During the last century or so, the scientist and the philosopher havebecome radically distinct individuals. They each have their separate fields ofenquiry, 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 thehistory of science or the history of philosophy. However, it is now generallyaccepted that the thinkers of the seventeenth century did not themselves drawany sharp line between physics and metaphysics. Physicists and philosopherswere 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 ahomogeneous plenum, only secondarily divided into corpuscles of varioussizes, or was it composed of discrete atoms moving in empty space? What sortof 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 asthey were reformulations of traditional, scholastic questions about the natureof individual substances: did they involve components other than matter andform (quiddity or haecceity, for example)? How did accidents inhere in them?Were there essentially distinct categories of substance? On the other hand, thenew questions were at the same time scientific, since they concerned the fun-damental categories which were to be accepted as appropriate for the scientificdescription of nature. One might say that what distinguishes modern sciencefrom 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 matterwhich have been discussed since the very beginning of philosophy.

If we now turn to Leibniz in particular, his concept of a monad can beseen 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 theultimate elements of matter were material atoms, then these atoms wouldhave a finite size and a particular shape. Consequently they would not be trulyelementary, since they would have specifiable parts. One could then raiseprecisely the same questions about the structure and cohesion of these parts asone 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, thenthey would be mathematical points. In that case, they would not be materialat all, since it is at least part of the definition of matter that it has spatialdimensions. Besides, points could not be constitutive of matter, since noamount of adding points to points will yield anything other than a point.

However, long before Leibniz finalised his theory of monads, he haddeveloped the infinitesimal calculus. One might therefore have expected himto use its key concept of an infinitesimal to solve the problem of the size ofmonads. Describing them as infinitesimals would enable him to say that theywere smaller than any material atom, but larger than a mere mathematicalpoint – just as an infinitesimal line is shorter than any specifiable line, butlong enough to have a gradient.

There are many passages where Leibniz himself encourages this sort ofinterpretation. 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. Andthere must be simple substances, because there are compounds; for the com-pound is nothing but a collection or aggregate of simples. Now where thereare 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 elementsof things.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 fautqu’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, nyfigure, ny divisibilité possible. Et ces Monades sont les veritables Atomes de la Nature, et en unmot les Elemens des choses”.

And a little later he says:

Monads, however, must have some qualities, otherwise they would not bebeings at all. Monadology § 8, GP VI, 608: “Cependant il faut que les Monades ayent quelques quali-tés, autrement ce ne seroient pas même des Etres.”

In other words, monads are not mere points, but they are the ultimate compo-nents of infinitely divisible, and indeed infinitely divided matter. So whyshould he not say that they are infinitesimal? Why does he need to endowthem with spiritual qualities in order to prevent them from collapsing intomere mathematical points?

Leibniz was absolutely explicit that monads were not to be interpreted asreal infinitesimals. For instance, in a letter to Varignon of 20 June 1702, hewrote:

Between you and me, I think Fontenelle... was joking when he said hewould derive metaphysical elements from our calculus. To tell the truth, Imyself am far from convinced that our infinites and infinitesimals should beconsidered as anything other than ideals, or well-founded fictions... I believeI can prove that there do not, and never could exist, any infinitely smallthings.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 petitsautrement que comme des choses ideales ou comme des fictions bien fondées ... je ne croispoint qu’il y en ait, ny même qu’il y en puisse avoir d’infiniment petites et c’est ce que je croispouvoir demonstrer.”

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 bepossible, and indeed probable. My main reason was that there is no reasonwhy God should have willed the existence only of this degree of infinity ororder of magnitude, which constitutes our objects, proportioned to our intel-lects. I can easily conceive that, in the smallest particle of dust, there can exista world, in which all things have the same relative proportions as in this largeworld; and, on the other hand, that our world might be nothing but a particleof dust in another, infinitely larger world. This way of conceiving things canbe continued upwards and downwards indefinitely.GM III, 503-4: “nec ego infinitos infinitorum gradus pro certo asserui, sed conjecturastantum 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 existerecum tamen facile concipere possim, in minimo pulvisculo posse existere Mundum, in quoomnia 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.”

Bernoulli then went on to argue that the inhabitants of a smaller world wouldhave 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, thehypothesis of worlds within worlds is very close to a number of Leibniziandoctrines, 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 theycan be conceived.

Leibniz accepted that there were indeed worlds within worlds, but hedenied that there were any infinitesimal worlds. He also implicitly acceptedthat if infinitesimals were conceivable, then it would not only be possible forthem 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 actuallycut 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 toare smaller and smaller portions, but of the same order of magnitude... Thisis 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 theiractual existence.GM III, 524: “Etsi enim concedam, nullam esse portionem materiae, quae non actu sitsecta, non tamen ideo devenitur ad elementa insecabilia, aut ad minimas portiones, imo nec adinfinite 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 infiniteparva, nedum ultima. Si talia, de quibus inter nos agitur, ... infinite parva possibilia esse con-cederem, etiam crederem esse”.

In subsequent letters, Bernoulli insisted that we must have a concept ofthe infinitesimal, on the grounds that infinite series have infinitely manyterms; 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 ofsomething it is sufficient that we have a mathematical notation for it; and thatit 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 adistinction within mathematics between quantities that are “real”, and thosethat are merely “imaginary”. Real quantities are conceivable, and can have acounterpart in reality; whereas imaginary quantities are contradictory fictions.They may be useful or well-founded fictions, and play an essential role inmathematical reasoning; but there can be no mental concept corresponding tothem, let alone any external reality. They are purely formal devices for thepurpose of symbolic manipulation.

In the passage just quoted above, Leibniz implies a distinction betweentwo different degrees of inconceivability: “atomic elements”, “portions of aminimum size”, or “elemental animals” are even more out of the questionthan infinitesimals. He explains this in a letter of late August 1698:

Many years ago I proved beyond any doubt that the number or multitudeof all numbers implies a contradiction, if taken as a unitary whole. I think thatthe same is true of the largest number, and of the smallest number, or thelowest of all fractions. The same has to be said about these, as about thefastest 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 numberone, can be cut into parts or fractions. I admit that the impossibility of ourinfinitesimals does not follow directly from this, since a maximum is not thesame 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 inreasoning – unlike the maximum, the unbounded, and the minimum… So,even though I am convinced that every part of matter is actually subdividedinto further parts, I do not think it follows from this that there exists anyinfinitely small portion of matter. Still less do I admit that it follows thatthere is any absolutely minimum portion of matter.GM III, 535-6: “Sane ante multos annos demonstravi, numerum seu multitudinemomnium numerorum contradictionem implicare, si ut unum totum sumatur. Idem de numeromaximo et numero minimo, seu fractione omnium infima. Et de his dicendum, quod de motucelerrimo, 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 maximointerminatoque, 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 portiomateriae infinite parva, et minus adhuc sequi concedo, quod ulla detur portio omnino mini-ma”.

As for Bernoulli’s point that all the terms of an infinite series mustactually exist (at least in the realm of mathematical concepts), Leibniz writesas 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 fromthis that there also exists an infinitieth term. I, on the other hand, think thatit only follows from this that any specifiable finite fraction, however small,actually exists.GM III, 536: “Ponamus in linea actu dari, 1/2, 1/4, 1/8, 1/16, 1/32 etc. omnesqueseriei hujus terminos actu existere; hinc infers dari et infinitesimum, sed ego nihil aliud hincputo sequi, quam actu dari quamvis fractionem finitam assignabilem cujuscunque parvitatis.”

Here I should apologise for coining the term “infinitieth” to translate theLatin infinitesimus. When Leibniz means “infinitesimal” in the sense of “infi-nitely small”, he always uses the adjectival phrase infinite parvus. When heneeds a noun, he has to use expressions like quantitas infinite parva, or portionesinfinite parvae.E. g. GM III, 524. Infinitesimus refers equally to infinitely small and infinitelylarge 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 January1699:

If there are ten terms, then there is a tenth; but it is debatable whether itfollows from this that, if there are infinitely many terms, then there is aninfinitieth one. Someone might say that an inference from the finite to theinfinite is invalid in this case. When it is said that there are infinitely manyterms, it is not being said that there is some specific number of them, but thatthere are more than any specific number. It could equally well be arguedthat, since among any ten numbers there is a highest number, which is alsothe largest of those numbers, it follows that among all numbers there is ahighest number, which is also the largest of all numbers. But I think thatsuch a number implies a contradiction. Besides, you yourself give no answerto my objection, when I pointed out that it is possible to make sense of aninfinite series consisting only of finite numbers. It is obvious that, even if wesuppose your series consisting of infinite as well as finite terms, on thisassumption it is still possible to conceive the part of the series which consistsonly of the finite terms, leaving out the remaining part comprising the infiniteterms. This series would consist only of finite terms, and it would itself beinfinite, but it would have no infinitieth term.GM III, 566: “Dubitari potest an sequatur: Positis terminis decem, datur decimus:ergo positis terminis infinitis, datur infinitesimus. Dicet enim fortasse aliquis, argumentum definito ad infinitum hic non valere: et cum dicitur dari infinita, non dicitur dari eorum numerumterminatum, 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 exmeris 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.”

From a modern perspective, Leibniz’s attitude may seem inexcusably tim-id. The invention of the infinitesimal calculus was one of the most significantmathematical advances of his day; and yet he was not prepared to apply itsfundamental concept to an understanding of the real world. He is perhapsreminiscent of those twentieth-century physicists who accepted the usefulnessof relativity theory and quantum mechanics on pragmatic grounds, but refusedto 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 ofhis theory of monads with the support of a mathematical concept. He quiteproperly had an overriding respect for logic; and he believed that logical con-sistency was a necessary condition of scientific truth. Unlike some modernphysicists, 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 conceptof infinity is a theological as well as a mathematical concept, and he had nowish to deny that God was in some sense a real infinite whole. But as far asmathematics and science were concerned, the concept of a completed wholewas 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, alimit, or a boundary. As long as the term “infinite” was taken simply asmeaning “having no limit”, there was no problem; and Leibniz always used itin this sense. The problem arose only with the introduction of the notion ofa limit – for example, the last term of an infinite series,Although certain convergent infinite series can be said to “tend towards a limit”, theyare still unlimited in the sense that there is no limit to the number of their terms. the highest or low-est number, or the smallest part of a continuous line. But there could be noconcept 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 weretherefore possible in the real world. He held that there were many things ofwhich we had perfectly clear conceptions, but which could not exist as such inreality. In particular, no specifiable geometrical figure could exist in reality,since all physical objects were distinguished by greater or lesser deviationsfrom perfection. Geometrical figures were, in his phrase, “incomplete no-tions”. Similarly, the laws of mechanics and dynamics were incomplete, sincethey left out of account the infinitely complex means by which energy wastransferred from one object to another. To assume that there were forces innature exactly corresponding to the abstract terms of the laws of motion wasan unwarranted reification which explained nothing – for example, Newton’spostulation of a gravitational force as a substitute for an account of what reallyhappened, in terms of the transfer of energy between intervening particles ofmatter. Thus Leibniz wrote as follows to de Volder:

These and innumerable other such considerations clearly show that thetrue notions of things are utterly subverted by this new philosophy, whichmakes substances purely out of material or passive elements. Things that aredifferent must differ in some respect, or have in themselves some specifiablediversity. It is amazing that mankind has failed to take advantage of such anaxiom, along with many others. But people are usually not interested in rea-sons, since they are fully satisfied with pictorial imagery. This is why thecause of true philosophy has been harmed by the introduction of so manymonstrous 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 merelypassive, motion taken in the mathematical sense, etc. In these cases peoplecan imagine things as being distinct without having any point of distinction –for example two equal parts of a straight line. This is possible because astraight line is something incomplete and abstract, which is what we need toconcern ourselves with for theoretical purposes; but in reality any givenstraight line is distinguished from any other by what it consists of. This iswhy it is impossible for two real objects to have exactly the same size andshape simultaneously. Even things which are in different positions mustexpress their positions, or the things surrounding them; so it is not the case, asis 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 arenothing other than the incomplete mental constructs of philosophers whohave not penetrated deeply enough into the natures of things.Winter 1702/3, GP II, 249-250: “Haec et innumera hujusmodi satis indicant verasrerum 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 haberediversitatem, mirumque est manifestissimum hoc axioma cum tot aliis ab hominibus adhibitumnon fuisse. Sed vulgo homines imaginationi satisfacere contenti rationes non curant, hinc totmonstra introducta contra veram philosophiam. Scilicet non nisi incompletas abstractasqueadhibuere 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 lineaerectae partes aequales, quia scilicet linea recta aliquid incompletum abstractumque est, quoddoctrinae 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 simulaequalia 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.”