Table of Contents
1. Life
Hans Reichenbach, born on September 26th 1891 in
Hamburg, Germany, was a leading philosopher of science,
a founder of the Berlin circle, and a proponent of
logical positivism (also known as neopositivism or
logical empiricism). He studied physics, mathematics and
philosophy at Berlin, Erlangen, Gottingen and Munich in
1910s. Among his teachers were the neo-Kantian
philosopher Ernst Cassirer, the mathematician David
Hilbert, and the physicists Max Planck, Max Born and
Albert Einstein. Reichenbach received his degree in
philosophy from the University at Erlangen in 1915; his
dissertation on the theory of probability was published
in 1916. He attended Einstein's lectures on the theory
of relativity at Berlin in 1917-20; at that time
Reichenbach chose the theory of relativity as the first
subject for his own philosophical research. He became a
professor at Polytechnic at Stuttgart in 1920. In the
same year he published his first book on the
philosophical implications of the theory of relativity,
The theory of relativity and a priori knowledge,
in which Reichenbach criticized Kantian theory of
synthetic a priori. In the following years he published
three books on the philosophical meaning of the theory
of relativity: Axiomatization of the theory of
relativity (1924), From Copernicus to Einstein
(1927) and The philosophy of space and time
(1928); the last in a sense states logical positivism's
view on the theory of relativity. In 1926 Reichenbach
became a professor of philosophy of physics at the
University at Berlin. His methods of teaching philosophy
were something of a novelty; students found him easy to
approach (this fact was uncommon in German
universities); his courses were open to discussion and
debate. In 1928 he founded the Berlin circle (named Die
Gesellschaft fur empirische Philosophie, "Society for
empirical philosophy"). Among the members of the Berlin
circle were Carl Gustav Hempel, Richard von Mises, David
Hilbert and Kurt Grelling. In 1930 Reichenbach and
Carnap undertook the editorship of the journal
Erkenntnis ("Knowledge").
In 1933 Adolf Hitler became Chancellor of Germany. In
the same year Reichenbach emigrated to Turkey, where he
became chief of the Department of Philosophy at the
University at Istanbul. In Turkey Reichenbach promoted a
shift in philosophy course; he introduced
interdisciplinary seminars and courses on scientific
subjects. In 1935 he published The theory of
probability.
In 1938 he moved to the United States, where he
became a professor at the University of California at
Los Angeles; in the same year was published
Experience and prediction. Reichenbach's work on
quantum mechanics was published in 1944 (Philosophic
foundations of quantum mechanics). Afterwards he
wrote two popular books: Elements of symbolic logic
(1947) and The rise of scientific philosophy
(1951). In 1949 he contributed an essay on The
philosophical significance of the theory of relativity
to Albert Einstein: philosopher-scientist edit by
Paul Arthur Schillp. Reichenbach died on April 9th 1953
at Los Angeles, California, while he was working on the
philosophy of time. Two books Nomological statements
and admissible operations (1954) and The
direction of time (1956) were published
posthumously.
2. The Philosophy of Space and
Time and the Philosophical Meaning of the Theory of
Relativity
a. Space
Euclidean geometry is based on the set of axioms
stated by Greek mathematician Euclid who developed
geometry into an axiomatic system, in which every
theorem is derivable from the axioms. Euclid's work
revealed that the truth of geometry depends on the truth
of axioms and therefore the question arose whether the
axioms were true. Many Euclidean axioms were
self-evident, but the axiom of parallels, which states
that there is one and only one parallel to a given line
through a given point, was considered not self-evident,
and many mathematicians tried to derive it from the
other axioms. Eventually it was proved the axiom of
parallels is not a logical consequence of the remainder.
As a result of this research non-Euclidean geometries
were discovered and mathematicians became aware of the
existence of a plurality of geometries, namely:
Euclidean geometry, in which the axiom of parallels
is true;
geometry of Bolyai and Lobachevsky, also known as
hyperbolic geometry, in which there is an infinite
number of parallels to the given line through the given
point (Janos Bolyai b 1802 d 1860, Hungarian
mathematician, published in 1832 the first account of a
non-Euclidean geometry; Nikolay Lobachevsky b 1793 d
1856, Russian mathematician, independently discovered
hyperbolic geometry);
elliptical geometry, in which there exist no
parallel.
In Reichenbach opinion, it must be realized that
there are two different kinds of geometry, namely
mathematical geometry and physical geometry.
Mathematical geometry, a branch of mathematics, is a
purely formal system and it does not deal with the truth
of axioms, but with the proof of theorems, ie it only
search for the consequences of axioms. Physical geometry
is concerned with the real geometry, ie the geometry
which is true in our physical world: it searches for the
truth (or falsity) of axioms, using the methods of
empirical science: experiments, measurements, etc; it is
a branch of physics.
How can physicists discover the geometry of the real
world? Look at the following example, which Reichenbach
analyses in The philosophy of space and time.
Two-dimensional intelligent beings live in a
two-dimensional world, on the surface of a sphere, but
they do not know where they live; in their opinion, they
might live on a plane, a sphere or whatever surface. How
can they discover where they live? They could use some
mathematical properties that characterize a geometry;
for example, in Euclidean geometry the ratio of the
circumference of a circle to its diameter equals pi
(3.14...) while in elliptical geometry the ratio is
variable and it is less than pi; also in hyperbolic
geometry the ratio is variable but greater than pi.
Therefore they could measure the circumference and the
diameter of a circle; if the ratio equals pi the surface
is a plane; if the ratio is less than pi the surface is
a sphere. Thus they could discover where they live with
the help of such measurements. This method, invented by
Gauss (Karl Friedrich Gauss, b 1777 d 1855, German
mathematician, was the first to discover a non-Euclidean
geometry although he did not published his work) is
suitable for a two-dimensional world. Riemann (Bernhard
Riemann, b 1826 d 1866, German mathematician, developed
both the elliptical geometry and the generalized theory
of metric space in any number of dimension which
Einstein used in his general theory of relativity)
invented a method suitable for a three-dimensional
world. There is no reason in principle why physicists
could not use Riemann's method to discover the geometry
of our world.
Riemann's method is based on physical measurements.
Reichenbach carefully examines the epistemological
implications of measuring geometrical entities. The
empirical measurement of geometrical entities depends on
physical objects or physical processes corresponding to
geometrical concepts. The process of establishing such
correlation is called a co-ordinative definition.
Usually, a definition is a statement that gives the
exact meaning of a concept; this kind of definition is
called an explicit definition. There is another kind of
definition, namely the co-ordinative definition; it is
not a statement, but an ostensive definition. The
co-ordinative definition of a concept is a correlation
between a real object or a physical process and the
concept itself. Some geometrical entities cannot be
defined by an explicit definition but they require a
co-ordinative definition. For example, the unit of
length, ie the metre, is defined by a co-ordinative
definition; the physical object corresponding to the
metre is the standard rod in Paris (Museum of weights
and measures in Paris houses the units of measure for
International System of Units). Another example is the
definition of straight line which is co-ordinated with a
physical process, namely the path of a light ray.
What is the philosophical meaning of a co-ordinative
definition? Reichenbach proposes the following problem,
discussed in The philosophy of space and time. A
measuring rod is moved from one point of space (say A)
to another point (say B). When the measuring rod is in
B, is its length altered? Many physical circumstances
can alter the length, eg if temperature in A differs
from temperature in B. In this example, we can discover
whether the temperature is the same by means of a
metallic rod and a wooden rod which are of equal length
when they are in A. Move the two rods to B: if their
length becomes different then the temperature is also
different, otherwise the temperature is the same. This
method is suitable because temperature is a
differential force, ie a force that produces
different effects on different substances. But there are
universal forces, which produce the same effect
on all type of matter. The best known universal force is
gravity: its effect is the same on all bodies and
therefore all bodies fall with the same acceleration.
Now suppose a universal force alters the length of the
measuring rods when they are moved from A to B; in this
instance, we do not observe any difference between the
measuring rods and we cannot know whether the length is
altered. Consequently, if a rod stays in A and the other
is moved to B where a universal force alters its length,
we cannot know their length is different. So we must
acknowledge that there is not any way of knowing whether
the length of two measuring rods, which are equal when
they are in the same point of space, is the same when
the two rods are in two different points of space. We
can define the two rods equal in length if all
differential forces are eliminated and disregard
universal forces. But we can adopt a different
definition, of course. Thus we must accept - Reichenbach
says - that the geometrical form of a body is not an
absolute fact, but depends on a co-ordinative definition.
There is an astonish consequence of this fact. If a
geometry G was proved to be the real geometry by a set
of measurements, we could arbitrarily choose a different
geometry G' and adopt a different set of co-ordinative
definitions so that G' would become the real geometry.
This is the principle of relativity of geometry,
which Reichenbach examines, from a mathematical point of
view, in Axiomatization of the theory of relativity
and, from a philosophical point of view, in The
philosophy of space and time. This principle states
that all geometrical systems are equivalent; it
falsifies alleged a priori character of Euclidean
geometry and thus it falsifies the Kantian philosophy of
space too.
At a first glance, the principle of relativity of
geometry proves it is not possible to discover the real
geometry of our world. This is true if we limit
ourselves to metric relationships. Metric relationships
are geometric properties of bodies depending on
distances, angles, areas, etc; examples of metric
relationships are "the ratio of circumference to
diameter equals pi" and "the volume of A is greater than
the volume of B". But we can study not only distances,
angles, areas but also the order of space, the
topology of space, ie way in which the points of
space are placed in relation to one another; an example
of a topological relationship is "point A is between
point B and C". A consequence of the principle of
relativity of geometry is, for instance, that a plane
and a sphere are equivalent with respect to metric. From
a topological point of view, a sphere and a plane are
not equivalent (in topology, two geometrical objects are
equivalent if and only if there is a continuous
transformation that assign to every point of the first
object a unique point of the second and vice versa;
there is not any transformation of this kind between a
sphere and a plane). What is the philosophical
significance of topology?
Reichenbach examines the following example (The
philosophy of space and time). Measurements of
space, performed by a two-dimensional being, suggest
that he lives on a sphere, but, in spite of such
measurements, he believes he lives on a plane. There is
not any difficult, when he limits himself to metric
relationships: he could adopt appropriate co-ordinative
definitions and those measurements would become
compatible with a plane. But the surface of a sphere is
a finite surface and he might do a round-the-world tour,
that is he could walk along a straight line from a point
A and eventually he would arrive to the point A itself.
Really this is impossible on a plane and he therefore
should assert that this last point is not the point A,
but a different point B which, in all other respects, is
identical to A. Now there are two possibilities: (i) he
changes his theory and acknowledges that he lives on a
sphere or (ii) he maintains his position, but he needs
to explain why point B is identical to A although A and
B are different and distant points of space; he could
accomplish his task only fabricating a fictitious theory
of pre-established harmony: everything that occurs in A,
immediately occurs in B.
Reichenbach says the second possibility entails an
anomaly in the law of causality. If we assume normal
causality, topology become an empirical theory and we
can discover the geometry of the real world. This
example is another falsification of Kantian theory of
synthetic a priori. Kant believed both the Euclidean
geometry and the law of causality were a priori. But if
Euclidean geometry were an a priori truth, normal
causality might be false; if normal causality were an a
priori truth, Euclidean geometry might be false. We
arbitrarily can choose the geometry or we arbitrarily
can choose the causality; but we cannot choose both.
Thus the most important implication of the philosophical
analysis of topology is that the theory of space
depends on normal causality.
b. Time
Normal causality is the main principle that underlies
not only the theory of space but also the theory of
time. The solution to the problem of an empirical theory
of space was found when we acknowledged the priority of
topological relationships over metric relationships.
Also in the philosophy of time we must recognize the
priority of topology. We must distinguish between two
different concepts which are fundamental to the theory
of time, namely the order of time and the
direction of time. Time order is definable by means
of causality (see The philosophy of space and time).
The definition is: event A occurs before event B
(and, of course, event B occurs after event A) if
event A can produce a physical effect on event B.
When can event A affect event B? The theory of
relativity states that it is required a finite time for
an effect to go from event A to event B. The required
time is finite because the velocity of light is a speed
limit for all material particles, messages or effects
and the velocity of light is finite. Suppose A and B are
two events occurring in point PA and PB. Event A can
affect event B if a light pulse emitted from PA when
event A occurs reaches the point PB before event B
occurs. If the light pulse reaches point PB when event B
already occurred, event A cannot affect event B. If
event A cannot affect event B and event B cannot affect
event A, the order of the two events is indefinite and
we could arbitrarily choose the event that occurs first
or we might define the two event simultaneous; therefore
simultaneity depends on a definition.
Reichenbach examines the consistency of this
definition. Suppose an event A occurs before an event B
and, from another point of view, the event A occurs
after the event B. In this circumstance there is a
closed causal chain so that the event A produces an
effect on the event B and the event B produces an effect
on the event A. The definition is consistent only if we
assume that there are not closed causal chains: the
order of time depends on normal causality.
Reichenbach asserts that the relativity of
simultaneity is independent from the relativity of
motion. The relativity of simultaneity is due to the
finite velocity of causal propagation. So it is a
mistake - Reichenbach asserts in The philosophy of
space and time and From Copernicus to Einstein
- to derive the relativity of simultaneity from the
relative motion of observers. Reichenbach also cautions
against a possible misunderstanding of the multiplicity
of observers in some expositions of the theory of
relativity: observers are used only for convenience; the
relativity of simultaneity has nothing to do with the
relativity of observers. We must recognize - Reichenbach
asserts - that the theory of an absolute simultaneity is
a consistent theory although it is a wrong one. Absolute
simultaneity and absolute time does not exist, but they
are clever concepts.
Reichenbach also faces the problem of the
direction of time. All mechanical processes are
reversible: if f(t) is a solution of the equations of
classical mechanics then f(-t) is also an admissible
solution; also in the theory of relativity f(-t) is an
admissible solution. Thus neither theory gives a
consistent definition of the direction of time. In fact
the direction of time is definable only by means of
irreversible processes, ie processes that are
characterized by an increase of entropy. But the
definition is not straightforward. The second law of
thermodynamics, which states the principle of increase
of entropy, is a statistical law, not a deterministic
law. Really the elementary processes of statistical
thermodynamics are reversible, because they are
controlled by the laws of classical mechanics. In fact
all macroscopic processes are also reversible, in a
sense: every upgrade of entropy is naturally followed by
a corresponding downgrade; we cannot control the
downgrade and thus we cannot reverse the process. But
statistical thermodynamics asserts that after a large
amount of time the entropy will diminish to the initial
value. In an isolated system, in an infinite time, there
are as many downgrades as upgrades of the entropy. Thus
if we observe two states A and B, and the entropy of B
is greater than the entropy of A, we cannot assert that
B is later than A. But if we consider not an isolated
system, but many isolated systems, we realized that the
probability that we observe a decrease of entropy is
less than the probability we observe an increase of
entropy. We can therefore use many-system probabilities
to define a direction of time. Reichenbach asserts that
it is possible to define an entropy for the whole
universe and the statistical theory proves that the
entropy of the universe first increases and then
decreases; thus we can define a direction of time only
for sections of time, not for the whole time.
Reichenbach notes that this theory of time was stated in
19th century by Boltzmann (Ludwig Boltzmann, b 1844 d
1906, Austrian physicist, formulated the statistical
theory of entropy).
c. The Special Theory of Relativity
The special theory of relativity gives an unified
theory of space and time in the absence of gravitational
field. One example of the necessity of an unified theory
of space and time is the length contraction, an
effect predicted by the theory; this effect shows that
the length of a moving rod depends on simultaneity. The
special theory of relativity states that the length of a
rod measured using a metre that is at rest with respect
to the rod is different from the length measured using a
metre which is moving with respect to the rod. In the
first instance we measure the length of the rod by means
of the well-known method used by classical mechanics.
But we use a different method when the measuring rod is
not at rest with respect to the metre. We measure the
length of the moving rod by means of the distance
between the two points occupied at a given time by the
two ends of the moving rod, ie we mark the simultaneous
positions of the two ends and we measure the distance
between those positions; thus this method depend on the
definition of simultaneity, which also depends on a
definition. It must be acknowledged that the length of a
moving rod is a matter of definition, but the length
contraction is a genuine physical hypothesis confirmed
by experiments. We must also recognize the priority of
time over space: the ability to measure time is a
requisite for the theory of space. Therefore only an
unified theory of space and time is suitable. In spite
of the necessity for an unified theory of space and
time, Reichenbach states (in The philosophy of space
and time) that space and time are different concepts
which remain distinct in the theory of relativity. The
real space is three-dimensional and the real
time is one-dimensional: the four-dimensional space-time
used in the theory of relativity is a mathematical
artefact. Also the mathematical formulation of the
special theory of relativity acknowledges the difference
between space and time: the equation that defines the
metric is dx^2 + dy^2 + dz^2 - dt^2 = ds^2 and the time
coordinate is distinguishable from the space coordinates
by the negative sign. How can we know the space is
three-dimensional? and how can we recognize the
difference between a real space and a mathematical
space?
A physical effect is not immediately transmitted from
one point to another distant point but it passes through
every point between the source and the destination. This
principle is known as the principle of local action
and it denies the existence of action at a distance. In
three-dimensional space the principle of local action is
true while in a four-dimensional space it is false, so
we can recognize that the real space is
three-dimensional. We can also distinguish between a
mathematical space and the real space because in a
mathematical space the principle of local action is
false. Reichenbach says that the truth of the principle
of local action is an empirical fact, not an a priori
truth: it could be false. But if this principle is true
then there is only one n-dimensional space in which it
is true; this n-dimensional space is the real space and
n is the number of the dimensions of space. So we
recognize that the real space is three-dimensional while
the four-dimensional space used in the theory of
relativity is a mathematical space, not a real one. We
also recognize that the unified theory of space and
time depends on normal causality.
Among the results of the special theory of relativity
is time dilation: the period of a moving clock is
greater than the period of a clock at rest and therefore
the moving clock slows. Time dilation is an empirical
hypothesis and Reichenbach says its physical meaning is
that a clock does not measure the time coordinate but it
measures the interval, ie the space-time distance
between two events. In classical mechanics space is
Euclidean and Pythagoras' theorem gives the distance ds
between two points: ds^2 = dx^2 + dy^2 + dz^2; x,y,z are
the space coordinates. The distance ds is measured by
rod. Time is an independent coordinate and is measured
by clock. The mathematical formulation of the special
theory of relativity uses a four-dimensional space-time
known as the Minkowski space (mathematician
Hermann Minkowski, b 1864 d 1909, gave a mathematical
formulation of Einstein's special theory of relativity),
in which three coordinates are the space coordinates and
one coordinate is the time coordinate. The distance ds
between two points of Minkowski space is: ds^2 = dx^2 +
dy^2 + dz^2 - dt^2; t is the time coordinate and ds (or
ds^2) is the interval. A positive (negative) ds^2 is
called a spacelike (timelike) interval. Suppose A and B
are two events, interval ds^2 is negative and S is an
inertial frame of reference moving with constant
velocity v so that both events A and B occurs at the
origin O of S, and suppose there is a clock in O; the
time measured by the clock, called characteristic
time, equals the interval ds. When the interval is
positive, there is an inertial frame of reference S'
with respect to which the two events are simultaneous;
in this instance, the interval ds is realized by a
measuring rod with the two ends coinciding with the
events A and B and at rest with respect to S'. Time
dilation shows an important difference between the
special theory of relativity and classical mechanics;
the special theory asserts that clocks and rods measure
the interval while classical mechanics asserts they
measure coordinates.
I briefly mention also Reichenbach's view on the
velocity of light. He asserts that there is no way of
measuring the velocity of light and proving it is
constant, because the measurement of the velocity of
light requires the definition of simultaneity which
depends on the speed of light. Einstein - Reichenbach
says - does not prove the speed of light is constant,
but the special theory of relativity assumes it is
constant, ie it is constant by definition.
d. The General Theory of Relativity
Newton's second law of motion states that the
acceleration a of a body is proportional to the
force F applied, so that F = m * a,
where m is the inertial mass which represents the
resistance to acceleration (force and acceleration are
vectors and I use bold face as indicator of vector).
Newton's law of gravitation asserts that every particle
attracts every other particle with a force F
proportional to the product of gravitational masses:
F = G (m * m') / r^2; r is the
distance between the two particles, m and m' are the
gravitational mass which represent the response to the
gravitational force. In classical mechanics,
gravitational mass and inertial mass are equivalent;
this principle of equivalence accounts for the
law of free fall which states that the acceleration of
every falling body is the same. The principle of
equivalence is one of the principle of the general
theory of relativity and its consequences are very
important.
Suppose a physicist is into a closed elevator and he
observers a body attached to a spring; he find the
spring is stretched. There are two different although
equivalent explanations.
First explanation. The body is attracted by the
Earth and the gravitational force accounts for the
stretching of the spring.
Second explanation. The elevator is in empty space
so there is not any gravitational force, but the
elevator is accelerated and the inertia of the body
causes the stretching of the spring.
The two explanation are indistinguishable because of
the equivalence between gravitational and inertial mass.
This thought experiment shows that an accelerated frames
of reference can simulate a gravitational field. Now
suppose that in another thought experiment the body does
not exert any force on the spring. Also in this instance
there are two explanations.
First explanation. The elevator is at rest in empty
space so there is not any force.
Second explanation. The elevator is free falling in
a gravitational field so its acceleration equals
gravitational acceleration; the body is falling but also
the spring, the elevator and the physicist are falling
with the same acceleration and therefore they are
relatively at rest and there is not any force.
The consequence of this second thought experiment is
that a gravitational field can be eliminated by means of
an accelerated frame of reference. The theory of general
relativity states that free falling accelerated frames
of reference are inertial systems. Reichenbach says that
this hypothesis is not a consequence of the principle of
equivalence; it is a genuine physical hypothesis which
goes beyond experience. There is an important
consequence of this hypothesis. The special theory of
relativity is true in inertial frames of reference, so
in every inertial system the motion of a light ray is
represented by a straight line. But the general theory
of relativity states that a free falling frame of
reference is an inertial system, so the light moves in a
straight line with respect to this frame of reference;
with respect to a frame of reference which is at rest on
Earth (in this system there is a gravitational field)
the light rays are curved. The consequence is that light
is curved by gravity. Another consequence of the
hypothesis that a free falling frame of reference is an
inertial system is the time dilation in the presence of
a gravitational field.
The general theory of relativity gives an unified
theory of space, time and gravitation; it requires a
non-Euclidean four-dimensional geometry, known as
Riemannian geometry. Reichenbach explains the main
properties of this kind of geometry and the main
differences between Euclidean geometry and Riemannian
geometry. In Euclidean geometry the distance between two
points is given by a simple function of coordinates;
also in Minkowski four-dimensional space-time the
interval is calculable by means of coordinates. In
Euclidean geometry the coordinates have both a metric
and topological significance; this is true also in the
special theory of relativity. In Riemannian geometry the
four coordinates perform a topological function, not a
metric one. This means that we cannot calculate the
distance between two points by means of coordinates. The
metric functions is performed by the metric tensor
g; it is a mathematical entity represented by 16
components. The geometry of four-dimensional space-time
depends on the metric tensor g; for example, if
the components of g are
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
1 |
then the geometry is a Minkowski geometry (ie the
geometry of the special theory). Thus the tensor g
expresses the geometry. But g is determined by
the gravitational field, because the metric tensor also
expresses the acceleration of the frame of reference and
the effects of an acceleration are equivalent to the
effects of a gravitational field. The metric tensor g
expresses both the physical geometry and the
gravitational field. The consequence is astonishingly:
the geometry of the universe is produced by
gravitational fields. Therefore the general theory
of relativity does not reduce gravitation to geometry;
on the contrary, geometry is based on gravitation. The
properties of space and time are empirical properties
caused by gravitational fields.
e. The Reality of Space and Time
Reichenbach asserts (in The philosophy of space
and time) that the reality of space and time is
an unquestionable result of the epistemological analysis
of the theory of relativity. With respect to the
problem of reality, space and time are not different
from the other physical concepts. But the reality of
space and time does not imply the concept of an absolute
space and time. Space and time are relational concepts
and we can study their properties because of the
existence of physical objects, eg clocks, that realize
relationships between space-time entities. Reichenbach
also emphasizes the causal theory of space and time:
causality is the basis of both philosophical and
physical theory of space and time.
3. Quantum Mechanics
a. Interpretation of Quantum
Physics: Part I
The main thesis of Reichenbach's work on quantum
mechanics (Philosophic foundations of quantum
mechanics) is that there is not any exhaustive
interpretation of quantum mechanics which is free from
causal anomalies. A causal anomaly is a violation of
the principle of local action; this principle states
that the action at a distance does not exist. We have
found the principle of local action and causal anomalies
in Reichenbach's philosophy of space and time.
Two main interpretations of quantum mechanics are
involved with the wave-particle duality. Wave
interpretation states that atomic entities are waves
or things that resemble waves; it grew out of the
discovery of the wave-like nature of light and it is
supported by many experiments, for example the two-slit
experiment. In this experiment a beam of electrons is
direct towards a screen with two slits and an
interference pattern is produced behind the screen,
showing that electrons act as waves. The corpuscolar
interpretation regards atomic entities as particles;
it is supported by a long standing tradition and by the
fact that atomic entities show corpuscular properties,
eg mass and momentum. Both wave and corpuscular
interpretation entail causal anomalies. For example
corpuscular interpretation cannot fully explain the
two-slit experiment. An electron acting as a particle
goes through only one slit and its behaviour is
independent of the existence of another slit in a
different point of space. In fact, if one slit is open
and the other is close, the interference pattern is not
produced: electrons behave as if they were informed
whether the other slit is open. But wave interpretation
cannot fully explain a slightly different experiment. An
electron can be localized by a detector put near a slit
and the electron is detected as particle. However for
every event in quantum realm there is an interpretation
by means of particles or waves but there is not a unique
interpretation for all events. Both corpuscular and wave
interpretation are not verifiable; they are not matter
of experience but they are matter of definition.
There are two models that are free of causal
anomalies; they are restricted interpretations, ie they
exclude the admissibility of certain statements. One is
Bohr-Heisenberg interpretation (Niels Bohr, b
1885 d 1962, Danish physicist winner of Nobel prize in
1922, gave the first account of the quantum theory of
atoms; Werner Karl Heisenberg, b 1901 d 1976, German
physicist winner of Nobel prize in 1932, formulated
matrix mechanics and proved the principle of
indeterminacy according to which there is no way of
measuring both position and momentum of atomic
particles). This interpretation states that speaking
about values of not measured physical quantities is
meaningless. In the two-slit experiment, when the two
slits are open and electrons interfere with themselves,
the position of electrons cannot be measured; thus a
statement about the position of electrons is meaningless
and the particle interpretation is forbidden. There are
two main faults - Reichenbach says - in Bohr-Heisenbergh
interpretation: (i) Heisenberg indeterminacy principle
becomes a meta-statement on the semantics of the
language of physics and (ii) it implies the presence of
meaningless statements in physics.
The other interpretation depends on three-valued
logic, ie a formal system that acknowledges three truth
values: true, false and indeterminate.
b. Mathematical Formulation of
Quantum Mechanics
Reichenbach carefully examines and explains the
mathematical formulation of quantum mechanics. It is
based on the notion of quantum operator; a
quantum operator is a mathematical entity corresponding
to a given classical quantity. For example, the quantum
operator energy correspond to the energy in
classical physics. A quantum operator can only assume
discrete values while the corresponding classical
quantity assumes continuous values. Note that an
operator is not a function; it indicates a set of
operation to be performed on a function.
Let U be a classical quantity; U depends on position
Q and momentum P, that is U=F[Q,P]
(position and momentum are vectors and I use bold face
as indicator of vector; I use square brackets to show
that a function depends on given quantities). The
quantum operator corresponding to U is called Uop and is
defined by the following statements.
1. For every function F[Q], substitute
'multiply by F[Q]' to 'F[Q]'.
2. Substitute 'multiply the first partial derivative
with respect to Q by C' to 'P', where
C=h/(2*pi*i), h is the Planck constant, pi equals
3.14..., i is the square root of -1.
3. Substitute 'multiply the second partial
derivative with respect to Q by C^2' to 'P',
where C=h/(2*pi*i), h is the Planck constant, pi equals
3.14..., i is the square root of -1.
c. Examples of Quantum Operators
Let T be the kinetic energy; in classical mechanics,
the kinetic energy is given by the ratio of the square
of momentum P to twice the mass m, that is T=P^2
/ 2m. Quantum operator Top is given by Top=C^2 * (1/2m)
* D" (I use symbol D' to indicate the first
partial derivative with respect to position and D" to
indicate the second partial derivative with respect to
position).
Let H be the mechanical energy, ie the sum of the
kinetic energy T and the potential energy V: H=T+V[Q];
therefore Hop=Top+Vop=C^2 * (1/2m) * D" + V[Q].
If F is a given function, the result (indicated by Hop
F) of performing the operations described by operator
Hop on function F is C^2 * (1/2m) * D" F + V * F.
Classical and Quantum Physical
Quantities; Schrodinger Equations
Quantum operators are useful to describe quantum
systems; they transform physical quantities defined in
classical mechanics into quantum quantities. Let U and
Uop be a physical quantity and the corresponding
operator; the very simple rule is
(E1) Uop F = U * F.
In equation E1 the function F is a parameter and the
function U is the variable; functions F satisfying
equation E1 are called eigenfunctions. When F is
an eigenfunction, the variable U satisfying equation E1
is called an eigenvalue. Usually eigenvalues do
not belong to a continuous interval but they are
discrete values and they represent the admissible values
of quantity U. The first Schrodinger equation can be
derived from equation E1 substituting the energy H to
the general function U.
(S1) Hop F = H * F that is
(S1) C^2 * (1/2m) * D" F + V * F = H * F.
The physical meaning of first Schrodinger equations
is that the energy H of an atomic particle, eg an
electron, can only assume values satisfying the
equation; these values are discrete and belong to a set
of fixed values. A given function F satisfying equation
S1 is a wave function and describe a stationary state.
The amplitude of the wave function F gives the
probability to find the particle in a given point of
space. The second Schrodinger equation is:
(S2) Hop PSI = (ih/2*pi) * PSI'
where PSI is a linear combination of wave functions
and PSI' is the first partial derivative with respect to
time. Equation S2 describe a quantum system by means of
function PSI; this function is the infinite sum of
eigenfunctions.
(S3) PSI = K1 * F1 + K2 * F2 + K3 * F3 + K4 * F4 +
...
where Kn is a series of coefficients and Fn is the
series of eigenfunctions satisfying equation E1. The
square of coefficient Kn gives the probability that the
system is in the state described by Fn, ie the square of
Kn is the probability that the value of U equals the
eigenvalue corresponding to Fn. The second Schrodinger
equations is a deterministic equation, ie if we know the
wave function PSI in a given time t, we can calculate
PSI in every time. Note that PSI does not fully describe
the quantum system; it only gives the probability (by
means of coefficients Kn) that the energy of the quantum
system equals a specific value. Suppose a measurement of
U gives the value Un, which is the eigenvalue
corresponding to the eigenfunction Fn; then PSI = Fn. A
measurement of U therefore changes the function PSI so
that PSI = Fn, for an appropriate eigenfunction Fn.
d. Heisenberg Indeterminacy Principle
Let Pop and Qop the quantum operator
corresponding to momentum and position. It is easy to
verify that for every function F
(H) Pop Qop F - Qop Pop
F = C * F
and the equation H is a mathematical formulation of
Heisenberg indeterminacy principle. The proof of
equation H is straightforward.
Pop Qop F - Qop Pop F =
Pop (Q * F) - Qop (C * D'
F) =
C * (D' (Q * F) - Q * (D'
F) =
C * (D' Q * F + Q * D' F - Q
* D' F) =
C * F
Reichenbach explains the physical meaning of equation
H. Equation H proves that the eigenvalues of position
and momentum are different. Now suppose a physicist
measures both position and momentum of a particle; let
Fp be the eigenfunction corresponding to the measured
momentum and Fq be the eigenfunction corresponding to
the measured position. From the measurement of position:
PSI = Fp; from the measurement of momentum: PSI = Fq.
Therefore Fp = Fq and the eigenvalues are the same; but
the eigenvalues are different. So position and momentum
of a particle cannot be simultaneously measured.
Reichenbach asserts that Heisenberg indeterminacy
principle is not due to the alleged interference an
observer exerts on particles (the explanation of
indeterminacy principle in terms of an interference is
due to Heisenberg). This principle is an objective law
of nature, and it can be stated without reference to
observers.
e. The Interpretation of Quantum
Physics: Part II
After the mathematical formulation of quantum
mechanics, Reichenbach states the basic assumption of
the different interpretation of quantum mechanics.
Corpuscolar interpretation relies on the following
definition. If a measurement of U equals Um, then Um
is the values of U not only at the time of measurement
but also immediately before and immediately after.
If a physicist measures the position of an electron and
immediately after its momentum, than he know both
position and momentum of the electron. In this
interpretation atomic particles have both momentum and
position, so they are real particles; a physicist can
also measure both momentum and position. The knowledge
of both position and momentum is unusable because of the
difference between the eigenfunctions: if PSI equals the
eigenfunction "position" the knowledge of momentum is
totally unused while if PSI equals the eigenfunction
"momentum" the knowledge of position is totally unused.
Wave interpretation states that the value
of a measured quantity exists after the measurement but
before the measurement the quantity assumes
simultaneously all possible values. The effect of
the measurement is the collapse of wave function.
Bohr-Heisenberg interpretation asserts that
the value of a physical quantity exists only after the
measurement; a statement about this value before the
measurement is therefore meaningless.
The interpretation based on three-valued logic
states that a statement about a not measured physical
quantity can be neither true nor false: it can be
indeterminate. The following tables show the
properties of logical connectives in the three-valued
logic suggested by Reichenbach (symbols used in these
tables differ from symbols used by Reichenbach).
negation: cyclic (-) diametrical (?) complete (^))
A |
-A |
?A |
^A |
T |
I |
F |
I |
I |
F |
I |
T |
F |
T |
T |
T |
or (v) and (&)
implication: standard (>) alternative (#) quasi (*)
equivalence: standard (=) alternative (<=>)
A |
B |
(AvB) |
(A&B) |
(A>B) |
(A#B) |
(A*B) |
(A=B) |
(A<=>B) |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
I |
T |
I |
I |
F |
I |
I |
F |
T |
F |
T |
F |
F |
F |
F |
F |
F |
I |
T |
T |
I |
T |
T |
I |
I |
F |
I |
I |
I |
I |
T |
T |
I |
T |
T |
I |
F |
I |
F |
I |
T |
I |
I |
F |
F |
T |
T |
F |
T |
T |
I |
F |
F |
F |
I |
I |
F |
T |
T |
I |
I |
F |
F |
F |
F |
F |
T |
T |
I |
T |
T |
Suppose P is the statement "the momentum of the
particle is p" and Q is the statement "the position of
the particle is q"; then Heisenberg indeterminacy
principle is expressed by the following statement:
(Pv-P) # --Q. The following table is the truth-table of
this sentence.
P |
Q |
-P |
Pv-P |
-Q |
--Q |
(Pv-P) # --Q |
T |
T |
I |
T |
I |
F |
F |
T |
I |
I |
T |
F |
T |
T |
T |
F |
I |
T |
T |
I |
F |
I |
T |
F |
I |
I |
F |
T |
I |
I |
F |
I |
F |
T |
T |
I |
F |
F |
I |
T |
I |
T |
F |
T |
T |
T |
I |
F |
F |
F |
I |
T |
T |
F |
T |
T |
F |
F |
T |
T |
T |
I |
F |
The truth of (Pv-P) # --Q implies that the situations
described in 1st, 3rd, 7th and 9th row of the
truth-table are forbidden. Reichenbach explains how the
three-valued interpretation hides causal anomalies. Look
at the two-slit experiment. Suppose the two slits are
open and the interference pattern is produced. Let P(A)
be the probability that an electron goes through the
first slit; let P(B) be the probability that an electron
goes through the second slit; let P(A,C) be the
probability that an electron gone through the first slit
hits the screen in point C; let P(B,C) be the
probability that an electron gone through the second
slit hits the screen in point C; let P(C) the
probability that an electron hits the screen in point C.
Corpuscular interpretation suggests that
(E2) P(C)=P(A)*P(A,C)+P(B)*P(B,C)
In fact P(C) is not given by equation E2: this is the
origin of causal anomalies. Equation E2 can be expressed
by the following statement: (AvB)#C, where A is "the
electron goes through the first slit", B is "the
electron goes through the second slit" and C is E2. We
know that (i) if an electron goes through the first slit
then it does not go through the second slit and vice
versa, ie A # -B and B # -A; (ii) if an electron does
not go through a slit then it goes through the other
slit, ie -A # B and -B # A. In classical logic, (i) and
(ii) imply AvB, ie [(A # -B)&(B # -A)&(-A # B)&(-B # A)]
# AvB is true (look at the following table).
A |
B |
[((A # -B) |
& |
(B # -A)) |
& |
((-A # B) |
& |
(-B # A))] |
# |
AvB |
F |
F |
F T T |
T |
F T T |
F |
T F F |
F |
T F F |
T |
F |
The truth-table is restricted to one combination of
truth-values because in the other combinations the
consequence AvB is true and the statement Z # (AvB) is
true for all Z. In corpuscular interpretation of
two-slit experiment the statement (A # -B)&(B # -A)&(-A
# B)&(-B # A) is true; in classical logic the statement
[(A # -B)&(B# -A)&(-A # B)&(-B # A)] # AvB is true and
thus also AvB is true; therefore E2 is true. But E2 does
not give the correct formula for the probability and so
there is a causal anomaly. In three-valued logic, (i)
and (ii) do not imply AvB; this fact is proved by means
of the following table.
A |
B |
[((A # -B) |
& |
(B # -A)) |
& |
((-A # B) |
& |
(-B # A))] |
# |
AvB |
I |
I |
I T F |
T |
I T F |
T |
F T I |
T |
F T I |
F |
I |
Thus we cannot assert E2 and there is not any causal
anomaly.
4. Reichenbach's Epistemology
a. The Structure of Science and
the Verifiability Principle
A scientific theory is a formal system which requires
a physical interpretation by means of co-ordinative
definitions. Reichenbach's philosophical research on the
theory of relativity and quantum mechanics implicitly
depends on this view. For example, the distinction
between mathematical geometry and physical geometry
entails the distinction between a purely formal system
and a system interpreted by means of definitions.
Co-ordinative definitions are true by convention and
cannot be verified, but they are not meaningless; in
fact scientific theories require them to acquire an
empirical significance. The acknowledgement of the
existence of meaningful and not verifiable sentences is
very important for a right interpretation of the
epistemology of logical positivism. The verifiability
principle is often regarded as the most important
principle of logical positivism; it states that the
meaning of a sentence is its method of verification and
a sentence which cannot be verified is meaningless.
According to this principle, co-ordinative definitions
might be meaningless; on the contrary, in Reichenbach
opinion, they are not only meaningful but also required
by scientific theories. Note that Reichenbach explicit
agrees with verifiability principle. In 'The
philosophical significance of the theory of relativity'
(1949) he says that the meaning of a sentence is
reducible to its method of verification; he also says
that a physicist can fully understand the Michelson's
experiment only if he adopts the verifiability theory of
meaning. In the same essay, Reichenbach says that the
logic foundation of the theory of relativity is the
discovery that many problems are not verifiable; these
problems can be solved by means of co-ordinative
definitions. Thus co-ordinative definitions are
meaningful and not verifiable. So we must acknowledge
that Reichenbach agrees with the verifiability principle
and, at the same time, asserts that in scientific
theories there are meaningful sentences, namely
co-ordinative definitions, that are not verifiable. Why
these sentences are not meaningless? Because they belong
to scientific theories that are verifiable. For example,
Reichenbach states that (i) the Euclidean geometry is
not verifiable, (ii) the co-ordinative definitions of
geometrical entities are not verifiable but (iii) the
Euclidean geometry plus the co-ordinative definitions of
geometrical entities is verifiable. The theory must
be verifiable, the individual statements belonging to
the theory can be not verifiable.
b. Conventionalism vs. Empiricism
In Reichenbach opinion, among the purposes of the
philosophy of science is the search for a distinction
between empirical and conventional sentences. The
separation of empirical from conventional sentences is
not only possible but also necessary for a full
understanding of scientific theories. Philosophical
research on modern science clearly shows that
conventional elements are present in scientific
knowledge. The description of our world is not uniquely
determined by observations, but there is a plurality of
equivalent descriptions; for example, we can use
different geometry for describing the same space. But
conventionalism is in error. For example,
conventionalism states that we can always adopt the
Euclidean geometry by means of appropriate definitions.
But if we adopt a set of definitions so that the
geometry on the Earth is Euclidean, it is possible that
in another point of the universe the same set of
definitions entails a non-Euclidean geometry; so we can
discover an objective difference between different
points of space. Note that Reichenbach does not state
that scientific knowledge can be proved by means of
experience. On the contrary, he asserts that scientific
theories are based on physical hypotheses which are not
a logical consequence of experiments, eg the general
theory of relativity is based on Einstein's hypothesis
that free falling frames of reference are inertial
systems; we cannot prove this hypothesis, but we can
verify its consequences. Scientific theories cannot
be proved, but we can test their forecasts.
c. Causality
Causality plays a central role in Reichenbach's
philosophy of science. Reichenbach uses the theory of
causality as a key to provide access to modern physics
and understanding of the philosophical significance of
both the theory of relativity and quantum mechanics.
According to Reichenbach, the causal theory of space and
time is the basis for both the theory of relativity and
the philosophy of space and time. In the theory of
relativity it is always possible to choose a set of
co-ordinative definitions satisfying normal causality.
Therefore different geometrical systems are not
equivalent and they can be divided into two groups, one
group satisfying normal causality while the other
entails causal anomalies. Only geometrical systems
belonging to the first group are admissible. It is the
experience that decides whether a given geometry belongs
to the first group; thus conventionalism's view on
geometry is wrong. In quantum mechanics there is not any
set of co-ordinative definitions which is free from
causal anomalies and satisfies classical logic. In fact,
a three-valued logic is required to give an
interpretation satisfying normal causality.
d. Science and Philosophy
First of all, we must acknowledge his scientific
seriousness and physical-mathematical skill. His deep
knowledge of modern physics is unquestionable.
Reichenbach's positive attitude towards scientific
knowledge was influenced not only by his teachers but
also by his own philosophical views. In his opinion,
modern physics is concerned with problems that, until
the late 19th century, were regarded as philosophical
problems, eg the nature of space and time, the source of
gravitation, the real extent of causality. In 17th and
18th century - Reichenbach says - philosophers were
usually interested in science and many of them were also
mathematicians and physicists, eg Descartes and Leibniz;
Kant's epistemology was based on scientific knowledge.
But since 18th science became extraneous to philosophy.
Nowadays - Reichenbach wrote in 1928 - there is an
almost complete separation of philosophy from physical
sciences; philosophical researches into epistemology are
fruitless, because of this separation. On the other
hand, scientists cannot explicitly help the progress of
epistemology: they are too much involved in technical
researches. There is only one way to overcome this
difficulty: philosophers, who are not concerned with
technical subjects but deal with genuine philosophical
problems, must dedicate themselves to the philosophical
analysis of modern physics, so they can clearly express
the implicit philosophical content of scientific
theories. In fact, modern physics is rich in
philosophical consequences: there is more philosophy in
Einstein's work than in many philosophical systems.
5. Bibliography
Reichenbach's Main Works, arranged in
Chronological Order..
1916 Der Begriff der Wahrscheinlichkeit fur die
mathematische Darstellung der Wirklichkeit,
dissertation, Erlangen, 1915
1920 Relativitatstheorie und Erkenntnis apriori
(English translation The theory of relativity and a
priori knowledge, Berkeley : University of
California Press, 1965)
1921 'Bericht uber eine Axiomatik der Einsteinschen
Raum-Zeit-Lehre' in Phys. Zeitschr., 22
1922 'Der gegenwartige Stand der
Relativitatsdiskussion' in Logos, X (English
translation 'The present state of the discussion on
relativity' in Modern philosophy of science :
selected essays by Hans Reichenbach, London :
Routledge & Kegan Paul ; New York : Humanities press,
1959)
1924 Axiomatik der relativistischen
Raum-Zeit-Lehre (English translation
Axiomatization of the theory of relativity, Berkeley
: University of California Press, 1969)
1924 'Die Bewegungslehre bei Newton, Leibniz und
Huyghens' in Kantstudien, 29 (English translation
'The theory of motion according to Newton, Leibniz, and
Huyghens' in Modern philosophy of science : selected
essays by Hans Reichenbach, London : Routledge &
Kegan Paul ; New York : Humanities press, 1959)
1925 'Die Kausal-strukture der Welt und der
Unterschied von Vergangenheit und Zukunft' in
Sitzungsber d. Bayer. Akad. d. Wiss.,
math-naturwiss.
1927 Von Kopernikus bis Einstein. Der Wandel
unseres Weltbildes (English translation From
Copernicus to Einstein, New York : Alliance book
corp., 1942)
1928 Philosophie der Raum-Zeit-Lehre (English
translation The philosophy of space and time, New
York : Dover Publications, 1958)
1929 'Stetige Wahrscheinlichkeits folgen' in
Zeitschr. f. Physik, 53
1929 'Ziele und Wege der physikalische Erkenntnis' in
Handbuch der Physik ed. by Hans Geiger and Karl
Scheel, Bd IV, Berlin : Julius Springer
1930 Atom und kosmos. Das physikalische Weltbild
der Gegenwart (English translation Atom and
cosmos; the world of modern physics, London : G.
Allen & Unwin, ltd., 1932)
1931 Ziele und Wege der heutigen Naturphilosophie
(English translation 'Aims and methods of modern
philosophy of nature' in Modern philosophy of science
: selected essays, Westport : Greenwood Press, 1959)
1933 'Kant und die Naturwissenschaft', Die
Naturwissenschaften, 33-34
1935 Wahrscheinlichkeitslehre : eine Untersuchung
uber die logischen und mathematischen Grundlagen der
Wahrscheinlichkeitsrechnung (English translation
The theory of probability, an inquiry into the logical
and mathematical foundations of the calculus of
probability, Berkeley : University of California
Press, 1948)
1938 Experience and prediction: an analysis of the
foundations and the structure of knowledge, Chicago
: University of Chicago Press
1944 Philosophic foundations of quantum mechanics,
Berkeley and Los Angeles : University of California
press
1947 Elements of symbolic logic, New York,
Macmillan Co.
1948 Philosophy and physics, 'Faculty research
lectures, 1946', Berkeley, Univ. of California Press
1949 'The philosophical significance of the theory of
relativity' in Albert Einstein: philosopher-scientist,
edit by P. A. Schillp, Evanston : The Library of Living
Philosophers
1951 The rise of scientific philosophy,
Berkeley : University of California Press
1953 'Les fondaments logiques de la mechanique des
quanta' in Annales de l'Istitut Henri Poincare',
Tome XIII Fasc II
1954 Nomological statements and admissible
operations, Amsterdam : Nort Holland Publishing
Company
1956 The direction of time, Berkeley :
University of California Press
Collected works (in German).
Gesammelte Werke : in 9 Banden ; herausgegeben von
Andreas Kamlah und Maria Reichenbach, Wiesbaden :
Vieweg
1977 Bd. 1: Der Aufstieg der wissenschaftlichen
Philosophie
1977 Bd. 2: Philosophie der Raum-Zeit-Lehre
1979 Bd. 3: Die philosophische Bedeutung der
Relativitatstheorie
1983 Bd. 4: Erfahrung und Prognose : eine Analyse
der Grundlagen und der Struktur der Erkenntnis
1989 Bd. 5: Philosophische Grundlagen der
Quantenmechanik und Wahrscheinlichkeit
1994 Bd. 7: Wahrscheinlichkeitslehre : eine
Untersuchung uber die logischen und mathematischen
Grundlagen der Wahrscheinlichkeitsrechnung
Other sources.
1959 Modern philosophy of science : selected
essays by Hans Reichenbach, London : Routledge &
Kegan Paul ; New York : Humanities press
1959 Modern philosophy of science : selected
essays by Hans Reichenbach, Westport, Conn. :
Greenwood Press
1978 Selected writings, 1909-1953 : with a
selection of biographical and autobiographical sketches,
'Vienna circle collection', Dordrecht ; Boston : D.
Reidel Pub.
1979 Hans Reichenbach, logical empiricist,
'Synthese library', Dordrecht ; Boston : D. Reidel Pub.
1991 Erkenntnis orientated : a centennial volume
for Rudolf Carnap and Hans Reichenbach, Dordrecht ;
Boston : Kluwer Academic Publishers
1991 Logic, language, and the structure of
scientific theories : proceedings of the
Carnap-Reichenbach centennial, University of Konstanz,
21-24 May 1991, Pittsburgh : University of
Pittsburgh Press ; [Konstanz] : Universitasverlag
Konstanz
Erkenntnis was published between 1930 and 1940. Its
name was Erkenntnis - im Auftrage der Gesellschaft
fur empirische Philosophie, Berlin und des Vereins Ernst
Mach in Wien, hrsg. v. R. Carnap und H. Reichenbach
(Knowledge - in agreement with Society for empirical
philosophy, Berlin and Ernst Mach Association at Vienna,
edit by R. Carnap and H. Reichenbach). In 1939-40 its
name changed into The Journal of unified science
(Erkenntnis), edit by O. Neurath, R. Carnap, Charles
Morris, published by University of Chicago Press.
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