HIGHWAYS AND BY-WAYS
Practical Aims of Science.--Pure Search for Truth.--Retrospective
Considerations.--The Practical Side of Kepler.--A Saying of
Kant.--Mathematics as a Criterion of Truth.--Deductive and Inductive
Methods.--Conceptual and Perceptual Knowledge.--Happiness and the
Pleasures of Theory.--Achievements of Science and Works of Art.--Ethical
Results.--Minor Questions.
AGAIN we chanced to refer to the great subject: Can or should
theoretical science also pursue practical aims?
It is impossible to overrate the importance of this question. It haunts
us daily and often enough looms up threateningly on the horizon of
mankind. Observe what form the discussions of educated people take when
the finest and most sublime achievements of mind are being debated: one
talks of the wonders of research in the remotest corners of astronomy
where the structures of world-wide star-systems are being investigated;
we hear observations about the theories that aim at tracing the
cosmogonic development of universes from the original chaos of countless
ages ago. We hear mention of exalted sciences, the Theory of Functions
and Numbers, whose founders and representatives are just as remarkable
in propounding problems as in solving them, and inevitably the following
question obtrudes itself: Of what use is it, ultimately? What can one do
with it? Can it be admitted that theoretical science has an object of
its own, or have we at least the right to maintain the hope that, sooner
or later, it will bring us a real "Utility" expressible in practical
terms?
And just as the devotees of pure art have framed the expression, "L'art
pour l'art," so Einstein proclaims that science is its own object,
"Science for its own sake!" It carries its aims absolutely in itself and
must not, through aiming at other purposes, stray from its own highways.
"It is my inner conviction," said he, "that the development of science
itself seeks in the main to satisfy the longing for pure knowledge,
which, psychologically, asserts itself as religious feeling."
"To yourself, Professor, the practical aspect seems comparatively
insignificant?"
"I did not say that, and it was not implied in the question. We must not
lose sight of our premises. As long as I am interested in working along
lines of research--this was the assumption--the practical aspect, that
is, every practical result that is found simultaneously or arises out of
it later, is a matter of complete indifference to me."
Far be it from me, even in thought, to wish to question this confession
of faith, particularly as the fact that it comes from a searcher of the
truth gives it the more weight. Yet a certain uneasiness has crept over
me because voices have recently made themselves heard that demand for
science a totally different tendency. They arise not only from the
public at large, but also from academic circles. Just a short time ago
I read an exposition by a well-known scientist, W. Wien, in which he
indulged in a violent polemic against the view that purely scientific
objects are alone valid. Professor Wien addressed himself particularly
to German physicists, reproaching them with underestimating technical
science, and with regarding it as a "lowering of status" when a
physicist enters into practical life.
To this Einstein remarked: "I do not know at whom this reproach is
aimed, but I venture to think that my own attitude can never have given
rise to an attack of this kind. For I make no divisions of rank, and
recognize no higher and no lower status. I affirm only what is the
nature of science herself, and the objects according to which she,
objectively, has to direct her gaze. Whatever further orientation
individual investigators may seek for themselves depends on the
determining conditions of life of each, although these conditions do not
serve as a means for deducing the main lines of research. The accusation
that I am unwarranted in putting forward this view will, I hope, not be
levelled at me, for my connexions with practice are manifold enough, and
up to the present moment I have often collaborated with practical
physicists...."
"As I have regretfully observed when you were obliged to interrupt a
conversation with me to give an audience to impatient persons seeking
advice in technical matters!"
"My own associations with the world of practice are not, indeed, of
recent date. My own parents originally wanted me to become a technical
scientist, and I was expected to choose this profession to earn my
livelihood. I was not, however, sympathetically inclined to it, for even
at an early age these practical aims were to me, on the whole,
indifferent and depressing. My idea of human culture did not coincide
with the current view, that cultural development is to be measured in
terms of technical progress. Doubts, indeed, arose in me as to whether
technical improvements and advances would actually contribute to the
well-being of mankind. I must add that, later, when I came into actual
touch with technical science, my opinion became somewhat modified, for
the reason that, here too, pleasures of theory often visited me."
The true position is probably that the technical worker who does not
merely think out improvements for machines, but occupies himself with
inventions on a higher plane, never ceases to feel himself a theorist,
since his achievements are dependent for their inspiration on the fruits
of theory. The practical results of to-day are rooted in the theoretical
results of decades ago, and what is nowadays regarded as an idea of pure
research may in later decades acquire practical value. Whether it
actually becomes of value, or not, is of little account in judging the
idea. At any rate experience has shown that the beginning of theoretical
investigations hardly ever gives us the chance of making
prognostications. We spoke of the discoveries of Volta, Ampère, and
Faraday. When these were first known, the world might have asked: Why
have they been disclosed? To what can they be applied? Of what use are
they? Nowadays we know the answers that still lay hidden at that time,
and we proudly point to modern dynamos. But does a dynamo really
represent the significance of these discoveries? Would the importance
and rank of Volta, Ampère, and Faraday be less if the dynamo had not
come into existence? Only an out-and-out materialist would affirm this,
and, strictly speaking, the question should not even be raised. For it
is in a sense equivalent to wishing to judge of the importance and
significance of the Polar Star from its usefulness to the navigator on
the earth's surface in finding his bearings. We may put the question
(although only in the spirit of psychological curiosity, and without
expecting much elucidation): Would these discoverers have been
particularly happy if they had divined the far-reaching consequences of
their work? Did they, indeed, in the course of their abstract
researches, have a pre-vision of the future dominated by the dynamo?
Einstein refused to answer this in the decisive negative. He left room,
if ever so little, for doubts--that is, he considered that, in all
probability, these three discoverers had no presentiment of these
consequences, and even if they had in a dream caught a glimpse of our
present electrical age, their zest for discovery, their "pleasure in
theory," could scarcely have been increased; for they were discoverers
by nature, who, swept along by their own spirits, did not need to wait
to satisfy the desires of practical application.
In Einstein's opinion, the presentiment that a discovery may have
practical applications in the future may react on pure research. He
quoted bacteriology as a proof of this. In the series of eminent
bacteriologists, ranging from Spallanzani to Schwann and Pasteur, there
were certainly some whose desire for knowledge was directed primarily
towards discovering purely scientific relationships. Pasteur himself
started from the theoretical question of the creation of life, that is,
from the problem of the origin of organic creatures from inorganic
matter without the medium of parent organisms. As a pan-spermist he took
up a negative attitude, that is, he tried to prove that it is impossible
to discover a bridge between organic and inorganic matter. Yet he
doubtless knew that his theoretical efforts stretched out into practical
regions, and he may easily have foreseen that they would exert a very
important influence on Medicine and Hygiene, although he could not
measure its full extent. In this case, then, we cannot fail to recognize
that a certain connexion between the desire for pure knowledge and the
impulse to apply it practically is possible, serviceable, and justified
in itself.
An influence in the opposite direction is also possible, and when,
during the course of our conversation, we went in search of examples, we
came across one of great interest. It shows us that a question may arise
out of ordinary practice that may open up an immense field of pure
knowledge, nay, it may lead to a science of very wide scope. As this
example is not well known, I shall mention it here; I do so with
additional pleasure as the scientist involved is one of those whom
Einstein quotes most frequently and for whom he has the greatest
admiration, namely, Johannes Kepler. First we have the surprising fact
that Kepler, who, even when at the height of his fame, was not free from
care, was once the possessor of some money. In the year 1615, his
blessed year of fortune, the great astronomer owned a comfortable home
in Linz, and even dared to conceive the idea of placing some well-filled
casks in his cellar; nay, more, he was in a position to publish a new
scientific work at his own expense, and thus appear as his own
publisher.
This production of Kepler and his casks of wine are directly connected,
as we see clearly from the title: _Doliometrie_, literally, "The
Measurement of Casks." But the title of the work gives not the slightest
hint of its importance. For these investigations relating to wine-casks
actually became the foundation of a science of sovereign power, the
_Infinitesimal Calculus_.
What was Kepler's aim? It was something entirely practical, and directed
to a definite purpose, quite independent of "pleasures of theory," to
repeat Einstein's expression. His problem was a question of economy, of
using material sparingly and appropriately, in accordance with the
requirements of the careful head of a house. How must such a cask be
constructed from a minimum of wood to give the greatest cubical content?
His deliberations began by regarding wine as the precious content
enclosed by a figure in space, and then conceiving the cask as
representing a particular class of "bodies of revolution," that is, of
figures in space that may be regarded as produced by the revolution of a
curved line about an axis. At this point he at first endeavoured to gain
a complete survey of the question. He varied the boards along the sides,
the staves, and formed successively ninety-two such bodies of
revolution, some of which he named after the fruits which they resembled
in shape, as, for example, apple-shaped, lemon-shaped, olive-shaped
bodies. He started out by measuring casks, and the final result was that
his work, _Doliometrie_, became the source of all future cubatures or
measurements of volume.
Now we come to the deciding point. What conditions has the limiting
surface of such a cask-like body of revolution to fulfil, if the body is
to have a maximum volume? An epochal discovery here came to light. The
practical head of the house soars up into the sublime realms of the
theory of magnitudes. Kepler discovered the conception of changes in
functions, and their peculiarities at the maximum point. (He did not, of
course, use these modern terms.) By this means, long before Newton and
Leibniz, he laid the foundations of Infinitesimal Calculus, which later
became the heart and soul of mathematics, of astronomy, of theoretical
physics, and of technical science, in so far as it is founded in
mechanical relations.
On the other hand, Einstein who now, three hundred years later, has set
up his differential equations, and, with them, a new world-system,
stands before us as a pure discoverer, devoid of practical aims. But in
these equations there are elements of analysis that once came to light
in a happy idyll. This event did not come out of the grey obscurity of
abstraction, but out of a region of earthly happiness, when a ray of
light found its way into Kepler's gloomy existence. No poet has yet
expressed this curious complex of events in a ballad, telling how Truth,
the only object of Science, was pressed out of the grape, and how
Practice, inspired by the inquiry of a cooper, found its way to a Theory
that stretches to the confines of the Universe.
II
The conversation touched on famous expressions, words carved in stone,
in particular a saying of Kant which seeks to fix the foundation and the
limits of knowledge. "Every science of Nature," the great philosopher of
Königsberg had said, "contains just as much Truth as it contains
mathematics." And since, ultimately. Nature includes everything--for a
demarcation between physical and mental science no longer seems
possible--then, if we follow Kant, we should have to regard mathematics
as the sole measure of science.
It is certainly not yet possible to enter into a discussion on this
point with historians, medical or legal practitioners. They would be
justified in refusing it, since, in their subjects, "truth" is not the
sole factor, and because we cannot see at present how the conception of
a comprehensive mathematical truth is to find a place in them. But when
we question a physicist on this point, who unceasingly uses mathematics
as his chief instrument, we should surely expect him to answer with an
unconditional affirmative. At least, I should not have been surprised if
Einstein had answered in this way, and if he had indeed claimed its
validity for every branch of science.
But Einstein considered this quotation to be true only conditionally, in
that he accepted it as a principle, but did not regard it as universal.
That is, he does not recognize mathematics as the only test of truth.
"The sovereignty of mathematics," said Einstein, "is based on very
simple assumptions; it is rooted in the conception of magnitude itself.
Its dominant position is due to the fact that it gives us much more
delicate means of distinguishing between infinitely varied possibilities
than any other method of thought that expresses itself in language and
is restricted to the use of words. The greater the field taken into
consideration, the clearer does this become; but even in such a narrow
range as 1 to 100, an estimate such as 27 is incomparably more exact
than can be expressed in words in any other way. If we think of a series
of sensations, ranging from pleasure to pain, or from sweet to bitter,
we find that words leave us in an uncertain, confused state, and we do
not succeed in fixing on a point of the series with the same precision
as we above fixed on the 27 out of the 100. But when the theory of
magnitude plays a part in the question, as, for example, in a series of
tones, whose vibrations exhibit a mathematical sequence, we immediately
attain a much higher order of precision by using numbers...."
That is why there is a sort of scientific pleasure in the sequence of
tones, so my thoughts ran on. Leibniz remarks that "Music is the
pleasure of the human soul, which arises from counting without knowing
that it is counting." Here Pythagoras' "Number is the essence of all
things" is verified. As soon as we arrive at the stage at which we feel
the psychological essence of number, we fall into a sort of ecstasy,
because, in our subconscious minds, we experience not only the pleasure
of sense but also the underlying truth.
Einstein resumed: "Kant's remark is correct in the sense that it sets up
two things in clear contradiction to one another. On the one hand, he
has in view the fruits of knowledge of ordinary life, in which our
ordinary perceptions and experiences are intermingled and cannot be
disentangled by inductive methods and deductive considerations. Opposed
to these, and to be regarded of higher rank, are the properly scientific
constructions—that is, such in which we find a neat differentiation of
connected thoughts that are based on regular foundations and that form
the links of a chain of deduction. Whenever our science succeeds in
detaching this logically ordered knowledge from its sense-sources, it
has a mathematical character, and the amount of truth contained in it
will accordingly be determined by Kant's criterion. But Kant demands too
much when he asks us to apply this scale to all attainable knowledge of
science. It would seem advisable to draw limitations if his remark is to
serve as a regulative measure. A great part of biological science will
in future still be obliged to make its way independently of purely
mathematical considerations."
"Your reflections, Professor, would then also apply to the saying of
Galilei: The book of Nature lies open before us, but is written in
letters other than those of our alphabet; its characters are composed of
triangles, quadrilaterals, circles, and spheres."
"With all due honour to the beauty of this observation, I cannot refrain
from doubting its universal validity. If we were to accept it
unconditionally we should have to regard the paths of all research as
purely mathematical, and this would exclude certain very important
possibilities, above all, certain forms of intuition that have shown
themselves to be extremely fruitful. Thus, according to Galilei's
interpretation, the book of Nature would have been illegible for Goethe,
for his spirit was entirely non-mathematical, indeed anti-mathematical.
But he possessed a particular form of intuition that expressed itself as
a feeling which put him into direct contact with Nature, with the result
that he obtained a clearer vision than many an exact investigator."
"Do you then consider intuitive gifts to be separable at all in form and
in kind?"
"It would be pedantic to seek to establish a fundamental difference,
even if we may regard the non-mathematical intuition of Goethe as a very
striking case. Moreover, as I have often emphasized, all great
achievements of science start from intuitive knowledge, namely, in
axioms, from which deductions are then made. It is possible to arrive at
such axioms only if we gain a true survey of thought-complexes that are
not yet logically ordered; so that, in general, intuition is the
necessary condition for the discovery of such axioms. And it cannot be
denied that, in the great majority of minds with a mathematical
tendency, this intuition exhibits itself as a characteristic of their
creative power."
"From these remarks it would appear that you value deduction
considerably higher than induction. Perhaps in using these catchwords I
am expressing myself a little vaguely; it seems to me that great things
have been achieved, too, by using inductive processes."
"Let us first define what each of these terms means. Deduction is the
derivation of the particular from the general, whereas induction is the
process of deriving the general from the particular case. Now, quote any
example of a brilliant achievement, which you feel illustrates the power
of the inductive method. Of whatever kind your example may be, you will
soon become aware of the difference in the significance of the two
processes."
"For me the most perfect example of induction is given by certain
reasoning of Euclid. The question was whether there is a finite or an
infinite number of primes (that is, numbers that cannot be divided
without leaving a remainder except by unity). Euclid found an elegant
proof that the total number is infinite by the following strictly
inductive reasoning. If the total number were finite there would have to
be a _greatest_ prime. Let us call it _n_, and then form the product of
all primes up to _n_ and including it, finally adding one, thus:
2 x 3 x 5 x 7 x 11 x 13 ... _n_, plus 1. This new number, say Y, is
certainly greater than _n_, and now there are two possibilities, either
_n_ is prime or it is not prime.
"If it is not prime, it must be divisible by some existing prime. But
the primes up to and including _n_ cannot divide exactly into Y, as
there is always a remainder, namely, 1. Hence Y must be divisible by an
existing prime X greater than _n_. This contradicts the assumption that
_n_ is the greatest prime, for X is shown to be greater than _n_.
"Secondly, if Y _is_ a prime, it immediately follows that _n_ cannot be
the greatest prime, for Y is greater than _n_. Hence, however great may
be any prime that we may assume, there will always be one that is
greater, and even if we do not succeed in expressing it in figures, we
see that it must certainly exist. Thus by studying carefully a
particular case--the prime _n_, which was assumed to be the greatest
possible one--we have arrived at a general theorem which states that
there is no limit to the number of primes. Is not that, too, a triumph
of intuition?"
"Certainly," said Einstein. "But you must not overlook the fact that a
theorem of this kind cannot be ranked with a theorem of a fundamentally
axiomatic character. The one you have discussed has been derived by a
clever process of reasoning, but it does not exhibit the characteristic
of a momentous discovery. This theorem of Euclid can be imagined absent
from science without the content of truth in science being essentially
effected. Compare with it a theorem of axiomatic significance, such as
Galilei's Law of Inertia, or Newton's Law of Gravitation. Theorems such
as the latter are characterized by being starting-points of knowledge
that are inexhaustible in the consequences that may be deduced from
them. Your question, earlier, as to whether I consider the deductive
method superior to the inductive, was not formulated in correct terms.
To this I answered above that the inductive method as a means of
discovering general truths usually appears over-estimated. The proper
form of the question is: Which truths are of the higher order, those
that are found inductively, or those that lead to further deduction?
There can scarcely be doubt about the answer."
"No, that is certainly true. If I understand your meaning rightly, the
answer may be expressed by an allegory. Intuition of the highest order
creates treasure-mines, those of lesser degree individual articles of
value that are significant in themselves, although they cannot be
compared with the inestimable value of the mines. The fact that the
highest intuition is found in minds with a mathematical trend makes it
appear possible that Kant's remark may gain more and more credence in
the future. It already applies in a measure to subjects to which it
seemed inapplicable during Kant's lifetime, for example, in Psychology,
in which the relations between stimulus and response have been
established mathematically only since the Weber-Fechner Law was set up;
and also, since the time of Quetelet, in Moral Science and Sociology, we
learn from mathematical methods of statistics and probability that even
Man as an active being is subjected to mechanical causality. At any rate
it seems manifest that Kant's remark, that in every science there is
just as much truth as there is mathematics, has received additional
support in recent times."
"That may be admitted," concluded Einstein, "without recognizing his
remark as an axiom. It is still far removed from making possible
unassailable deductions, and will never quite succeed in doing so; yet
it may claim equal significance as a beautifully expressed idea with
that of Pythagoras, which asserts number to be the nature of all
things."
III
"The lines of demarcation between 'conceptual knowledge' (_Erkennen_)
and 'perceptual knowledge' (_Kennen_) are being drawn more and more
closely nowadays. The former is regarded as being the exclusive
possession of the highly developed human mind, and the latter as being
characteristic of the lower intelligence of other living creatures. Is
this not a pronounced case of anthropomorphism, and does it not mislead
us to form opinions that we should at once disown if we succeed in
stepping out of our human frames even for a moment?"
"We have to rest satisfied with anthropomorphism once and for all,"
answered Einstein, "and there is no sense in wishing to escape from it,
for the arguments about anthropomorphism are necessarily also diffused
with it, itself. We are thus moving in a circle if we imagine we can
deduce something outside of human knowledge. As soon as we have argued
around the circle, we find ourselves again at the starting-point, and so
we are compelled to mark clear lines of division between instinctive
knowledge, derived directly by perception, from conceptual knowledge,
derived by processes of abstraction and reflection; in this way we award
the palm of supremacy to the human mind."
"But what if the following contradiction were to assert itself? Suppose
that the logical 'circle' is not a circle at all, but a spiral, so that
the final point of the argument lies just a trifle above the initial
point. I feel instinctively that such apparently fruitless circuitous
arguments might finally lead to a definite piece of knowledge. For
example, a certain insect, the ichneumon-fly, although devoid of a
knowledge of science in our sense, infallibly plants its sting in a
definite point in the rings of a caterpillar, at just the point that
serves its purpose of paralysing the caterpillar without killing it. It
acts instinctively, and it is open to me to interpret this occurrence in
other words. The fly discloses that it 'knows' the anatomy of the
foreign creature, although it has no conceptual knowledge of it in our
sense. But it immediately follows from this analogy that, from the point
of view of the fly, its _perceptual_ intelligence stands higher than our
_conceptual_ intelligence--that is, by changing the perspective, I am
led to declare the anatomical knowledge of the fly to be of higher rank
than the analogous knowledge of the most learned anatomist. In the same
way I might persuade myself that the mathematics of a bird of passage
stands above the cartographic knowledge of any human explorer. The
migratory bird that flies from the interior of Africa in a straight fine
to its nest in Mecklenburg must have something in the nature of a
co-ordinate system in its organism. The real reason that we assign a
higher position to our conceptual knowledge is that we are equally proud
of our intelligence as of our science; this is perhaps a deception
depending on some compromise, a sort of illicit deal in which the mind
draws bills of exchange on science, and, as a return, science meets its
obligations by paying in cheques drawn on the mind!"
I must confess that these hazardous suggestions received no welcome from
Einstein, and were not even met with the friendly smile with which he
usually accompanies his refutations. Nor do I disguise from myself that
the question of conceptual or perceptual knowledge can in no way serve
as a basis of proof; we may at most base certain conjectures on the
difference of these types of knowledge, conjectures that suggest in
words what eludes our clear comprehension. Einstein's refusal to allow
this possibility certainly rests on much firmer ground than the somewhat
Bergsonian views that I tried to present. Perhaps they are of a
hair-splitting nature, and deal with things lying on different planes;
and are deduced by unjustifiably altering the perspective with a sort of
sophistic somersault; perhaps I may be reproached with seeking, like
Münchhausen, to reach a higher standpoint without having a support from
which to start. Yet how is it that I find it impossible to free myself
from this chain of thought? No reason is forthcoming, for it is a purely
metaphysical question, and there has never yet been a clear system of
metaphysics free from ambiguities and sophism.
Let us rather confine ourselves to the conceptual intelligence
characteristic of human beings, with which, according to Einstein, so
many pleasures of theory are available. I asked him whether he would
recognize differences of degree in these pleasures, dependent on their
intensities. Although I rightly felt that he would answer in the
affirmative, his answer took a totally different turn from what I had
expected. It was, indeed, a great surprise, for in the matter of
happiness of spirit he expressed a view, according to which he--a great
discoverer!--does not regard Science as the deepest source of happiness!
"Personally," said Einstein, "I experience the greatest degree of
pleasure in getting contact with works of Art. They furnish me with
happy feelings of an intensity such as I cannot derive from other
realms."
"This is indeed a remarkable revelation, Professor!" I exclaimed. "Not
that I have ever doubted your receptivity for products of art, for I
have often enough observed how you are affected by good music, and with
what interest you yourself practise music. But even at such moments when
you gave yourself up to the pleasures of the Muses, and were soaring in
regions far removed from the earth, I used to say to myself: This is a
delightful arabesque in Einstein's existence; but I should never have
surmised that you regard this decorative side-issue as the greatest
source of happiness. But your confession seems to go further, perhaps
even beyond music?"
"At the moment I was thinking particularly of literature."
"Do you mean literature in general? Or had you a definite writer in
mind, when you were speaking of the felicitous effect of works of art?"
"I meant it generally, but if you ask in whom I am most interested at
present, I must answer: Dostojewski!" He repeated the name several times
with increasing emphasis. And, as if to deal a mortal blow at every
conceivable objection, he added: "Dostojewski gives me more than any
scientist, more than Gauss!"
"If, Professor," said I, after a pause that may easily be accounted
for--"if you mention in the same breath the names of two such powerful
but essentially different intellects, you open the way to a discussion
that cannot be settled by a mere positive assertion. It is possible to
admire intensely Dostojewski as one who moulds personalities and who
analyses the inner struggles of the soul, and yet to deny him perpetual
fame. This depends on individual judgment, and, as for my own, I believe
that Dostojewski, in spite of his direct artistic appeal, will not have
his name perpetuated through the centuries like that of many another
member of Parnassus. It seems to me to be a more important matter
whether a common measure can be found for Art and Discovery at all.
Perhaps the test of how far a work can be replaced may be regarded as
valid for each. When you say that Dostojewski gives you more than Gauss,
this probably corresponds with the feeling that without Dostojewski you
would have no 'Karamasoffs' and hence would lack a certain life-value
that cannot be replaced. But if Gauss had failed to produce one of his
fundamental theorems of Algebra, probably some other Gauss would have
appeared, who would have achieved this result. According to this, then,
our instinct increases the value of a work of art, as we feel that we
are dependent on one being alone for its creation."
"But this is only to be admitted conditionally," said Einstein, "for the
best that Gauss has given us was likewise an exclusive production. If he
had not created his geometry of surfaces, which served Riemann as a
basis, it is scarcely conceivable that anyone else would have discovered
it. I do not hesitate to confess that to a certain extent a similar
pleasure may be found by absorbing ourselves in questions of pure
geometry."
"Perhaps we may use a different characteristic as a means of
comparison," I suggested, "namely, the permanency of the impression
produced on the subject receiving it. For example, a fine piece of music
never loses its influence. We can listen to the first movement of
Beethoven's Ninth Symphony a hundred times, and, although we know at
every beat what will follow, the state of pleasure continues unweakened;
indeed, it might rather be said that the expectation of pleasure
increases from one hearing to the next."
"This characteristic, too," answered Einstein, "cannot be claimed as the
exclusive property of works of art. Its existence cannot be doubted,
inasmuch as it belongs to every eminent example of art. Yet we encounter
it outside the realm of art, too, in great advances of science, with
which we never cease occupying ourselves, and yet the impression
continues unweakened."
"Do you include among them the impressions that a discoverer experiences
when he reviews in his mind the progress due to his own efforts?"
"Naturally, and these, indeed, quite particularly; and if this question
were put to me directly, I should answer unhesitatingly that I find
pleasure in reflecting on my own discoveries, and never experience
feelings of weariness in passing over them again. So that, to return to
our original thesis, we must adopt a new basis of value if we wish to
account for the fact that the greatest degree of happiness is to be
expected of a work of art. It is the moral impression, the feeling of
elevation, that takes hold of me when the work of art is presented. And
I was thinking of these ethical factors when I gave preference to
Dostojewskis works. There is no need for me to carry out a literary
analysis, nor to enter on a search for psychological subtleties, for all
investigations of this kind fail to penetrate to the heart of a work
such as "The Karamasoffs." This can be grasped only by means of the
feelings, that find satisfaction in passing through trying and difficult
circumstances, and that become intensified to exultation when the author
offers the reader ethical satisfaction. Yes, that is the right
expression, 'ethical satisfaction'! I can find no other words for it."
His whole face lit up, and I was deeply touched by his expression. At
that moment it seemed to me that he had drawn the last veil from his
soul to allow me to share in his ecstasy. Was that the same physicist
who interprets the events of the world in terms of mathematics, and
whose equations encompass phenomena from electrons to universes? If so,
it was a different soul; one which gave utterance, like that of Faust,
to the words:
"And when in the feeling wholly blest thou art,
Call it then what thou wilt.
Call it Bliss! Heart! Love! God!
I have no name for it!
Feeling is all in all!
Name is but sound and reek,
A mist round the glow of heaven!"
And, certainly, the book need not have been one of Dostojewski's to
excite this feeling in him. He chose the latter to give expression to a
mood that may change according to what he reads, but undergoes no
fluctuations in its ethical foundation. From other occasions we know how
little ethics, that is conducted along systematic lines, signifies to
him, and that he does not even include it in the sciences. But at the
same time we see now that his inner life is dominated entirely by the
ethical principle. His deep love of Art is characterized by it, and
receives full satisfaction from the source of ethical joy of which Art
is the centre.
IV
During the autumn of 1918 Einstein was feeling indisposed, and, on the
advice of his doctor, did not leave his bed. When I entered his room, I
saw at once that there was no reason for alarm, for pieces of paper
covered with mysterious symbols were lying about, and he was absorbed in
making additions to some of them. Nevertheless, I considered it my duty
to treat him as a patient under medical care, and did not conceal my
intention of leaving him after having inquired about his condition. But
he would not accept my visit as a mere call to ascertain his progress
towards recovery, and insisted that I should remain with him a while, to
converse about amusing little problems as usual.
I pointed out to him that there were two objections to this, the first
being that he was unwell, and the second that I was intruding on his
work.
"How illogical!" he answered. "If I interrupt my work to chat with you,
I am putting aside exactly what the doctor would deny me if I were to
allow him. So, let us make a start. You have probably some conundrum
weighing on your mind."
"That may not be far wrong. I have been troubled by something in
connexion with Kepler's second law. It almost robbed me of my night's
sleep. My thoughts kept returning to a certain question, and I should
like to know whether there is any sense in the question itself at all."
"Let us hear it!"
"The law in question states that every planet in describing its elliptic
path, sweeps out with its radius vector equal sectorial areas in equal
intervals of time. But this seems only half a law, for the radius
vectors are only considered drawn from the one focus of the ellipse,
namely, the gravitational centre. Now, another focus exists, that may be
situated in space somewhere, perhaps far away in totally empty regions,
if we assume the orbit to be very eccentric. My question is: What form
does this law take if the radius vectors are drawn from this second
focus and if the corresponding sectorial areas are considered, instead
of these quantities being referred to the first focus exclusively?"
"This question is not devoid of sense, but it serves no useful purpose.
It may be solved analytically, but would probably lead to very
complicated expressions, that would be of no interest for celestial
mechanics. For the second focus is only a constructive addition, that
has nothing real in space corresponding to it. What else is troubling
you?"
"My next difficulty is a little problem that sounds quite simple and yet
is sufficiently awkward to make one rack one's brains. It was suggested
to me by an engineer who certainly has a keen mind for such things, and
yet, as far as I could judge, he did not get a solution for it. It
concerns the position of the hands of a clock."
"You surely are not referring to the children's puzzle of how often and
when both hands coincide in position?"
"By no means. As I said just now, it is really quite perplexing. Let us
assume the position of the hands at twelve o'clock, when both hands
coincide. If they are now interchanged, we still have a possible
position of the hands, giving an actual time. But, in another case, say,
exactly six o'clock, we get a false position of the hands, if we
interchange them, for on a normal clock it is impossible for the large
hand to be on the six whilst the small hand is on the twelve. The
question is now: When and how often are the two hands situated so that
when they are interchanged, the new position gives a possible time on
the clock?"
"There, you see," said Einstein, "that is just the right kind of
distraction for an invalid. It is quite interesting, and not too easy.
But I am afraid the pleasure will not be of great duration, for I
already see a way to solve it."
Supporting himself on his elbow, he sketched a diagram on a sheet of
paper that gave a clear picture of the conditions of the problem. I can
no longer recollect how he arrived at the terms of his equation. At any
rate, the result soon came to hand in a time not much longer than I had
taken to enunciate the problem to him. It was a so-called indeterminate
(Diophantic) equation between two unknowns, that was to be satisfied by
simple integers only. He showed that the desired position of the hands
was possible 143 times in 12 hours, an equal interval separating each
successive position; that is, starting from twelve o'clock, the two
hands may be interchanged every 5 minutes ²⁄₁₄₃ seconds, and yet give a
possible time.
* * * * * * * *
I mention this little episode, which is insignificant in itself, merely
to give an example of how a great discoverer, too, finds amusement in
such distractions. In Einstein's case this tendency to practise his
ingenuity on unimportant trifles is so much the more pronounced from the
fact that he requires an outlet for his virtuosity in calculation, and
gratefully welcomes every suggestion that helps him to relieve his
mental tension. Similar characteristics are reported of the great Euler,
as well as of Fermat, whereas many another eminent mathematician feels
decidedly unhappy if he drifts within reach of the realm of actual
numerical calculation. In my mind's eye I still see Ernst Kummer, the
splendid savant (who, in his time, conferred distinction on Berlin
University by his very presence), suffering agonies whenever ordinary
arithmetical tables threatened to appear in the working-out of his
formulae. As a matter of fact, these two things, a mastery over
mathematics and a talent for ingenious calculation, are to be considered
as quite independent, even if we now and then find them present in the
same person.
In the case of Einstein this tendency is a symptom of an incredible
universality of spirit. It moreover presents itself in the pleasantest
forms, and a character-sketch of Einstein would be incomplete if this
trait were not mentioned. Every problem which is in any way amusing
excites in him a willing interest and enthusiasm. I once directed our
conversation to the so-called _Scherenschnitte_. These are made from
long strips of paper or canvas, the ends of which are caused to overlap
a little and then pasted together, but instead of being fixed so that a
flat wheel results, which rolls on one side of the strip, the strip is
twisted one or more times before the ends are fastened together. If now
the strip is cut lengthwise right along its centre, various unexpected
results occur, depending on the number of twists that have been made
before pasting.
Some very complex geometrical difficulties are involved in these
problems. This is shown by the fact that learned mathematicians have
written extensive disquisitions on these curious constructions (for
example, Dr. Dingeldey's book, published by Teubner, Leipzig). Einstein
had never taken notice of these wonders of the scissors, but when I
began to form these strips, to paste them, and to cut them, he
immediately became interested in the underlying problem, and predicted
in a flash what puzzling chain constructions would result in each case,
with a certainty that would lead one to imagine that he had spent days
at it. On another occasion a space-problem dealing with dress came up
for discussion: Can a properly dressed man divest himself of his
waistcoat without first taking off his coat? One would not have dared to
confront Copernicus or Laplace with such a problem. Einstein at once
attacked it with enthusiasm, as if it were an exercise in mechanics, the
body being the object; he solved it in a trice, practically, with a
little energetic manipulation, much to the amazement and joy of the
beholder, who asked himself: Is this the same Einstein who developed the
work of Copernicus and Newton? A little later, perhaps, the conversation
centres around some serious point drawn from politics, political
economy, sociology, or jurisprudence. Whatever it may be, he knows how
to spin out the suggested thread, to establish contact with his partner
in conversation, to open up his own perspectives without ever insisting
on his point of view, always stimulating and showing a ready sympathy
for the subject of discussion and for all the ideas which it
crystallizes, the prototype of the scientist, in the mouth of whom
Terence put the words: "I am a human being; nothing that is human is
alien to me!"