On 09 Mar 2014, at 21:46, LizR wrote:
On 10 March 2014 02:15, Edgar L. Owen <edgaro...@att.net> wrote:
Russell,
Yes, but that is crazy because it assumes all theories are equally
valid with which I disagree. Science selects theories based on which
best explain the observable universe.
This is true. David Deutsch argues for this view convincingly in
"The Fabric of Reality". (Russell and Brent are not disputing this
view as a practical approach, I think, they are just pointing out
that there are metaphysical assumptions built into it....unless they
correct me on this.)
Therefore it is reasonable to assume that theories DO reflect actual
reality. They are not just made up by humans willy nilly....
Not willy nilly, certainly. However the assumption that they reflect
an actual reality is only an assumption, partly because it's
impossible to prove and partly because, in any case, all theories
are open to revision. (This is why people keep asking you for some
testable predictions of p-time and Bruno for testable predictions of
comp, for example.)
And that is why I keep answering that there are there. The
*observable* is given by Z1*, or qZ1* (a whole sort of quantum
"mathematics), so it is enough to compare the propositions of Z1* with
the quantum logics of the physicist, to test comp+Theatetetus, and to
abandon it or improve it.
And then what I say, is not a proposition of a theory, but something
derived from an hypothesis, already made by many if not most
scientists, unfortunately made often in the materialist context, which
leads to eliminativism of the person, when correct.
Hmm... Ad we are not yet close to Z1*, especially that the main
shortcut was in the post that you did not comment. I have also teach
this to some students here, and it heps me to understand that this is
not so simple ...
Yet, I will just reprint it below, to not lost it (!), and called it
"the real things", an obligatory passage which unfortunately, when
done with all details is very long, as we have to explain to "a very
dumb machine", the not so simple functioning of that "very dumb
machine".
I slightly correct it, I think it will be an evolving post. Don't
comment it, as long as it looks "chinese".
====================== T H E R E A L T H I N G
=================================
And don't worry, at some point I will have to re-explained all this,
to what some people might take as a very dumb machine, which indeed
believes only few axioms of elementary arithmetic.
That will be the real thing. Some modal logics will impose themselves
there, including the one corresponding to alternating consistent
extensions, whose measure should provide the physical laws.
The theory of everything, here, is classical first order logic + the
following formula:
0 ≠ s(x)
s(x) = s(y) -> x = y
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
or
0 ≠ (x + 1)
((x + 1) = (y + 1)) -> x = y
x + 0 = x
x + (y + 1) = (x + y) + 1
x * 0 = 0
x * (y + 1) = (x * y) + x
The hard task will consist in defining an "observer" using only the
theory above. It will be defined to be a sound extension believer of
the axioms above, + some amount of induction axioms, of the type:
(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x), with F(x) being a formula in
the arithmetical language (with "0, s, +, *).
We have to explain to a dumb machine, which understands only 0, s(0),
s(s(0)), ... and can only add and multiply, but yet can reason in
classical logic, the very functioning of such a dumb machine.
There is no miracle. To define the variables, we can use the letter x,
y, ..., it works well for many human people, but the dumb machine
understands only 0, s(0), s(s(0)), so we will have to decide to say
something like let the variable be defined by 0, s(s(0)),
s(s(s(s(0)))), that is, the even number, so we will defined in
arithmetic, the variable by the even numbers.
Variable(x) <-> even(x) <-> Ey(2*y = x)
And about "&", "->" "t", and even what about "(", and ")" ?
Well, again, there is no magic, you have to chose particular odd
numbers (to not confuse them from variable) to represent them.
That is both logic and polite.
And then, how about finite sequences of symbols like "0≠s(x)"?
They too must be defined in terms of number relations, and in this
case a simple way, if we allow ourselves the use of exponentiation, is
given by the uniqueness of prime decomposition. If g(0), g(≠), ...
represents the particular odd number symbol for "0", "≠", etc. then
you can represent "0≠s(x)" by
2^g(0)*3^g(≠)*5^g(s)*7^g(()*11^g(x)*13^g()).
Or better: 3^g(0)*5^g(≠)*7^g(s)*11^g(()*13^g(x)*17^g()). To avoid the
confusion with the variable, we start from the prime number 3, if not
we would get an even number which represents already a variable.
Then the theory itself can be defined or represented, as a number,
being a finite sequences of the number corresponding to the axioms
above.
We will have to defined in arithmetic what we mean by a valid proof. A
proof is itself a finite (or infinite) sequences of application of
inference rules, making proof "easy" to check (and hard to find in
many domains). So we can define in arithmetic a predicate b(x, y)
true when y is a proof (in the dumb number language) of x.
Then provable(x) can be defined by EyB(x, y). It is a Turing complete
sigma_1 arithmetical predicate, a Löbian once it get few induction
axioms.
That "provable(x)", or "believable(x)", or "assertable(x)" by the
modest believer in the axiom above, is an arithmetical modality.
Solovay theorem shows that a modal logic G characterize completely and
soundly (and the logicians' sense) the logic of that provability.
G characterized actually what the machine (theory, believer) can prove
about this, and Solovay second theorem provides a logic G* which get
completely and soundly the truth about the machine. Amazingly enough.
The main axiom of G is the Löb formula []([]p->p) -> []p. We will talk
about it later.
That provability is the "[]" of the modal logic G. It is 3p self-
reference. 1p self-reference(s) will be obtained by weakening or
strengthening of the "[]" in G. The 8 hypostases, and infinitely many
others, are consequences of incompleteness, some of them should gives
the core physical observable (mainly the []p & <>t nuances), on the
sigma_1 sentences.
Modal logic can be used to describe quantum logic, (by results from
Goldblatt and others), and on the sigma_1 sentences (on the UD, if you
want), the "observable" (modal nuance) obeys the right modal laws
needed for an arithmetical quantization, which should normally defined
the core invariant laws obeyed by the measure "one" on the consistent
extensions.
Bruno
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