites)?
In music, the reciprocal might be a musical
'retrograde series'. In philosophy, the
reciprocal of 'everything is a concept' is 'for
every concept there is one thing alone'.
This is an odd reciprocal, more of a necessary
implication to make the first statement
work. There is no such necessary in finding
the reciprocal of the principle of identity: [it
is more like a leap into analysis, since there is
no necessary reason for all true propositions
being analytical]. This, however was a cry
from Leibniz [of triumph?] since it leads to
infinite analysis, and therefore to the technique
and so on.
An implication of the notion that for every
concept there is only one thing means there can
never be two absolutely identical things or that
'every difference is conceptual in the last
instance'[that is, related to a concept we have of
the thing?] The implications are delirious [no two
drops of water can be identical, for example], and
contradict classical logic, where the concept is
general to encompass a lot of things. We can
generalize, but if we analyze deeply enough, we
will find that concepts relate to individual
things only. This is the 'principle of
indiscernibles'. If we take the second
formulation, that every difference is conceptual,
we only gain knowledge through concepts, the ratio
cognoscendi, 'the reason as reason for
knowing'.
What about other kinds of differences, such as
numerical differences? What about
differences in space and time that locate
particular objects, this chair rather than that
one? What about 'distinctions of figure and
of movement', or extension and movement—the
dimensions that objects occupy, or the speed at
which they extend? For Leibniz, these are
only matters of appearance expressing conceptual
difference—two drops of water 'do not have the
same concept'. This is a new way to think
about individuation. Leibniz is then faced
with the task of showing how bodies take on these
particular appearances. One way he does this
when discussing Descartes is to suggest that
dimensions of mass and velocity are only relative,
not enduring principles, that they express
something deeper. The same goes for
extension, which is not sufficient as a substance
in itself: he argues that it is force that
manifests itself in figures and extension, force
'is the true concept'.
This also connects to the earlier work on trying
to develop a mathematics of forces raised to
different powers. Specifically, Leibniz is
going to argue that force is velocity squared,
involving continuous change. Again, he is
consistent to the principle of indiscernibles and
says 'there are no two similar or identical
forces', or that force is [only an abstract]
concept. Force is not the same as movement,
but 'is the reason for movement'. Mass times
velocity squared is a formula for force not
movement. Everything will move towards
concept of force, including number and the
conventional dimensions [expressed in the
development of abstract functions for numerical or
conventional algebra in the calculus?].
This leads to a fourth principle, this time
expressed as a law—the law of continuity.
This is more fundamental than a principle.
The vulgar formulation says there is no
discontinuity in nature. One scholarly
alternative says that 'if two causes get as close
as one would like, to the point of only differing
by difference decreasing to infinity, the effects
must differ in like manner'. This is really
a way of arguing with Descartes and the view that
if two bodies have the same mass and velocity
meet, the one with a greater mass or velocity will
carry off the other [the argument seems to be
instead that differences will decrease to
infinity, and there would not be contradictory
effects, as Descartes had predicted, either
repulsion or continuity of the dominant mass].
But there is another scholarly formulation, that
'in a given case, the concept of the case ends in
the opposite case', the 'pure statement of
continuity', with examples where concepts of
movement end with concepts of rest, given that
rest is infinitely small movement [argued above
from discussing infinitesimals]. As an
alternative, we can say that singularities extend
themselves into a series of ordinaries up until
the neighbourhood of the next singularity.
This is a way of explaining 'the composition of
the continuous'. But doesn't this principle
contradict the one about discernibles? Not
for Leibniz, apparently. The problem is that
the principle of indiscernibles refers to
determined differences, 'an assignable difference
in the concept' itself. The principle of
continuity, however refers to progress through
vanishing differences, on assignable
differences. We need to invoke yet another
form of reason, the ratio fiendi, 'the
reason for becoming', through continuity.
It is necessary again to remember the strange idea
that for Leibniz, concepts [meaning notions in
this case] are single words expressing the whole
world. In normal philosophy, concepts are
not a single world but a proposition or a complex,
with verbs, implying movement showing the
way concept expresses itself. For
Leibniz, God has created worlds in which
particular individuals do things, but we only
express this world in individual notions.
The world does not exist in itself, but only in
these notions. Yet we can perform an
abstraction—the world is going to be a complex
curve with singular and ordinary points, with
singular as extending into ordinaries and so
on. God chooses in order to produce maximum
continuity. This is how the world is going
to be subjected to the law of continuity, although
it is not expressed like that. Each monad
can only grasp a small number of singularities,
according to its point of view, a small number
drawn from the curve of the world produces an
individual notion. Different individuals
occupy different points of view and therefore
encompass some singularities rather than
another. [And then, irritatingly, the
transcript peters out].
To finish, it is naive to think that philosophers
simply hold opposite views or critique each
other. We can look at the opposition
between Kant and Leibniz. First we have to
properly organize the oppositions.
Leibniz argues that all propositions are
analytical and that this is essential to the
progress of knowledge. In the first case,
it's not worth arguing about, and philosophy has
long preceded by trying to discover what's
included in the concept. Kant argues for
synthetic propositions, 'in which one of the terms
is not contained in the concept of the other', and
says that knowledge only exist through the
synthetic propositions. For Kant, synthesis means
leaving the concept behind in order to pay tribute
to it something else, to go beyond the concept How
could we decide between the two? It is clear
they had both imply certain conceptions of
knowledge. For Leibniz, knowledge is based
on a particular model of perception or 'passion',
something that needs to be apprehended: for Kant,
we have to pursue a particular knowledgeact to
leave the concept behind. How could we decide
between them? It is obviously not a matter
of finding facts to support one or the other:
philosophical propositions do not depend on the
verification of facts. In classic
philosophy, there were two questions—quid facti,
what was derived from fact and quid juris,
what was derived from principle, leading to de
jure questions.
Leibniz never denies that phenomena are synthetic,
that it is possible to encounter qualities of them
from experience [the example is this straight line
is white] an a posteriori synthesis.
But this is based on experience, not
knowledge. To know, we must refer
propositions to a principle, something that is
'the universal and the necessary' such as that are
straight line is the shortest path from one point
to another: we know this a priori, we do
not need to wait for experience to confirm it [we
have defined it as such]. So the issue is
whether a priori propositions are
analytical or synthetic.
Kant says a priori propositions are
synthetic, that we leave behind the concept of
straight line, to add something to it, a content
about it being the shortest path. Leibniz
would say that the concept already has this
content. For Kant, we have to actually go
ahead to draw lines and compare them, in other
words to work with a synthesis, which includes
extending the notion of straight lines to reach
curved lines used in our comparison: when we say
the straight line is the shortest path, we are
actually comparing straight to curved lines.
Obviously, Leibniz is going to disagree, because
he has already found, in differential calculus,
that the straight line is the limit of curved
ones: we have an analytic process, although it
tends to infinity. At this stage, it is
simply a difference in terminology: what Leibniz
calls the difference between finite and infinite
analysis, Kant is going to call the difference
between analysis and synthesis. It's easy to see
why 'good sense' would see philosophical disputes
as interminable and irrelevant. However, it
is not just a matter of choosing different words,
and there are deeper oppositions.
In Leibniz's principle of indiscernibles., any
difference is ultimately a conceptual one—if two
things differ, they cannot have the same concept,
and every differences conceptual. This is
why we need to analyze concepts, especially to
analyze the differences, and this is well
developed in the mathematics. Kant, however
argues that there are two determinations: one is
conceptual, so that the concept represents what
the thing is, a lion is an animal that roars, for
example. But there are also 'spatiotemporal
determinations', which are irreducible to
conceptual analysis.
Take the example of our two hands. For
Leibniz, they differ through the concept.
For Kant the issue is whether we are developing an
interesting proposition or a platitude.
Again, an interest in a proposition cannot be
derived from a concept: all the interesting
propositions are not found in principle.
Spatial determinations cannot be reduced to the
concept either—the interesting thing about hands
is that they are either right or left, and this is
nonconceptual. Dimensions are important in
all sorts of interesting ways, because they are
now objects to be 'superposed' [one hand placed on
top of the other]. The same goes for time
which is equally irreducible. For Kant
knowing involves synthesizing conceptual and
spatio temporal determinations—he actually called
this aesthetic judgement. Here, we have
moved beyond just arguing that knowledge involves
something more than is contained in a concept—we
do this not to affirm another concept, but to
leave concepts behind altogether.
What has happened here, really, is that Kant has
affirmed irreducibility, and he has actually done
this by a process of changing 'radically the
traditional definition of space and time'.
Here again, he departs from Leibniz. Leibniz
argued that determinations of space and time are
reducible, using different definitions to
Kant. The old concept of space saw the
matter as one of 'the order of coexistence
is or the order of simultaneities', while time
'was defined as the order of successions'.
Leibniz wants to push these ideas by asking about
possible coexistences and possible successions,
and this is going to lead to compossibility.
For Kant, there are problems with these
traditional definitions, for example because
coexistence already implies something about time,
but time is not only about succession, but
something which refers to the same time.
Here, Kant is anticipating a later theory that
says simultaneity is a matter of time and not
space [I don't know if he means Einstein, or
whether this is going to link with his own
interest in movement which he identifies with the
work of Bergson
among others].
Kant is going to go on to say that time actually
has three 'modalities'—permanence, succession, and
simultaneity of coexistence. Time cannot be
defined by privileging succession, nor space
through coexistence. This change is
significant, because Leibniz's view that you can
understand by analyzing concepts depends on the
notion of succession, and succession even affects
space and time, so 'there is no longer any
difference between spatiotemporal differences and
conceptual differences'.
For Kant, space is a form not a substance, that
can be defined by the things that coexist, in
exteriority. We can see time as a form of
interiority, and this is not just subjective
interiority. Again this is to become much
more important in later philosophy.
Classically, the issue was extension and how it
might relate to thought, how bodies might
relate to souls. In modern philosophy, it is
the relation between thought and time. For
Kant, it is a matter of how we relate to an
outside. Kant's notion of space is an open one, an
aesthetic space because we are no longer confined
to logical analysis of the concept, a Romantic
space 'because it is the space of overflows', and
the space of poetry.
The difference between Leibniz and Kant is a
difference between the end of the 17th and the
start of the 18th century, when an awful lot of
other things were changing as well, including the
development of Newtonian science, and the French
revolution. We can even see that the French
revolution finds expression in 'the order of
philosophical concepts'.
[another missing part of the lecture, because the
tape seems to have ended]
There is another change that Kant brought about
for classical philosophy, 'concerning the concept
of the phenomenon'. It used to be the case
that phenomenon was simply seen as appearance,
sometimes something that was accessible to the
senses. It was usually coupled with the
notion of essence. As with Plato, this
sometimes led to the notion of a duality between
appearance in essence, senses and intellect, two
worlds, where the world of appearances was a
prison, confining our senses and our
intellect. Kant uses the word phenomenon
differently, to mean 'apparition, that which
appears insofar as it appears'. Here, there
is an implication that what appears does so in
space and time. Apparitions are not related
to essences, but to conditions which make them
appear, and we have to understand their meaning
like this. This is another revolutionary
idea, that what appears no longer refers to
essences, but to conditions. Concepts don't
relate to essences either but to the meaning
of the apparition [requiring us to analyze
conditions?].
The focus on the meaning of the apparitions
produces the new discipline of phenomenology,
developed as an autonomous discipline by
Hegel. Because there is only a single world,
with no other worldly essences, this is also a
break between philosophy and theology.
Husserl was going to develop a different
approach and 'invent a form of logic proper to
phenomenology'. Kant was unable to break
quite as decisively, and preserved 'the
distinction between the phenomenon and the thing
in itself'. Leibniz, on the other hand, had
developed some of the debates about essence and
appearance to develop 'a theory of symbolization',
and this was also necessary as a preparation for
Kant: 'the phenomenon symbolizes with essence… [it
is] no longer… appearance with essence'.
There are also new 'disturbances' with
subjectivity. Leibniz maintained an alliance
with theology, with God having a crucial role in
creation. Already there were attempts
to bypass the old God of the Word and to begin
with infinity, but early attempts to reject
creationism were hard to develop. Infinity
was one way to extend the possibilities, including
the infinitessimal.
Descartes uses the notion of subjectivity with his
concept of the thinking subject, arguing that a
thought must relate to a subject. We can
think of the thinking subject as created by God,
implying that 'the thinking subject is substance,
is a thing… A thinking thing', for
Descartes. It would be possible to
substitute the thinking subject for God as an
infinite thinking subject, but Leibniz and Kant
did not do this. For Kant, thinking subjects
were not substances, but pure forms, 'form of the
apparition of everything that appears'. To
say that 'I think' is an empty form 'that
conditions every apparition'. This does
replace the notion of the divine, however, but
this only opens a new problem how is the world
founded? Leibniz has a particular answer—he
'spoke the discourse of God', founding the
existing world [showing us how to understand
it?]. This is also the role of classical
art, to create 'in the situation of God', 'by
organizing milieus'. The romantic artist is
not interested in milieu, but territory, leaving
territory, heading for 'the bottomless', the
ultimate founding. In philosophy, the issue
is apparition and the conditions of apparition,
once a form of heroic thought.
Kant says that the thinking subject is a form that
conditions apparition, or the finite ego.
And this breakthrough 'depends greatly on the
reform' [the Reformation?]. This is a finite
foundation itself, not something traceable to an
infinite, not a limitation of it, or more of 'an
overcoming' of the problem, and the finite is to
become infinite, to go beyond itself. We can
see the connections with Hegel and even
Nietzsche. There is an implication for the
problem of identity in Leibniz—now, A is A only if
A exists, and 'if there is nothing, A is not
A'. Identity becomes hypothetical.
Kant wants to go beyond the hypothetical to the
thetic—and we only do that by explaining that 'A
is A because it is thought', that identity of the
thinking subject founds the identity of things,
although we have to remember that the thinking
subject is the finite ego. What we get is
the idea that 'ego equals ego', a synthetic
identity, meaning that ego gets its identity by
thinking of itself as the condition of everything
that appears in space and time. 'Hence the
synthetic identity of the finite ego replaces the
infinite analytic identity of God'.
What would it mean to be Leibnizian today?
Kant's new form of thinking established 'a kind of
radically new conceptual aggregate', but there are
still things in it which are not explained, for
example, the 'exact relation… between the
condition of the phenomenon itself insofar as it
appears' [how do apparitions turn into formed up
phenomena?] How does thought condition
phenomena? Everything depends on what the
form is of thinking. Kant simply has to
assert this quality of the thinking subject as a
fact of reason, despite his official attempt to
make it follow quid juris [all
philosophers do this, it seems to me, smuggling in
the empirical, usually via some banal
assertion]. It follows from some notion that
our faculties must be in harmony, especially
between passive sensibility and active
thought. The idea of God ultimately
guarantees this harmony, so God appears after all
'behind our backs'. This is a problem for
postKantians.
Instead, we could show that conditions of
apparition are genetic elements in some way.
This could proceed through the idea that the
finite has some internal mechanism that leads it
to try and overcome itself and reach the
infinite. Kant did not develop this idea,
because he was not particularly interested in the
infinite. It is this act of overcoming
itself that produces the world of apparitions, as
a genetic process not just the conditioning
one. In some ways, this involves a return to
Leibniz, however: all the elements to explain a
genesis are present in Leibniz, including the
notion of an unconscious of thought [at the level
of perception rather than apperception?].
This unconscious could 'contain the differentials
of what appears in thought', and this would be
developed by Fichte and then Hegel.
Philosophy involves creating concepts, and this is
as creative as art: it also occurs 'in
correspondence with other modes of
creation'. We need concepts as some
spiritual accompaniment to a material
existence. The old concepts will still prove
serviceable as long as we locate them 'within new
conceptual coordinates'
We use our 'philosophical sensibility' to judge
the consistency of concepts. Philosophy has
a history which is not separate, and 'nothing,
never is anyone overcome'. It is the
concepts that we do not create that limit
us. Philosophers are no longer seen as
heroes—that belonged to the Romantic era.
Philosophers no longer attempt to construct or
found an entire world. There is a 'kind of
continuous flow [with] twists and turns', and we
no longer head in that direction.
NB the blog also attracted some comments.
One person found the whole thing as 'all silly
thoughts'. Another commentator tried to
analyze the person who had made a comment as
intolerant and neurotic..
back to Deleuze page

