PROBLEMS
Questions of the Future.--Problem of Three Bodies.--Conception of
Approximation.--Object of Mechanics.--Simplicity of Description.--Limits
of Proof.--Reflections about the Circle.--From the History of
Errors.--Causalities.--Relativity on a Physiological Basis.--Physicists
as Philosophers.
WE spoke of the objects and problems of science in general, and touched
on certain recurrent questions with which reputed men of science are
confronted from time to time, so that we may ascertain their opinions
about immediate as well as more remote aims, and about worthy objects
and those within reach.
"Such stimuli," said Einstein, "may be quite interesting inasmuch as
they sharpen the appetite of the public for the works of investigators,
and give the latter the opportunity of making wider circles acquainted
with their plans. Yet the value of their suggestions must not be
overrated, when they are directed at giving trustworthy information
about the future lines of development of science. Every scientist, in
working out his own research, gravitates to particular points on the
boundary which separates the known from the unknown, and becomes
inclined to take his particular perspective from these points. It must
not, however, be expected that these individual aspects will form a
complete picture, and will indicate the only paths along which science
can or will advance."
"May I suggest, Professor," I answered, "that we select certain answers
that have been given to these recurrent questions for discussion? I have
brought along a whole series of them; it would be of value to know what
attitude you take up towards some of the statements that have been made
about future possibilities."
Einstein acquiesced, and so I read out a number of expressions of
opinion, given by eminent authorities, particularly in natural science
and mathematics. They came under the heading, "The Future Revolution of
Science." At the outset we encountered arguments by Bailhaud, the
director of the Paris Observatory; he dealt with the so-called "Problem
of Three Bodies," and with "The Finitude or Infinitude of the Universe."
Einstein elucidated these questions as follows. The celebrated Problem
of Three Bodies is a special case of the general problem of Many Bodies,
the object of which is to discover the exact paths of the heavenly
bodies. If we suppose that the planets and the comets are subject only
to the attraction of the central body, the sun, then their paths would
be exactly those given by Kepler's Laws--that is, they would move about
the central body, or, more precisely, about the common centre of gravity
in perfectly elliptical orbits. The same result would happen if we
regard the orbit of a moon to depend solely on its parent planet. But
this assumption is not in agreement with reality, since all the bodies
of our system are also subject to their mutual attraction depending on
their masses and distances. Consequently we have the so-called
disturbances, perturbations, and divergences from the ideal paths; and
the problem of ascertaining these disturbances is essentially identical
with the Problem of Three Bodies. Regarded from the point of view of
pure mechanics, this problem may be considered solved in so far as we
are able to write down the equations of motion. But, in addition to this
purely mechanical process, there is a mathematical problem which has not
been completely solved--that is to say, the integral expressions that
occur in it can be calculated only approximately. This makes no
difference to the practical calculation, since the degree of
approximation, according to the present methods, may be carried as far
as we wish. The error may be reduced to any desirable extent, so that it
is probably wrong to expect new revelations on this point from future
upheavals in physics. We read on and discovered that several of the
scientists mentioned did not stop at expecting all advances of the
future from pure theory. They had visions of an optimum of happiness, to
gain which the increase of knowledge alone did not suffice. Thus the
celebrated Swedish astrophysicist Svante Arrhenius had summarized his
judgment in a few lines: "After the stupendous progress that has been
made in the physical and chemical sciences in recent times, it seems to
me that the moment has come for attacking the most important problems of
mankind with full success, namely, those of biology, and in particular
of the art of healing, with the weapons that are furnished by the
arsenal of the exact sciences." And the mathematician, Emile Picard,
Membre de l'Académie, expressed himself in still more hopeful terms:
"There is no doubt but that the discoveries which the human race is
awaiting with impatience are those that are seeking to eliminate
sickness and the decrepitude of old age. Injections giving immunity
against all diseases, an elixir of life (_une eau de Jouvence_) for
persons of advancing age--these are the discoveries that are longed for
by every one. There are also sciences that are to be termed 'moral,'
from which we are impatiently expecting that guidance which will
diminish the hate which seems to be increasing from day to day among the
nations. That would be a splendid discovery."
"These are, indeed, noble and inspiring words," said I. "It shows how
deeply rooted is the demand for ethical values in human nature, when
even a mathematician, whose intellectual interests are directed
primarily towards exact results, ranks the discoveries of ethics above
all others."
Einstein answered: "We must carefully distinguish between what we wish
for in general and what we have to investigate as belonging to the world
of knowledge. The question under consideration is not one of wishes and
feelings, but was unmistakably aimed at the advances and revolutions in
the realm of science. It does not come within the scope of science at
all to make moral discoveries! Its one aim is rather the Truth. Ethics
is a science about moral values, but not a science to discover moral
'truths.' Ethics, conceived as a science in the usual way, can therefore
serve to discover or to promote truth only indirectly. To illustrate my
point of view I shall quote an example taken from a totally different
field; it is merely to serve as an analogy. Let us consider the game of
chess. Its value and its meaning is not to be sought in scientific
factors, but in something entirely different, in a struggle which takes
place according to definite rules. But even chess, inasmuch as it
sharpens the intellect, may exhibit an indirect value for promoting
truth. It may, for instance, suggest examples in permutations, which may
contain mathematical, that is, purely scientific, truths. I certainly
do not deny that there is an ethical factor in all genuine sciences. For
being occupied with things for the sake of truth alone emancipates and
ennobles the mind."
"This ennobling effect," I interposed, "should surely show itself in a
moderation of the passions which were mentioned in the above expression
of opinion. With Picard we should expect above all things to see a
diminution in the feelings of hate between peoples, the tragic
consequences of which we have experienced."
Einstein smiled, and, with a touch of sarcasm, said, "Hate is presumably
a privilege of the 'cultured,' who have the time and the energy for it,
and who are not the slaves of care." His tone indicated clearly that he
used the generic term "cultured" to denote the Philistines of culture,
its snobbish satellites, but not those whose intensive work aimed at
increasing and deepening the fields of culture. In general he maintained
his view that it is an illusion to expect "discoveries" in the realm of
ethics, since every real discovery belonged alone to the sphere of truth
in which the division only into right and wrong, not that into good and
evil, holds good.
This led us to the old question of Pilate: What is Truth? In seeking an
answer to this question Einstein first called special attention to the
conception of "approximation," which plays a great part in the actual
search for truth, inasmuch as every physical truth, expressed in
measures and numbers, always leaves some remainder, that marks its
distance from the unattainable truth of reality. This conception, which
manifests itself so prominently in the relation of Einstein's own
researches to the older, so-called classical, mechanics, will be
developed here according to his line of thought as far as I can
recollect from a number of conversations.
Let us suppose that we overhear two people arguing about the shape of
the earth's surface. The one affirms that it is an unlimited plane,
whilst the other maintains that it is a sphere. We should not hesitate a
moment to say that the first is in error, and that the second gives the
true answer. As long as the question was to be decided in favour of a
"Plane or a Sphere," the sphere would represent the absolute truth. Yet
it would be only relative, for these two statements are contradictory
only between themselves, but will no longer be so if a third assertion
is made which opposes a new alternative to "sphere."
If this alternative objection is actually raised, the third person would
be quite justified in saying that the "sphere" explanation is wrong. For
the conception "sphere" requires that all diameters be equal, whereas we
know that they are not so, since the distance from pole to pole has been
proved to be smaller than that between opposite points on the equator.
The earth is an ellipsoid of rotation, and this truth is absolute in the
face of the errors which are expressed by the terms, plane and sphere.
It would again have to be added that this absoluteness would stand only
as long as this contradiction is regarded as being one between a
definite sphere and a definite ellipsoid. If, as in the case of the
earth, there are quite different diameters in the equatorial and the
diametral planes, then there is complete contradiction between the two
statements, and as the supporter of the ellipsoid is right, the one who
supported the sphere must now give in, although he previously triumphed
over his first opponent. His statement was true compared with the
latter, but showed itself to be an error when compared with the
statement of the third person.
This does not run counter to the laws of elementary logic. One of these,
somewhat inadequately called the Law of Contradiction, states that two
directly contrary statements--_e.g._ this figure is a circle, and this
figure is not a circle--cannot both be true simultaneously. The truth of
the one implies necessarily the falseness of the other. As this cannot
be disputed, it follows in our case that we cannot have been confronted
with contradictory judgments at all concerning the figure of the earth.
This is to be understood in a geometrical sense. The sphere does not
entirely contradict the ellipsoid, since it is a limiting case of the
latter: and the plane is likewise a limiting case of the sphere, as well
as of the surface of ellipsoids.
But we are not concerned with purely geometrical considerations, for the
earth is a definite body, and not a limiting configuration derived from
abstraction. We are here dealing with measurable quantities, whose
difference can be proved, and hence we must have one of the disputants
proclaiming the absolute truth, whilst the other proclaims an absolute
error. This, however, again is incompatible with our result that the
second person is right in the one case and wrong in the other.
The logical Law of Contradiction overcomes the dilemma in the simplest
way. None of these assertions contains the truth, hence none of these
judgments allows the falseness of the others to be deduced. Only this
may be said, that there is a fraction of truth in each judgment. The
true shape of the earth is given by the plane to a first, the sphere to
a second, the ellipsoid of rotation to a third, degree of approximation:
we reserve the right of further approximations, each of which in
succession approaches a higher degree of correctness, but none attains
the absolute truth.
This reflection on a particular case may be generalized, and remains
when we extend it to our attempts at grasping the states, changes, and
occurrences of Nature. Whenever we talk of physical laws, we must bear
in mind that we are dealing with human processes of thought, that are
subjected to a succession of judgments, courts of appeal, as it were,
excluding, however, a final court beyond which no appeal is possible.
Each new experience in the course of natural phenomena may render
necessary a new trial before a higher court, whose duty is then to give
a more definite or different form to the law formulated by us, so as to
attain a still higher degree of approximation to the truth.
If we call to mind some of the most valuable statements made by modern
investigators about the nature of natural laws, we recognize that they
are all connected by a single thread of thought, namely, that even in
the most certain law there is left a remainder that has not been
accounted for, and that obliges us to consider a greater approximation
to the truth as possible, even if a final stage is not attainable.
Mechanics furnishes us with the expression of its laws in equations,
whose importance Robert Kirchhoff explained in 1874 by a definition that
has been considered conclusive by scientists. According to him, it is
the object of mechanics to describe completely (and not to explain) in
the simplest manner the motions that occur in Nature.
The postulate of simplicity is derived from the fundamental view of
science as an economy of thought. It expresses the will of man's mind to
arrive at a maximum of result by using a minimum of effort, and to
express the greatest sum of experience by using the smallest number of
symbols. Let us consider two simple examples quoted by Mach. No human
brain is capable of grasping all the possible circumstances of bodies
falling freely, and it may well be doubted whether even a supernatural
mind like that imagined by Laplace could succeed in doing so. But if we
take note of Galilei's Law for Falling Bodies and the value of the
acceleration due to gravity, which is quite an easy matter, we are
equipped for all cases, and have a compendious formula, accessible to
any ordinary mind, that allows us to picture to ourselves all possible
motions of falling bodies. In the same way no memory in the world could
retain all the different cases of the refraction of light. Instead of
trying to do the impossible task of grasping this infinite abundance, we
simply take note of the sine law, and the indices of refraction of the
two media in question; this enables us to picture any possible case of
refraction, or to complete it, since we are free to relieve our memories
entirely by having the constants in a book. Thus we have here natural
laws that give us a comprehensive yet abbreviated statement of facts,
and satisfy the postulate of simplicity to a high degree.
But these facts are built up on experiences, and it is not impossible
that some new unexpected experience will reveal a new fact, which is not
sufficiently taken into account in the law. This would compel us to
correct the expression for the law, and to seek a closer approximation
for the enlarged number of facts.
The Law of Inertia, according to our human standard, seems unsurpassable
in simplicity and completeness; it seems to us fundamental. But this
law, which prescribes uniform rectilinear motion to a body subject to no
external forces, selects only one possibility out of an infinite number
as being valid for us. It does not seem evident to a child, and it is
easy to imagine a good scholar in some branch of knowledge other than
physics, to whom it would likewise not seem evident. For it is by no
means necessary a priori that a body will move at all when all forces
are absent. If the law were self-evident, it would not need to have been
discovered by Galilei in 1638. Nevertheless, it appears to us, now, to
be absolutely self-evident, and we can scarcely imagine that it can ever
be otherwise. This is simply because we are bound to the current set of
ideas that cannot extend beyond the sum of sense-data and experiences
that have been inculcated into us by heredity and environment. At a very
distant date in the future the average mind may surpass that of Galilei
to the same extent as Galilei's surpasses that of a child, or of a
Papuan native. And of all the infinite possibilities one may occur to a
Galilei of the distant future, which, when formulated as a law, may
serve to describe motions of a body subject to no forces better than the
law of inertia, proposed in 1638.
These reflections are not mere hallucinations, but have to do with
scientific occurrences that we have observed in the twentieth century.
Newton's equation that gives the Law of Attraction is beyond doubt a
model of simplicity, and it would have occurred to no thinking person of
even the last generation to doubt its accuracy. The easily grasped
expression k (m.m^1⁄r^2) apparently expresses truth in a law which is
valid for all eternity. In this expression, he denotes a gravitational
constant, that is, a quantity which is invariable in the whole universe;
_m_ and _m_^1 are two masses that act attractively on one another; and
_r_ is the distance between them. But Newton has been followed by
Einstein, who has proved that this expression represents only an
approximate value, that leaves a small remainder as an error that may be
detected if the greatest refinement be made in our methods of
observation. The equations that have been set up by Einstein represent
the approximation that is to be considered final for the present, and
that may remain valid for thousands of years. They are certainly very
complicated, being included in a system of differential equations of
awe-inspiring length, and we may feel tempted to object with the
question: how do they agree with Kirchhoff's postulate that the simplest
description of the motions must be sought? But this objection falls to
the ground if we look carefully into the question. For simplicity
consists not merely in being brief or in excluding difficulty from a
formula, but rather in asserting the simplest relation to the universe
as a whole, which is independent of all systems of reference. When this
independence is proved--and in Einstein's case it is so--the complicated
aspect of the formula disappears entirely in the light of the higher
simplicity and unity of the world-system that presents itself--a
world-system that is directed in conformity with the one fundamental law
of general relativity as well in the motion of the electrons as in
motion of the most distant stars. With regard to the other postulate,
that of completeness, _i.e._ absolute accuracy, we have been furnished
with proofs that have rightly excited the wonder of the present
generation. But are we then to recognize the Principle of Approximation
in every direction? Is there then nothing that can be proved rigorously,
nothing that is unconditionally valid in the form of knowledge that
corresponds exactly to truth?
We are led to think of mathematical theorems, which, when they have once
been proved, are evident to the same degree as the axioms from which
they have been derived, by virtue of logic which cannot be disputed
since a contradiction leads to absurdity. It has been said that
mathematics _est scientia eorum_, _qui per se clara sunt_, that is, is
the science of what is self-evident.
But here again doubts arise. If we should get to know only a single
case, in which the self-evident came to grief, the road to further
doubts becomes open. Such a case will now be quoted.
As we know, a tangent is a straight line, which makes contact with a
curve at two coincident (or infinitely near) points without actually
cutting the curve. The simplest case of this is the perpendicular at the
extremity of a radius of a circle. And it agrees fully with what our
feeling leads us to expect when it is stated that every curved line that
is "continuous," that is, which discloses no break and no sudden bend,
has a tangent at every point. Analysis, which treats plane curves as
equations in two variables, gives the direction of the tangent in terms
of the differential coefficient, and declares accordingly that every
continuous function has a differential coefficient, that is, may be
differentiated, at every point. The one statement amounts to the same as
the other, since there must be an equivalent graphical picture
corresponding to every functional expression.
But this apparently rudimentary theorem involves an error, which was not
discovered before the year 1875. The theory of curves has been in
existence for centuries, but it occurred to no one to doubt the general
validity of this theorem of tangents. It was regarded as self-evident,
as a mathematical intuition. And certainly neither Newton, nor Leibniz,
nor Bernoulli, not to mention the mathematicians of olden times, even
dreamed that a continuous curve without a tangent, or a continuous
function without a differential coefficient, was possible.
Moreover, a proof of the theorem had been accepted. It appeared in
text-books, and was often to be heard in lecture rooms; nor was a shadow
of a doubt suggested. For it was not merely a _demonstratio ad oculos_,
but it appeared directly to our sense of intuition. And we may safely
say that up to the present day no one has ever been able to _imagine_ a
continuously curved line which has no tangent; no one has been able to
picture even one point of such a curve at which no tangent could be
drawn.
Nevertheless, scientists appeared who began to entertain doubts. In the
case of Riemann and Schwarz these doubts assumed a concrete form, in
that they proved that certain functions are refractory at certain
points. But Weierstrass was the first to make a real breach in the old
belief that was so firmly rooted. He set up a function that is
continuous at every point, but differentiable at no point. The graphical
picture would thus have to be a continuous curve having no tangent at
all.
What is the appearance of such a configuration? We do not know, nor
shall we presumably ever get to know. During a conversation in which
this problem of Weierstrass arose, Einstein said that such a curve lay
beyond the power of imagination. It must be remarked that, although the
mathematical expression of the Weierstrass function is not exactly
simple, it is not inordinately complex. Moreover, seeing that one such
function (or curve) exists, others will soon be added to it (Poincaré
mentions that Darboux actually gave other examples even in the same year
that the first was discovered); there will, indeed, be found an infinite
number of them. We may go still further, and say that, corresponding to
each curve that has tangents, there are an infinite number that have no
tangents, so that the former form the exception and not the rule. This
is an overwhelming confession that shakes the foundations of our
mathematical convictions, yet there is no escape.
How may we apply the principle of "approximation" to these
considerations? May we say that the theorem that was believed earlier is
an approximation to a mathematical truth?
This is possible only conditionally, in a certain extremely limited
sense, namely, if we picture to ourselves that point in the development
of science at which the conception and properties of tangents first
began to be investigated. Compared with this stage of science, the above
theorem denotes a first approximation to the truth, in spite of its
incorrectness; for it makes us acquainted with a great abundance of
curves that are very important for us and that exhibit tangents at every
point. This knowledge brings us a step nearer to the more approximate
truth given by Weierstrass's example. In the distant future, the earnest
student will learn this theorem only as a curious anecdote, just as we
hear of certain astrological and alchemistic fallacies. He will learn,
in addition, other theorems that are looked on as proved by us of the
present day, although actually they were proved only approximately. For
what does it mean when Gauss, for example, repudiated certain proofs of
earlier algebraists as being "not sufficiently rigorous," and replaced
them by more rigorous proofs? It signifies no more than that, in
mathematics, too, what appears to one investigator as flawless, strict,
and evident, is found by another to have gaps and weaknesses. Absolute
correctness belongs only to identities, tautologies, that are absolutely
true in themselves, but cannot bear fruit. Thus at the foundation of
every theorem and of every proof there is an incommensurable element of
dogma, and in all of them taken together there is the dogma of
infallibility that can never be proved nor disproved.
It must appear extremely interesting that, at first sight, this example
of the tangent has its equivalent in Nature herself, namely, in
molecular motions the investigation of which is again largely due to
Einstein.
Jean Perrin, the author of the famous book, _Atoms_, describes, in the
introduction, the connexion between this mysterious mathematical fact
and results that are visible and may be shown by experiment, to which we
have been led by the study of certain milky-looking (colloidal) liquids.
If, for example, we look at one of those white flakes, which we get by
mixing soap solution with common salt, we at first see its surface
sharply outlined, but the nearer we approach to it, the more indistinct
the outline becomes. The eye gradually finds it impossible to draw a
tangent to a point of the surface; a straight line which, viewed
superficially, seems to run tangentially, is found on closer examination
to be oblique or even perpendicular to the surface. No microscope
succeeds in dispelling this uncertainty. On the contrary, whenever the
magnification is increased, new unevennesses seem to appear, and we
never succeed in arriving at a continuous picture. Such a flake
furnishes us with a model for the general conception of a function which
has no differential coefficient. When, with the help of the microscope,
we observe the so-called Brownian movement, which is molecular by
nature, we have a parallel to the curve which has no tangent, and the
observer is left only with the idea of a function devoid of a
differential coefficient.... We find ourselves obliged, ultimately, to
give up the hope of discovering homogeneity at all in studying matter.
The farther we penetrate into its secrets, the more we see that it,
matter, is spongy by nature and infinitely complex; all indications tend
to show that closer examination will reveal only more discontinuities.
I have not yet had an opportunity of seeing these Brownian movements
under the microscope, but I must mention that Einstein has repeatedly
spoken to me of them with great enthusiasm, of an objective kind, as it
were, for he betrayed neither by word nor by look that he himself has
done research leading to definite laws that have a recognized place in
the history of molecular theory.
As soon as we approach the question of molecular irregularities we
recognize that, when we earlier spoke of the figure of the earth in
discussing the principle of "approximation," we were still very far from
the limit that may be imagined. We had set up the three stages:
plane--sphere--ellipsoid of revolution, as relative geometrical steps,
beyond which there must be still further geometrical approximations. If
we imagine all differences of level due to mountains and valleys to be
eliminated, for example, and if we suppose the earth's surface to
consist entirely of liquid, undisturbed by the slightest breath of wind,
even then, the ellipsoid is by no means the final description. For now
the discontinuities from molecule to molecule begin, the infinite number
of configurations without tangents, the macroscopic parallels of what
the white flake soap solution showed as microscopically, and no
conceivable geometry would ever be adequate to grasp these phenomena. We
arrive at a never-to-be-completed list of functions which can never be
described either in words or in symbolic expressions of analysis.
But even if the ultimate geometrical truth is hidden behind the veils of
Maya,[6] we are yet left with the consolation that the method of
approximation, even when applied to a relatively modest degree, produces
remarkable results in the realm of numbers. Let us consider for a moment
in the simple figure of a circle the ratio between the circumference and
the radius.
[Footnote 6: Maya = appearance.]
As we know, this ratio is constant, and is called in honour of the man
who first gave a trustworthy value for it, Ludolf's number, namely, π
(pi). Thus it makes no difference whether we consider a circle as small
as a wedding-ring, or as large as a circus arena, or even one the radius
of which is as great as the distance of Sirius. And it makes just as
little difference what happens to the circle whilst it is being
measured; the above ratio must remain constant.
But here, too, a contradiction makes itself heard, issuing from one
section of modern science. It calls to mind the saying of Dove that when
professors are not quite sure about a thing they always preface their
remarks with the phrase: "it is well known that" ... We should be well
advised in avoiding this method of expression altogether, for even when
we feel quite sure, the ghost of the unknown lurks behind what we fain
would call well known.
The theorem that all circles without exception are subject to the same
measure-relation belongs _a priori_ to the synthetic judgments. But
fields of thought have been discovered in which the _a priori_ has lost
its power. Mathematics--once a quintessence of synthetic statements _a
priori_--is now regarded as being dependent on physical conditions.
Physical conditions, however, are empirical and subject to change.
Therefore, since the _a priori_ is not subject to change, we encounter a
discrepancy. It leads to the question: Is the Euclidean geometry with
which we are familiar the only possible geometry? Or, in particular: Is
π the only possible measure-relation?
Einstein replies in the negative. He not only shows how another geometry
is possible, but he also discloses what once seemed inconceivable,
namely, that if we wish to describe the course of the phenomena of
Nature exactly by means of the simplest laws, it is not only impossible
to do so with the help of Euclidean geometry alone, but that we have to
use a different geometry at every point of the world, dependent on the
physical condition at that point.
From the comparatively simple example of two systems rotating relatively
to one another, Einstein shows that the peripheral measurement of a
rotating circle, as viewed from the other system, exhibits a peculiarity
which does not accompany the radial measurement. For, according to the
theory of relativity, the length of a measuring rod is to be regarded as
being dependent on its orientation. In the case quoted, the rod
undergoes a relative contraction only when applied along the
circumference, so that we count more steps than when we measure the
circumference of the same circle at rest, that is, in non-rotation.
Since the radius remains constant in each case, we get a relatively
greater value for π, which shows that we are no longer using Euclidean
geometry.
Yet, formerly, before such considerations could even be conceived in
dreams, this π was regarded as absolutely established and immutable;
and observers used every possible means of determining its value as
accurately as possible.
In Byzantium there lived during the eleventh and twelfth centuries a
learned scholar, Michael Psellus, whose fame as the "Foremost of
Philosophers" stretched far and wide, and whose mathematical researches
were regarded as worthy of great admiration. This grand master had
discovered by analytical and synthetical means that a circle is to be
regarded as the geometric mean between the circumscribed and the
inscribed square, which gives to the above quantity, as may easily be
calculated, the value √8, that is, 2.8284271.... In other words, the
length of the circumference is not even three times that of the radius.
We have the choice of regarding the result of Psellus as an
approximation, or as mere nonsense. Every schoolboy who, in a spirit of
fun, measures a circular object, say a top, with a piece of string,
arrives at a better result, but the contemporaries of Psellus accepted
this entirely wrong figure with credulous reverence, and continued to
burn incense at the feet of the famous master. It is all very well for
us of the present to call him a donkey. We have just as much right in
saying that mathematicians differ, not in their natures, but only in the
order of their brain functions. If a man like Psellus missed the mark by
so much, it is possible that men like Fermat or Lagrange may also have
erred occasionally or even consistently.
No heavenly power will give us a definite assurance to the contrary, and
all of us may be just as false in our judgment of accepted celebrities
as were the Byzantines eight hundred years ago in their estimate of
Psellus.
Whereas the latter had obtained a value "less than 3," there are learned
documents of about the same date that have been preserved, according to
which the value of π comes out as exactly 4. Compared with this
grandiose bungling, even the observations mentioned in the Old Testament
are models of refinement. For, as early as three thousand years ago, it
is stated of the mighty basin in the temple of Solomon (First Book of
Kings, chapter VII.): "And he made a molten sea, ten cubits from the one
brim to the other: it was round all about, and his height was five
cubits; and a line of thirty cubits did compass it round about." Thus π
here appears as 3, an approximation which no longer satisfied later
generations. The wise men of the Talmud went a step further, in saying 3
plus a little more; and this agrees roughly with the actual value.
The view became more and more deeply rooted that this π was a main
pillar of mathematical thought and calculation. The more the problem of
the quadrature of the circle seized on men's minds, the greater were the
efforts made to find the exact value of this "little more" of the
Talmud. Since 1770 we know that this is not possible, for π is not
rational, that is, it can be represented only as an infinite and
irregular (that is, non-repeating) decimal expression. It occupies,
further, a special rank as a transcendental quantity; this fact was
proved by Lindemann as late as 1882 for the first time. Yet, even
nowadays, there are incorrigible devotees of quadrature, who are still
hunting a solution because they cannot rid themselves of the
hallucination that such a simple figure as the circle must submit
ultimately to a constructive process.
The correct way was to carry out an even more accurate determination of
the decimal figures. The above-mentioned Ludolf van Ceulen got as far as
the 35th place of decimals; at the turn of the eighteenth century the
100th decimal place was reached. Since 1844, thanks to the lightning
calculator Dase, we have its value to the 200th decimal place, and this
should satisfy even the most extravagant demands. This number,
associated with the circle, is a classical example of how an
approximation that is expressible in figures of very small value gives
an order of accuracy that can be described only by using fantastic
illustrations.
If we take a circle of the size of the equator, and also multiply the
value of the diameter of the earth by π, we know that the latter result
will not be exactly equal to the former, and that there will always be a
small remainder. If this discrepancy were less than a metre, the order
of exactness would be extraordinarily high, for a metre is practically
insignificant compared with a mighty circle of the dimensions of the
earth's circumference.
Let us stipulate still greater accuracy. We demand that the error is to
be less than the thickness of the thinnest human hair. We find, then,
that we must take for π at most 15 places of decimals. Thus, if we use
π = 3.14159265358973, we are applying a means of calculation that
reduces the possible error in all measurements of circles on the earth
to a degree beyond the limits of human perception.
If we pass beyond the world out into celestial space, and consider
circles of the dimensions of a planetary orbit, nay, further, if we pass
on to the Milky Way or even to the limit of visible stars, to find space
for our circle, and if in this case we still reduce the discrepancy so
as to be less than any length that is observable under a microscope,
then the last given value of π still suffices. Yet we must not forget
the proviso: _semper aliquid haeret_, something unsolved still clings to
the problem.
Such numerical approximations, however instructive they may be,
nevertheless retain a comparatively playful character, and furnish only
a superficial analogy to the most important approximations that are
contained in our natural laws themselves. It is these, above all, that
manifest themselves so clearly in Einstein's life-work, and they bear
the same relation to the former as truth bears to correctness. Truth
comprises the greatest conceivable circle of ideas and passes far beyond
the sphere of correctness, which deals only with measure-relations, and
not with the things in themselves. If Einstein, as we learn,
emphatically declares truth to be the only object of science, he means
the strictly objective truth that is to be derived from Nature, the true
relationship of phenomena and occurrences, independently of whether
restless philosophy assigns a question mark to this ultimate
objectivity. A great discoverer in the realm of Nature cannot and dare
not proceed otherwise. For him there is behind the veil of Maya not a
phantom that finally vanishes, but something knowable, that becomes ever
clearer and more real as he detaches each successive veil in his process
of approximation.
During this conversation, when we were talking of the "Future of the
Sciences," Einstein gave his ideas free rein, shooting far ahead of the
views and prognostications of the above-mentioned scientists:
"Hitherto we have regarded physical laws only from the point of view of
_Causality_, inasmuch as we always start from a condition known at a
definite cross-section of time, that is, by taking a time-section of
phenomena in the universe, as, for example, a section corresponding to
the present moment. But, I believe," he added, with earnest emphasis,
"that the laws of Nature, the processes of Nature, exhibit a much higher
degree of uniformity of connexion than is contained in our
time-causality! This possibility suggests itself to me particularly as
the result of certain reflections concerning Planck's Quantum Theory.
The following may be conceived: What belongs to a definite cross-section
of time may in itself be entirely devoid of structure, that is, it might
contain everything that is physically conceivable, even such things (so
I understood him to say) as, in our ordinary physical thought, we
consider impossible of realization, for example, electrons of arbitrary
size, and having an arbitrary charge, iron of any specific gravity, etc.
By our causality we have adjusted our thought to a lower order of
structural limitations than seems realized in Nature. Real Nature is
much more limited than our laws imply. To use an allegory, if we regard
Nature as a poem, we are like children who discover the rhyme but not
the prosody and the rhythm." I interpret this as meaning that children
do not suspect the restrictions to which the form of the poem is
subject, and just as little do we, with our causality, divine the
restrictions which Nature imposes on occurrences and conditions even
when we regard them as governed by the natural laws we have found.
Thus a leading problem of science in the future will be to discover the
restrictions of Nature as compared with the apparent causality implied
in physical laws.
We have in this an example of the transcendental perspectives that are
opened up when we accompany Einstein on one of his excursions of
thought. In this case it is actually a question of ultimate things, of a
region of discovery of which we cannot yet form a conception, and it
appears doubtful whether the problems latent in it are to be treated by
making investigations into physical nature, or whether they are to be
allotted to speculative philosophy.
In the first place, Einstein's remark seems to aim at nothing less than
a revision of the conception of causality. However much has been done to
purify this conception and to make it clear, we have here, perhaps, a
new possibility of refining it by making a synthesis of scientific and
abstract philosophical views. We shall just touch very lightly and
superficially on the possibility of a synthesis giving us an avenue to
truth. Whoever has heard these words of Einstein, feels the need of
getting on to firm ground to rescue himself out of the turmoil of ideas
into which he has been plunged.
What is Causality? A physiological answer may be given by saying that it
is the irrepressible animal instinct, rooted in our brain-cells, that
compels us to connect together things that we have experienced and
imagined. Poets have defined Hunger and Love as the fundamental elements
of our social lifes; we need only add the thirst for causality to this
to complete the list of primary instincts. For this mental thirst is not
less intense than our bodily hunger, and is even greater in that it
never forsakes us for a moment. It is easier for the body to check
breathing than for the soul to still the question of the why and
wherefore, of the cause and effect, of the antecedent and consequent.
This ceaseless search for a connexion between occurrences has become
organized into a fixed and immovable form of thought, which remains
mysterious even when we imagine that we have eliminated all the mystery
from it. The relations that we seek and that we regard as being of an
elementary character are totally foreign to Nature herself. David Hume,
the first real, and at the same time the most penetrating, explorer into
this form of thought, said that, in the whole of Nature not a single
case of connexion is disclosed which we are able to grasp. All
happenings appear, in reality, disconnected and separate. One "follows
on" another, but we can never detect a connexion between them. They
appear "co-joined," but never "connected." And since we can form no idea
of what has never presented itself to our outer or inner perception, the
necessary conclusion seems to be that we have absolutely no idea of
causal connexions or causative forces, and that these expressions are
quite devoid of meaning, however much they may be used in philosophical
discussions or in ordinary life. This "Inquiry concerning Human
Understanding," with its atmosphere of resignation, has been elaborated
in manifold ways, particularly by Kant and the Kantians; for it is
impossible to take up a philosophic thread without entering on an
examination of the fundamental question concerning the existence of a
causality which lies outside our instinct for causality. It is also
inevitable that, whenever we start out in this direction, we encounter
the further question: What is Time? For causality directs itself to the
problem of succession, both of sensations and phenomena, consequently
the two questions are not only intimately connected, but are really only
different expressions of one and the same question. Time, which
according to Descartes and Spinoza is a _modus cogitandi_, not an
_affectio rerum_, and, according to Kant, is an _a priori_ form of
thought, dominates our intelligence with the same sovereign power, as
the imagined course of things: what we perceive in the corresponding act
of thought is regarded as temporal and causal, and impossible of further
analysis.
Now, the conception of time has been entirely revolutionized by Einstein
himself; and it may be expected that the conception of causality,
too--which, in accordance with custom, we still endow with a separate
existence--will also be affected by this revolution.
We thus approach a relativization of causality, and we may advance a
step further in this direction, if we call to mind the differences of
time-perception that Nature herself leaves open to us. It must be
clearly understood that we are not dealing at present with the
theoretical time of physics, in the sense of Einstein's theory, but with
something physiological that ultimately, however, resolves itself into a
relativization of time, and hence also of the causal connexions in time.
To do this, we have to follow the lines of reasoning developed by the
celebrated St. Petersburg academician, K. E. von Baer, and we need
extend it only very little to get at the heart of causality, if we start
from his address of 1860: "Which View of Living Nature is correct?" For
the human brain is a part of living nature, and hence the processes of
thought may also be conceived as expressions of life.
The starting-point is a figment, the fictitious character of which
vanishes as soon as we approach its results. The bridge of thought may
be destroyed later; it suffices to carry us temporarily, as long as it
lands us in safety on the other side.
The rapidity of perception, of the arbitrary motions, of intellectual
life seems in the case of various animals to be proportional
approximately to the rapidity of their pulse-beats. Since, for example,
the pulse of a rabbit beats four times as quickly as that of a bull, it
will, in the same interval of time, also perceive four times as quickly,
and will be able to execute four times as many acts of will, and will
experience four times as much as the bull. In the same astronomical
length of time the inner life and perceptual world, in the case of
various animals, including Man, will take place at different specific
rates, and it is on these rates that each of these living creatures
bases its subjective measure of time. Only when compared with our own
measure of time does an organic individual, say, a plant, appear as
something permanent in size and shape, at least within a short interval.
For we may look at it a hundred times and more in a minute, and yet
notice no external change in it. Now, if we suppose the pulse-beat, the
rate of perception, the external course of life, and the mental process
of Man, very considerably accelerated or retarded, the state of affairs
becomes greatly changed, and phenomena then occur, which we, fettered by
our physiological structure, should have to reject as being fantastic
and supernatural, although, on the supposition of a new structure they
would be quite logical and necessary. If we suppose human life from
childhood to old age to be compressed into a thousandth part of its
present duration, say, into a month, so that the pulse beats a thousand
times more quickly than occurs in our own experience, we should be able
to follow the course of a discharged bullet very exactly from point to
point with our eyes, more easily than we can at present observe the
flight of a butterfly. For now the motion of the bullet in a second will
be distributed among at least 1000 pulse-beats, and will induce at least
1000 perceptions, and accordingly, in comparison with our everyday
perception, it will appear 1000 times slower. If the duration of our
life were again to be reduced to a thousandth of its first reduced
value, that is, shortened to about forty minutes, then our flowers and
herbs would seem just as motionless and immutable as rocks and
mountains, in which we only infer the changes without having directly
observed them. We would in the course of our lifes see little more of
the growth and decay of a bud and a flower in full bloom than we at
present see of the geological changes in the earth's crust. The acts of
animals would be much too slow to be seen; at most, we could infer them
as we do the motions of the stars at present. If life were shortened
still further in the same way, light would cease to be an optical
occurrence to us. Instead of seeing the things on which light falls, we
should become aware of them as being audible, and what we at present
call tones and noises would long have ceased to have an effect on the
ear.
If, however, we let our fancy roam in the opposite direction, that is,
if, instead of compressing the duration of human life, we expand it
enormously, what a different picture of the world would present itself!
If, for example, the pulse-beat, and hence the rate of perception, were
to be made a thousand times slower, so that the average human life would
be spread out over, say, 80,000 years, and that we should experience in
one whole year only as much as we now experience in a third of a day,
then, in every four hours winter or any other season would pass by,
vegetation would spring up and as rapidly die. Many a growth would not
be perceptible, on account of its relative rapidity compared with the
rate of the pulse-beat. For example, a mushroom would suddenly come into
existence, like a newly formed spring. Day and night would alternate as
a light and a dark minute; and the sun would appear to fly over the
heavens like a fiery projectile. If we were again to make the duration
of human life a thousand times longer still, and hence the rate of life
a thousand times slower still, we should, during the whole of an
ordinary year, be able to have only 190 distinct perceptions, so that
the difference between day and night would vanish entirely, and the
sun's path would be a glowing circular band in the heavens, and all
changes of form that seem to us to happen quietly and regularly, and to
preserve a certain permanency, would melt together in the wild stream of
happening, engulfed in its onward rush.
Are we justified in opposing to this relative perception of time "our
own" time, which is something specific and dependent on our constitution
as human beings? Should we not rather adopt the view that this specific
time, adapted to our particular pulse-beat, gives only a very limited
picture of the world, which is conditioned and determined by the
limitations of our own definite intelligence? Is it, perhaps, only a
distorted picture, a caricature, of actual occurrences?
An intelligence infinitely superior to our own would no longer be
dependent on the separate sensations such as are presented to us with
the rhythm of the pulse. For such a mind there would be no metronomic
foundation in the sequence of occurrences, beyond what represents itself
as time to our understanding. He would be situated outside of time in
what Thomas Aquinas called the _nunc stans_, in the stationary present,
without a retrospect of the past and without expectation of a future.
Without the Before and the After, the occurrences of the world would
acquire the clearest and simplest meaning, like that given by an
equation of identity. What presents itself to us as a "succession" of
events would merge together into one whole, just as a succession of
numerical calculations become summarized in a rule of calculation, or as
a series of logical operations resolves into a logical self-evident
truth. If the mind conceived by Laplace actually existed, it would stand
above the necessity of introducing time as a quantity into its
world-equations, for time is a purely anthropomorphic quantity, produced
by our perception, and regulated by our own characteristic pulses.
Accordingly, the conception of causality, too, which is indissolubly
connected with time, must be regarded as anthropomorphic, as something
that we read into, and not out of Nature. We should at least have to
recognize that if there is a causality outside ourselves, then we can
learn only a minimum about it, and even this only in a world displaced
or distorted by the accidental rate of our pulse-beat.
Let us now repeat Einstein's assertion "that the laws of Nature, the
processes of Nature, exhibit a much higher degree of uniformity of
connexion than is contained in our time-causality! It is possible that
what belongs to a definite cross-section of time may in itself be
entirely devoid of structure, that is, it might contain everything that
is physically conceivable, even such things as, in our ordinary physical
thought, we consider impossible of realization, for example, iron of any
arbitrary specific gravity." It seems to me that the non-physicist will,
perhaps, gain a clearer insight into these highly significant words of
Einstein, now that he has received the assistance of these physiological
considerations. It must be granted that the philosophic grounds of
Einstein are quite different and lie much deeper than those of von Baer,
who starts from organic functions and ends by arriving at a mysterious
relativity that is yet consistent in itself. Nevertheless, there is one
point of contact, inasmuch as in each case possibilities that lie
apparently _extra naturam_ are suggested.
Einstein says: "Hitherto we have regarded physical laws only from the
point of view of _causality_, inasmuch as we always start from a
condition known at a definite cross-section of time, as, for example, a
section corresponding to the present moment." At our own risk an easy
paraphrase of his words will be attempted:
The time-section of the present contains for us the sum of all previous
experiences, out of which the necessary course of our thought sifts out
the category of causality.
What is not present in experience cannot appear in our causality. Let us
consider for a moment Hume's example of the Indian who has never known
ice. Without being told, and if he is dependent only on his own
sensations, he would never learn that water freezes in cold climates.
The influence of cold on water is not gradual, corresponding to an
increase of cold, and not one that may be anticipated in all its
consequences, but at the freezing-point water, which a moment before was
a very mobile liquid, passes into a very rigid solid. The causality of
the Indian cannot account for this. If we tell him of this phenomenon,
he has two courses open to him. Either he refuses to believe it--and
this would be quite natural, since rigid water is to him as meaningless
as is a square circle to us. Or else he believes the story, and then his
list of categories incurs a break, passing through the middle of
causality. He has then to reconcile himself to the assumption that
something that is meaningless to him and that stands outside the
connexion of cause and effect is possible of realization. Up to that
moment, in his time-section of the present, there was no room for it in
his causality. To Torricelli the conception of liquid air, which we have
been able to prepare only since 1883, would have appeared impossible and
incompatible with his causality.
So there is no room in our causality for the idea of iron with the
specific gravity of air, or with one several times that of gold. For,
reasoning along the lines of our causality, we should conclude that a
substance that is so light or so heavy may, indeed, exhibit chemical
relationship with iron, but it would not itself be sufficiently defined
by the term _iron_.
Now Einstein also said: "Real Nature is much more limited (or bound)
than our laws imply." A sceptic might be disposed to take these
statements separately in order to construe a contradiction out of them.
For, if there are limiting conditions in Nature, which are foreign to
the views expressed in our laws, how would it then be possible for
phenomena, which cannot be imagined, to become realized? If Nature can
do this, surely she must have more liberty than we seek to impose on
her. This apparent contradiction vanishes if we treat the conception of
structural design or uniformity as something distinct from the measure
of all experience up to the present. This would give us the following
interpretation:
Out of the manifold of occurrences that are possible in mechanical
Nature, real Nature selects a very closely defined manifold. Thus the
true laws imply a much greater degree of limitation than those known to
us. For example, the laws known to us at present would not be affected
if we should discover electrons of arbitrary size or iron of arbitrary
specific weight. But Nature realizes only electrons of a quite definite
size and iron of a definite specific weight.
* * * * * * * *
Let us bear in mind that in aiming at ultimate truths we have no final
courts of appeal. Nor are the latter to be assumed even when, in
pursuing a theory, we encounter a difficulty, which at first exhibits
all the signs of a direct conceptual contradiction. It should rather be
realized that a fiction containing an initial but only provisional
contradiction serves as a starting-point for just those investigations
that are most subtle and that have far-reaching consequences. We should
have no Infinitesimal Calculus, no Algebra, no Atomic Theory, no Theory
of Gravitation if, to avoid all initial contradictions, we surrender the
fiction of differentials, of imaginary quantities, of the atom, of
action at a distance. In short, it may, indeed, be said that not only
knowledge, but also life, the holding together of people by convention,
law, and duty, would become impossible if we did not accept the fiction
of free will, which directly contradicts the determinate character of
all happening, including actions and motives, which, physically, alone
seems recognizable.
Fiction (not to be confused with hypothesis) and anthropomorphism, in
spite of their inner inconsistency, are the two poles about which our
thoughts and our lives revolve. And no doctrine will ever soar to such
heights that it will be able to deny completely its origin from these
roots of all thought. The Archimedean thought-centre of the universe,
which would enable us to lift the world out of its hinges, is
unattainable, because it does not exist at all.
Is this also to apply to the new physics, whose results are to be
regarded as the last word in scientific knowledge? Many a hypercritical
thinker might be led away by the current of the preceding statement, and
feel disposed to answer in the affirmative, were it not that, here too,
a contradiction intrudes itself. This is expressed in the fact that not
one of the present-day philosophers is in a position to pursue the
threads of this theoretical fabric to their hidden ends.
Thus we arrive at a parting of the ways. Whoever aims at becoming
thoroughly familiar with Einstein's new world-system finds that the
study of the theory claims so much attention that there is scarcely a
possibility left of proceeding to an ultimate philosophical analysis.
And whoever is absorbed only by the desire of making philosophic
investigations soon enough arrives at border-lines of thought, at which
his conscience warns him to beware of insufficient scientific knowledge.
He will be attacked by doubts as to whether he has properly understood
the theory. And he will be confronted with the question whether he is
justified in drawing ultimate philosophical conclusions before he has
mastered all the mathematical details.
As far as can be judged at present, only one thinker has, so far, had
sufficiently wide knowledge to enable him to correlate the physical
theory methodically with the theory of knowledge. I mean Professor
Moritz Schlick of Rostock, who has set out his ideas systematically in
his book _Erkenntnislehre_, which is extraordinary in itself and in its
great scope; it takes us beyond Kant. In Schlick's opinion Einstein's
theory furnishes us with the key to new and unexpected chambers of
thought; it is a wonderful instrument for opening up new avenues, and
would appear more wonderful still if we could use this instrument
without having recourse to anthropomorphism. This limitation may lead to
a Utopia, or may entail a _circulus vitiosus_. But we have one
philosophy nowadays which applies to what cannot be fulfilled "AS IF" it
really is capable of fulfilment. Among the disciples of Vaihinger, the
founder of the As-If-doctrine of thought, we, however, notice the
tendency to follow anthropomorphic and fictitious paths also in his
field of thought.
From numerous utterances of Einstein, I have gathered that he himself
does hot give his unqualified approval to all attempts at unravelling
the ultimate problems by means of philosophy, that is, by using
metaphysics alone. He does not deprecate these endeavours, but even
expresses admiration for some of the newer works, as for that of
Schlick, yet he sees certain obstacles in the purely philosophical
methods, that at least restrain him from taking a systematic interest in
them. This reluctant acceptance of, and doubt in, the processes of
philosophy, that has never forsaken the exact investigator, this
suspicious attitude which scents traces of sophistic and scholastic
machinations in all metaphysical arguments, also asserts itself in him
in a noticeable form. He feels the absence of rigour and of consistency
of direction, which is a guarantee of progress in passing from one
result to another, in the method of thought of those who are pure
philosophers: and he deplores the spongy and murky appearance of certain
expressions of thought, which, it must be admitted, form a poor contrast
to the completeness and the crystal clearness of mathematico-physical
reasoning. There was an inscription on the portals of the Athenian
Academy which stated that entrance was forbidden to all who had had no
mathematical training; we may imagine next to it an academy of pure
transcendental philosophy, bearing the inscription: No exact research
allowed! I believe that this clear-cut distinction would tally with
Einstein's view.
In the case of the great Ernst Mach, for whom Einstein has intense
admiration, we observe a similar attitude, or we may say that, in the
language of allegory, he sang openly the same refrain in another key. He
never ceased reiterating that he was properly "no philosopher at all,
but only an investigator of Nature." At the beginning of the
introduction to one of his works we read his confession: "Without in the
slightest degree being a philosopher, or even wishing to be one ...";
and some lines further on he calls himself sarcastically "a mere amateur
sportsman" in philosophical regions. Yet, Mach's initial remark is
followed by a remarkable result, for the book in question, _Knowledge
and Error_ (Erkenntnis und Irrtum), is to be reckoned among the most
important works in philosophical literature; and he himself, the amateur
sportsman, who did not even desire to be called a philosopher, accepted
in 1895 the post of Professor of Philosophy at Vienna University. It was
merely his timidity in the face of the philosophical fraternity that had
made him emphasize repeatedly the distinction between his own work and
that of the philosophers, whereas in his heart he had nourished a
passion for Philosophy, the first mother of Science. And in my opinion
such a moment may arrive for even the most rigorous investigator when he
succumbs to the siren strains from the shores of philosophy.
As far as Einstein himself is concerned, I cannot venture on a
prognostication. Even though he belongs to the category and rank of
Descartes, Pascal, d'Alembert, and Leibniz, in whom Mathematics and
speculative Philosophy are intermingled, he is yet characterized by such
a pronounced individuality, that it is quite inadmissible to draw
conclusions about him from others. He has no need to experience a day of
Damascus, for he carries the gospel of salvation in himself, and it
radiates from him. One thing seems possible, in my opinion, namely, that
Einstein will occasionally roam into the neighbouring realm merely from
æsthetic motives. Although the means of philosophy are nebulous and
more indefinite than those of exact science, which are almost glaringly
distinct, philosophy itself for this reason is the more closely related
to Art. And a theory that applies to the whole universe must assuredly
contain many germs that may come to life if subjected to the methods of
Art. The connecting link between Kant and Schiller shows in what sense
this is to be understood. Even at present there are indications in Art
which tend to show that it is ready to establish points of contact with
Knowledge. In France symphonic poems were written on the measure
relations of the circle, and on logarithms: these are at present only
curiosities, but may in future become models. At a much later date,
perhaps, the four-dimensional universe may become ripe for treatment by
such methods of Art. On the way to this goal there is the treatment with
the symbolic, non-rigorous, and semi-poetic means of expression used by
Philosophy. Many will use their efforts to achieve this, and perhaps
they will come within closer range of success, if Einstein himself lends
a helping hand. It will not be possible to arrive at new physical truths
by following this path, but those that are actually known will be traced
more readily to the great mainstream of philosophy. To fathom the
secrets of the world is the work of a recluse, but to make it
comprehensible to a wide circle, a preacher is necessary, who uses the
beautiful methods of philosophical rhetoric. Cosmos denotes the World
and its Ornamentation; its creator, Demiurge, is a master who fashions
his forms along the lines of Art.
Thus we have learned what Einstein regards as the sole purpose of
Science, namely, the search after Truth. For him, the latter is
something absolute in itself, and the possibility of getting nearer to
it is as great as the impossibility of deriving results of scientific
use from, say, ethical discoveries. For ethics is a field which is
haunted by the conceptual ghosts, and the manner of treatment, _ordine
geometrico_, that Spinoza wished to apply to it, is reserved for
physics. Einstein leaves the inverse philosophical query: "Is not Truth
in itself only something that we have constructed in imagination?" to
those who find pleasure in sauntering along paths of thought that are
totally unconnected, whereas he himself advances in a straight line with
the consciousness that even if the goal is unattainable, he will at
least not lose the right direction!