Transcription in infinitum: Leibniz’s approaches to infinity.
Adrian Mackenzie
(Australasian Association for Philosophy Conference
Melbourne, AAP July 1999)
Introduction
The general context of this paper has two facets. On the one hand, stands the
reception of Leibniz in European philosophy over the last hundred years or
so. On the other hand, stands Leibniz as a possible resource for approaching
certain difficulties that beset European philosophy as it tries to deal with
contemporary technological and scientific knowledges and practices.
I would like to move between these two facets by first sketching for you how
Leibniz has been treated over the last sixty or seventy years in mainstream
continental philosophy. Some of the principal figures in European
philosophy, ranging from Husserl through Heidegger to Derrida, Lyotard,
Serres and Deleuze, have commented on Leibniz. With the exceptions of
Serres and Deleuze, Leibniz has been regarded - I am speaking in broad terms
- by all these philosophers as an example of the impasses and dead-ends
which metaphysical thought encounters today. Obviously, I can’t deal with
all their work. Deleuze is an exception that I will return to. After sketching
how Leibniz exemplifies metaphysical thought for Heidegger, Lyotard and
Derrida, I would like to then sketch out how a different approach to Leibniz,
partially present in Deleuze’s work, might assign a different significance to
Leibniz’s thought today.
transcriptum ad infinitum
What is the meaning of the title "transcriptum ad infinitum"? The title both
refers to a preoccupation with notions of infinity, the infinite, and the
indefinite that runs through Leibniz’s work, and with the notion of writing,
and marks to which notions of the infinite are often joined in Leibniz’s
thought. My interpretation of both European philosophy’s reception of
Leibniz, and my alternative to that reception, will focus on notions of the
infinite as they intersect with Leibniz’s understanding of writing, marks and
traces.
First, a preliminary on notions of the infinite in Leibniz’s thought. A central
element of nearly every major reading of Leibniz undertaken in European
philosophy in the last few centuries responds to the difficult question of
infinity. To put Leibniz into a broader historical perspective, we need to take
into the shift that Kant introduced specifically in response to Leibniz. Kant
condemned the bad speculative infinities of Leibniz’s metaphysics on the
basis that they confused the limits betweent the sensible and the intelligible.
The idea of infinity for Kant can only be regulative, it cannot condition actual
knowledge. Infinity of any kind can never become an object of knowledge,
given the sensible conditions under which human knowledge is possible. For
Kant, we cannot say "the universe is infinite" because no sensible intuition
gives us an infinite magnitude as a sensed object.
If Kant’s critical revolution governed most understandings of Leibniz during
the late eighteenth and nineteenth centuries, at least during the second half of
this century, most readings of Leibniz today in European philosophy are
dominated by the interpretation that Heidegger assigned to him. The
background against which Leibniz is read during this century, beginning
with Husserl in his Cartesian Meditations, is undoubtedly complex and varied,
but Heidegger more than anyone else has developed an account of the
Leibniz as the exemplar of the completion or accomplishment of metaphysics
in modern technology. Heidegger wrote two books on Leibniz. The first, The
Metaphysical Foundations of Logic from 1928 takes up the question of Leibniz
as a the founder of modern logic, and asks what thinking logically means in
relation to the radical finitude of existence. For the purposes of this sketch, I
will more or less leave that work aside in order to concentrate briefly on the
second, which is more richly linked into recent and contemporary European
philosophy.
The second book, The Principle of Reason, is based on a series of lectures that
Heidegger gave in 1955-56, and explicitly views Leibniz through the lens of
the principle of sufficient reason, usually formulated as nihil est sine ratione,
"nothing is without reason." Starting from this principle, Heidegger will
derive his diagnosis of the predicament of humans faced with everexpanding technological possibilities. The principle of sufficient reason,
which is implicit in most Western thought since the sixth century B.C.,
Leibniz calls the principium reddendae rationis sufficientis, and doing so, he
makes it into his principle of principles, the most fundamental and supreme
principle of reason.
I cannot track down here the nuances of Heidegger’s reading of the principle
of sufficient reason. Instead I will only indicate in what ways, according to
Heidegger, Leibniz’s principle of sufficient reason becomes decisive for the
information age and modernity in general. At the close of his book,
Heidegger explicitly links the principle to the "construction of thinking
machines and for the building of frameworks for large calculations." For him
1
it underlies the prevalence of technological frameworks for the organisation
of human action and interaction. The dynamism of modern science and
technology are driven for Heidegger by a set of demands and requirements
explicitly articulated through the principle of sufficient reason.
What are requirements? They are implied in three questions that Heidegger
asks about the principle. The principle is usually understood to mean, as
Leibniz himself points out, that nothing happens without a cause. What is at
stake in the name that Leibniz gives it? Heidegger’s reading particularly
concentrates on the meaning of two words in the principium "reddendae
rationis" sufficiens, to render reasons, and asks three questions: (i) why is a
reason always a rendered reason? (ii) why must a reason always be rendered,
or made explicitly? (iii) to whom or what is a reason rendered? His answers
2
will be, based on his reading of Leibniz, that a reason is a rendered reason
because truth is only true if a reason can be given for it. "Truth is always ... a
true proposition, that is, a correct judgment." Within the structure of
3
judgment, reason is what supports the connection between subject and
predicate. Nothing occurs that one could not render a reason for. The reason
must be made explicit otherwise a judgment remains unjustified or
unaccounted for. Finally, the site where the account is rendered will be the
human subject or rational soul. Such a being relates to the world by
rendering to the world to itself in terms of representations or judgments
whose connection with each other determines the world as object. The I who
judges or represents is the one to whom reasons, here understood as the
connection of representations, are rendered.
The reasons given back to the representing subject set up a series of
connections between objects. These connections constitute sufficient reasons
when they allow the object to cross a certain threshold between non-existence
and existence. In order to be sufficient reasons, the reasons given an object
secure the object or bring it to a standstill for the purposes of cognition. This
can only happen if the reasons given so thoroughly comprehend or determine
the object of cognition that for all times and places, it will be the same object.
1
@#b000400 (124)
2
@#b000400 (118)
3
@#b000400 (118)
This, according to Heidegger, is precisely what happens in technoscientific
calculation and representation:
The completeness of the reasons to be rendered–perfectio–is what originally
guarantees that something is, in the literal sense, firmly established–secured
in its stance–as an object for human cognition.
4
The principle of reason is the fundamental principle of Rational cognition in
the sense of a reckoning that securely establishes something.
5
Now given that this principle, is in Heidegger’s view, the grand fundamental
principle of cognition, not only for Leibniz but for "the age of Western history
we call ’modernity,’" it must be at work in modern technology. Hence:
6
Without really understanding it, we know today that modern technology
intractably presses toward bringing its contrivances and products to an allembracing, greatest-possible perfection. This perfection consists in the
completeness of the calculably secure establishing of objects, in the
completeness of reckoning with them and with the securing of the
calculability of possibilities for reckoning. ... Modern technology pushes
towards the greatest possible perfection. Perfection is based on the
calculability of objects. The calculability of objects presupposes the
unqualified validity of the principium rationis. It is in this way that the
authority characteristic of the principle of reason determines the essence of the
modern, technological age.
7
A few points to note from this long citation. The perfection that Heidegger
stresses is understood in terms of completeness. The main trend or dynamic
active in technology is towards completeness. What is to be completed here?
Basically the process of calculation itself. Technology as the completion
(Vollendung) of metaphysics tends towards the ’greatest possible perfection’ in
calculability of objects. All modern thought, insofar as it is regulated by the
principle of sufficient reason, would participate in enhancing the calculability
of objects. Heidegger names information (understood in the sense of the
mathematical theory of information developed post World War II) as the
keyword for the contemporary manifestation of the principle of sufficient
reason.
4
@#b000400 (120)
5
@#b000400 (121)
6
@#b000400 (121)
7
@#b000400 (121)
The limit, the completion: moving on from Heidegger
What is the limit of calculability, the point at which completion would occur?
Leibniz has an account of this that Heidegger discusses in passing (although
he gives a lot more attention to it in the earlier 1928 lectures on Leibniz’s
logic). The limit of calculability, and of reason, is, of course, embodied in God,
that being who is both most rational, and most perfect by virtue of the perfect
or complete set of reasons that it can give. The supreme being completes the
series of reasons that no finite reason can encompass all at once. The kinds of
infinity that inhere in God according to Leibniz’s metaphysics rest on the
notion of perfect existence, an existence that gives itself sufficient reason by
being perfectly rational. Leibniz writes" "it may be that there is only one thing
which is conceived through itself, namely God himself." (We could say that
8
God is like an operating system that boots itself up, and furnishes the
framework for all events to occur.) God both puts an end to the infinite
recursion of sufficient reason by providing a foundation, and presents that
foundation as a complete set of reasons, or perfect rationality.
The connection that Heidegger works to establish between Leibniz’s
metaphysics and modern technology pivots on the substitution of modern
technology, understood as the outgrowth of the principle of sufficient reason,
for the limit point that God provided in Leibniz’s theory. This substitution has
continued in readings of Leibniz in recent philosophy. The philosopher JeanLuc Nancy specifically declares that "Leibniz shows the first clear
consciousness of modern technology, as no longer deus ex machina, but
machina ex deum." The prime example here, however, would probably be
9
Lyotard who often invokes Leibnizian notions in discussing contemporary
thought. For instance, Lyotard advances what he calls a "Leibnizian
hypothesis" in describing how time today is organised by technoscientific
accumulation of information:
There is good reason to assume two extreme limits to the capacity to
synthesize a multiplicity of information, the one minimal, the other maximal.
Such is the major intuition which guides Leibniz’s work, and in particular,
the Monadology. God is the absolute monad to the extent that he conserves
in complete retention the totality of information constituting the world. ... As
8
@#b000402a (2)
9
Nancy, [FULL REF NEEDED HERE]
consummate archivist, God is outside time, and this is one of the grounds of
modern Western metaphysics.
... [O]n the side of the other limit ... [o]ne can imagine a being incapable of
recording and using part information by inserting it between the event and
its effect: a being, then, which could only convey or transmit the ’bits’ of
information as they are received. ... This is the being that Leibniz calls a
’material point’
10
Between these two limits found in Leibniz’s metaphysics, God and the bare
material point, modern technology definitely, for Lyotard tends towards the
maximal limit of retention of information. It tends towards becoming the
complete monad, God, retaining more and more data, becoming more and
more capable of stabilising the event in advance by laying down a series of
connections between representations. It does so by ordering or complicating
arrangements of ’material points’ into a ’big monad’, a computer "much more
complete, much more capable or programming, of neutralizing the event and
storing it."
11
Leibniz, technology and writing
There are two points to be made here for my purposes. Firstly, there are
undoubtedly good grounds to link Leibniz’s thought with technoscientific
thought. I have no argument with the general association that Heidegger,
Lyotard and others make between Leibniz and technology. Leibniz’s writings
are full of technological metaphors and ideas, his contributions to
mathematics, physics, engineering, law and many other fields are wellknown. To give one example among thousands scattered through his texts of
the technological conceptuality that underlies his thinking, Leibniz writes in
1702,
In fact the world is ... a machine, each part of which is composed of a truly
infinite number of devices.
12
And immediately, giving the limit term for this world under as composed of
machines all the way down, Leibniz writes of God the archivist,
10
Lyotard, "Time Today" 60-61
11
Lyotard, "Domus and the megalopolis" 198-199
12
@#b000401p (244)
But it is also true that the one who made it, and governs it, is of a yet more
infinite perfection, since he encompasses an infinity of possible worlds, from
which he selected the one that pleased him.
13
Echoes of this view of what exists are still readily to be found today, for
instance, in the view proposed by physicists that the universe be considered
as a computational process.
More specifically, and this is my second point, there are good grounds to see
how that linkage between technology and metaphysics passes through the
nexus of notions of infinity and Leibniz’s notions of writing. (Hence, again the
transcriptum ad infinitum of my title.) The notion of writing is important not
just in terms of the quasi-computational projects Leibniz undertook in
relation to coding thought in order to allow it to become automatic. His
various projects for a mathesis universalis, a characteristica universalis or
combinatoria all share the notion that systems of marks could somehow
enhance the finitude of human reason, which by contrast, with divine reason,
is obliged to work in time or successively rather than instantaneously.
For instance, in the text "Of Universal Synthesis and Analysis," Leibniz
undertakes to develop a system for creating inventories of ideas. The
combinatory system he comes up with would allow both the analysis of
existing ideas down to their primitive components (in order to see whether
the internal connections of primitive ideas are valid) and synthesis and
testing of new ideas by producing new combinations of symbols. In both
cases, the techniques rely on substituting symbols for concepts, and
manipulating systems of symbols. (Is it any accident then, that when he
comes to illustrate the difference between synthesis and analysis, Leibniz
makes use of technological inventions as his example? )
14
When technology is substituted for God in recent rereadings of Leibniz, it is
precisely on the grounds that modern technology increasingly orients itself
around systems of writing, and that it works to reduce the time of writing. It
tends to reduce the time of calculation or time of writing beneath the
threshold of consciousness.
Leibniz and deconstruction of the sign
The significance of the notion of writing in Leibniz’s thought brings me to a
final example of the reading of Leibniz in recent European philosophy, this
13
@#b000401p (244)
14
@#b000402c (10)
time Derrida’s positioning of Leibniz’ understanding of writing within the
history of Western metaphysics.
It is a well-rehearsed tenet of deconstruction that in the Western tradition of
metaphysical thought, written signs are secondary or auxiliary in relation to
self-present thought. For deconstruction, what is called thinking in that
tradition understands itself as "the being-before-oneself of knowledge in
consciousness". (This is the same structure of representation coupled to
15
reason that Heidegger addresses in his reading of the principle of sufficient
reason. ) Correlatively, what epitomises metaphysics and the metaphysical
epoch for deconstruction is the secondary status it attributes to the sign,
particularly the written sign. Metaphysical thought maintains that signs,
while they may be instrumental in economising the processes of reason, are
themselves never necessary to thought. Thought is not founded in any way
on signs, marks or writing, because it is already able to represent itself to
itself without taking a detour through signs or marks. Metaphysics stabilises
an account of what exists on the basis of the primacy of living thought, on
thought that presents itself to itself (e.g. Descartes’ cogito ergo sum) without
recourse to marks which carry passivity, forgetfulness and exteriority.
Abundant passages from Derrida’s work and those influenced by his work
make these points.
Deconstruction has by and large assigned Leibniz a central role as a
rationalist proponent of just this metaphysics of self-presence. It is no surprise
that a so-called "rationalist" philosophy such as Leibniz’s, which ostensibly
accords an indispensable and absolute priority to reason, would receive scant
attention in deconstructive philosophy. This judgment on Leibniz seems to
readily place his thought amongst the most chronically logocentric
metaphysics of presence. Furthermore, Leibniz’s projects for writing systems
such as a universal characteristic and combinatory have been seen as still
16
subordinate to the metaphysics of presence. For example, Derrida writes of
Leibniz’s project to set up "characteristic numbers" in order to code notions:
That is why, appearances to the contrary, and in spite of all the seduction that
it can legitimately exercise on our epoch, the Leibnizian project of a universal
15
J. Derrida, Speech and Phenomena and Other Essays On Husserls Theory of Signs, tr. David B.
Allison, Evanston 1973, p. 102
16
Gottfried Wilhelm Leibniz, “Preface to a Universal Characteristic”, G.W. Leibniz
Philosophical Essays, eds.Garber, Daniel & Roger Ariew, Indianapolis & Cambridge1989, pp.
6-8
characteristic that is not essentially phonetic does not interrupt logocentrism
in any way.
17
Translated, this means that Leibniz remains too metaphysical, and his
thought (at least as represented by his projects for writing systems) only
drives the completion of metaphysics through technology even harder. I see
this dismissal of Leibniz’s project as a little hasty.
What interests me in Leibniz’s understanding of the relation between signs
and infinity is precisely the status of incompleteness that he assigns to
writing. All the recent readings of Leibniz that I have been pointing regard
him as a metaphysician who substructures his system by recourse to an
infinite term, God. God guarantees an ultimate limit, or a series unanalysable
infinite primitives that support all the contingent, derived combinations that
come into existence as individual substances (ranging from the ’bare’ monads
of matter through to monads with memory and reason). What I will argue, by
contrast, is that whenever the recourse to infinite terms occurs, it is linked to
notions of marking or writing that undercut any notion of the infinite as
immediately present. Rather, infinity is constructed transcriptively. It is not
present all at once.
This implies that a certain level, the deconstructive move and certain
pathways in Leibniz’s thought intersect. In particular, Leibniz’s approaches to
the problems of the infinite can be understood as attempts to think through
the limits of reason, not in order to demonstrate the power of reason to render
the world fully knowable as rational existence, nor in order to reach the limits
at which reason must recoil in favour of an affirmation of enigmas, but in
order to continuously transform the basis on which thinking unfolds itself.
18
This possibility surely remains of current interest.
Approaches to the infinite
Leibniz has at least four different ways of talking about, or approaching
infinity. Each of them stands in a different relation to marks or inscription. I
will move through them quickly, although they should probably be
interpreted at greater length in order to establish the reading more securely.
The first task here is to ask: what are these four different ways of speaking
17
Jacques Derrida, Of Grammatology, tr. Gayatri Chakravorty Spivak, Baltimore & London
1974 p. 78
18
A similar theme has been taken up elsewhere, but without attention to Leibniz’s concept of
the mark: see Nathalie Chouchan &Frank Burbage, Leibniz et l’infini, Paris 1993, p. 55
about infinity? The second will be to show that their relation to each other is
unstable, and therefore, interesting. This will be the basis of my
"transcription" argument. (So I’m undertaking an analyis which makes quasiarbitrary cuts.)
(i) Infinite substance
The classic and well-known picture of substance in Leibniz’s metaphysics
shows the monads, ranging from angels down to mere points, or naked
monads, pulsating material points that retain nothing from moment to
moment. Every monad, no matter what its place in ranking of monads,
contains infinity. For these reason, as everyone knows, monads have no
windows. They don’t need to communicate. At most, they reflect on
themselves.
Individual substances are defined as those entities possessing "a conception
so complete that the concept shall be sufficient for the understanding of it and
for the deduction of all the predicates of which the substance is or may
become the subject." As this concept is complete, the concept of individual
19
substance implies the inclusion of "folds to infinity". Leibniz writes in the
Monadology: "[N]othing can limit it [the monad] to represent merely a part of
things. ... It is not in the object represented that the Monads are limited, but in
the modifications of their knowledge of the object. In a confused way they
reach out to infinity or to the whole." The expression "folds to infinity"
20
implies something important for our purposes. The requirement that the
concept of an individual be complete means that the concept of any
individual substance must, in principle, allow the deduction of any event in
the world at any time. But this perception would only be fully available to
one who unfolded everything fully and all at once. Within monads,
everything is not fully available. Individuality in fact comes from what is
encoded: "a soul can, however, read in itself only what is there represented
distinctly. It cannot all at once open up all its folds, because they extend to
infinity". The way in which a world is included within the perceptions–the
passing states which "represent a plurality in a unity" – of an individual
21
22
substance varies according to the way in which these "folds to infinity" are
19
G.W. Leibniz Discourse on Metaphysics tr. G. Montgomery La Salle, 1902 §8, p. 13
20
G.W. Leibniz,"Monadology", §60 in Discourse on Metaphysics tr. G. Montgomery La Salle, p.
264
21
G.W. Leibniz,"Monadology", §61 in Discourse on Metaphysics, p. 265
22
G.W. Leibniz,"Monadology", §14 in Discourse on Metaphysics, p. 253
organised or ordered. The condition under a world can be included within
the concept of an individual, is understood in terms of "reading". Already
marks are involved in the understanding of how finite individual substances
include a world.
Rather than an unfounded application of an abstract idea of infinity to the
notion of substance, the inclusion of folds to infinity brings a particular form
of perspectival limitation or ordering that Leibniz articulates as the operations
of folding and reading. It limits and differentiates rather than abstracting and
homogenising. This folding differentiates each substance from every other
substance in the same world. To be an individual substance is to express
infinity by possessing zones of unreadability, domains where the complexity
of marks becomes too deep to be read simultaneously. This is the basis for
Leibniz’s perspectivalism. Infinity is not the outer limit for the totality. In this
respect it is present in every perception as the unanalysed, undeciphered
components of the perception. (Leibniz regards every conscious perception as
floating on a stream of unconscious perceptions. He was an early proponent
of the unconscious.)
(ii) 2nd approach: auto-inclusive infinities
When it comes to a substance that could be fully read, there is only one
candidate, God, and only God knows how to read God, so to speak. God
forms an auto-inclusive infinity.
We have already glimpsed the role that God plays in relation to reason, and
hence to what exists supported by reasons or causes. If the infinity that
pertains to monads is the innumerable folds of perception they harbour,
which they unify without being able to express fully, what distinguishes God,
"the Necessary Being", from individual substances? The distinction can be
made on the basis of Leibniz’s second approach to infinity. God, while also a
unity, indeed "the primary Unity, or original simple substance", is not a
23
substance distinguished by unreadable, compressed marks. God is substance
without limits, contradications or negations: "where there are no bounds, that
is to say in God, perfection is absolutely infinite". There is a tendency to
24
construct God on the model of the rational soul, which Leibniz warns against.
Although Leibniz frequently speaks of God’s will and understanding, or of
23
G.W. Leibniz,"Monadology", §47, p. 261
24
G.W. Leibniz,"Monadology",§41, p. 260
God as the divine architect and monarch, he also marks these capacities as
25
anthropomorphic projections: "he humanises himself, ... he is prepared to
tolerate anthropomophisms, and ... he enters into a society with us, as a
prince with his subjects." For Leibniz, when we try to think beyond the
26
limitations of individual substance, we tend to rely on a set of
anthropomorphic projections and extrapolations. The habit of speaking of
God’s justice, will or understanding, for instance, must stem from these
projections. God entails infinity in a different way, a way which individual,
finite and embodied reason finds elusive and difficult to articulate without
anthropomorphisms.
Even the most rational monad is flooded by confused and obscure
perceptions that cannot be reduced to clear and distinct ideas. This because
"at every moment there is in us an infinity of perceptions, unaccompanied by
awareness or reflection." The series of reasons involved in fully clarifying
27
any contingent event or proposition is infinite. Here the infinities associated
with God provides a buffer or absolute reassurance for finite reason. God
saves finite or embodied perception, for instance, from trying to iteratively
analysing contingent reasons without limit or horizon:
In the case of a contingent truth, ... one never arrives at a demonstration or an
identity, even though the resolution of each term is continued indefinitely. In
such cases it is only God, who comprehends the infinite at once, and can
understand a priori the perfect reason for contingency.
28
Although we can’t read our contingent perceptions apriori, God can.
However, this view of God remains oriented by the anthropomorphic
projections that Leibniz accepts as pragmatic precautions, as a prosthesis to
stabilise thought against a loss of balance.
But there is another function served by God which Leibniz, again not
accidentally, accesses in terms of writing systems. Leibniz writes:
25
e.g. G.W. Leibniz, "Principles of Nature and of Grace", §15, Gottfried Wilhelm Leibniz
Philosophical Writings, tr. M. Morris, & ed. G.H.R. Parkinson, London 1973, p. 202.
26
G.W. Leibniz Discourse on Metaphysics §36, pp. 61-62.
27
G.W. Leibniz, "Preface" New Essays on Human Understanding, tr. & ed. Peter Remnant &
Jonathan Bennett, Cambridge 1981, p. 53.
28
G.W. Leibniz "Necessary and Contingent Truths", Gottfried Wilhelm Leibniz Philosophical
Writings, p. 97.
It may be that there is only one thing which is conceived through itself, namely God
himself, and besides this there is nothing, or privation. This is made clear by an
admirable simile. ...
(0)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
0
1
10
11
100
101
110
111
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
1000
1001
1010
1011
1100
1101
1110
1111
... The immense advantages of this system I do not touch on at present; it is enough
to have noted in what a wonderful way all numbers are thus expressed by unity and
by nothing
29
Again, the conception of the infinite involves marks and their combination.
In this simile, the binary number 1 marks a unity that cannot be analysed, and
"0" the absence of that unity. If God provides a limit to analysis, it is because
God can be understood as a primitive concept, like the mark "1." Such a
concept cannot be analysed because it is, as Leibniz explains, it "has no marks,
it is its own sign." God is attribute rendered absolute or infinite as mark that
30
marks itself. In God, not only knowledge, but all attributes such as
immensity, power and will become perfect, or "absolutely infinite" solely in
the sense that they include themselves. In his book on Leibniz, Gilles Deleuze
writes, "every form that can be thought of as infinite by itself would be
identical to itself, capable of being raised directly to infinity, by itself, and not
by means of a cause".
31
What relates these absolute, auto-inclusive infinities to other infinities? In the
New Essays Leibniz writes that "the genuine infinite is not a ’modification’; it is
the absolute; and indeed it is precisely by modifying it that one limits oneself
and forms a finite ... These absolutes are nothing but the attributes of God."
32
Each of the attributes of God is an absolute. If the attributes are absolute,
relative ideas that are formed by modifying or limiting absolute attributes are
not themselves genuine infinities. For instance, rather than speak of the
infinity of spatial extension, Leibniz speaks of the absolute immensity of God.
Infinite space would be a relative idea formed by a limitation of a divine
attribute.
29
@#b000402a (2)
30
@#b000402b (7)
31
G. Deleuze, The Fold. Leibniz and the Baroque, p. 44.
32
G.W. Leibniz New Essays on Human Understanding, Bk II, ch. xvii §§2-3, p. 158
These absolutes can be thought of as ’self-identical’, in contrast to the monad’s
perceptions which always await further resolution. Whereas the mode of
infinity at stake in individual substance concerned the "folds to infinity"
understood as particular orderings and combinations of marks involved in
the inclusion of all predicates in a complete concept, the infinities involved in
the notion of the necessary being God are infinities of "auto-inclusion". They
exist by including only themselves, without contradiction or relation to any
other absolute attribute. If monads are infinity included within unifying
limits, what Leibniz calls "God" is the ensemble of absolute attributes without
contradiction or even relation, infinities without limit.
(iii) World as innumerable infinity
If we move across to another node on the graph, this time the world
Leibniz describes in the Discourse on Metaphysics how the order of the world
can be understood by analogy to a line that joins together a scattering of
points on a page. A line joins together distinct points in a continuous order.
For any given line, or any given ordering of points, there must exist a
function that can generate that line. Leibniz offers a computationally difficult
example: "there is no instance of a face whose contour does not form part of a
geometric line and which can not be traced entire by a certain mathematical
motion." But like a geometrical curve, the world understood as a continuum,
33
does not contain a numerable set of points. and like a set of points, any
number of different lines can drawn connecting them.
Again, it is difficult to overlook the connection here between the conception
of innumberable infinity and the renewed recourse to notions of marks
implied by the "scattering of points on a sheet of paper helter skelter."
34
Starting with a highly compressed text traced on the interior membrane of the
monad, we moved on to read the primitive ultimately binary marks of that
text as the absolute attributes of God. Now, viewing the world as a collection
of substances, the perspective moves around to regard the world as a line
connecting metaphysical points. The scale of the writing is now at the level of
the contour of the mark, exemplified by the contour of face understood as one
part of a geometrical line.
Again, this is no completed infinity. The world composes no whole. Although
it has "parts", they do not compose a unified totality or whole. The
33
@#b000403a (10)
34
@#b000403a (10)
appearance that the world has of composing of whole made up parts is a
phenomena stemming from the ordered perceptions of individual substances.
For Leibniz, unity never arises merely from a collection or association of
parts: "we have to admit that this unity that collections have is merely a
respect or relation, whose foundation lies in what is the case within each of
the individual substances taken alone." Certainly parts can group together to
35
form aggregates or clusters, to mark particular regular contours or
symmetries, but where there are only parts, there is no unity except in those
parts themselves. Although the world has many levels of order, many
different layers of patterning and complexity, it is not a substance and it has
no unity. Instead, the existing world emerges as the greatest possibility or
"amount of essence": "out of the infinite combinations of possibles, and the
infinite possible series, that one exists by whose means the greatest possible
amount of essence or possibility is brought into existence."
36
There are a number of points to be drawn from this statement. Firstly, the
world is understood as "the chain of states, or series of things whose
aggregate constitutes the world". The world is a chain or series which
37
composes an aggregate, not a unity. This means that it cannot include infinity
in either of the previously mentioned modes. It can neither include it from a
singular point of view, under limits, nor can it include infinity absolutely,
without limits, as perfection.
Secondly, the states of the world compose a series which is incomplete and
incompletable in that the series itself does not contain any unconditioned
reason why this world should exist. In terms of the function that generates
the line connecting points on the page, we might speak of the existing order
as given by a function which maximises some value, but reaches no absolute
value. The world, as Leibniz explains by way of illustration, is one solution to
a massive "tiling problem" in which the task is to realise the maximum
capacity or degree of order to be found in the infinite combinations of
possibles given minimum expenditure on metaphysical constructs or
determinations (such as being prevails over non-being). It is only by
38
reference to an infinite number of other possible combinations that the variety
35
36
Leibniz, New Essays, Bk II, ch xii, p.146
@#b000402n (138) Leibniz, "On the Ultimate Origination of Things", Philosophical Writings,
p. 138
37
Leibniz, "On the Ultimate Origination of Things", Philosophical Writings, p. 137
38
Leibniz, "On the Ultimate Origination of Things", Philosophical Writings, p. 138
of things within the series of this world could be calculated and determined
as of maximum consistency.
Thirdly, the actual world contains the most ordered variety possible, yet
without being unified or unifiable. Even in God’s understanding only allows
comparison and evaluation of possibilities, not perfect unification of the
existing world. The world contains an infinite quantitative multiplicity of
simple substances whose relations with each other entail an inevitable degree
of disunity, decomposition and disharmony. The order of this world, even in
its most ordered configurations, is not evidence of the world as a perfect
totality or whole, or even of an enumerable collection of things. As Bourbage
and Chouchan remark, "the world, or ’aggregate of finite things,’ would
neither be able to ’constitute a true whole’ nor be determined by a numerical
quantity." Although the pre-established harmony means that all substances
39
do belong in an ordered fashion to the same world for Leibniz, there is no
way of enumerating these substances as elements of a finite set.
(iv) Matter and texture: the infinite labyrinth of the continuum
The problem of how to move from the world understood as an innumerable
aggregate of individual substances to the world as a continuum opens onto a
final approach to infinity: the labyrinth of matter and nature. Again a shift of
perspective is involved here. The maximising function which ordered points
in a series as the face or contour of a world is now remapped, or transcribed
into a fluid and none homogeneous continuum. No longer linking scattered
points on the page, we are now scratching the surface of the page, so to speak.
Nature is regulated by the principle of continuity: "all nature is a plenum"
Leibniz writes. As a plenum, it is filled with material bodies. The question
40
will be for Leibniz how we should understand the texture of the continuum.
Strictly speaking, the plenum of nature is an aggregate mass. From this
41
perspective, a material thing consists of the innumerable metaphysical points
or individual substances of which it is composed, and it appears to undergo
changes – divisions, motions, collisions – by virtue of the accretion and
departure of those unities from particular regions or zones of aggregation.
Bodies change, they are in perpetual flux, and this flux reflects the unlimited
39
Nathalie Chouchan &Frank Burbage, Leibniz et l’infini, pp. 70-71.
40
G.W. Leibniz, "Principles of Nature and of Grace Founded on Reason", Philosophical
Writings, p.195.
41
G.W. Leibniz, "Letter to de Volder, 20 June 1703", G.W. Leibniz Philosophical Essays, p. 177.
passivity of matter. Material bodies affect each other so pervasively, for
Leibniz, that "each corpuscule is acted on by all the bodies in the universe."
42
From these considerations the labyrinth of the material continuum emerges:
"this could not happen unless matter were everywhere divisible, and indeed
actually divided ad infinitum." The divisibility of the continuum of matter
43
means that another approach to infinity is required: infinity approached
through repeated division.
It would be tempting to represent this divisibility in mathematical terms as
the infinitesimal dx that Leibniz developes in the differential calculus. Such a
representation would imply that the continuum could be understood as a
homogeneous geometrical continuum. Against this, two points are made by
Leibniz. Firstly, "mathematical points ... are nothing but extremities of the
extended and modifications out of which it is certain that nothing continuous
could be compounded." In other words, the idea of the point should be
44
understood as an ideal limit. Secondly, Leibniz’s description of the texture of
the continuum in the introduction to the New Essays Concerning Human
Understanding gives a different picture to that of homogeneous divisibility:
Space should be considered of as full of a matter originally fluid, susceptible of
any division, and submitted indeed actually to divisions and subdivisions ad
infinitum; with this difference, however, that it is divisible and divided
unequally in different places on account of motions in it which are more or
less harmonious.
45
The continuum contains variations in division that pass down and up
through the layers of aggregation and ordering that compose living and nonliving bodies.
Leibniz’s use of the expression "labyrinth of the continuum" reflects this
ordering of the continuum. In distinction to the homogeneous extension of
matter in Descartes’ metaphysics, Leibniz poses the variations and thresholds
in divisibility which compose organic and inorganic aggregates as the bases
42
G.W. Leibniz, "Metaphysical Consequences of the Principle of Reason", Philosophical
Writing, p.176.
43
G.W. Leibniz, "Metaphysical Consequences of the Principle of Reason", Philosophical
Writing, p.176..
44
G.W. Leibniz, "New System and Explanation of the New System", Philosophical Writings, p.
116.
45
G.W. Leibniz, "Preface", New Essays on Human Understanding, p. 59
of the ensembles of bodies which appear in natural phenomena. However,
even given these variations in texture which correspond to differentiated
sequences of relations between monads, does not this account of the
continuum as divisible ad infinitum still leave this approach to infinity as only
a possible infinity rather than something really existing?
Leibniz’s judgment is definite on this point: the divisibility of the continuum
of matter is actual not potential: "each portion of matter is not only infinitely
divisible, as the ancients recognised, but is also actually subdivided without
limit, each part into further parts, of which each one has some motion of its
own." With this actual infinity of division, there is no arbitrarily chosed
46
portion of matter, no matter how small, which would not contain an
unlimited further ordering of parts. From this, it follows that any portion of
matter bears traces of the rest of the continuum.
46
G.W. Leibniz,"Monadology", §65, p..266.