robin durie - the mathematical basis of bergson' philosophy

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The Mathematical

Basis of Bergson'sPhilosophyRobin Durie

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aUniversity of Exeter

Published online: 21 Oct 2014.

To cite this article:Robin Durie (2004) The Mathematical Basis of

Bergson's Philosophy, Journal of the British Society for Phenomenology,

35:1, 54-67, DOI: 10.1080/00071773.2004.11007422

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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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robin durie - the mathematical basis of bergson' philosophy

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