On 20 Dec 2017, at 23:14, Brent Meeker wrote:
On 12/20/2017 5:09 AM, Bruno Marchal wrote:
On 19 Dec 2017, at 23:03, Brent Meeker wrote:
On 12/19/2017 9:47 AM, Bruno Marchal wrote:
On 18 Dec 2017, at 07:48, Brent Meeker wrote:
On 12/17/2017 8:06 AM, Bruno Marchal wrote:
On 15 Dec 2017, at 22:19, Brent Meeker wrote:
On 12/15/2017 9:24 AM, Bruno Marchal wrote:
that the statistics of the observable, in arithmetic from
inside, have to "interfere" to make Digital Mechanism making
sense in cognitive science, so MW-appearances is not bizarre
at all: it has to be like that. Eventually, the "negative
amplitude of probability" comes from the self-referential
constraints (the logic of []p & <>p on p sigma_1, for those
who have studied a little bit).
Can you explicate this.
Usually, notions like necessity, certainty, probability 1, etc.
are assumed to obey []p -> p. This implies also []~p -> ~p, and
thus p -> <>p, and so, if we have []p -> p, we have [] -> <>p
(in classical normal modal logics).
Then provability, and even more "formal provability" was
considered as as *the* closer notion to knowledge we could hope
for,
Something a mathematician or logician might dream, but not a
mistake any physicist would ever make. Knowledge is
correspondence with reality, not deducibility
from axioms.
Which reality?
Since Gödel we do distinguish correspondence with the
arithmetical reality and deducibilty from axioms. We know that
*all* effective theories can only scratch the arithmetical truth.
You seem to identify reality with physical reality. That is a
strong physicalist axiom. When doing metaphysics with the
scientific method, especially on the mind-body problem, it is
better to be more neutral.
I identify reality with what we can empirically agree on.
That is close to Aristotle metaphysics. It does not work with
Mechanism.
But even without mechanism, I prefer to be metaphysically neutral,
and identify reality with whatever is the reason why we can agree
on an empirical reality. It does not need to be the empirical
reality itself, as there is no evidence for that up to now.
That's the epitome of question begging: "What you gonna take as
evidence for empirical reality? Your lyin' eyes or my beautiful
theory?"
We must distinguish between the numbers that we measure, the relations
that we infer, the metaphysical interpretation of those relations, etc.
We just need to be clear of what we assume at the start. You seem to
assume an Aristotelian God (Primary Matter), and I assumed a
Pythagorean God (elementary arithmetic).
Note that you assume also, at the meta level though, elementary
arithmetic, to get predictions in the frame of your metaphysics. But
you need an identiy thesis (à-la mind-brain) which does not work when
assuming Mechanism (without adding Ptolemaic ad hoc metaphysical
epicycles).
Would the logic S4Grz1, Z1* and X1* be different from quantum logic
(soemthing we can measure) then we would have an evidence that the
reality is the empirical reality, but up to now, it fits, and so
there is no evidences that the physical empirical reality is the
fundamental one.
Quantum mechanics is entirely based on empirical observation.
Nobody proposed to derive it from meta-physics.
Yes, but with mechanism we have no choice, except for hiding the mind-
body problem under the rug (a bad habit since a very long time).
and so it came as a shock that no ("rich enough") theory can
prove its own consistency. This means for example that neither
ZF nor PA can prove ~[]f, that is []f -> f,
This seems to me incorrectly rely on []f->f being equivalent to
~f->~[]f and ~f=t. I know that is standard first order logic,
but in this case we're talking about the whole infinite set of
expressible propositions. It's not so clear to me that you can
rely on the law of the excluded middle over this set.
We limit ourself to correct machine, by construction. It does not
matter how they are implemented below their substitution level,
and this is only what correct machine can prove on themselves at
their correct substitution level, and any higher order correct 3p
description.
That is all what we need to extract the "correct physics". No
need to interview machines which believe they are Napoleons. I
mean it is premature to invoke them in the fabric of the physical
reality (despite it is unclear what is the part of possible lie
at play here, cf Descarte's malin démons)
and so such machine cannot prove generally []p ->, and
provability, for them, cannot works as a predicate for
knowledge, and is at most a (hopefully correct) belief.
Now, this makes also possible to retrieve a classical notion of
knowledge, by defining, for all arithmetical proposition p, the
knowledge of p by []p & true(p).
I'm not impressed.
You should!
The beauty is that "Bp & p" leads to an explanation of why the
machine get suck in infinities when trying to know who she is.
from the machine's view, this looks quite like a soul, or subject
of consciousness, which "of course" cannot justify any 3p account
of him. from its point of view, the doppelganger is a
construction which proves that he is not a machine, and that the
doppelganger is an impostor! The beauty of "Bp & p" is that it
says "no" to the doctor! The machine's elementary first insight
is that she is no machines at all, and she is right from that
points of view, as G* can justify.
Unfortunately, we cannot define true(p) in arithmetic (Tarski),
nor can we define knowledge at all (Thomason, Scott-Montague).
But for eaxh arithmetical p, we can still mimic knowledge by
[]p & p,
Since you can't define knowledge, how can you say you can mimic
it?
All (serious) philosophers agree that knowledge is well
axiomatized by the modal logic T and S4 (T + Bp -> BBp).
I've had Edmund Gettier over for dinner and he definitely does not
agree with this idea of knowledge. If I'm right in assuming that
T means "true".
Oops. Sorry. No T is the theory
B(p -> q) -> (Bp -> Bq)
Bp -> p
Plus classical logic (modus ponens rule and classical tautologies)
plus the rule p/Bp.
(I use t and f for the truth value).
I suspect Gettier reason is close to Gerson or ... Socrates. It is
a non obvious fact that incompleteness refutes the common critics
of the standard theory of knowledge.
Gettier has a problem of epistemology named after him:
https://en.wikipedia.org/wiki/Gettier_problem
Gettier argument are equivalent of the use of dream argument to
demolish the Theatetus' notion of knoweldge, and somehow, those
argument are valid, and Socrates/Plato already thought about that.
And it is here that Mechanism might seem rather shocking, as we have
to abandon such kind of knowledge, and admit, as the dream argument
warned us, that any knowledge is not completely justifiable. The
realtion between "you" and "truth" are private, and the first person
knowledge of the machine M will not beend definable by the machine M.
I like very much Gettier argument, and it is valid, but he asks much
to much to "justified true belief". In fact, he asks too much on
"justification", which is the own under the joug of Gödel's theorem
when we assume mechanism.
"Bp & p", applied to the K4 reasoner (close to full self-
referential ability) gives S4, and is called "the standard theory
of knowledge" by both scholars in antic philosophy, and in
artificial intelligence (a rare agreement in philosophy).
What the machine cannot do is to define itself as a knower. That
is why she will be unable to recognize itself at any substitution
level, and that is why she will have to trust the doctor, or
prey, because nobody can tell her who she is, and which
computations support her in arithmetic.
for each p, and this lead to a way to associate canonically a
knower to the machine-prover. It obeys to a knowledge logic
(with []p -> p becoming trivial). That logic is captured
soundly and completely by the logic S4Grz (already described in
many posts).
Similarly, the logic G of arithmetical self-reference cannot be
a logic of probability one, due to the fact that []p does not
imply <>p (which would again contradict incompleteness). It
entails in the Kripke semantics that each world can access to a
cult-de-sac world in which []p is always true, despite there is
no worlds accessible to verify such facts.
But why should we accept that as a good model of inference? It
does not make intuitive sense to say []p is true in some world
where p is neither true nor even possible. What would be an
example of such a world given a proposition like "7 is prime."?
"7 si prime" is true in all worlds/models-of-löbian-machine. But
"provable(0 = 1)" is true only in the cul-de-sac world
(corresponding to alterated state of consciousness/non-standard
model (say)). To avoid them we have to define a new box [im]p
=[]p & <>t; to ensure the "cup of coffee" certainty in the WM
duplication experience.
But you wrote: " It entails in the Kripke semantics that each
world can access to a cul-de-sac world in which []p is always
true" I take it that p is a variable, which can take the value "7
is prime" so []p is true but "there is no worlds accessible to
verify such facts."
Yes, that is correct. In the cul-de-sac world []"7 is prime" is
true, but it cannot be verified from there, as no worlds can be
accessed from a cul-de-sac world. But there are plenty of other
worlds where it can be verified, so I am not sure of your point.
Also, there are many semantics.
That's seems to me a bad conclusion and a reason to reject this
modal logic.
But it is imposed by incompleteness. G *is* the modal logic for
*all* self-referentially correct entities, machines or even some
gods, physical or mathematical, or whatever. You cannot reject it
as much as you can reject 2+2=4. G main *axiom* relates only that
Löb's theorem is provable by all (enough rich) self-referentially
correct machine.
Keep in mind that I am not proposing anything new. If mechanism is
correct, physics is really a consequences of the following
assumption, besides Mechanism:
0 ≠ (x + 1)
((x + 1) = (y + 1)) -> x = y
x = 0 v Ey(x = y + 1)
x + 0 = x
x + (y + 1) = (x + y) + 1
x * 0 = 0
x * (y + 1) = (x * y) + x
We get a logic of probability by ensuring that "we are not in a
cul-de-sac world",
But isn't that equivalent to saying "anything is possible"?
On the contrary. It is a way to avoid "anything is necessary". In
the cul-de-sac world, everything is necessary, and nothing is
possible.
But in normal, rational logic a thing cannot be necessary unless
it is possible.
In alethic philosophy. Yes. In epistemological (knowledge) too.
That is why it came as a shock to discover that probability allows
[]f -> f being false.
In standard logic []f -> everything.
In standard theory of knowledge. Yes.
But in correct self-reference, we have only []f -> [](everything).
Well, in the mundane life too. If your plumber tell you one day that
the bill is 10$, and someday after that the bill is 18$, the world
does not disappear, nor do white rabbits appear in the living room and
playing trumpets.
What difference does probability make (I think Kolmogorov assumed
standard logic)?
Well, your actual state is related to an infinity of machine state
whose existence are Sigma_1 arithmetical, and prediction are on your
consistent computational extensions (by the First Person
Indeterminacy), so physics is a calculus of probability (or
credibility) on those extensions.
ISTM you have created a modal logic which is just word salad.
It is a consequence of 2+2=4 & Co. That logic has been discovered,
and is not something we have to add to RA+induction.
Are you saying that axioms of Robinson Arithemetic plus induction
don't need any other other rules of logical inference.
RA *without* induction gives the "ontology".
That is enough to prove the existence of numbers "believing" in RA
+induction, like PA, ZF, even you and me with mechanism. Then, yes, RA
+induction is enough to get the whole propositional theology and
propositional physics, but to get on higher order level, by
incompleteness, the machine have to add infinities of axioms and/or
inference rules.
It something which is probable in RA+induction, or in ZF, etc.
<of course, I meant provable here, not probable, as you have seen>
It is the unique correct logic for self-reference. Nobody has
invented it. It is a main discovery of the 20th century.
What is this discovery called? Where is this proved...and how is it
proved with second order logic?
Well, Gödel discovered 99% of this in his 1931 paper, but it was made
100% clear (but unreadable and expensive) by Hilbert and Bernays in
their "Grundlagen" later. The big clarification came with the
discovery of the universal machine by Turing and Church, and well many
others. Post anticipated the whole thing (including immaterialism!) in
1922).
A culminating point is when Solovay proves that G and G* are the modal
logic of "effective self-reference" (not just machine's one, also many
non-machine mathematical entities).
RA is mainly due to the work of Tarski, Robinson and Mostowski. They
made clear that Gödel's proof shows not just incompleteness of
arithmetic, but the incompleteteness of all their consistent
extensions. PA and ZF, and all Löbian numbers are not just incomplete,
they are incomplete-able.
I eventually decided to bought the nice treatise by Hajek and Pudlak,
"the Metamathematics of first order arithmetical theories", where it
is shown that RA (called Q, which is the original name) can interpret
many theories capable of proving a large part of the metamathematics
of PA.
Q is typically Turing universal, but not Löbian. PA is Löbian, and
the book attempt to see, somehow, where Löbianity starts.
A good book also is Torkel Franzen "Inexhaustibility", but all this
require studying a bit of mathematical logic, especially Hakek and
Pudlak, which requires a bit of model theory.
What is the discovery called? It is the mathematical discovery of the
computer in arithmetic, and of its many thought on the matter ...
Bruno
Brent
I recognize that technical language may overwrite new meanings on
words, but you've given no hint as to why that should be done.
You are proving hints that you have not seriously study logic.
Do you not agree that formal logic is just a way of making
ordinary discourse precise and consistent...not to twist it's
meaning.
Exactly. But the point is that the logic of provability/consistency
seems indeed twisted, a bit like QM is counter-intuitive. The
arithmetical reality just kick back when our intuitive
conception is wrong.
Bf is verified "trivially" in the end-world, because they can't
access to any world. (alpha R beta -> beta verfies f) is always
true because alpha R beta is always false when alpha is an end-
world. Of course, end-world are consistent: from Bf you can't
derive f.
which is the main default assumption need in probability
calculus. In that case, you can justify, for example, that when
you are duplicated in Washington and Moscow, the probability of
getting a cup of coffee is one, when the protocol ensure the
offering of coffee at both place: []p in that case means "p is
true in all accessible words, and there is at least one".
So, by incompleteness, [] & <>t provides a "probability one"
notion, not reducible to simple provability ([]p).
Then, by step 8, we are in arithmetic (in the model of
arithmetic, "model" in the logician's sense), and we translate
computationalism by restricting the accessible "p" to the
leaves of the universal dovetailing. By Gödel+Church-Turing-
Kleene we can represent those "leaves" by the semi-computable
predicates: the sigma_1 sentences. When we do this, we have to
add the axiom "p->[]p" to G. This gives G1 (and G1*). It is
enough thanks to a proof by Visser. For the logic of the
nuances brought by incompleteness, like []p & p, and []p & <>t,
it gives the logic S4Grz1 and the logic Z1*. Then, we can
extract an arithmetical interpretation of intuitionist logic
from S4 (in a usual well known way), and, a bit less well
known, we can extract a minimal quantum logic from B, and then
from Z1* which is very close to B, using a "reverse" Goldblatt
transform (as Goldblatt showed how the modal logic B (main
axioms []p -> p, p -> []<>p, and NOT= []p -> [][]p) is a modal
version of minimal quantum logic.
I don't see that you have explicated negative amplitude of
probability:
Can you build a quantum logic with only positive amplitude of
probability?
Answer: yes that does exist, for dimension 2, but with Gleason
theorem, quantum logic + dim bigger than 3 entails "negative
amplitude of probability".
No you can't. But you claimed that your theory implied negative
amplitides of probability. Now you turn it around and say your
theory must imply negative probability amplitudes since otherwise
it couldn't produce QM. It's another, "My theory must produce QM,
physics, consciousness, etc. because otherwise it would be wrong."
?
Not at all. It is proved. If you know a formula of quantum physics
which is not recovered by the "material hypostases",
OK, show us how to "recover" Schroedinger's equation of the harmonic
oscillator?
then please show it to us, as this could refute mechanism. The
point is that nobody has yet found it, ad as such, Mechanism is the
only theory explaining both matter and consciousness, instead of
doing only 3p prediction, and eliminating consciousness.
You must study the theory, Brent. I thought you did understood that
G is a modal logic derived from arithmetic.
No. I and Liz followed your lessons on modal logic, although it was
not clear that they derived from arithmetic. Arithmetic has it's
own axioms and rules of inference.
That is the kind of stuff I have been asked to withdraw, because it
is too much simple (but that was perhaps only malevolent intention,
I can't know).
I am not a "philosopher". All I say has been verified by many
peers. There is nothing to disagree with (if you agree with 2+2=4
and co, and mechanism). I refer only to proved statements and
definitions.
No, you also make interpretations, such as B="provable" is also a
model for "believes". And invalid inferences such as, "If I can
show that X explains Y, then it must be false that Z explains Y, and
advocates of 'Z explains Y' are just invoking magic."
Brent
Bruno
Brent
Bruno
"Eventually, the "negative amplitude of probability" comes from
the self-referential constraints (the logic of []p & <>p on p
sigma_1, for those who have studied a little bit). "
Brent
Note that here "[] and "<>" are arithmetical predicate. We do
not assume more than Q, and use only internal
interpretabilities of the observer-machines.
This is explained in most of my papers, but the details are in
the long french text "Conscience et Mécanisme".
Bruno
Brent
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