Throughout the history of science new
discoveries and new ideas have always caused scientific disputes, have led to
polemical publications criticizing the new ideas, and such criticism has often
been helpful in their development; but these controversies have never before
reached that degree of violence which they attained after the discovery of the
theory of relativity and in a lesser degree after quantum theory. In both cases
the scientific problems have finally become connected with political issues,
and some scientists have taken recourse to political methods to carry their
views through. This violent reaction on the recent development of modern
physics can only be understood when one realizes that here the foundations of
physics have started moving; and that this motion has caused the feeling that
the ground would be cut from science. At the same time it probably means that one
has not yet found the correct language with which to speak about the new
situation and that the incorrect statements published here and there in the
enthusiasm about the new discoveries have caused all kinds of misunderstanding.
This is indeed a fundamental problem. The improved experimental technique of
our time brings into the scope of science new aspects of nature which cannot be
described in terms of the common concepts. But in what language, then, should
they be described? The first language that emerges from the process of
scientific clarification is in theoretical physics usually a mathematical
language, the mathematical scheme, which allows one to predict the results of
experiments.
The physicist may be satisfied when he has the mathematical scheme
and knows how to use it for the interpretation of the experiments. But he has
to speak about his results also to nonphysicists who will not be satisfied
unless some explanation is given in plain
language, understandable to anybody. Even for the physicist the description in
plain language will be a criterion of the degree of understanding that has been
reached. To what extent is such a description at all possible? Can one speak
about the atom itself? This is a problem of language as much as of physics, and
therefore some remarks are necessary concerning language in general and
scientific language specifically.
Language
was formed during the prehistoric age among the human race as a means for
communication and as a basis for thinking. We know little about the various
steps in its formation; but language now contains a great number of concepts
which are a suitable tool for more or less unambiguous communication about
events in daily life. These concepts are acquired gradually without critical
analysis by using the language, and after having used a word sufficiently often
we think that we more or less know what it means. It is of course a well-known
fact that the words are not so clearly defined as they seem to be at first
sight and that they have only a limited range of applicability. For instance,
we can speak about a piece of iron or a piece of wood, but we cannot speak
about a piece of water. The word piece' does not apply to liquid substances.
Or, to mention another example: In discussions about the limitations of
concepts, Bohr likes to tell the following story: A little boy goes into a
grocer's shop with a penny in his hand and asks: "Could I have a penny's
worth of mixed sweets?" The grocer takes two sweets and hands them to the
boy saying: "Here you have two sweets. You can do the mixing
yourself."'' A more serious example of the problematic relation between
words and concepts is the fact that the words red' and 'green' are used even by
people who are colorblind, though the ranges of applicability of these terms
must be quite different for them from what they are for other people.
This intrinsic
uncertainty of the meaning of words was of course recognized very early and has
brought about the need for definitions, or - as the word 'definition' says -
for the setting of boundaries that determine where the word is to be used and
where not. But definitions can be given only with the help of other concepts,
and so one will finally have to rely on some concepts that are taken as they
are, unanalyzed and undefined.
In
Greek philosophy the problem of the concepts in language has been a major theme
since Socrates, whose life was - if we can follow Plato's artistic
representation in his dialogues - a continuous discussion about the content of
the concepts in language and about the limitations in modes of expression. In
order to obtain a solid basis for scientific thinking, Aristotle in his logic
started to analyze the forms of language, the formal structure of conclusions
and deductions independent of their content. In this way he reached a degree of
abstraction and precision that had been unknown up to that time in Greek
philosophy and he thereby contributed immensely to the clarification, to the
establishment of order in our methods of thought. He actually created the basis
for the scientific language.
On the other
hand, this logical analysis of language again involves the danger of an
oversimplification. In logic the attention is drawn to very special structures,
unambiguous connections between premises and deductions, simple patterns of
reasoning, and all the other structures of language are neglected. These other
structures may arise from associations between certain meanings of words; for
instance, a secondary meaning of a word which passes only vaguely through the
mind when the word is heard may contribute essentially to the content of a
sentence. The fact that every word may cause many only half-conscious movements
in our mind can be used to represent some part of reality in the language much
more clearly than by the use of the logical patterns. Therefore, the poets have
often objected to this emphasis in language and in thinking on the logical
pattern, which - if I interpret their opinions correctly - can make language
less suitable for its purpose. We may recall for instance the words in Goethe's
Faust which Mephistopheles speaks to the
young student (quoted from the translation by Anna Swanwick):
Waste not your
time, so fast it flies;
Method will teach you time to win;
Hence, my young
friend, I would advise,
With college logic to begin.
Then will your mind be so well brac'd,
In Spanish boots so tightly lac'd,
That on 'twill
circumspectly creep,
Thought's beaten track securely keep,
Nor will it,
ignis-fatuus like,
Into the path
of error strike.
Then many a
day they'll teach you how
The mind's spontaneous acts, till now
As eating and
as drinking free,
Require a
process; — one, two, three! In truth the subtle web of thought
Is like the weaver's fabric wrought,
One treadle moves a thousand lines,
Swift dart the
shuttles to and fro,
Unseen the threads unnumber'd flow,
A thousand knots one
stroke combines.
Then forward steps your sage to show,
And prove to you it must
be so;
The first being so, and so the second.
The third and fourth deduc'd we
see;
And if there were no first and second,
Nor third nor fourth would ever be.
This, scholars of all countries prize,
Yet 'mong themselves no weavers rise.
Who would describe and study aught alive,
Seeks first the living spirit thence to drive:
Then are the lifeless fragments in his hand,
There only fails, alas! — the
spirit-band.
In
other sciences the situation may be somewhat similar in so far as rather
precise definitions are also required; for instance, in law. But here the
number of links in the chain of conclusions need not be very great, complete
precision is not needed, and rather precise definitions in terms of ordinary
language are sufficient.
In
theoretical physics we try to understand groups of phenomena by introducing
mathematical symbols that can be correlated with facts, namely, with the
results of measurements. For the symbols we use names that visualize their
correlation with the measurement.
Thus the symbols are attached to the
language. Then the symbols are interconnected by a rigorous system of
definitions and axioms, and finally the natural laws are expressed as equations
between the symbols. The infinite variety of solutions of these equations then
corresponds to the infinite variety of particular phenomena that are possible
in this part of nature. In this way the mathematical scheme represents the
group of phenomena so far as the correlation between the symbols and the measurements
goes. It is this correlation which permits the expression of natural laws in
the terms of common language, since our experiments consisting of actions and
observations can always be described in ordinary language.
Still, in the
process of expansion of scientific knowledge the language also expands; new
terms are introduced and the old ones are applied in a wider field or
differently from ordinary language. Terms such as '
energy,"electricity,"entropy' are obvious examples. In this way we
develop a scientific language which may be called a natural extension of
ordinary language adapted to the added fields of scientific knowledge.
During
the past century a number of new concepts have been introduced in physics, and
in some cases it has taken considerable time before the scientists have really
grown accustomed to their use. The term 'electromagnetic field,' for instance,
which was to some extent already present in Faraday's work and which later
formed the basis of Maxwell's theory, was not easily accepted by the
physicists, who directed their attention primarily to the mechanical motion of
matter. The introduction of the concept really involved a change in scientific
ideas as well, and such changes
are not easily accomplished.
Still,
all the concepts introduced up to the end of the last century formed a
perfectly consistent set applicable to a wide field of experience, and,
together with the former concepts, formed a language which not only the
scientists but also the technicians and engineers could successfully apply in
their work. To the underlying fundamental ideas of this language belonged the
assumptions that the order of events in time is entirely independent of their
order in space, that Euclidean geometry is valid in real space, and that the
events happen' in space and time independently of whether they are observed or
not. It was not denied that every observation had some influence on the phenomenon
to be observed but it was generally assumed that by doing the experiments
cautiously this influence could be made arbitrarily small. This seemed in fact
a necessary condition for the ideal of objectivity which was considered as the
basis of all natural science.
Into this
rather peaceful state of physics broke quantum theory and the theory of special
relativity as a sudden, at first slow and then gradually increasing, movement
in the foundations of natural science. The first violent discussions developed
around the problems of space and time raised by the theory of relativity. How
should one speak about the new situation? Should one consider the Lorentz
contraction of moving bodies as a real contraction or only as an apparent
contraction? Should one say that the structure of space and time was really
different from what it had been assumed to be or should one only say that the
experimental results could be connected mathematically in a way corresponding
to this new structure, while space and time, being the universal
and necessary mode in which things appear to us, remain what they had always
been? The real problem behind these many controversies was the fact that no
language existed in which one could speak consistently about the new situation.
The ordinary language was based upon the old concepts of space and time and
this language offered the only unambiguous means of communication about the
setting up and the results of the measurements. Yet the experiments showed that
the old concepts could not be applied everywhere.
The
obvious starting point for the interpretation of the theory of relativity was
therefore the fact that in the limiting case of small velocities (small
compared with the velocity of light) the new theory was practically identical with
the old one. Therefore, in this part of the theory it was obvious in which way
the mathematical symbols had to be correlated with the measurements and with
the terms of ordinary language; actually it was only through this correlation
that the Lorentz transformation had been found. There was no ambiguity about
the meaning of the words and the symbols in this region. In fact this
correlation was already sufficient for the application of the theory to the
whole field of experimental research connected with the problem of relativity.
Therefore, the controversial questions about the ' real' or the ' apparent'
Lorentz contraction, or about the definition of the word simultaneous' etc.,
did not concern the facts but rather the language.
With
regard to the language, on the other hand, one has gradually recognized that
one should perhaps not insist too much on certain principles. It is always
difficult to find general convincing criteria for which terms should be used in
the language and how they should be used. One should simply wait for the
development of the language, which adjusts itself after some time to the new
situation. Actually in the theory of special relativity this adjustment has
already taken place to a large extent during the past fifty years. The distinction
between 'real' and 'apparent' contraction, for instance, has simply
disappeared. The word 'simultaneous' is used in line with the definition given
by Einstein, while for the wider definition discussed in an earlier chapter the
term 'at a space-like distance' is commonly used, etc.
In the
theory of general relativity the idea of a non-Euclidean geometry in real space
was strongly contradicted by some philosophers who
pointed out that our whole method of setting up the experiments already
presupposed Euclidean geometry.
In fact
if. a mechanic tries to prepare a perfectly plane surface, he can do it in the
following way. He first prepares three surfaces of, roughly, the same size
which are, roughly, plane. Then he tries to bring any two of the three surfaces
into contact by putting them against each other in different relative
positions. The degree to which this contact is possible on the whole surface is
a measure of the degree of accuracy with which the surfaces can be called '
plane.' He will be satisfied with his three surfaces only if the contact
between any two of them is complete everywhere. If this happens one can prove
mathematically that Euclidean geometry holds on the three surfaces. In this
way, it was argued, Euclidean geometry is just made correct by our own measures.
From
the point of view of general relativity, of course, one can answer that this
argument proves the validity of Euclidean geometry only in small dimensons, in
the dimensions of our experimental equipment. The accuracy with which it holds
in this region is so high that the above process for getting plane surfaces can
always be carried out. The extremely slight deviations from Euclidean geometry
which still exist in this region will not be realized since the surfaces are
made of material which is not strictly rigid but allows for very small
deformations and since the concept of contact' cannot be defined with complete
precision. For surfaces on a cosmic scale the process that has been described
would just not work; but this is not a problem of experimental physics.
In the
theory of general relativity the language by which we describe the general laws
actually now follows the scientific language of the mathematicians, and for the
description of the experiments them-selves we can use the ordinary concepts,
since Euclidean geometry is valid with sufficient accuracy in small dimensions.
The most
difficult problem, however, concerning the use of the language arises in
quantum theory. Here we have at first no simple guide for correlating the
mathematical symbols with concepts of ordinary language; and the only thing we
know from the start is the fact that our common concepts cannot be applied to
the structure of the atoms. Again the obvious starting point for the physical
interpretation of the formalism seems to be the fact that the mathematical
scheme of quantum mechanics approaches that of classical mechanics in
dimensions which are large as compared to the size of the atoms. But even this
statement must be made with some reservations. Even in large dimensions there
are many solutions of the quantum-theoretical equations to which no analogous
solutions can be found in classical physics. In these solutions the phenomenon
of the ' interference of probabilities' would show up, as was discussed in the
earlier chapters; it does not exist in classical physics. Therefore, even in
the limit of large dimensions the correlation between the mathematical symbols,
the measurements, and the ordinary concepts is by no means trivial. In order to
get to such an unambiguous correlation one must take another feature of the
problem into account. It must be observed that the system which is treated by
the methods of quantum. mechanics is in fact a part of a much bigger system
(eventually the whole world); it is interacting with this bigger system; and
one must add that the microscopic properties of the bigger system are (at least
to a large extent)
unknown. This statement is undoubtedly a correct description of the actual
situation. Since the system could not be the object of measurements and of
theoretical investigations, it would in fact not belong to the world of
phenomena if it had no interactions with such a bigger system of which the
observer is a part. The interaction with the bigger system with its undefined
microscopic properties then introduces a new statistical element into the
description - both the quantum-theoretical and the classical one - of the
system under consideration. _ In the limiting case of the large dimensions this
statistical element destroys the effects of the 'interference of
probabilities' in such a manner that now the quantum-mechanical scheme really
approaches the classical one in the limit. Therefore, at this point the
correlation between the mathematical symbols of quantum theory and the concepts
of ordinary language is unambiguous, and this correlation suffices for the
interpretation of the experiments.
The remaining problems again concern the
language rather than the facts, since it belongs to the concept ' fact' that it
can be described in ordinary language.
But the
problems of language here are really serious. We wish to speak in some way
about the structure of the atoms and not only about the 'facts' - the latter
being, for instance, the black spots on a photographic plate or the water
droplets in a cloud chamber. But we cannot speak about the atoms in ordinary
language.
The
analysis can now be carried further in two entirely different ways. We can
either ask which language concerning the atoms has actually developed among the
physicists in the thirty years that have elapsed since the formulation of
quantum mechanics. Or we can describe the attempts for defining a precise
scientific language that corresponds to the mathematical scheme.
In answer to
the first question one may say that the concept of complementarity introduced
by Bohr into the interpretation of quantum theory has encouraged the physicists
to use an ambiguous rather than an unambiguous language, to use the classical
concepts in a somewhat vague manner in conformity with the principle of
uncertainty, to apply alternatively different classical concepts which would
lead to contradictions if used simultaneously. In this way one speaks about
electronic orbits, about matter waves and charge density, about energy and
momentum, etc., always conscious of the fact that these concepts have only a
very limited range of applicability. When this vague and unsystematic use of
the language leads into difficulties, the physicist has to withdraw into the
mathematical scheme and its unambiguous correlation with the experimental facts.
This
use of the language is in many ways quite satisfactory, since it reminds us of
a similar use of the language in daily life or in poetry. We realize that the
situation of complementarity is not confined to the atomic world alone; we meet
it when we reflect about a decision and the motives for our decision or when we
have the choice between enjoying music and analyzing its structure. On the
other hand, when the classical concepts are used in this manner, they always
retain a certain vagueness, they acquire in their relation to ' reality' only
the same statistical significance as the concepts of classical thermodynamics
in its statistical interpretation. Therefore, a short discussion of these
statistical concepts of thermodynamics may be useful.
The concept
'temperature' in classical thermodynamics seems to describe an objective
feature of reality, an objective property of matter. In daily life it is quite
easy to define with the help of a thermometer what we mean by stating that a
piece of matter has a certain temperature. But when we try to define what the
temperature of an atom could mean we are, even in classical physics, in a much
more difficult position. Actually we cannot correlate this concept 'temperature
of the atom' with a well-defined property of the atom but have to connect it at
least partly with our insufficient knowledge of it. We can correlate the value
of the temperature with certain statistical expectations about the properties
of the atom, but it seems rather doubtful whether an expectation should be
called objective. The concept 'temperature of the atom' is not much better
defined than the concept 'mixing' in the story about the boy who bought mixed
sweets.
In a similar
way in quantum theory all the classical concepts are, when applied to the atom,
just as well and just as little defined as the 'temperature of the atom'; they
are correlated with statistical expectations; only in rare cases may the
expectation become the equivalent of certainty. Again, as in classical
thermodynamics, it is difficult to
call the expectation objective. One might perhaps call it an objective tendency
or possibility, a 'potentia' in the sense of Aristotelian
philosophy. In fact, I believe that the language actually used by physicists
when they speak about atomic events produces in their minds similar notions as
the concept potentia.' So the physicists have gradually become accustomed to
considering the electronic orbits, etc., not as reality but rather as a kind of
'potentia.' The language has already adjusted itself, at least to some extent,
to this true situation. But it is not a precise language in which one could use
the normal logical patterns; it is a language that produces pictures in our
mind, but together with them the notion that the pictures have only a vague
connection with reality, that they represent only a tendency toward
reality.
The
vagueness of this language in use among the physicists has therefore led to
attempts to define a different precise language which follows definite logical
patterns in complete conformity with the mathematical scheme of quantum theory.
The result of these attempts by Birkhoff and Neumann and more recently by
Weizsacker can be stated by saying that the mathematical scheme of quantum
theory can be interpreted as an extension or modification of classical logic.
It is especially one fundamental principle of classical logic which seems to
require a modification.
In classical logic it is assumed that, if a statement
has any meaning at all, either the statement or the negation of the statement
must be correct. Of here is a table' or ' here is not a table,' either the
first or the second statement must be correct. 'Tertium non datur,' a third
possibility does not exist. It may be that we do not know whether the statement
or its negation is correct; but in 'reality' one of the. two is
correct.
In quantum
theory this law tertium non datur' is to be modified. Against any modification
of this fundamental principle one can of course at once argue that the
principle is assumed in common language and that we have to speak at least
about our eventual modification of logic in the natural language. Therefore, it
would be a self-contradiction to describe in natural language a logical scheme
that does not apply to natural language. There, however, Weizsacker points out
that one may distinguish various levels of language.
One
level refers to the objects — for instance, to the atoms or the electrons. A
second level refers to statements about objects. A third level may refer to
statements about statements about objects, etc. It would then be possible to
have different logical patterns at the different levels. It is true that
finally we have to go back to the natural language and thereby to the classical
logical patterns. But Weizsacker suggests that classical logic may be in a
similar manner a priori to quantum logic, as classical physics is to quantum
theory. Classical logic would then be contained as a kind of limiting case in
quantum logic, but the latter would constitute the more general logical
pattern.
The
possible modification of the classical logical pattern shall, then, first refer
to the level concerning the objects. Let us consider an atom moving in a closed
box which is divided by a wall into two equal parts. The wall may have a very
small hole so that the atom can go through. Then the atom can, according to
classical logic, be either in the left half of the box or in the right half.
There is no third possibility: tertium non datur.'
In quantum theory, however,
we have to admit — if we use the words 'atom' and box' at all — that there are
other possibilities which are in a strange way mixtures of the two former
possibilities. This is necessary for explaining the results of our experiments.
We could, for instance, observe light that has been scattered by the atom. We
could perform three experiments: first the atom is (for instance, by closing
the hole in the wall) confined to the left half of the box, and the intensity
distribution of the scattered light is measured; then it is confined to the
right half and again the scattered light is measured; and finally the atom can
move freely in the whole box and again the intensity distribution of the
scattered light is measured. If the atom would always be in either the left
half or the right half of the box, the final intensity distribution should be a
mixture (according to the fraction of time spent by the atom in each of the two
parts) of the two former intensity distributions. But this is in general not
true experimentally. The real intensity distribution is modified by the
'interference of probabilities'; this has been discussed before.
In order to
cope with this situation Weizsacker has introduced the concept 'degree of
truth.' For any simple statement in an alternative like 'The atom
is in the left (or in the right) half of the box' a complex number is defined
as a measure for its 'degree of truth.' If the number is 1, it means that the
statement is true; if the number is o, it means that it is false. But other
values are possible. The absolute square of the complex number gives the
probability for the statement's being true; the sum of the two probabilities
referring to the two parts in the alternative (either ' left' or right' in our
case) must be unity. But each pair of complex numbers referring to the two
parts of the alternative represents, according to Weizsacker's definitions, a
'statement' which is certainly true if the numbers have just these values; the
two numbers, for instance, are sufficient for determining the intensity
distribution of scattered light in our experiment. If one allows the use of the
term statement' in this way one can introduce the term complementarity' by the
following definition: Each statement that is not identical with either of the
two alternative statements — in our case with the statements: 'the atom is in
the left half' or the atom is in the right half of the box' — is called
complementary to these statements. For each complementary statement the
question whether the atom is left or right is not decided. But the term 'not
decided' is by no means equivalent to the term not known.' 'Not known' would
mean that the atom is 'really' l eft or right, only we do not know where it is.
But 'not decided' indicates a different situation, expressible only by a complementary
statement.
This
general logical pattern, the details of which cannot be described here,
corresponds precisely to the mathematical formalism of quantum theory. It forms
the basis of a precise language that can be used to describe the structure of
the atom. But the application of such a language raises a number of difficult
problems of which we shall discuss only two here: the relation between the
different 'levels' of language and the consequences for the underlying
ontology.
In
classical logic the relation between the different levels of language is a
one-to-one correspondence. The two statements, 'The atom is in the left half'
and It is true that the atom is in the left half,' belong logically to different
levels. In classical logic these statements are completely equivalent, i.e.,
they are either both true or both false. It is not possible that the one is
true and the other false. But in the logical
pattern of complementarity this relation is more complicated. The correctness
or incorrectness of the first statement still implies the correctness or
incorrectness of the second statement. But the incorrectness of the second
statement does not imply the incorrectness of the first statement. If the
second statement is incorrect, it may be undecided whether the atom is in the
left half; the atom need not necessarily be in the right half. There is still
complete equivalence between the two levels of language with respect to the
correctness of a statement, but not with respect to the incorrectness. From
this connection one can understand the persistence of the classical laws in
quantum theory: wherever a definite result can be derived in a given experiment
by the application of the classical laws the result will also follow from
quantum theory, and it will hold experimentally.
The
final aim of Weizsacker's attempt is to apply the modified logical patterns
also in the higher levels of language, but these questions cannot be discussed
here.
The
other problem concerns the ontology that underlies the modified logical
patterns.
If the pair of complex numbers represents a 'statement' in the sense
just described, there should exist a 'state' or a 'situation' in nature in
which the statement is correct. We will use the word state' in this connection.
The 'states' corresponding to complementary statements are then called
coexistent states' by Weizsacker. This term 'coexistent' describes the
situation correctly; it would in fact be difficult to call them 'different
states,' since every state contains to some extent also the other 'coexistent
states.' This concept of 'state' would then form a first definition concerning
the ontology of quantum theory. One sees at once that this use of the word
'state,' especially the term 'coexistent state,' is so different from the usual
materialistic ontology that one may doubt whether one is using a convenient
terminology. On the other hand, if one considers the word ' state' as
describing some potentiality rather than a reality — one may even simply
replace the term 'state' by the term 'potentiality' — then the concept of
coexistent potentialities' is quite plausible, since one potentiality may
involve or overlap other potentialities.
All these
difficult definitions and distinctions can be avoided if one confines the
language to the description of facts, i.e., experimental results.
However, if one wishes to speak about the atomic particles themselves one must
either use the mathematical scheme as the only supplement to natural language
or one must combine it with a language that makes use of a modified logic or of
no well-defined logic at all. In the experiments about atomic events we have to
do with things and facts, with phenomena that are just as real as any phenomena
in daily life. But the atoms or the elementary particles themselves are not as
real; they form a world of potentialities or possibilities rather than one of
things or facts.
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