ARE THERE REAL INFINITESIMALS IN LEIBNIZ’S METAPHYSICS?
George MacDonald Ross
ARE THERE REAL INFINITESIMALS IN LEIBNIZ’S METAPHYSICS?
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During the last century or so, the scientist and the philosopher have

become radically distinct individuals. They each have their separate fields of

enquiry, and they operate within different traditions, and in effectively sepa-

rate institutions. Because of the present gulf between science and philosophy,

there is a certain pressure on historians to decide whether they are doing the

history of science or the history of philosophy. However, it is now generally

accepted that the thinkers of the seventeenth century did not themselves draw

any sharp line between physics and metaphysics. Physicists and philosophers

were often the same people: sometimes working on detailed physical prob-

lems, and at other times on more general and abstract issues.

One of the main concerns of the philosopher-scientists of the early mod-

ern period was to describe the ultimate constitution of matter. Was it a

homogeneous plenum, only secondarily divided into corpuscles of various

sizes, or was it composed of discrete atoms moving in empty space? What sort

of qualities did matter possess in itself? Was force or energy a distinct ontolo-

gical category? Such questions were at least partly metaphysical, in so far as

they were reformulations of traditional, scholastic questions about the nature

of individual substances: did they involve components other than matter and

form (quiddity or haecceity, for example)? How did accidents inhere in them?

Were there essentially distinct categories of substance? On the other hand, the

new questions were at the same time scientific, since they concerned the fun-

damental categories which were to be accepted as appropriate for the scientific

description of nature. One might say that what distinguishes modern science

from pre-science is not its freedom from metaphysics, but its assuming a parti-

cular type of solution to the metaphysical questions about the nature of matter

which have been discussed since the very beginning of philosophy.

If we now turn to Leibniz in particular, his concept of a monad can be

seen as a solution to the typically physical-cum-metaphysical problem of the

“labyrinth of the continuum”, as he called it. The problem was this: if the

ultimate elements of matter were material atoms, then these atoms would

have a finite size and a particular shape. Consequently they would not be truly

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elementary, since they would have specifiable parts. One could then raise

precisely the same questions about the structure and cohesion of these parts as

one could about the parts of any macroscopic object. If, on the other hand,

the ultimate atoms were so small that they had no size, shape, or parts, then

they would be mathematical points. In that case, they would not be material

at all, since it is at least part of the definition of matter that it has spatial

dimensions. Besides, points could not be constitutive of matter, since no

amount of adding points to points will yield anything other than a point.

However, long before Leibniz finalised his theory of monads, he had

developed the infinitesimal calculus. One might therefore have expected him

to use its key concept of an infinitesimal to solve the problem of the size of

monads. Describing them as infinitesimals would enable him to say that they

were smaller than any material atom, but larger than a mere mathematical

point – just as an infinitesimal line is shorter than any specifiable line, but

long enough to have a gradient.

There are many passages where Leibniz himself encourages this sort of

interpretation. For example, the beginning of the Monadology:

The monad of which we shall speak here, is nothing but a simple sub-

stance which enters into compounds; simple, that is to say, without parts. And

there must be simple substances, because there are compounds; for the com-

pound is nothing but a collection or aggregate of simples. Now where there

are no parts, there neither extension, nor shape, nor divisibility is possible.

And these monads are the true atoms of nature and, in a word, the elements

of things.1

And a little later he says:

Monads, however, must have some qualities, otherwise they would not be

beings at all.2

In other words, monads are not mere points, but they are the ultimate compo-

nents of infinitely divisible, and indeed infinitely divided matter. So why

should he not say that they are infinitesimal? Why does he need to endow

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them with spiritual qualities in order to prevent them from collapsing into

mere mathematical points?

Leibniz was absolutely explicit that monads were not to be interpreted as

real infinitesimals. For instance, in a letter to Varignon of 20 June 1702, he

wrote:

Between you and me, I think Fontenelle... was joking when he said he

would derive metaphysical elements from our calculus. To tell the truth, I

myself am far from convinced that our infinites and infinitesimals should be

considered as anything other than ideals, or well-founded fictions... I believe

I can prove that there do not, and never could exist, any infinitely small

things.3

Leibniz’s reasons for denying the reality of infinitesimals come out partic-

ularly clearly in his correspondence with Johann Bernoulli. In a letter to Leib-

niz of 5 July 1698, Bernoulli had written as follows:

I did not definitely assert that there are infinitely many degrees of infini-

ties; I merely made certain conjectures, by virtue of which I deemed this to be

possible, and indeed probable. My main reason was that there is no reason

why God should have willed the existence only of this degree of infinity or

order of magnitude, which constitutes our objects, proportioned to our intel-

lects. I can easily conceive that, in the smallest particle of dust, there can exist

a world, in which all things have the same relative proportions as in this large

world; and, on the other hand, that our world might be nothing but a particle

of dust in another, infinitely larger world. This way of conceiving things can

be continued upwards and downwards indefinitely.4

Bernoulli then went on to argue that the inhabitants of a smaller world would

have as much and as little reason as us to suppose that their order of magni-

tude was the only one.

Two points in particular need to be noted about this passage. First, the

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hypothesis of worlds within worlds is very close to a number of Leibnizian

doctrines, such as that animals are composed of smaller animals ad infinitum,

and that animals do not die, but shrink to a microscopic size. Secondly, Ber-

noulli bases his claim that these worlds could really exist on the fact that they

can be conceived.

Leibniz accepted that there were indeed worlds within worlds, but he

denied that there were any infinitesimal worlds. He also implicitly accepted

that if infinitesimals were conceivable, then it would not only be possible for

them to exist, but they would actually exist. As he wrote on 29 July 1698:

Even if I accept that there is no portion of matter which is not actually

cut up into parts, this does not automatically lead to atomic elements, or por-

tions of a minimum size, and not even to infinitely small ones. All it leads to

are smaller and smaller portions, but of the same order of magnitude... This

is also the sense in which I readily agree that miniature animals always con-

tain smaller animals, without there having to be any infinitely small animals,

let alone elemental ones. If I were to admit the possibility of the sort of...

infinitely small things we are talking about, I would also believe in their

actual existence.5

In subsequent letters, Bernoulli insisted that we must have a concept of

the infinitesimal, on the grounds that infinite series have infinitely many

terms; and these must include an infinitesimal term, and others following it.

In other words, Bernoulli seems to have held that for us to have a concept of

something it is sufficient that we have a mathematical notation for it; and that

it is then only a contingent question whether there is in fact any reality corre-

sponding to the concept. Leibniz, on the other hand, wanted to maintain a

distinction within mathematics between quantities that are “real”, and those

that are merely “imaginary”. Real quantities are conceivable, and can have a

counterpart in reality; whereas imaginary quantities are contradictory fictions.

They may be useful or well-founded fictions, and play an essential role in

mathematical reasoning; but there can be no mental concept corresponding to

them, let alone any external reality. They are purely formal devices for the

purpose of symbolic manipulation.

In the passage just quoted above, Leibniz implies a distinction between

two different degrees of inconceivability: “atomic elements”, “portions of a

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minimum size”, or “elemental animals” are even more out of the question

than infinitesimals. He explains this in a letter of late August 1698:

Many years ago I proved beyond any doubt that the number or multitude

of all numbers implies a contradiction, if taken as a unitary whole. I think that

the same is true of the largest number, and of the smallest number, or the

lowest of all fractions. The same has to be said about these, as about the

fastest motion and the such-like. … But just as there is no numerical ele-

ment, or minimum fraction of the number one, or minimum number, similar-

ly there is no minimum line, or linear element, since a line, like the number

one, can be cut into parts or fractions. I admit that the impossibility of our

infinitesimals does not follow directly from this, since a maximum is not the

same thing as an infinity, and a minimum is not the same thing as an infini-

tesimal. And our infinitesimals can at least be used in the calculus and in

reasoning – unlike the maximum, the unbounded, and the minimum… So,

even though I am convinced that every part of matter is actually subdivided

into further parts, I do not think it follows from this that there exists any

infinitely small portion of matter. Still less do I admit that it follows that

there is any absolutely minimum portion of matter.6

As for Bernoulli’s point that all the terms of an infinite series must

actually exist (at least in the realm of mathematical concepts), Leibniz writes

as follows:

Let us suppose that in a line there actually are its 1/2, 1/4, 1/8, 1/16,

1/32, etc., and that all the terms of this series actually exist. You infer from

this that there also exists an infinitieth term. I, on the other hand, think that

it only follows from this that any specifiable finite fraction, however small,

actually exists.7

Here I should apologise for coining the term “infinitieth” to translate the

Latin infinitesimus. When Leibniz means “infinitesimal” in the sense of “infi-

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nitely small”, he always uses the adjectival phrase infinite parvus. When he

needs a noun, he has to use expressions like quantitas infinite parva, or portiones

infinite parvae
.8 Infinitesimus refers equally to infinitely small and infinitely

large numbers. In the present instance, the distinction is not a crucial one,

since the infinitieth term of a descending series will also be the first infinitesi-

mal quantity.

Leibniz defended his position in greater detail in his letter of 13 January

1699:

If there are ten terms, then there is a tenth; but it is debatable whether it

follows from this that, if there are infinitely many terms, then there is an

infinitieth one. Someone might say that an inference from the finite to the

infinite is invalid in this case. When it is said that there are infinitely many

terms, it is not being said that there is some specific number of them, but that

there are more than any specific number. It could equally well be argued

that, since among any ten numbers there is a highest number, which is also

the largest of those numbers, it follows that among all numbers there is a

highest number, which is also the largest of all numbers. But I think that

such a number implies a contradiction. Besides, you yourself give no answer

to my objection, when I pointed out that it is possible to make sense of an

infinite series consisting only of finite numbers. It is obvious that, even if we

suppose your series consisting of infinite as well as finite terms, on this

assumption it is still possible to conceive the part of the series which consists

only of the finite terms, leaving out the remaining part comprising the infinite

terms. This series would consist only of finite terms, and it would itself be

infinite, but it would have no infinitieth term.9

From a modern perspective, Leibniz’s attitude may seem inexcusably tim-

id. The invention of the infinitesimal calculus was one of the most significant

mathematical advances of his day; and yet he was not prepared to apply its

fundamental concept to an understanding of the real world. He is perhaps

reminiscent of those twentieth-century physicists who accepted the usefulness

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of relativity theory and quantum mechanics on pragmatic grounds, but refused

to let the new physics alter their Euclidean, materialist, and determinist pic-

ture of physical reality. Instead of revising his notion of what was conceiv-

able or possible, Leibniz belittled the concept of the infinitesimal as a mere

“useful fiction”.

However, Leibniz of all people could hardly have pretended that infini-

tesimals were logically satisfactory, simply to provide the stranger aspects of

his theory of monads with the support of a mathematical concept. He quite

properly had an overriding respect for logic; and he believed that logical con-

sistency was a necessary condition of scientific truth. Unlike some modern

physicists, he was not prepared to amend his logic in order to suit the conve-

nience of scientific theory.

In the case of infinity, Leibniz had a special difficulty, in that the concept

of infinity is a theological as well as a mathematical concept, and he had no

wish to deny that God was in some sense a real infinite whole. But as far as

mathematics and science were concerned, the concept of a completed whole

was something with a beginning and an end, or with a two- or three-dimen-

sional boundary. But the word “infinite” meant lacking a finis – an end, a

limit, or a boundary. As long as the term “infinite” was taken simply as

meaning “having no limit”, there was no problem; and Leibniz always used it

in this sense. The problem arose only with the introduction of the notion of

a limit – for example, the last term of an infinite series,10 the highest or low-

est number, or the smallest part of a continuous line. But there could be no

concept involving a limit to the unlimited.

However, even if Leibniz had accepted that infinitesimals were conceiv-

able, he still would not necessarily have agreed with Bernoulli that they were

therefore possible in the real world. He held that there were many things of

which we had perfectly clear conceptions, but which could not exist as such in

reality. In particular, no specifiable geometrical figure could exist in reality,

since all physical objects were distinguished by greater or lesser deviations

from perfection. Geometrical figures were, in his phrase, “incomplete no-

tions”. Similarly, the laws of mechanics and dynamics were incomplete, since

they left out of account the infinitely complex means by which energy was

transferred from one object to another. To assume that there were forces in

nature exactly corresponding to the abstract terms of the laws of motion was

an unwarranted reification which explained nothing – for example, Newton’s

postulation of a gravitational force as a substitute for an account of what really

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happened, in terms of the transfer of energy between intervening particles of

matter. Thus Leibniz wrote as follows to de Volder:

These and innumerable other such considerations clearly show that the

true notions of things are utterly subverted by this new philosophy, which

makes substances purely out of material or passive elements. Things that are

different must differ in some respect, or have in themselves some specifiable

diversity. It is amazing that mankind has failed to take advantage of such an

axiom, along with many others. But people are usually not interested in rea-

sons, since they are fully satisfied with pictorial imagery. This is why the

cause of true philosophy has been harmed by the introduction of so many

monstrous ideas. To be specific, people have restricted themselves to incom-

plete and abstract notions, or mathematical ones, which are mind-dependent,

and cannot exist as such in the real world. Examples are the notions of time,

space (or the pure subject-matter of geometry), matter in so far as it is merely

passive, motion taken in the mathematical sense, etc. In these cases people

can imagine things as being distinct without having any point of distinction –

for example two equal parts of a straight line. This is possible because a

straight line is something incomplete and abstract, which is what we need to

concern ourselves with for theoretical purposes; but in reality any given

straight line is distinguished from any other by what it consists of. This is

why it is impossible for two real objects to have exactly the same size and

shape simultaneously. Even things which are in different positions must

express their positions, or the things surrounding them; so it is not the case, as

is popularly imagined, that things are distinguishable by their position alone,

or solely by an extrinsic characterisation. Consequently, fashionable concep-

tions of physical substances, such as the atoms of the Democriteans or the

“perfect globules” of the Cartesians, cannot exist as such in reality, and are

nothing other than the incomplete mental constructs of philosophers who

have not penetrated deeply enough into the natures of things.11

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There is in fact an ambiguity in the notion of reality as Leibniz applies it

to mathematical concepts. In one sense, even straightforward geometrical

concepts, such as the concept of a perfect circle, are “unreal”, since there are

no realities exactly corresponding to them. They are entia rationis, or “mental

entities”, or “incomplete things”. In another sense, all logically coherent

mathematical concepts are “real”, as contrasted with “imaginary” ones, which

contain a contradiction, and therefore cannot properly be concepts at all. Of

these last, some are useless, like the notion of the highest number; whereas

others, such as the notion of the square root of minus one, or of the limit of

an infinite series, or of an infinitesimal quantity, are at least useful at the level

of symbolic manipulation.

The question now is: whether Leibniz’s insistence that infinitesimals are

only imaginary is consistent with his doctrine that everything in the material

world is not merely infinitely divisible, but actually subdivided to infinity.

This puts into even sharper relief the earlier question of why he did not avail

himself of the concept of the infinitesimal in expounding his monadology. As

Bernoulli put it in a letter of 16 August 1698:

You admit that any finite portion of matter is already in fact divided up

into an infinite number of parts; and yet you deny that any of these parts can

be infinitely small. How is this consistent? If no part is infinitely small, it

follows that each one is finite; and if each one is finite, it follows that all of

them taken together will constitute an infinite magnitude – contrary to the

original hypothesis.12

Leibniz’s reply has two dimensions to it. At the mathematical level, he

claims that Bernoulli is wrong to assume that there must be a smallest finite

part:

I would accept [the validity of your conclusion] if there were some finite

portion which was smaller than all the rest, or at least no bigger than any

other. In that case, assuming more such portions than any given number, I

agree that this gives rise to a quantity larger than any given quantity. But it is

universally accepted that, however small a part you specify, there is always a

smaller, finite part.13

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Leibniz’s reply is perhaps over-compressed. His point is that what is

meant by “infinitely divided” is that however small a part you may specify,

there always actually exists a smaller, finite part. Consequently, there can

never be a completed infinity of finite parts to generate the contradiction. It

is precisely because the actual division of matter proceeds to infinity that you

can never arrive at a real infinitesimal.

The second dimension to Leibniz’s reply is metaphysical rather than

mathematical. He writes:

By “monad” I mean a genuinely unitary substance, in other words, one

that is not an aggregate of substances. Matter taken in itself, or mass, or what

you could call primary matter, is not a substance, or even an aggregate of

substances, but something incomplete. Secondary matter, or bulk, is not a sub-

stance, but substances; in the same way as a flock or a pond of fish is not a

single substance, but only an animal or fish is. Even though an organic body,

such as an animal’s or mine, is in its turn composed of innumerable sub-

stances, these are not parts of the animal or of me. But if there were no

souls, or something analogous to them, then there would be no self, no mon-

ads, no real units, and therefore no substantial compounds. It would even

follow that everything in the physical world was nothing but images. From

this it can readily be concluded that there is no part of matter in which there

do not exist monads.14

Not surprisingly, Bernoulli in his reply of 6 September 169815 shows

himself completely perplexed by Leibniz’s abrupt move to arcane metaphysics

and hints of phenomenalism. Much the same happened to other long-term

correspondents such as de Volder and des Bosses, when Leibniz suddenly

revealed his underlying metaphysical theory. Almost as if he regretted his

indiscretion, his replies to Bernoulli’s questions, in his letter of 20 September

1698, are extremely cryptic. Thus he says:

Neither you, nor I, nor anyone else are compounds of the parts of our

bodies. You are worried that matter might be compounded out of non-quan-

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tities. My answer is that it is no more a compound of souls than it is a com-

pound of points.16

Unlike Bernoulli, we do at least have the benefit of Leibniz’s other writ-

ings, in the light of which we can make sense of these remarks, and relate

them to the question of real infinitesimals. What Leibniz is saying is that

there are two dimensions to reality: the spiritual and the material. The spir-

itual realm consists of an infinity of monads, which are indivisible unities.

The material realm consists of compound bodies, which are infinitely divided

into smaller bodies. But this realm is nothing other than the world as mani-

fested in perception. If it were not for its relation to the more fully real spir-

itual realm, it would be nothing but images.17

As with the chorismos between the noumenal and phenomenal realms in

Plato, Kant, and other such philosophers, Leibniz has the problem of relating

the two realms closely enough to allow the one to support the other, while at

the same time preserving their ontological heterogeneity. When matter is

explained as grounded in monads, it is tempting to made the monads

infinitesimal parts of matter. This is the temptation to which Leibniz himself

seems to have succumbed in passages such as the opening of the Monadology,

quoted above. One might compare the temptation to see Kant’s noumena

merely as things-in-themselves, lying just behind the appearances, like Locke’s

substratum substances.

However, as Leibniz himself makes abundantly clear on other occasions,

monads are not parts of matter, but requisites, of which matter is the resultant.

For example, he wrote to de Volder as follows:

Even if you take a piece of matter as an aggregate consisting of a plurality

of substances, you can still conceive it as containing a single pre-eminent sub-

stance, provided the piece of matter constitutes an organic body animated by

its own principal entelechy. But on my theory, all that is needed in addition

to entelechy in order to make up a monad or complete simple substance, is a

quantity of primitive passive power proportional to the total amount of matter

in the organic body. The other, subordinate monads situated in the organs of

the body do not constitute parts of its matter, but are immediate requisites of it.

136

These monads together with the principal monad make up the organic physi-

cal substance, or animal or plant.18

Precisely what it means for monads to be requisites rather than parts of

matter is a long story, which has to be pieced together from hints in widely

scattered texts, and which I shall not go into here.19 As Leibniz said to des

Bosses in his letter of 24 April 1709:

Meanwhile, I do not think it appropriate for us to consider souls as in

points. Perhaps someone might say that they are in space only by virtue of

their actions – this would be to speak in terms of the old system of causal

influence. It would be better to say (speaking in terms of the new system of

pre-established harmony) that they are in space by virtue of correspondence,

and that they are therefore in the whole organic body which they animate.

On the other hand, I do not deny a certain real metaphysical union between

soul and organic body (as I also said in my reply to Tournemine), by virtue of

which the soul can be said to be really in the body. But because this union

cannot be explained on the basis of phenomena, and does not bring about any

change in them, I cannot clarify any further what it essentially consists in.

Suffice to say, it has something to do with correspondence.20

In a postscript to the same letter, Leibniz admits that during an early

period in the development of his thought, he failed to make a proper distinc-

tion between the two realms, and believed that it was possible for physical

objects to be generated out of monads as their elementary parts. But he soon

realised that this was a mistake:

Many years ago, when my philosophy was not yet sufficiently mature, I

located souls in points, and so I thought that the multiplication of souls

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could be explained through their splitting off from the parent stock, since a

multiplicity of points can be generated out of a single one, just as the

apexes of a multiplicity of triangles can be generated by division out of the

apex of a single triangle. But after closer deliberation, I realised that this

not only landed us with innumerable difficulties, but that there was also

here a certain category mistake, so to speak. Souls are not to be assigned

spatial predicates, and their unity or multiplicity is not to be subsumed

under the category of quantity, but of substance – that is, not from points,

but from the primitive power of acting. But the activity proper to the soul

is perception, and the unity of the perceiver depends on the interconnec-

tion of perceptions, whereby later perceptions are derived from preceding

ones.21

In Leibniz’s philosophy, it is as if there were two separate, infinite

escalators. The one moves downwards from the material world as we

know it by means of sensory experience. Everything is divided into small-

er and smaller components; but the escalator has no bottom where we

might arrive at infinitesimal or minimal components. The other escalator

moves upwards from the lowest grade of monad to higher and higher

grades of more and more dominant monads; but if the escalator has any

top, it ends up, not in the material world of experience, but in God. Leib-

niz’s earlier model suggested that the two escalators were really only one.

We knew about the monadic realm through our knowledge of our own

souls, and about the material realm through sense experience; and we could

assume that the upward and downward escalators somehow met in the mid-

dle. According to his later model, they never meet, but at every level there

are cross connections. Our souls and bodies are somewhere in the middle.

We are intuitively aware of the connections between our spiritual and

physical selves. They are mutually interdependent; but it would be a gross

category mistake to say that the soul was a part of the body. We will not

arrive at souls by dividing matter; nor will we arrive at matter by multiply-

ing souls.

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At the beginning of this paper, I suggested that it was natural for us to

expect monads to be infinitesimal constituents of matter. But why should

this be natural, if Leibniz’s philosophy was in fact so different? The fault is

partly Leibniz’s own, in using the term “monad” for his basic substances,

and in describing them as “spiritual atoms”. The implication is that they

are units out of which everything else is composed. But in fact he started

using the term “monad” only many years after developing his mature philo-

sophical system. Certainly he had always held that genuine substances must

be unities, since he accepted the scholastic equivalence of ens and unum; and

he often referred to them simply as unitates. However, his preferred terms

for his elementary substances were entelechia and forma.

By his choice of vocabulary, Leibniz was implicitly referring back to

the Platonic/Aristotelian conception of substance. Given the differences

between Plato and Aristotle, it may seem odd to talk of a single Pla-

tonic/Aristotelian conception; but ever since his university days, Leibniz

had always seen the true philosophy as lying in a reconciliation or “harmo-

ny” of Platonism and Aristotelianism. On the question of the nature of

substance, there is indeed an element common to Plato and Aristotle which

distances them from the majority of the “modern” philosophers of the sev-

enteenth century. The modern philosophers were almost unanimous in

rejecting the concept of form, and in seeing substance as an intrinsically

formless substratum underlying the qualities of individual things, whether

material or immaterial. Plato and Aristotle, on the other hand, agreed that

individual substances were complete beings, sharing in both matter and

form. For Aristotle, matter and form were equally indispensible, and co-

existed in each individual substance. For Plato, matter had a more shadowy

existence, and what reality it had was dependent on form. Forms were

ontologically prior, and one of the cruxes of Platonic interpretation is the

question of precisely how material objects “share in” the non-temporal,

non-spatial forms.

Most of the time Leibniz sounds more an Aristotelian than a Platonist.

The term entelechia is Aristotle’s, and Leibniz repeatedly insisted that all indi-

vidual substances, other than God, must intrinsically possess matter as well

as form. Yet he himself claimed to be more a Platonist than an Aristote-

lian. For example, in the New Essays, he wrote as follows of his relation-

ship with Locke:

Actually, although the author of the Essay says thousands of excellent

things for which I congratulate him, our systems are very different. His

has a closer connection with Aristotle, and mine with Plato, even though

we both differ on many points from these two ancient philosophers. He is

more accessible to the popular understanding, whereas I am sometimes

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compelled to be a little more acroamatic and more abstract. This is a dis-

advantage to me, especially when one is writing in a living language.22

So how is this to be reconciled with Leibniz’s apparent Aristotelianism ?

As we have just seen, he contrasts his own “acroamatic” or esoteric style

of philosophy with that of Aristotle and Locke. Elsewhere, he describes the

latter style as “exoteric”, and allies himself with Plato as an esoteric philoso-

pher:

So Plato’s innate notions, which he disguised with the term “reminis-

cence”, are far preferable to the tabula rasa of Aristotle, Locke, and other mod-

erns, who philosophise exoterically.23

That is closer to popular ideas, as is usual with Aristotle, whereas Plato

goes deeper.24

My metaphysics is a little more Platonic than his [Locke’s]; but this is

also why it is not so in tune with the general taste.25

In short, Leibniz was acutely aware of different levels at which it was

appropriate to address different audiences. When addressing the uninitiated,

who would be incapable of appreciating the more arcane secrets of existence,

he was prepared to speak as an Aristotelian. He would assume the reality of

the every-day objects of perception, but would insist that they possessed form

as well as matter. Even so, this was already a bold and unfashionable stance,

since he emphasised that, just as in human beings the form of the body is the

soul, so in all other substances the form is soul-like. Again, although the

Aristotelian form is in a sense a component of a substance, it is not a part which

can be arrived at by the subdivision of matter.

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However, despite Leibniz’s belief in the greater accessibility of Aristote-

lianism, the formulation of his philosophy in Aristotelian terms was not only

radically different from the current Cartesian and atomist orthodoxies, but it

left much that was obscure. It is hardly surprising that his correspondents

failed to understand him. In particular, the Aristotelian formulation of his

position could provide no satisfactory account of the nature of the primary

matter informed by entelechy.

On the other hand, when he spoke Platonically, he could make form

ontologically prior, and reduce matter to the realm of appearance and non-

being, thus eliminating the problem of its ultimate nature. Of course, Leib-

niz’s forms were not the forms of Plato. They were not abstractions, but per-

ceivers; and as perceivers, they had to have perceptions in order to exist as

such. It was through their perceptions that they were chained to the realm of

matter – of appearance, passivity, imperfection, and non-being. Without mat-

ter, they would be identical with God, who is the only being to know every-

thing as it really is, to be pure activity, perfect, and completely devoid of non-

being. In his more mystical moments, he used the generation of all numbers

from 1 and 0 in binary arithmetic as an analogy for God’s creation of the

universe by progressive admixtures of non-being with his own being:

All creatures come from God and nothing – their being from God, and

their unbeing from nothing. (The same is true of numbers in a wonderful

way – and the being of things is like that of numbers). No creature can be

without unbeing, otherwise it would be God himself. Even the angels and

the souls of the dead participate in unbeing.26

Ultimately, therefore, as with Plato, matter was unreal. However, it was

“well-founded” in that every part of the phenomenal realm of matter con-

tained an infinity of organic bodies, and there was a non-temporal, non-spatial

form corresponding to every organic body. Exoterically, we could speak as if

the material realm were full of substances; but esoterically we would know

that it was mere appearance.

Again, as with Plato, there is the problem of the chorismos between the

noumenal and the phenomenal – between the realm of real forms, and that of

merely apparent matter. As I have already said, it has not been my concern

in this paper to explain how Leibniz believed that he could bridge the gap.

Here, the important point has been that he believed there was a gap.

141

Leibniz’s insistence on a radical difference of kind between monads and

matter shows that, despite my introductory remarks, he did after all preserve a

sharp distinction between the domains of physics and of metaphysics. Physics

was concerned with the material realm of phenomena, where everything had

to be explained in terms of efficient causes; and it could never reach real sub-

stances. Metaphysics was concerned with the spiritual realm of monads,

where everything had to be explained in terms of final causes; and it could

never yield results which would come into conflict with the scientific know-

ledge of matter. Parallel to this distinction was that between mental or imag-

inary, and real concepts. If Leibniz had allowed an imaginary concept, such as

that of an infinitesimal, to have any counterpart in reality, then this would

have obliterated a distinction which lay at the very roots of his philosophy. It

is therefore no wonder that he resisted the temptation to make his metaphysi-

cal system more accessible, at least to some mathematicians, by “deriving

metaphysical elements from his calculus.”27

1.
Monadology §§ 1-3, GP VI, 607: “La Monade, dont nous parlerons icy, n’est autre chose,

qu’une substance simple, qui entre dans les composés; simple, c’est à dire, sans parties. Et il faut

qu’il y ait des substances simples, puisqu’il y a des composés; car le composé n’est autre chose,

qu’un amas, ou aggregatum des simples. Or là, où il n’y a point de parties, il n’y a ny étendue, ny

figure, ny divisibilité possible. Et ces Monades sont les veritables Atomes de la Nature, et en un

mot les Elemens des choses”.
2.
Monadology § 8, GP VI, 608: “Cependant il faut que les Monades ayent quelques quali-

tés, autrement ce ne seroient pas même des Etres.”
3.
GM IV, 110: “Entre nous je crois que Mons. de Fontenelle... en a voulu railler,

lorsqu’il a dit qu’il vouloit faire des elemens metaphysiques de nostre calcul. Pour dire le vray,

je ne suis pas trop persuadé moy même, qu’il faut considerer nos infinis et infiniment petits

autrement que comme des choses ideales ou comme des fictions bien fondées ... je ne crois

point qu’il y en ait, ny même qu’il y en puisse avoir d’infiniment petites et c’est ce que je crois

pouvoir demonstrer.”
4.
GM III, 503-4: “nec ego infinitos infinitorum gradus pro certo asserui, sed conjecturas

tantum adduxi, quibus rem possibilem et probabilem esse statui. Et quidem rationem praeci-

puam hujus esse quod nulla sit ratio, cur Deus hunc tantum infinitatis gradum seu hoc quantita-

tum genus, quae nostra faciunt objecta nostroque intellectui proportionata, voluisset existere

cum tamen facile concipere possim, in minimo pulvisculo posse existere Mundum, in quo

omnia proportionata sunt huic magno, et contra nostrum mundum nihil aliud esse, quam pul-

visculum alius infinities majoris; atque hunc conceptum continuari posse ascendendo et descen-

dendo sine fine.”
5.
GM III, 524: “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... Sic etiam semper ani-

malcula in animalculis dari facile concedo; et tamen necesse non est dari animalcula infinite

parva, nedum ultima. Si talia, de quibus inter nos agitur, ... infinite parva possibilia esse con-

cederem, etiam crederem esse”.
6.
GM III, 535-6: “Sane ante multos annos demonstravi, numerum seu multitudinem

omnium numerorum contradictionem implicare, 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… Quemadmodum autem non datur Elementum Numericum seu mini-

ma pars unitatis, vel minimum in Numeris, ita nec datur linea minima, seu elementum lineale;

linea enim, ut Unitas, secari potest in partes vel fractiones. Interim fateor, cum aliud sit maxi-

mum ab infinito et minimum ab infinite parvo, non hinc statim refutari possibilitatem nostro-

rum infinite parvorum. Et saltem in calculo et ratiocinatione adhiberi possunt, quod de maximo

interminatoque, itemque de minimo non licet… Etsi igitur pro certo habeam, quamlibet par-

tem materiae esse rursus actu subdivisam, non ideo tamen hinc sequi puto, quod detur portio

materiae infinite parva, et minus adhuc sequi concedo, quod ulla detur portio omnino mini-

ma”.
7.
GM III, 536: “Ponamus in linea actu dari, 1/2, 1/4, 1/8, 1/16, 1/32 etc. omnesque

seriei hujus terminos actu existere; hinc infers dari et infinitesimum, sed ego nihil aliud hinc

puto sequi, quam actu dari quamvis fractionem finitam assignabilem cujuscunque parvitatis.”
8.
E. g. GM III, 524.
9.
GM III, 566: “Dubitari potest an sequatur: Positis terminis decem, datur decimus:

ergo positis terminis infinitis, datur infinitesimus. Dicet enim fortasse aliquis, argumentum de

finito ad infinitum hic non valere: et cum dicitur dari infinita, non dicitur dari eorum numerum

terminatum, sed dari plura quovis numero terminato. Et pari jure conclusurum iri: Inter nume-

ros decem datur ultimus, qui et maximus eorum; ergo et inter omnes numeros datur ultimus,

qui et maximus omnium numerorum; qualem tamen numerum puto implicare contradictionem.

Ipse quoque non respondes meae objectioni, cum monueram posse intelligi seriem infinitam ex

meris numeris finitis constantem. Manifestum enim est, etsi poneretur Tecum series ex (magni-

tudine) finitis pariter et infinitis (numeris) constans, hoc posito posse intelligi partem ejus con-

stantem ex meris (magnitudine) finitis, reliqua parte (magnitudine) infìnitos complectente omis-

sa. Haec autem series ex meris (magnitudine) finitis esset quidem et ipse (multitudine) infinita,

sed tamen nullum haberet terminum infinitesimum.”
10.
Although certain convergent infinite series can be said to “tend towards a limit”, they

are still unlimited in the sense that there is no limit to the number of their terms.
11.
Winter 1702/3, GP II, 249-250: “Haec et innumera hujusmodi satis indicant veras

rerum notiones per novam illam philosophiam quae ex solis materialibus sive passivis substan-

tias format, plane subverti. Quae differunt, debent aliquo differre seu in se assignabilem habere

diversitatem, mirumque est manifestissimum hoc axioma cum tot aliis ab hominibus adhibitum

non fuisse. Sed vulgo homines imaginationi satisfacere contenti rationes non curant, hinc tot

monstra introducta contra veram philosophiam. Scilicet non nisi incompletas abstractasque

adhibuere notiones sive mathematicas, quas cogitatio sustinet sed quas nudas non agnoscit natu-

ra, ut temporis, item spatii seu extensi pure mathematici, massae mere passivae, motus mathe-

matice sumti etc. ubi fingere possunt homines diversa sine diversitate, exempli gratia duas lineae

rectae partes aequales, quia scilicet linea recta aliquid incompletum abstractumque est, quod

doctrinae causa spectare oportet; at in natura quaelibet recta a qualibet alia [Gerhardt omits

‘alia’] distinguitur contentis. Hinc fieri nequit in natura ut duo corpora sint perfecte simul

aequalia et similia. Etiam quae loco differunt, oportet locum suum, id est ambientia exprimere,

atque adeo non tantum loco seu sola extrinseca denominatione distingui, ut vulgo talia conci-

piunt. Hinc corpora vulgari modo sumta, veluti Atomi Democriticorum, globuli perfecti Carte-

sianorum, dari non possunt in natura, neque aliud sunt quicquam quam incompletae cogita-

tiones philosophorum non satis rerum naturas inspicientium.”
12.
GM III, 529: “Concedis materiae portionem finitam actu jam divisam esse in partes

numero infinitas, et tamen negas aliquam istarum particularum posse esse infinite exiguam:

quomodo haec cohaerent? Nam, si nulla est infinite exigua, ergo singulae sunt finitae; si singu-

lae sunt finitae, ergo omnes simul sumtae constituent magnitudinem infinitam, contra hypothe-

sin.”
13.
GM III, 536: “Hanc consequentiam ... concederem si aliqua daretur [particula] finita,

quae minor esset caeteris omnibus, vel certe nulla alia major; tunc enim fateor talibus assumtis,

pluribus quam est datus numerus quivis, oriri quantitatem majorem data quavis. Sed constat,

quavis parte aliam minorem finitam dari.”
14.
GM III, 537: “Per Monadem intelligo substantiam vere unam, quae scilicet non sit

aggregatum substantiarum. Materia ipsa per se, seu moles, quam materiam primam vocare pos-

sis, non est substantia; imo nec aggregatum substantiarum, sed aliquid incompletum. Materia

secunda, seu Massa, non est substantia, sed substantiae; ita non grex, sed animal; non piscina,

sed piscis, substantia una est. Etsi autem corpus animalis, vel meum organicum, rursus ex sub-

stantiis innumeris componatur, eae tamen partes animalis vel mei non sunt. Sed si nullae essent

animae, vel his analoga, tunc nullum esset Ego, nullae monades, nullae reales unitates, nullae-

que adeo multitudines substantiales forent; imo omnia in corporibus non nisi phasmata essent.

Hinc facile judicatur, nullam esse materiae partem, in qua Monades non existant.”
15.
GM III, 539-540.
16.
GM III, 542: “Neque enim ego, Tu, ille componimur ex partibus corporis nostri. Ve-

reris ne materia componatur ex non-quantis. Respondeo, non magis eam componi ex animabus,

quam ex punctis.”
17.
I discuss this relationship in greater detail in: G. MacDonald Ross, Leibniz (“Past

Masters”), Oxford 1984, pp. 88-95, and in G. MacDonald Ross, Leibniz’s Phenomenalism and the

Construction of Matter
, “Studia Leibnitiana”, Sonderheft XIII (1984), pp. 26-36.
18.
Winter 1702/3, GP II, 252: “Si massam sumas pro aggregato plures continente sub-

stantias, potes tamen in ea concipere unam substantiam praeeminentem, si quidem corpus

organicum ea massa constituat, sua [Gerhardt has ‘seu’ instead of ‘si quidem... sua’] entelechia

primaria animatum. Caeterum in Monada seu substantiam simplicem completam cum Entele-

chia non conjungo nisi vim passivam primitivam relatam ad totam massam corporis organici,

cujus quidem partem non faciunt reliquae monades subordinatae in organis positae, ad eam

tamen requiruntur immediate, et cum primaria Monade concurrunt ad substantiam corpoream

organicam, seu animal plantamve.”
19.
See note 17, above.
20.
GP II, 370-371: “Interim non puto convenire, ut animas tanquam in punctis

consideremus. Fortasse aliquis diceret, eas non esse in loco nisi per operationem, nempe loquendo se-

cundum vetus systema influxus, vel potius (secundum novum systema harmoniae praestabilitae)

esse in loco per corresponsionem, atque ita esse in toto corpore organico quod animant. Non nego

interim unionem quandam realem metaphysicam inter Animam et Corpus organicum (ut Tourneminio

etiam respondi), secundum quam dici possit, animam vere esse in corpore. Sed quia ea res ex

Phaenomenis explicari non potest, nec quicquam in iis variat, ideo in quo formaliter consistat,

ultra distincte explicare non possum. Sufficit corresponsioni esse alligatam.”
21.
GP II, 372: “Ante multos annos, cum nondum satis matura esset philosophia mea,

locabam Animas in punctis, et ita putabam multiplicationem animarum per Traducem explicari

posse, dum ex uno puncto fieri possunt plura, ut ex apice trianguli unius per divisionem fieri

possunt apices plurium triangulorum. Sed factus consideratior, deprehendi non tantum ita nos in

difficultates innumeras indui, sed etiam esse hic quandam, ut sic dicam, μετάβασιν εἰς ἄλλο γένος.

Neque animabus assignanda esse quae ad extensionem pertinent, unitatemque earum aut multi-

tudinem sumendam non ex praedicamento quantitatis, sed ex praedicamento substantiae, id est

non ex punctis, sed ex vi primitiva operandi. Operatio autem animae propria est perceptio, et

unitatem percipientis facit perceptionum nexus, secundum quem sequentes ex praecedentibus

derivantur.”
22.
GP V, 41-42: “En effect, quoyque l’Auteur de l’Essay dise mille belles choses où j’ap-

plaudis, nos systèmes different beaucoup. Le sien a plus de rapport à Aristote, et le mien à

Platon, quoyque nous nous eloignions en bien des choses l’un et l’autre de la doctrine de ces

deux anciens. Il est plus populaire, et moy je suis forcé quelquefois d’estre un peu plus acroamati-

que
et plus abstrait, ce qui n’est pas un avantage à moy sur tout quand on écrit dans une langue

vivante.”
23.
Letter to Hansch, 25 July 1707, E, 446: “Longe ergo praeferendae sunt Platonis Notitiae

innatae, quos reminiscentiae nomine velavit, tabulae rasae Aristotelis et Lockii aliorumque recen-

tiorum, qui ἐξωτερικῶς philosophantur.”
24.
Discours de Métaphysique § 26, GP IV, 452: “Cela s’accorde d’avantage avec les notions

populaires, comme c’est la maniéré d’Aristote, au lieu que Platon va plus au fond.”
25.
Letter to Burnett, 8/18 May 1697, GP III, 204: “Ma Métaphysique est un peu plus

Platonicienne que la sienne; mais c’est aussi pour cela qu’elle n’est pas si conforme au goust

general.” For a more detailed discussion of Leibniz’s Platonism, see G. MacDonald Ross,

Leibniz and Renaissance Neoplatonism, in A. Heinekamp (éd.), Leibniz et la Renaissance, Wiesbaden

1983, pp. 125-134.
26.
Von der wahren Theologia Mystica, GD I, 411: “Alle Geschöpfe sind von Gott und Nichts;

ihr Selbstwesen von Gott, ihr Unwesen von Nichts. (Solches weisen auch die Zahlen auf eine

wunderbare Weise, und die Wesen der Dinge sind gleich den Zahlen). Kein Geschöpf kann

ohne Unwesen sein; sonst wäre es Gott. Die Engel und Heiligen müssens haben.”
27.
Cf. Note 3, above.


George MacDonald Ross . :

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