Consider lines. Any line, say from a to b;
this line segment is either divisible or indivisible.
If divisible, then it is divisible either into a finite number of parts
or into an infinite number of parts.
Now these parts will either have magnitude or lack magnitude.
If the line segment is divisible into a finite number of parts,
lacking magnitude, the line segment cannot be reconstituted.
If the line segment is divisible into an infinite number of parts,
lacking magnitude, the same result follows.
And if the line segment is divisible into an infinite number of parts,
and each part has magnitude, then the reconstituted line segment will be
longer than the original line.
And if the line segment is divisible into an finite number of parts,
and each part has magnitude, then the reconstituted line segment will be
shorter than the original line segment.
Hence, the line segment is not divisible.
Zeno argued that being is indivisible, for the opposite claim that being is divisible leads to a logical absurdity. Suppose that being is divisible; this means that it will be composed of parts that either may have magnitude or may not. If the parts lack magnitude, then even an infinite number of parts cannot produce something with magnitude. If, on the other hand, being has parts with magnitude, then the infinite number of parts will produce a being that will occupy space larger than the space occupied by the original being. And a finite number of parts will produce a being that will occupy less space than the space occupied by the original being. Hence, being must not be divisible. From this indivisiblity of being, Zeno argued that the concept of space is also contradictory and must be given up since being is not divisible.
Zeno also argued that reality is changeless and the appearance of motion is an illusion. He argued: between two points on the path of a moving object, however near they may be each other, there lies at least one other point, and, therefore, an infinite number or points. Since the movement from one point to another takes time, however little, and since an infinite number of intervals must be crossed in order to get from one point to another, a movement between two points would take an infinite time. Therefore, the motion along a continuous path, or indeed any motion is impossible.
As can be seen in the above, the problem of continuity is intimately related to the problem of infinity. The word "infinity" comes from the Latin in ("not") and finis ("limit"); hence, without limit or boundary or end. Some have claimed that the concept of the infinite is logically prior to the concept of the finite, inspite of the fact that etymologically the term "infinite" is gained by the negation of the term "finite". Very early the conception of the infinite has been associated with a series of numbers, magnitudes, space and time. Some have claimed that the endlessness of series is basic to the conception of infinity. But if the predicates "finite" and "infinite" is applied to the concept of being rather than to a series, the conception changes back to its root meaning: finite being is limited in extent, properties, etc. and infinite being means without limit, and absolute in all these respects.
"A thing is continuous when any two sucessive parts, the limits, at which they touch, are one and the same, and are, as the word implies, held together."That is, the "continuous" is that subdivision of the contiguous whose touching limits are one and the same, and are contained in each other. This implies, as he points out, that nothing continuous can be composed of indivisibles; that is, the continuous is infinitely divisible. As Aristotle says, "divisible into divisibles that are infinitely divisible" (Physics 231b); that is, continuity implies infinite divisiblity. On this basis, Aristotle rejected the atomic theory of Leucippus and Democritus (5th B.C.). To the objection that an infinite number of divisible elements is impossible, Aristotle's reply was to reject the concept of the infinite as an actual or complete infinite totality. For Aristotle, a class can be only potentially infinite. Its membership can be increased without limit, but there can be no complete totality given. According to Aristotle, one of the mistakes of Zeno in his paradoxes of time and motion is that he did not distinguish between actual and potential infinities. Aristotle definition of continuity required the concept of a potential infinity. The "continuous" is that subdivision of the contiguous whose limits are one and the same, and contained in each other. This implies that the parts of the continuous are infinitely divisible. The idea of dividing of a line segment into an infinite number of divisible elements can not actually be done; a line does not actually consist of an infinite number of unextended points. It is only "potentially infinite." Potential infinity is the statement of a capacity; it applies to that which can be infinitely divided, augmented, or diminished. But the infinitely divisible is not divided into an actual infinite series. For Aristotle, the mathematical concepts of continuity and infinity are abstractions from perceptual sense experience.
Although Aristotle's conception of continuity remained dominant until the middle of the nineteenth century, it was not unanimously accepted. Philosophers of the Platonic tradition, including theologians of the Augustinian tradition, regarded the concept of an actual and infinite totalities as legitimate. They were not bother by the inapplicablity of such a concept to sense experience, since for them mathematical concepts are not an abstraction from, or a description of, sense experience, but a description of the reality apprenhended by reason.
Against Aristotle, medieval Schoolman following Augustine believed that time had a beginning. Between eternity, the Res Tota Simul, and time, is the Aevum, or everlastingness, of heavenly bodies and of angels.
A triangle is defined by the positions of its three vertices, but these three vertices now become three groups of coordinates. A line is a continuous set of points -- a continuous set of coordinates. Consider the algebraic equation y = 2x, which allows y to be calculated from given values of x. If x is 1, then y is 2; if x is 2, then y is 4. Or to put it another way, the pair of numbers (1,2), (2,4), (3, 6), (4, 8), (5,10), etc. are all solutions to the equation y = 2x. But these pairs of numbers can also be thought of as coordinates in a plane. They define a set of points, and when these points are connected they produce a straight line. This line is therefore a representation of the algebraic equation y = 2x. Algebra is about numbers but is also about points, curves, and shapes in space. Algebra is about geometry, and geometry is about algebra.
At one stroke, Descartes was able to link two great fields of mathematics. A line becomes an algebraic equation. The intersection of two lines become the solution of two simultaneous equations. Theorems in geometry and manipulations in space become transformations within algebra. The symmetries of the triangle, square, and pentagon are now properties of algebraic transformations. Anything that can be done in geometry can be done in algebra. Descartes had shown how to reduce geometry to algebra. And since much of physics is concerned with paths in space, spatial relationships, and symmetries, these also become problems of algebra. Suddenly much of physics is translated into algebra. Indeed one of Newton's great acts of genius was to extend this algebra by inventing the calculus.
"The properties of magnitude by which no part of them is the smallest possible,In the Kantian view, the emphasis is laid on the wholeness and the unity of space which exists prior to the points; the latter are merely ideal limits which are never actually attained. According to Bertrand Russell, the oppositie is true; the points are constitutive parts of space. In this view, space is regarded as an infinite aggregate of all dimensionless points. [2] But philosophically speaking, there is hardly any significant difference between the two views; the infinite divisibilty of space is accepted by both. The only difference is that while Kant space is divisible without limit, for Russell it is actually divided into an infinite number of parts. But sice the concept of unextended point and the infinite divisibility of space are correlative, the difference between the views of Kant and Russell is one rather of emphasis than of substance. Russell's view seemed to express more completely and more explicitly the trends of classical thought, for which the concept of the pointlike geometrical position was of fundamental importance.
that is, by which no part are simple, is called continuity.
Space and time are quanta continua, because no part of them can be give save as enclosed between limits (points or instants),
and therefore only in such fashion that this part is itself again a space or a time. Space therefore consistes only of spaces, times soley of times. Space therefore consists only of spaces, time soley of times. Points and instants are only limits, that is, mere positions which limit space and time." [1]
The calculus is a formal way of relating properties in space that are infinitesmally close to each other. By assuming that space is continuous, it is possible to relate distance objects through an infinite but continuous series of these baby steps. Thanks to the calculus, Newton's laws could now be expressed in terms of algebra, and what are known as differential equations. (Differential equations are ways of relating velocities, accelerations, and other rates of change that are determined at different [dimensional] coordinate points in space and are assumed to be continuous.) From now on, physics was formally wedded to coordinates and all of mathematics that flowed from them.
Classical modern science understood space as a homogeneous medium existing objectively and independently of it physical content, whose rigid and timeless structure had been described by the postulates, axioms and theorems of Euclidean geometry. This space, that was self-sufficent and independence of the matter which it contains, was clearly defined by Issac Newton (1642-1727) in his Principia:
"Absolute space, in its own nature, without regard to anything external, remains always similar and immovable."Newton was not the first to formulate this definition of Absolute Space, even though it is commonly held that he formulated it. Pierre Gassendi, Henry More, and several Renaisance philosophers, Telesio, Patrizzi, Bruno, and Campanella used it. They rejected Aristotle's concept of space as the plenum, as occupied space, but retained the immutability of space, considering space as the void, or empty space, and matter as occupying space. With the Greek Atomists, they stressed this separability of space and matter, holding that space is absolute and independent of what occupies it.
Closely related to the independence and immutability of space is its homogeneity ("the quality of being the same in nature or kind"). In fact, its independence and immutability logically comes from its homogeneity. Since no place in space is different from any other place, space is independent of what is at any place in space. Though the content at any place in space may change, the underlying space is immutable. Though Newton historically does not start with the homogeneity of space in his definition of Absolute Space, he silently assumed it, though it is not mentioned by him explicitly. The tacit assumption of the homogeneity of space was made as soon as space was separated from its physical content; and this was done by the Greek atomist. In their veiw, all qualitative diversity in the physical world was due to the various position, shapes, and motion of the atoms, not to the instrinsic differentation of space itself, as belived by Aristotle and his followers. Aristotle's idea of "natural place" for the different elements implied the qualitative heterogeneity of space. This feature was explicitly rejected by the modern seventeenth century atomist. In a conscious return to the ancient Greek ideas of the atomist, the homogeneity of space was explicitly expressed.
Two other features of Absolute Space follow directly from its homogeneity:
its infinity and mathematical continuity (infinite divisibility).
Space has no limit, since any boundary of space must be located in space,
and cannot be the end of space. Aristotle's elaborate proofs that outside the
sphere of the fixed stars there is no space are therefore unconvincing.
The homogeneity of space also implies its infinite divisibility. Space is
homogeneious because it is made up of points that are all alike. And all
of these points are so related to each other such that between any two points
there is always another point between them. This juxtaposition of points
means that no matter how small a spatial segment may be, it is always
possible to divide it into two further segments, and so infinitium.
In other words, no matter how small the segment may be, there must be a
segment separating two points, each of which are external to each other.
To claim that certain intervals of space are indivisible means that it is
impossible to discern within them any juxtaposition parts; but since
juxtaposition is the very essence of spatiality, this would mean that such
segments are themselves devoid of spatiality. Thus this thesis of the
indivisible spatial intervals is self-destructive; while it denies the
possibility of "zero lengths" (points), it at the same time reintroduces their
existence when it speaks of atomic intervals separating two very near points.
With respect to matter, Absolute Space had the same attributes that had traditionally been assigned to the Supreme Being by the scholastics. This was observed by Henry More in his Enchiridion Metphysicum (1671):
"Unum, Simplex, Immobile, Aeternium, Completum, Independens, A se existens, Per se subsistens, Incorruptibile, Necessarium, Immensum, Increatum, Incircumscriptum, Incomprehensibile, Omnipresens, Incorporeum, Omnia permeans et complectans, Ens per essentiam, Ens actu, Purus actus."That this divinization of space infuenced Newton's philosophy of Nature is well known. But he regarded this Absolute Space as an attribute of God, the sensorsium Dei, by which the divine omnipresence as well as the divine knowledge of the totality of things is made possible. Of course, Newton believed that God created everything in that Absolute Space, the heavens and the earth. The Absolute Space is eternal but not the things in Absolute Space. They were created by God. But the logical, if not the temporal, priority of space to its physical content was a dogma that few dared doubt, especially in England. For Newton as for Gassendi and More, this priority was temporal as well; Absolute Space, being the attribute of God, naturally had to exist prior the creation of the world. There was nothing absurd about this belief, although the coeternity of space and matter was equally compatible with the Newtonian laws of physics. But in France where a rejection of Roman Catholic theology and Thomistic philosophy took place, this second alternative, logically simpler, was adopted. The co-eternity of space and matter appeared to be logically simpler and more elegant view of reality than the arbitrary creation of matter at a definite date in the past. This lead many philosophers to return to ancient atomistic materialism or the pantheism of Spinoza, which appeared to be more satisfactory than the synthesis of Lucretian metaphysics and Christian theism as found in Gassendi and Newton.
["One, simple, immovable, eternal, complete, independent, existing by itself, existing through itself, incorruptible, necessary, measureless, uncreated, unbounded, incomprehensible, omnipresent, incorporeal, all-prevading and all-embracing, Being in essence, Being in act, Pure act."]
While Newton's flat backdrop of absolute space did not survive Einstein's revolution, the idea of coordinates was carried over. Admittedly they were no longer attached to the rigid rectangular grid proposed by Descartes but to a grid that could respond to the presence of matter. Nevertheless, the basic notions of continuity and infinite divisibility remained.
But against this view it may be objected that it is still possible to speak about instants when the spatial distance is zero, as in the case of a single world line. Not only is each event constituting a world line simultaneous with itself, but there is nothing in the relativity theory which would forbid one to regard a world line as mathematically continuous, that is, as consisting of a mathematically continuous succession of durationless events. In fact, the assumption about the pointlike character of world events is made either silently or explicitly by the majority of relativists.
"can be divided arbitrarily into elements of infinitely small volume, each containing an infinitely small quantity of matter, and each being subject to the action of elements having a volume infinitely near to their own. In this way the mechanics of continuous media can be formed, known as the Theory of Elasticity for solids and as Hydrodynamics for fluids." [3]But the conviction of most physicists is that in reality the solids and fluids consists of atoms or molecules in motion, which the obtuseness of our senses prevents us from perceiving this ultimately corpuscular structure of matter; the continuous character of them is illusory.
This is what may be called the corpuscular-kinetic view of matter. Its first formulation appeared in the early Greek atomism. Its basic premises have hardly changed through the ages. Although atomism suffered a temporary (by no means complete) eclipse during the Middle Ages, it reasserted itself with renewed vigor in the century of Gassendi and Newton and since then has exerted a persistent and fascinating influence on the imagination of physicists, at least until the end of the nineteenth century. In the twentieth century, its attractiveness has weakened, but not altogether been destroyed.
This problem of continuity has particularly been in the forefront of Modern Physics in the theory of light. The ancient philosophers, and later Newton and the majority of 18th century scientists, had adopted an corpuscular or atomic theory of light; but in the 17th century the Dutch physicist Huygens proposed a totally different conception of light. In his view, light was an undulation progagated in a continuous medium, the aether, the latter being supposed to penetrated every material object and to fill all regions of space which appears void to us. The discovery of the phenomena of interference and diffraction made by Young and confirmed by Fresnel and, later, of the development of the mathematical theory of waves by Fresnel and its verification by experiment in 1801, brought about, in the first half of the 19th century, a complete abandonment of the discontinuous view of light and the general adoption of the continuous Wave Theory of Light. But now in the first half of the 20th century Planck's discovery of the Law of Black Body Radiation (1900) and Einstein's explanation of the photo-electric effect (1905), the discontinuous view of light has returned in the Quantum Theory of Radiation. This theory assumed the existence of light-corpuscles, or photons. Unlike the corpuscles of matter, the energy of these photons are defined by the frequency of the radiation and not by their mass. But like the particles of matter they preserve their individuality in moving through space. This hypothesis was confirmed by experiment, particularly by X-ray scattering discoveried in 1922 by the American physicist H. A. Compton. But since the continuity interpretation of light has been confirmed by interference and diffraction experiments, the present theory of light is dualistic, holding that light has both a continuous and discontinuous aspect. Thus light has a dual nature; it shows wave properties in some situations and particle properties in other. That is, when a light ray exchanges energy with matter, the exchange can be explained on the assumption that a photon is absorbed (or emitted) by matter; on the other hand, if we wish to explain the propagation of light through space, then we can fall back on the assumption that light is waves. A further elaboration of this theory of propagation is that light is a cloud of photons whose density is proportional to the intensity of this wave.
"In this way, therefore, a sort of synthesis of these two ancient rival theories are reached, so that we are enabled to explain interference phenomena as well as the photo-electric effect...." [4]
"in the theory of matter, as in the theory of radiation, it was essential to consider corpuscles and waves simultaneously, if it were desired to reach a single theory, permitting of the simultaneous interpretation of the properties of Light and of those of Matter. It then becomes clear at once that, in order to predict the motion of particles, it was necessary to construct a new Mechanics -- a Wave Mechanics, as it is called today -- a theory closely related to that dealing with wave phenomena, and one in which the motion of a corpuscle is inferred from the motion in space of a wave. In this way there will be, for example, light corpuscles, photons; but their motion will be connected with that of Fresnel's wave, and will provide an explanation of the light phenomena of interference and diffraction. Meanwhile it will no longer be possible to consider the material particles, electrons and protons, in isolation; it will, on the contrary, have to be assumed in each case that they are accompanied by a wave which is bound up with their own motion." [5]De Broglie was even able to predict the wavelength of the associated wave belonging to an electron having a given velocity.
The dynamical variables used in classical mechanics, such as position and momentum, do not have definite values in Quantum Mechanics. Instead they are described by a quantity called a "wave function" into which is encoded probabilistic information about position, momenta, energies, etc. Thus in quantum mechanics the motion of particles is not deterministic, as in classical mechanics, but probablistic. The wave function for a particular system is found by solving the Schrodinger equation.
In the case of a single point particle, the wave function may be thought of as an oscilating field spread throughout physical space. At each point in this space it has an amplitude and a wavelength. The square of the amplitude is proportional to the probability of finding the particle at that position; the wavelength, for a constant amplitude wave function, is related to the momentum of the particle. The particle will therefore have a definite position if the wave function is tightly bunched about a particular point in space; and it will have definite momentum if the wavelength and amplitude of the wave function are uniform throughout all of space. Typical wave functions for a system will not be of either of these types and there will be a certain amount of indefiniteness, or uncertainty, in both position and momentum. In particular, because of the mutually exclusive types of wave functions required for definite position and definite momentum, position and momentum cannot be definite simultaneously. This is known as the Heisenberg's Uncertainty Principle (HUP), and is an elementary consequence of the wavelike nature of particles. In a "coherent" state, which is a compromise between definite position and definite momentum, there is uncertainty in both position and momentum. This means that the laws of physics are no longer deterministic and phenomena that they describe are no longer subject to a rigorous determinism; they only obey the laws of probability. Heisenberg's Principle of Uncertainity gave an exact formulation to this fact.
The quantum theory did not dispense with a space created out of dimensionless points. Schrodinger's wave funcion is a differential equation that uses essentially the same mathematics that was developed by Newton. The generalization of quantum theory, called quantum field theory, also relies on coordinates, for the quantum field is defined at each point in space and is a continuous function of Cartesian coordinates. The dimensionless point remains the basic paradigm of modern physics. Quantum field theory is plagued with such problems as the infinite results that are found when certain properites are calculated. Some physicists believe that the orign of the problems lies in the assumption that the quantum field is defined right down to infinitesimally short distances. Quantum theory seems to be demanding a new mathematical language for space-time.
String theory attempts to get rid of the problem of infinities by getting
rid of particle interactions that occur at a single point. Take a look at
the Heisenberg's Uncertainty Principle:
ΔpxΔx ≥ h/2π,
where Δpx is the uncertainty of the momentum of a
particle moving in the x direction, and Δx is the
uncertainty in the position of the particle in the x direction, and
h is Planck's constant, the atom of action
(h = 6.625 × 10-34 joule-sec).
This principle says that it is impossible to determine simultanesously the
velocity (or momentum) and the position of an electron or any other
microphysical particle; that is, the more accurate the determination of
its velocity (or momentum) is, the hazier its position becomes and vice versa.
Now if the uncertainty of the momentum px blows up, that is, Δpx → ∞, this has been interpreted to imply that Δx → 0 and to mean that if the uncertainty of the momentum p in the x direction is very large (infinite), then the uncertainty of the position in x direction will be very small (zero) distance. Or to put it another way, pointlike interactions (zero distances) imply infinite momentum. This leads in Quantum Field Theory to loop integrals and infinities in calculations. The existence of these infinities caused some physicists to wonder if there was a basic flaw in the foundations of Quantum Field Theory. Could these infinities be somehow related to the prevailing idea of infinite divisibity of space-time and the use of dimensionless points as the building blocks of geometry? So in string theory, a point particle is replaced by a one-dimensional string. That is, where in the old quantum theory, a particle is a mathematical point, with no extension, in string theory, the particles are strings, with extension in one dimension. This gets rid of infinities. That is, the Δx does not go all the way to zero but instead cuts off at some small, but nonzero value. This means that there will be a large, but finite value of the momentum and hence Δpx does not become infinitely large. Instead the uncertainty of the momentum goes to a large, but finite value and the loop integrals can be gotten rid of. Now in order to get a cutoff by the length of the string, the uncertainty relation must be modified.
But the uncertainty relation does not need to be modified and the particles need not be replaced with one-dimensional strings. The Δx cannot be zero in the uncertainty relation, because the product of Δx and Δpx is always greater than or equal to h/2π and Planck's constant h is never zero (h = 6.625 × 10-34 joule-sec). In fact, the uncertainity relation implies that the x dimension of space has a finite quantum value. Neither Δx nor Δpx can have zero value since their product is equal to h/2π which is non-zero. Thus the uncertainty relation does not need to be modified to include a term which can serve to fix a minimum distance for Δx. And strings are not needed to get rid of the infinities.
A similar relation holds for all physical quantities whose products have
the same dimensions as Planck's constant, h.
We can obtain another equally important version of the uncertainty principle
that was proposed by Einstein, if we multiply Δpx by
vx and divide Δx by vx
(since vx = Δx/Δt):
ΔEΔt ≥ h/2π.
where ΔE and Δt represent the uncertainty of energy
and of time. This shows that the uncertainty of an energy measurement depends
on the time available to make it. Thus if an atom remains in an excited state
for a very short time, the precise energy of the state may be
determined very accurately.
For practical purposes, and when considered macroscopically, space and time are continuous: the duration of chronons is so insignificant that they may safely be equated with durationless instants; similarly, the difference between mathematical points and spatial regions of the radius of 10-13 cm is entirely negligible on our macroscopic scale. This, however, does not make the difference between the classical continuous space and time and its modern atomistic counteparts less radical.
But speculation about the nature of discrete "chronon" and "hodon" on the part of physicists have been contradictory, or at least stated in a self-contradictory language. When they claim that time consists of chronons succeeding each other, and when they claim that the duration of each individual chronon is 4.48 × 10-24 second, what do they assert except that the minimum intervals of time are bounded by two successive instants, one of which succeeds the other after the time interval specified? The concept of chronon seems to imply its own boundaries; and as these boundaries are instantaneous in nature, the concept of instant is surreptitiously introduced by the very theory which purports to eliminate it. A similar consideration can be said about the atomization of space.
In answer to the above argument, of course nothing is gained if a theory introduces in a disguised way the very concept which it overtly eliminates. But it needs to be recognized that it is almost impossible to discuss concepts in which the language involved assumes the concepts that it attempts to refute and replace. What is needed is an extensive and systematic revision of our intellectual habits associated with the traditional ideas of space and time.
The first thing that must be recognized is that these early theories of atomistic space and time spoke separately of chronons and hodons, as if they if they were atoms of space and atoms of time, betraying a prerelativistic state of mind. Before the theories of relativity, it seemed legitimate to treat space and time separately because their separation was one of the basic assumptions classical physics. The impossibility of separating space and time in the special theory of relativity is the reason for giving up the concept of absolute simultaneity or, what is the same, of purely spatial distance. By asserting the existence of the hodon, they were claiming that there is a purely spatial distance approximately equal to 10-13 cm; in other words, they were separating space from time on the microscopic level, although it was precisely on this level that the consequences of relativity were so spectacularly confirmed.
On the other hand, the assumption of hodonless or spaceless chronons does not seem to contradict directly the relativity theory, in which the existence of the infinitely tenuous world lines (that is, world lines without any spatial extent) was freely assumed. However, on closer inspection, even this assumption is incompatible, if not with the letter, then at least with the spirit, of relativity. The assumption of extensionless points, whose infinite continuous aggregates would constitute space, was merely another way of saying that space is infinitely divisible.
But there is no static space in the relativity theory. We have seen that the theory admits only successive timelike connections between events; there are no purely spacelike world lines as long as we take the relativity of simultaneity seriously. Thus the assertion of the spatially extensionless world lines is equivalent to the assertion either that there are purely spatial distances which are infinitely divisible or that the spatiotemporal distances themselves are infinitely divisible. As the first assertion is excluded, we have to consider only the second one. But to postulate the mathematical continuity of timelike world lines is contrary to the chronon theory. For this theory assumes that all timelike world lines, whether those of material particles or those of photons, are not divisible ad infinitum. We shall now see how the probability of this theory is strengthened by the converging empirical evidence which necessated the wave-mechanical theory of matter.
Thus in the light of the foregoing considerations, the assumption of the atomicity of space is superfluous because the existence of hodon is merely a certain aspect of the reality of the chronotopic (spatiotemporal) pulsations. While the chronon measure the minimum duration of events constituting a single world line, the hodon measures the minimum time necessary for the interaction of two independent world lines. Everything which had been said about the necessity of redefining spatiality can be repeated here; the only difference is that it is now being applied on the microcosmic scale. There are no instantaneous purely geometrical connections either in the macrocosm or in the microcosm; on either scale these connections should be replaced by chronogeometrical ones. On either scale, the concept of spatial distance is redefined in terms of causal independence. But while the interval of independence between, for instance, the world line of earth and that of Neptune is eight hours, it is equal to the duration of two chronons in the case of two microscopic "particles" when their "distance" is minimum.
For all practical purposes, this tiny interval may be disregarded; in other words, the temporal link between microphysical events can be regarded as instantaneous and the corresponding world lines as infinitely close. The relativistic picture of the world as a four-dimensional continuum of pointlike events is appoximately true on a macroscopic scale, but becomes seriously inadequate when microscopic relations are considered. But while the "pulsatinal" character of the world lines is incompatible only with what may be called "textbook relativity," it is entirely consistent with the basic assumptions of the theory.
In order to avoid a self-contradictory formulation of the pulsational character of space-time, we have to make a serious effort to get rid of all spatial associations with which our classical concept of time is tinged. The theory of chronons, though outwardly denying the existence of instants, really assumes their existence. What does the alleged existence of chronons mean if not the assertion that two successive instants are separated by an interval of the order of 10-24 second?
But the self-contradictory statement in the chronon theory is due to the fact that we are trying to translate the pulsational character of world lines into visual and geometrical terms. In our imagination, we represent the flux of time by an already drawn geometrical line on which we may distinguish an unlimited number of points; hence our belief in the infinite divisibility of time. The chronon theory does not basically depart from this habit of spatialization; it merely substitutes, for the zero intevals, intervals of finite length. But again these intervals are imaginatively represented by geometrical segments; and as the concept of linear segments naturally implies the existence of its pointlike boundaries, the concept of the instant, verbally eliminated, reappears in the very act by which it is denied. What is overlooked by both those who assert and those who deny the existence of chronons is that it is impossible to reconstruct any temporal process out of static geometrical elements, whether these elements are dimensionless points or segments of finite length.
The spatial picture of time is inadequate in a triple sense:
(1) because of the essential incompleteness of time,
(2) because it is wrongly suggested that time, like a geometrical line,
is without transversal extension, and
(3) because it is wrongly suggests the infinite divisibilty of time.
The last two errors led respectively to the concepts of extensionless
and infinitely divisible world lines, infinitely close to each other.
To such a view the idea of chronotopic pulsation is radically opposed,
but we have to be on guard not to slip back into spatializing fallacies
when we try to state this theory.
The difficulty which the chronon and hodon theory faces is analogous: it is extremely difficult to formulate this theory without surreptitiously introducing the concept of extensionless boundaries. Our language is so thoroughly molded by the intellectual habits created by infinitesimal calculus that we continue to speak of instants and points even when we are trying to deny them. Yet even some outstanding mathematicians have now begun to realize that the very concepts of point and instants may not be legitimate because the infinite divisibility of space and time, which two concepts presuppose, may be an unwarranted extrapolation of our limited macroscopic experience.
The continuity (infinite divisibility) of time faces a situation analogous to that confronting the concept of space. The whole concept of spatiotemporal continuity, which was so wonderfully fruitful on the macroscopic and even on the molecular level, apparently loses its applicability on the electronic and quantic level. This accounts for the simultaneous appearance of the "hodon" and "chronon" hypothesis. The shortcomings of these hypotheses -- in particular their ad hoc character, their pre-relativistic separation of space and time, their surreptitious assumption of the very concepts of points and instants which they purport to eliminate -- should nevertheless not blind us to increasing evidence that the concept of infinite divisibility of space and time is, to use the words of Erwin Schrodinger, "an enormous extrapolation" of what is macroscopically accessible to us.
In view of the close union between time-space and its physical content, the traditional concepts of matter and motion were both transformed -- and it is no exaggeration to say that they were transformed beyond recognition.
[1] Immanuel Kant, Critique of Pure Reason,
translated by Norman Kemp Smith
(Humanities Press, 1950), p. 204.
[2] Bertrand Russell, Principles of Mathematics
(New York, W. W. Norton & Company, 1903; 2nd edition, 1938), pp. 442-4.
[3] Louis De Broglie, Matter and Light: The New Physics
(New York, W. W. Norton & Company, Inc., 1939.
Reprinted by Dover Publications), p. 220.