Re: A scary theory about IS

[Bruno Marchal]

On 23 Sep 2015, at 01:18, Brent Meeker wrote:




On 9/22/2015 2:55 PM, John Mikes wrote:
Bruno, I am at a loss with your explanation. I lived the active  
first 50 years of my life in Europe and never heard about such  
'liberalism' (for a short time was even connected to the Hungarian  
Liberal Democratic Party).
"Liberal" was in no connection with right/wrong, or even right/ 
left, only pointed to some freedom of action in the political  
arena. And the other thing:


Democracy IMO is an oxymoron, the full "demos" cannot exercise it's  
full "cratos" for ruling, becuase every person has different aims,  
goals, interests, etc. Those, who call a "majority-rule" a  
democracy are establishing a minority whose interests are trampled  
down by the so called "majority" which is not even so sure, to BE a  
majority indeed. Voting is cheating, candidates LIE in the campaign  
and the voters compromise their (real?) interests for the least  
controversial lies. What is even worse: the "elected" persons don't  
even follow their own lies later on in practice. They go after  
their (untold???) interest. Impeachment is difficult.


One word about 'capitalism' - with a caveat not to fall into  
Marxist traps:
it is the open exploitation of the power of wealth over the have- 
nots, be it by employment, marketing, or production policy. Not the  
"haves" - mind you, but the oligarchs, super-wealthy owners,  
political donors, etc. etc. established since Adam Smith. Growth is  
NOT maintainable with the limited resources existing. And a

(cut-throat?) Competition as life? thanks, but no thanks. .
Do you mean cooperative and collaborating goodwilling people dead?


Capitalism is the use of money to make more money.


OK. In fact the number e (2.71828...) has been discovered when Neper  
discovered how money tend to grow once we let a market free. Bank are  
then institution in which people can let grow the money without taking  
action, (which are supposed to be done by the bankers). It reflects  
that money can be used to invest in things which will bring back more  
money. The laws is a simple self-bootstrapping type of differential  
equation: dC = KCdt, and so C = Ke^Bt, with B a parameter depending on  
the economical situation. That entails grows and expansion, but that  
is already the case with self-dividing amoeba, and the fact that the  
repeated mutiplications lead to exponentials.


This is natural, and unless a tyranny, cannot be avoided. But that  
does not mean that people are free to use money to lie and create huge  
amount of money based on lies. That is just stealing everyone.





 If you don't like it, what freedom will you take away to prevent  
investing money to make things of value and hence more money?  Who  
will decide on which freedoms will be forbidden?


Exactly. I am for universal allocation, but also for the free  
enterprise. But again, free enterprise does not mean enterprise freely  
based on deluding the population about their need. False advertising,  
like defamation, should be illegal and rather severely punished.


Bruno



Brent

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Re: A scary theory about IS

[Bruno Marchal]

On 22 Sep 2015, at 23:55, John Mikes wrote:

Bruno, I am at a loss with your explanation. I lived the active  
first 50 years of my life in Europe and never heard about such  
'liberalism' (for a short time was even connected to the Hungarian  
Liberal Democratic Party).


 In my counntry the right party has the name "parti libéral", for  
example. Liberal means "open to free markets".

May be that is only in West Europa.


"Liberal" was in no connection with right/wrong, or even right/left,  
only pointed to some freedom of action in the political arena. And  
the other thing:


Democracy IMO is an oxymoron, the full "demos" cannot exercise it's  
full "cratos" for ruling,



Democracy means, for me, presence of election. It can be partial, like  
in the beginning where woman did not have the right to vote, or like  
in the antic greece were election was for the educated class, and not  
for slaves, or it is "universal", meaning everyone can vote. In some  
country it is "everyone *must* vote (in Belgium election are  
obligatory).


Then a democracy can be corrupted, and/or under the influence of  
corporatism, and/or sick etc. Democracy is not the final state of  
politics, it is the prerequisite of having a representative politics.  
If the main powers (mainly justice and press) are not independent, a  
democracy can be de facto a tyranny disguised into democracy. I think  
that is the case today (since prohibition).


It is the like the Islamic bill or right, which is a copy of the  
universal definition except that they have added "as long as it  
verifies the Charia" for each principle (which of course changes the  
very idea). The same with Obama who signed a text which respect the  
human right except for a category or people, but something have to be  
universal to make sense. The human right applies to all humans, or  
there is no more human right at all.





becuase every person has different aims, goals, interests, etc.  
Those, who call a "majority-rule" a democracy are establishing a  
minority whose interests are trampled down by the so called  
"majority" which is not even so sure, to BE a majority indeed.  
Voting is cheating, candidates LIE in the campaign and the voters  
compromise their (real?) interests for the least controversial lies.  
What is even worse: the "elected" persons don't even follow their  
own lies later on in practice. They go after their (untold???)  
interest. Impeachment is difficult.


Yes, but that is because our democracies are sick. It is not because a  
car is broken that a car is not supposed to be driven.


And I wish the minority having no power, but a problem with more than  
two parties is that the minority can have tremendous infuence. Indeed  
the minority will often makes the difference when the majorities  
disagree. But the french Condorcet has already studied the  
impossibility of satisfying everybody by a voting procedure, and  
democracies can evolve, and we can change the rules, ... unless the  
system has been corrupted.


It is not because we can die of cancer that we are not alive. It is  
the same with democracy, they can get sick, and the election might no  
more represent what the people desire.
I am opposed to referendum and participative democracies, because this  
can give all the power to the media.





One word about 'capitalism' - with a caveat not to fall into Marxist  
traps:
it is the open exploitation of the power of wealth over the have- 
nots, be it by employment, marketing, or production policy. Not the  
"haves" - mind you, but the oligarchs, super-wealthy owners,  
political donors, etc. etc. established since Adam Smith.


I disagree, even if in practice, some facts can lead to the  
perversion. But the non-democratic ruling consist in putting the  
perversion right at the beginning. Free-market is a win-win game, when  
it is not perverted by a minority (like today). The rich needs, in  
that case, to enrich the poor, as they have interest to make the poor  
into clients. It works, in the sense that most democracies have much  
less people starving than tyrannies. Now, when the rich exploit the  
poor, it means that the democracy is not functioning.







Growth is NOT maintainable with the limited resources existing.


Computer science provides a non limited resources.




And a
(cut-throat?) Competition as life? thanks, but no thanks. .


I would defend the universal allocation, and the right of laziness,  
but for this, we have to be lucid and realist on the economical  
difficulties which can rise in that case.


I personally hate competition, but it is right, and without  
progressing in politics, instead of regressing, it is a necessity.





Do you mean cooperative and collaborating goodwilling people dead?


On the contrary, without a democracy, cooperation is quickly  
impossible. It leads to the ruling of the strongest, like with mafia.  
democracy, I think, is the main tool for cooperation. people do my  

Re: Undecidability of the Spectral Gap

[smitra]

They are considering the limit if an infinite system size and then 
asking if you get a continuous spectrum or if the ground state is 
separated by a gap from the excited states in that limit. That this is 
not decidable is perhaps surprising, but this has nothing to do with 
Nature not being computable. Turing gave an example of a physical system 
that exhibits the same phenomena in an appropriate limit a long time 
ago.


Saibal


On 23-09-2015 06:26, Bruno Marchal wrote:

On 22 Sep 2015, at 19:27, Brent Meeker wrote:




On 9/22/2015 5:17 AM, Bruno Marchal wrote:


On 22 Sep 2015, at 00:29, Brent Meeker wrote:


A fascinating application of computability theory to physics:

Undecidability of the Spectral Gap
Toby Cubitt,  David Perez-Garcia,  and Michael M. Wolf

The spectral gap—the difference in energy between the ground state  
and the first excited state—is one of the most important prop-
erties of a quantum many-body system. Quantum phase transitions  
occur when the spectral gap vanishes and the system becomes
critical. Much of physicsis concerned with understanding the phase  
diagrams of quantum systems, and some of the most challenging
and long-standing open problems in theoretical physics concern the  
spectral gap, 1–3 such as the Haldane conjecture 4 that the Heisen-
berg chain is gapped for integer spin, proving existence of a  
gapped topological spin liquid phase, 2,3 or the Yang-Mills gap  
conjecture 5
(one of the Millennium Prize problems). These problems are all  
particular cases of the general spectral gap problem: Given a quan-
tum many-body Hamiltonian, is the system it describes gapped or  
gapless? Here we show that this problem is undecidable, in the
same sense as the Halting Problem was proven to be undecidable by  
Turing.


I guess he means unsolvable.

"undecidable" is relative to a theory. Unsolvable or uncomputable  is 
absolute and does not depend on any theory. It means that there  is 
no alogorithm to do some task, like computing some function or  
deciding some set.






6 A consequence of this is that the spectral gap of certain
quantum many-body Hamiltonians is not determined by the axioms of  
mathematics,


? (that does not make sense)



much as Gödels incompleteness theorem implies
that certain theorems are mathematically unprovable.


Gödel proved only that all theories are undecidable when it comes  to 
proving propositions in some domain (like natural numbers).


It makes no sense to say that some mathematical proposition are  
unprovable. there always some theories which can prove them: just  
keep such proposition as axioms, for example. PA (or ZF, ...)  cannot 
prove that PA is consistent, but PA + consistent(PA) can  prove that 
PA is consistent, trivially. More interestingly: PA +  epsilon-zero 
is well founded can also prove that PA is consistent.




We extend these results to prove undecidability of other low  
temperature prop-

erties, such as correlation functions.


Well, I guess again that they talk only about unsolvability, not  
undecidability.



The proof hinges on simple quantum many-body models that exhibit  
highly unusual physics in

the thermodynamic limit.


I will take a look when I have more time. I might need to revise a  
bit the quantum many-body problem for that!


In QM, already the 0-body problem is Turing complete (when you need  
at least three bodies to have Turing completeness for classical  
physics), so it is hard to be astonished, but I don't judge the  
paper (above using a vocabulary which is confusing when you know  the 
difference between computing and proving). The main difference  is 
that in proof theory, the result are dependent of the theory  (they 
are not absolute), where in computability, the results are  absolute 
if we assume Church-thesis (which is accepted by virtually  all 
experts in the domain).


But I think their motivation was to show that nature may perform  
hyper-Turing computation.


? I am not sure.



 So they would not assume Church-Turing.


? You need Church thesis to define "hyper-Turing" computation. In fact
 you need a stronger version of Church's thesis, a sort of
hyper-Turing  Church thesis (the hyper-arithmetical Church's thesis).

Without Church's thesis, computation is not definable, still less
hyper-computation.

Bruno





Brent

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Re: What day is it?

[smitra]

On 16-09-2015 16:24, Bruno Marchal wrote:

On 15 Sep 2015, at 22:33, smitra wrote:


On 12-09-2015 10:26, Bruno Marchal wrote:

On 11 Sep 2015, at 18:17, smitra wrote:
It seems to me that COMP should lead to MWI plus a preferred  basis  
where the latter derives from well defined computational  states.  
Many of the problems with the MWI should not arise here,  they are 
an  artifact of the theory never defining what an  observation is,  
appealing to ad hoc intuitive notions that are  never formulated 
from  within the theory itself.
The notion that the environment plays a fundamental role should  be  
rejected on physical grounds, it just explains the effective  
physics  we observe just like air resistance explains why Newton's  
laws were  not all that obvious to people who lived many centuries  
ago.
The only way you can explain Newton's law to students is by  letting 
 them contemplate a perfect vacuum. It doesn't matter here  how  
physically unrealistic that perfect vacuum is or isn't. The  same is 
 true for quantum mechanics. You'll never make process if  you 
always  invoke the environment and environment induced  decoherence 
to try to  define fundamental concepts, because Nature  cannot 
possibly work  that way on the fundamental level.
Instead, within quantum mechanics (i.e. if we forget about the   
desire to derive QM from COMP or some other deeper theory)  defining 
 observers as computations,  means that they should be  represented 
as  operators of the form:

sum over input of |output>t (& p). The FPI is not algorithmic, even if the
distribution of probability is algorithmic with the simple protocol,
but even the simple FPI is no more algorithmic on the universal
dovetailing (or on the sigma_1 propositions) as we cannot recognize
them as such algorithmically. This makers very nice that the
propositional logic of observable is decidable (and close to a Quantum
 Logic).

Note that the quanta appears at the star-level (in X1* minus X1),
making quanta into special case of qualia, which is coherent with
Everett's superposition of collection of people (first person plural)
and with the idea that the "absolute 3p reality is a multi-dreams (and
 note a many-worlds). This shows that even with the Everett "MWI", we
don't have any world: only sharable first person experiences. If the
quanta would have appeared in Z1 or X1 (and not in the proper star-
extension) a notion of apparent global physical reality would have
made sense, but it looks we lost this.

I recall the 8 povs or "hypostases";

1) p   (truth of p)
2) [0]p = []p = bewesibar('p'), with p a (sigma_1) arithmetical  
sentence.

3) [1]p = []p & p  (the knower, or soul or inner god)
4) [2]p = []p & <>t (the observer, gambler, ...)
5) [3]p = [2]p & p  (the "senser").

which can be put in this diagram:

   1=1*
22*
   3=3*

44*
55*

We have that
   1=1*
22*
   3=3*
is the basic propositional theory of mind/soul  (1= One, 2 =
Intellect, 3 = Soul)
and
44*
55*
gives the "two sorts of matter": 4 = intelligible matter, 5 = sensible  
matter.


Quantizations appear at 3*, 4*, and 5*. That suggests 3 sorts of
logics structuring (slightly?) differently the physical reality. I
guess that 3* is "heaven physics" (the physics of the soul which has
not yet felt), and 4*, like 5*, are the physics of "earth", when we
sin in the bet on the non justifiable "Reality" (<>t).

So we can not only test mechanism, but we can test if we are in heaven
 or not :)

 No need to take this too much seriously. A lot of research needs to
be pursued to clarify all this. At the quantified logical level, we
know 

Mandela effect?

[Brent Meeker]

Never heard of it before.

http://www.skeptic.com/insight/the-mandela-effect/

Brent

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Re: A scary theory about IS

[Brent Meeker]

On 9/23/2015 3:15 AM, Bruno Marchal wrote:


On 22 Sep 2015, at 23:55, John Mikes wrote:

Bruno, I am at a loss with your explanation. I lived the active first 
50 years of my life in Europe and never heard about such 'liberalism' 
(for a short time was even connected to the Hungarian Liberal 
Democratic Party).


 In my counntry the right party has the name "parti libéral", for 
example. Liberal means "open to free markets".

May be that is only in West Europa.


"Liberal" was in no connection with right/wrong, or even right/left, 
only pointed to some freedom of action in the political arena. And 
the other thing:


Democracy IMO is an oxymoron, the full "demos" cannot exercise it's 
full "cratos" for ruling,



Democracy means, for me, presence of election. It can be partial, like 
in the beginning where woman did not have the right to vote, or like 
in the antic greece were election was for the educated class, and not 
for slaves, or it is "universal", meaning everyone can vote. In some 
country it is "everyone *must* vote (in Belgium election are obligatory).


Then a democracy can be corrupted, and/or under the influence of 
corporatism, and/or sick etc. Democracy is not the final state of 
politics, it is the prerequisite of having a representative politics. 
If the main powers (mainly justice and press) are not independent, a 
democracy can be de facto a tyranny disguised into democracy. I think 
that is the case today (since prohibition).


It is the like the Islamic bill or right, which is a copy of the 
universal definition except that they have added "as long as it 
verifies the Charia" for each principle (which of course changes the 
very idea). The same with Obama who signed a text which respect the 
human right except for a category or people, but something have to be 
universal to make sense. The human right applies to all humans, or 
there is no more human right at all.




Democracy is necessary but not sufficient for good government. Supposing 
that democracy is enough was the mistake of George W. Bush and the 
neo-conservatives.   They thought that if we just held elections in Iraq 
all would be well.  But there must be limitations on government, 
constitutional restraints and traditional restraints.  Otherwise 
whomever has the majority assumes that democracy means they can oppress 
the minority.






becuase every person has different aims, goals, interests, etc. 
Those, who call a "majority-rule" a democracy are establishing a 
minority whose interests are trampled down by the so called 
"majority" which is not even so sure, to BE a majority indeed. Voting 
is cheating, candidates LIE in the campaign and the voters compromise 
their (real?) interests for the least controversial lies. What is 
even worse: the "elected" persons don't even follow their own lies 
later on in practice. They go after their (untold???) interest. 
Impeachment is difficult.


Yes, but that is because our democracies are sick. It is not because a 
car is broken that a car is not supposed to be driven.


And I wish the minority having no power, but a problem with more than 
two parties is that the minority can have tremendous infuence. Indeed 
the minority will often makes the difference when the majorities 
disagree. But the french Condorcet has already studied the 
impossibility of satisfying everybody by a voting procedure, and 
democracies can evolve, and we can change the rules, ... unless the 
system has been corrupted.


It is not because we can die of cancer that we are not alive. It is 
the same with democracy, they can get sick, and the election might no 
more represent what the people desire.
I am opposed to referendum and participative democracies, because this 
can give all the power to the media.





One word about 'capitalism' - with a caveat not to fall into Marxist 
traps:
it is the open exploitation of the power of wealth over the 
have-nots, be it by employment, marketing, or production policy. Not 
the "haves" - mind you, but the oligarchs, super-wealthy owners, 
political donors, etc. etc. established since Adam Smith.


I disagree, even if in practice, some facts can lead to the 
perversion. But the non-democratic ruling consist in putting the 
perversion right at the beginning. Free-market is a win-win game, when 
it is not perverted by a minority (like today). The rich needs, in 
that case, to enrich the poor, as they have interest to make the poor 
into clients. It works, in the sense that most democracies have much 
less people starving than tyrannies. Now, when the rich exploit the 
poor, it means that the democracy is not functioning.







Growth is NOT maintainable with the limited resources existing.


Computer science provides a non limited resources.




And a
(cut-throat?) Competition as life? thanks, but no thanks. .


I would defend the universal allocation, and the right of laziness, 
but for this, we have to be lucid and realist on the economical 
difficulties which 

Re: What day is it?

[Jason Resch]

On Tue, Sep 8, 2015 at 8:45 PM, Bruce Kellett 
wrote:

> On 9/09/2015 1:29 pm, Jason Resch wrote:
>
> On Tue, Sep 8, 2015 at 9:44 PM, Bruce Kellett 
> wrote:
>
>> I presume you mean that the world is duplicated on each toss, with one
>> branch showing each outcome. We are back to the dreaded "person
>> duplication" problem. My opinion on this is that on such a duplication, two
>> new persons are created, so the probability that the original person will
>> see either heads or tails is precisely zero, because that person no longer
>> exists after the duplication.
>>
>
> So if some aliens create a copy of you in Andromeda, then you cease to
> exist as a person?
>
>
> Since I might know if they gathered the requisite information, it is not
> an issue.
>

I don't see how this follows. Are you saying you would cease to exist as a
person if a duplicate of you arose far away in this universe, or that you
would not cease to exist as a person?


>
> Note: according to current comological models, space is infinite and
> uniform, which means infinite copies of you exist (though very far away).
>
> Such models make really quite strong assumptions about initial conditions.
>

This all follows from thw concordance model of cosmology, which is the
"standard model" in cosmology. See:
http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf



> You might well have an infinity of worlds with our present cosmology, but
> they might all be copies of some bland, boring model with no intelligent
> life.
>

I don't think you grasp the implications of infinity. If there are infinite
worlds, there is effectively 100% probability that an infinite number of
them will be identical to this entire Earth as you see it.

Pi has infinite digits. Any sequence, however long, the encoding of any
documentary, can be found in the digits of Pi, and moreover, it recurs an
infinite number of times.

I think you are trying to avoid answering my question.

Jason


> An infinite number of universes does not imply an indefinite number of
> copies of every particular universe. There are sets of zero measure, after
> all. So again, I don't think there is anything here to concern us. Even if
> there are exact duplicates, they are in principle non-communicating, so are
> operationally different persons.
>
> Bruce
>
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Re: Undecidability of the Spectral Gap

[Bruno Marchal]

On 23 Sep 2015, at 06:51, Brent Meeker wrote:




On 9/22/2015 9:26 PM, Bruno Marchal wrote:


On 22 Sep 2015, at 19:27, Brent Meeker wrote:




On 9/22/2015 5:17 AM, Bruno Marchal wrote:


On 22 Sep 2015, at 00:29, Brent Meeker wrote:


A fascinating application of computability theory to physics:

Undecidability of the Spectral Gap
Toby Cubitt,  David Perez-Garcia,  and Michael M. Wolf

The spectral gap—the difference in energy between the ground  
state and the first excited state—is one of the most important  
prop-
erties of a quantum many-body system. Quantum phase transitions  
occur when the spectral gap vanishes and the system becomes
critical. Much of physicsis concerned with understanding the  
phase diagrams of quantum systems, and some of the most  
challenging
and long-standing open problems in theoretical physics concern  
the spectral gap, 1–3 such as the Haldane conjecture 4 that the  
Heisen-
berg chain is gapped for integer spin, proving existence of a  
gapped topological spin liquid phase, 2,3 or the Yang-Mills gap  
conjecture 5
(one of the Millennium Prize problems). These problems are all  
particular cases of the general spectral gap problem: Given a  
quan-
tum many-body Hamiltonian, is the system it describes gapped or  
gapless? Here we show that this problem is undecidable, in the
same sense as the Halting Problem was proven to be undecidable  
by Turing.


I guess he means unsolvable.

"undecidable" is relative to a theory. Unsolvable or uncomputable  
is absolute and does not depend on any theory. It means that  
there is no alogorithm to do some task, like computing some  
function or deciding some set.


Yes, and it would be quite surprising to find there is no algorithm  
to compute whether or not a Hamiltonian system has a mass gap -  
since it is presumably a fact of nature whether it does or not.


Not sure that this entails the existence of an algorithm. In the  
arithmetical reality, many facts exists with provably no algorithm to  
decide them.



This may point to nature doing hyper-Turing computation or there may  
be some aspect of nature that has been overlooked.  Either way it's  
an interesting development.


I am not sure the paper alludes to hyper-Turing computation. Despite  
his quite bad vocabulary, the paper is correct on Church thesis, which  
it accepts, and concerns just insolubility, I mean unsolvability in  
theoretical physics. Like such result exist in topology, group theory,  
etc.

What amaze me is the technic and some results by Kitaev.











6 A consequence of this is that the spectral gap of certain
quantum many-body Hamiltonians is not determined by the axioms  
of mathematics,


? (that does not make sense)



much as Gödels incompleteness theorem implies
that certain theorems are mathematically unprovable.


Gödel proved only that all theories are undecidable when it comes  
to proving propositions in some domain (like natural numbers).


It makes no sense to say that some mathematical proposition are  
unprovable. there always some theories which can prove them: just  
keep such proposition as axioms, for example. PA (or ZF, ...)  
cannot prove that PA is consistent, but PA + consistent(PA) can  
prove that PA is consistent, trivially. More interestingly: PA +  
epsilon-zero is well founded can also prove that PA is consistent.


That doesn't help if nature somehow computes whether or not there is  
a mass gap, but you can't.


If one universal system can compute something, all the other universal  
system, including you and me, can do the computation, if we are given  
the time.






Simply adding an axiom may contradict nature so then you have a  
provable proposition, but it's empirically false.


OK. But in this case I was assuming the addition of a true axiom. This  
makes many propositions which were true but undecidable in the theory  
becoming decidable. "True" means "satisfied by the domain under  
scrutiny" (with or without the theory or ourself knowing it).











We extend these results to prove undecidability of other low  
temperature prop-

erties, such as correlation functions.


Well, I guess again that they talk only about unsolvability, not  
undecidability.



The proof hinges on simple quantum many-body models that exhibit  
highly unusual physics in

the thermodynamic limit.


I will take a look when I have more time. I might need to revise  
a bit the quantum many-body problem for that!


In QM, already the 0-body problem is Turing complete (when you  
need at least three bodies to have Turing completeness for  
classical physics), so it is hard to be astonished, but I don't  
judge the paper (above using a vocabulary which is confusing when  
you know the difference between computing and proving). The main  
difference is that in proof theory, the result are dependent of  
the theory (they are not absolute), where in computability, the  
results are absolute if we assume Church-thesis (which is  

Re: Undecidability of the Spectral Gap

[Brent Meeker]

On 9/23/2015 12:19 PM, Bruno Marchal wrote:


On 23 Sep 2015, at 06:51, Brent Meeker wrote:




On 9/22/2015 9:26 PM, Bruno Marchal wrote:


On 22 Sep 2015, at 19:27, Brent Meeker wrote:




On 9/22/2015 5:17 AM, Bruno Marchal wrote:


On 22 Sep 2015, at 00:29, Brent Meeker wrote:


A fascinating application of computability theory to physics:

Undecidability of the Spectral Gap
Toby Cubitt,  David Perez-Garcia,  and Michael M. Wolf

The spectral gap—the difference in energy between the ground 
state and the first excited state—is one of the most important prop-
erties of a quantum many-body system. Quantum phase transitions 
occur when the spectral gap vanishes and the system becomes
critical. Much of physicsis concerned with understanding the 
phase diagrams of quantum systems, and some of the most challenging
and long-standing open problems in theoretical physics concern 
the spectral gap, 1–3 such as the Haldane conjecture 4 that the 
Heisen-
berg chain is gapped for integer spin, proving existence of a 
gapped topological spin liquid phase, 2,3 or the Yang-Mills gap 
conjecture 5
(one of the Millennium Prize problems). These problems are all 
particular cases of the general spectral gap problem: Given a quan-
tum many-body Hamiltonian, is the system it describes gapped or 
gapless? Here we show that this problem is undecidable, in the
same sense as the Halting Problem was proven to be undecidable by 
Turing.


I guess he means unsolvable.

"undecidable" is relative to a theory. Unsolvable or uncomputable 
is absolute and does not depend on any theory. It means that there 
is no alogorithm to do some task, like computing some function or 
deciding some set.


Yes, and it would be quite surprising to find there is no algorithm 
to compute whether or not a Hamiltonian system has a mass gap - since 
it is presumably a fact of nature whether it does or not.


Not sure that this entails the existence of an algorithm. In the 
arithmetical reality, many facts exists with provably no algorithm to 
decide them.


But many scientists implicitly assume that reality is computable, that 
there is an algorithm for deciding how the state of the universe 
evolves.  And so they reject the idea that reality is isomorphic to 
arithmetic.  If the mass gap is not computable that's very surprising.  
But I'm afraid it depends on systems being potentially infinite, which 
is dubious.


Brent

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The Axiom Of Choice and ComputationalismT

[John Clark]

It seems to me the debate I’v  been having with Bruno, the one about
Arithmetic being able to perform calculations all by itself without the
help of matter that obeys the laws of physics, comes down to the Axiom Of
Choice. I would humbly propose that maybe just maybe mathematics is
everything EXCEPT for the Axiom of Choice and physics is mathematics PLUS​ the
Axiom of Choice ​If this is true then for something to be really real and
not just sorta real physics must be able to calculate (choose) it.

The Axiom of Choice says that if you have an infinite number of bins with
two or more different types of things in them then you can always create a
new bin containing exactly one item from each bin. Bertrand Russell gave
this example: “To choose one sock from each of infinitely many pairs
of socks requires the Axiom of Choice, but for shoes the Axiom is not
needed.” With shoes you could have a finite number of rules (just one in
this case)  that would work, always pick the left shoe from each bin, but
no corresponding finite number of rules exists for socks so you’d have to
invoke the Axiom of Choice. This may have some relevance to the following
question: If it exceeds the computational power of the entire universe to
calculate (choose) does the 423rd prime number greater than 10^100^100
really exist or only sorta exist?

To create a bin containing all the integers the Axiom of Choice is not
needed, the 8 Zermelo-Fraenkel Axioms are enough; thus you could create a
bin containing all the integers and only the integers {1,2,3,4...} , you
can also create bins with {2,3,4,5...} and another with {3,4,5,6...} etc. A
finite number of rules (just 8) can create such bins (sets) . But what
about a bin that contains all the prime numbers and only the prime numbers?
Without the Axiom of Choice there is no rule of finite length that would
allow you to choose one and only one prime number from all the bins I
listed above and use them to come up with a new bin containing all the
prime numbers and nothing but the prime numbers.

Godel proved in 1938 that if you assume the Axiom of Choice is true then it
will cause no contradictions in Zermelo-Fraenkel or in arithmetic, and Paul
Cohen proved in 1963 that if you assume the the Axiom of Choice is false it
will cause no contradictions in Zermelo-Fraenkel or in arithmetic. In other
words the Axiom of Choice is independent of arithmetic and independent of
the Zermelo-Fraenkel Axioms.

The Axiom of Choice has always been far more controversial than the 8
Zermelo-Fraenkel Axioms, and mathematicians are reluctant to use it in
their proofs unless they have to, in fact it’s almost as controversial as
Euclid’s Fifth Postulate. As I’ve stated it the Axiom seems intuitively
true, almost bland; but the trouble is that you can state the same thing in
a different way that is absolutely equivalent but when stated that way it
seems intuitively false. For example, the Axiom of Choice can also be
stated as "every set can be well ordered” and that seems false; “well
ordered” means it has a least element, it’s easy to see that the set of
positive integers is well ordered but how would you well order the real
numbers? Mathematicians think it’s ugly for the Axiom Of Choice to produce
a set as if by magic with no instructions on how to actually build it.

Also if the the Axiom Of Choice is true then the Banach-Tarski construction
(sometimes called paradox) can be done. If you cut up a solid sphere and
then put all the pieces back together in a way specified by Banach and
Tarski you can create TWO solid spheres of a size equal to the original
single sphere. This can’t happen in the real physical world so does this
fact work against my idea that Physics is arithmetic plus the Axiom Of
Choice? Maybe not because maybe it does happen in the real physical world.
We know from astronomical observation that space is expanding, new space is
being created, and maybe Banach-Tarski is how physics does it.

  John K Clark

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Re: Undecidability of the Spectral Gap

[Brent Meeker]

Right.  All physical systems are finite, so they are necessarily 
computable.  The gap in the limit is undecidable like the halting 
problem is undecidable; you have to run the computation to find out and 
no matter how big you have made the system you cannot be sure that 
making it a little bigger won't change it from gapped to gapless.  At 
least that's how I understand the result.


Brent

On 9/23/2015 8:38 AM, smitra wrote:
They are considering the limit if an infinite system size and then 
asking if you get a continuous spectrum or if the ground state is 
separated by a gap from the excited states in that limit. That this is 
not decidable is perhaps surprising, but this has nothing to do with 
Nature not being computable. Turing gave an example of a physical 
system that exhibits the same phenomena in an appropriate limit a long 
time ago.


Saibal


On 23-09-2015 06:26, Bruno Marchal wrote:

On 22 Sep 2015, at 19:27, Brent Meeker wrote:




On 9/22/2015 5:17 AM, Bruno Marchal wrote:


On 22 Sep 2015, at 00:29, Brent Meeker wrote:


A fascinating application of computability theory to physics:

Undecidability of the Spectral Gap
Toby Cubitt,  David Perez-Garcia,  and Michael M. Wolf

The spectral gap—the difference in energy between the ground 
state  and the first excited state—is one of the most important prop-
erties of a quantum many-body system. Quantum phase transitions  
occur when the spectral gap vanishes and the system becomes
critical. Much of physicsis concerned with understanding the 
phase  diagrams of quantum systems, and some of the most challenging
and long-standing open problems in theoretical physics concern 
the  spectral gap, 1–3 such as the Haldane conjecture 4 that the 
Heisen-
berg chain is gapped for integer spin, proving existence of a  
gapped topological spin liquid phase, 2,3 or the Yang-Mills gap  
conjecture 5
(one of the Millennium Prize problems). These problems are all  
particular cases of the general spectral gap problem: Given a quan-
tum many-body Hamiltonian, is the system it describes gapped or  
gapless? Here we show that this problem is undecidable, in the
same sense as the Halting Problem was proven to be undecidable by  
Turing.


I guess he means unsolvable.

"undecidable" is relative to a theory. Unsolvable or uncomputable  
is absolute and does not depend on any theory. It means that there  
is no alogorithm to do some task, like computing some function or  
deciding some set.






6 A consequence of this is that the spectral gap of certain
quantum many-body Hamiltonians is not determined by the axioms of  
mathematics,


? (that does not make sense)



much as Gödels incompleteness theorem implies
that certain theorems are mathematically unprovable.


Gödel proved only that all theories are undecidable when it comes  
to proving propositions in some domain (like natural numbers).


It makes no sense to say that some mathematical proposition are  
unprovable. there always some theories which can prove them: just  
keep such proposition as axioms, for example. PA (or ZF, ...)  
cannot prove that PA is consistent, but PA + consistent(PA) can  
prove that PA is consistent, trivially. More interestingly: PA +  
epsilon-zero is well founded can also prove that PA is consistent.




We extend these results to prove undecidability of other low  
temperature prop-

erties, such as correlation functions.


Well, I guess again that they talk only about unsolvability, not  
undecidability.



The proof hinges on simple quantum many-body models that exhibit  
highly unusual physics in

the thermodynamic limit.


I will take a look when I have more time. I might need to revise a  
bit the quantum many-body problem for that!


In QM, already the 0-body problem is Turing complete (when you 
need  at least three bodies to have Turing completeness for 
classical  physics), so it is hard to be astonished, but I don't 
judge the  paper (above using a vocabulary which is confusing when 
you know  the difference between computing and proving). The main 
difference  is that in proof theory, the result are dependent of 
the theory  (they are not absolute), where in computability, the 
results are  absolute if we assume Church-thesis (which is accepted 
by virtually all experts in the domain).


But I think their motivation was to show that nature may perform  
hyper-Turing computation.


? I am not sure.



 So they would not assume Church-Turing.


? You need Church thesis to define "hyper-Turing" computation. In fact
 you need a stronger version of Church's thesis, a sort of
hyper-Turing  Church thesis (the hyper-arithmetical Church's thesis).

Without Church's thesis, computation is not definable, still less
hyper-computation.

Bruno





Brent

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Re: What day is it?

[Bruce Kellett]

On 24/09/2015 4:02 am, Jason Resch wrote:


On Tue, Sep 8, 2015 at 8:45 PM, Bruce Kellett 
> wrote:


On 9/09/2015 1:29 pm, Jason Resch wrote:

On Tue, Sep 8, 2015 at 9:44 PM, Bruce Kellett
> wrote:

I presume you mean that the world is duplicated on each toss,
with one branch showing each outcome. We are back to the
dreaded "person duplication" problem. My opinion on this is
that on such a duplication, two new persons are created, so
the probability that the original person will see either
heads or tails is precisely zero, because that person no
longer exists after the duplication.


So if some aliens create a copy of you in Andromeda, then you
cease to exist as a person?


Since I might know if they gathered the requisite information, it
is not an issue.


I don't see how this follows. Are you saying you would cease to exist 
as a person if a duplicate of you arose far away in this universe, or 
that you would not cease to exist as a person?


The closest continuer account of personal identity would have no 
difficulty with this. The remote "copy" is purely a matter of chance, 
which has no physical or causal connection with you, so is not a 
continuer in the required sense.



Note: according to current comological models, space is infinite
and uniform, which means infinite copies of you exist (though
very far away).

Such models make really quite strong assumptions about initial
conditions.


This all follows from thw concordance model of cosmology, which is the 
"standard model" in cosmology. See:

http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf

You might well have an infinity of worlds with our present
cosmology, but they might all be copies of some bland, boring
model with no intelligent life.


I don't think you grasp the implications of infinity. If there are 
infinite worlds, there is effectively 100% probability that an 
infinite number of them will be identical to this entire Earth as you 
see it.


As I said, that assumes some regular distribution over initial 
conditions --  condition for which we have no evidence whatsoever. So 
our universe - and our particular personal existences - might be unique, 
even in an infinite universe. There can be universes of zero probability 
measure.


Pi has infinite digits. Any sequence, however long, the encoding of 
any documentary, can be found in the digits of Pi, and moreover, it 
recurs an infinite number of times.


That, too, is an unproved assumption about uniformity --essentially the 
assumption that pi is a normal number. And that has never been proven.


Bruce



I think you are trying to avoid answering my question.

Jason


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Re: 1P/3P CONFUSION again and again

[John Clark]

On Mon, Sep 21, 2015  Bruno Marchal  wrote:


> ​> ​
> the existence of particular computations and emulations of computations by
> other computations can be proved already in Robinson Arithmetic.
>

​I don't want proof of computations, I want computations!​

​>​
> There is a continuous and a diecrete quantum teleportation technic
>

​I don't know what that means. But I do know that Quantum Mechanics can't
​deal with distances smaller than 1.6*10^-35 meters; if distances smaller
than that exist then Quantum Mechanics will need a *MAJOR *overhaul.


> ​>>​
>> ​I'm just playing ​devil's advocate
>> ​,​
>> ​ ​
>> ​
>> ​unlike
>>  you I don't claim to have proven anything
>> ​.​
>>
>
> ​> ​
> Proving is my job. That is what I do. That is what mathematician does, in
> math or in applied theoretical field. When I say that RA proves the
> existence of the terminating computations, I am saying a standrd result.
>

​Very standard indeed! Every mathematician knows that some computations
terminate, and some computations don't terminate, and for some computations
there is no way to know if they terminate or not and all you can do is
watch it and see. ​


> ​> ​
> You oppose this by introducing a notion of physical computation, which you
> have not yet define.
>

​I can provide something​

​much *much* better than a definition, I can give A EXAMPLE.
 ​

> ​> ​
> even if physics is quite important. the fundamental science is theoretical
> computer science
>

​I do admit that sometimes physics papers about entropy and Black Holes
look a lot like papers in computer science or information theory.

 John K Clark ​

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Re: What day is it?

[Jason Resch]

On Wed, Sep 23, 2015 at 4:20 PM, Bruce Kellett 
wrote:

> On 24/09/2015 4:02 am, Jason Resch wrote:
>
>
> On Tue, Sep 8, 2015 at 8:45 PM, Bruce Kellett 
> wrote:
>
>> On 9/09/2015 1:29 pm, Jason Resch wrote:
>>
>> On Tue, Sep 8, 2015 at 9:44 PM, Bruce Kellett 
>> wrote:
>>
>>> I presume you mean that the world is duplicated on each toss, with one
>>> branch showing each outcome. We are back to the dreaded "person
>>> duplication" problem. My opinion on this is that on such a duplication, two
>>> new persons are created, so the probability that the original person will
>>> see either heads or tails is precisely zero, because that person no longer
>>> exists after the duplication.
>>>
>>
>> So if some aliens create a copy of you in Andromeda, then you cease to
>> exist as a person?
>>
>>
>> Since I might know if they gathered the requisite information, it is not
>> an issue.
>>
>
> I don't see how this follows. Are you saying you would cease to exist as a
> person if a duplicate of you arose far away in this universe, or that you
> would not cease to exist as a person?
>
>
> The closest continuer account of personal identity would have no
> difficulty with this.
>

It might not, but closest continuer theory makes no sense and appears to be
an ad hoc way to escape what otherwise clear conclusions from non-dualist
theories of mind.

If you run an identical computer program on a different computer, one on
mars and one on the moon, why say the one on the moon the only one that is
identical to the program last run on Earth?

What if the two copies are an identical number of Plank lengths away? Or
what if many are all run on a sphere whose center is where the last
instance ran?

Closest continuer theory has no theoretical justification. The only reason
it even exists is that some find the idea that they are not unique to be
too upsetting. Closet continuer theory purports to offer a way to guarantee
uniqueness of the individual (at least until you consider ties by equally
close continuations).



> The remote "copy" is purely a matter of chance, which has no physical or
> causal connection with you, so is not a continuer in the required sense.
>
> Note: according to current comological models, space is infinite and
>> uniform, which means infinite copies of you exist (though very far away).
>>
>> Such models make really quite strong assumptions about initial conditions.
>>
>
> This all follows from thw concordance model of cosmology, which is the
> "standard model" in cosmology. See:
> http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf
>
>
>
>> You might well have an infinity of worlds with our present cosmology, but
>> they might all be copies of some bland, boring model with no intelligent
>> life.
>>
>
> I don't think you grasp the implications of infinity. If there are
> infinite worlds, there is effectively 100% probability that an infinite
> number of them will be identical to this entire Earth as you see it.
>
>
> As I said, that assumes some regular distribution over initial conditions
> --  condition for which we have no evidence whatsoever.
>

All current observations are consistent with the uniformity of the
universe. At large scales the universe is very homogenous, and it is
believed that early quantum fluctuations (which are effectively random)
shaped the clumping of matter.


> So our universe - and our particular personal existences - might be
> unique, even in an infinite universe. There can be universes of zero
> probability measure.
>
> Pi has infinite digits. Any sequence, however long, the encoding of any
> documentary, can be found in the digits of Pi, and moreover, it recurs an
> infinite number of times.
>
>
> That, too, is an unproved assumption about uniformity --essentially the
> assumption that pi is a normal number. And that has never been proven.
>
>
It doesn't have to be normal, it just has to be irrational (no repeating
pattern) which is proven. In any event, I was just using Pi to illustrate
that when there is an infinite extension (without a trivial repetition) the
same sequences will recur. You need to adopt non-standard cosmological
theories to say this implication does not apply in the case for our
universe.

Jason



> Bruce
>
>
> I think you are trying to avoid answering my question.
>
> Jason
>
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