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
abovementioned 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 magnitude, (and the same numerical value).
(This question is based on Euclid's conception of the
number as a line segment.) For, if it has the same
magnitude, i.e. the same corresponding length,
then not only will there be no accelerated motion, but
there will be no uniform motion either. In fact, there
will be no motion at all (because the cosmos of
uniform motion must necessarily
make a transition, by means
of an inertial system, into
a stationary cosmos). However, conversely, if time
and space are not considered, either arithmetically
(numerically) or geometrically (according size and its
representation), as being the same but as the different,
then there will be no continuity of motion.
"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 onethird
and onefourth of the first arc and onethird and
onefourth 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.4445, p.2122)
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 selfidentical 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. 4749, p.2223). 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 nonbeing 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 nonbeing
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 nonbeing 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 nonbeing with its being ...
True nothingness has no true properties ... being
in itself excludes nonbeing” (par.5255,
p.2226).
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
nonbeing. 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. (NonEuclidean 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 continuumcontinuity 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 t1 and t2 are not conceived as
points but as lengths, i.e. t1= a, t2
= b. Einstein would have written simply, like
Boscovich, dT = t1  t2, 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 = t1  t2,
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
tn of the tortoise is always in advance of the tm
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
nonexisting limit to connect quantities, Boscovich makes
an indiscernible continuum of them, thereby contradicting
experience. Boscovich connected time intervals in the
same way. But, these nonexisting 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 nonuniform 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.3940.) 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.138140
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 m1 and m2 can have the same spatial
coordinates but two different time coordinates (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 m1 and m2 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
stumblingblock. 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 nonexistence
(nonbeing) 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 coexist . 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 allembracing. (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 biform 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
wellknown, 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.
