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The Mathematical
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Journal of the British Society for Phenomenology Vol. 35, No. I, January 2004
THE MATHEMATICAL BASIS OF BERGSON S
PHILOSOPHY
ROBIN DURIE
To create
a
healthy philosophy you should renounce metaphysics but be
a good
mathematician.
Russell, Lecture,
935
The original impetus for the divergence in the trajectories
of
analytic and
continental philosophy during the 20th century has, in recent years, become
a subject
of
scrutiny for both traditions. Most writers are agreed that the two
traditions share a more or less common point
of
departure, represented by
the shared concerns
of
Husserl and Frege at the start
of
the century. After
this, it is argued that Husserl s transcendental tum in Ideas (1913) marks the
moment at which the continental tradition begins to veer away from the
analytic tradition. Subsequently, a fundamental commitment to interiority,
and an affirmation
of
the irreducibility
of
subjectivity, are seen to be
hallmarks
of
the continental tradition. In contradistinction, the analytic
tradition has striven for objectivity. The methodological model to which it
has aspired in seeking to attain this objective has been scientific, and
specifically, mathematical, reflected in the central role accorded
to
logic by
the analytic tradition. From this perspective, it could therefore be argued
that the two traditions have been developing in opposition a twofold legacy
of Cartesianism - on the one hand, the analytic tradition seeking objectivity
by means
of
a contemporary reinterpretation
of
the
more geometri o
delineated in the Reply to the Second Set of Objections , while, on the
other hand, the continental tradition re-enacts the subjective turn of
Descartes
Meditations.
Arguments of this nature highlight the fact that much
of
the suspicion of,
and indeed antipathy towards, the
style
of
continental philosophy stem from
a
conviction
that continental
philosophy fails
to
engage
with the
mathematical or logical grounds of an adequate philosophical methodology.
t
is
just such a conviction that underpins the single text which did as much
as any other to provoke analytic philosophy s disenchantment with the
practices of its continental European sibling, namely, Russell s address to
The Heretics at Cambridge, on the evening of 11th March, 1912. This lecture
was subsequently
published
simultaneously
by
Open
Court as The
Philosophy of Bergson in volume 22 of The Monist (July, 1912), and as a
separate pamphlet bearing the same title. A revised version of the paper was
then included in Russell s
History
of
Western Philosophy
(1945).
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Developing the perspective outlined in my opening remarks, I will proceed
to demonstrate how Russell bases his attack on Bergson on mathematical
grounds. I will then seek to assess the extent to which Russell s attack on the
inadequacy
of
the mathematical basis of Bergson s philosophy - and hence,
as a consequence, on Bergson s philosophy as a whole - is valid. In
conclusion, I propose to indicate how a sufficient appreciation
of
the
mathematical basis for Bergson s philosophy not only serves to reveal the
gratuity of attacks such as those of Russell, but also opens up new avenues for
appreciating the contemporary significance of
Bergsonism.
That mathematics will indeed provide Russell with the grounds for his
attack on Bergson is evident from the opening criticisms he directs towards
Bergson s theories of
space and time. Russell begins by correctly identifying
the fundamental role played by the notion
of
separateness in Bergson s
work. For Bergson argues that the tendency of the intellect is
to
view things
as separated from one another, and matter
so viewed is that which is
separated into distinct things.
2
As a consequence:
things which really interpenetrate each other are seen [by the intellect] as separate solid
units: the extra-spatial degrades itself into spatiality, which is nothing but separateness. Thus
all intellect, since it separates, tends to geometry; and logic, which deals with concepts that
lie wholly outside each other,
is really an outcome of geometry.
This identification
of
separateness and spatiality is then extended to the
concept
of
number. Russell states that, in the first chapter
of
Time and Free
Will,
Bergson contends that
greater
and
less
imply space, since he regards
the greater as essentially that which contains the less. Russell then claims
that, in the second chapter
of
Time and Free Will, Bergson maintains the
same thesis as regards number.
4
Taking as his cue Bergson s definition
of
number-
that Number may be defined in general as a collection of units, or,
speaking more exactly,
as
the synthesis
of
the one and the multiple [TFW
75/ E
51] -Russell proceeds to argue that this definition, taken alongside
Bergson s identification
of
number with spatiality and separateness,
suffice[s] to show ... that Bergson does not know what number is, and has
himself no clear idea of it.
5
Russell s argument
is
straightforward. Bergson s thesis of number is, he
points out, in fact a thesis regarding various collections, to which numbers
may be applied. t is not, however, a thesis about these numbers themselves,
nor indeed about number as a general concept, a concept, namely, which
refers to a property common to any particular numbers. Whether or not
collections do indeed involve spatiality and separateness, it is most certainly
the case that both particular numbers
p r
se and the general concept
of
number do not. This is the case simply because both particular numbers and
the general concept of number are purely abstract, and as such, do not
involve any recourse either to space or to separateness.
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But what of actual collections, such as the collections of the months of the
year, the signs of the zodiac, and the apostles, to all of which the number 2
can be applied - do these provide sufficient grounds for supporting
Bergson s contention that every plurality of separate units involves space?
Russell thinks not, and refers to some of Bergson s own examples, such as
that
of
hearing a clock strike 12. Far from depending upon our rang[ing] the
strokes of a clock in an imaginary space, Russell claims, most people count
them without any spatial auxiliary. This being the case, it demonstrates that
spatialisation, and hence separateness, is not necessary for conceiving
collections. Russell therefore concludes that Bergson s recourse to number,
and hence mathematics, is wholly spurious, commenting that as regards
mathematics, he has deliberately preferred traditional errors in interpretation
to the more modem views which have prevailed among mathematicians for
the last half century.
8
There is, then, from Russell s point
of
view, a way
of
doing philosophy
which trades in problems that emerge as a consequence of philosophers
adherence to assumptions which themselves follow as a consequence of
these philosophers' failure
to understand
and
adopt
the findings of
contemporary mathematics. This very failure, we could then argue, is
perpetuated by the tradition of continental philosophy, whereas the analytic
tradition begins to describe its own distinctive trajectory from the moment
that it overcomes this failure.
Whether or not we might wish to give credence to such an account, it
nevertheless behoves us to reflect on the legitimacy of Russell s portrayal of
Bergson. What I now aim to show is that, in fact, Bergson s position is far
more nuanced than Russell allows, and indeed, that it is so precisely because
of
his keen
awareness
of
certain
theoretical developments
in
the
mathematics
of
the last half century. In fact, this should not come as a
surprise, since, in 1877, Bergson won first prize in mathematics for the
Concours General, and, with the publication the following year
of
his plane
solution of Pascal in
Nouvelles Annates
e
Mathematique,
it was assumed
that his academic future lay in geometry - indeed, it was expected that he
would enter the Ecole Normale to study mathematics.
9
The first clue we come across suggesting that all is not quite
as
Russell
makes out is to be found in the long footnote that Bergson appends to the
chapter heading which is itself immediately followed in the body
of
the text
by the definition of number cited by Russell. In this footnote, Bergson
admonishes a certain F Pillon for failing to distinguish between time as
quality and time as quantity, between the multiplicity
of
juxtaposition and
that of interpenetration [penetration mutuelle]. Bergson then writes that it
is the chief aim of the present chapter to establish ... this vital distinction.
He
goes
on
to
conclude the
footnote by
observing that 'the verb
to
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distinguish [distinguer] has two meanings,' meanings which philosophers
have tended to confuse [TFW75-6/CE 51-2]. Let us note at the outset that this
notion
of
the multiplicity
is
the same
s
that which
is
cited in Bergson's
definition
of
number, namely, 'the synthesis
of
the one and the multiple.'
1
On the basis of this note, therefore, I propose to argue that the key to
understanding
Bergson s
philosophy in general, and his discussion
of
number in particular, lies in an appreciation
of
the way he develops, hand in
hand, a 'theory of multiplicities' and a 'theory of distinctions.' In making
such a claim, I am in part taking
my cue from the way in which Deleuze
chose to characterise Bergson's methodology in his pioneering work
of
the
1950s and 60s. Deleuze consistently argued that the decisive methodological
concept
in Bergson s
philosophy
is that
of
multiplicity .
Thus,
in
Bergsonism,
he writes that the concept of multiplicity 'is essential from the
perspective of the elaboration of the method,' and he subsequently draws
attention to the fact that multiplicity constitutes the 'fundamental theme
of
[Bergson's] encounter with Einstein.'
12
Taking the arguments
of
Russell's
article alongside Bergson s apparently long-standing opposition to the
mathematisation
of
duration, it is clearly
of
the utmost importance to reflect
critically on the lineage of
Bergson s
appropriation of the concept of
multiplicity, which, s Deleuze has argued, derives in the first instance from
the German mathematician Riemann.
The point of departure for
Riemann s
thesis is the observation that
classical Euclidean geometry, s a deductive science, begins from a series of
assumptions about the nature of space and the basic constructions within
space, such as points and lines. These assumptions are expressed, as
Riemann notes, in a series of definitions 'which are merely nominal.' The
'essential determinations [wesentliche Bestimmungen] pertaining
to
these
assumptions are in fact only set out in the axioms. But this approach begs the
question of 'whether and how far' the connection between the assumptions
about the nature
of
space, and the assumptions about the basic constructions
within such a space, is necessary, and indeed,
of
'whether, a priori, it is
possible.' Riemann argues that this blind spot is a consequence of the fact
that the source of classical geometry is space
s
it
is
experienced in everyday
life. Geometry should, instead, begin from 'the general notion
of
multiply
extended magnitudes [der allgemeine egriff mehrfach ausgedehnter
Grossen],
what we now know
s
'n-dimensional spaces.'
Riemann therefore sets himself the first
of
two tasks, namely that of
'constructing the notion of a multiply extended magnitude out of general
notions
of
magnitude.' He specifies in tum that any such magnitude-notions
are themselves possible 'only where there is an antecedent general notion
which
admits
of
different
ways of
determination [verschiedene
Bestimmungsweisen].
t
is precisely such a general notion, capable of
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these magnitudes functions
as
a measure
[Mafistab]
for the other. However,
since the corollary of this is that, with regard to the compared magnitudes, it
is only the more and the less that can be determined, rather than the how
much, and that, as a consequence, such measured magnitudes cannot be
expressed in terms of a unit, then it must be concluded that the principle of
the
metric relations
[Princip der Mafiverhiiltnisse]
of
a
continuous
multiplicity must come from outside. In the case of a discrete multiplicity,
on the other hand, the principle of its metric relations is given in the notion
of it, precisely to the extent that its divisions of magnitude can be compared
with regard to quantity by means of a counting, the unit of which thereby
expresses the measured magnitude.
6
Today, there is absolute agreement regarding the classic status of
Riemann s
text within
the
history
of
mathematics. Equally, it
is
acknowledged that the significance of
Riemann s
thinking wasn t fully
appreciated until Einstein was able to utilise the mathematical apparatus that
had been developed on the
basis
of Riemann s
text
to give rigorous
expression to the physical ideas underpinning his theory of relativity.
Bearing this in mind, let us now recall Russell s conclusion to his critique
of Bergson, namely, that Bergson s recourse to number, and hence
mathematics, is wholly spurious, and that as regards mathematics, he has
deliberately preferred traditional errors in interpretation to the more modem
views which have prevailed among mathematicians for the last half century.
The implication of Russell s comment is that Bergson is ignorant specifically
about the developments in number theory which had taken place between
around 1860 and 1910. The traditional errors to which Russell refers would
appear
to apply to the traditional
philosophical theories of number
-
Platonism,
which holds that numbers
are
abstract entities existing
independently
of
the mind; psychologism, which claims that numbers are in
fact ideas in the mind (a theory initially held by Husserl in his
Philosophy
of
Arithmetic (1893), but then devastatingly undermined in the Prolegomena to
his
Logical Investigations
(1901)); and intuitionism, which argued that
number was a product of a process such as counting. Russell s emphasis on
the role
of
counting in his critique
of
Bergson suggests that he takes Bergson
to be an intuitionist. These traditional theories were ultimately supplanted by
the development of set-theory, on the basis of which numbers were viewed
as sets, such that the axioms
of
the theory of, for example, natural numbers -
i.e., arithmetic -were true to the extent that they were true of specific sets. In
turn,
the properties of numbers,
the
relations between
them, and the
operations which could be applied to them, were understood as properties of,
relations between, and operations on sets. This set-theoretical approach to
number was initiated by Cantor, on the basis
of
developments in pure
mathematics that had taken place during the latter half
of
the 19th century.
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In order to quantify anything, it
is
necessary that a number be applied to
what is to be quantified. Then when comparing objects to which numbers
have been so applied, we are able to determine which of the objects is the
greater on the basis of the unequal spaces occupied by each of the objects
numbered. Specifically, we are able to make this comparison because
of
the
fact that we call that space the greater which contains the other.
[TFW 2/(E
5] This early remark
is
the basis for Russell s claim that Bergson contends
that
greater
and
less
imply space, since he regards the greater as essentially
that which contains the less.
7
But is Russell correct to go on
to
claim that
Bergson thereby identifies number with space, an identification which would
betray that Bergson does not know what number is, thereby undermining
his philosophical position to the extent that it is founded on a theory of
number?
Certainly, Bergson wants to argue that the relation
of
container to
contained does not apply to intensive states (we will return to his reasons for
arguing in this way shortly). But it
is
the precise way in which he formulates
this point that is instructive: he writes that intensities ... are unable
to
be
superposed on each other ne sont pas choses superposables]. This failure
inevitably returns us to the issue, therefore, of whether indeed an intensity
can be assimilated to a magnitude
[grandeur]. [TFW 2/(E
6] Bergson s
point here is to draw attention to the fact that people tend to claim that,
although intensity cannot be measured whereas extensity can, nevertheless,
both intensity and extensity can be assimilated to magnitudes, and as a
consequence, we are justified when we talk in terms of being more or less
happy, more or less sad, etc. Since there must be something common to
these two forms of magnitude, people are therefore led to assume that
intensity and extensity do not differ in kind, but are, rather, two species of
quantity.
[TFW 3/(E
6] Bergson s language
is
instructive in the context of
our present discussion because, when he argues that intensities cannot be
superposed he is directly echoing
Riemann s
stipulation that measure
consists in the superposition of the magnitudes to be compared. To be sure,
this verbal resonance will require more in the way of supportive evidence if
we wish to establish that Bergson does indeed have Riemann in mind here.
However, if he is indeed referring implicitly to Riemann, then his argument
is of the utmost significance. For Riemann is distinguishing comparison with
regard
to
quantity
in
the case
of
discrete multiplicities
-
where the
comparison is achieved by counting - and continuous multiplicities - where
the comparison is achieved by measuring. In the case of each multiplicity,
that is, both
comparison
with
regard
to quantity, and assimilation to
magnitude, can be accomplished. But Bergson is wanting to argue that
neither measure nor assimilation to a magnitude is appropriate in the case of
intensity.
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Why then does Bergson seek to argue in this way, and why should we
continue to insist on the connection to Riemann? In order to answer these
questions, it is necessary to ascertain in what the essence
of
quantity
consists. What
is
at stake in the more or the less with regard to quantity
is to
be discerned in that fact that, in order to speak in this way
of
the more or
less, the initially given quantity must be able to increase or diminish. But if a
quantity
x
increases, and we subsequently wish to compare the initial
quantity x with the final quantity, to determine which is the greater, then we
must be able to
divide
or separate out, the initial quantity
x
from the final
quantity. [T W
3/ E
6] Furthermore,
if
the comparison
is to
be worthwhile,
then during this process
of
division, the quantity x must not change. This
seems to me to be the true import of Bergson s argument with regard to
separation - namely, that it is not spatial separateness that is significant,
which is what Russell wishes to claim but rather the capacity for
separability
that is definitive of quantity. From this perspective, therefore,
we can begin to argue that, contrary to Russell s claim, Bergson is not
advancing a thesis about number as such, but rather about the conditions of
possibility for counting (where countability is in tum to be understood as
condition for either measuring or assimilation to a magnitude). Thus, the
condition of possibility for counting is that the elements of a multiplicity be
denumerable - and, correlatively,
if
the elements are non-denumerable,
then they cannot be counted. In turn, the condition of denumerability is
precisely that the elements of the multiplicity be
distinct
or discontinuous.
Now, the distinctness of elements in a multiplicity does not, as such, require
that they be laid out in space,
as
Russell would have Bergson argue. Rather,
distinctness, or discontinuity, constitutes the condition for separability. As a
consequence, Bergson should be understood as arguing that the property of
distinctness which characterises the elements
of
a discrete multiplicity is also
a property that characterises space.
Thus, Bergson will argue that intensities are not quantifiable precisely
because they do not possess this capacity for divisibility or separability.
Now, if our preceding argument is correct, then this way of interpreting
Bergson has a more fundamental significance. For it is tempting to read
Bergson as propounding a series of more or less metaphysical dualisms -
such as duration/space quality/quantity intensity/extensity
continuous/discrete - between which he then seeks, as we have seen, to
establish an equivalence. The criticism which is then levelled against
Bergson is that none
of
these concepts is rigorously defined; rather,
Bergson s style of writing consists in a tendency to conflate concepts at
crucial junctures, explaining one in terms
of
the other. The problem would
then be that at no point does Bergson determine the justification for
establishing the equivalence between the concepts, with the consequence that
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his style
of
arguing is at best circular. The significance
of
the interpretation
that I am seeking to advance is that there is, in fact, a rigorous principle
underpinning all
of
these dualisms, a principle that is strictly defined, that
establishes a dualism which does not simply reduce one of the terms
to
the
negation of the other, and which, moreover, has a mathematical provenance
in Riemann s theory
of
multiplicities.
Before we can elucidate this principle, we must first return to the issue of
why it is that intensities are non-denumerable, despite our tendency to talk of
feeling, for instance, more or less happy. For thus far, we have merely
established that distinctness, and hence separability, is the condition for
counting, and thus measurement or assimilation to a magnitude. We have yet
to demonstrate that distinctness, or discontinuity, does not pertain to
intensity
as
such. In fact, Bergson s discussion allots to intensity something
akin to a midway position between purely qualitative affective sensations,
and extensive quantities, and it will be in this way that he is able to account
for our tendency to apply to intensities concepts more properly limited to
extensity, while at the same time maintaining that psychic phenomena [are]
in themselves pure quality or qualitative multiplicity.
[TFW 224/ E
146-7]
Bergson argues that where affective sensations have an external cause, a
cause for which quantity is an appropriate property, there is a tendency for
the sensation to be experienced as an effect
of
this cause, in such a way that
the sensation signifies its cause, becomes a sign for its quantitative cause,
with the consequence that the quantity appropriate to the cause is transferred
to the sensation. t is just such a sensation, experienced as a sign representing
its quantitative cause, that Bergson means by the notion of intensity.
[TFW
70, 90, 224/ E 49, 61, 146-7] As Bergson writes, we associate the idea of a
certain quantity of cause with a certain quality of effect; and finally . . .
we
transfer the idea into the sensation, the quantity
of
the cause into the quality
of the effect. [TFW 42/ E 31] t is in this way that intensity comes to
be
a
property
of
sensation.
[TFW
7/(E
9]
Moreover, it is also in this way that
Bergson
provides
an account
with
regard to
Riemann of
how this
multiplicity, which, to the extent that its elements are not discontinuous, does
not bear its own metric within itself, comes to secure the principle of its
metric relations from the outside. In this respect, therefore, despite Bergson
disagreeing with Riemann over whether the elements
of
a non-discrete
multiplicity can be measured he nevertheless maintains the principle set out
by Riemann that the possibility of quantitatively expressing the relations
between elements of a non-discontinuous multiplicity must be transferred
into the multiplicity from outside.
When we turn to purely affective sensations, we find Bergson arguing that
they are non-denumerable in themselves, and thus have to attain the property
of intensity
if
they are to be assimilated to a magnitude, because changes in
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sensations correspond less to variations
of
degree than to differences of state
or of nature. [TFW 17/(E 15] There is a qualitative difference rather than a
quantitative difference, a difference in nature rather than a difference in
degree, between irritation
and
anger, amusement
and
happiness,
apprehension and fear, and so forth. For Bergson, fundamental psychic
phenomena such as affective sensations are not
things
which can be
juxtaposed. Rather, an affective sensation alter[s] [modifie1 the shade
of
a
thousand perceptions or memories, and in this sense, it penetrates fpenetre]
them. [translation modified;
TFW
8-9/(E 1 ] A sensation, Bergson suggests,
should not be compared to a note which can grow louder or softer, but rather,
to
the note which gives the tone
to
the other instruments in an orchestra.
[TFW 35/ E
26] As we shall see presently, the precise terms in which
Bergson makes this distinction, between juxtaposition and penetration,
are of the utmost significance. For now, let us note that Bergson underscores
that affective sensations which thus penetrate conscious experiences, but
which cannot be juxtaposed, are indivisible. [TFW 32/ E 24] On this basis,
we can confirm once again that for Bergson, quantitative difference, or
difference in degree,
is founded on the discontinuity of the elements of the
discrete multiplicity, a discontinuity which in tum entails that these elements
are divisible. On the other hand, qualitative difference, or difference in kind,
would be founded in the non-discontinuity of the elements of what Riemann
had called the continuous multiplicity, and that this non-discontinuity entails
that the elements are indivisible. In what, then, does qualitative difference
consist,
if it is different from divisibility?
In the discussion of number, with which Chapter 2
of
Time
and
Free Will
begins, and upon which Russell lays so much critical stress, the key theme is
not, I propose, space as such. Rather, it is this principle
of
divisibility that we
have
been
discussing.
Whether
one follows
Bergson s
approach, and
conceives of a number as a collection
of
units, [TFW 75/ E 51] or whether
one follows Russell, and conceives of
a number x as the set
of
all sets with x
elements, a basic condition is that one is able to conceive
of
the collection or
set both as a unit, or whole, and as a sum [that] covers a multiplicity
of
parts
which can be considered separately. [TFW
76/ E 52]
1
8
As a multiplicity
of
units or elements, the collection or set is divisible; as a unit, it is indivisible,
in the same sense as the units or elements comprising the collection or set
must themselves
remain indivisible (although,
when
considered
in
themselves, rather than as elements
of
the collection or set, these units or
elements may well be infinitely divisible).
19
The divisibility of the units or
elements consists, as
we know, in the discontinuity, and hence, separability
or ability to be juxtaposed,
of
the units or elements. But, as we noted in
passing above, it
is
not just the discontinuity of the units or elements that
enables this divisibility. As Bergson specifies: It is not enough to say that
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number is a collection
of
units; we must add that these units are identical
with one another. .. [or if] they are all different from one another ... we agree
in that case to neglect their individual differences and to take into account
only what they have in common.
[TFW
76 CE 52] This shared identity of the
units within a collection corresponds to the common property that is shared
by members belonging
to
a set. What, then,
is
of
fundamental importance
is
that, in a process of division, made possible by the discontinuity between the
elements, the identity of the elements so divided is maintained. The elements
do not suffer any change in themselves as they undergo the process of
division. This
is
precisely because the process of division consists in nothing
other than a separation of parts from one another, a separability the condition
of
which is, as we have seen, founded in the fact that the multiplicity is
discontinuous.
On the basis of this argument, we can begin to determine in what non
divisibility will consist. Let us imagine, for the sake
of
argument, dividing a
unit which
is
conceived as such ultimately rather than provisionally. Since,
from this perspective, it does not consist in separable parts, what would be
the effect of
such a division?
t
would, of necessity, change the nature
of
the
unit essentially. It would create something other than the original unit,
something that would, as a consequence, be unable to continue fulfilling its
role as a unit with regard to the constitution
of
the number. This
is
precisely
what Bergson has in mind, I propose, when he talks in terms
of
a change in
nature, or a difference in kind.
The principle, therefore, that underpins the various dualisms between
which Bergson seeks
to
establish an equivalence, and which, furthermore,
determines the precise nature of his recourse to the Riemannian theory of
multiplicities, consists precisely in the difference between the two senses
of
to distinguish to which Bergson, as we noted above, draws attention in the
crucial footnote which supports the opening to Chapter 2 of
Time
and
Free
Will. [TFW
75-6/CE 51-2] The defining characteristic pertaining to the
Riemannian discrete multiplicity, the elements
of
which are discontinuous,
is
that distinguishing the elements is a process that leaves these elements
unchanged. The defining characteristic
pertaining
to the Riemannian
continuous multiplicity, the elements of which are continuous, is that
distinguishing the elements is a process that changes the elements. Thus,
more
properly called
this latter sense of
distinction consists
not in
quantitative division, but rather in qualitative
differentiation.
[emphasis
added; T W
95/CE
64] The
difference
between the two Riemannian
multiplicities as they are taken up by, and function for, Bergson, therefore,
consists in the difference between the nature
of
difference which determines
each multiplicity, division on the one hand, differentiation on the other. As
Bergson writes: the multiplicity of
conscious states, regarded in its original
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4.
Russell (1992), 328; referring
to
Bergson, Time and Free Will, translated by F.L. Pogson
(London: Macmillan, 1910), 78-9; (Euvres, 53-4. Hereafter cited in the body of the text
as
TFW/(E, followed by the relevant page numbers.
5.
Russell (1992), 328.
6.
Russell (1992), 329.
7.
Russell (1992), 329.
8.
Russell (1992), 330-1. Following this passage, Russell proceeds to attack Bergson s
refutation of Zeno s paradoxes, paradoxes which purport to demonstrate that change is
impossible.
9.
Cf. the Chronology of Life and Works in Bergson, Key Writings, edited by Keith Ansell
Pearson John Mullarkey (London: Continuum, 2002), viii.
10.
For the record, Russell s citation
of
this definition necessarily overlooks this identity,
following, as it does, Pogson s authorised translation of l un et du multiple as the one
and the many. Russell (1992), 328.
II. Gilles Deleuze, Bergsonism, translated by Hugh Tomlinson Barbara Habberjam (New
York: Zone Books, 1991), 39; Le bergsonisme (Paris: Presses Universitaires de France,
1966), 31.
12. De euze, Bergsonism, Afterword to the English translation, 117.
13. All citations from: Bernhard Riemann, On the Hypotheses which lie at the Foundations of
Geometry , translated by W.K. Clifford, Nature, vol. 8 no.
183
(1873),
14;
Gesammelte
mathematische Werke und wissenschaftlicher Nachlass (Leipzig: Teubner, 1876), 254.
14. Riemann (1873), 14/(1876) 254.
15.
Riemann (1873), 14-15/(1876) 254-5.
16.
Riemann (1873), 17/{1876) 257. The consequence
of
this argument, returning
to
the issue
of the simplest matters of fact from which the measure-relations of space may be
determined, matters
of
fact would form the hypotheses for the geometrical system
of
Euclid, is that
the
reality which underlies space [das
Raume
zu runde liegende
Wirkliche]
must form a discrete multiplicity, or we must seek the ground
[Grund] of
its
metric relations outside it, in binding forces [bindene Kraften] which act upon it.
17. Russell (1992), 328.
18.
This condition goes
to
the heart of Cantor s use of set theory in his remarkable work in the
second half
of the 19th century- in particular, given the significance of the role played by
infinite sets in Cantor s work the question which arises is the extent to which an infinite set
can indeed be grasped as a whole.
19.
This indivisibility arises as a consequence
of
their being conceived as ultimate units,
whereas the divisibility would arise as a consequence of their being conceived as
provisional units. [T W 81/(E 55]
20. Thus Bergson writes:
'In
a word, pure duration might well be nothing but a succession of
qualitative changes, which melt into and permeate one another, without precise outlines,
without any tendency to extemalise themselves in relation to one another, without any
affiliation with number: it would be pure heterogeneity. [TFW 104/(E 70]
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