Re: Why is Church's thesis a Miracle?

[agrayson2000]

On Thursday, August 23, 2018 at 7:02:21 PM UTC, Bruno Marchal wrote:
>
> Grayson, 
>
> Let me explain you something crazy but absolutely important to understand 
> about the set of all computable function from N to N. 
>
> It is true that later, we will be able to identify that with the 
> computable real numbers, but that is another story. 
>
> So what is a computable function from N to N? 
>
> It is a function from N to N, i.e. an association to each natural numbers 
> to some other or not natural number. 
>

*If you include "or not a natural number", you have a range R (or possibly 
domain), the real numbers, which is not what you assert as the definition 
of the function at hand. Incidentally, I am traveling by car for a few 
days, so it will take some time to study your latest posts. I plan to study 
Cantor's theorem on the Internet and compare it with you proof. AG*

>
> The notion admits generalisation, like the function from N x N to N, that 
> is, the function with many variables. 
>
> But what means computable? 
>
> It means you can explain, in a finite time, how to compute its values and 
> this in a finite time for each value. 
>
> You can explain to who? 
>
> To the dumbest people in the room. OK, we will come back on this one, but 
> at first the notion of computable seems to be epistemological, and depends 
> on the ability of the subject. How could we dream to define mathematically 
> that notion. 
>
> Gödel already used a version of Cantor Diagonal to show that all theories 
> (rich enough to axiomatise elementary arithmetic) are essentially 
> undecidable, and that there were no universal provability predicate. And 
> Traski, but also Gödel proves a similar result for the undefinability of 
> truth. 
>
> So how to hope for this? If only a Miracle! 
>
> Let me show you the miracle. Probably not in one post. First I will show 
> you that the class or set or collection of the computable  functions is NOT 
> computable, and in fact it does not admit a universal computable function! 
> No universal machine in that class and for this class! 
>
> Take a rest, and then we start. If you understand this, you will 
> understand something which I think is truly amazing. 
>
> I need somme lemma. 
>
> Take an alphabet, A = {a, b}. You can build words with the element of the 
> alphabet, like a, b aa, ab, … aabbabbaabbb, etc. Let A* be the set of all 
> finite words on A. 
>
> As long as the alphabet is finite, the set of words will be enumerable. 
> OK? I mean there is a bijection (a one-one and onto function) from the set 
> of words and N. To see it, it is enough to order them lexicographically: 
> that is by length, and alphabetically for those who have the same length. 
>
> That gives for {a, b}, with 0 on the empty word. 
>
> 1- a 
> 2)  b 
> 3) aa 
> 4) ab 
> 5)  ba 
> 6) bb 
> 7) aaa 
> 8) aab 
> ... 
>
> So we can enumerate the finite words. BTW, this shows immediately that the 
> rational numbers are enumerable, just enumerate their description in 
> English, like#one#on#two, or thirty#five#on#twenty#four, … 
>
> Usually the diagonal is used to show that something is impossible. Take 
> A** the set of infinite words! 
>
> It can be shown easily (but I will pass) that there is a bijection between 
> A** and the real number, and the set of subset of N, and the set of 
> functions of N to N (or to any finite non empty set). 
>
> Let me just show, or remind, you how the Cantor diagonal shows that there 
> is no bijection between N and A**, with A = {0, 1}. 
>
> If there was a bijection between N and the set of all infinite sequence, 
> you would have a matrix like 
>
> 0  -0100111010.. 
> 1   -   1101100… 
> .2  -   1011011... 
> … 
>
> That  is, for all i 
>
> i   -   a_i1 a_i2  a_i3 … 
>
> But once that bijection is supposed to exist, it is easy to find a 
> sequence of one and zero which is not in the sequence: it is the diagonal 
> sequence: 
>
> (flip_00)(flip_11)(flip_22)(flip a_33)(flip a_44)(flip a_55) ... 
>
> It is the diagonal in the correspondence above, but where each 0 have flip 
> to 1, and each 1 have flip to zero (by subtracting them from 1). 
>
> That sequence cannot be in the list above, because if that was the case it 
> would correspond to number k, 
> And (a_kk) would be equal to flip (a_kk). CQFD. OK? 
>
> Do you see that, Grayson? 
>
> I have to go. Sorry. It is easier on on board with the chalk.But you have 
> to understand the passage above, so tell me if it is OK. 
>
> I guess some have an idea of the miracle I am talking about, but perhaps 
> they don’t realise the true nature of the creative bomb here. The term 
> “miracle” comes from Gödel, and refer to the fact that despite the set of 
> computable functions from N to N is not computable, the set of all 
> computable  functions from subset (including N!) of N to N, will be 
> computable, but then at the price of non cont

Re: Church-Turing Thesis

[agrayson2000]

On Friday, August 24, 2018 at 12:25:03 PM UTC, telmo_menezes wrote:
>
> On 23 August 2018 at 06:31,  > wrote: 
> > 
> > 
> > On Thursday, August 23, 2018 at 2:01:24 AM UTC, Jason wrote: 
> >> 
> >> 
> >> 
> >> On Wed, Aug 22, 2018 at 4:43 PM  wrote: 
> >>> 
> >>> 
> >>> 
> >>> On Tuesday, August 21, 2018 at 3:22:04 PM UTC, Jason wrote: 
>  
>  
>  
>  On Tue, Aug 21, 2018 at 1:16 AM  wrote: 
> > 
> > I've been looking at the Wiki article on this topic. I find that I 
> > really don't understand what it is, or why it's important. Maybe a 
> few 
> > succinct words from the usual suspects can be of help. TIA. 
> > 
> > 
>  
>  
>  Bruno provided a great definition and background of the Church-Turing 
>  Thesis. I will try to answer why it is important and comes up often 
> in our 
>  discussion. 
>  
>  
>  The Church-Turing thesis says that anything that is computable is 
>  computable by any computer.  In other words, there is nothing that 
> the 
>  computer in your cell phone can't compute, that your laptop or that a 
> super 
>  computer (or even a quantum computer) can.  It just comes down to 
> having 
>  enough time and memory. 
>  
>  This is why you don't need to buy a new phone with new hardware every 
>  time you want to install a new app.  Regardless of the type of CPU in 
> your 
>  phone, it can be extended in its power of what it might compute only 
> given 
>  some new software.  It is in this sense that computers are 
> "Universal", they 
>  are universal in the same sense that of a universal remote, or in the 
> sense 
>  that a record player is a universal sound imitating device.  A record 
> player 
>  might emulate the sounds of an orchestra, Britney Spears, whale 
> songs, etc., 
>  all it needs is the appropriate record and it can produce the sound. 
>  
>  In the same sense, all a Turing Machine (computer) needs to imitate 
> (or 
>  emulate) the right program or function is the right software. 
>  Because of 
>  this, anything that can be described in software, be it a brain 
> emulation, 
>  an AI, a virtual environment, a virtual machine or operating system, 
> can 
>  never know what hardware is running it, because the Church-Turing 
> thesis 
>  says that any computer is capable of running it. 
>  
>  This is why if consciousness is computable (the computational theory 
> of 
>  mind) we cannot know what is computing us (e.g. we could be in a 
> matrix type 
>  simulation for all we know).  The other implication is that if 
> computations 
>  exist in mathematics (and they do), then we exist within mathematics. 
>  Mathematics (or at least the part necessary to describe computations) 
>  becomes the fundamental science of what we experience and what is 
> possible 
>  to experience or what we may predict about our future experiences 
> (physics). 
>  
>  
>  Jason 
> >>> 
> >>> 
> >>> If someone digitizes (emulates) the Mona Lisa, is this equivalent to 
> the 
> >>> Mona Lisa? 
> >> 
> >> 
> >> If you digitize a person and put the digitized Mona Lisa before them, 
> it 
> >> is equivalent to the real Mona Lisa to that person, at least as far as 
> they 
> >> can tell. 
> >> 
> >> 
> >>> 
> >>> Can you write a function which is not computable? AG 
> >>> 
> >>> 
> >> 
> >> If by not computable you mean it never returns, then this is easy: 
> >> 
> >> function foo(): 
> >>   while (true) 
> >>   { 
> >>  // loop forever 
> >>   } 
> >> 
> >> There are also programs for which no one knows if they are computable 
> or 
> >> not.  If you can prove whether or not this function ever completes, you 
> will 
> >> be world famous, and may even earn a million dollars (though I think 
> the 
> >> prize has been retracted, it might be oferred again): 
> >> 
> >> Step 1: Set X = 4 
> >> Step 2: Set R = 0 
> >> Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set R 
> = 1 
> >> Step 4: If R = 1, Set X = X + 2 and go to Step 2 
> >> Step 5: If R = 0, print X and halt 
> >> 
> >> All you have to prove is the computer either never gets to step 5 or 
> that 
> >> it does get to step 5.  Mathematicians have been working on a related 
> >> problem for 300 years, no one has solved it yet. 
> >> 
> >> 
> >> Jason 
> > 
> > 
> > I was asking about a well-defined mathematical function that can be 
> written 
> > in closed form, or possibly as an infinite series. I believe that all 
> such 
> > functions are computable. I was not discussing subroutines that might 
> never 
> > terminate. If all well defined mathematical functions are computable, 
> why 
> > did computability become a big deal? AG 
>
> It is not true that all well-defined functions are computable. You 
> have already been given examples by Jason and John of well-defined 
> mathematical functions that are non-computable. 
>
> 

Re: Why is Church's thesis a Miracle?

[Bruno Marchal]

Grayson, people,

Unlike the combinators (which is in part something asked by some of my 
students), in they thread, I will proceed only if at least one of you asses he 
understood the theorems and proofs.

Grayson, it seems you are not aware of Cantor Theorem. I do not need that 
theorem. I gave a proof just to illustrate the use of the diagonal method. I 
give a different proof here. I show that the set of functions from N to N (all 
of them, computable or not) is NOT enumerable. It means that there is no 
bijection between that set of functions with the set N (N = {0, 1, 2, 3, …}.

Obviously, there is no bijections between any finite set and N. OK?

But up to Cantor, people thought it would be obvious that there is a bijection 
between any infinite sets. Galilee and Gauss saw the natural bijection, sending 
n to (2 x n), between N and the set of even natural numbers:

0 ———  0
1 ———  2
2 ———  4
3 ———  6
4 ———  8
5 ———  10
6 ———  12
...

That is a bijection between N and 2xN (where “2xN” is a (common) notation for 
the set of even number.).

Seeing this both Galilee and Gauss concluded that infinite objects are too much 
weird to be accepted as proper mathematical citizens. 

Eventually, in case you are very patient, I can explain that Mechanism 
ultimately side with Galilee and Gauss, and the “modern” intuitionist on this, 
yet without endangering the existence of "Cantor Paradise”, as Hilbert called 
the set theoretical view of the whole of mathematics envisaged by Cantor.

Cantor will also, and at first, show that there is a bijection between many 
infinite set. I use (A x B) for the et of all couples (x, y) with x in A and y 
in B. NxN is the set of all couples of natural numbers. OK. “x” is associative, 
so we can consider NxNxN, the set of all triples. All those sets have been 
shown enumerable by Cantor, and that can be shown easily (with or without the 
theorem below, that the set of all finite words build on a finite alphabet is 
enumerable. 

He showed that the rational numbers are enumerable, but eventually will 
discover that some sets are infinite, but so much big that there were no 
bijection possible between them and N. He discovered the existence of infinite 
set which were not enumerable.

I illustrate with the set of all functions from N to N. It is written N^N, 
because you can see by yourself that if A and B are finite sets, and if #A is 
the number of elements in A, then the number of elements in A^B, that is #(A^B) 
is equal to (#A)^(#B).


Cantor theorem.  There is no bijection between N and N^N.

Proof, by reduction ad absurd. Suppose that there is a bijection b between N 
and the set of all functions from N to N. That means that there is a 
enumeration of those functions: f_0, f_1, f_2, …. 

Now define g by g(n) = f_n(n) + 1. That means, g, applied on n has the well 
defined value of the nth function in the list (of all functions) applied on the 
number n, 

But if all functions are in the list, it means that g is equal to some f_k in 
the list. OK?

That means, g(n) = f_k(n) for all n.

But g has been defined, on each n, by g(n) = f_n(n) + 1. In particular, on the 
number k, we have both 

 g(k) = f_k(k)

 g(k) = f_k(k) + 1

By Leibniz identity rule, this entails

f_k(k) = f_k(k) + 1

As all f_k are functions from N to N, f_k(k) is a number, and we can subtract 
it at the left and right side of the equation above, and we get:

0 = 1. 

Contradiction. So there cannot be any bijection between N and N^N.


OK?

No need of philosophy. At Cantor’s time, that reasoning was pretty courageous. 
Intuitive set theory was full of paradoxes, many found by Cantor. Frege first 
quasi-formal theory was shown inconsistent. Today that proof can be formalised 
in many different set theories. Those theories are interesting, but that does 
not mean we have to believe in them. Eventually mathematical logic will show 
that cardinality (the size of a set) is a relative notion. 

But it is good to understand Cantor use of the diagonal at least the main idea. 


I suggest, to better see this, that you draw a finite approximation of the 
matrix of all f_i(j) which gives the values of all functions on all arguments, 
so that to see that the diagonal terms f_i(i) describes a geometrical diagonal, 
and that the contradiction is obtained at the crossing of that diagonal with 
the horizontal of the values of the function g = f_k.

Ask anything. 

The next proof will lead to more concrete things, like the discovery of the 
universal machines---the creative bombs, God’s terrible children (they do put 
some mess in Plato Heaven). 

Grayson, you can also tell me that you are not interested. No problem with 
that. Just avoid insulting me when I suggest that the physical reality might be 
of a type of dream(s), because I justify this from the mathematical study of 
the universal machine, in the context of a precise hypothesis in the cognitive 
science. 
It is almost obvious when you assu

Re: Church-Turing Thesis

[Bruno Marchal]

> On 24 Aug 2018, at 14:24, Telmo Menezes  wrote:
> 
> On 23 August 2018 at 06:31,   wrote:
>> 
>> 
>> On Thursday, August 23, 2018 at 2:01:24 AM UTC, Jason wrote:
>>> 
>>> 
>>> 
>>> On Wed, Aug 22, 2018 at 4:43 PM  wrote:
 
 
 
 On Tuesday, August 21, 2018 at 3:22:04 PM UTC, Jason wrote:
> 
> 
> 
> On Tue, Aug 21, 2018 at 1:16 AM  wrote:
>> 
>> I've been looking at the Wiki article on this topic. I find that I
>> really don't understand what it is, or why it's important. Maybe a few
>> succinct words from the usual suspects can be of help. TIA.
>> 
>> 
> 
> 
> Bruno provided a great definition and background of the Church-Turing
> Thesis. I will try to answer why it is important and comes up often in our
> discussion.
> 
> 
> The Church-Turing thesis says that anything that is computable is
> computable by any computer.  In other words, there is nothing that the
> computer in your cell phone can't compute, that your laptop or that a 
> super
> computer (or even a quantum computer) can.  It just comes down to having
> enough time and memory.
> 
> This is why you don't need to buy a new phone with new hardware every
> time you want to install a new app.  Regardless of the type of CPU in your
> phone, it can be extended in its power of what it might compute only given
> some new software.  It is in this sense that computers are "Universal", 
> they
> are universal in the same sense that of a universal remote, or in the 
> sense
> that a record player is a universal sound imitating device.  A record 
> player
> might emulate the sounds of an orchestra, Britney Spears, whale songs, 
> etc.,
> all it needs is the appropriate record and it can produce the sound.
> 
> In the same sense, all a Turing Machine (computer) needs to imitate (or
> emulate) the right program or function is the right software.  Because of
> this, anything that can be described in software, be it a brain emulation,
> an AI, a virtual environment, a virtual machine or operating system, can
> never know what hardware is running it, because the Church-Turing thesis
> says that any computer is capable of running it.
> 
> This is why if consciousness is computable (the computational theory of
> mind) we cannot know what is computing us (e.g. we could be in a matrix 
> type
> simulation for all we know).  The other implication is that if 
> computations
> exist in mathematics (and they do), then we exist within mathematics.
> Mathematics (or at least the part necessary to describe computations)
> becomes the fundamental science of what we experience and what is possible
> to experience or what we may predict about our future experiences 
> (physics).
> 
> 
> Jason
 
 
 If someone digitizes (emulates) the Mona Lisa, is this equivalent to the
 Mona Lisa?
>>> 
>>> 
>>> If you digitize a person and put the digitized Mona Lisa before them, it
>>> is equivalent to the real Mona Lisa to that person, at least as far as they
>>> can tell.
>>> 
>>> 
 
 Can you write a function which is not computable? AG
 
 
>>> 
>>> If by not computable you mean it never returns, then this is easy:
>>> 
>>> function foo():
>>>  while (true)
>>>  {
>>> // loop forever
>>>  }
>>> 
>>> There are also programs for which no one knows if they are computable or
>>> not.  If you can prove whether or not this function ever completes, you will
>>> be world famous, and may even earn a million dollars (though I think the
>>> prize has been retracted, it might be oferred again):
>>> 
>>> Step 1: Set X = 4
>>> Step 2: Set R = 0
>>> Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set R = 1
>>> Step 4: If R = 1, Set X = X + 2 and go to Step 2
>>> Step 5: If R = 0, print X and halt
>>> 
>>> All you have to prove is the computer either never gets to step 5 or that
>>> it does get to step 5.  Mathematicians have been working on a related
>>> problem for 300 years, no one has solved it yet.
>>> 
>>> 
>>> Jason
>> 
>> 
>> I was asking about a well-defined mathematical function that can be written
>> in closed form, or possibly as an infinite series. I believe that all such
>> functions are computable. I was not discussing subroutines that might never
>> terminate. If all well defined mathematical functions are computable, why
>> did computability become a big deal? AG
> 
> It is not true that all well-defined functions are computable. You
> have already been given examples by Jason and John of well-defined
> mathematical functions that are non-computable.
> 
> You seem to confuse "well-defined" with "written in closed form". The
> latter is not even well-defined (heheh) because it hangs on the idea
> of a set of "well-known" functions, and people already have different
> ideas on what that set inclu

Re: Church-Turing Thesis

[Telmo Menezes]

On 23 August 2018 at 06:31,   wrote:
>
>
> On Thursday, August 23, 2018 at 2:01:24 AM UTC, Jason wrote:
>>
>>
>>
>> On Wed, Aug 22, 2018 at 4:43 PM  wrote:
>>>
>>>
>>>
>>> On Tuesday, August 21, 2018 at 3:22:04 PM UTC, Jason wrote:



 On Tue, Aug 21, 2018 at 1:16 AM  wrote:
>
> I've been looking at the Wiki article on this topic. I find that I
> really don't understand what it is, or why it's important. Maybe a few
> succinct words from the usual suspects can be of help. TIA.
>
>


 Bruno provided a great definition and background of the Church-Turing
 Thesis. I will try to answer why it is important and comes up often in our
 discussion.


 The Church-Turing thesis says that anything that is computable is
 computable by any computer.  In other words, there is nothing that the
 computer in your cell phone can't compute, that your laptop or that a super
 computer (or even a quantum computer) can.  It just comes down to having
 enough time and memory.

 This is why you don't need to buy a new phone with new hardware every
 time you want to install a new app.  Regardless of the type of CPU in your
 phone, it can be extended in its power of what it might compute only given
 some new software.  It is in this sense that computers are "Universal", 
 they
 are universal in the same sense that of a universal remote, or in the sense
 that a record player is a universal sound imitating device.  A record 
 player
 might emulate the sounds of an orchestra, Britney Spears, whale songs, 
 etc.,
 all it needs is the appropriate record and it can produce the sound.

 In the same sense, all a Turing Machine (computer) needs to imitate (or
 emulate) the right program or function is the right software.  Because of
 this, anything that can be described in software, be it a brain emulation,
 an AI, a virtual environment, a virtual machine or operating system, can
 never know what hardware is running it, because the Church-Turing thesis
 says that any computer is capable of running it.

 This is why if consciousness is computable (the computational theory of
 mind) we cannot know what is computing us (e.g. we could be in a matrix 
 type
 simulation for all we know).  The other implication is that if computations
 exist in mathematics (and they do), then we exist within mathematics.
 Mathematics (or at least the part necessary to describe computations)
 becomes the fundamental science of what we experience and what is possible
 to experience or what we may predict about our future experiences 
 (physics).


 Jason
>>>
>>>
>>> If someone digitizes (emulates) the Mona Lisa, is this equivalent to the
>>> Mona Lisa?
>>
>>
>> If you digitize a person and put the digitized Mona Lisa before them, it
>> is equivalent to the real Mona Lisa to that person, at least as far as they
>> can tell.
>>
>>
>>>
>>> Can you write a function which is not computable? AG
>>>
>>>
>>
>> If by not computable you mean it never returns, then this is easy:
>>
>> function foo():
>>   while (true)
>>   {
>>  // loop forever
>>   }
>>
>> There are also programs for which no one knows if they are computable or
>> not.  If you can prove whether or not this function ever completes, you will
>> be world famous, and may even earn a million dollars (though I think the
>> prize has been retracted, it might be oferred again):
>>
>> Step 1: Set X = 4
>> Step 2: Set R = 0
>> Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set R = 1
>> Step 4: If R = 1, Set X = X + 2 and go to Step 2
>> Step 5: If R = 0, print X and halt
>>
>> All you have to prove is the computer either never gets to step 5 or that
>> it does get to step 5.  Mathematicians have been working on a related
>> problem for 300 years, no one has solved it yet.
>>
>>
>> Jason
>
>
> I was asking about a well-defined mathematical function that can be written
> in closed form, or possibly as an infinite series. I believe that all such
> functions are computable. I was not discussing subroutines that might never
> terminate. If all well defined mathematical functions are computable, why
> did computability become a big deal? AG

It is not true that all well-defined functions are computable. You
have already been given examples by Jason and John of well-defined
mathematical functions that are non-computable.

You seem to confuse "well-defined" with "written in closed form". The
latter is not even well-defined (heheh) because it hangs on the idea
of a set of "well-known" functions, and people already have different
ideas on what that set includes. Having well-known representations
such as sin(x) or e^x, or even x + y does not magically make the
related computations non-algorithmic. How do you think you learned how
to add, subtract, multiply and divide in basic school? Those were
algo

Re: Church-Turing Thesis

[Bruno Marchal]

> On 24 Aug 2018, at 00:53, agrayson2...@gmail.com wrote:
> 
> 
> 
> On Thursday, August 23, 2018 at 5:55:33 PM UTC, agrays...@gmail.com wrote:
> 
> 
> On Thursday, August 23, 2018 at 3:28:13 PM UTC, Brent wrote:
> Why don't we all chip in an buy Alan a computer so he can look stuff up on 
> Wikipedia.
> 
> Brent
> 
> I will when you have the courtesy to explain your contradictory statements 
> about the instantaneous, infinite extent of the wf. Oh BTW, with your big 
> brain, I suppose it never occurred to you that I wanted to hear Bruno's 
> definition, which if experience is worth anything, could be wildly DIFFERENT 
> from Wiki. While you assess all that, why don't you go fuck yourself, and 
> then tell us how it felt. OK? AG
> 
> FWIW, comparing Bruno's description with Wiki, which was my intent, confirms, 
> at least for me, that the postulates of QM are easier to understand, even 
> though many of the defining functions of a Turing Machine are known to those 
> who have programmed modern computers, notwithstanding that the latter use 
> random access memory. I don't see why the use of RAM is decisively important 
> in distinguishing a Turing Machine from how modern computers are designed. AG 
>  


You are right. You can see a von Neumann computer as a Turing machine. As the 
set of symbols, and state are arbitrary, you can even see a brain as a Turing 
machine. That is why eventually Gödel accepted Turing’s argument for the 
Church’s thesis.

Note that the wikipedia is often bad on logic or on computation, ...not 
mentioning computationalism. 

Have you understood Cantor’s diagonal proof of the non enumerability of the set 
of infinite sequence, in my yesterday post. If not I can explain again, with 
different notation. There is a real surprise at the end of that thread, you 
will see (I think and hope).

Bruno




> 
> 
> On 8/22/2018 5:58 PM, John Clark wrote:
>> On Wed, Aug 22, 2018 at 8:26 PM, > wrote:
>> 
>> >> Yes, the Busy Beaver Function is not computable. We know that:
>> 
>> BB(1) =1
>> BB(2) =6
>> BB(3) =21
>> BB(4) =107
>> 
>> > You haven't *written* the function, just its alleged values for 1,2,3,4.  
>> > What is the function? 
>>  
>> 
>> Starting with a all zero tape BB(N) is the number of operations any N state 
>> Turing Machine performs after it writes the largest number of 1's and then 
>> halts. It is very important that it halt, some machines will go on forever 
>> but they don't count. For example we know for sure that BB(5) is at least 
>> 47,176,870 because one 5 state Turing Machine has been found that halts 
>> after it goes through 47,176,870 operations (and prints 4098 1’s on the 
>> tape), but there are 28 other 5 state machines displaying non-regular 
>> behavior that are well past 47,176,870 operations and 4098 1's. If one of 
>> them eventually halts then that larger number of operations will be BB(5), 
>> if none of them ever halts then 47,176,870 really is BB(5); but the trouble 
>> is we'll never be able to know it’s 47,176,870 because we'll never know that 
>> none of those other 28 5 state machines will never halt because the Halting 
>> problem is insolvable.
>> 
>> John K Clark 
>> 
>> 
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Re: : Many-minds interpretation?

[Bruno Marchal]

> On 23 Aug 2018, at 19:49, agrayson2...@gmail.com wrote:
> 
> 
> 
> On Thursday, August 23, 2018 at 3:16:24 PM UTC, Bruno Marchal wrote:
> 
>> On 23 Aug 2018, at 02:05, Bruce Kellett > > wrote:
>> 
>> From: Bruno Marchal 
 On 22 Aug 2018, at 01:54, Bruce Kellett >>> > wrote:
 
 From: Bruno Marchal >
 
> The other sort of infinity, the one which I think you disagree with, is 
> typical for the  superposition of tensor products, like the singlet state 
> ud - du. Before measurement Alice has the same probability of finding u, 
> or d for any measurement she can do in any direction. Both Alice and Bob 
> are maximally ignorant of their possible measurement results. The MW on 
> this, or a MW way to interpret this, to keep the rotational symmetry, is 
> that we have an infinity of couples Alice+Bob, with each couple being 
> correlated.  If not, some implicit assumption is made on u and d, like it 
> is a preferred base.
 
 But the problems with any such suggestion are obvious. Firstly, Alice does 
 not choose her measurement angle in that way, so there is no 
 super-superposition created. Secondly, this construction does not restore 
 the rotational symmetry in any case. You might have an infinite number of 
 Alices, measuring the singlet at all possible angles, but that 
 multi-multiverse is not rotationally symmetric either! All it needs is for 
 Alice number 7,234,826 to poke her tongue out and the rotational symmetry 
 is lost! Of course, you could add yet more multiverses to cover every 
 possible deviation of Alice from the stationary state. But the process 
 rapidly becomes ridiculous.
 
 So this Rube Goldberg construction of additional multiverses of 
 superpositions does not actually   restore stable 
 rotational symmetry. So why propose such a construction? William of Ockham 
 will rise out of his grave to haunt you for such pointless extravagance of 
 entities!
>>> 
>>> Alice destroys the rotational symmetry in all its universe. Not of the 
>>> whole wave, where Alice does not exist as a determinate subsystem.
>> 
>> I can't really parse this. The point is that when Alice interacts with the 
>> singlet with her magnet she destroys the rotational symmetry of the state. 
>> This symmetry is not restored by considering and large system, or the whole 
>> wave. If anything, enlarging the context in this way simply lessens any 
>> symmetry that might remain.
>> 
>> I think what you have in mind is a situation such as arises if you shine a 
>> light through a small aperture. The photon emerges as a spherical wave, with 
>> the rotational symmetry of such a (hemi-)spherical wave. If there is a 
>> hemispherical screen downstream, the photon will interact with the screen at 
>> some single point. If you consider only one branch of the SWE evolution, 
>> this interaction point breaks the rotational symmetry. But if you consider 
>> all branches of the wave function together, there is a branch for every 
>> single point at which the photon can hit the screen, so that the symmetry is 
>> preserved in the wave function as a whole -- over the ensemble of all 
>> branches. But that is a situation in which the environment with which the 
>> photon interacts is itself symmetrical. If the screen, rather than being a 
>> smooth equidistant hemisphere, is just the rough walls of the laboratory, 
>> there is no symmetry in the points at which the photon can hit the walls, 
>> and the rotational symmetry is lost, even in the wave function as a whole, 
>> even by considering the superposition of all possible branches.
>> 
>> The take away message from this is that the symmetry of the original system 
>> can be lost by interaction with a non-symmetrical environment. The boundary 
>> conditions of the total system may not have the symmetries of the original 
>> state. So loss of symmetry is ubiquitous in the universe, even for 
>> Everettian no-collapse quantum mechanics. If you introduce a non-symmetrical 
>> interaction into the system, the symmetry is lost. That is all that is 
>> happening with the measurement of the spin projection of the singlet state 
>> by Alice. Your idiosyncratic interpretation of the tensor product, and your 
>> insistence the the symmetry be preserved regardless of the non-symmetrical 
>> environment, are just misguided. There is no need to try to preserve 
>> symmetry given non-symmetrical boundary conditions.
>> 
>> Since the symmetry is broken, the singlet state no longer exists in its 
>> original form, and the state that Bob measured is affected by the 
>> measurement Alice makes. There is no more to it than this. If Alice and Bob 
>> are space-like separated, there are some interpretational issues with this 
>> instantaneous influence at a distance.
> 
> Nice to hear that. It was basically my point.We have never disagreed except 
> on some definitio