The Natural Science according to the Tractatus: a view from the “Triumph” of Quantum Physics.
Abstract
We
take a look at quantum physics to explore Wittgenstein’s thought
about natural science. The main aspects revised are the metaphor of
the grid, his ideas about probability, isomorphism between reality
and theories and his ideas about causality. [38 words]
6.44:
Not how
the world is is the mystical, but that
it is.
A
branch to some degree neglected of Wittgenstein’s first philosophy
is his thought about the causal nexus, scientific laws and natural
science in general. Although it is arguable, the Tractatus’
treatment of the topic is the most familiar to non-twenty century
philosophical traditions and perhaps, because of it, the easiest to
explore without having in mind much of the proposition’s theory of
the first Ludwig.
Nevertheless,
the lenience in the exploration of the issue gets return in the
unique conclusions: from regular empirical postulates and
suppositions, Wittgenstein manages to produce a concise philosophy of
science that challenges common sense. The corpus of such is a dozen
or fewer statements intertwined with some about the nature of
“elemental propositions” and logical deduction. So the mixture,
with the flavor of eighteen century epistemology, it is also composed
by more Wittgenstein-like ideas. I propose to the reader a traveling
around them on the back of one of the central polemics in particle
physics of the mid twenty century: the EPR argument vs. the Bell test
experiments. It is worth already to state that the discussion is
solved in favor of the experiments and what is considered the
Copenhagen interpretation of quantum physics. Hence, the contribution
of the put together of these Wittgenstanian thoughts
to the debate is the polishing of the considerations about reality
that some physicist developed for the dispute; it is a philosophical
endeavor, a work of elucidation using Wittgenstein.
Einstein,
Podolsky and Rosen begun their famous paperi
with some considerations about the nature of physical theories and
their relation to “objective reality”; we will do the same. The
reader will find, as a first part, an account of the main aspects of
natural science according to our philosophical fellow, a large
part—“Uncertainty”—and
then two concise expositions: “EPR argument” and “the Bell
Tests”. All of these will be accompanied by Wittgenstanian
observations, clarifications and conclusions.
The total natural science
All
true propositions make up natural science (4.11ii),
but they are related in a particular way. Wittgenstein uses a couple
of similes to expose the main concepts in his theory of natural
science: the pointing at a black spot (4.063) and the covering by a
net of a white spotted surface (6.341). Anyone can feel a resemblance
between the two, so let us carry on with this intuition to explore
the concepts: Truth and System.
In
order to describe a pencil drawing we can, as kids do to copy itiii,
trace a fine grid over it and say what square of the grid is black
and which is white. Of course, in order to do so, we need to use
words as “white”, “black”, “square”, etc., in a proper
way; we are obligated to know what a “black square” is in the
case. Regular English speakers will not have any problem complying
with such instructions because they understand the conditions a
square has to present to be called black or white, but in more
complicated situations the simile acquires its illustrative shine. At
this stage, the drawing represents observable reality, and our grid,
a regular way to produce interlocked propositions. A deductive
system, with axioms and rules to produce theorems, could play that
roleiv.
In fact it does, only if we aloud mathematical rules to transform the
axioms. Let me establish that the saying about the blackness of each
square represents the use of such axioms and theorems. Then, if this
is so, to what is equivalent in real life our knowledge of what is to
be a “black square”? Do we have to know in advance how thing
should look like when described by a new theory?
Although
we do not think our every day reality, for example, in terms of
classical physics, we know that such and such must be the case if
classical physics describes physical reality correctly. When we
predict an observation or set an experiment to prove some theoretical
statement, we are establishing the conditions “the black square”
has to present in order to be called that way. If elucidations of the
correct way to use such expressions are a way to grant them meaningv,
then such experiments are a way to grant meaning to theories. We
could say then if they are true or not, if things behave like they
say or not. We will see that David Bohm and John Bell had to work
into a more “testable” version of the original EPR argument even
when it was clear enough. They needed to specify how things should
look like if the grid of quantum mechanics could be used to describe
physical reality. Their contributions were in fact more important
because depending on the result of the experiments they inspired,
quantum mechanics could prove itself as a more accurate
way to describe reality.
The
grid represents the systems of propositions used to describe facts
but a very important feature of such grid is missing in the latest
accounts. Its shape is arbitrary in the sense that any shape,
triangular, hexagonal, etc., could serve to the purpose of depicting
the drawing. What the system of quantum-mechanical propositions
accomplishes, if correct, is a simpler but more accurate description
of reality, in particular, of sub-atomic particle behavior. The fact
that we could depict them with such system is not really the
important thing, since any mesh of any shape could; what is important
is that the behavior of such particles can be completely
depicted with the desired level of detail in a uniform way by such
system.
Uncertainty
Most
popular accounts of quantum mechanics give the leading part in it to
Heisenberg relations of uncertainty. Einstein had his share of guilt
in such popularity for he constantly searched ways to avoid the
principle of uncertainty derived from such relations. The principle
mainly says that: “It is impossible to know both the exact position
and exact momentum of an object at the same time.”vi
We can describe the position of a particle in space by giving its
three spatial coordinates and the momentum of such—its tendency to
keep moving in a particular direction—, by calculating the product
of its velocity and mass. Such descriptions can be random if we give
any values to the magnitudes required: we can describe whatever
particlevii.
But what we really want is the momentum and position of this or that
particle. We must measure the magnitudes in order to know the real
position and momentum. In classical physics there are no problems in
doing so.
Our every day theory of regular size objects
portraits them as cohesive conglomerates of smaller particles. These
too should have the same properties as the things they composed; they
ought to be solid chunks of matter. Classical physics expected the
same, so the early portrays of the smallest pieces of matter
presented them as tiny spheres. But by the 1930s this was not the
case. Louis de Broglie realized that one of the most important models
of atomic structure—the Bohr model—could be explained if
electrons where more wave-like. In the original model, electrons
surrounded the positive atom nucleus in a very strange way. Unlike
regular size materials they did not move in a continuous manner,
meaning that they jumped from place to place without any intermediate
steps. Also, only certain orbits where permitted for electrons to
surround the nucleus. This “quantized” behavior suggested to de
Broglie the idea of periodicity. The appearance of an electron in the
first orbital could be seeing as the crest of a wave of matter that,
according to the metaphor, should reappear at the wavelength in the
next orbital.viii
In the other hand, de Broglie’s mathematical development of the
wave nature of the electron resembles a previous work done by
Einstein; the descriptions of the electrons wavelength have the same
form as the photons—a light particle—wavelength. So de Broglie
produced the symmetrical complement of light theory: as light
sometimes—an electromagnetic wave—behaves as a particle, also
electrons behave as waves. But: “In water waves, the quantity that
varies periodically is the height of the water surface. In sound
waves, it is pressure. In light waves, electric and magnetic field
vary. What is it that varies in the case of matter waves?”ix
The answer is probability.
Probability
can be thought at least in two ways. The first, held for example by
Wittgenstein, portrays probability as a manner to approximate
descriptions to their most accurate form without knowing all the
aspects of a fact. A description bears certainty when things occur as
portrayed by it. The probability statement does not assert the
existence or nor existence of a fact. It asserts the likelihood of
happening that a fact gives to another. This likelihood depends on
the logical form of the statement, v. g., a conditional: if this is
the case, then (is probably that) this should be too. Probability can
be shown by simply calculating the truth tables of the statementsx.
But something has to be missing here if we are using logical
properties of statements not to obtain certainty. In a pragmatist
view, probability works because, independently of our ignorance of
finicky situations, we can predict with a very desirable rate of
success the occurrence or non occurrence of a fact. It is important
to stress that probability should be desirable in order for us to use
it because, as Einstein immortalized this view, is just a measure of
our ignorancexi.
The
other view was explicitly developed for quantum phenomena so let us
continue where we left. A wave of matter has a length and amplitude.
De Broglie’s work calculated the first, but other wave function was
needed to calculate its amplitude. In general we use trigonometric
functions of the wave’s velocity, angular frequency and wave number
to calculate it, but that method implies that we are talking about
infinitely long uniform waves. Electrons perhaps are not spheres but
their certainly are not infinitely long strings of matter, so in
order to describe the wave associated with a moving electron we need
to see it as a wave group. A very common example of wave groups are
what in sound engineering is called beats.xii
When sound waves of different length but identical amplitude
superpose, their crest and troughs interfere, canceling each other
when contraries coincide and amplifying when else. The result of the
latter is the sound pitch produced at regular intervals –the beat.
A wave related to a moving electron is a superimposition of infinite
different waves with similar amplitudes but different lengths and
velocities. This is the reason of quantum uncertainty. The lengths of
the different waves cannot vary absolutely if the wave group is to
exist. They have to approximate to the length of one
arbitrarily-chosen main wave. The more they do the more chances the
amplitude of them will amplify the “beat” of the wave group.
The relation with momentum and position was
inspired by Einstein. Once the complete wave function was developedxiii,
Max Born use it to interpret the amplitude of a wave group as Albert
the amplitude of his waves of light: as a square rout of the
probability of finding a photon. If we want to see the electron as a
particle we will probably found it experimentally in the region where
the wave amplitude of a wave group is most sharp. It is just
probable to find the electron there because wave groups are beats.
Although composed by a large crest, the other superposing waves
produce little other crests around the beat, so wherever the
amplitude is not equal to zero, the particle could be found.
The wavelength of a wave of matter is equal to the
Planck constant—the basic unit of quantization—divided by the
particle’s momentum. If we do not know the later we need to measure
the wavelength experimentally, then the momentum will be the Planck
constant divided by the wavelength. If we want to know the momentum
of the particle associated with a wave group, we need to know its
length, but since wave groups are beats, they do not have a sharply
defined length. If we want this to be, we can “construct” a wave
group with waves that have increasingly different wavelengths. We can
then establish the length of the wave group, but the picks of the
beats will shorten, distributing the probability of finding the
particle over the more equal wave amplitudes. Since a wave is made of
length and amplitude, we cannot have a wave group with both sharply
defined amplitude and length. We cannot know
both the particle’s position and momentum at the same time for, if
electrons are waves, the sharply defined momentum and position do not
exist.
EPR
argument
The
EPR argument can be seen as a much elaborated way to avoid the
uncertainty principle. Einstein’s main motivation was that the wavy
nature of electrons is provided by probability in the sense that our
representation
of the probability of finding the particle and knowing its momentum
can take the form of a wave; but this was just
a way of depicting our chances for
discovery. An electron was a punctual thing that had a position and a
momentum, for, if this were not the case, waves of what composed the
electron?
The
argument runs in this way: The supposition that the Ψ
function of a quantum state contains a complete description of the
physical reality of the state, leads us to a contradiction. “At
first sigh this assumption is entirely reasonable, for the
information obtainable from a wave function seems to correspond
exactly to what can be measured without altering the state of the
system.”xiv
But when we see the behavior of a pair of particles (a particle and
its “coordinate”), the psi
function gives us precise values for things that shouldn’t be
known—a violation of the principle of uncertainty. This values
where important for Einstein and company for “[i]f, without in any
way disturbing a system, we can predict with certainty (I.E., with
probability equal to unity) the value of a physical quantity, then
there exists an element of physical reality corresponding to this
physical quantity.”xv
Einstein and company believed that a complete theory should behave
like this: “every element of the physical reality must have a
counter part in the physical theory”.xvi
They subscribed themselves to a form of Wittgenstein’s pictorial
theory of the proposition: we make ourselves pictures of the facts by
representing every element of reality with an element in our
pictures, plus a way in which they are combinedxvii.
The interesting part of this relationship is that Wittgensteinan
theory of isomorphic propositions, and hence theories, is not
violated while EPR argument fails. That is to say: quantum mechanics
is complete even when violating Einstein and company expectations.
The
properties of a quantum state that fulfills the EPR conditions for
completeness and reality are those quantizedxviii.
This is not terribly important unless we consider that identical
systems should bear the same quantized value for measurements if all
are in the same state described by the pertinent wave function.
The
EPR property that does not fulfils the conditions for completeness
and reality is the momentum of the coordinate of a particle. The
momentum for the particle, not his coordinate, is quantized in the
sense we want, so it has “physical reality”. Its coordinate
momentum does not have the same quantized value; we can only use
probability predictions to know it. As it turns out all possible
values are equally probable. If we perform a measurement, the
original wave function that we were using to calculate if the
properties were quantized will change as a result. So there is no way
to know the coordinate value.
We
have seen a similar preclusion of knowledge before. The relation
between the momentum of a particle and his coordinate could be
represented by operators that do not commute (AB≠BA),
so the position and the momentum, so all the other uncertainty
relations. Things come for Einstein and company to the point of
saying that either the two properties denoted by A
and B do
not have simultaneous reality or quantum mechanics is incomplete.
For, if this is not the case, we can construct a system of particles
that has exact values for two noncommuting operators.xix
The
Bell Test
The
complete development of the later part of the EPR argument consist in
the construction of a wave function that describes the relation
between two particles. This function aloud us to perform a
measurement of a quality A
of the first particle to which a wave function φ
with some value x
of the other particle corresponds. The same function says that if we
perform another measurement B
of a quality of the first particle, then a wave function υ with some
value y
should come out. The interesting thing is that the first measurement
can be performed while the particles are interacting and the second
when they are not. A
and B were
proven to be noncommuting operators in the original paper. Quantities
x and y
could be measured without interfering with the system and with
absolute precision.
John Bell developed a regular form to account the
sort of quantities required by the EPR argument to work: the values
for all properties of very well-defined quantum states and the exact
speed of its effects. If an alternative theory of quantum mechanics
could be right, then Bell’s expectations—the sort of interactions
that will come out if all the properties in the known quantum states
have exact values—should happened. But if this was the case, a lot
of phenomena that can be predicted with quantum mechanics will not be
by the alternative theory. Although he did not know the exact
configuration of the Einstein’s alternative theory, he knew
something of its form.
Of course in order to know what alternative was
right, we should perform experiments. If the Bell expectations—called
Bells inequities—happened, then for example, we could know the y
value of the particle coordinate based on the mathematical
predictions. If Bells inequities were violated, then we will not know
the value of y
in advanced but only until measured. We need to know experimentally
that number in either case to see who was right anyway, but quantum
mechanics predicted just a probability of getting things right where
the other theories should bear the same result over and over again.
The most convincing Bell Test for Inequalities Experiment was
performed by A. Aspect in 1982 at Orsay, Paris. A couple of
particles—in the case photons—were interacting for a while and
then dispersed on contrary directions. The first particle
polarization was measured and then the other’s via a polarization
changer. If the measurement provides us with information about the
others particle polarization, then both particles should outcome with
the same polarization. The experiment proved that the outcome for the
other particle’s polarizations obeys not an exact relation but just
a probabilistic one even when the polarization devices were displayed
after the particles dispersed. The probabilistic rate was the one
predicted by quantum mechanics: particles are made of waves of
probability.
Conclusion:
Causality and Description.
A
way to see the counter-intuitive results of the Bell tests is
imagining the first particle sharing its momentum information with
the other to change it and avoid getting the same result. The
experiment is design in a way that the speed of such information
exchange should be faster than the light’s. Any physical process
propagates its effects at a lower speed so something else should be
happening. Einstein and company believed that such process was
violating causality laws. How the coordinate does know the particle
momentum? But “causality laws” are the laws of natural science.
There is no such thing as a law of causality that describes the
general features of causal processes; we know that something has to
be in order for other thing to be tooxx.
This empty statement is the form of natural laws but is not violated
by the predictions of quantum mechanics. The
later says what a state the particle has to have in order for the
coordinate to be undetermined. The
coordinate properties are in a wave state, they do not have sharply
defined values but not because we do not know them, but simply
because waves do not have them. Einstein and company expected that
the coordinate was as much particle as the particle itself, but not
always our predictions are correct. We estimated that the finer grid
we could create will give us a precise account of particles
behavior; it does give us a precise account and much more of the
world is said by a correct description that contradicts our
expectations.
References
-Wittgenstein
L., (1922),
Tractatus Logico-Philosophicus,
Routledge & Kegan Paul, Tr. C. K. Odgen.
-Einstein,
A. & Podolsky, B. & Rosen, N. (1935) “Can
Quantum-Mechanical Description of Reality Be Considered Complete?”
Physical Review,
47, 777-780.
-Beisler
A., (2003), Concepts of modern physics,
New York, McGraw-Hill.
i
Cf. Einstein, A. & Podolsky, B. & Rosen, N. (1935) “Can
Quantum-Mechanical Description of Reality Be Considered Complete?”
Physical Review,
47, 777-780.
ii
The numbers of course are those of Wittgenstein’s numeration of
the Tractatus propositions. I use the infamous Ogden translation for
quoting.
iii
The copying of the drawing done by the kids also is a way to
describe it, but the grid here is only a tool to improve a different
kind of mechanism for description, namely, copying. In Wittgenstein
image that method is replaced by recording the color of squares.
iv
An axiom is a statement established true without prove and a theorem
is a statement derived from axioms by logical transformations.
Ordinarily, deductive systems do not change, via the
transformations, the amount of information put in the axioms; it is
very likely that physics seems as an ever growing system because is
not completely formalized, i. e., it has not being put into a
deductive system form.
v
4.063 “I must have determined under what conditions I call "p"
true, and thereby I determine the sense of the proposition.”
vii
6.3432: We must not forget that the description of the world by
mechanics is always quite general. There is, for example, never any
mention of particular
material points in it, but always only of some
points or other.
viii
Things actually turn out to be more complex than that but de Broglie
hypothesis can work to calculate the electron orbital circumference
in Bohr’s model: 2πr=nλ.
The circumference is equal to a multiple of the electrons wavelength
(λ).
ix
Beisler A. 2003: 95.
pq(p or q) &
not (p &q)VV VFFVFV
VVVFVF
VVVFFF
FFVF
xi
Cf.: 5.1-5.156
xii
In this exposition I follow Beisler A. (2003) “Wave properties of
particles”, IN Concepts
of modern physics,
New York, McGraw-Hill
xiii
The famous Schrödinger’s wave function Ψ
can give us more information than just the amplitude of the
electron’s wave: linear momentum, angular momentum and energy of
the electron. Ψ can
be considered the general formula for wave displacement but due to
its mathematical structure is often consider as physically
meaningless.
xv
Einstein, A. et al. 1935: 777.
xvi
Ibid.
xvii
Cf. 2
xviii
This means that a measurement of a physical property G
could only take the values Gn.
This is the case only if the wave functions of the system be such
that the G
operator transformations over a wave function n
will be equal to the product of the value Gn
and the wave function n.
Cf. Beisler A. (2003) “Quantum mechanics”, IN Concepts
of modern physics,
New York, McGraw-Hill
xix
Cf. Einstein, A. & Podolsky, B. & Rosen, N. (1935) No. 2, IN
“Can Quantum-Mechanical Description of Reality Be Considered
Complete?”
Physical Review,
47, 777-780.
xx
Cf. 6.3-6.361
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