AS IMPERFECTION IN THE DIFFERENTIAL CALCULUS

Velimir Abramovic

(Excerpt from the study “The Problem of Continuity in the Natural Philosophy of Leibniz and Boscovich”, Author: Velimir Abramovic, Scienza e Storia No. 14, (Rivista del Centro Internazionale di Storia dello Spatio e dell Tempo), Brugine (Padua) , CISST , MMI)

We shall try here to draw more closely together the questions, problems and difficulties which Leibniz encountered as a mathematician with those he met as a philosopher.

It should be emphasized immediately that Leibniz was not consistent in his comprehension of the fundamentals of the infinitesimal calculus. In the first two treatises (published in Acta eruditorum in 1684 and 1686), Leibniz started from the assumption of quantities which are actual infinitely small quantities while later he tried to eliminate them and to present his infinitesimal as an immeasurably small finite quantity1.

The basic principle of the infinitesimal calculus was derived from the method of exhaustion: a tangent is taken as the limit of a secant and a curved-line figure as a limit of a straight-line figure. However, the infinitesimal calculus not only assumes infinitely small quantities of the first order; it also implies the assumption of the existence of an infinite series of infinitely small quantities. In other words, the infinitesimals into which a finite quantity is divided are not simply small indivisible unities, they are also capable of being infinitely divided into smaller infinitesimals, i.e. infinitely small quantities of the second order. Consequently, there are also infinitely small quantities of the third and higher orders. In order to enable something to tend towards a limit, Leibniz assumed that "... an infinitely small quantity of the first order tends towards a finite quantity, and, generally, any infinitely small quantity of a higher order tends to zero when compared to an infinitely small quantity of a lower order, which is infinitely large in comparison to it." (Leibniz, Acta ..., 1684). Leibniz called the infinitely small quantity a differential and the limiting differential value between two quantities a differential coefficient2. By means of this notion, Leibniz simultaneously discovered the functional character of the differential calculus: two quantities depend on one another when the variation of one (the dependent variable) depends fully on the variation of the other one (the so-called independent variable). Accordingly, if the increment in the independent variable is infinitely small, the increment in

the dependent variable will also be infinitely small. Of course, this is valid only for continuous functions where the ratio of two infinitely small increments may be taken as a differential coefficient of these quantities.

If the functional dependence between two quantities with a known differential coefficient has to be found, then it is necessary to use the integral calculus, which is complementary to the differential calculus3.

Leibniz denoted the differential by dx. If x is the variable, the second-order differential would be ddx, the third-order, dddx, etc. He denoted the integral by å (Summa), the same sign which is still being used today, and took any finite quantity to be an integrated differential, i.e. an integral. In this way, by integrating, it became possible to express arithmetically the quadrature of any curved-line figure since any quantity can be considered as the sum of differentials. Accordingly, a curved-line figure can be considered as the limit of a straight-line figure.

Strictly speaking, the differential calculus is neither Leibniz's nor Newton's discovery. It was known and applied a long time ago by Greek philosophers and mathematicians. It could be stated more accurately that Leibniz and Newton recognized a need for this method in their time and translated ancient Greek ideas into modern mathematical and logical language. However, let us leave aside the task of the historians and return to the concept of infinitesimals. The concept of differentials, which exerted such a crucial influence on the creation of Leibniz's monadological doctrine, followed directly from the concept of infinitesimals and, in spite of all its deficiencies and contradictions concerning the natural order and especially the functioning of Nature (i.e. motion), incorporated the concept of Leibniz's monad. The aforementioned failure of Newton to avoid the assumption of infinitely small quantities by introducing time and velocity is by no means insignificant. We shall now demonstrate that it necessarily results from an assumption of the continuity of motion. (N.B. Leibniz also failed to solve this problem. He simply considered the infinitesimal as a "quantum" of space, i.e. of time4, and only seemingly clarified the problem in this way. The problem, however, reappeared in the twentieth century in an even more distinct form in discussions on quantum mechanics and in Heisenberg's uncertainty principle. By introducing the concept of differentials instead of infinitesimals, Leibniz did the same as Eudoxus in ancient Greece: he considered proportions instead of measures since measuring failed because of infinite divisibility. From that time on, the problem has remained essentially unsolved. In contrast to Einstein, who used the differential calculus, Heisenberg, by using matrix mechanics (i.e. matrices), returned to the atomistic mathematical ideas of ancient Greece.

Let us first consider more thoroughly the basis of Leibniz's principle of continuity:

"... The general principle of order has its origin in the infinite; ... If, in a series of given or assumed elements, the difference between two quantities can be diminished without limit, then the difference must necessarily become smaller than any small quantity, which exists in the both given and assumed series. Or ...: If, in a series of given quantities, two quantities continuously approach one another, so that finally one of them coincides with the other, then the same process must necessarily result in a corresponding series. It depends ... on a more general principle: One ordered series in the given has its corresponding series in the assumed" (G. W. Leibniz, Izabrani spisi (Selected works) pp.18-19).

Leibniz illustrated this general statement by the behaviour of points on the projection of a circle; the motion of the points on the projection depends directly on the motion of the points on the circle which is being projected. However, as we shall soon see, his example does not prove continuity but synchronicity. Further, with regard to velocity, a strict discontinuity is caused by the conservation of synchronicity.

"If a circle intersects a straight line at two points, a projection of these two points will intersect the projection of the circle (e.g. an ellipse or a hyperbola) at two points. And, the secant of the circle can be varied such that it moves more and more away from the centre and such that the points of intersection approach one another more and more until they finally coincide; in that case, the secant will go more and more out of the circle and become a tangent. Then, the projected points

of intersection of the straight line and the circle (i.e. the points of intersection of the projected straight line and projected circle) must also approach one another and, finally, when both points of intersection become one, they will also coincide. Hence, as soon as the first straight line becomes a tangent to the circle, its projection will also become a tangent to the corresponding cross-section of a cone. In this way, an important scientific proposition about the cross-sections of a cone can be proved without circumvention and without the use of figures (i.e. by spiritual insight itself). It will not be a proof, as it would otherwise be, for an individual cross-section of a cone, but a general proof." (Ibid., III On the principle of continuity, p.19-20).

Extremely serious objections could be raised to Leibniz's comprehension of the above example, especially with respect to Leibniz's subsequent criticism of Descartes' conceptions in physics, which were derived from this example. For example, if we designate the points of intersection of the secant and the circle by a and b, the point where the secant becomes a tangent by C, and the corresponding points on the ellipses (i.e. on the [ series of parallel ] projections of the circle) by (A1, A2, A3 ... B1, B2, B3 ...) and (C1, C2, C3...), respectively, and we observe the velocities, times and distances traversed by these points (in the same way that the people in the Plato's cave observed shadows), we shall be able to deduce the following, assuming a parallel series of elliptic projections of the given circle:

a) the time (T) which the points A and B take to coincide at a common point C (i.e. the time in which the secant becomes a tangent) is equal to the corresponding times on the projections of the circle, i.e. T = T1, T2, T3 ... . This means that the motions of all these points are synchronous;

b) the distances on the projections form an increasing series, inversely proportional to the series of their distances from the light source. That is, the distances are ordered so that AC < A1C1 < A2C2 < A3C3..... , BC < B1C1 < B2C2 < B3C3 ... , and the projection A1B1C1 (or AnCnBn ) is the most remote from ACB, so that in the general case we can state that AnCnBn > ACB (i.e. An-1Cn-1Bn-1 ³ ACB ) where An-1Cn-1Bn-1 is the first term in a convergent series of projected distances;

c) since the points A and B projected into the series of points A1, A2, A3 ...An and B1, B2, B3 ...Bn traverse unequal distances to C and C 1, C2, C 3... C n, respectively, in the same time, and since they move with a uniform motion, a series of characteristic velocities V, V1, V2, V3 ... Vn will be formed.

There is no doubt that the velocities obtained in this way form a discontinuous series, each term of which depends not only on the velocity of the motion of the points on the circle but also on the distance of the projection of the circle from the light source. Leibniz's assumption of continuity would only be valid in the situation where an absolutely dense series of projections could be formed. However, such a case could not be considered as a projection because it would be a situation where any extension is ignored.

This assumption of Leibniz's can be checked more simply and understandably on any projection of a circle which is not identical to it. The problem of simultaneously traversing different distances (i.e. a discontinuity in velocities) always appears. Further, the same problem can be studied on concentric circles where the points on concentric circles have to move at different velocities in order to maintain the same relative position between each other and to preserve the condition T = Tn. The velocity of the points will increase proportionally to their distance from the centre of the concentric circles. (Hence, the Special theory of relativity is not valid for rotational motion: the relative translational motion of two coordinate systems, with light traveling between them as an informer, might be comprehended as the cause of our measurement of the shortening of length in such systems. However, in systems in rotational motion it is quite the contrary. The different velocities of so called material points in such systems are needed to preserve the proportional differences in lengths between them. In other words, while the shortening of length (and consequently, the asynchronicity, derived from the formula for uniform motion, C = s/t) can easily be deduced from the distance between two systems in translational motion (uniform motion, in Einstein's case) provided that we are informed about this motion by light, which requires some time (t) to travel from one system to another, the synchronicity in rotational motions presupposes not only the equality of the traversed distances in different systems but, above all, the difference of velocities, ( i.e. the existence of a characteristic velocity of each ‘material point’). Finally, let us consider the simplest case, the case of the mathematical pendulum. The series of points formed by the circular motion behaves identically to the series of points on the concentric circles which have a common centre at the point at which the pendulum is fixed. During a single swing each of these points traverses a different arc length, i.e. in the same time they traverse different distances, and they move at different velocities. Since their angular momenta are the same (because they are moving on the peripheries of concentric circles), it may be concluded that the role of different velocities in rotational motion is opposite to that which Einstein discovered in translational motion, i.e. it is to conserve space and time relations5.

Starting from the example analysed above and the statements allegedly proved by it, Leibniz drew very significant, but at the same time erroneous conclusions: "If we now transfer the same principle to physics, we could e.g. consider the resting state as an infinitely slow motion. Accordingly, what is generally valid for velocity or slowness, must equally be valid for the resting state, being the highest degree of slowness. ... The rule for the resting state must be so formulated that it can be conceived of as a sort of corollary and special case of the law of motion. If this requirement has not been fulfilled, it would be the most reliable indication that the established rules are defective and inconsistent" (Ibid., p.20). It was exactly this opinion of Leibniz, that the law of rest must be formulated as a specific case of the law of motion, which Einstein attempted to deal with by introducing the concept of an "inertial system". However, as was already well known to ancient Greek mathematicians, philosophers and mechanists (i.e. long before Leibniz and Einstein), the problem is not how to bring two moving bodies into a state of relative rest, but how to stop a body or move it relative to another body without a leap. As early as the beginning of this century, many of Einstein's critics stated that although two bodies can be relatively at rest and simultaneously both in absolute motion, it is certain that these bodies can move relatively (to one another) while one of them is presumably at absolute rest.

Leibniz developed his thesis further: "Likewise, equality can be considered as an infinitely small inequality, as a certain difference which is smaller than any supposed quantity. By ignoring this possibility, even Descartes himself, in spite of his genius, was under an illusion when he determined the natural laws. ... let us see how he violated the above principle. Take, for example, his first and second rules of motion, as presented in his Principles of philosophy: I claim that one rule disproves the other. His second rule reads: If two bodies, B and C collide with an equal velocity, and B is larger than C, then C will return with its previous velocity in the opposite direction (to the place where it came from) while B will continue its motion; accordingly, both of them will move forward in the direction of body B. However, according to the first rule, if the bodies B and C are equal, and if their velocities are the same, after colliding they will be rejected with their initial velocity." And Leibniz continues: "Such a contradiction between the cases of equality and inequality would not be reasonable; the inequality between the bodies can be diminished more and more until it will finally be so minute that it could hardly be conceived; consequently, the difference between the assumptions of equality and inequality will be less than the smallest quantity. In such a case, according to our principle and to the natural requirements processes of the mind, the difference between effects or consequences, which correspond to the assumed conditions, must continuously decrease until, finally, it is inconceivably small. However, if the second rule were correct, just as the first rule, then the result would be the opposite. Since, according to that rule, any quite insignificant increase in body B, which was previously equal to C, would not cause, as would be assumed, an insignificantly small change in the effect, which would only increase gradually, but would at once result in a maximal change of effect: its result would be that B, previously rejected with its total velocity, would now move forward in the same direction. In other words, in a great leap it would pass from one extreme case to the other. Common sense, however, requires that B, after an insignificant increase in its size, and accordingly in its force, would be rejected at first to a less significant extent; that is, if the increase and the excess are insignificant or almost equal to zero, the rejections will be changed to a very small and insignificant extent." (Ibid., p.21.)

Let us analyse Leibniz's criticism of Descartes' propositions about collisions from the viewpoint of his understanding of continuity, infinitesimal calculus and the concept of differentials, all of which are so interrelated that they result from Leibniz's understanding of infinite divisibility. Essentially, he reproached Descartes because of his violation of the principle of continuity. The question could be raised, however, of whether Leibniz, taking into account his own solutions, had the right to do so? The leap in space (and in time, conceived customarily as a length, i.e. pictured as an equivalent of space) necessarily performed by the colliding body which stops or changes its direction of motion, is not understood by Leibniz as a leap, but rather as an infinitesimal which tends towards zero. Even if we accept his idea of the gradual cessation of motion of the colliding body, the problem remains, since it does not make any difference (it depends completely on perception) whether the body, if it does make a leap, makes a large or small leap. Descartes simply did not consider the size of the leap (and this is essentially more consistent), while Leibniz pretended to have solved the problem by considering leaps of diminishing magnitude and introducing the infinitesimal as the initial term of a series of presumably continuously decreasing distances. Of course, it is the old problem which stems from the ill-conceived assumption that space is infinitely divisible; essentially it is nothing other than the resurrection of Zeno's paradox. To the unresolved problem of infinitesimals, Leibniz adds the concept of the differential as the representative of infinitesimal relations, thereby exiting from the area of natural philosophy and entering into the mighty empire of formal mathematics. To the present day, neither Leibniz nor anyone else has answered the question of what is the magnitude of the infinitesimal of space which, divided by the infinitesimal of time, will produce the zero differential of velocity. In other words, how can it be possible that the quotient formed by two quantities that both represent finite extensions is equal to zero; or, put even more simply: up to the present day, nobody has shown, either mathematically or physically, that a body can be started or stopped continuously, i.e. without a leap, even though a very small one. (If infinite divisibility is only potential -- a necessary assumption in infinitesimal calculus since otherwise division would never be finished and infinitesimal calculus would always lead to a non-terminating number -- then the infinitesimal and, consequently, the ratio of differentials of space and time can never be equal to zero. Hence, a moving body, according to the officially adopted mathematical interpretation can never cease to move, even relatively. Unintentionally, we have arrived at another important assumption of the differential calculus: the assumption of perpetual motion taken in its absolute sense. This assumption is deduced from the fact that neither infinitesimals nor differentials can be equal to zero. On the other hand, stillness follows from an assumption of infinite divisibility, again in its absolute sense, i.e. motion is extinguished as such. This real problem is circumvented by admitting, sometimes unconsciously, the two-fold nature of the infinitesimal: it can be a quantity (i.e. larger than zero) and simultaneously, it can be no quantity (i.e. equal to zero). The same applies to the differential. This illustrates how the assumptions of infinite divisibility and perpetual motion collide in the application of the differential calculus to physics).

By introducing a potential for infinite divisibility into the solution of Zeno's arguments against motion, Aristotle made the following two implicit assumptions:

a) a moving body must finally cease to move, since space it is not actually infinite; and

b) a body at rest must start moving in space exclusively in a leap, since space is not actually infinitely divisible, i.e. it is not continuous. (According to Aristotle, only the limits, the perases are continuous, so that the moving body, in performing a motion continuously, must in fact leap over extensions and "jump" from one limit to another; if we, however, suppose that Aristotle in fact considered that the limits of space extensions (since they are adjoining) build the uniform continuum and, hence, that the motion of bodies in such a cosmos is continuous, we would obtain two very clear contradictions: (1) a body in a continuum where the limits connect exclusively could not be distinguished from them and, consequently, it could not be determined what is a body and what is not; and (2) motion and rest in such a cosmos would be equal in the sense of being absolutely identical, since the relative character of the state of rest would be eliminated by the assumed uniform continuity).

Let us consider now the consequences of actually assuming infinite divisibility (of course, in the sense of the above considerations of the application of differentials to physics, i.e. to motion):

a) a body at rest can never be started (Zeno's proof); and

b) a body in motion can never stop (for the same reason that it cannot be started if at rest; in other words, the infinite divisibility of space means that, in order to traverse the smallest distance which separates it from the point at which it will stop, a body would perpetually travel according to the series l/1 + 1/2 +1/4 + 1/8 + 1/16 + 1/32 + ... to infinity. This is a series with an infinite number of terms, which must be infinitely summed to obtain the result of two).

In fact, the assumption of actual infinite divisibility not only denies the possibility of a body at rest being started but also, in the same way, the possibility of stopping a body in motion. Consequently, motion and rest diverge theoretically to such an extent that, under this assumption or any idea which subsumes it, it is no longer possible to establish a causal relationship between motion and rest . Our conclusion is -- and we shall discuss it in detail when considering the same problem in Boscovich's theory of natural philosophy -- that motion is exclusively of a discontinuous nature (i.e. it occurs in leaps) and, accordingly, space must be discrete.

From an analysis of the consequences of accepting the idea of infinite divisibility, either potential or actual, it follows that Leibniz's (and Boscovich's, as we shall see later) law of continuity (lex continui) contradicts an understanding of continuity itself. Moreover, the continuum could not have been established on the sole basis of the law of continuity since it cannot be constructed from parts (i.e. from discretums)6 and since Leibniz's law of continuity regulates the continuity of successive discrete parts (a body approaching a certain limit has to traverse a certain distance not only in space but also in time, i.e. it will traverse the distance in a future period)7. This means that the concept of the continuum is transferred from the discretum conceived empirically to the continuum cognited intuitively; in other words it is an empirical law, derived by incomplete induction.

The mathematical concept of infinitesimals and the physical concept of atoms also collide, since the idea of infinite divisibility (i.e. Leibniz's differential) is in disagreement with Leibniz' own statement that aggregates can be decomposed into simple substances (atoms) -- elements of things, as he called them. From the idea of infinite divisibility, which is assumed in the concept of the differential, it follows that aggregates can be divided into subaggregates, which can be divided further. However, that which is composed, always remains composed even after infinite division. On the other hand, physical atomic theory assumes explicitly that there exists a limit to divisibility, i.e. there are some final parts of aggregates which must have extension. To claim that an extensive aggregate can be divided into indivisible parts, as Leibniz stated, would be the same as claiming that the process of infinite division has been finished and that we have actually, after much labour, reached zero. This is an obvious inconsistency between mathematical and physical atomism. Leibniz deliberately attempted, by using indistinctly (i.e. ambiguously) defined concepts, to introduce mathematical atomism into physics. In other words, is the question "does the division of a physical aggregate lead one to monads (i.e. mathematical atoms)?" the same as the question "does the division of a line lead one to points?"? (Just as points are not parts of a physical length, so monads cannot be parts of an extensive physical aggregate.) And, if by dividing an extension perpetually we obtain only new extensions, so by aggregating inextensions we always obtain an inextension. Division (or aggregation), taken as the relationship between extension and inextension does not make a transition from one to the other possible. A far better conceptual solution is the one in Euclid's geometry (already mentioned) where the inextensive is conceived as the absolute limit of the extensive, i.e. all discreteness and extension is encompassed by one inextension -- the continuum or the point which is being infinitely multiplied while essentially remaining the same, not only qualitatively but also quantitatively. Here, in Euclid's third definition -- the extremities of a line are points -- the natural, undistorted principle of the identity of the indiscernible is expressed. That is, if the point as an inextension is the limit of all extensions, then a) it cannot divide them (and hence Archimedes' observation, i.e. his formulation of the experience of mathematicians and physicists, that "the continuum is the sum of indivisible (but extensive - V.A.) parts" is valid). Accordingly, every length begins and ends in characteristic points and cannot be further divided regardless of its size. Further, any length shorter or longer than the given one is a new part of the continuum, which is also indivisible.

Such an interpretation would connect the role of the point in Euclidean geometry with Euclid's own conviction that his elements are real. And, b) since inextension is the universal limit of all extensions, considered from the viewpoint of extension, all inextensions would coincide, thereby "forming" (this expression is inadequate and should be taken only conditionally) the uniform continuum.

Finally, let us consider the mathematical rationale for the inextension of Leibniz's monads. The geometrical interpretation of integrals leads to the conclusion that the elements being integrated are extensive (in contrast to the concept of mathematical atoms), just like aggregates (physical, i.e. geometrical) because of the following. If we integrate inextensive elements, then the integral (analogous to the physical aggregate) will have to be inextensive, i.e. the geometrical figures would not have an explainable basis for their dimensionality. It would contradict the assumption that dx/dy can never be equal to zero. Hence, considered statically, the concept of differentials is inconsistent with the statements of mathematical atomistic theory because dx/dy ¹ 0 and mathematical atoms are inextensive. Considered dynamically (with an emphasis on the continuous) and conceived as the mathematical expression of a physical process, the limiting differential series is not only inconsistent with the assumptions of physical atomic theory (since, being based on infinite divisibility, it excludes the possibility that any initiated process will ever be finished, even relatively; i.e. it denies the attainability of any goal projected forward in space and time), it is also opposed to its own meta-assumption of a prior given infinity: to demonstrate how a moving body stops by using a differential convergent series, it must be conceded that the expression dx/dy = 1 becomes dx/dy = 0, i.e. in geometry, conceding dx/dy = 0 (the transition of the side of a polygon into the arc of a circle) in order to enable the transformation of dx/dy = A (the side of polygon) into dx/dy = B, where B corresponds quantitatively and qualitatively to the arc above the infinitely small side of a polygon which has an infinitely large number of sides, i.e. when the circle and the inscribed polygon finally become equal. Of course, it should be emphasized that there is no reason for such logical leaps and conceptual alterations in the mathematical operation of differential series. (It is interesting to observe that the integrative method, from which the differential calculus originated, implicitly denies the dialectic conviction of "modern times" that the transition of a quantity into a quality is possible)8.

On the other hand, the concept of integrals is consistent with physical atomic theory, i.e. with the concept of extensive atoms, and is opposed to the idea of mathematical atomism, i.e. inextensive atoms of extension. The integration of inextensive monads does not explain the existence of extensive matter; an integral of inextensive elements (whether mathematical or physical) is inextensive itself (only one such integral -- the physical continuum -- is a real existing integral). The integral of extensive elements is itself extensive, in both physics and mathematics. And, as already mentioned, inextensive atoms cannot be obtained by differentiating an extensive aggregate.

The above contradiction could, in our opinion, have been resolved by assuming that a part is potentially a whole, i. e. the discretum is potentially the continuum, in accordance with the fundamentals of Euclid's geometry. Consequently, the continuum itself could potentially be differentiated into extensive parts, remaining as their absolute limit. Only a sole, physical continuum would be actual. (This conception would be opposed to Aristotle's doctrine of potential infinity).

Accordingly, differentiation of the continuum would be potential, but the limits of the parts would remain actual. Consequently, integration of extensive parts, with regard to their actual limits, could also be conceived as actual, since integration is nothing other than the integration of limits. Hence, the actual integral of potentially separated actual limits must be extensive9.

There is no doubt that the task of ontology is to discover which law generates the relatively independent parts of the uniform physical continuum and which consequence of that law determines the differences between these parts. The discovery of that law would mean that Einstein's dream about the foundation of physics becoming a real ontological science would be fulfilled.

Proposition 16: " We experience in ourselves the multitude in a simple substance when we ascertain that even the smallest thought, of which we are conscious, comprehends the diversity of the represented content. Hence, all who accept that the soul is a simple substance, must accept the multitude in a monad; ..." Ibid., p.51). In referring to the experience of the multitude in a simple substance, which does not enter into the definition of a monad, Leibniz deviated from consistency and jumped without warning from the assumed actuality of monads to their virtuality, (if the monad is assumed to be a physical atom and hence indivisible, i.e. actually simple, then the multitude inside it, in the absence of an additional definition, can be only virtual, or potential).

Proposition 8: "... Monads must have some peculiarities, otherwise they would not even be beings. For, if simple substances did not differ from one another in their peculiarities, there would be no possibility of determining changes in things since that which is in a compound can only originate from simple constituents; accordingly, monads without qualities would be indiscernible from one another -- as they also do not differ quantitatively -- and consequently, assuming the fullness of space, every place would always, in motion, accept only a content equivalent to what it had before and one state of affairs would be indiscernible from another." (Ibid., p.50.) If one monad does not quantitatively differ from another, then it is not clear why they should be qualitatively distinguishable if each monad has been defined in the same way. The possession of qualities as accidental properties, by which monads could be distinguished, would necessarily require a change in the definition of the monad, since a) a monad with more than one quality would no longer be simple, and b) distinguishing a multitude of qualities in a monad, regardless of the fact that they are internal characteristics, would introduce the concept of parts into the monad itself.

(At this point, we should remind ourselves of Aristotle's concept of individuals, which are, according to him, self-identical and simple -- corresponding to Euclid's concept of unity, i.e. to any one which is one. Even if we accept that one can correspond to individual as it is self-identical, it can by no means be accepted that the individual is qualitatively simple because the individual is a form of essence and, hence, is qualitatively compound. Leibniz makes completely the same assumptions in ascribing qualities to his quantity-less monads. In fact, quantity as a form of essence is also a quality (or characteristic) of the individual, like its very essence. Therefore, pure quantity (without content, as pure form) is impossible, while pure quality actually exists as pure essence (pure unity) without form, i.e. as the physical continuum. Since quantity is impossible without quality, and since quality without quantity is identical to pure essence, Leibniz's monads, imagined as quantity-less carriers of a multitude of qualities, are also impossible in the same way that Aristotle's qualitatively simple individual is impossible. It is the same problem as considering how the parts of the continuum attain independence. If we start from the statement that the continuum is pure quality, while quantities are potential aspects of that quality, then quantity as a univocal notion does not exist at all, but is only the name for a derived quality; in other words, Aristotle's individual, as a derived quality, is not simple).

According to statements (a) and (b) above, the absence of quantitative differences between monads implies an absence of qualitative differences, since a qualitative distinction between quantitatively equal monads would imply differences in their internal structures (i.e. the differentiation of forms within a monad). As we have already concluded, pure continuous homogeneous essence is possible without form, but pure form is not possible. (Form is not expressed, as such, if it is not a limit of the discretum and, as we have demonstrated, form must originate from the continuum). Leibniz's concept of continuous change is neither sufficiently explained nor thoroughly and precisely discussed. (If continuity is related to essence and change to the form of that essence, then monads cannot be varied for the simple reason that, according to their definition, they have no form.)

Propositions 11 and 12: "It follows further that natural changes in monads result from some internal principle, since an external factor could not exert an influence on its interior." (Ibid., p.50.) However, "apart from the principle of change, there should be some detail to that which is being changed to constitute, so to say, a peculiarity and the diversity of simple substances." (Ibid., p.50.) At this point, Leibniz elaborates the concept of the multitude in unity, a particularity of variation in each simple substance; i.e. in proposition 10 he claims: "I take as granted that every created being, and consequently a created monad, is subject to change and, what is more, that change is continuous in each of them." Does a specific possess a form or not? Is a monad something specific? If a monad is something specific, as Leibniz thought, and if it has no form, as it should be treated judging from the above, how can it be distinguished from a simple substance (which is, according to Leibniz, also simple and formless)? Further, as we have already noted, not only does the monad have no windows, it cannot have any walls either. Consequently, it is not closed in itself because it has no exterior. But, according to its definition, the monad has no interior, no form and no limit. How is it possible then to speak about an internal principle of monads?

Proposition 13: "This detail (mentioned in proposition 12 - V.A.) should compose the multitude in unity or in a simple substance: since every natural change occurs gradually, something is being changed and something remains. Consequently, in a simple substance there should be a multitude of affinities and relations, although there are no parts." (Ibid., p.50.) In this proposition, the concept of gradual change in a monad, which has no form and hence no internal limits, is not clear. Gradualness, as an idea, can be conceived only through the concept of previously separated discretums.

Proposition 14: "A temporary state which includes and assumes the multitude in unity and in a simple substance is nothing other than that which is called perception ..." (Ibid., p.50). Any temporary state presumes a difference between two states. However, there is not only no difference between monads and, hence, the multitude in unity is impossible, but, as there is no closed form in a monad, Leibniz's perception is also impossible since there are no forms by means of which any change could be ascertained.

"Hence the simplicity of a substance does not restrict the multitude of modifications which must coexist in a simple substance; and they must consists of varieties (dans la variété) of relations with things outside. It is just the same as at one centre or point; in spite of its simplicity, there is an infinity of angles made of straight lines which converge upon it" (Gerh. VI, 598). In Leibniz's opinion, straight lines and angles come out of the point but, as the point has no dimensions, this statement is itself not quite clear. It seems that this conviction, originating from everyday experience, can only be theoretically affirmed by erroneous reasoning, i.e. an inaccurate interpretation of Euclid's first and third definitions. Even Greek mathematical atomic theory fails in this case because, if the ends of a line are points, it is clear that the line is contained in the point and not vice versa. Consequently, the statement that straight lines and angles come out of the point must be proved separately.

In assuming the variability of monads in proposition 9, Leibniz simply postulates: "Each monad must be different from any other monad because two beings in nature are never completely identical; if this were not so, it would not be possible to find any internal differences or any differences based on internal determinations" (G. W. Leibniz, Ibid., p.50). This is inexplicable since the differences, which we perceive in nature as the relationships between things (i.e. the lack of identity of things), are by no means identical with the differences between monads. In fact, we do not perceive monads at all. Consequently, the concept of the variability of monads, based only on a statement about their individuality which contradicts the initial definition of a monad, is not valid in itself (and it cannot be substantiated by claiming that "there are never two beings in nature completely equal one to another")12.

Proposition 43: "God is not only the origin of existences but also the origin of essences.... For God's mind is the realm of eternal truth, or the ideas upon which it depends, and hence without him there would be nothing real in possibilities, and not only would there be nothing existing but also nothing possible". (Ibid., p.55). The assumption of a God who serves as the origin of a number of essences accords with Leibniz's postulate about the pluralism of simple substances. However, the question of whether various essences really exist remains, i.e. is the essence of a ultimately different from the essence of b? If we adopt Aristotle's viewpoint that the limit is indivisible and that it unites, then the essence or essences of all the substances, both simple and compound, would be united into one and the same essence. Accordingly, the proposal that God is the origin of the multiplicity of essences is a hypermetaphysical hypothesis and cannot be proved in our universe.

Proposition 61: "... Since all is a plenum - meaning that all matter is connected together - and since, in the plenum, every motion exerts an effect upon distant bodies in proportion to their distance and since, therefore, each body is not only interconnected with other bodies in its close vicinity and somehow feels everything that happens in these bodies, but also, through their mediation, is affected by all the processes which take place in yet other bodies which are in contact with these bodies in their close environments, it follows that interconnection extends to any distance. Accordingly, every body feels everything that happens in the universe; hence the one who sees everything is able to read everything that happens in the universe in each individual, including what has occurred and what will occur, and to recognize in the present that which is temporally or spatially remote..." (Ibid. p.58) . This famous proposition of Leibniz confirms Spinoza's sentence "Omnis determinatio est negatio", since Leibniz, emphasizing general interconnectedness in space (which is to him the natural order of coexistent phenomena), denies the correctness of his comprehension of time -- "time is the natural order of the non-simultaneous appearances". Cosmic spaces, divided into layers by time, cannot coexist because of succession, i.e. they cannot all exist in the same space in the same cosmos. Accordingly, it is necessary to assume that worlds with no interconnection other than time also exist. It follows that these worlds, ordered according to time, should also possess their own independent spaces. In other words, they would not be able to coexist, i.e. they would not be able to influence each other in the same space. By means of proposition 61 (i.e. general interconnectedness), Leibniz denies not only the assumption of an order based on time (i.e. the existence of an a priori future and past) but also his assumption of the pluralism of simple substances (that each monad is an individual world); general interconnectedness only results from the assumption that time is reduced to the present moment in which all matter coexists in the same space, filling it completely. The generally interconnected cosmos is the only one13.

Although, in our opinion, there can be no details, specifics or varieties in the simple, Leibniz held that, with respect to combinations of monads in a series, (mathematical) operations are laws relating to the continuation of series: "legem continuationis seriei operationum suarum". He established the connection between an infinite series, the law of continuity and substance in the following way: "From the derivative force, which represents any term of a series, the whole series can be reconstructed, i.e. the primitive force, f(x), dx or x2: that is the law of the series and that is substance"14.

Accordingly, the integral, as the law related to the terms of a series, is at the same time a unique law of substances. This proposition implies difficulties, which have already been mentioned, in understanding the integral and its application: from the point of view of continuum, the infinite series cannot be considered as an infinite continuation of discretums (since the assumption of the infinity of a series must be preceded by an assumption of the infinity of the continuum) and the series itself must be discrete. (The same difficulty remains to the present day, e.g. in the wave, i.e. quantum mechanics: it is not possible to derive the wave function of light from spectroscopic observations and measurements, i.e. by inductive generalization, since the observed substance is always discrete whereas the wave function is always continuous. This problem, together with other similar problems will be discussed in the concluding treatise). However, since the continuum itself has no limits (not even in its interior), the assumption of a continuum excludes the possibility of the existence of a series with separate terms in Leibniz's sense of the concept, i.e. as actual parts of the integral of substance. Obviously, a law which could differentiate series in the continuum would necessarily differentiate the terms of these series into sets of equal terms, thus producing sets of the next higher order and of even higher orders; hence, the application of Leibniz's law of the series means that a continuous series becomes impossible. The natural numbers are an example: Let law a differentiate the continuum into a series of equal unities. If we leave the series at that level of differentiation, there will be no limits (discontinuities) between the unities, and, essentially, the continuum will still not be differentiated. However, these unities associate themselves according to the same law into the sets 2, 3, 4, 5 ... etc., thereby resulting in discontinuities and, in fact, a series is being formed at this moment. But the series, according to Leibniz, must always be a series of aggregates, of agglomerated unities (because of the properties of the limits of discretums), otherwise it would not be an actual series. A series of equal parts is only possible virtually, as the basis for the formation of a real, actual series of a higher order. Accordingly, the formation of the series of natural numbers can be expressed by a simple law: n +1.

Leibniz's approach to the problem implies the existence of a law which would be more general than the law of a series, i.e. a law which would determine the divisibility of the continuum itself, in itself; since if the integral expresses the law of a series (i.e. how monads associate into aggregates), then the integral of an integral (which we shall call the crown integral) would express the divisibility of the continuum itself.

It could be concluded that Leibniz's combinatorics of substances would be substantiated, both mathematically and ontologically if, instead of assuming the pluralism of simple substances, we assumed their unity, i.e. if we introduced:

a) a law of the division of the continuum into different unities which would simultaneously represent the initial terms of an infinite series of identical unities;

b) sub-laws (as consequences of the basic law) organizing the unities within each series of identical unities;

1. inter-series laws which would be valid: (1) within subseries of each initial series of identical unities; and (2) between series and subseries whose initial unities are quantitatively different.

Otherwise, Leibniz's concept of the pluralism of simple substances would remain as an unproved projection onto the continuum. Further, by deriving the world from a combination of the parts of an actual infinity, the problem of overcoming an intrinsic inherent infinity (i.e. the impossibility of reaching the last term in the series of a mathematical progression) is solved by using the law which gives rise to the infinite series. Is there not really a serious contradiction in holding that infinity is not all-encompassing, so that, to conceive it, we must add some more infinity to infinity? The aporia really results from the assumption that it is the substance that is actual instead of the laws. It results from incomplete human reasoning from experience to cognition, as though substance were controlling the laws of nature and not vice versa.

Finally, let us conclude: Leibniz's attempt (idea) to construct an infinite series from previously differentiated terms (monads) eliminates time from mathematics and, consequently, makes it impossible to give an exact physical interpretation of basic mathematical concepts (i.e. the point, the straight line, zero, infinity, unity, etc.).

In his study Über Leibnizens Methode der direkten Differentiation publ. in Isis, 1934, B. Petronijevi} revealed that Leibniz's infinitesimal calculus was in the beginning exclusively based on infinitely small actual quantities.

2 The differential quotient corresponds to Newton's fluxia and the differentials to his "moments".

3 The dispute between Newton and Leibniz about the discovery of the infinitesimal calculus blazed up with such intensity that even indecent words were used, indicating the extent to which it was important to both of them. With regard to this issue, Petronijevi} is fully on Leibniz's side. We shall only quote his conclusions without going into the arguments about his historical evidence which would require an investigation in the European archives. "Leibniz made his great discovery in 1676, fully independently of Newton. ... Shortly afterwards, there appeared a criticism of Newton's Principia in the Acta eruditorum in which Newton was sharply criticized for his assumption that gravitation is a force acting at a distance. The complete identity of Newton's infinitesimal method with that of Leibniz was emphasized. (This review was not signed, but it was written by Leibniz himself.) It was also emphasized that the fact that Newton himself conceded priority to Leibniz. ... Leibniz not only published his discovery earlier than Newton, ... but his discovery is not quite identical to that of Newton. It is only identical in its results, i.e. both methods have one and the same goal but Leibniz's method is more rational than that of Newton. Newton made his discovery 1666 and 1667, i.e. much earlier than Leibniz. However, in his method he wanted to circumvent (unsuccessfully) the assumption of infinitesimally small quantities by introducing the concepts of time and velocity, while Leibniz had not considered these quantities in the process of creation but as existing and complete, and as the sums of infinitely small quantities. The independence of Leibniz's discovery has been proved, in spite of a contrary decision by the Commission of the Royal Society, by the subsequent development of mathematics: the infinitesimal method was developed and perfected by Leibniz's supporters and not by those of Newton. ... (More details in: B. Petronijevi}, Über Leibnizens ..., Ibid., pp.7-15, and also: B. Petronijevi}, Istorija novije fil., Ibid., pp.215-216).

4 The dualistic nature of Leibniz's infinitesimal lies in the fact that it is considered as a limited infinity, i.e. it is both actually and potentially unlimited at the same moment. This vagueness was inherited from Aristotle.

5 In addition to incomplete conclusions about the motions of points and the consequences of these motions, Einstein also took over the concept of vis viva (mv2) from Leibniz. To Einstein, vis viva is total energy of a physical system (mc2), which is undetermined and without form, just like Leibniz's vis viva.

6 If infinite divisibility existed in nature, it would also be logical that the continuum could easily be constructed from an infinite number of parts resulting from such a division. However, in that case, because of its assumption, it would be prior to itself: both in its decomposition into the continuum and in its composition into the continuum; it has, in fact, already been assumed.

7 To Leibniz, "Tempus est ordo existendi eorum quae non sunt simul" (Time is the order of the existence of that which is not simultaneous", and "Spatium est ordo coexistendi" (Space is the order of coexistences). The criticism of Leibniz's "lex continui" is appropriate to his comprehension of space and time. Nowadays there are different opinions about it, but the problem is no longer studied in its original form. However, we are convinced that philosophy, in seeking an ontological origin in common with that of mathematics, will have return at least once more to ancient Greece.

8 This issue could be elaborated more extensively but it would be beyond the scope of this study. The law of the transition of quantity into quality and vice versa was formulated by priests and alchemists in ancient Egypt, in the same form as it was adopted by the Marxists; it also appears in a great number of cryptographs of the mystics (Neoplatonists) of the early Middle Ages. It is generally considered that the law originated from the great magician Hermes Trismegistos (the emerald tables of occultism) and it can also be found in the papers of Maimonides and Albertus Magnus. Paracelsus, among others, also mentioned it.

1. The derivation of parts as potential elements of the actual physical continuum would be in accordance with numerous philosophies which attempted to explain the world completely, e.g. Plotin's idea of emanation or Hegel's idea of nature as alienated absolute spirit. Because, to establish somehow the relationship between the integration of inextensive elements and the extensive integral, we have to conceive the extensive elementary discretum as a potential part of the actual inextensive continuum. Only then does it become clear why the continuum cannot be composed of parts, i.e. why it cannot be decomposed into parts. What could the limits of the continuum be if not the continuum itself?

10 Die philosophischen Schriften von G. W. Leibniz, herausgegeben von C. J. Gerhardt, Berlin, 1875-90.

11 Of course, this expression should be taken conditionally, since inextension, in our opinion, cannot be related to unity. This must always be emphasized because it is the most serious error of mathematicians and philosophers. For example, in Cantor's series of natural numbers, Cantor equates the number (i.e. quantity) with the inextensive geometrical point.

12 Although Leibniz's concept of force should be studied separately, this concept, with the exception of the term mv2 which expresses the continuum of living force (i.e. energy of no particular sort -- neither thermal, nor atomic, nor electrical, etc.), is not crucial to his concept of continuity. However, it should be noted that Leibniz, probably drew his idea about the internal active principle in monads from the proposition: praedicatum inest subjecto. He held that the pure forms of the ancient philosophers, the entelechies, are nothing other than forces. Leibniz originally began the twelfth proposition of the Monadology as follows: "And generally it can be stated that force is nothing other than the principle of change" (Robert Lata, Monadology, Oxford, 1898, p.233) but, later on, he altered the initial text.

13 Imaginatively, the monad can also be conceived as "a centre which expresses an infinite circumference" although, of course, numerous objections could be made to this idea. Relieved from the concept of the pluralism of substances, but adopting Leibniz's concept of general interconnectedness, Fichte wrote: "At any moment of its duration, Nature is a connected whole: at any moment, each part must be what it is because all other parts are what they are. ... Accordingly, my connection with the totality of Nature is what determines what I was, what I am and what I shall be ... (Werke, II, 178. ) To mark this universal interconnection of each part with the whole, Leibniz usually uses the terms "conspirantia" i.e. "tout est conspirant".

14 This is again confirmation of that which we have already concluded and that which Leibniz elaborated in greater detail in his Brevis demonstratio, i.e. it seems that mv2 is Leibniz's relationship for transforming matter into the continuum . D'Alembert used Leibniz's relationship in his Tractat on dynamics (1743) in the form of mv2/2, defining in this way a measure of the work done by a living force. Later, the concepts of potential and kinetic energy are derived from Leibniz's vis viva