Antonio Lamarra


Exactly forty years ago, on October 26th 1946, in R. B. Braithwaite’s

study in King’s College, Cambridge, K. Popper gave a lecture to the members

of the Moral Sciences Club, among them L. Wittgenstein and B. Russell, on

whether philosophical problems existed. Popper had gone to Cambridge with

the aim of provoking Wittgenstein to defend his thesis that authentic philo-

sophical problems did not exist, they were merely “linguistic perplexities”.

However, the lecture did not have the effect of encouraging a rational discus-

sion. Braithwaite’s study soon became the scene of a memorable verbal battle

between Popper and Wittgenstein, who, when the diatribe reached its climax,

angrily left the room slamming the door1. Pressed by Wittgenstein to furnish

examples of philosophical problems, Popper, who had prepared a list of them,

had quoted, amongst others, the problem of whether potential infinite and

actual infinite exist. Wittgenstein had replied that this was a mathematical

problem. Whatever our opinion may be today, there is no doubt that, seen

historically within the framework of seventeenth-century European culture,

the idea of infinite constituted not only an arduous mathematical problem, but

also, and above all, raised embarrassing philosophical questions. This was not

only because of the surviving theological implications of the idea, which natu-

rally continued to carry weight, but also because of the decisive and unprece-

dented fact that the infinite, until then banished from the realm of any ration-

al description of the world, had become in the space of a few decades one of

the cardinal tenets of the new scientific view of the physical universe. The

introduction of the infinite into mathematics had constituted one of the funda-

mental premises for the consequent prodigious development of this discipline,

so much so that, according to L. Couturat, it is precisely in the role assigned to

the idea of infinite that we recognise “la différence essentielle qui sépare la


science moderne de la science antique”2. Such a change could not but be

reflected in the philosophy of the period, also because some of the major pro-

tagonists of the new season of mathematical studies, such as Pascal, Descartes

and Leibniz, were themselves eminent philosophers. Leibniz is typical from

this point of view. His philosophy is so permeated with concepts and meth-

odological principles which come from mathematics that it becomes complete-

ly incomprehensible when separated from the broader problematic matrix

which often nourishes it. The infinite not only constitutes one of the central

ideas of his philosophy, but also one of the poles on which his meditations

rest: “Mes meditations fondamentales roulent sur deux choses, sçavoir sur

l’unité et sur l’infini”, Leibniz wrote to Princess Electress Sophie in 16963.

Within this whole framework the comparison between the opposite ideas of

Leibniz and Locke on the problem of the infinite acquires a special interest.

E. Cassirer stated with reference to this conflict of ideas: “There are not many

points in which the difference between ancient and modern thought is seen so

clearly as in this problem which, apart from its mathematical and gnoseologi-

cal significance, is also important as a general historical symptom”4.

Locke’s idea of infinite, if understood in the literal sense of the word,

belongs to the domain of ideas of quantity, that is to those ideas that imply the

concept of discrete part, such as the ideas of space, time and number5. Locke

concedes that there may be other possible concepts of the infinite, but sees in

a quantitative idea of the infinite the only clear paradigm of an idea that

diversely can only be applied figuratively in the domain of ideas of quality, at

least by the human intellect and its “weak and narrow thoughts”6. As well

as being a quantitative idea, the infinite is also a purely negative notion. This

notion derives from an awareness of the mind’s unlimited power to construct

ever-increasing numerical and spatial-temporal quantities, starting from the


simple ideas which it receives through the immediate experience of sensation

and reflection7. Although led to assume that it is able to increase at will any

given idea of quantity, the mind cannot, however, possess positive ideas of

infinite number, space or time, since every actual mental image is necessarily

determined and therefore finite8. Thus the Lockean idea of infinity may be

seen to derive from the subjective capacity for arranging the actual ideas of

number, space and time in indefinitely increasing series (or in indefinitely

decreasing series, which is analogous). Hidden beneath the apparent simplici-

ty of these theses lies one of the most complex and conflictual chapters in

Locke’s Essay Concerning Human Understanding.

The complexity and the limits of Locke’s theory of the infinite derive from a

common origin. In dealing with the problems connected with the idea of infi-

nite, Locke seems more concerned with demonstrating the empirical origin of

the concept, than with carefully analysing its logical nature and cognitive impor-

tance. And, on the other hand, in a cultural climate continually pervaded by the

new image of the world created by both mathematics and modern physics, the

idea of infinite might constitute one of the most embarrassing counter-examples

available to the adversaries of Lockean empiricism9. It is therefore hardly sur-

prising that Locke, in the second book of the Essay, dedicates the whole of the

seventeenth chapter to this idea of infinite and that he also directs his greatest

efforts towards demonstrating how even this idea, so remote “from any Object of

Sense, or Operation of our Mind”10, derives – just like every other concept –

from the simple ideas the mind receives through sensation and reflection. This

primary concern of Locke’s, however, is not lacking in consequences for his anal-

ysis of infinite. Firstly it greatly limits the scope of his idea of infinite, and

secondly in classifying infinity, with space, time and number, among the ideas of

mode it increases its complexity11. In order to clarify the difficulties involved in

classifying infinity as a mode, it might be useful to recall the classification of ideas

proposed by Locke in the second book of his Essay.


Locke’s distinction between simple and complex ideas is a familiar one.

The former, the real material of all knowledge, reach the mind via sensation

and reflection. The latter, on the contrary, derive from the fundamental

operations of composition, comparison and decomposition, which the mind

performs on the data furnished by sensation and reflection, in exercising its

own autonomous power over them. While simple ideas can only be given,

complex ideas can only be produced. Therefore, with regard to the simple

constituent elements of knowledge the mind is purely passive, while it uses

them to carry out its own activity and forge all kinds of complex ideas. How-

ever great the number and variety of complex ideas, Locke classifies them into

three fundamental types: modes, substances and relations. In order to be

included among the ideas of mode infinity belongs to the group of complex

ideas that “contain not in them the supposition of subsisting by themselves”

but that must be considered “Dependences on, or Affections of Substances”12.

Furthermore, in so far as it is a simple mode, the idea of infinity, just like

those of space, time and number, should be derivable from variations and

combinations of one and the same simple idea13. However, we immediately

note that, if infinity is seen as a negative idea and consequently it is impossible

for the mind to possess positive ideas of number, space or time, one excludes a

priori the possibility that the idea of infinity represents a mode attributable to

some substance. And in point of fact any such reference to substances is com-

pletely absent from Locke’s argument. His analysis, however, cannot fail to

take into consideration the complex problem constituted by the relations

between the ideas of space, time and number.

Infinity is presented by Locke as a complex idea of mode, which is itself

distinguished from those of space, time and number. Locke uses these collec-

tive nouns to cover the different modes which may derive from the simple

ideas of distance, duration and unity14. However, when we move on to con-

sider the idea of infinite, we realize immediately that this idea, in contrast to

the previous modes, is not generated by a specific simple idea, but is defined

by Locke rather as a possible attribute of the ideas of space, time and number.

Finite and infinite, indeed, in so far as they are modes of quantity, are attribut-

able in the literal sense of the word only to those things which are composed


of parts and are capable of increase or diminution by the addition or subtrac-

tion of any minimal part15.

This first Lockean definition, which is clearly aimed at freeing the idea of

infinite from the metaphysical and theological implications typical of the Chris-

tian philosophical tradition (and which are not extraneous to the new Newton-

ian cosmology) is, however, inadequate for defining the ideas of finite and infi-

nite as simple modes. Indeed, here these ideas appear to be possible predicates

of a plurality of complex ideas of mode and therefore seem to call to mind the

concept of abstract idea rather than that of simple mode16. Furthermore, as

Locke’s analysis proceeds the more it becomes evident that it is impossible to

put the ideas of space, time and number on the same plane as regards their

relations with the idea of infinite. While, from the point of view of psychologi-

cal evidence, this idea seems to originate from modalities of representation of

space and time, the analysis of the ideas of eternity and infinite space involves

the recognition of the very close connection which – on the logical plane –

exists between the idea of number and the idea of infinite. As regards both

spatial and temporal representations the idea of infinite is connected with the

mind’s power to add unities indefinitely and thus to continue to consider ever

greater representations without there being any reason why this process should

cease17. It is precisely this lack of any positive reason why the progressive

increase of its representations should cease that leads the mind to conceive the

idea of infinite space and infinite time18. The simple ideas of distance and

duration, however, taken in isolation and considered in relation to the actual

operations of the mind, would not by any means be sufficient to explain the

transformation of a determinate spatial or temporal representation into the idea

of infinite distance or duration, if the idea of number did not intervene in the

process. This is therefore the real logical source of Locke’s idea of infinite and

only by its application to the ideas of unities of distance and duration can the

mind forge the idea of infinite with regard to time and space: the idea of infini-


ty” is nothing but the Infinity of Number applied to determinate parts, of which

we have in our Minds the distinct Ideas”19.

On the basis of this new definition the infinity of space and time consists

therefore in the indefinite enlargement, regulated by the progression of simple

modes of number, of determinate ideas of distance and duration. In this pro-

cess, which is still eminently psychological, what permits this passage from the

indefinite to the infinite is only the infinity which we cannot but attribute to

the series of numbers. But if we ask ourselves yet again what the justification

for this idea of infinity of number derives from, in the end we are sent back to

our subjective internal awareness of being continually able to repeat the funda-

mental logical operation of forming the number, without any external obstacle

bringing this process to an end. Given any positive (and therefore finite

number), however large, we know that we can always increase it by at least

one unity and furthermore that we can repeat this operation endlessly and

thus create ever larger numbers20. The idea of this endless progression which

accompanies and characterises the extension of a series of numbers, is not

therefore an element belonging to this series that is effectively known or

knowable like the others. Neither can it be considered an idea of reflection,

generated in the mind by its proceeding along the series of numbers, at least if

we go by the original meaning Locke gives to the ideas of reflection21. Here

it is not a question of a series of operations actually carried out by the mind,

but of the assumption that a determinate logical operation can be repeated

endlessly. Without such an assumption infinity of number could not even be

thought of, but the idea of infinity is already obviously contained within it.


The idea of infinite is not represented and is not representable as a positively

perceptible idea, but it must already be present in the mind in order that the

mind may conceive the negative ideas of infinite number, time and space.

We may here stop a moment to complain of the evident circularity of Locke’s

argument and, nevertheless, as regards the gnoseological principles of Locke’s

thought, the greatest paradox to which his analysis of the idea of infinite gives

rise consists rather in its appeal – as to a fundamental – to an assumption that

cannot be empirically proved. This assumption, however, must be accepted.

Without it the idea of infinite would not only be negative but completely vain

and chimerical and Locke’s philosophy would acquire greater systematic rigour

at the price of completely delegitimising an idea, such as that of infinity, indis-

solubly linked to the new line of modern scientific thought. On the other

hand – it must be said – Locke does not seem at all aware that his theses are

contradictory. Having denied the possibility of the idea of qualitative infinity,

and demonstrated the negativity of every actual idea of infinity possessed by

the mind, quite possibly he believes he has achieved his aim by deriving the

idea of infinity from modes such as number, time and space, whose empirical

origin he had already previously demonstrated22.

Leibniz’s philosphy, on the contrary, does not question that the idea of

infinite belongs to the heritage of ideas that the mind brings with it and which

are independent of the experience of the senses. Leibniz considered the infi-

nite an innate idea which cannot be derived by means of induction. His criti-

cism of Du Tertre’s Réfutation of Malebranche’s system, published in Paris in

1715, could also be applied to Locke’s arguments. If one claims to explain

our knowledge of the infinite by referring to the mind’s ability to repeat a

determinate logical operation endlessly, the very explanation presupposes what

it is aimed at explaining23. But Leibniz’s criticism of Locke is not specifically


concerned with the problem of the origin of the idea of infinite. What was

for the English philosopher a primary problem was for Leibniz a secondary

one compared with the need to clarify his own thought on the logical nature

of the idea and its great metaphysical implications.

It is difficult to imagine a greater contrast between two theories than that

between Locke and Leibniz on the idea of infinite. There is an abyss between

the two positions: the apparent agreement on particular theses disappears as

soon as these are situated in their respective theoretical constructs.

One of Locke’s greatest breaks with medieval tradition and the view of the

infinite held by most of his contemporaries was his drastic reduction of the idea

of infinite to a mere quantitative aspect. Locke’s analysis consciously aimed at

rendering unsubstantial the many and often quite subtle distinctions developed

during the previous centuries in favour of the one potential infinite or rather –

as the medieval philosophers termed it – the one syncategorematic infinite.

Leibniz’s view is diametrically opposed to this. In his eyes the infinite is seen

as quite a complex concept, which requires rigorous distinctions, without which

the intellect would lose itself as in a labyrinth. To Locke’s radical reductionism

he therefore immediately opposes a multiplicity of meanings attached to the

concept, according to the different levels of reality at which it is applied.

Leibniz first distinguishes a meaning of the concept, which, for the

moment, we may call ontological, by means of which it is possible to speak of

an infinity of things in the world. The infinite in the logical abstract sense is

different from this, it is the mathematical infinite, according to which infinity

may not be predicated for any quantity in general. Finally one may speak of a

rigorous metaphysical and logical meaning of the concept thanks to which it is

recognised to be deeply rooted in the idea of absolute:


Une notion des plus importantes est celle du Fini et de l’Infini qui sont regardées

comme des Modes de la Quantité


A proprement parler il est vray qu’il y a une infinité de choses, c’est à dire qu’il

y en a tousjours plus qu’on n’en peut assigner. Mais il n’y a point de nombre

infini ny de ligne ou autre quantité infinie, si on les prend pour des veritables

Touts, comme il est aisé de demonstrer. Les écoles ont voulu ou dû dire cela, en

admettant un infini syncategorematique, comme elles parlent, et non pas l’in-

fini categorematique. Le vray infini à la rigueur n’est que dans l’absolu qui est

anterieur à toute composition, et n’est point formé par l’addition des parties24.


In this passage actual infinite and potential infinite are opposed both as

regards their referential content (on the one hand things and on the other

geometrical and mathematical entities) and as regards the different relation

presupposed in them between the denotated wholes and their real or possible

constituent parts. There is, indeed, no doubt that in his brief reference to the

“infinité des choses” Leibniz is referring precisely to his thesis that the actual

infinite is a principal characteristic of physical and metaphysical reality. This

contrasts sharply with a line of thought which from Aristotle onwards had

dominated Western philosophy, and according to which nature is averse to the

infinite. Leibniz indeed, maintains the presence of the actual infinite in the


Je suis tellement pour l’infini actuel, qu’au lieu d’admettre que la nature l’ab-

horre, comme l’on dit vulgairement, je tiens qu’elle l’affecte partout, pour

mieux marquer les perfections de son auteur. Ainsi je crois qu’il n’y a aucune

partie de la matiere qui ne soit je ne dit pas divisible, mais actuellement divi-

sée, et par consequent, la moindre particelle doit estre considérée comme un

monde plein d’une infinité de creatures différentes25.

On the other hand, nature, the physical world of matter and of real

beings, “qui fait entrer l’infini en tout ce qu’elle fait”26 does not possess this

characteristic, except in so far as it is a phenomenon through which a meta-

physical reality is manifested, which is constituted in its turn by an actual

infinity of monads or simple substances. In this case too, as is typical of Leib-

niz, the metaphysical universe and the phenomenal universe are not insolubly

in antithesis to one another. On the contrary, they constitute two aspects of

the same reality, distinct but correlated by functional relations. If the world

of nature must be recognised as a phenomenon, it does not for this reason

degenerate into a chimerical or illusory world, since, as it is a phenomenon

bene fundatum, it is nothing else but the way in which the universe of surround-

ing substances is represented – and thus becomes knowable – by every individ-

ual monad27.


The fact that the metaphysical world of substances (and, consequently the

physical world of bodies) is actually infinite does not, however, imply that

there exists an infinite number of monads or of bodies which may be

expressed positively. What distinguishes the actual infinite for Leibniz is pre-

cisely its non-numerability. The absence of limits and the fullness of the

world as a result of which we may state “qu’il y a une infinité des choses”

simply mean “qu’il y en a toujours plus qu’on n’en peut assigner”28. Leibniz

also repeats this concept in a short essay, which is, perhaps, his last philosophi-

cal work:

Et non obstant mon Calcul Infinitesimal, je n’admets point de veritable nombre

infini, quoyque je confesse que la multitude des choses passe tout nombre fini,

ou plustôt tout nombre29.

Leibniz’s denial of infinite number now permits us to return to the text of

the Nouveaux essais and to the opposition between the actual infinite and poten-

tial or syncategorematic infinite. The distinction between these two different

concepts of the infinite is of prime importance for the understanding of Leib-

niz’s text. Indeed it refers back to that distinction between actual and ideal,

without which one is destined to lose oneself in the antinomies of the laby-

rinth de compositione continui30. Confusing these two levels of reality would mean

ruinously confusing the concrete with the abstract, the real with the possible,

the discrete with the continuous, the determinate with the indeterminate and

the definite with the indefinite31.


Leibniz’s approach to the metaphysical and ontological problems connect-

ed with the idea of infinite presupposes and requires the solution of the prob-

lems that originated in mathematics from the idea of continuum. Without a

preliminary close examination of the paradoxes of the continuum in the field

of geometry, the possibility of a rigorous metaphysics is precluded. Therefore,

echoing the Platonic motto “medeis ageometretos eisito”, Leibniz states:

Nam filum labyrintho de compositione continui deque maximo et mini-

mo ac indesignabili atque infinito non nisi geometria praebere potest, ad

metaphysicam vero solidam nemo veniet, nisi qui illac transiverit32.

The solution to the problems posed by the concept of continuum implies

answering two, in way, specular questions. Firstly, whether given a continu-

ous entity the ultimate constituent (or indivisible) elements may be distin-

guished and secondly, on the other hand, whether given indivisible elements

may constitute a continuous whole. In both cases Leibniz’s answer is nega-

tive : a continuous quantity, in so far as it is an actually given infinite, does not

possess ultimately real components (nor is it constituted by indivisible minimal

parts);33 neither can an infinite series of indivisible data constitute a continu-

ous whole34. In the first case, that which is given as actual is the whole,


which therefore can only be ideally and potentially subdivided into parts.

While in the second case, it is the parts that are actual and logically precede

the whole, which results from their composition. The first is the case of con-

tinuous quantity and potential infinite, the second is the case of discrete quan-

tity and actual infinite. The continuum belongs to the field of the ideal and

of abstract physical-matemathical concepts, the discrete to the field of the

actual and of metaphysical reality:

Dans l’idéal ou continu le tout est anterieur aux parties, comme l’unité

Arithmetique est anterieure aux fractions qui la partagent, et qu’on y peut

assigner arbitrairement, les parties ne sont que potentielles; mais dans le reel

le simple est anterieur aux assemblages, les parties sont actuelles, sont avant le


Continuous quantities, therefore, in so far as they are ideal, must be

assigned to the realm of the possible and not to that of the real, since – not

having actual constituents – they imply the concept of indefinite part36. Thus,

whereas in mathematics what is given, the continuous whole, cannot be con-

ceived as being composed of indivisible elements, which in this case are only

theoretical constructs, ideal limits of an infinite possibility of subdivisions;

vice versa in the case of metaphysical reality only the ultimate constituents,


the monads, are actual, but although they are infinite in number they cannot

give rise to a continuous whole, because any plurality of monads, as it is an

aggregate, possesses a purely phenomenal reality and unity37. Indeed, accord-

ing to Leibniz, one cannot even strictly speak of a real or actual whole, but

only of ideal aggregates composed of real components. Any group of monads

we consider does not constitute a real unitary whole since only the simple

substances that compose it possess the attributes of reality and unity. Howev-

er, when we move from the real to the ideal the situation is in a sense

reversed. In this field the whole, yet again because of its ideal nature, pos-

sesses a logical priority in relation to the parts. It thus appears to knowledge

as a definite concept which does not presuppose any actually given multitude

of parts or primary elements. Thus the ideal whole only virtually possesses

possible and indeterminate parts.

Therefore a characteristic difference sharply distinguishes the actual from

the ideal: the different relation that exists in them between the concepts of

whole and part. While in Locke space, time and number referred back –

because of their very constitution – to the concept of elementary part (in the

final analysis, the simple idea from which they were generated), in Leibniz

that concept is precluded. Space and time are conceived neither as substances

nor – as Locke says – as modes which may refer to substances, but as pure

relational ideas in which only relations of order, coexistence or succession are

expressed38. In so far as they are relational concepts, the expression of rela-

tions of order between actual substances, they possess an ideal reality and, like

number, do not possess real constituent elements39. In reply to Foucher’s

objections to the Systeme Nouveau, Leibniz writes:


L’etendue ou l’espace, et les surfaces, lignes et points qu’on y peut conce-

voir, ne sont que des rapports d’ordre, ou des ordres de coexistence, tant pour

l’existent effectif que pour le possible qu’on pourrait y mettre à la place de ce

qui est. Ainsi ils n’ont point des principes composans, non plus que le


Naturally this does not mean that space, time and number are quantities

that are indivisible in parts, but that these parts are only possible and ideal,

not actual. Thus, for example, the numerical unity is divisible into increas-

ingly smaller fractions and the line into increasingly smaller segments, without

the process of subdivision coming to an end, because neither the number nor

the line – being abstract concepts – can be conceived as a whole, composed of

ultimate elements. And this is due to the fact that the way in which they may

be subdivided is completely indefinite:

Hinc numerus, Hora, Linea, Motus seu gradus velocitatis, et alia huius-

modi Quanta idealia seu entia Mathematica revera non sunt aggregata ex par-

tibus, cum plane indefinitum sit quo in illis modo quis partes assignari velit,

quod vel ideo sic intelligi necesse est, cum nihil aliud significent quam illam

ipsam meram possibilitatem partes quomodocunque assignandi41.

What applies to mathematical elements in general also applies to space

and time, which are continuous quantities, and their lack of real constituent

elements is the basis of their divisibility into arbitrarily defined ideal parts:

Nam spatium, perinde ac tempus ordo est quidam nempe (pro spatio)

coëxistendi, qui non actualia tantum, sed et possibilia complectitur. Unde

indefinitum est quiddam, ut omne continuum cujus partes non sunt actu, sed

pro arbitrio accipi possunt, aeque ut partes unitatis seu fractiones42.

This indefinite divisibility attributable to relational concepts does not pre-

vent one from arriving at logically primitive notions of them, too. The resolu-

tion in notions must not be confused with the division into parts, nor must

logical complexity be confused with size. The parts are not necessarily sim-

pler than the whole, even if they are smaller. The best example of this can be

seen in the numerical unity, which we can quite rightly consider a primitive

concept of number (and therefore not further resolvable), while – from the


quantitative point of view – it may be infinitely subdivided into smaller frac-

tions, which are logically more complex. “Les parties ne sont pas tousjours

plus simples que le tout, quoyqu’elles soyent tousjours moindres que le tout”,

Leibniz concludes in a letter to Bourguet43; and he writes to Des Bosses:

Ens et unum converti Tecum sentio; Unitatemque esse principium nu-

meri, si rationes spectes, seu prioritatem naturae, non si magnitudinem, nam

habemus fractiones, unitate utique minores in infinitum44.

This greater logical complexity to which the division into parts of a rela-

tional quantitative concept may lead excludes the possibility of the concept

itself resulting from the aggregation of parts. The numerical unity is not the

result of the sum of the infinite fractions into which it may be subdivided,

neither is a given segment the result of the infinite parts its subdivision may

produce. These processes of subdivision are infinite, but not because the con-

stituent parts of which the original whole is composed are infinite in number.

These quantitative indivisible or minimal parts do not exist in the field of

numbers nor in the continuous quantities of geometry45. In both cases the

given whole logically precedes the purely possible parts into which it may be

subdivided. The basis of infinite divisibility, and, conversely, of the infinite

possibility of increasing a given quantity without ever reaching an absolute

maximum, lies, according to Leibniz, in the constant existence of a generative


On this point, too, he is in complete disagreement with Locke’s view of

the problem. Whereas in the Essay the generation of infinitely large or infi-

nitely small ideas of quantity was explained by the lack of a reason why the

mind should set a limit on the progressive increase or decrease of its represen-

tations, Leibniz, on the contrary, requires the permanence of a positive reason,

of a constituent legality. To Locke’s arguments explaining the emergence of

the idea of infinite space Leibniz replies:


Il est bon d’ajouter que c’est parce qu’on voit que la même raison subsiste


The permanent existence of a psychological possibility of representation is

not enough; what is required as a basis for the concept of infinite is a constant

generative law that, remaining identical during the process of the formation of

the concept, guarantees its possibility (and, therefore, according to Leibniz, its

reality) independently of its representation in sense images. This basis is

clearly seen by Leibniz in the concept of similarity:

Prenons une ligne droite et prolongeons la, en sorte qu’elle soit double de

la premiere. Il est clair que la seconde estant parfaitement semblable à la

premiere, peut estre doublée de même, pour avoir la 3me qui est encor sembla-

ble aux precedentes; et la même raison ayant tousjours lieu, il n’est jamais

possible qu’on soit arresté; ainsi la ligne peut estre prolongée à l’infini. De

sort que la consideration de l’infini vient de celle de la similitude ou de la

même raison, et son origine est la même avec celles des vérités universelles et


Naturally the concept of similarity is not to be understood in a generally

intuitive sense, but in the strictly logical sense according to which two objects

are said to be similar, if they come under the same definition (“Similia sunt,

quorum species seu definitio est eadem”)48. Thus in Leibniz’s example the

rational guarantee that the operation of reduplication of a given segment may

be repeated indefinitely lies in the fact that in every phase of the line’s con-

struction process the same logical conditions which generate the whole pro-

cess, are repeated unchanged. The same explanation is given for the opposite

operation of the infinite subdivision of a given straight line:

Continuum in infinitum divisibile est. Idque in Linea Recta vel ex eo

constat, quod pars ejus est similis toti. Itaque cum dividi possit, poterit et

pars partis. Puncta non sunt partes continui, sed extremitates, nec magis

minima datur pars lineae, quam minima fractio Unitatis49.

Thus the same rational certainty which assures us of the inexistence of

assignable limits to the processes of addition and subdivision into parts of a


given quantity, necessarily implies the exclusion of quantitative maximums

and minimums. This is therefore the field of the potential infinite in which

both the continuous quantities (particularly those of space and time) and the

numerical quantities converge. According to Leibniz the concept of infinite

must be recognised as being logically innate in the mind and therefore not

logically inferred from the experience of the senses, but on a par with univer-

sal and necessary truths50.

On the other hand, though his concept of potential infinite is similar to

the Scholastic concept of syncategorematic infinite, his concept of actual infi-

nite, which in his argument is opposed to potential infinite, may not be con-

fused with the categorematic infinite which the Scholastics opposed to the syn-

categorematic infinite. There is no room for the categorematic infinite, that

is for the idea of an infinite whole resulting from the composition of actual

parts, in Leibniz’s system: it is a concept that implies contradiction51. Thus

the concept of an absolute space (but one could say the same thing of time and

number), of a space that is infinitely large and composed of parts, must be

rejected because, in so far as it implies the categorematic infinite, it is a con-

tradictory concept:

Mais on se trompe en voulant s’imaginer un espace absolu qui soit un

tout infini composé de parties; il n’y a rien de tel, c’est une notion qui impli-

que contradiction, et ces touts infinis, et leur opposés infiniment petits ne sont

de mise que dans le calcul des Geometres, tout comme les racines imaginaires

de l’Algebre52.

The infinitely large and its opposite, the infinitely small, understood as

quantities actually given, would imply different paradoxical consequences

among which – as Galileo had already observed – the rejection of the axiom

according to which the whole is greater than the part53.

It is true that even in the seventeenth century there were some who had

hypothesised that this axiom was not valid if applied to the field of the infi-

nite. But this is not the path that Leibniz takes. As he sees it, given that, if

the infinite is a true whole, it should undoubtedly be greater than its parts, it

follows that, if this condition is not always satisfied, one must not abandon the


axiom, but the idea that the infinite is a true whole. In other words, in pass-

ing from finite to infinite the apparently immediate correlation between the

concepts of whole and constituent part must be denied. And this is precisely

what happens in Leibniz’s philosophy as a consequence of the distinction

between real and ideal, which – as has already been seen – implies both a

different logical priority of the two concepts and their different characterisa-

tion in modal terms. This distinction serves to render possible in Lebniz’s

philosophy the compatibility of two concepts otherwise difficult to reconcile:

that of the continuum, which characterises his mathematics and that of unity,

which characterises his metaphysics.

Furthermore Leibniz is led to deny real unity to the concept of a whole

composed of infinite parts also for metaphysical reasons. Because of the meta-

physical equivalence of ens and unum, if the physical and metaphysical universe

possessed real unity, it would be an entity, a living organism composed of

simple substances and God could be thought of as its soul.54 In view of the

expressive relation between physical phenomenon and metaphysical reality

postulated in Leibniz’s philosophy, he would seem to be on the path to Pan-

theism and to Spinoza. And yet, in theory, Leibniz, too, could have accepted

the thesis of those who denied absolute validity to the axiom according to

which the whole is greater than the part. His very philosophy could easily

have offered an example of this. In Leibniz’s metaphysical universe the part

is not smaller than the whole. The system is self-reflective, in it every part

may be in biunique correspondence with the whole; indeed every monad is a

representation of the entire universe and is as large and permanent as the


chaque substance simple est un miroir du même Univers, aussi durable et

aussi ample que luy55.

The actual infinite in Leibniz’s metaphysics seems therefore to comply

with the definition of the actual infinite introduced into mathematics by Can-

tor and Dedekind two hundred years later.


One is tempted to think that if, as R. Aaron saw it, Locke’s concept of

the infinite constituted a challenge to contemporary rationalistic philoso-

phies56, however – from a more general point of view – the very contrast

between Locke and Leibniz on this subject may be seen as an even greater

challenge. It is the challenge which the idea of infinite has continued to pre-

sent to Western philosophy ever since the dawn of Greek philosophy.

See K. R. Popper, Unended Quest, Germ. Trans, by F. Griese and K. R. Popper, Ausgangs-

punkte. Meine intellektuelle Entwicklung
, Hamburg, Hoffmann und Campe 1979, chap. 26, pp. 175-

L. Couturat, De l’infini mathématique, Paris 1896, nouveau tirage, Paris 1973, p. 265.

On the other hand one knows the decisive importance that the concept of infinite assumed in

the historiographic interpretation of the Renaissance supported by A. Koyré and characterized

by the emphasis placed on showing the consequences of the idea, of Greek origin, of a finite

and hierarchically ordered cosmos. The very useful volume by P. Zellini, Breve storia dell’infinito,

Milano, Adelphi 1980, is collocated in a rather different perspective, which is yet not clearly

made explicit by the author.
GP VII, p. 542.
E. Cassirer, Leibniz’ System in seinen wissenschaftlichen Grundlagen, Marburg 1902, chap. 4, 1,

p. 199 (my translation).
J. Locke, An Essay Concerning Human Understanding, edited by P. H. Nidditch, Oxford,

Clarendon Press 1975, reprinted with corrections 1979, II, xvii, I, p. 209. In the following

notes I shall use the abbreviation E.
Ibid., p. 210.
See E, II, xvii, pp. 12-20.
“For our idea of Infinity being, as I think, an endless growing Idea, but the Idea of any

Quantity the Mind has, being at that time terminated in that Idea, (for be it as great as it will, it

can be no greater than it is)…”, E, II, xvii, 7, p. 213. “Whatsoever positive ideas we have in

our Minds of any Space, Duration, or Number, let them be never so great, they are still finite”,

E, II, xvii, 8, p. 214; cf. E, II, xvii, 13, pp. 216-217.
See R. I. Aaron, John Locke, third edition, Oxford, Clarendon Press 1971, reprinted

with corrections 1973, pp. 154-156.
E, II, xvii, 22, p. 223.
In Aaron’s opinion, the classification of the ideas of space, time, number and infinity in

the Essay is “somewhat artificial” (R. I. Aaron, op. cit., p. 155).
E, II, xii, 4, p. 165.
“Of these Modes, there are two sorts, which deserve distinct consideration. First,

there are some which are only variations, or different combinations of the same simple Idea,

without the mixture of any other, as a dozen, or score; which are nothing but the Ideas of so

many distinct Unites added together, and these I call simple Modes, as being contained within

the bounds of one simple Idea”, E, II, xii, 5, p. 165.
See respectively E, II, xiii, 4; xiv, I; xvi, 1-2.
E, II, xvii, I p. 209. Cf. also E, II, xvii, 6.
“ Locke seems to soon renounce the logical equivalence of finite and infinite as simple

modes. Indeed, he defines the idea of finite as a simple idea furnished by sensation, rather than

a complex idea compounded by the mind in II, xvii, 2 (p. 210). From this moment on, anyway,

Locke examines only the problems connected with the idea of infinite.
See E, II, xvii, 3 and 5.
A very interesting criticism of Locke’s negative concept of infinite was developed by

E. Cassirer in his Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Bd. 2,

Berlin 1907, pp. 172-176. In particular, Cassirer pointed out that Locke’s definition of infinite,

if accepted, would require a new concept of reflection, completely different from that previous-

ly proposed by Locke in his Essay.
E, II, xvii, 10, p. 215. Cf. ibid., II, xvi, 8, p. 209: “This farther is observable in Num-

ber, That it is that, which the Mind makes use of in measuring all things, that by us are measur-

able, which principally are Expansion and Duration; and our Idea of Infinity, even when

applied to those, seems to be nothing, but the Infinity of Number”.
“Though it be hard, I think, to find any one so absurd, as to say, he has the positive

Idea of an actual infinite Number; the Infinity whereof lies only in a Power still of adding any

Combination of Unites to any former Number, and that as long, and as much as one will; the

like also being in the Infinity of Space and Duration, which Power leaves always to the Mind

room for endless Additions; […] For, I think, it is evident, that the Addition of finite things

together (as are all lengths, whereof we have the positive Ideas) can never otherwise produce

the Idea of infinite, than as Number does; which consisting of Additions of finite Unites one to

another, suggests the Idea of Infinite, only by a Power, we find we have of still increasing the

Sum, and adding more of the same kind, without coming one jot nearer the end of such Pro-

gression”, E, II, xvii, 13, pp. 216-217.
“By Reflection then, in the following part of this Discourse, I would be understood to

mean, that notice which the Mind takes of its own Operations, and the manner of them, by

reason whereof, there come to be Ideas of these Operations in the Understanding”, E, II, i, 4,

p. 105.
See E, II, xvii, 22, p. 223: “I pretend not to treat of them [i.e., of duration, space, number,

] in their full Latitude: it suffices to my Design, to shew, how the Mind receives them,

such as they are, from Sensation and Reflection; And how even the Idea we have of Infinity,

how remote soever it may seem to be from any Object of Sense, or Operation of our Mind, has

nevertheless, as all our other Ideas, its Original there”.
See Leibniz to Rémond (4. XI. 1715), in GP III, pp. 658-659 : “L’Auteur adjoute (Tom. 1.

p. 307) que dans la pretendue connoissance de l’infini, l’esprit voit seulement, que les longueurs

peuvent etre mises bout-à-bout et repetées tant qu’on voudra. Fort bien, mais cet auteur pouvoit

considérer, que c’est déjà connoistre l’infini, que de connoistre que cette repetition se peut tous-

jours faire”. The whole letter is a long commentary on Du Tertre’s book, Réfutation d’un nouveau

système de Métaphysique, proposé par le P. M… Auteur de la Recherche de la Vérité
, which Leibniz had

received from Rémond. A critical edition of Leibniz’s letter and further information on Du

Tertre’s criticism to Malebranche are contained in A. Robinet, Malebranche et Leibniz. Relations

, Paris, Vrin 1955, pp. 465-467 and pp. 478-482.
Nouveaux essais sur l’entendement humain, II, xvii, 1 (A, p. 157). In the following notes I

shall use the abbreviation NE.
Leibniz to Foucher (January 1692), in GP I, p. 416. In June 1692 part of this letter was

published in the “Journal des Savans” (2. VI. 1692, pp. 247-249).
GM II, p. 219 (Leibniz to De l’Hospital).
See Leibniz to de Volder (30. VI. 1704), in GP II, p. 268: “Accurate autem loquendo

materia non componitur ex unitatibus constitutivis, sed ex iis resultat, cum materia seu massa

extensa non sit nisi phaenomenon fundatum in rebus, ut iris aut parhelion, realitasque omnis

non sit nisi unitatum. Phaenomena igitur semper dividi possunt in phaenomena minora quae

aliis subtilioribus animalibus apparere possent, nec unquam pervenietur ad minima phaenome-

na. Unitates vero substantiales non sunt partes, sed fundamenta phaenomenorum”. Cf. GP

VII, pp. 467-468.

NE II, xvii, 1 (A, p. 157).

GP VI, p. 629. Cf. GP I, p. 338; GM III, p. 575.
See GP IV, pp. 491-492 (Remarques sur les Objections de M. Foucher): «Et c’est la confusion

de l’ideal et de l’actuel qui a tout embrouillé et fait le labyrinthe de compositione continui.

Ceux qui composent la ligne de points, ont cherché des premiers elemens dans les choses ideales

ou rapports tout autrement qu’il ne falloit; et ceux qui ont trouvé que les rapports comme le

nombre ou l’espace (qui comprend l’ordre ou rapport des choses coexistentes possibles) ne sau-

roient estre formé par l’assemblage des points, ont eu tort pour la pluspart de nier les premiers

elemens des realités substantielles, si elles n’avoient point d’unités primitives ou comme s’il n’y

avoit point de substances simples. Cependant le nombre et la ligne ne sont point des choses

chimeriques, quoyqu’il n’y ait point de telle composition, car ce sont des rapports qui renfer-

ment des verités eternelles, sur lesquelles se reglent les phenomenes de la nature».

See Leibniz to de Volder (19. I. 1706), in GP II, p. 282: «Patet etiam ex iis quae dixi, in

Actualibus non esse nisi discretam Quantitatem, nempe multitudinem monadum seu substantia-

rum simplicium, quovis quidem numero majorem in quocunque sensibili aggregato seu phae-

nomenis respondente. Sed continua Quantitas est aliquid ideale, quod ad possibilia et actualia,

qua possibilia, pertinet. Continuum nempe involvit partes indeterminatas, cum tamen in actua-

libus nihil sit indefinitum, quippe in quibus quaecunque divisio fieri potest, facta est. Actualia

componuntur ut numerus ex unitatibus, idealia ut numerus ex fractionibus: partes actu sunt in

toto reali, non in ideali. Nos vero idealia cum substantiis realibus confundentes, dum in possi-

bilium ordine partes actuales, et in actualium aggregato partes indeterminatas quaerimus, in

labyrinthum continui contradictionesque inexplicabiles nos ipsi induimus». Cf. GP VII, p. 468.

GM VII, p. 326. With regard to the relations between Leibniz’s concept of infinite

and the problems of the continuum, the pages dedicated by B. Russell to the labyrinth of the

continuum in his A critical exposition of the philosophy of Leibniz, Cambridge 1900, chap. 9, pp. 108-

117 are still very interesting. See also N. Rescher, Leibniz’ Conception of Quantity, Number, and

, in “The Philosophical Review”, 1955 (64), pp. 108-114, and Leibniz. An Introduction to

his Philosophy
, Oxford 1979, pp. 99-104.
See Leibniz to Joh. Bernoulli (29. VII. 1698), in GM III, p. 524: «Itaque mirari etiam non

debes, quod dubito, an revera detur quantitas infinite parva, aut infinite magna utrinque termi-

nata. Etsi enim concedam, nullam esse portionem materiae, quae non actu sit secta, non tamen

ideo devenitur ad elementa insecabilia, aut ad minimas portiones, imo nec ad infinite parvas,

sed tantum ad minores perpetuo, et tamen ordinarias; similiter ut ad majores perpetuo in augen-

do acceditur. Sic etiam semper animalcula in animalculis facile concedo; et tamen necesse non

est dari animalcula infinite parva, nedum ultima». Cf. ibid., pp. 535-536: «Quemadmodum

autem non datur Elementum Numericum seu minima pars unitatis, vel minimum in Numeris,

ita nec datur linea minima, seu elementum lineale»; GP II, p. 276 (Leibniz to de Volder): «… etsi

in extensione Mathematica, qua possibilia intelliguntur, nulla sit divisio actualis nec partes nisi

quas cogitando facimus, nec prima Elementa, non magis quam inter numeros fractos minimus

datur velut Elementum caeterorum».
See GP IV, p. 492 (Remarques sur les Objections de M. Foucher): «Mais dans les realités où il

n’entre que des divisions faites actuellement, le tout n’est qu’un résultat ou assemblage, comme

un trouppeau de moutons; il est vray que le nombre des substances simples qui entrent dans

une masse quelque petite qu’elle soit, est infini, [...] et quoyque cela aille à l’infini, il est mani-

feste, qu’au bout du compte tout revient à ces unités, le reste ou les resultats n’estant que des

phenomenes bien fondés».
GP III, p. 622. Anyway, the opposition between (ideal) continuum and (real) discrete is

not an obstacle for rational knowledge: «Cependant on peut dire en general que toute la conti-

nuité est une chose ideale et qu’il n’y a jamais rien dans la nature, qui ait des parties parfaite-

ment uniformes, mais en recompense le reel ne laisse pas de se gouverner parfaitement par

l’idéal et l’abstrait, et il se trouve que les regles du fini reussissent dans l’infini, comme s’il y

avait des atomes (c’est à dire des elemens assignables de la nature), quoyqu’il n’y en ait point la

matiere estant actuellement sousdivisée sans fin; et que vice versa les regles de l’infini reussis-

sent dans le fini, comme s’il y avoit des infiniment petits metaphysiques, quoyqu’on n’en ait

point besoin; et que la division de la matiere ne parvienne jamais à les parcelles infiniment

petites: c’est par ce que tout se gouverne par raison, et qu’autrement il n’y aurait point de

science ny regle, ce qui ne seroit point conforme avec la nature du souverain principe», GM IV,

pp. 93-94 (Leibniz to Varignon, 2. II. 1702). Cf. GP II, pp. 282-283 (Leibniz to de Volder, 19. I.

1706): «Interim scientia continuorum hoc est possibilium continet aeternas veritates, quae ab

actualibus nunquam violantur, cum differentia semper sit minor quavis assignabili data. Neque

aliam in phaenomenis habemus aut optare debemus notam realitatis, quam quod inter se pariter

et veritatibus aeternis respondent».
See GP II, p. 282: «In realibus non nisi discreta est quantitas, id est multitudo ex uni-

tatibus veris resultans; continua quantitas quae non apparens sed exacta sit, pertinet ad idealia

et possibilitates, cum indefinitum aliquid sive indeterminatum involvat, quod non patitur

actualis natura rerum».

See NE, II, xii, 7 (A, p. 146): «Cette unité de l’idée des Aggregés est très veritable,

mais dans le fond il faut avouer que cette unité de collections n’est qu’un rapport ou une rela-

tion dont le fondement est dans ce qui se trouve en chacune des substances singulieres à part.

Ainsi ces Estres par Aggregation n’ont point d’autre unité achevée que la mentale et par conse-

quent leur entité aussi est en quelque façon mentale, ou de phenomene, comme celle de l’arc en

ciel». Cf. ibid. II, xxi, 72 (A, pp. 210-211).

See Leibniz to Tolomei (17. XII. 1705), in GP VII, p. 467: «Mihi olim meditanti visum

est non aliter illo Labyrintho continui exiri posse, quam ipsum quidem spatium perinde ac tem-

pus commune non accipiendo pro alio quam quodam ordine compossibilium vel simultaneorum

vel successivorum, a quo nec Tu abhorrere videris. Quicquid a toto reali abscindi potest, ei actu

inesse Tecum putavi, non aeque quod a possibili seu ideali, uti numerus non potest intelligi ex

omnibus possibilium fractionibus conflatus nec fingi ultima minimave fractio sive simplex et

aequabilis sive inflata aut alia quavis ratione utcunque sumta. Porro lineae sectiones unitatis

sectionibus per fractiones proportione respondent».
See Leibniz to de Volder (30. VI. 1704), in GP II, p. 268: «Quod primum attinet, eo ipso

quod corpus mathematicum non potest resolvi in prima constitutiva, id utique non esse reale

colligitur, sed mentale quiddam nec aliud designans quam possibilitatem partium, non aliquid

actuale. Nempe linea mathematica se habet ut unitas arithmetica, et utrobique partes non sunt

nisi possibiles et prorsus indefinitae; et non magis linea est aggregatum linearum in quas secari

potest, quam unitas est aggregatum fractionum in quas potest discerpi».
GP IV, p. 491.
GP II, p. 276.
GP II, p. 379.
Leibniz to Bourguet (5. VIII. 1715), in GP III, p. 583: «Quand j’ay dit que l’unité n’est

plus resoluble, j’entends qu’elle ne sauroit avoir des parties dont la notion soit plus simple

qu’elle. L’unité est divisible, mais elle n’est pas resoluble; car les fractions qui sont les parties de

l’unité, ont des notions moins simples, parce que les nombres entiers (moins simples que l’uni-

té) entrent tousjours dans les notions des fractions. Plusieurs qui ont philosophé en Mathemati-

que sur le Point et sur l’Unité, se sont embrouillés, faute de distinguer entre la Resolution en

Notions et la Division en parties. Les parties ne sont pas tousjours plus simples que le tout,

quoyqu’elles soyent tousjours moindres que le tout».
GP II, p. 300 (Leibniz to Des Bosses, 14. II. 1706).
See GP II, p. 276.
NE, II, xvii, 3 (A, p. 158).
Ibidem. Cf. GP I, p. 215: «distinguere me inter heterogenea et homogenea. In hetero-

geneis ex. gr. numeris finitum non esse negationem infiniti. Nam numerus aliquis finitus consti-

tuitur, non abscindendo eum a toto aggregato infinitarum unitatum, sed formando eum ex

aggregato unitatum finitarum. Secus est in spatio seu extensione, nam spatium totum est unum

homogeneum, et demonstrari potest, ipsum non finiri, quia, cum per omnia sibi similes sint ejus

partes, impossibile est, inveniri rationem, cur alicubi finiatur».
GM VII, p. 276.
GP II, p. 300.
NE, II, xvii, 3 (A, p. 158).
See GP II, pp. 314-315: «Datur infinitum syncategorematicum seu potentia passiva

partes habens, possibilitas scilicet ulterioris in dividendo, multiplicando, subtrahendo, addendo

progressus. Datur et infinitum hypercategorematicum seu potestativum, potentia activa habens

quasi partes, eminenter, non formaliter aut actu. Id infinitum est ipse Deus. Sed non datur

infinitum categorematicum seu habens actu partes infinitas formaliter».
NE, II, xvii, 3 (A, p. 158).
See GP I, pp. 338-339.
See GM III, p. 535 (Leibniz to Joh. Bernoulli): «Uti Dn. Volderus, ita olim Gregorius a

S. Vincentio alicubi dixit, in infinito non habere locum Axioma, quod Totum sit majus parte.

Sed mihi videtur alterutrum dicendum, vel infinitum revera non esse unum totum, vel infini-

tum, si totum sit, et tamen non sit majus sua parte, esse aliquid absurdum. Sane ante multos

annos demonstravi, numerum seu multitudinem omnium numerorum contradictionem impli-

care, si ut unum totum sumatur. Idem de numero maximo et numero minimo, seu fractione

omnium infima. Et de his dicendum, quod de motu celerrimo, et similibus. Etiam Universum

non est unum totum, nec concipi debet ut animal cujus anima Deus, uti Veteres faciebant».
GP III, p. 623.
.R. Aaron, op. cit., p. 168.

Antonio Lamarra . Date:

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