© Velimir Abramovic

GEOMETRY, TIME AND THE LAW OF CONTINUITY

(*Theoria Philosophiae Naturalis* ...^{1} and Boscovich’s synthesis of the continuous and the discontinuous;

Criticisms of Boscovich's concepts of motion space and structure of matter)

Author: Velimir Abramovic

"The law of continuity ... consists of ... that each quantity, in experiencing a transition from one magnitude into another, must pass through all the intermediary quantities of the same sort. This is usually expressed by stating that the transition occurs through intermediate stages. Maupertuis considered these stages to be a number of minute additions occurring in a moment of time. He thought that the law must be violated at the same time for it is violated equally by the smallest leap or the greatest leap, since the concepts of small and great are quite relative. He was right if the term 'stage' denotes an instantaneous increment in any quantity. Really, it should be so conceived that particular states correspond to particular instants, whereas increases or decreases correspond exclusively to very small continuous intervals of time" (par. 32, p.13).

The basic contradiction which follows from such a formulation of the law of continuity, which was recognised by Boscovich himself, is *the distinction between continuity as an attribute of the small inetrval of time (tempusculum) and continuity as an attribute of the point, i.e. of the indivisible limit*. It follows that there are at least two sorts of continuity: extensive and inextensive. For, if it is assumed that the tempusculum has an internal continuity, then division of the tempusculum is not possible without the internal continuity of the point which divides it (i.e. at least the potential finiteness of the point); ultimately, it follows that there can** **be** **no division, i.e. it is as hypothetical as the existence of the tempusculum itself.

Now let us consider another incompleteness (which is not only attributable to Boscovich) in the mathematical expression of the law of continuity. If we simplify Boscovich's derivation and reduce it to an obvious, elementary example, we obtain the following: Let the function y = f (x) increase or decrease in the interval between the point A and the point B. Obviously, for every x ¹
0, (x not equal to zero), the function y is either increasing or decreasing in leaps; as x is an independent variable, it can take any numerical value, including successive values. However, the series of numbers, e.g. the** **integers, is a discrete series and, if any of these numbers is added to x (e.g. as an exponent) or if x is replaced by it, then** **y,** **being** **a** **function** **of** **x,** **will** **also** **make** **a** **leap,** **either** **in** **its** **gradient** **or** **its** **value** **or** **both**. **(In** **fact,** **any** **change** **in** **x** **inevitably** **results** **not** **only** **in** **a** **change** **in** **the** **gradient** **of** **the** **function** **but** **also** **in** **a** **change** **in** **the** **value** **of** **y,** **a** **fact** **which** **is** **usually** **neglected**.**) Moreover, infinitely small increments in the value of the function (like any other increments) will be traversed at an infinitely high velocity. Only continuous changes in the value of x would not lead to leaps in the value of y , but in that case (a) x would be an infinite number or zero, and (b) the assumption that the numerical value of x varies continuously would make x and y equal. In other words, their functional relationship would be substituted by their identity; consequently, the abscissa and the ordinate would coincide and, for x = 0, the whole Cartesian coordinate system would be reduced to its origin and become the point.

This apparently peculiar behaviour of a simple function has its roots in an inadequate and deficient comprehension of the essence of number. Moreover, if we talk about induction in physics, then we must take into account the fact that the act of measurement is not a heuristic process; it only indicates a higher causal relationship, which has to be assumed, and, provided that it was correctly assumed, it should be possible to deduce a natural law from this relationship which will subsequently be confirmed by measurements.

As we have explained, a decrease, or increase, in a function is not - as usually conceived - a manifestation of the law of continuity, because the sum of the infinite number of possible distances between the points *A* and *B* (for Boscovich, this is the "transition through all intermediary stages", for Leibniz, a "transition through all intermediate quantities") is never equal to the finite distance *AB*. (*An infinite sum of finite distances is never a finite distance * because of the assumption of infinite divisibility implicit in the concept of an infinite sum; on the other hand, a finite distance can be obtained only as the result of adding a finite number of finite distances.) However, if we adopt Leibniz's conception (adopted tacitly at the present time) that infinitesimal distances are not equal to zero, then the situation will be even more peculiar, conflicting even more with above-mentioned conceptions of the consequences of the law of continuity: a function decreases or increases from one point to another in leaps, passing small finite distances at an infinite velocity in zero time. Obviously, this problem was well known to Boscovich. To avoid it, he declared his tempusculum to be continuous time, although he represented it as a segment of a straight line which is, according to its definition, a discretum. In this way, Boscovich succeeded in matching each variation in distance (i.e. in space) to an interval of elapsed time, i.e. by means of a corresponding tempusculum. Essentially, for Boscovich, it follows that all bodies move with the** **same** **velocity because, for him, *s */*t* = 1 const. (*s* = distance, *t* = tempusculum), i.e. the magnitudes of time and space are always equal. In the case of accelerated motion, i.e. when s t, i.e. the distance is not (numerically) equal to the time, the problem of traversing small finite distances at an infinitely high velocity (i.e. a velocity proportional to distance) again arises.

If it were consistently deduced from his law of continuity, Boscovich's cosmos would be completely deprived of motion (disregarding** **the** **existence** **of various directions and consequent variations in perspective, i.e. excluding the suggestion that motion is possible as a perception, i.e. only apparently) and would constitute an inertial system in which differences between the relative velocities of bodies would be equal to zero.

Consistently induced, Boscovich's cosmos would be congruent with Newton's idea that infinite continuous space and infinite continuous time are cosmic fundamentals in which matter floats. But here, as with Newton, the problem of the double continuum arises: either both time and space are finite or they are not phenomena of the same order. Neither Newton nor Boscovich could offer a solution to this problem. Finally, if we hold the conception, which today is generally accepted, that in traversing the distance from *A* to *B* one only passes through points, then the distance from *A* to *B* would not have any length since points have no dimension.

"Moments of time are represented by points and continuous time by a line segment. ... In the same way that points in geometry are indivisible limits of continuous segments of a line, but not parts of the line, so, in time, one should distinguish parts of continuous time, which correspond to parts of the line and which are similarly continuous, from instants, which are the indivisible limits of these parts and correspond to points" (*Ibid*., p.14).

The introduction of the concept of the *tempusculum* creates great difficulties in comprehending accelerated motion. It does not matter whether or not it has been assumed that a body traverses infinitely short distances in infinitely short intervals of time or finite distances in finite times, but *whether and in which sense time is considered as equal to the corresponding distance, and whether it is essentially the same as distance*, i.e. the same

"Individual states correspond to instants and any increment, however minor, corresponds to a minute interval of continuous time. ... However, leaving aside these ambiguities, the essential point** **is** **that** **the** **addition** **of** **increments** **occurs** **not** **in** **a** **moment** **of** **time but in a continuous extremely small interval of time which is a part of continuous time. ... In time, there is no such moment of time which is so close to the previous moment that it would be the** **coming** **moment; either they constitute one and the same moment or there is a continuous minute interval of time between them which is infinitely divisible into intermediate moments. Similarly, there is absolutely no continuously variable state of quantity which is so close to the previous state that it would be the next ** **state. However, the difference between these states should be ascribed to the continuous interval of time which has elapsed between them. Therefore, if the law of variation (i.e. the nature of the curve which expresses it) is given and if any increment, however minor, is given, then it will be possible to determine that minor continuous interval of time in which that increment has occurred." (*Ibid*., p.15).

The problems involved in comprehending continuity and, an associated issue, the role of mathematics in physics are also evident in the following:

a) Any increment in* x* in the function *y *=* f *(*x*) is always determined transcendentally, whether the value is chosen by a scientist or derived from nature if the function is the expression of a valid natural law.

b) The phrases "minor increment" and "minor continuous interval of time" indicate Boscovich's doubts about the concepts of the finite and the continuous (he even mentioned "the ambiguity of the concept of an** **intermediate stage", but didn't refer to this in further considerations) because the increment, however minor, is nothing other than a kind of leap, as we have previously shown. (The strength of this argument is increased even more by Boscovich's emphasis on the relativity of quantities.)

Boscovich's propositions concerning geometry are inconsistent. He held that "geometry does not recognise any leaps" (par. 39, p.16) while, at the same time, arithmeticising geometry. (It had to be well known to him, for it is an ancient truth, that arithmetic is based on the concept of the discretum, i.e. on the concept of the leap. In fact, in geometry, the act of opening compasses represents nothing other than a leap. This "leap" determines the radius of a circle or the length of a side of a square and corresponds to any finite value of *x *in algebra. For, if there were no leaps in geometry, the concept of length, i.e. of number, would be superfluous and all operations would have to be performed exclusively through the infinite^{2}.

In the preceding section on Leibniz, we have already remarked that the law of continuity conceived as a law involving the connection of intermediate quantities (i.e. the connection of quantities through continuous limits, i.e. points, as occurs, for example, in the composition of a straight line from its segments) is not sufficiently precise because it implies the existence of discretums continued to infinity, i.e. a discretum which has no outer limit and is therefore not a discretum. The continuum can be the limit of a discretum and its content in a sense (the line segment is continuous within its limits)^{3}, but it cannot be the set of internal limits of a discretum which is continued to infinity and formed by the infinite interconnection of finite quantities because that would lead to conceptual confusion. Therefore, Boscovich's conception of continuity as the** **continuous** **limit** **of** **successive** **intermediate** **stages, or, more distinctly, his conception of continuity as the perfect adhesive for the seamless merging of segments of space and time is, as a whole, not real; it is essentially mechanistic and there is no discussion of the basic ontological assumption that the continuum exists as a elemental entity. Moreover, from a theological viewpoint, Boscovich's conception of the law of continuity is a Manichaean dualism since it assumes a division of the wholeness of God's world into phenomenal discretums which obey his law of continuity and something additional; something unknown, potential and noumenal, in which these discretums exist. Being founded too superficially, Boscovich's law of continuity generates a division into continua and discretums, which leaves them ununified, in spite of the fact that a unification should have been achieved in any complete theory of continuity. Finally, if we consider the continuum itself in relation to this law of continuity, the discretum appears to have an unreal basis, and the question arises of why discretums differentiate and become independent in the continuum. On the other hand, the law of continuity as applied to discretums does not provide answers to the questions of where the continuum originated and how it was created. However, in our opinion, a corresponding law of discontinuity, starting from the continuum, would have the potential to provide an answer to the question of the origin of discretums, which we, human beings, perceive everywhere in the world.

The following two examples, one geometrical and one physical, illustrate Boscovich's conception of continuity at its best.

"A geometrical example of the first sort where we omit intermediate magnitudes. ... We form segments of equal length, AC, CE and EG, on the abscissa of a curve (Fig. 9) and raise the ordinates AB, CD, EF and GH. The areas BACD, DCEF and FEGH resemble the terms of a continuous series, such that there is a direct transition from the area BACD to the area DCEF and from DCEF on to FEGH; hence the second area differs from the first area by a certain quantity, just as the third does from the second. That is, if we make the lengths CI and EK equal to the lengths BA and CD, and if we transfer the arc BD to IK, the area DIKF will be the incremental amount by which the second area is greater than the first. It seems as if the whole of this increment appeared completely at once, such that it was not possible to observe half or any part of this increment at any moment. It is as if the transition from the first to the second area occurred without intermediate stages. However, we have here omitted the intermediate stages which preserve the continuity. Hence, if we move *ac,* which is equal to *AC,* continuously from *AC * to *CE*, then the magnitude of the area BACD will pass through all the intermediate areas *bacd *to reach the magnitude *DCEF* without any abrupt leap and without any violation of the law of continuity.

Fig. 9

"The physical example of the terrestrial day and its** **associated oscillations; The law of** **continuity** **occurs (i.e. the previous example - V.A.) everywhere that the beginning of a second quantity is separated by an interval from the beginning of a first quantity, whether the beginning is situated immediately after the end or is separated from it by some interval. That is the case in physical examples: if we conceive the day as the interval between two successive sunsets, or from sunrise till sunset, then, at some time of the year, one day will differ from the next by a great number of seconds, and it will seem like a leap, without any intermediate day which would differ by less. Take, for example, a parallel on the Earth's surface which is a continuous sequence of all the places with the same geographical latitude. At individual places, the day will have a particular duration, and the beginning and end of all those days will change continuously until the parallel returns to the place where it was the previous day, which is the first place in that continuous sequence and the last place in next one. The magnitudes of all these days change continuously without any leap. It is we who make leaps by leaving out the intermediate days, not nature. A similar response** **could be given in all the other cases in which beginnings and ends flow continuously but we observe them in leaps. For example, a pendulum oscillating in the air: any particular oscillation is separated from the previous one by a certain quantity, but both the beginning and end of that oscillation** **are separated from the beginning and end of the previous oscillation by a certain interval of time; the intermediate stages in the intermediate sequence between the first oscillation and the second would be those which would be obtained if we, after dividing the arcs of the first and second oscillations into equal numbers of parts, took the distances, or the times required for them to be traversed, which lie between the ends of all the proportional parts of the arcs, such as between one-third and one-fourth of the first arc and one-third and one-fourth of the second arc. This approach can be easily applied to all cases of that sort and it can always be directly proved that no violation of the law of continuity occurs anywhere." (*Ibid.*, par.44-45, p.21-22)

In both the examples we have quoted, it is characteristic of Boscovich to ascertain the action of the law of continuity in motion *a posteriori*, after the motion has been completed. (That is, in fact, a consequence of his chosen method of proof by induction.) But Boscovich does not explicate the essence of continuous motion by means of these examples, for none of the motions in the above examples is continuous. Both in the case of the earth's rotation and in the motion of the line segment in the geometrical example, there was a failure to either demonstrate or prove that the motion from one point to another was continuous both in time and space (we take the distance from one point to another as an interruption in continuity).

The problems are very similar to, if not the same as, those encountered in the discussion of Leibniz's pluralism of substances. For, if we assume that infinite divisibility of the discretum is possible for a certain distance (e.g. between *a* and *b*), then that distance can be divided into an infinite number of points which potentially coincide. In this way, if the infinite number of points obtained by an infinite division of the finite (determined) discretum were fused together, the result will be only one point. If we assume that the discretum is potentially the continuum because of its property of infinite divisibility, then we must assume that the actual continuum, in** **fact,** **exists** **prior to such a discretum. A direct consequence of the fusion of an infinite number of points into only one point (i.e. of the fusion of the points obtained by infinite division of a certain finite discretum) would be a denial of the actual existence of the discretum itself. Therefore, we consider that infinite divisibility is incompatible with the concept of the discretum and, even if infinite** **divisibility** **exists in general, it cannot be applied to the** **discretum.

Another problem lies in the precise relationship between a length and a point. It is impossible to determine the position of a point on a line segment since the point has no dimensions. Similarly, the division of a line segment by points is actually impossible since points, having no dimensions, cannot separate the line's parts (as a dimensionless limit of the parts, the point connects them instead of separating them; to separate them it would have to possess a certain length).

In our opinion, it is only by conceiving of the discretum as potential that a way will be left open to discriminate illusion from reality, the modal (virtual) from the actual, the real. However, as the existence of parts is essentially real because they have been derived from a real, actual totality (i.e. the continuum), it follows that there** **are** **degrees** **of** **reality **--** reality cannot be ascribed equally and** **in** **the** **same** **sense to all phenomena**. **The** **degree of reality (which is in fact a basic quality) decreases from the simply identical (i.e. from the dimensionless continuity, which is the most real)^{4} through the identical (finite parts of the continuum that are indivisible extensions or units) to the most complex phenomena. The maximum achievable by human knowledge and understanding of the cosmos, as it appears at the present time, is an awareness of the continuum and its internal emanations^{5}. (It is the task of science to discover the laws applicable to these emanations, i.e. the laws of self-identical discretums, and it is the task of technology to exploit them.)

After these geometrical and physical proofs we shall also cite Boscovich's metaphysical proof of the law of continuity:

"The continuum only has one limit, as in geometry, ... this results from the very nature of continuity ... as Aristotle himself remarked ... and in it there must be a common limit which connects the preceding with the succeeding and, therefore, it must be indivisible, for that is a property of the limit ... A surface which separates two bodies has no thickness ... therefore the immediate transition from one side to the other occurs in it.... Two consecutive** **continuous indivisible and inextensive points cannot exist without some mutual interpenetration and some merging ... Likewise, this must be the case with time, so that between a previous continuous time and that which immediately follows there is only one moment, which is the indivisible limit of both and, therefore, ... there cannot be two consecutive** **connected moments, but between them there must always be a continuous time which is divisible to infinity" (par. 47-49, p.22-23). In defending the idea of the tempusculum as a continuous moment of time (to prevent the contraction of the whole world and the cosmos into one point, for if two points coincide, everything will collapse), Boscovich argued further: "If the line of motion were somewhere interrupted, would the moment of time, in which the motion would have taken place, at the first point of the second part of the motion line follow the moment, in which the motion would have taken place, at the last point of the first part of the motion line, or would the** **first** **moment be the same as** **the** **second** **moment or would** **it** **precede** **it? In the first and the third cases, there would be some continuous time between these moments which would be infinitely divisible into other intermediate moments, since two moments of time, conceived in the sense I conceive them, cannot be continuously sequential ... Therefore, in the first case, the body would have been nowhere during all these infinite intermediate moments; in the second case, it would have been in two places at the same moment and, hence, it would have been replicated; in the third case, replication would have occurred, not only with regard to two moments, but also with regard to all the intermediate moments in which the body would have occupied more** **than** **one** **place. However, since an existent body cannot be without being somewhere and since it cannot be in several places simultaneously, the change in route and that sudden leap cannot occur ... and the distance of one body from another cannot be varied in leaps ... for it would be at two distances at the same time ....

"The objection which results from being and non-being merging during creation or annihilation:^{6} ... The creation or annihilation of any thing is impossible. If the end of a preceding series has to be merged with the beginning of the following series, in the very transition from *non-being* into *being,* or vice versa, both will have to be merged into one and, hence, both will simultaneously be and not be, and that is absurd. This is the answer. The real limited series, which exists, must have real transitional and final points which, similarly, actually exist, and not points which are nothing and do not possess the properties which the series requires. Therefore, if a series of real states is followed by another series of real states and if they were not connected by a common limit, then there would be two states at the same moment and these states would be two limits of the same series. And, since non-being is in fact the same as nothing, such a series would not require any final limit. It would be immediately and directly excluded by *being* itself. Therefore, in the first and last moments of that continuous duration in which the thing exists, it will actually exist and will not, at the same time, merge its *non-being* with its *being *... True nothingness has no true properties ... *being* in itself excludes *non-being*” (par.52-55, p.22-26).

In our opinion, the principal value of Boscovich's arguments is that he has demonstrated that the discretum actually exists. However, we do not this is acceptable as it would follow that the continuum is virtual, i.e. it is non-being. Let us now consider, what kind of further** **considerations are possible after Boscovich.

If the limit between a beginning and an end is considered to be continuous, then: either (a) there is no difference between the beginning and the end, i.e. they are identical, or (b) the limit itself is discrete.

Boscovich's conceptions of the interval of time, presented as the *tempusculum*, and of the *momentum,* as a** **break in the course of time, both present great difficulties. It is obvious that two momentums cannot immediately follow without merging into one and the same momentum. However, the concept of a momentum conceived as the limit between two tempusculums in which two different events occur entails an even deeper inconsistency, especially if it is considered from the point of view of a physical interpretation of Boscovich's geometrical examples. In other words, the momentum merges the beginning of one event with the end of another so perfectly that they become one and the same event, which is contrary to the assumption of two different events. This can be represented geometrically in the following way (as a consequence of Boscovich's comprehension of time as comprising** **points, i.e. *momentums,* and line segments, i.e. *tempusculums*):

Consider a line segment *AB* with a point *C * at** **the** **mid** **point** **of it (Fig. 10).

Fig. 10.

Since the point *C* has no dimensions, its position on the line segment *AB* cannot be determined without reference to a line segment *DC* (equal in length to the line segment *AC *or *CB, *i.e. *DC* = *AC* or *DC* = *CB*). It is only with reference to the line segment *DC,* the limit of which is the point *C*, that it is possible to determine the position of C on the line segment *AB*. All things considered, it is not in the point's nature to exist independently, but only to exist as the limit of a line segment. Further, if a point is conceived as being independent, then it is no longer the limit of a line segment but is instead the unique actual primordial point, i.e. the natural model for the unique actual continuum (the idea of which is identical to the** **concept** **of** **the** **point, since it is impossible to imagine a point which has parts)^{7}.

Euclid's geometry is an example of the best and most consistently performed process of deduction in science. At each stage in Euclid's derivation of the whole of geometry from the first definition, i.e. from the point, it is possible to confirm the validity of his postulates and axioms by induction. (Non-Euclidean geometries are in fact based on parallel assumptions and not on a rejection of Euclid's geometry.)

If we accept that induction should not be used for drawing independent conclusions, for** **which** **we** **have** **more** **thorough** **scientific** **justifications, but for testing deduced expectations, then we can easily accept the suggestion that Boscovich's argument against motion in leaps (the argument of the replication of bodies) is not valid because of the aforementioned reason that it did not involve a conception of the essence of Euclid's idea of the line and its relationship to the point (i.e. the continuum-continuity ontological connection).

For Boscovich, the *tempusculum* is merely a line segment. It is not correct to say that he conceived it as a representation of continuous time. According to Euclid, a line segment is only interrupted by its limits and *nowhere else *(this means that, although it is not divisible, its hypothetical parts must, existentially, be other line segments). Its continuity means that it is comparable to some extent with the idea of a tempusculum but, on the other hand, in the final analysis, because of its limitations it is an inadequate visualisation of continuous time and is unsuitable for representing all that Boscovich intended by the term tempusculum. In addition, according to that same source, Euclid, the point is clearly not discontinuous, even though Boscovich takes it as a model for the* momentum*, i.e. as a visualisation of discrete time. It was by inverting the true meanings and properties of the point and line segment that Boscovich covered up the insufficiencies in his inductive and indirect proof that there is no motion in leaps. (Fig. 11)

Fig. 11

To Boscovich, the extension *CD* exists in the tempusculum *AB *but, since the momentums *A* and *B* are moments at two different times (separated, e.g., by two hours), the two ends of the line segment *CD* do not exist simultaneously and, in fact, the whole of the line segment *CD* does not exist as such in relation to Boscovich's tempusculum. By introducing the tempusculum, Boscovich reduced space to a point and ascribed extension to time. It is obvious that, of the whole line segment *CD,* only one point can coexist simultaneously with itself (if the tempusculum *AB* is valid for it) because any other point on the line segment *CD* corresponds to some other time. Consequently, if it is postulated that a moment of time, *AB*,* *corresponds to some point (but only one point) on the extension *CD*, then this will not only result in the elimination of the extension of the line segment, but also of its continuation in time (so that again we obtain the unique and uniform continuum).

In order to exist simultaneously with itself, the extension *CD,* with respect to time, must ** **be** **defined** **in** **relation to a momentum and not to a tempusculum (Fig. 12).

Fig. 12

If the extremities of the line segment *CD * do not exist simultaneously, i.e. at the momentum *H*, its extension is not possible as such, at least not in the sense in which we see it (as simultaneous to itself). In taking over (most probably from Boscovich) this inverted comprehension of Euclid's definition, Einstein, in his *Special theory of relativity,* introduced the concept of local time (like Boscovich) which differs from one point of space to another, thereby eliminating both extension and continuation in time, just as Boscovich did, and disintegrating the whole cosmos into points of space and points of time which cannot establish any interrelationship** **(no extension in space or continuation in time); hence they coincide, thus forming that singular primordial point in which there are absolutely no distinctions and where Einstein's process of inductive reasoning** **(like that of Boscovich) had necessarily to cease completely.

Consequently, if ** **space** **is** **to** **be** **extensive,** **the** **“leap”** ***CD*** **must** **exist** **in** **a** **corresponding continuous time, i.e. in the momentum *H*. In other words:

a) The tempusculum (i.e. the discretum of time) *T* = *dt*, but only if *t*1 and *t*2 are not conceived as points but as lengths, i.e. *t*1= *a*, *t*2 = *b*. Einstein would have written simply, like Boscovich, *dT* = *t*1 - * t*2, thus subtracting time lengths, which could in turn be expressed, as Einstein expressed them, as changes in extension in space. Physically, subtracting tempusculums is not the same as subtracting momentums, and this difference has been extensively overlooked. The quotient of two equal tempusculums is simply one and the same momentum, which can correspond to any spatial extension, provided that the two points at its extremities exist simultaneously (otherwise the extension does not exist). The quotient of two momentums is, in fact, the subtraction of tempusculums; hence, it is implicitly assumed that there is a tempusculum between two momentums which is equivalent to the spatial extension which is, through the operation *dt *= *t*1 - *t*2, traversed by the motion (or otherwise delineated). Of course, according to this conception, two momentums cannot belong to the same moment of time as, in that case, they would coincide. (This is the meaning of Boscovich's prohibition of sequential momentums.) Accordingly, algorithms involving tempusculums are only correct if their spatial implications are considered simultaneously and are taken into account (particularly because the length of a tempusculum is a property of space and not of time); otherwise, reality would be interpreted** **according** **to** **a** **time** **perspective, even though we know that a person is not smaller because he is more distant from us^{8}. To conclude: *dT* = *Ta* - *Tb* , provided that *a* and *b* are corresponding lengths in space.

b) According to the above, it follows that operations with momentums and corresponding equal tempusculums are senseless because the result is always the same, i.e. the continuity (one point). Momentums are not independent; it is not possible to determine a momentum by itself, since there is no reference quantity. Therefore, two momentums are always simultaneous (like the extremities of an extension) and they can only be taken in twos; as** **a** **single,** **isolated** **momentum,** **being** **inextensive,** **is** **not** **sustainable** **because** **there** **is** **no** **line** **segment,** **corresponding** **to** **the** **tempusculum,** **with** **only** **one** **extremity (Fig. 13). Accordingly, it is incorrect to conceive the momentums as a series of independent moments: t1, t2, t3, ... tn (the assumed direction of time is from t1 to tn), but exclusively as a series of the extremities of a number of tempusculums.

Why should motion occur exclusively in leaps? If the whole extension *AB* (the volume of the body *AB*) must correspond with a momentum in order to exist simultaneously with itself (i.e. to exist at all), then each extension of the distance traversed by the extension of the body *AB* in its motion must also have its own corresponding momentum, and this momentum must be the same momentum which corresponds to the body, otherwise the extension *AB* in its continuous motion through space would make leaps in time.

Boscovich's conception is illustrated in Fig. 13, bearing in mind that the tempusculum has been transferred from space and is, hence, inadequate for representing time^{9}.

Fig. 13

*HH*' - the virtual tempusculum (i.e. continuous time); the momentum corresponding** **to the extension of the body, *H*, and the momentum corresponding** **to the extension of the distance, *H,*' exist** **simultaneously.

*AB* - the extension of the body.

*BC* - the extension of the distance, i.e. the space traversed by the body *AB* in motion.

With reference to the above diagram, the difficulties in Boscovich's conception of motion in relation to time and space can be eliminated in the following way (Fig. 14) :

(The transitions of point *A* of the body *AB* * *in* *momentums *q, l, s* ... occur in leaps)

Fig. 14

If we accept that the extension *AB*, in order to exist simultaneously with itself, must correspond with a particular momentum (i.e. the fact that one point in time can simultaneously correspond with several points in space explains the existence of lengths and spatial distance, in general), then it is clear that any other extension, which is traversed by the motion of the body *AB*, also has the same corresponding momentum. Therefore, the extension AB will make spatial leaps in motion (even though these may be small and we may perceive the motion as continuous) and, in this way, time will preserve its continuity. In this conception, time is disintegrated (but only apparently) into points, each of which corresponds to at least two points in space, but all the points in space in fact coincide and form a singular point in time which is applicable to the whole of cosmic space (which exists simultaneously with itself). This accords with our perception that space exists simultaneously with us, i.e. with matter, and with our failure to perceive the entity time. (Since time is a point, it is simply the ratio of line segments in space, and the determination of time itself becomes simply the determination of the position of a point on a line segment with the help of a reference line segment, one end of which is that point. Of course, this is only valid on the assumption that space and, consequently, all lengths are potentialities of the continuum. In fact, the limit of all extensions is the primordial point, i.e. the continuum which is uniform and physically infinite, and, because it forms the limit of our discrete cosmos, seems infinite to us.)

Boscovich's conclusion that the movement of a body in leaps means that it would have to be simultaneously present in two places at the same time is not valid unless the role of local time has been made numerically equivalent to the role of absolute time (Newton's time), which is what Boscovich did, as we have seen. In contrast, if the entirety of time is conceived as only one momentum, then all potentialities are in fact only local projections of that momentum, i.e. points on lengths between which the body makes leaps, i.e. moves (the limits of leaps are represented by the ratios of lengths, i.e. of line segments)^{10}.

In order to avoid leaps in the motion of a body in space, Boscovich introduced the tempusculum and transferred the leaps from space to time.

A body, which is assumed to traverse a distance in space continuously, must, in zero time, pass through an appropriate** **tempusculum, which, according to Boscovich, has a certain time length. If it is claimed, however, in order to defend the concept of the tempusculum as a finite length of time, that a body, during its motion through space and time, exclusively passes through points, then it will not be able to either traverse a distance in space nor pass through an interval in time but, as it must pass through a spatial and temporal continuum in which measurement is meaningless, will simply cease to move. Let us again consider Zeno's famous paradox in *Achilles and the tortoise* from this point of view:

1. According to Boscovich's understanding: (a) space is not quantized, it is continuous and infinitely divisible, and (b) time is quantized (I consider here a corrected interpretation of the *tempusculum* as a quantum), continuous (*momentum*) and infinitely divisible. In infinitely divisible space and time, Achilles can never overtake the tortoise which started before him, since the *t*n of the tortoise is always in advance of the *t*m of Achilles. Achilles can only pass this tempusculum with the help of a time machine. Hence, according to Boscovich, Zeno's paradoxes are valid, but motion also exists.

2. Let us recall here that Leibniz considered time and space as series of successions, i.e. series of coexistences, and that this conception lies essentially beyond ** **our** **actual** **understanding of these phenomena. On the other hand, his concept of the infinitesimal as a quantity which, no matter how small, is still larger than zero, and his concept of the convergence of series enabled Leibniz to satisfactorily solve the problem of *Achilles and the tortoise* on the basis of experience while, at the same time, introducing new, even more complicated and paradoxical relationships.

Let us, however, assume the following: space is quantized by certain extensions which are characteristic of bodies, i.e. it is not divisible at all (only this or that quantum can be taken as a whole), and time is similar to spatial quanta, i.e. the projection of a smaller quantum onto a larger one (this concerns, of course, local time because all space quanta are simultaneous with respect to the primordial momentum). Hence:

a quantum of distance > Achilles' step > the tortoise's step.

Regardless of the starting order, Achilles' local time, which corresponds to his space quantum (i.e. to his step), will always be longer than that of the tortoise and he will overtake it, even if they move with the same velocity, but different steps.

To Zeno, space and time are infinitely divisible and continuous. Therefore, Achilles and the tortoise could not even move, let alone compete in a race. (It would be interesting to know Zeno's views on the human perception of motion.)^{11}

"... the limit must be the one and only indivisible common limit, just as the instant is the one and only indivisible limit between the continuous preceding and following times" (p.28, par.62). By using a non-existing limit to connect quantities, Boscovich makes an indiscernible continuum of them, thereby contradicting experience. Boscovich connected time intervals in the same way. But, these non-existing limits do not separate. (Boscovich himself emphasised several times that: "a real nothing has no real properties.")

Boscovich distinguished actual velocity, which is related to uniform motion, and potential velocity, which is an inclination towards an actual velocity. He insisted on this difference and it led him to the conclusion that force acts at a distance. ("to determine the actual velocity of a non-uniform motion, mechanics usually use the concept of a completely insignificant distance traversed in an infinitely short interval of time, in which they consider the motion to be uniform ... It may be concluded that direct contact cannot occur at different velocities. ... Accordingly, there will be a force which has an effect even when two bodies are not yet in direct contact ..." (p.33).)^{12}

The problem of a force acting at a distance (e.g. gravity, but all forces in fact act the same way) must be solved by an interpretation of the concept of distance, of space. For it is only formally and apparently that an infinite increase in the intensity of a repulsive force enables the infinitely close** **approach of two bodies in accordance with the idea of the infinite divisibility of space. In reality, two bodies approaching one another will stop at a distance which is precisely determined by the increase in the repulsive force; it follows that, for these bodies, there are points in space which cannot be crossed by continuous motion in the same direction. (Either space is infinitely divisible or Boscovich's repulsive force is not asymptotic. This latter alternative corresponds to the concepts of descretized and quantized space.)

Because real space, for Boscovich, is divided into a finite number of distances which are fixed by the action of forces on points of matter (so that further progression in the same direction becomes impossible), it follows that the motion of bodies through space occurs exclusively in leaps. (For a given magnitude of force, the body will pass through points which are more or less distant from one another, i.e. the density of space varies according to the velocity of the body and, in any case, motion is discontinuous.) On the other hand, uniform motion does not exist in nature because space itself exerts a frictional effect such that actual inertial motion, in fact, experiences a negative acceleration. On thorough analysis, Boscovich's force curve implies that space is descretized or even quantized (or, at least, Boscovich had to assume that it was):

*A,B,C* ...= Points of transition from -*F* into +*F * and vice versa.

*O* = Origin of the coordinates.

Fig.15

The points at which the force curve crosses the abscissa and changes its direction are in fact the points at which the quanta of the space^{13} of given bodies (i.e. the quanta of given forces, expressed by their corresponding segments on the abscissa) begin and end and at which the body must correspondingly change its direction of motion. If any definite value is ascribed to a force, space consequently becomes quantized^{14}.

"81. Since the repulsive force is infinitely augmented by an infinite diminution in distance, it becomes absolutely clear that any one part of matter cannot be contiguous with another part because the repulsive force separates these parts from one another. It follows, as a necessary consequence, that the primary elements of matter are quite simple and that they are not composed of contiguous parts" (p.37). The continuity of force, as conceived by Boscovich, connects matter to the same extent that it causes** **space** **to** **be** **discontinuous (matter is nonspatial according to both Boscovich and Leibniz). Moreover, Boscovich makes no actual distinctions between a point of time, a point of matter and a point of space; therefore, it is not clear why matter is, to him, exclusively discrete (as he apodictically claims) whereas space and time can be both continuous and discrete, as required. (The contradiction is even starker because Boscovich excludes the virtual extension of matter.)

"... so that two points of matter never connect the same point of space with two moments in time, whereas numerous pairs of points of matter connect the same point of time with two points of space; in this way they coexist." (p.39-40.) Boscovich assumed that a point of matter never returns to any point of space which was once occupied by another point of matter. (This is, in fact, a modification of Descartes' conception of space (p.138-140 of this study).)

In our opinion, two points of matter can connect the same point of space with two moments in time. In other words, the bodies *m*1 and *m*2 can have the same spatial coordinates but two different time co-ordinates (Fig. 16).

Fig. 16

This corresponds to Leibniz's definition of time as "the order of existence of that which is not simultaneous". This is in contrast to Boscovich's opinion that parallel worlds^{15} differ only in the way in which they cross the abscissa, i.e. that they exist simultaneously in different spaces. Depending on the way in which the ordinate is crossed (and Boscovich had not considered this case), it is possible to obtain occurrences at different times at one and the same place. Boscovich, however, considered only one possibility; he adopted the essence of Leibniz's conception of space as the order of coexistence.

To Boscovich, the only possible case is the one in which the bodies *m*1 and *m*2 have the same time coordinates but two different spatial coordinates (Fig. 17).

Fig. 17

Although each point of matter could arbitrarily have its own time in the same space, Boscovich rejected this idea, emphasising that it was improbable that a point of matter could return to the same point of space in which it (or some other point) had already been. His refutation was based on the assumption of the irreversibility of space (i.e. its irreversible mobility). However, he made no attempt to prove that space was actually in motion (and it cannot be proved by probability).

"... if the primary elements of matter are a number of solid parts, composed of parts or perhaps only virtually extensive, then, in a continuous motion from the vacuum through such a particle, an instantaneous leap would occur from the zero density of vacuum to the actual density which is found when that particle occupies space. However, there is no such leap if these elements are simple, inextensive and distant from one another. For the whole continuum is then simply composed of a vacuum, and the transition from continuous vacuum to continuous vacuum occurs in continuous motion through a simple point. That point of matter occupies only one point of space, and this point of space is the indivisible limit between preceding and subsequent space. It neither stops the moving body which arrived there in a continuous motion, nor does this body make a transition to it from any point of space which is in immediate proximity to it because there is no such point, as we have already said; but there is a transition from continuous vacuum to continuous vacuum through that point of space which is occupied by the point of matter" (p.40, par.88).

For Boscovich, as for Leibniz, the *point* is the main stumbling-block. A *point *of matter cannot divide the continuum simply because its definition would not allow it, and the point has not been redefined since Euclid. It is possible to conceive of discretums connected by points in a continuous (to Boscovich, discontinuous) vacuum, but from this standpoint (i.e. from the continuum), it is not possible to reconstruct the world of perceptions by means of general** **deductions without introducing a *law of discontinuity*. Boscovich, therefore, did not make deductions because he could not go back beyond the continuum. Eager to prove the continuity of phenomena, Boscovich introduced contradictory concepts (i.e. he operated using some sort of discretely continuous entities) and overlooked the fact that only recursive induction (i.e. deduction) can be verified.

* In conclusion*:

“**Time is Continuity**” - ontological definition of Time.

The Theory of Time asks for a new rigorous method:

the exact **physical interpretation of Ontology and Mathematics**.

- The end of excerpt -

**Footnotes**

^{1}Full title: *Theoria philosophiae naturalis; reducta ad unicam legem virium in natura existentium*, auctore P. Rogerio Josepho Boscovich, Societatis Jesu, nunc ab ipso perpolita, et aucta, Ac a plurimis praecedentium editionum mendis expurgata. Editio Veneta Prima, ipso auctore presente, et corrigente, Venetiis, MDCCLXIII, Ex Typographia Remondiniana, Superiorum permissu, ac privilegio. (Theory of natural philosophy reduced to a single law of the forces existing in nature etc.)

^{2}Generally, the problem lies not in the nature of geometry (let us consider it as real, as Euclid did) nor in the nature of physics (which has always been generally considered as real) but simply in human conceptions, in the human comprehension of their natures, i.e. of their unique nature.

^{3}The problem of the actual and the virtual in the relationship of the continuum to the discretum would again arise here.

^{4}Although the comparative forms of the adjective real are irregular, I have used the word to express the thought most appropriately.

^{5}It is interesting here to quote Descartes fascinating viewpoint: "The concept of *the infinite* which I have is, for me, prior to the concept of *the finite*, since I conceive of infinite Being using the very concept of *Being* or *that which is*, irrespective of whether it is finite or infinite, and to conceive of Being as finite, I have to subtract something from the general concept of Being, which, accordingly, must have preceded it." (Ed. Adam et Tannery, V, p.356.) Descartes' statement, although very elegant and essentially correct, includes some contradictions since it first claims that the infinite is in the subject (i.e. in Descartes, in the philosopher) and, only after that, that it precedes the finite. This means that it precedes the subject, which is finite and complete. This creates some confusion, as Descartes begins with the concept of infinity, or better, the idea of the infinite, and ends with actual infinite being. By using a sequence of subjective concepts (the infinite prior to the finite), he establishes a sequence for the creation of objective phenomena (again the infinite prior to the finite), a reasoning which is not necessarily consistent; accordingly, we agree with his conclusion but not with its derivation.

^{6}This proof is, in fact, implicitly included in all previous proofs. We quote it here, not only for information, but also to point out the ontological difficulties in Boscovich's conception of the continuum, which must presuppose the non-existence (non-being) of the continuum in order to prove the existence (being) of the discretum, which is a prerequisite for the existence of bodies.

^{7}The point, although single, is not conceptually equivalent to unity (since it has no form whereas unity does). Euclid correctly represents unities as line segments. In that sense, Boscovich's indirect proof (p.23, par.50) is, in our opinion, correct, but his solution to the problem of the continuity of motion and occurrences is such that it is not possible to make them correspond with our perceptions and to use them to explain why things and events in the world are distinguishable at all. In fact:

(1) The continuum of time and the continuum of space cannot exist separately in the same representation of one cosmos since continua cannot co-exist . It follows, as we have already mentioned, that time and space are not phenomena with the same level of complexity, i.e. they are not continuous entities of the same sort;

(2) The conception of the point (i.e. an object which makes a continuous connection between lines, i.e. a volume both in space and time) leads, in the next step, to the conception of a divided cosmos in which time and space are changed into an implicitly assumed basic medium, about which there can be no attempt to know anything.

(3) In our opinion, it is not possible to reach the exact and complete truth by induction, simply because human experience is not all-embracing. (In a finite human lifetime, it is not possible to investigate the whole abundance of cosmic variations. It clearly indicates that the heuristic power of "total induction" is just an illusion of the human discriminative mind.)

^{8}This issue will be discussed in detail in the concluding treatise.

^{9}According to Boscovich, time and space are parallel, infinite and continuous, and they consist of reciprocally corresponding points, i.e. a point of time corresponds to a point of space, and vice versa.

^{10}In Heizenberg's *uncertainty relation* precise determinations of the momentum and position of a particle at any specified time or place exclude one another, because a change of position does not result from the action of a force but from motion occurring in leaps. Otherwise, it would be possible to express the particle's momentum as an "space quantum" , (discretum or extension) , and it would be possible to determine simultaneously both momentum and position. The ratio of the length of a body to the distance it has to traverse projects the primordial point of time into the local point of time corresponding to the extension of the distance to be traversed (the extension of the distance to be traversed is always larger than the extension of the body in motion), therefore the position can be changed without the momentum being changed. Determination of the particle's momentum is nothing other than an observation of the dissimilarity between the extension of the body and the extension of the distance traversed, whereas the determination of position is a determination of their momentary similarity. And since the extension of the body and the extension of the distance traversed cannot be simultaneously both similar and dissimilar, their simultaneous determination is impossible.

^{11}At this point, we could have taken the opportunity to review in detail the previously presented bi-form finitistic solution used by B. Petronijevic to solve Zeno's problems but we shall leave it for the next section.

^{12}It is believed that the infinite divisibility of space makes the continuity of motion possible. However, according to Zeno -and we agree with him on this issue- this would eliminate motion. On the other hand, the theory of the continuity of force is itself contrary to the concept of the infinite divisibility of space, because it follows that, in dividing space, we do not divide the forces acting in it. Therefore, according to Boscovich, it must be assumed that force, if it continuous, does not act through space.

^{13}The magnitude of space quanta is not determined, although any actual space quantum is not divisible. If we consider the physical model of the light quantum (i.e. the electromagnetic wave, EMW) as a quantum of space and, if we accept, as is well-known, that light has a maximum velocity and does not expend any energy in its motion (i.e. no force has to be applied to attain or maintain the velocity of the light quantum), then it is probable that the time necessary to emit one EMW is equal to zero. This means that the total period of oscillation can be expressed as 1/T = 1, i.e. each EMW (light quantum) appears instantaneously along its whole length. This is not only valid for light but also for the entire electromagnetic spectrum (radio waves, "gravity waves", etc.), regardless of the wavelength. The reason for this behaviour has still not been discovered, i.e. the law governing this phenomenon has not yet been explained.

^{14}It is Boscovich's theory of force which gives an excellent explanation of the fact that the action of a force is simply a consequence of an, as yet, undiscovered natural law. (To Boscovich, the action is always and only *actio in distans*).

^{15}This will soon be discussed in detail.