KABUL
ORALBAY KURMANBAYULY
KAZAKHISTAN
THE
PROBLEM OF CONTINUITY
Проблема преемственностей
Сабақтастықтағы проблемалар
Annotation
The word
"continuity" is from the Latin continere meaning "to
hold together". The problem of continuity arises with respect to number,
lines, planes and solids. For example, a line is said to be continuous if
it can be infinitely divisible. This concept of the infinite divisibility of
space can be traced back to Greek thought. As we will see below, it underlies
the arguments of Zeno the Eleatic against the possibility of motion; it is
implicit in the first postulate of Euclid, according to which it is possible to
draw a straight line between any pair of points, no matter how near
or far apart they may be. This excludes the possibility of "holes" in
space. It is more explicit in the tenth proposition of the first book of
Euclid, which proves the possibility of bisecting any straight-line segment;
this excludes the possibility of any atomic length. In this respect, the modern
period added hardly anything execpt a greater explicitness of formulation.
Аннотация
Слово "преемственность" происходит от латинского
"continere" означает "держаться вместе". Проблема преемственностией возникает в отношении чисел, линий и плоскостей. Например, линия, говорят, продолжительной, если
она может быть бесконечно длинными. Это
понятие бесконечной делимости пространства можно проследить в греческой мысли.
Как мы увидим ниже, лежит в основе аргументов Зенона Элейская против
возможности движения; подразумевается, в первую постулат Евклида, согласно
которой можно провести прямую линию между любыми двумя точками, независимо от
того, насколько близко или далеко друг от друга они могут быть. Это исключает
возможность "дыр" в космосе. Это более четких в десятом предложение
первой книги Евклида, который доказывает возможность деления пополам любой
отрезок прямой; это исключает возможность любого атомного длина. В этом плане
современный период добавил почти ничего, кроме большей эксплицитности
разработки.
Аңдатпа
"Сабақтастық"
сөзі латынының
"continere" деген сөзінен
шыққан, "бірге болу,
бірігу" дегенді білдіреді. Сабақтастықта проблема сандар мен
сызықтар, және жазықтықтарға қатысты
туындайды. Мысалы, сызық, егер ол ұзақ болса жалғасып
жатуды айтады. Бұл ұғымды кеңістіктің шексіз
бөлінгіштік байқауға болады дейді грек ойларында.
Төменнен көретіндеріңіздей, Зенона Элейскаяның қозғалысқа қарсы
мүмкіндігі дәлелдерінің негізінде, бірінші жорамал
Евклиданың кез келген екі
нүкте арасындағы желіні бір-бірінен жақын немесе алыс болуы
мүмкін екендігіне қарамастан тікелей өткізуге болады.
Евклидтің бірінші кітабының оныншы сөйлемінде барынша
анық көрсетілгендей бұлкез келген атомның ұзындығын
екіге бөлу мүмкіндігін растайды. Бұл жоспарда қазіргі
дамыған кезеңде эксплицитті әзірліктерден басқа дерлік
ештеңе қоспады.
The problem of
continuity was raised by Zeno of Elea (490-430 B.C.), a disciple of
Parmenides (c.515-c.450 B.C.) and one of the most prominent Eleatic
philosophers. Zeno is known for his skillful defense of the Parmenides'
doctrines. Aristotle regarded him as the inventor of the dialectic, the
method of refutation and deductive proof. With this method, Zeno supported
Parmenides' view of reality as one changeless, indivisible being by developing
the paradoxes of space, time, motion, and change, which he believed to be
implicit in the commonsense view of the world. The following paradox is
characteristic of Zeno's method and was reported in Aristotle's Physics,
VI, 9.
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.
Atomism. The
atomic theory of Leucippus (5th B.C.), and Democritus (460-370 B.C.), a
follower Leucippus, rejected the concept of continuity and held that the
ultimate constituents of things to be spatial entities that are not further
divisible. They held that all things consists of many (pluralism) indivisible
things called atoms (Greek: a, "not", and tomos,
"cutable", hence "not able to cut"). Each atom is
absolutely homogenous, indivisible and immutable, just like the One of
Parmenides. They are separate by empty space or void which Parmenides denied.
The objects of ordinary life are thus compounded of these entities.
Aristotle. Aristotle
(384-322 B.C.) attempts to resolve the problem of continuity by defining
continuity in the following words:
"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.
Augustine. Augustine
(A.D. 354-430), confessing that he knows what time is when no one asks, and
that he does not know what it is when anyone does ask, held that God is
timeless and has created the world with time, having a beginning. He was fully
aware of the paradox of this view. All that we really know about time is that
time is in us, measuring time as before and after, the memory of the past and
the anticipation of the future. Augustine as young man found himself incapable
to understand Christian theism until he had learned and discovered in
Neoplatonism the basis for the existence of immaterial entities (the Ideas). He
finally concluded that these eternal ideas, or rationes aeternas, existed
in the mind of God. From Neoplatonism he derived his basic understanding of
God. With regard to God's nature, Augustine with Newoplatonism affirms God's
perfection, eternity, infinity, incomprehensibility, simplicity and unity.
Augustine's understanding of God as "the inexhausible light" or
"intelligible light" was derived from the Neoplatonic understanding
that intelligible light is original, and physical light is derivative, the
former causing the latter. The Neoplatonic theory of emanation was based on the
metaphor of light radiating and enlightening all. Augustine rejected the
Neoplatonic theory of emanations and held to the Christian doctrine of
creation ex nihilo ("out of nothing") at the moment chosen
by God. The world and time thus had a definite beginning. But what God wills to
create is determined by what God's knowledge has determined to be good. God's
intellect is the primary motive to create. From Neoplatonism, Augustine also
derived his conception of God as timeless. The One is unchanging, therefore,
timeless. God is not only eternal, having no beginning nor end, but He is
without time, no past nor future, but just an eternal "now". In this
eternal "Now", God sees all the past, present, future of the world
that He will and has created. According to Augustine, eternity is motionless,
no succession; everything is present at once; there is no past nor future. Time
was created by God out of nothing, ex nihilo. It is not an independent
principle nor a being, the Receptacle, nor non-Being, the void.
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.
Descartes. As
a physicist, the French philosopher, Rene Descartes (1596-1650), rejected the notions
of space as a void and argued that all is matter, so that even the motion of
the planets can be described by vortices in a background plenum (A plenum is
literally a full space, the opposite of a vacuum), a view strongly rejected
later by Newton. But since Descartes was also a mathematician, he was concerned
with giving a measure and order to this plenum. By a stroke of genius, he hit
upon a method to label or specify a point anywhere in space. His invention was
the coordinate grid and its system of coordinates. With two axes a point can be
located on a page, a two dimensional space. And with three coordinates a point
can be located in three-dimensional space. Corresponding to each point, there
is a trio of numbers: points and numbers become compllementary ways of
discussing space.
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.
Leibniz. Leibniz
(1646-1716) distinguished sharply between mathematical and physical continua.
He found a law of continuity in both thought and reality, in geometry and in
nature. He fashioned it on the model of the mathematical infinite, according to
which there are no breaks in the series, "but everything takes place by
degrees." Because of this law, rationality applies to reality, and other
laws are possible. As a result of this law, it can be expected a fulness or
plenitude in nature, such that there will be no discontinuous changes. That is,
there will be a graded hierarchy of monads, from those which are infinitely
close to insentience to the monad exemplifying perfect being, that is, God. In
Leibniz' language of perception the hierarchy runs from monads whose
perceptions are almost totally confused to the monad all of whose perceptions
are distinct; and in his language of appetency or propensity, the hierarchy
runs from the almost totally passive to actus purus [pure act].
Immanuel
Kant. The following statement of Kant (1724-1804), whose philosophy was so
thoroughly imbued with the spirit of classical physics and Euclideam geometry,
is fairly typical.
"The properties of
magnitude by which no part of them is the smallest possible, 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]
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.
Newton. The
creation of the differential and integral calculus by Newton (1642-1727) and
independently by Leibniz led to a new stage in understanding the philosophical
problem of continuity. Leibniz' conception of mathematical continuity contains,
implicitly or explicitly, the definition of a continuous function, as found in
standard twentieth century calculus textbooks. Newton gave many clear analyses
of continuous and differentiable functions. In his Principia, Newton
clearly states that the ultimate ratios in which quantities vanish are not
ratios of the ultimate quantities, but limits to which the ratios of decreasing
quantities approach. That is, the derivative of y with respect
to x, dy/dx, is not the ratio of two infinitesmals, but is the limit
of a series of difference quotient.
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."
["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."]
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.
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.
Relativity. Einstein's
theory of Relativity threw strong doubts upon the existence of the continuous
medium, the aether; it had to be abandoned, being replaced by electromagnetic
fields. But the concept of spatiotemporal continuity seemly remained unaffected
by relativity theory. Relativistic space-time was defined as a
four-dimensiona continuum of pointlike events; there are no doubts
about theinfinite divisibility of spatiotemporal regions. The world lines
of Minkowski were considered to be continuous as the trajectories of classical
physics. This is not surprising: the relativity theory was by its nature a macroscopic
theory whose main object was the macrostructure of the universe. Its most
revolutionary consequences concerned astronomy and cosmology, that is, the
structure of the universe at large; the question whether the concept of
spactiotemporal continuity is legitimate does not arise on the macroscopic
level. The non-Euclidean character of space-time is unimportant even on the
planetary scale, and its curvature may be even more safely disregarded on the
atomic level.
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.
Modern
Mathematics. As we saw above, the problem of continuity arises in
mathematics with respect to numbers, lines, planes, solids, and series. In
modern mathematics, the idea of compact or dense order, asserting that between
any two distinct members of a class or arguments of a function there is always
a third, was introduced in part by Julius William Richard Dedekind (1831-1916)
and Georg Cantor (1845-1918). In set theory, a continuum is a nondenumerable infinite
set.
Modern Physics. In
physics, the problem of continuity arises with respect to matter, energy, and
fields. Is matter continuous or is it composed of discrete particles called
atoms? Are energy and fields continuous or are they composed of discrete units
called quanta? Modern physics has answered that matter, energy and fields are
discontinuous. But physics also treats these phenomena with various theories of
continuity. When a liquid and a solid is considered in these theories it is
assumed that the material substances"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]
Neils Bohr. This
quantum theory of light had a profound influence on the Theory of Matter, when
it was realized that the motion of material particles on a very small scale did
not move according to the laws of classical Mechanics. As the result of the
experiments of the British physicist Lord Rutherford in 1911, the Danish
physicist Niels Bohr (1885-1962) in 1913 developed the theory that atoms
consists of electrons and protons. The atoms of an elementary substance was
shown by Rutherford's experiments to consists of a central nucleus with
electrons revolving around the nucleus. The nucleus has a positive charge equal
to a whole number N times the charge of the proton, and
with N negatively charged electrons revolving around the nucleus. So
the entire atom is electrically neutral, since the charge on the proton is
equal to the charge on the electron but with opposite sign. Almost the entire
mass of the atom is concentrated in the nucleus, the protons being more massive
than the electrons. The hydrogen atom is the simplest of the atoms, and
consists of nucleus of a single proton around which a single electron revolved.
The atoms of one element are differentiated from that of another element by the
number N of protons in their nucleus. The electrons revolve about the
nucleus in a kind of miniature solar system, with the nucleus as the sun and
the electrons as the planets. Niels Bohr developed an explanation how
the electrons revolved about the nucleus by borrowing from the quantum theory
previously developed by Planck. By quantizing the angular momentum of the
electrons as they revolved about the nucleus, Bohr was able to restrict the
electrons to certain orbits about the nucleus. This allowed him to quantized
the radiation emitted or absorbed by the atom. Thus he was able explain the
spectrum of an elementary substance which are composed of atoms.
De Broglie. In
September, 1923, Louis De Broglie (1892-1987) presented two papers that became
his doctoral dissertation in which he proposed that "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 thesimultaneous 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.
Quantum
Mechanics. One of De Broglie's thesis examiners knew Einstein and passed
the thesis to him, who in turned recommended it to another colleague, Erwin
Schrodinger (1887-1961). Few people paid attention to the thesis, but
Schrodinger changed all that. He developed De Broglie's ideas mathematically
and published in March, 1926, a single equation, now called Schrodinger's
Equation, purporting to explain all aspects of the behavior of electrons in
terms of De Broglie's waves. Thus was born a new branch of physics,
called Quantum Mechanics.
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. Quantum mechanics and general relativity were the major
developments in theoretical physics in the twentieth century. Unifying them
into a single theoretical theory has proven to be extremely challeging, if not
impossible. This is because the resulting quantum theories are plague by
infinities that result from the fact that interactions take place at a single
mathematical point (zero distance scale). By spreading out the interactions,
string theory offers the hope of developing not only a unified theory of
particle physics, but a finite theory of quantum gravity.
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.
Atoms of Time and
Space. It may be objected that Heisenberg's Principle does not necessarily
lead to the concept of the minimum spatiotemporal atomicity, although it is
compatible with it. It is theoretically conceivable that while Δpx is increased without limit, Δx will
approach zero, similarly, for ΔE → ∞, Δt would be equal to zero. Although many physicists have claimed
that the Heisenberg's Principle is mathematically compatible with the existence
of pointlike positions and mathematical instants which the principle of
spatiotemporal continuity requires, the principle in its mathematical
formulation does not allow for that. Planck's constant being non-zero does not
allow for that Δx or Δt to be equal to zero. The minimum possible temporal interval Δt would be equal to l0/c, wherel0 is the minimum length
and c is velocity of light. The estimated numerical values of chronon
and hodon was found by Levi, Pokrovski, Beck, and others during the period
between the two World Wars. In that period, the new names "chronon"
and "hodon" were invented for designating the atoms of time and of
space, respectively. The value computed for the chronon was naturally extremely
small. According to J. J. Thomson, it is of the order of 10-21 second,
while according R. Levi it is 4.48 × 10-24 seconds. The computed
magnitude of the hodon is of the same order as the radius of the classical
electron which is 10-13 cm.
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 spatialdistance. 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.
Spatiotemporal
Pulsations. But if we accept the fusion of space and time even on the
subatomic level, then it is evident that no separation of hodon and chronon is
possible; they are complementary aspects of a single elementary entity which
may be called a pulsation of time-space. Thus there is no chronon without
a hodon and vice versa. To postulate a timeless (that is, a chrononless)
hodon would mean that instantaneous cuts of four-dimensional processes are
possible, at least on the atomic level, that there are absolutely simultaneous
events within atoms. More specifically, it would mean that there are within the
atoms couples of events interacting with infinite speed; for we have seen that
absolutely simultaneous events would lie on a world line of
any instantaneous physical action. All these assumptions (which are
really one assumption in several forms) are contrary to the special theory of
relativity and thus their plausibility is very small.
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,
arenot 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
absence of the beginning of time was one of the most cherished dogmas of
classical thought; it mattered little whether it was interpreted theologically
as the infinity of the divine duration (Newton) or naturalistically as the
beginningless cosmic duration (Giordano Bruno and others). Yet even this dogma
is now being challenged by some cosmogonic theories, in particular by the theory
of the expanding universe. In fact, unless we amend this theory by some
additional assumptions, the denial of the beginningless past follows from it.
(It is possible to avoid this consequence by assuming successive periods of
expansion and contraction, but this would be precisely an amending assumption.)
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.
5.
Ibid., pp. 46-47.