Re: Rovelli on EPR

Eugene Stefanovich <eugenev@xxxxxxxxxxxx> writes:

rof@xxxxxxxxxxxx wrote:

Unfortunately I have not been able to find a copy of Piron's book,
either for sale on the web or at the libraries of two major
American universities.

You can ask your library to get this book through interlibrary loan.
This is a small book with powerful ideas. Definitely worth the effort.
Otherwise, you can look for Piron's journal articles in 1960's and
1970's, e.g.,

C. Piron, Helv. Phys. Acta 37 (1964), 439.

Thanks; I'll get hold of a copy soon.

From what you say above, though, it sounds
like Piron does more or less what the others do, namely introduce
a set of reasonable axioms, and then say that quantum mechanics is
one of a set of systems which satisfy those axioms.

He says that QM is, basically, a unique system that satisfies those
axioms.

Well, in that case, he and I are likely to disagree about what
constitutes a reasonable axiom. If the introduction of an
axiom is justified by an appeal to a failure of what you
call classical logic, then I have to regard that as an
indication that the axioms are chosen merely because
they give the desired result.

After all, if I send a mechanic to find out why a mechanical
failure occurred, and he returns to tell me that the failure
happened because the laws of logic failed, I would not find
his explanation plausible. If a physicist gives me the same
excuse, then I do not find it any more plausible than when
the mechanic says it.

In my opinion, classical logic developed by Aristotle
and Boole refers only to propositions about classical objects.
For quantum objects we need to take into account the statistical nature
of measurements and indeterminism. This requires a change in the rules
of logic.


When we find that we need to use probability, then we have found
that we need to use probability. It doesn't mean that logic
needs to be revised. The statisticians would be surprised to
hear that one has to discard logic itself in order to do
statistics.

Note that quantum probabilities are quite different from classical
probabilities. Classical probabilities arise in a classical mixed
state. Quantum probabilities are present in both pure quantum state
(a ray in the Hilbert space) and in the mixed quantum state
(the density operator).

There are two sides from which probability can be viewed. One can
consider only the mathematical representation of probability,
which is a field of pure mathematics, or one can consider the
physics involved in unpredictable events. If there is any
non-circular distinction between "classical" probability
and "quantum" probability, then it is a purely mathematical
distinction, and it concerns the symbols which we use when
we calculate the probabilities.

Doesn't Feynman's two-slit experiment defies the rules of classical
logic?

This is not the case, but before examining why it isn't, I think
it is important to understand why no experimental result can
ever "defy the rules of classical logic".

Logic deals with propositions, implication, the relations of
and, or, and not, and objects and predicates. We start with
a set of propositions, which are given (the data), and then
we proceed to deduce new propositions. In this way, we use
logic to increase the set of propositions which we know to
be true. That is all that logic does, and it does not make
predictions about the results of experiments, because
the notion of a prediction, or an experiment, is foreign
to logic.

Pure mathematics is built upon logic. The theorems of
mathematics are only acceptable to us because all of
the inferences which are used in the proofs come
from the rules of inference of logic. Each line of
a valid mathematical theorem is a proposition, and,
in annotated proofs, the rule of logic which allows
that proposition to be asserted should be listed
beside the proposition. That way we know that the
theorems of mathematics are acceptable. If somebody
tells us that logic is wrong, then he is saying
that the theorems of mathematics are no longer valid.
The statement that logic is wrong is, therefore,
absurd.

Logic makes no predictions about the results of experiments.
The discipline of logic ignores the content of the
propositions with which it deals. Propositions about
experiments or what will or did happen are no different
from those about set theory or legal procedure, as far
as logic is concerned. If you want to specialise in
propositions about experiments and make predictions
about the results of experiments, you need a different
discipline, namely physics.

For this reason, if a prediction about the result of
an experiment is incorrect, it is not logic which has
failed; the person who made the prediction has failed.
If you make a prediction, you must have some procedure
for making predictions. Logic does not give you that
procedure; you must find it elsewhere. If that procedure
doesn't work, don't blame logic.

With regard to the two-slit experiment, here is what
you can do with logic. You can write down a set of
propositions which say things like "A spot was observed
at point X", and so on, and deduce what you can from
those propositions. You will not be able to derive
a contradiction if you merely describe what was observed.

According to these rules we should admit that the electron
passes through both slits, which is nonsense.

The statement that the electron passes through both slits
is not a statement about what was observed. It is a story
that people like to tell when talking about the two-slit
experiment, but it has no relationship to either the
experimental facts or the mathematical formalism. In quantum
mechanics, the wavefunction is not something which waves
around in physical space; it is a function on configuration
space, so it lives in an abstract mathematical space which
looks nothing like the physical world.

A person might mathematically examine the propagation of the
wavefunction in configuration space. Such a person is qualified to
tell us about the solution to a differential equation, and we respect
him as a mathematician. However, he is not qualified to tell us
what transpires in physical space, and if he says that the electron
passed through both slits then he is talking about matters which
he knows nothing about. At best he is telling us a fictional
story inspired by true events (where the true events constitute
the interference pattern). I suspect this story is repeated so
frequently merely because people think that it's a cool thing
to say.

Of course, this experiment
can be described by invoking the formalism of quantum mechanics, i.e.,
wave functions and the Hilbert space and al that.
The contribution of Birkhoff,
von Neumann and others was to recognize that at a deeper level this
formalism amounts to the change of the rules of logic.

I'm afraid that for the reasons that I describe above the rules
of logic rest on a secured foundation, established forever. Birkhoff
and von Neumann may have established that a particular symbolic
system has something to do with incompatible measurements, but
the rules of logic remain in force when we want to prove theorems
about orthomodular lattices. Logic cannot be declared
invalid without removing the foundation upon which the theory of
orthomodular lattices is built. Logic has not changed; it is
merely being applied to a particular system.

Quantum logic says that all classical axioms are still OK,
except the axiom of distributivity. This axiom wasn't very intuitive
in the classical system anyway. Quantum logic uses the "orthomodular
law" instead. The distributivity axiom is a particular case of the
"orthomodular law".

Well, the axiom which you claim isn't intuitive is the one which
says that if two statements are not both true, then at least one
of them is false.

Are we talking about the same distributive law? ...

My mistake; I confused distributive with de Morgan. The distributive
laws say that x AND (y OR z) = (x AND y) OR (x AND z), and the
same thing with AND and OR swapped. The distributive law is, of
course, absolutely true, and has not been found to be deficient
in any way. If one takes the time to understand what it asserts,
one can clearly see its truth. There can be no counterexample
to it because any counterexample would be self-contradictory.

If x, y, and z are propositions abot classical objects, like
books or cars, then I fully agree with you - the distributive
law is valid. However, I am not sure if it is still true
for propositions about the quantum electron. Of course, in the quantum
case, one needs to first define the meaning of operations 'and'
and 'or'.

It is absolutely and completely true for all propositions about
all objects. Suppose that there were three propositions, x, y,
and z, such that the distributive law is false. That is, suppose
x AND (y OR z) is true, while (x AND y) OR (x AND z) is false.

This is a contradiction. If we accept the first supposition,
then we really accept that x is true and that either y or z is
true. If the first option (y) is right then x AND y is true.
If the section option is right then x AND z is true. This
contradicts the second supposition, namely that (x AND y) OR (x AND z)
is false.

Now, a person might respond to me by saying that I have assumed
above that the laws of "classical" logic apply, and that this is
precisely what is in question so I may not assume it. However, one
must understand that I am not saying "If we follow these rules then
we get this conclusion". I am saying that these rules must inevitably
be followed and we must inevitably be led to the conclusion that
the distributive law always holds. It is as simple as saying that
if x is true and x implies y then y is true. It follows from the
truth tables. That is, it follows from the definitions of AND, OR,
NOT, and so on.

You cannot keep the same definitions of AND, OR and NOT and
have a counterexample to the distributive law. This is why
you say:

Of course, in the quantum
case, one needs to first define the meaning of operations 'and'
and 'or'.

But if we put something different in place of "and", then we should
not call it "and" unless we are being poetic or want to deliberately
cause confusion. If we replace both "and" and "or" with some
completely different operations, whose actions must be discovered
by experiment, then we are not talking about logic any more,
and we should not call it logic because it isn't.

I suppose one could be sceptical enough to doubt
that axiom, but I do not think that "if proposition x implies
proposition y, then x and y are compatible" is more intuitive.

I may agree that this axiom is not very intuitive, but it
ought to be true, because it leads directly to the formalism of
quantum mechanics which has been verified by experiment an uncountable
number of times.

When you say "leads directly", I suspect you are assuming that
throwing away logic is acceptable.

So what has happened is that the term "weakest experimentally
verifiable proposition implying two given propositions" was renamed
"meet", and something similar happened with "join". Then it was
observed that "meet" and "join" have algebraic properties similar
in some respects to the logical AND and OR. From this observation
the whimsical and metaphorical name "quantum logic" was chosen.
This name was heard by the masses, who interpreted it as a failure
of "classical" logic. Now we have a population who think that logic
itself is wrong and needs to be modified.

You seem to suggest that logic is independent on physical experience.
I don't think so. I think that Aristotle's and Boole's postulates
look so obvious to us simply because we never meet quantum objects in
our everyday life.

No; the rules of propositional logic follow from the definitions.
You can only object to the rules of logic if you object to the
procedure of considering the consequences of definitions to be
true.

This principle - that the consequences of definitions are true,
is a fundamental principle which mathematics and physics both
assume. You cannot reject logic without rejecting mathematics
and physics too. A mathematician is not qualified to discover
that logic is false. A physicist is not qualified to discover
that mathematics is false, or that logic is false.

With regard to experience, logic does not say anything about
what we will or won't experience. If you write down a set
of propositions describing any experience you have had,
regardless of what type of objects you interact with in
everyday life, you will not be able to derive a contradiction.

If you start from a set of propositions and derive a contradiction,
then one or more of those propositions is not true. Hence,
if you write down a set of propositions describing your
experiences and derive a contradiction from them, then some
of those propositions were false and you shouldn't have written
them down.

For quantum objects, two properties may not be
measurable simultaneously and measurements performed in an ensemble
of identically prepared systems may not be reproducible. From classical
standpoint these are pretty unusual features.

No; there is no such concept as "unusual" in logic. Logic does
not state that all properties are simultaneously measurable.
It doesn't say anything about what is or isn't measurable. Hence,
the discovery that something is or isn't measurable is not
pretty unusual for logic.

R.

.

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