It was here in this part of the world, on the coast of the Aegean Sea,
that the philosophers Leucippus and Democritus pondered about the
structure of matter, and down there in the marketplace, where twilight
is now falling, that Socrates disputed about the basic difficulties in
our modes of expression and Plato taught that the Idea, the form, was
the truly fundamental pattern behind the phenomena. The problems first
formulated in this country two and a half thousand years ago have
occupied the human mind almost unceasingly ever since and have been
discussed again and again in the course of history whenever new
developments have altered the light in which the old lines of thought appeared.
If I endeavor today to take up some of the old problems concerning the
structure of matter and the concept of natural law, it is because the
development of atomic physics in out own day has radically altered our
whole outlook on nature and the structure of matter. It is perhaps not
an improper exaggeration to maintain that some of the old problems have
quite recently found a clear and final solution. So it is permissible
today to speak about this new and perhaps conclusive answer to questions
that were formulated here thousands of years ago.
There is, however, yet another reason for renewing consideration of
these problems. The philosophy of materialism, developed in antiquity by
Leucippus and Democritus, has been the subject of many discussions since
the rise of modern science in the seventeenth century and, in the form
of dialectical materialism, has been one of the moving forces in the
political changes of the nineteenth and twentieth centuries. If
philosophical ideas about the structure of matter have been able to play
such a role in human life, if in European society they have operated
almost like an explosive and may yet perhaps do so in other parts of the
world, it is even more important to know what our present scientific
knowledge has to say about this philosophy. To put it in rather general
and precise terms, we may hope that a philosophical analysis of recent
scientific developments will contribute to a replacement of conflicting
dogmatic opinions about the basic problems we have broached, by a sober
readjustment to a new situation, which, in itself, can even now be
regarded as a revolution in human life on this earth. But even aside
from this influence of science upon our time, it may be of interest to
compare the philosophical discussions in ancient Greece with the
findings of experimental science and modern atomic physics. If I may
already anticipate at this point the outcome of such a comparison; it
seems that, in spite of the tremendous success that the concept of the
atom has achieved in modern science, Plato was very much nearer to the
truth about the structure of matter than Leucippus or Democritus. But it
will doubtless be necessary to begin by repeating some of the most
important arguments adduced in the ancient discussions about matter and
life, being and becoming, before we can enter into the findings of
modern science.
The Concept of Matter in Ancient Philosophy
At the beginning of Greek philosophy there stood the dilemma of the
“one” and the “many.” We know that there is an ever-changing variety of
phenomena appearing to our senses. Yet we believe that ultimately it
should be possible to trace them back somehow to some one principle.
The founders of atomism, Leucippus and Democritus, tried to avoid the
difficulty by assuming the atom to be eternal and indestructible, the
only thing really existing. All other things exist only because they are
composed of atoms. The antithesis of “being” and “non being” in the
philosophy of Parmenides is here coarsened into that between the “full”
and the “void.” Being is not only one; it can be repeated infinitely
many times. Being is indestructible, and therefore the atom, too, is
indestructible. The void, the empty space between the atoms, allows for
position and motion, and thus for properties of the atom, whereas by
definition, as it were, pure being can have no other property than that
of existence. This latter part of the doctrine of Leucippus and
Democritus is at once its strength and its weakness. On the one hand, it
provides an immediate explanation of the different aggregate states of
matter, such as ice, water, and steam, since the atoms may lie densely
packed and in order beside each other, or be caught in disorder and
irregular motion, or finally be separated at fairly large relative
intervals in space. This part of the atomic hypothesis was therefore to
prove exceedingly fruitful at a later stage. One the other hand, the
atom becomes in this fashion a mere building block of matter; its
properties, position, and motion in space turn it into something quite
different from what was meant by the original concept of “being.” The
atoms can even have a finite extension, and here we have finally lost
the only convincing argument for their indivisibility. If the atom has
spacial properties, why should it not be divided? At least its
indivisibility then becomes a physical, not a fundamental property. We
can now again ask questions about the structure of the atom, and we run
the risk of losing all the simplicity we had hoped to find among the
smallest parts of matter. We get the impression, therefore, that in its
original form the atomic hypothesis was not sufficiently subtle to
explain what the philosophers really wished to understand: the simple
element in the phenomena and in the structure of matter.
Still, the atomic hypothesis does go a large part of the way in the
right direction. The whole multiplicity of diverse phenomena, the man
observed properties of matter, can be reduced to the position and motion
of the atoms. Properties such as smell or color or taste are not present
in atoms. But their position and motion can evoke these properties
indirectly. Position and motion seem to be much simpler concepts than
the empirical qualities of taste, smell, or color. But then it naturally
remains to ask what determines the position and motion of the atoms. The
Greek philosophers did not attempt at this point to formulate a law of
nature; the modern concept of natural law did not fit into their way of
thought. Yet they seem to have thought of some kind of causal
description or determinism, since they spoke of necessity, of cause and effect.
The intention of the atomic hypothesis had been to point the way from
the “many” to the “one,” to formulate the underlying principle, the
material cause, by virtue of which all phenomena can be understood. The
atoms could be regarded as the material cause, but only a general law
determining their positions and velocities could actually play the part
of the fundamental principle. However, when the Greek philosophers
discussed the laws of nature, their thoughts were directed to static
forms, geometrical symmetries, rather than to processes of space and
time. The circular orbits of the planets, the regular geometrical
solids, appeared to be the permanent structures of the world. The modern
idea, that the position and velocity of the atom at a given time could
be uniquely connected by a mathematical law with its position and
velocity at a later time, did not fit into the pattern of thought of
that era since it employs the concept of time in a manner that arose
only out the the thinking of a much later epoch.
When Plato himself took up the problems raised by Leucippus and
Democritus, he adopted the idea of smallest units of matter, but he took
the strongest exception to the tendency of that philosophy to suppose
the atoms to be the foundation of all existence, the only truly existing
material objects. Plato’s atoms were not strictly material, being
thought of as geometrical forms, the regular solids of the
mathematicians. These bodies, in keeping with the starting point of his
idealistic philosophy, were in some sense the Ideas underlying the
structure of matter and characterizing the physical behavior of the
elements to which they belonged. The cube, for example, was the smallest
particle of the element earth and thereby symbolized at the same time
the earth’s stability. The tetrahedron, with its sharp point,
represented the smallest particle of the element fire. The icosahedron,
which comes closest among the regular solids to a sphere, stood for the
mobility of the element water. In this way the regular solids were able
to serve as symbols for certain tendencies in the physical behavior of matter.
But they were not strictly atoms, not indivisible basic units like those
of the materialist philosophy. Plato regarded then as composed from the
triangles forming their surfaces; therefore, by exchanging triangles,
these smallest particles could be commuted into each other. Thus two
atoms of air, for example, and one of fire could be compounded into an
atom of water. In this was Plato was able to escape the problem of the
indefinite divisibility of matter. For as two-dimensional surfaces the
triangles were not bodies, not matter any longer; hance matter could not
be further divided ad infinitum. At the lower end, therefore, in the
realm, that is, of minimal spatial dimensions, the concept of matter is
resolved into that of mathematical form. This form determines the
behavior, first of the smallest parts of matter, then of matter itself.
To a certain extent it replaces the natural law of later physics; for
without making explicit references to the course of time, it
characterizes the tendencies in the behavior of matter. One might say,
perhaps, that the fundamental tendencies were represented by the
geometrical shape of the smallest units, while the finer details of
those tendencies found expression in the relative position and velocity
of these units.
This whole description fits exactly into the central ideas of Plato’s
idealist philosophy. The structure underlying the phenomena is not given
by material objects like the atoms of Democritus but by the form that
determines the material objects. The Ideas are more fundamental then the
objects. And since the smallest parts of matter have to be the objects
whereby the simplicity of the world becomes visible, whereby we
approximate to the “one” and the “unity” of the world, the Ideas can be
described mathematically—they are simply mathematical forms. The saying
“God is a mathematician,” which in this form assuredly derives from a
later period of philosophy, has its origin in this passage from the
Platonic philosophy.
The importance of this step in philosophical thought can hardly be
reckoned too highly. It can be seen as the decisive beginning of the
mathematical science of nature, and hence be made responsible also for
the later technical applications that have altered the whole picture of
the world. By this step it is also first established what the term
“understanding” is to mean. Among all the possible forms of
understanding. Whereas all language, indeed, al art and all poetry in
some way mediate understanding, it is here maintained that only the
employment of a precise, logically consistent language, a language so
far capable of formalization that proofs become possible, can lead to
true understanding. One feels the strength of the impression made upon
the Greek philosophers by the persuasive force of logical and
mathematical arguments. They are obviously overwhelmed by this force.
But perhaps they surrendered too early at this point.
The Answer of Modern Science to the Old Problems
If we trace the history of physics from Newton to the present day, we
see that, despite the interest in details, very general laws of mature
have been formulated on several occasions. The nineteenth century saw an
exact working out of the statistical theory of heat. The theories of
electromagnetism and special relativity have proved susceptible of
combination into a very general group of natural laws containing
statements not only about electrical phenomena but also about the
structure of space and time. In our own century, the mathematical
formulation of the quantum theory has led to an understanding of the
outer shells of chemical atoms, and thus of the chemical properties of
matter generally. The relations and connections between these different
laws, especially between relativity and quantum theory, are not yet
fully explained. But the latest developments in particle physics permit
one to hope that these relations may be satisfactorily analyzed in the
relatively near future. We are thus already in a position to consider
what answers can be given by this whole scientific development to the
questions of the old philosophers.
During the nineteenth century, the development of chemistry and the
theory of heat conformed very closely tot he ideas first put forward by
Leucippus and Democritus. A revival of the materialist philosophy in its
modern form, that of dialectical materialism, was this a natural
counterpart to the impressive advances made during this period in
chemistry and physics. The concept of the atom had proved exceptionally
fruitful in the explanation of chemical bonding and the physical
behavior of gases. It was soon, however, that the particles called atoms
by the chemist were composed of still smaller units. But these smaller
units, the electrons, followed by the atomic nuclei and finally the
elementary particles, protons and neutrons, also still seemed to be
atoms from the standpoint of the materialist philosophy. The fact that,
at least indirectly, one can actually see a single elementary
particle—in a cloud chamber, say, or a bubble chamber—supports the view
that the smallest units of matter are real physical objects, existing in
the same sense that stones or flowers do.
But the inherent difficulties of the materialist theory of the atom,
which had become apparent even in the ancient discussions about smallest
particles, have also appeared very clearly in the development of physics
during the present century.
This difficulty relates to the question whether the smallest units are
ordinary physical objects, whether they exist in the same way as stones
or flowers. Here, the development of quantum theory some forty years ago
has created a complete change in the situation. The mathematically
formulated laws of quantum theory show clearly that our ordinary
intuitive concepts cannot be unambiguously applied to the smallest
particles. All the words or concepts we use to describe ordinary
physical objects, such as position, velocity, color, size, and so on,
become indefinite and problematic if we try to use then of elementary
particles. I cannot enter here into the details of this problem, which
has been discussed so frequently in recent years. But it is important to
realize that, while the behavior of the smallest particles cannot be
unambiguously described in ordinary language, the language of
mathematics is still adequate for a clear-cut account of what is going on.
During the coming years, the high-energy accelerators will bring to
light many further interesting details about the behavior of elementary
particles. But I am inclined to think that the answer just considered to
the old philosophical problems will turn out to be final. If this is so,
does this answer confirm the views of Democritus or Plato?
I think that on this point modern physics has definitely decided for
Plato. For the smallest units of matter are, in fact, not physical
objects in the ordinary sense of the word; they are forms, structures
or—in Plato’s sense—Ideas, which can be unambiguously spoken of only in
the language of mathematics. Democritus and Plato both had hoped that in
the smallest units of matter they would be approaching the “one,” the
unitary principle that governs the course of the world. Plato was
convinced that this principle can be expressed and understood only in
mathematical form. The central problem of theoretical physics nowadays
is the mathematical formulation of the natural law underlying the
behavior of elementary particles. From the experimental situation we
infer that a satisfactory theory of the elementary particles must at the
same time be a theory of physics in general, and hence, of everything
else belonging to this physics.
In this way, a program could be carried out that in modern times was
first proposed by Einstein: a unified theory of matter—and hence,
simultaneously, a quantum theory of matter—could be formulated, which
might serve quite generally as a foundation for physics. We do not yet
know whether the mathematical forms proposed for this unifying principle
are already adequate or will have to be replaced by forms more abstract
still. But our present knowledge of the elementary particles is
certainly enough for us to say what the main content of this law has to
be. It must essentially set forth a small number of fundamental symmetry
properties in nature, which have been known empirically for some years;
in addition to these symmetries, it must contain the principle of
causality as understood in relativity theory. The most important of the
symmetries are the so-called “Lorentz group” of space and time, and the
so-called “isospin group,” which has to do with the electric charge on
the elementary particles. There are also other symmetries, but of these
I shall say nothing here. Relativistic causality is connected with the
Lorentz group but must be considered an independent principle.
This situation reminds us at once of the symmetrical bodies introduced
by Plato to represent the fundamental structures of matter. Plato’s
symmetries were not yet the correct ones, but he was right in believing
that ultimately, at the heart of nature, among the smallest units of
matter, we find mathematical symmetries. It was an unbelievable
achievement of the ancient philosophers to have asked the right
questions. But, lacking all knowledge of the empirical details, we could
not have expected them to find answers that were correct in detail as well.
Consequences for the Evolution of Human Thought in Our Own Day
The search for the “one,” for the ultimate source of all understanding,
has doubtless played a similar role in the origin of both religion and
science. But the scientific method that was developed in the sixteenth
and seventeenth centuries, the interest in those details which can be
tested by experiment, has for a long time pointed science along a
different path. It is not surprising that this attitude should have led
to a conflict between science and religion, as soon as a law
contradicted, in some particular and perhaps very important detail, the
general picture, the mode and manner, in which the facts had been spoken
of in religion. This conflict, which began in modern times with the
celebrated trial of Galileo, has been discussed often enough, and I need
not repeat this discussion here. One may recall that, even in ancient
Greece, Socrates was condemned to death because his teachings seemed to
contradict the traditional religion. In the nineteenth century, this
conflict reached its peak in the attempt of some philosophers to replace
traditional Christianity by a scientific philosophy, based upon a
materialist version of the Hegelian dialectic. It might be said that, in
directing their gaze upon a materialistic interpretation of the “one,”
the scientists were attempting to find their way back again to this
“one” from the multitude of details.
If modern science has something to contribute to this problem, it is not
by deciding for or against one of these doctrines; for example, as was
possibly believed in the nineteenth century, by coming down in favor of
materialism and against the Christian philosophy, or, as I now believe,
in favor of Plato’s idealism and against the materialism of Democritus.
On the contrary, the chief profit we can derive in these problems from
the progress of modern science is to learn how cautious we have to be
with language and with the meaning of words. I would therefore like to
devote the last part of my address to a few remarks about the problem of
language in modern science and ancient philosophy.
If we may take our cue at this point from Plato’s dialogues, the
unavoidable limitations of our means of expression were already a
central theme in the philosophy of Socrates; one might even say that his
whole life was a constant battle with these limitations. Socrates never
wearied of explaining to his countrymen, here on the streets of Athens.
That they did not know exactly what they meant by the words they were
employing. The story goes that one of Socrates’ opponents, a sophist who
was annoyed at Socrates’ constant reference to this insufficiency of
language, criticized him and said: “But Socrates, this is a bore; you
are always saying the same about the same.” Socrates replied: “But you
sophists, who are so clever, perhaps never say the same about the same.”
The reason for laying such stress on this problem of language was
doubtless that Socrates was aware, on the one hand, of how many
misunderstandings can be engendered by a careless use of language, how
important it is to use precise terms and to elucidate concepts before
employing them. One the other hand, he probably also realized that this
may ultimately be an insoluble task. The situation confronting us in our
attempt to “understand” may drive us to conclude that our existing means
of expression do not allow for a clear and unambiguous description of
the facts.
The tension between the demand for complete clarity and the inevitable
inadequacy of existing concepts has been especially marked in modern
science. In atomic physics we make use of a highly developed
mathematical language that satisfies all the requirements in regard to
clarity and precision. At the same time, we recognize that we cannot
describe atomic phenomena without ambiguity in any ordinary language; we
cannot, for example, speak unambiguously about the behavior of an
electron in the interior of an atom. It would premature, however, to
insist that we should avoid the difficulty by confining ourselves to the
use of mathematical language. This is no genuine way out, since we do
not know how far the mathematical language can be applied to phenomena.
In the last resort, even science must rely upon ordinary language, since
it is the only language in which we can be sure of really grasping the phenomena.
This situation throws some light on the tension between the scientific
method, on the one hand, and the relation of society to the “one,” the
fundamental principle behind phenomena, on the other. It seems obvious
that this latter relation cannot and should not be expressed in a
precise and highly sophisticated language whose applicability to the
real world may be very restricted. The only thing that will do for this
purpose is the natural language everyone can understand. Reliable
results in science, however, can be secured only by unambiguous
statement; here we cannot do without the precision and clarity of an
abstract mathematical language.
The necessity of constantly shuttling between the two languages is,
unfortunately, a chronic source of misunderstandings, since in many
cases the same words are employed in both. The difficulty is
unavoidable. But it may yet be of some help always to bear in mind that
modern science is obliged to make use of both languages, that the same
word may have very different meanings in each of them, that different
criteria of truth apply, and that one should not, therefore, talk too
hastily of contradictions.
If we wish to approach the “one” in the terms of a precise scientific
language, we must turn our attention to that center of science described
by Plato, in which the fundamental mathematical symmetries are to be
found. In the concepts of this language we must be content with the
statement that “God is a mathematician”; for we have freely chosen to
confine our vision to that realm of being which can be understood in the
mathematical sense of the word “understanding,” which can be described
in rational terms.
Plato himself was not content with this restriction. Having pointed out
with the utmost clarity the possibilities and limitations of precise
language, he switched to the language of poetry, which evokes in the
hearer images conveying understanding of an altogether different kind. I
shall not seek to discuss here what this kind of understanding can
really mean. These images are probably connected with the unconscious
mental patterns the psychologists speak of as archetypes, forms of
strongly emotional character that, in some way, reflect the internal
structures of the world. But whatever the explanation for these other
forms of understanding, the language of images and likenesses is
probably the only way of approaching the “one” from more general
domains. If the harmony in a society rests on a common interpretation of
the “one,” the unitary principle behind the phenomena, then the language
of poetry may be more important here than the language of science.
