Re: “Could a Quantum Computer Have Subjective Experience?”

[Russell Standish]

Sorry for having gone dark, although maybe you relished the
respite. I've been travelling, and its not been all that convenient to
check and respond to emails.

On Sat, Jul 01, 2017 at 02:56:58PM +1000, Bruce Kellett wrote:
> On 1/07/2017 11:18 am, Russell Standish wrote:
> >To summarise, you are happy with everything up to (D.6), but think the
> >move to the linear superposition (D.7) is not justified, because you
> >say it cannot be an observer moment.
> 
> I would not say that I am happy with everything up to (D.7)! I find
> the notation confusing, and some of the manipulations do not seem to
> make operational sense. Your comments through these exchanges have
> certainly helped me to see what is going on but, as you know, I am
> fairly out of sympathy with the general approach anyway

I'm not asking for your sympathy. Actually, it is helpful if you
aren't sympathetic, as you're more likely to find a critical
flaw. Obviously, you need to be motivated enough to poke around in it, though.

> 
> >Maybe you are leaning on the fact that if A and B are projections,
> >then aA+bB is not in general a projection, since idempotency is not 
> >preserved?
> 
> That could be an issue, but it is not the main issue here. I think
> the concentration on projections and events/observations/outcomes is
> probably a mistake. You start with the concept of an observer moment
> as a set of possibilities consistent with what is known at time t.
> The elements of this set are infinite strings of bits encoding the
> information and possible continuations. This is fair enough, I
> suppose. But if you want to make contact with ordinary QM, you have
> to see this psi(t) essentially as a wave function (or equivalent).
> So it is this OM that is to be interpreted as a vector or ray in
> some space, and you have to establish that this is a linear space,
> with an defined inner product, so it is a Hilbert space.
> 

Establishing linearity is key. Establishing the resultant vector space
is a Hilbert space does follow fairly easily from Kolmogorov's axioms
(although its possible you have beef with those :).

> In this endeavour, concentration on projections as measurements is
> not actually helpful. These projections would correspond to
> operators on the vector space of OMs, and the existence and/or
> actions of operators is not actually going to help with establishing
> linearity. 

Observations are projections (in the general sense, not necessarily
linear space sense). This is why I focus on the projection operator
form of QM.

> The point here is that the action of an operator on a
> wave function does not actually change the wave function, it just
> gives a set of eigenfunctions in terms of which the original wave
> function can be expanded.

That is the case of a traditional observable - a full rank Hermitian
operator.

Going from a projection operator formalism to the traditional observer
formulation is mathematically quite trivial, however.

> 
> >However, I don't think that's what I'm relying on. Given a starting
> >vector ψ, the vector (αA+βB)ψ is going the the result of some
> >projection C, where Cψ=(αA+βB)ψ, modulo an arbitrary complex
> >factor, and so (αA+βB)ψ is still an observer moment.
> 
> Well, that might be the case for some linear operators, but it is
> not generally the case. One of the significant ways in which QM
> differs from classical mechanics is that some observations are
> mutually exclusive - the corresponding operators do not commute, so
> adding them does not result in another possible operator: (x + p) is
> not an operator in either x-space or p-space. In formal developments
> of QM, such as Landau and Lifschitz, or von Neumann, a lot of care
> is taken to distinguish between compatible and incompatible
> observations (commuting and non-commuting operators). I do not see
> how this could be incorporated in your approach -- just like the
> question of tensor product Hilbert spaces for different (commuting)
> operators.


S=X+P is definitely an observable, and corresponds to measuring the sum
of position and momentum crisply. You can work out the equivalent
Heisenberg uncertainty relations too
ΔsΔx ≥ ℏ and ΔsΔp ≥ ℏ
So if you measure with S, you will have uncertainty in both position
and momentum.

In general, any bounded hermitian operator is an observable of some
sort (one can create a machine, albeit weirdly wonderful in a Heath
Robisonesque way, that will measure that particular quantity).

The sum of two bounded hermitian operators is also a bounded hermition
operator. In fact a linear combination with real coefficients will
too, but not necessarily complex coeficients (since the hermitian
property may not be preserved).

All bets are off with unbounded operators, of course, but my attitude
is that unbounded operators are stictly unphysical, albeit sometimes
convenient for computational purposes.

> 
> 
> >Thinking along those lines some more, I'm incorrect to say that the
> >vector space V is 

Re: What lead to free-will denial?

[John Clark]

On Wed, Jul 5, 2017 at 11:51 AM, Bruno Marchal  wrote:

>
​>> ​
>> ​the study of the largest prime number seems rather silly to me.
>
>
> ​> ​
> It is just a logical point that you can't derive this from Robinson
> Arithmetic theory,
>

​That doesn't bode well for the future prospects of the Robinson Computer
Hardware Corporation now does it.​



> ​> ​
> a bit like the Euclid postulate on parallel has been shown non derivable
> from the other geometric axiom.
>

​But neither ​
Euclid
​'s Fifth P
ostulate
​ nor its absences ​leads to a world where logical contradictions can
exist, but a finite number that is the largest prime does.

​>> ​
>> ​A Turing machine can do ​Peano Arithmetic
>> ​ too.​ Nothing new in that.
>>
>
> ​> ​
> Of course not
> ​,
>

Of course not
​??


> ​
> ​
> I said that I wrote different program for a Löbian machine, and if you
> read Smullyan's book, you will find others, and they are all emulable by a
> Turing machine
>

​If it can emulate it then whatever a "​
Löbian machine
​" is and ​whatever it can do a
Turing machine
​ can do it too, including Peano Arithmetic.​ As I said, nothing new.


> ​> ​
> you seem to ignore the difference between a theory, or a set of "believed"
> propositions/sentences.
> ​ ​
> A (universal) Turing machine can compute (everything computable), but
> cannot prove anything.
>

​
As far back as 1956 a computer had proved
​ ​
38 theorems in Whitehead and Russell's Principia Mathematica,
​ and some of the proofs were judged by ​most to be more elegant than the
proofs
Russell
​ and ​
Whitehead
​ found. And the old machine used to do it was equivalent to a Turing
Machine, as are modern computers.​


​
>> ​>> ​
>> Turing did far more than define what his machine could do, he explained
>> exactly how to construct one in great detail, and that's why other people
>> gave it the name "Turing Machine".
>
>
> ​> ​
> Yes, he is the discoverer of the (mathematical) notion of computer, and he
> was interested in building one.
>

​It's true Turing built actually electronic devices but what I meant was in
in his original article about a long paper tape and marking pen and a
eraser etc he described exactly how to build one; he didn't expect anyone
would actually build a practical machine that way but he did prove it was
physically possible to do so. Where is your equivalent for a Löbian machine?
 ​

Martin Löb knew exactly what is a Löbian machine, although he would have
> called it a "sufficiently rich theory"


​
I know what departments in my big box hardware store to go to to buy light
detectors,
​
paper tape, erasers, marking pens, and gears and pulleys to build my Turing
Machine; but what department sells "sufficiently rich theory" and how heavy
is it, will I need help carrying it to my car?


> > Like I said, PA cannot prove its own consistency, but PA can prove that
> ZF can prove PA's consistency,
>

​That doesn't solve the problem, ​
​it just kicks the problem down the road. I
s ​
Zermelo–Fraenkel
​ consistent? ZF can't say, and even if it is there are true statements
that is can't prove.

​ John K Clark​

-- 
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.

Re: What lead to free-will denial?

[Bruno Marchal]

On 05 Jul 2017, at 16:09, John Clark wrote:

On Tue, Jul 4, 2017 at 2:26 PM, Bruno Marchal   
wrote:


​> ​In the field of mathematical logic, the theories are not used  
at all. The theories are the object of study.


​If ​mathematical logic​ is not to be a trivial subject then  
it's object of study should be the nature of reality,



?

The object of study of mathematical logic is mathematical reasoning  
and mathematical theories. Notbaly what can be derived and what cannot  
be derived in mathematical theories.





​that's why the study of the largest prime number seems rather  
silly to me.


It is just a logical point that you can't derive this from Robinson  
Arithmetic theory, a bit like the Euclid postulate on parallel has  
been shown non derivable from the other geometric axiom.








​>> ​I wish you's stop talking about that, even Google doesn't  
know what a​ ​"Löbian machine" is.


​> ​You don't believe in God but you talk like if Google was  
omniscient.


​It's not just Google, ​even Martin Löb​ ​never knew what  
a​ ​"Löbian machine" ​was,​​ and you're the only one who  
refers to it.​


Martin Löb knew exactly what is a Löbian machine, although he would  
have called it a "sufficiently rich theory" as it is often called. I  
use "machine" instead of "theory" because I limit myself to the study  
of machine, given my field of study (Mechanism).


Typically, RA is Turing universal, but not Löbian. PA, ZF, etc. are  
Löbian, as well as all their sound extensions.









​>> ​Turing explained exactly how to build one of his machines  
but I've never​ ​heard​ ​the construction details ​ 
on​​ ​just ​ ​how to manufacture one of your "Löbian  
machines".


​> ​I wrote the programs, it is easy. Read the books, or my  
papers.


​In other words you've explained this brilliantly elsewhere, or in  
yet other words I've stumped you.


I gave you the reference. here you show again that your goal is not to  
learn something, but to be negative.








​> ​Peano Arithmetic, is *the* exemple of a Löbian Machine.

​A Turing machine can do ​Peano Arithmetic​ too.​ Nothing new  
in that.


Of course not, you seem to ignore the difference between a theory, or  
a set of "believed" propositions/sentences.
A (universal) Turing machine can compute (everything computable), but  
cannot prove anything.
You might tell me that you were alluding to a (minimal) first-order  
logical specification of a Turing universal machine, in which case you  
will get a theory equivalent with RA, not with PA, which requires the  
(infinitely many) induction axioms.


PA proves much more than RA, and a fortiori than a (logical theory of)  
a Turing machine.






​> ​The most useful current definition is that a machine is  
Löbian if​ [blah blah blah]


​Turing did far more than define what his machine could do, he  
explained exactly how to construct one in great detail, and that's  
why other people gave it the name "Turing Machine".


Yes, he is the discoverer of the (mathematical) notion of computer,  
and he was interested in building one.





I can define what a perpetual motion machine is but I can't tell you  
how to build one, and that's why they're not called Clark Machines.  
If you know of a computing machine that can do things a Turing  
Machine can't and you can explain exactly how to build one then they  
should be called Marchal Machines, and I think Martin Löb would have  
agreed.



You seem to be ignorant in logic. With the Church-Turing-Post thesis,  
computable is universal, but provability, by incompletness is  
relative. All axiomatizable theories can be seen as machine, because  
their set of theorems is computably generable. Once such a theory can  
prove all true sigma_1 sentences, the theory becomes Universal, WITH  
RESPECT TO COMPUTABILITY, but the theory does not become universal  
with respect to PROVABILITY.


Above RA and PA, there is an infinity of more powerful machine with  
respect to arithmetical provability.


RA, by being Turing universal, can emulate PA, like all humans can  
emulate Einstein, but this does not mean that all humans are Einstein.


Like I said, PA cannot prove its own consistency, but PA can prove  
that ZF can prove PA's consistency, without understanding or being  
able to assess the proof (of course, by incompleteness).


With Mechanism, all believer, or theorem prover, are machine, but even  
with mechanism, not all machine are believer or theorem prover. A  
machine compute a function, a theorem prover, or a believer asserts  
propositions.




So I ask again, what parts from the hardware store do I need to buy  
to build one?


I said that I wrote different program for a Löbian machine, and if you  
read Smullyan's book, you will find others, and they are all emulable  
by a Turing machine, so you can use your computer. To implement my  
program(*), you need just to buy, or find on the net, a (good) LISP  
interpreter.


To implement a 

Re: What lead to free-will denial?

[John Clark]

On Tue, Jul 4, 2017 at 2:26 PM, Bruno Marchal  wrote:

​> ​
> In the field of mathematical logic, the theories are not used at all. The
> theories are the object of study.
>

​If ​
mathematical logic
​ is not to be a trivial subject then it's object of study should be the
nature of reality, ​that's why the study of the largest prime number seems
rather silly to me.

​>> ​
>> I wish you's stop talking about that, even Google doesn't know what a
>> ​ ​
>> "Löbian machine" is.
>
>
> ​> ​
> You don't believe in God but you talk like if Google was omniscient.
>

​It's not just Google, ​e
ven Martin Löb
​ ​
never knew what a
​ ​
"Löbian machine"
​was,​
​ and you're the only one who refers to it.​


> ​>> ​
>> Turing explained exactly how to build one of his machines but I've never
>> ​ ​
>> heard
>> ​ ​
>> the construction details
>> ​on​
>> ​ ​
>> just
>> ​ ​
>> how to manufacture one of your "Löbian machines".
>>
>
> ​> ​
> I wrote the programs, it is easy. Read the books, or my papers.
>

​In other words you've explained this brilliantly elsewhere, or in yet
other words I've stumped you.


> ​> ​
> Peano Arithmetic, is *the* exemple of a Löbian Machine.
>

​A Turing machine can do ​
Peano Arithmetic
​ too.​ Nothing new in that.

​> ​
> The most useful current definition is that a machine is Löbian if
> ​ [blah blah blah]
>

​Turing did far more than define what his machine could do, he explained
exactly how to construct one in great detail, and that's why other people
gave it the name "Turing Machine". I can define what a perpetual motion
machine is but I can't tell you how to build one, and that's why they're
not called Clark Machines. If you know of a computing machine that can do
things a Turing Machine can't and you can explain exactly how to build one
then they should be called Marchal Machines, and I think Martin Löb would
have agreed. So I ask again, what parts from the hardware store do I need
to buy to build one?

​>> ​
>> ​Nothing powerful enough to do arithmetic can prove it's own consistency;
>
>
> ​> ​
> Powerful to do enough of arithmetic, and being consistent. OK.
>

​And if it's not powerful enough to help a third grader with his homework
then it's not in my opinion very interesting, although there is no
disputing matters of taste.

​> ​
> The formula of Löb is the key formula of the Gödel-Löbian machine's
> theology.
>

​You really should buy a dictionary so you can look up that word.​

​

 John K Clark​







>

-- 
You received this message because you are subscribed to the Google Groups 
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.