Hans Reichenbach (1891—1953)

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.
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) (AB)
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. References and Further Reading

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.