Busy Beaver

[agrayson2000]

Is it not computable because after some value of its argument, its mapped 
value is greater than anything that can be written by a computer? AG

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Re: Why is Church's thesis a Miracle?

[Philip Thrift]

On Monday, September 10, 2018 at 4:11:46 AM UTC-5, Bruno Marchal wrote:
>
>
> I can argue that the Church-Turing thesis entails the falsity of the 
> physical Church-Turing thesis, even without postulating Mechanism.
> If we are machine, then we can exploit the computations which supports us 
> below our substitution level to mimic in real time processes which are not 
> rulable in real time by any classical computer.
> Might say more on this if asked. That has been explained already in this 
> list more than once. Deutsch assume a primitive physical reality. That is 
> coherent with the CT part of mechanism, but not the YD part (the yes 
> doctor, the first person indeterminacy). Mechanism is CT + YD. It leads to 
> a constructive reduction of the mind-body problem to a deduction of the 
> laws of physics from machine theology or machine self-reference if you 
> prefer. And its works, the physics deduced until now, even if quite modest, 
> is already enough quantum like to cast light on the origin of the measure 
> on the computation/sigma_1 sentences. Not yet that much as to be able to 
> derive Gleason theorem, though, but that is just complicated. To refute 
> mechanism, we should have a proof that such measure does not exist.
>
> Bruno
>
>
>  

I still think we just don't know what *computation* completely means in the 
material world (what I have called synthetic pancomputationalism [1]). In 
particular, possibly the brain (neural material) is not a digital or an 
analog or a hyper computer, but a novel kind of computer (cf. references in 
[2]).



[1]  
https://codicalist.wordpress.com/2018/04/07/synthetic-pancomputationalism/
[2] https://en.wikipedia.org/wiki/Gualtiero_Piccinini

- Philip Thrift

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Re: The codical-material universe

[Brent Meeker]

On 9/10/2018 9:34 AM, Philip Thrift wrote:



On Monday, September 10, 2018 at 3:25:53 AM UTC-5, Bruno Marchal wrote:



On 9 Sep 2018, at 13:06, Philip Thrift > wrote:



On Sunday, September 9, 2018 at 5:28:25 AM UTC-5, Bruno Marchal
wrote:



On 8 Sep 2018, at 23:53, John Clark  wrote:


Bruno MarchalWrote:

/> I cannot see primary matter.In fact I am not sure
what you mean by matter, or by “mathematical-material
universe”.[...] I have proven (40 years ago) that
materialism (the belief in some primary matter, or
physicalism) and Mechanism are incompatible./


If you don't know what "matter" means then you certainly
don't know what "primary matter" means, so what the hell did
you prove 40 years ago?


That if mechanism is true, the observable has to rely on a
sophisticated “sum” on all computations.

Matter = observable

Primary matter is the doctrine by Aristotle according to
which there is a primary physical universe, or a primary sort
of (non mathematical) reality from which those observable
would have emerge. With mechanism, it can be shown that the
laws pertaining on the observable have to be reduced to some
mode of arithmetical self-reference.




I'm not even going to ask what you think physicalism means
because any such answer has to include physics and physics
has to involve matter which you admit confuses you.


No, it does not confuse me. It is just shown inconsistent to
believe that we have to assume its existence. A realm is
primary if it cannot be reduced to some other field”. May
believe that biology is not primary, because it can be
reduced (apparently) to chemistry and physics. Similarly,
with Mechanism, physics is reducible to number theory or
Turing equivalent.






And for the same reason I'm not going to ask about
"Mechanism" , the reply would only contain yet more words
you can neither define nor give examples of.


Digital Mechanism  is the doctrine that there is a level of
description of our body such that we can survive with a
(physical) digital brain or body, if it faithfully represents
our body’s functionality at that description level.

Bruno



I seems /possible /to me that there could be a matter
decompiler/transporter/compiler that takes *me*, decompiles *me*
into some code, transports that code, and compiles that code into
a digital-technology-based "brain" in some sort of "body". And it
would be *me 2*. and "I" would exist again.

But if it never recompiled me into any kind of material output - 
I don't  think I would exist anymore.


How would you, or how would any universal machine, be able to
distinguish (without observable clue, by personal introspection)
if it has been recompiled in a physical reality or in a
number-theoretical reality imitating my brain below my
substitution level?
It seems to me that you need to give to matter some special role
in consciousness which cannot be recovered by anything
Turing-emulable, but then mechanism is false.But invoking
(primitive) Matter in this way seems arbitrary, and it
re-introduce the mind-body problem. It seems like adding something
difficult to avoid a consequence. If you survive only because the
physical stuff emulate correctly the computations associated to
your experience, then you will survive also in arithmetic, which
emulates all computations. Indeed the notion of “emulation” of a
machine by another has been discovered in arithmetic.

Bruno



If my (material) body was decompiled into some compressed code, that 
code was stored, and then later that code was compiled (with a 
biocompiler - https://en.wiktionary.org/wiki/biocompiler - for 
example) I might be able to check if my new body was different from my 
old body by comparing medical records.


If the person who compiled my code into my new body told me that my 
code  had been compiled - not into my previous material reality - but 
into a numerical reality. I'm not sure how that would change my life. 
I'd probably say, "Yeah, sure" and still be a materialist.


- pt


That's a point I've made before.  It's all very well to say that you 
could be replaced by an abstract machine (e.g. arithmetic) running your 
code; but to make that work there would also have to be an emulation of 
your environment, including its physics.  So then it's not clear that 
anything is different.  It becomes a metaphysical just-so story.


Brent

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Re: Why is Church's thesis a Miracle?

[Brent Meeker]

On 9/10/2018 2:11 AM, Bruno Marchal wrote:


On 9 Sep 2018, at 21:51, Philip Thrift > wrote:




On Sunday, September 9, 2018 at 10:04:20 AM UTC-5, John Clark wrote:

On Sun, Sep 9, 2018 at 6:44 AM Bruno Marchal > wrote:

>>Nobody on this planet uses the term "Löbian machine"
except you.

>/It is just a more precise version of what popular books
described by “sufficiently rich theory”./


There is nothing precise about homemade slang used by nobody but you.

/> There are many definition, but they are all equivalent./


And there is nothing profound about a definition, it's easy to
define a perpetual motion machine but that doesn't mean they
exist, I can define a Clark Machine as a machine that can solve
the halting problem but that doesn't mean I have the any idea how
to make one or can even show that such a thing could in principle
exist.

/>Any Turing complete theory of any universal machine, with
sufficiently strong induction axiom (like sigma_1 induction)
 constitute a Löbian machine. /


In the physical world induction is just a rule of thumb that
usually works pretty well most of the time, but it seldom works
perfectly and never works continuously, eventually it always fails.

>>Turing explained exactly precisely how to build one of
his machines but you have never given the slightest hint
of how to build a "Löbian machine" or even clearly
explained what it can compute that a Turing Machine can’t. 



>/?/

!

>/That means just that you need to go being step 3 in my thesis,/


Step 3? Ah yes I remember now, that's the one with wall to wall
personal pronouns without a single clear referent in the entire
bunch.

> /The notion of Löbian machine is easy to construct,/


The notion of a Perpetual Motion machine is also easy to
construct as is the Clark Machine that can solve the Halting
Problem, but Turing did far more than dream up a magical
universal calculating machine, he showed exactly how to make one.
But we're not as smart as Turing, I can't do that with my Clark
Machine and you can't do that with your Löbian machine.

/> and the mathematical reality is full of example of Löbian
machine, and Löbian god/


Löbian machine,  Löbian god, the propositional part of the
theology  tell me, have you ever wondered why so manypeople
fail to take you seriously?

/>A Lpobian machine is just a universal machine capable of
proving its own universality./


I have no trouble believing a universal machine is universal, but
no Turing Machine can in general prove it will halt and but no
machine of any sort, or anything else for that matter, can prove
its own consistency unless it is inconsistent.

> Why do you want it to be able to do what a god can do?


Odd question, who wouldn't want to do what a God can do? But if
God can solve the Halting Problem then He can also make a rock so
heavy He can't lift it.

>>How would things be different if "the propositional part
of the theology" were not decidable? 



>/Solovay theorem would be false, and the subject of machine
theology would be far more complex.
/


Idon't know if that's true or not because "machine theology" is
more of your homemade gibberish, just like "the propositional
part of the theology".

> /Note that the theology of machine has highly undecidable at
the first order level./


And I don't know if that is true or not either because "the
theology of machine" is yet more of your patented homemade baby
talk.

John K Clark




The only relevant "physical" theory I know about and is discussed 
widely is in terms of *relativistic computers* (which probably most 
think are forever merely fictional).



  Relativistic computers and the Turing barrier


https://www.sciencedirect.com/science/article/abs/pii/S0096300305008398

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.150.783=rep1=pdf


/We examine the current status of the physical version of the 
Church-Turing Thesis (PhCT for short) in view of latest developments 
in spacetime theory. This also amounts to investigating the status of 
hypercomputation in view of latest results on spacetime. We agree 
with [D. Deutsch, A. Ekert, R. Lupacchini, Machines, logic and 
quantum physics, Bulletin of Symbolic Logic 6 (3) (2000) 265–283] 
that PhCT is not only a conjecture of mathematics but rather a 
conjecture of a combination of theoretical physics, mathematics and, 
in some sense, cosmology. Since the idea of computability is 
intimately connected with the nature of time, relevance of spacetime 
theory seems to be unquestionable. We will see that recent 
developments in 

Re: The codical-material universe

[John Clark]

On Sun, Sep 9, 2018 at 6:28 AM Bruno Marchal  wrote:

*> Matter = observable*
>

Speed is observable, is speed matter? The qualia red is observable, is red
matter?


> > Primary matter is the doctrine by Aristotle
>

If I never hear another word about Aristotle I will not in any way feel
deprived.


> >A realm is primary if it cannot be reduced to some other field”.
>

OK

*>May believe that biology is not primary, because it can be reduced
> (apparently) to chemistry and physics. *
>

Yes


> > *with Mechanism, physics is reducible to number theory or Turing
> equivalent.*
>

No. You can't have a Turing Machine without a machine and you can't have a
number theory, or a theory of any sort, without a brain made of matter.

> *Digital Mechanism  is the doctrine that there is a level of description
> of our body such that we can survive with a (physical) digital brain or
> body, if it faithfully represents our body’s functionality at that
> description level.*
>

Digits are numbers so I guess you believe in Digital Mechanism, unless you
believe there is something special about the atoms that happen to occupy
your body right now so that they "cannot be reduced to some other field",
that is to say unless you believe the matter in your body is primary.

John K Clark

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Combinator 4 (Recursion)

[Bruno Marchal]

Hi,


All right, now the almost last fundamental result before showing the Turing 
Universality of the SK-combinators (or simply combinators).

(Except for an important but easy arithmetical interlude).

We have solved the following problem: to find a combinator A, i.e. a 
combination of S and K, such that, in virtue of the two laws (Kxy =x, Sxyz = 
xz(yz)), we have:

Axyz = xy(yxz)  (or any other combinations).

Indeed A is given by the ABF algorithm (or the others):

A = [x] [y] [z] xy(yxz).

The new problem of the day is, ‘---can you find a combinator A such that

Ax = x(Ax),

or

Axy = xAy(yA),

or similar?

You see the difficulty? The unknown combinator A appears at both sides of the 
equation. It is not clear at first sight if such a combinator A can even exist.
Such an equation is called a recursion equation. A is somehow defined in term 
of itself.

The fundamental result explained here is that the solution to the recursion 
equation always exists! 

So can we find it? I will again prove that such equations have always a 
solution by providing an algorithm which gives the solution (and proving that 
the algorithm is correct).

You will recognise a variant of the little song-question that I have used many 
times: “If Dx gives xx, what gives DD?”, which also appeared when we defined 
the Mocking Bird M (Mx gives xx). And we know the trouble when we apply M to 
itself, which is that MM gives a combinator without normal form, or unstable. 
The two laws keep being triggered and MM does not halt on some normal or stable 
combinator.

So, we might expect that the solution of the recursion equation have not 
necessarily a normal form (i.e. are stable). But that can be OK, given that our 
goal consists in implementing the computations, including the infinite 
computations (the processes) and what is an infinite  computation but an 
unstable combinators?


In Smullyan's book (How to Mock a Mocking Bird?) Smullyan is very quick on 
this, and he is right, because it is simpler that way. He made his hero, Craig, 
particularly “alert” that day, and Craig found very quickly two ways to solve 
the problem. One using a “sage bird”, and one “from scratch”.

The “sage birds” are known in the old literature as the “paradoxical 
combinators”, and are known in the modern literature as the “fixed point 
combinators”.

I have not talked much about them, and so I will give the method “from scratch” 
first. Then I will use it to define the sage, paradoxical, birds, and the 
relation with the fixed points theorems, and give the first method, which will 
be more efficacious and provide shorter solutions.

Instead of reasoning on arbitrary combinations, let us reason with the 
particular exercise above: to find A such that

Axy = xAy(yA).

I will first rename the variable x -> y, and y -> z. The reason for this is 
that I want x playing some role related to A.


Ayz = yAz(zA).  (This change nothing of course. OK?)

Then I replace A by x. And we know that there is a combinator A’, such that

A’xyz = yxz(zx).  (Which is the right hand side above with A replaced by x).

A’ has one more variable than A, and A, being replaced by x in the right hans 
side has become into an argument.

We know that such combinator A’ exist, because we can eliminate the variable:

A’ = [x][y][z]yxz(zx).

Then we give A’ to as sage bird, and that gives the (first) solution!

But wait! We don’t have yet studied the sage bird yet!

So, in the method from scratch, instead of A’, we use A’’, which is exactly 
like A’ except that it duplicates the variable x:

A’’xyz = y(xx)z(z(xx))

Of course A’’ exists too:

A’’ =  [x][y][z]y(xx)z(z(xx)). 

You can, or not, compute this at your time and convenience (if you are 
masochist), but you need only to understand that this expression defined some 
precise combinator (with extensional identity, that is up to syntactical 
differences).

Now, apply A’’ on itself, taken as first argument: you get

A’’A’’yz = y(A’’A'')z(z(A’’A'’))

And that solves the problem! The solution is A’’A’’, as you can see. 
With  A = A’’A’', we do have Ayz : yAz(zA), or, changing the variable again: 
Axy = xAy(yA), like we wanted.

Now the task of eliminating z, y and x in y(xx)z(z(xx)) is tedious. I will not 
illustrate it. 


So let us search a better solution, and for this we need to use Smullyan’s Sage 
Birds, or Curry's paradoxical combinators.


Definition. A combinator X is a fixed point of a combinator F if FX = X. 
(Smullyan says that F is fond of X).

Crazily enough all combinators have a fixed point. A student thought, at first, 
 that if that is true, a combinator should not be able to mimic a translation 
in the plane (say), as a translation is a typical transformation having no 
fixed point, unlike a rotation. But of course that does not follow. If a 
combinator mimic a translation in the plane, his fixed point will just denote 
an element which is not representing an element of the plane.

How is that possible? By the 

Re: The codical-material universe

[Philip Thrift]

On Monday, September 10, 2018 at 3:25:53 AM UTC-5, Bruno Marchal wrote:
>
>
> On 9 Sep 2018, at 13:06, Philip Thrift > 
> wrote:
>
>
>
> On Sunday, September 9, 2018 at 5:28:25 AM UTC-5, Bruno Marchal wrote:
>>
>>
>> On 8 Sep 2018, at 23:53, John Clark  wrote:
>>
>>
>> Bruno Marchal Wrote:
>>
>> *> I cannot see primary matter. In fact I am not sure what you mean by 
>>> matter, or by “mathematical-material universe”. [...] I have proven (40 
>>> years ago) that materialism (the belief in some primary matter, or 
>>> physicalism) and Mechanism are incompatible.*
>>
>>
>> If you don't know what "matter" means then you certainly don't know what 
>> "primary matter" means, so what the hell did you prove 40 years ago?  
>>
>>
>> That if mechanism is true, the observable has to rely on a sophisticated 
>> “sum” on all computations. 
>>
>> Matter = observable
>>
>> Primary matter is the doctrine by Aristotle according to which there is a 
>> primary physical universe, or a primary sort of (non mathematical) reality 
>> from which those observable would have emerge. With mechanism, it can be 
>> shown that the laws pertaining on the observable have to be reduced to some 
>> mode of arithmetical self-reference.
>>
>>
>>
>> I'm not even going to ask what you think physicalism means because any 
>> such answer has to include physics and physics has to involve matter which 
>> you admit confuses you. 
>>
>>
>> No, it does not confuse me. It is just shown inconsistent to believe that 
>> we have to assume its existence. A realm is primary if it cannot be reduced 
>> to some other field”. May believe that biology is not primary, because it 
>> can be reduced (apparently) to chemistry and physics. Similarly, with 
>> Mechanism, physics is reducible to number theory or Turing equivalent.
>>
>>
>>
>>
>>
>> And for the same reason I'm not going to ask about "Mechanism" , the 
>> reply would only contain yet more words you can neither define nor give 
>> examples of.
>>
>>
>> Digital Mechanism  is the doctrine that there is a level of description 
>> of our body such that we can survive with a (physical) digital brain or 
>> body, if it faithfully represents our body’s functionality at that 
>> description level.
>>
>> Bruno
>>
>>
>>
> I seems *possible *to me that there could be a matter 
> decompiler/transporter/compiler that takes *me*, decompiles *me* into 
> some code, transports that code, and compiles that code into a 
> digital-technology-based "brain" in some sort of "body". And it would be *me 
> 2*. and "I" would exist again.
>
> But if it never recompiled me into any kind of material output -  I don't  
> think I would exist anymore.
>
>
> How would you, or how would any universal machine, be able to distinguish 
> (without observable clue, by personal introspection) if it has been 
> recompiled in a physical reality or in a number-theoretical reality 
> imitating my brain below my substitution level?
> It seems to me that you need to give to matter some special role in 
> consciousness which cannot be recovered by anything Turing-emulable, but 
> then mechanism is false.But invoking (primitive) Matter in this way seems 
> arbitrary, and it re-introduce the mind-body problem. It seems like adding 
> something difficult to avoid a consequence. If you survive only because the 
> physical stuff emulate correctly the computations associated to your 
> experience, then you will survive also in arithmetic, which emulates all 
> computations. Indeed the notion of “emulation” of a machine by another has 
> been discovered in arithmetic.
>
> Bruno
>
>
 
 
If my (material) body was decompiled into some compressed code, that code 
was stored, and then later that code was compiled (with a biocompiler - 
https://en.wiktionary.org/wiki/biocompiler - for example) I might be able 
to check if my new body was different from my old body by comparing medical 
records.

If the person who compiled my code into my new body told me that my code  
had been compiled - not into my previous material reality - but into a 
numerical reality. I'm not sure how that would change my life. I'd probably 
say, "Yeah, sure" and still be a materialist.

- pt

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Re: Combinator 3 (a bit of Logic)

[Telmo Menezes]

Thanks Bruno!
Following so far with not problems.

On 5 September 2018 at 19:41, Bruno Marchal  wrote:
> Hi Jason, Telmo, Brent, others,
>
> Some of you might say “OK, combinators combine, and now we know that the
> combination of S and K, that is, the combinators, can do all combination”.
> The last post did prove the so-called combinatorial completeness.
>
> But that might still seem far from Turing universality.
>
> The post of today will make a simple step in that direction.
>
> I propose to implement elementary logic with the combinators. I expose a
> solution due to Barendrecht, which is extraordinarily elegant. It is also
> the one used by Smullyan in “To Mock a Mocking Bird”.
>
> I will first implement the control structure:
>
>if A then B else C
>
> As you can guess this is a pretty important step toward Turing Universality.
> A is supposed to be some “propositional combinators”, being true or false,
> on some argument(s) or not, and we want that if A is true, then the
> combinators B is trigged, and if A is false, then the combinator C should be
> trigged.
>
> For this we need some representation of the constant Boolean TRUE and FALSE.
>
> Barendrecht defined the constant TRUE by K. And he defined the constant
> false by KI. (I is of course the identity combinator, i.e. SKK). I will use
> t and f as usual for those boolean constants.
>
> So we define:
>
>  t = K
>
>  and
>
> f = KI
>
> Now, the implementation of
>
>if A then B else C
>
> Is simply
>
>   ABC
>
> Indeed, if A is true, A = t = K, and KBC = B, and if A is false, A = f = KI,
> then ABC = fBC = KIBC = C.
>
> (KIxy = (KIx)y = Iy = y)
>
> OK?
>
> Bold Summary: By choosing t = K, and f = KI, “if A then B else C” becomes
> ABC.
>
> This will be exploited later for programming, but right now, we can use this
> to implement elementary propositional logic.
>
> I will use minuscule for those logical combinators
>
> 1) conjunction: c("x and y” is written cxy)
>
> We want a combinator c such that cxy = t if both x and y are equal to t, and
> false in all other situations.
>
> But cxy, the conjunction of x and y, is really the same as "If x then y else
> f".
>
> Indeed, if x is false, “x & y” is false, and if x is true, “x & y” is given
> by the truth value of y.
>
> So:  cxy = 'if x then y else f' = xyf (cf the bold summary above), and c =
> [x][y] xyf, but that is Rfxy where R is the Robin (one of the combinator
> which does a circular permutation, that we have seen previously):
> Rxyz = yzx. Conclusion c = Rf.
>
> < To be sure we have seen that R = CC and that
> C = S(BBS)(KK),  (note the difference between c and C)
>
> R = CC =  S(BBS)(KK)(S(BBS)(KK)) but there are still B there, which we
> replace by its SK-implementation S(KS)K:
>
> R = S(S(KS)K(S(KS)K)S)(KK)(S(S(KS)K(S(KS)K)S)(KK))
>
> So the combinator c is Rf which is R(KI), and that gives, with I = SKK,
>
> c =  S(S(KS)K(S(KS)K)S)(KK)(S(S(KS)K(S(KS)K)S)(KK))(K(SKK))
>
> But I will write it simply c = Vf. I just wanted you to remember that c is
> truly a combinator, i.e.. a combination of K and S.>>
>
> Does it work? It should. No need to verify this with the long expression, as
> we have already verified that Vxyz = zxy, etc. We can test directly the
> truth table:
>
> ctt = Rftt = ttf = Ktf = t
> ctf = Rftf = tff = Kff = f
> cft = Rfft = ftf = KItf = f
> cff = Rfff = fff = KIff = f
>
> It works!
>
> 2) Disjunction:  d
>
> dxy = if x then t else y  (OK?)
>
> So dxy = xty. So d = [x][y]xty = Tty (with Txy = yx): Ttxy = xty indeed.
>
> So d = Tt
>
> Does it work? Let us verify:
>
> dtt = Tttt = ttt = Ktt = t
> dtf = Tttf = ttf = Ktf = t
> dft = Ttft = ftt = (KI)tt = t
> dff = Ttff = ftf = (KI)tf = f
>
> It works!
>
> 3) implication: i
>
> ixy should be false only if x is t and y is f, if you remember its truth
> table. ixy is basically the (negation of x) or y.
>
> So ixy = if x then y else t.   OK? So ixy = xyt, and i = [x][y]xyt, and that
> gives i = Rt.
> Vérification:
>
> itt = Rttt = ttt = Ktt = t
> itf = Rttf = tft = Kft = f
> ift = Rtft = ftt = KItt = t
> iff = Rtff = fft = KIft = t
>
> It works!
>
> 4) Negation : N (I use “N” instead of “n", as “n” is so much used for
> numbers …, who will soon appear)
>
> We know that (Not x) is the same as (x -> f), so Nx = ixf, and N =[x]ixf
> would do, but we can find it again directly by the fact that
>
> (Not x) = if x then f else t.   OK?
>
> So Nx = xft, and N = [x]xft, which gives N = Vft, with V being the Vireo,
> the other circular permuter combinator: Vxyz = zxy.
>
> Vérification:
>
> (Not t) = Vftt = tft = Kft = f.
> (Not f) = Vftf = fft = (KI)ft = t.
>
> It works!
>
> So we can do propositional logic!
>
> Summary:
>
> t = K
> f = KI
>
> “If x then y else z” = xyz
>
> c = Rf
> d = Tt
> i = Rt
> N = Vft
>
>
> Does this close the Turing Universality question? Some of my students
> thought so. The reason that they give is that with NOT and AND, or with NOT
> and OR, we are supposed to be able to build the Boolean 

Re: Why is Church's thesis a Miracle?

[Bruno Marchal]

> On 9 Sep 2018, at 21:51, Philip Thrift  wrote:
> 
> 
> 
> On Sunday, September 9, 2018 at 10:04:20 AM UTC-5, John Clark wrote:
> On Sun, Sep 9, 2018 at 6:44 AM Bruno Marchal > 
> wrote:
> 
> >>Nobody on this planet uses the term "Löbian machine" except you.
>  
> >It is just a more precise version of what popular books described by 
> >“sufficiently rich theory”.
> 
> There is nothing precise about homemade slang used by nobody but you.
> 
> > There are many definition, but they are all equivalent.
> 
> And there is nothing profound about a definition, it's easy to define a 
> perpetual motion machine but that doesn't mean they exist, I can define a 
> Clark Machine as a machine that can solve the halting problem but that 
> doesn't mean I have the any idea how to make one or can even show that such a 
> thing could in principle exist.
>  
> >Any Turing complete theory of any universal machine, with sufficiently 
> >strong induction axiom (like sigma_1 induction)  constitute a Löbian machine.
> 
> In the physical world induction is just a rule of thumb that usually works 
> pretty well most of the time, but it seldom works perfectly and never works 
> continuously, eventually it always fails.
> 
> >>Turing explained exactly precisely how to build one of his machines but you 
> >>have never given the slightest hint of how to build a "Löbian machine" or 
> >>even clearly explained what it can compute that a Turing Machine can’t.
> 
> >?
> ! 
> 
> >That means just that you need to go being step 3 in my thesis,
> 
> Step 3? Ah yes I remember now, that's the one with wall to wall personal 
> pronouns without a single clear referent in the entire bunch.
>  
> > The notion of Löbian machine is easy to construct,
> 
> The notion of a Perpetual Motion machine is also easy to construct as is the 
> Clark Machine that can solve the Halting Problem, but Turing did far more 
> than dream up a magical universal calculating machine, he showed exactly how 
> to make one. But we're not as smart as Turing, I can't do that with my Clark 
> Machine and you can't do that with your Löbian machine.
>  
> > and the mathematical reality is full of example of Löbian machine, and 
> > Löbian god
> 
> Löbian machine,  Löbian god, the propositional part of the theology  tell 
> me, have you ever wondered why so many people fail to take you seriously?
>  
> >A Lpobian machine is just a universal machine capable of proving its own 
> >universality.
> 
> I have no trouble believing a universal machine is universal, but no Turing 
> Machine can in general prove it will halt and but no machine of any sort, or 
> anything else for that matter, can prove its own consistency unless it is 
> inconsistent. 
> 
> > Why do you want it to be able to do what a god can do?
> 
> Odd question, who wouldn't want to do what a God can do? But if God can solve 
> the Halting Problem then He can also make a rock so heavy He can't lift it.
> 
> >>How would things be different if "the propositional part of the theology" 
> >>were not decidable? 
> 
> >Solovay theorem would be false, and the subject of machine theology would be 
> >far more complex. 
> 
> I don't know if that's true or not because "machine theology" is more of your 
> homemade gibberish, just like "the propositional part of the theology".
> 
> > Note that the theology of machine has highly undecidable at the first order 
> > level.
> 
> And I don't know if that is true or not either because "the theology of 
> machine" is yet more of your patented homemade baby talk. 
> 
> John K Clark
> 
> 
> 
> The only relevant "physical" theory I know about and is discussed widely is 
> in terms of relativistic computers (which probably most think are forever 
> merely fictional).
> 
> Relativistic computers and the Turing barrier
> 
> 
> 
> https://www.sciencedirect.com/science/article/abs/pii/S0096300305008398
> 
> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.150.783=rep1=pdf
> 
> 
> 
> We examine the current status of the physical version of the Church-Turing 
> Thesis (PhCT for short) in view of latest developments in spacetime theory. 
> This also amounts to investigating the status of hypercomputation in view of 
> latest results on spacetime. We agree with [D. Deutsch, A. Ekert, R. 
> Lupacchini, Machines, logic and quantum physics, Bulletin of Symbolic Logic 6 
> (3) (2000) 265–283] that PhCT is not only a conjecture of mathematics but 
> rather a conjecture of a combination of theoretical physics, mathematics and, 
> in some sense, cosmology. Since the idea of computability is intimately 
> connected with the nature of time, relevance of spacetime theory seems to be 
> unquestionable. We will see that recent developments in spacetime theory show 
> that temporal developments may exhibit features that traditionally seemed 
> impossible or absurd. We will see that recent results point in the direction 
> that the possibility of artificial systems computing non-Turing computable 

Re: Why is Church's thesis a Miracle?

[Bruno Marchal]

> On 9 Sep 2018, at 17:03, John Clark  wrote:
> 
> On Sun, Sep 9, 2018 at 6:44 AM Bruno Marchal  > wrote:
> 
> >>Nobody on this planet uses the term "Löbian machine" except you.
>  
> >It is just a more precise version of what popular books described by 
> >“sufficiently rich theory”.
> 
> There is nothing precise about homemade slang used by nobody but you.

Let us talk on ideas, not on people.

Take any Turing universal theory/machine, like Robinson arithmetic:

Classical logic +

0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x

Or Combinator theory: i.e. axiom of identity +

Kxy = x
Sxyz = xz(yz)

Both those theories/machines are Turing-complete/universal. They determine an 
entire Universal dovetailing. 
But none of them is Löbian. They are NOT “sufficiently rich”. But both becomes 
Löbian (verify and prove Löb’s formula []([]p->p)->[]p, and this obeys to the 
machine theology G*) once you add the induction axioms, i.e. the infinitely 
many axioms of induction: that is, with A an arbitrary formula (in the 
respective domain), the axioms:

If A(0) and if (n)(A(n) -> A(s(n))) then (n)A(n)

Or

If A(K) and A(S), and if (x)(A(x) & A(y) ->. A(xy)), then (x)A(x).




> 
> > There are many definition, but they are all equivalent.
> 
> And there is nothing profound about a definition, it's easy to define a 
> perpetual motion machine but that doesn't mean they exist, I can define a 
> Clark Machine as a machine that can solve the halting problem but that 
> doesn't mean I have the any idea how to make one or can even show that such a 
> thing could in principle exist.

Sure.



>  
> >Any Turing complete theory of any universal machine, with sufficiently 
> >strong induction axiom (like sigma_1 induction)  constitute a Löbian machine.
> 
> In the physical world induction is just a rule of thumb that usually works 
> pretty well most of the time, but it seldom works perfectly and never works 
> continuously, eventually it always fails.


?

You seem to confuse mathematical induction, and adductive inference.



> 
> >>Turing explained exactly precisely how to build one of his machines but you 
> >>have never given the slightest hint of how to build a "Löbian machine" or 
> >>even clearly explained what it can compute that a Turing Machine can’t.
> 
> >?
> ! 

I have given a lot of example. Peano arithmetic is a Löbian machine. 
Zermelo-Fraenkel Set Theory is a Löbian Machine, all humans, as as as they are 
arithmetically correct, are Löbian machine. I gave other examples in my long 
text, and two examples are given above. You can define a Löbian machine by any 
theory or machine on which Löb’s theorem is applicable (and then it can be 
shown that they will be aware of this). 

Here is another definition:

A machine is Turing universal iff for all sigma_1 proposition p -> []p is true.
A machine is Löbian iff for all sigma_1 proposition p -> []p is provable by the 
machine.




> 
> >That means just that you need to go being step 3 in my thesis,
> 
> Step 3? Ah yes I remember now, that's the one with wall to wall personal 
> pronouns without a single clear referent in the entire bunch.

No, you have agreed on each definition. You agreed that both the W-man and the 
H-man are honorable H-man survivor, and you did manage to take into account the 
first person/third person in some context (like in Everett).. It is you critic 
of step 3 which nobody understand. You are the only one person in the world 
that I know having taken so much time to get that extremely easy, if not 
obvious point. You did grasp it at repetition, and just adding something like 
it was obvious, but then never answer the step 4.





>  
> > The notion of Löbian machine is easy to construct,
> 
> The notion of a Perpetual Motion machine is also easy to construct as is the 
> Clark Machine that can solve the Halting Problem, but Turing did far more 
> than dream up a magical universal calculating machine, he showed exactly how 
> to make one.

Yes, it did that too, but that does not change the fact that his recovery was 
in pure mathematics at first. Then later it has been shown to be in already 
pure arithmetic.



> But we're not as smart as Turing, I can't do that with my Clark Machine and 
> you can't do that with your Löbian machine.

Nobody ever pretended anything like that. Universal machine suffer intrinsic 
limitation (which Brough all the incompleteness nuance on G* which build the 
machine theology), and the only difference with the Löbian machine is that they 
know this: they prove their incompleteness theorem, like Gödel foresaw already 
at the end of his 1931 paper (but that will be proved by Hilbert and Bernays in 
1937 for the first time, and extended in 1955 by Löb in an important way).




>  
> > and the mathematical reality is full of example of Löbian machine, and 
> > Löbian god
> 
> Löbian machine,  Löbian god, the propositional part of the 

Re: The codical-material universe

[Bruno Marchal]

> On 9 Sep 2018, at 13:06, Philip Thrift  wrote:
> 
> 
> 
> On Sunday, September 9, 2018 at 5:28:25 AM UTC-5, Bruno Marchal wrote:
> 
>> On 8 Sep 2018, at 23:53, John Clark > wrote:
>> 
>> 
>> Bruno Marchal Wrote:
>> 
>> > I cannot see primary matter. In fact I am not sure what you mean by 
>> > matter, or by “mathematical-material universe”. [...] I have proven (40 
>> > years ago) that materialism (the belief in some primary matter, or 
>> > physicalism) and Mechanism are incompatible.
>> 
>> If you don't know what "matter" means then you certainly don't know what 
>> "primary matter" means, so what the hell did you prove 40 years ago? 
> 
> That if mechanism is true, the observable has to rely on a sophisticated 
> “sum” on all computations. 
> 
> Matter = observable
> 
> Primary matter is the doctrine by Aristotle according to which there is a 
> primary physical universe, or a primary sort of (non mathematical) reality 
> from which those observable would have emerge. With mechanism, it can be 
> shown that the laws pertaining on the observable have to be reduced to some 
> mode of arithmetical self-reference.
> 
> 
> 
>> I'm not even going to ask what you think physicalism means because any such 
>> answer has to include physics and physics has to involve matter which you 
>> admit confuses you. 
> 
> No, it does not confuse me. It is just shown inconsistent to believe that we 
> have to assume its existence. A realm is primary if it cannot be reduced to 
> some other field”. May believe that biology is not primary, because it can be 
> reduced (apparently) to chemistry and physics. Similarly, with Mechanism, 
> physics is reducible to number theory or Turing equivalent.
> 
> 
> 
> 
> 
>> And for the same reason I'm not going to ask about "Mechanism" , the reply 
>> would only contain yet more words you can neither define nor give examples 
>> of.
> 
> Digital Mechanism  is the doctrine that there is a level of description of 
> our body such that we can survive with a (physical) digital brain or body, if 
> it faithfully represents our body’s functionality at that description level.
> 
> Bruno
> 
> 
> 
> I seems possible to me that there could be a matter 
> decompiler/transporter/compiler that takes me, decompiles me into some code, 
> transports that code, and compiles that code into a digital-technology-based 
> "brain" in some sort of "body". And it would be me 2. and "I" would exist 
> again.
> 
> But if it never recompiled me into any kind of material output -  I don't  
> think I would exist anymore.

How would you, or how would any universal machine, be able to distinguish 
(without observable clue, by personal introspection) if it has been recompiled 
in a physical reality or in a number-theoretical reality imitating my brain 
below my substitution level?
It seems to me that you need to give to matter some special role in 
consciousness which cannot be recovered by anything Turing-emulable, but then 
mechanism is false.But invoking (primitive) Matter in this way seems arbitrary, 
and it re-introduce the mind-body problem. It seems like adding something 
difficult to avoid a consequence. If you survive only because the physical 
stuff emulate correctly the computations associated to your experience, then 
you will survive also in arithmetic, which emulates all computations. Indeed 
the notion of “emulation” of a machine by another has been discovered in 
arithmetic.

Bruno




> 
> - pt
> 
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