Re: Are there real numbers that cannot be defined?

[Philip Thrift]

On Tuesday, March 5, 2019 at 12:23:28 AM UTC-6, Bruce wrote:
>
> On Tue, Mar 5, 2019 at 4:55 PM Russell Standish  > wrote:
>
>> On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote:
>> > On Tue, Mar 5, 2019 at 2:03 PM Russell Standish > > wrote:
>> > 
>> > You cannot represent n as a finite string for an arbitrary real 
>> number
>> > n. But you can for an arbitrary integer n.
>> > 
>> > 
>> > Sure. But that was not part of your definition of a 'computation'. The
>> > algorithm f(x): (r-1)+1 works for all reals r as well as for finite 
>> strings n.
>> > 
>> > Bruce
>>
>> I don't think it's 'my definition'. The usual meaning of computable
>> integer is that there exists a program that outputs it. For real
>> numbers, this is changed to a program exists that outputs a sequence
>> of numbers that converges to the real number in question. One could
>> also consider "spigot" programs for this purpose too - a program that
>> outputs the decimal (or binary say) expansion of the real number. It
>> is clear that this more relaxed definition is equivalent to the former
>> in the integer case.
>>
>
> It seems that you are relying on the idea of 'computable' as capable of 
> being calculated in a finite number of steps on a finite Turing machine. 
> That is fine; it then rules out functions such as (r-1)+1 for reals, since 
> these are not representable on a finite Turing machine. But it also renders 
> the concept of a computable number completely trivial, and all you are left 
> with for the Church-Turing thesis is the concept of computable functions 
> which are non-trivial in the sense that the function cannot depend ab 
> initio on the output number.
>
> Bruce
>


*What is computable *has its traditional computability theory 
[ https://en.wikipedia.org/wiki/Computability_theory ] answers, but outside 
of this, the more empirical, open-ended approaches.

First, there is the use of *futures and promises *[ 
https://en.wikipedia.org/wiki/Futures_and_promises 
].

Another, *persistent Turing machines* 
[ 
https://www.researchgate.net/publication/225181994_Persistent_Turing_Machines_as_a_Model_of_Interactive_Computation
 
].

Those are the two that come to mind right now.

- pt

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Re: Are there real numbers that cannot be defined?

[Bruce Kellett]

On Tue, Mar 5, 2019 at 4:55 PM Russell Standish 
wrote:

> On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote:
> > On Tue, Mar 5, 2019 at 2:03 PM Russell Standish 
> wrote:
> >
> > You cannot represent n as a finite string for an arbitrary real
> number
> > n. But you can for an arbitrary integer n.
> >
> >
> > Sure. But that was not part of your definition of a 'computation'. The
> > algorithm f(x): (r-1)+1 works for all reals r as well as for finite
> strings n.
> >
> > Bruce
>
> I don't think it's 'my definition'. The usual meaning of computable
> integer is that there exists a program that outputs it. For real
> numbers, this is changed to a program exists that outputs a sequence
> of numbers that converges to the real number in question. One could
> also consider "spigot" programs for this purpose too - a program that
> outputs the decimal (or binary say) expansion of the real number. It
> is clear that this more relaxed definition is equivalent to the former
> in the integer case.
>

It seems that you are relying on the idea of 'computable' as capable of
being calculated in a finite number of steps on a finite Turing machine.
That is fine; it then rules out functions such as (r-1)+1 for reals, since
these are not representable on a finite Turing machine. But it also renders
the concept of a computable number completely trivial, and all you are left
with for the Church-Turing thesis is the concept of computable functions
which are non-trivial in the sense that the function cannot depend ab
initio on the output number.

Bruce

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Re: Are there real numbers that cannot be defined?

[Russell Standish]

On Tue, Mar 05, 2019 at 02:22:05PM +1100, Bruce Kellett wrote:
> On Tue, Mar 5, 2019 at 2:03 PM Russell Standish  wrote:
> 
> On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote:
> >
> > My problem with your idea that the function: "(n-1)+1" is a valid
> computational
> > algorithm for n is that it makes all real numbers also computable, but
> the
> > notion of Turing computability applies only to the integers. We do not
> want a
> > definition of 'computable', that makes all reals computable.
> 
> You cannot represent n as a finite string for an arbitrary real number
> n. But you can for an arbitrary integer n.
> 
> 
> Sure. But that was not part of your definition of a 'computation'. The
> algorithm f(x): (r-1)+1 works for all reals r as well as for finite strings n.
> 
> Bruce

I don't think it's 'my definition'. The usual meaning of computable
integer is that there exists a program that outputs it. For real
numbers, this is changed to a program exists that outputs a sequence
of numbers that converges to the real number in question. One could
also consider "spigot" programs for this purpose too - a program that
outputs the decimal (or binary say) expansion of the real number. It
is clear that this more relaxed definition is equivalent to the former
in the integer case.

It is clear that all integers are computable according to the above,
and that nearly all real numbers aren't (by virtue of there only being
a countable number of programs).

Cheers

-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: Are there real numbers that cannot be defined?

[Bruce Kellett]

On Tue, Mar 5, 2019 at 2:03 PM Russell Standish 
wrote:

> On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote:
> >
> > My problem with your idea that the function: "(n-1)+1" is a valid
> computational
> > algorithm for n is that it makes all real numbers also computable, but
> the
> > notion of Turing computability applies only to the integers. We do not
> want a
> > definition of 'computable', that makes all reals computable.
>
> You cannot represent n as a finite string for an arbitrary real number
> n. But you can for an arbitrary integer n.
>

Sure. But that was not part of your definition of a 'computation'. The
algorithm f(x): (r-1)+1 works for all reals r as well as for finite strings
n.

Bruce

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Re: Are there real numbers that cannot be defined?

[Russell Standish]

On Tue, Mar 05, 2019 at 12:06:00PM +1100, Bruce Kellett wrote:
> 
> My problem with your idea that the function: "(n-1)+1" is a valid 
> computational
> algorithm for n is that it makes all real numbers also computable, but the
> notion of Turing computability applies only to the integers. We do not want a
> definition of 'computable', that makes all reals computable.

You cannot represent n as a finite string for an arbitrary real number
n. But you can for an arbitrary integer n.


-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: Are there real numbers that cannot be defined?

[Bruce Kellett]

On Tue, Mar 5, 2019 at 11:50 AM Russell Standish 
wrote:

> On Tue, Mar 05, 2019 at 10:42:00AM +1100, Bruce Kellett wrote:
> > On Tue, Mar 5, 2019 at 10:25 AM Russell Standish 
> wrote:
> >
> > On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote:
> > > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal 
> wrote:
> > >
> > >
> > > >> I don't follow you. If the 8000th BB number is
> unknowable then
> > it is
> > > certainly uncomputable
> > >
> > >
> > > > That is not true. All natural number n are computable. The
> program
> > is
> > > “output n”.
> > >
> > >
> > > I think you're being silly. You're saying if you already know that
> the
> > answer
> > > to a problem is n then you can write a program that will "compute"
> the
> > answer
> > > with just a "print n" command. But that's not computing that's just
> > printing.
> >
> > OK, so what about the program "print X+1", where X is the expansion
> of
> > the number BB(8000)-1?
> >
> > If that's not computing something, then I'm sure I can cook up
> > something more complicated to compute.
> >
> >
> > I think the trouble with that, or with variations of that idea, is that
> they
> > render the notion of 'computability' vacuous. In order to write such a
> program,
> > or concoct such an algorithm, you need to know the answer in advance.
> That is
> > fine, if you just want a program to compute the number 'n', 'n' being
> given in
> > advance. But that is no help in computing a number that can be defined,
> but is
> > not known in advance.
> >
> > So what people are really looking for here is a constructive notion of
> > computability -- anything else has a tendency to render the notion of
> > 'computability' trivial.
> >
>
> Not really, as it makes a distinction with respect to real
> numbers. All integers are computable, but no real numbers are except
> for a set of measure zero. And then there are well defined numbers
> that aren't computable, such as Chaitin's Omega.
>

My problem with your idea that the function: "(n-1)+1" is a valid
computational algorithm for n is that it makes all real numbers also
computable, but the notion of Turing computability applies only to the
integers. We do not want a definition of 'computable', that makes all reals
computable.

Bruce

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Re: Are there real numbers that cannot be defined?

[Russell Standish]

On Tue, Mar 05, 2019 at 10:42:00AM +1100, Bruce Kellett wrote:
> On Tue, Mar 5, 2019 at 10:25 AM Russell Standish  
> wrote:
> 
> On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote:
> > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  wrote:
> >
> >
> >         >> I don't follow you. If the 8000th BB number is unknowable 
> then
> it is
> >         certainly uncomputable
> >
> >
> >     > That is not true. All natural number n are computable. The program
> is
> >     “output n”.
> >
> >
> > I think you're being silly. You're saying if you already know that the
> answer
> > to a problem is n then you can write a program that will "compute" the
> answer
> > with just a "print n" command. But that's not computing that's just
> printing.
> 
> OK, so what about the program "print X+1", where X is the expansion of
> the number BB(8000)-1?
> 
> If that's not computing something, then I'm sure I can cook up
> something more complicated to compute.
> 
> 
> I think the trouble with that, or with variations of that idea, is that they
> render the notion of 'computability' vacuous. In order to write such a 
> program,
> or concoct such an algorithm, you need to know the answer in advance. That is
> fine, if you just want a program to compute the number 'n', 'n' being given in
> advance. But that is no help in computing a number that can be defined, but is
> not known in advance.
> 
> So what people are really looking for here is a constructive notion of
> computability -- anything else has a tendency to render the notion of
> 'computability' trivial.
>

Not really, as it makes a distinction with respect to real
numbers. All integers are computable, but no real numbers are except
for a set of measure zero. And then there are well defined numbers
that aren't computable, such as Chaitin's Omega.


-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: Are there real numbers that cannot be defined?

[Russell Standish]

On Mon, Mar 04, 2019 at 06:48:19PM -0500, John Clark wrote:
> 
> On Mon, Mar 4, 2019 at 6:25 PM Russell Standish  wrote:
>  
> 
> > OK, so what about the program "print X+1", where X is the expansion of
> the number BB(8000)-1?
> 
> 
> Well what about it? If you don't know what BB(8000) is, and you don't and
> neither does God, then neither of you will ever know what BB(8000)-1 is.
> 
> John K Clark

That is why I said the number is unknowable, rather than uncomputable.


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Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: Are there real numbers that cannot be defined?

[John Clark]

On Mon, Mar 4, 2019 at 6:25 PM Russell Standish 
wrote:


> >
> *OK, so what about the program "print X+1", where X is the expansion of
> the number BB(8000)-1?*
>

Well what about it? If you don't know what BB(8000) is, and you don't and
neither does God, then neither of you will ever know what BB(8000)-1 is.

John K Clark


>

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Re: Recommend this article, Even just for the Wheeler quote near the end

[Brent Meeker]

On 3/4/2019 3:54 AM, Bruno Marchal wrote:


On 3 Mar 2019, at 20:43, Brent Meeker > wrote:




On 3/3/2019 4:52 AM, Philip Thrift wrote:




Here's an example David Wallace presents (as an "outlandish" 
possibility): Suppose in *pi *(which is computable, so has a 
/program/ (a spigot one, in fact) that produces its digits. Suppose 
somewhere in that stream of digits is the Standard Model Equation


    (say written in LaTeX/Math but rendered here)
https://www.sciencealert.com/images/Screen_Shot_2016-08-03_at_3.20.12_pm.png

So what could this mean? (He sort of leaves it hanging.)



Nothing.  Given a suitable mapping the SM Lagrangian can be found in 
any sequence of symbols.  It's just a special case of the rock that 
computes everything.


Even if rock would exist in some primitive sense, which I doubt, they 
do not compute anything, except in a trivial sense the quantum state 
of the rock. A rock is not even a definable digital object.


It's an ostensively definable object...which is much better.

If someone want to convince me that a rock can compute everything, I 
will ask them to write a complier of the combinators, say, in the 
rock. I will ask an algorithm generating the phi_i associated to the rock.


There is no particular phi_i associated to the rock.  That's the point.  
The rock goes thru various states so there exists a mapping from that 
sequence of states to any computation with a similar number of states.  
Of course one may object that the actual computation is in the 
mapping...but that's because of our prejudice for increasing entropy.


Brent

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Re: Are there real numbers that cannot be defined?

[Bruce Kellett]

On Tue, Mar 5, 2019 at 10:25 AM Russell Standish 
wrote:

> On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote:
> > On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  wrote:
> >
> >
> > >> I don't follow you. If the 8000th BB number is unknowable
> then it is
> > certainly uncomputable
> >
> >
> > > That is not true. All natural number n are computable. The program
> is
> > “output n”.
> >
> >
> > I think you're being silly. You're saying if you already know that the
> answer
> > to a problem is n then you can write a program that will "compute" the
> answer
> > with just a "print n" command. But that's not computing that's just
> printing.
>
> OK, so what about the program "print X+1", where X is the expansion of
> the number BB(8000)-1?
>
> If that's not computing something, then I'm sure I can cook up
> something more complicated to compute.
>

I think the trouble with that, or with variations of that idea, is that
they render the notion of 'computability' vacuous. In order to write such a
program, or concoct such an algorithm, you need to know the answer in
advance. That is fine, if you just want a program to compute the number
'n', 'n' being given in advance. But that is no help in computing a number
that can be defined, but is not known in advance.

So what people are really looking for here is a constructive notion of
computability -- anything else has a tendency to render the notion of
'computability' trivial.

Bruce

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Re: Are there real numbers that cannot be defined?

[Russell Standish]

On Mon, Mar 04, 2019 at 05:31:00PM -0500, John Clark wrote:
> On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  wrote:
> 
> 
> >> I don't follow you. If the 8000th BB number is unknowable then it 
> is
> certainly uncomputable
> 
> 
> > That is not true. All natural number n are computable. The program is
> “output n”.
> 
> 
> I think you're being silly. You're saying if you already know that the answer
> to a problem is n then you can write a program that will "compute" the answer
> with just a "print n" command. But that's not computing that's just printing.

OK, so what about the program "print X+1", where X is the expansion of
the number BB(8000)-1?

If that's not computing something, then I'm sure I can cook up
something more complicated to compute.


-- 


Dr Russell StandishPhone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellowhpco...@hpcoders.com.au
Economics, Kingston University http://www.hpcoders.com.au


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Re: Are there real numbers that cannot be defined?

[John Clark]

On Mon, Mar 4, 2019 at 11:04 AM Bruno Marchal  wrote:

>> I don't follow you. If the 8000th BB number is unknowable then it is
>> certainly uncomputable
>
>
> *> That is not true. All natural number n are computable. The program is
> “output n”.*
>

I think you're being silly. You're saying if you already know that the
answer to a problem is n then you can write a program that will "compute"
the answer with just a "print n" command. But that's not computing that's
just printing.

Incidentally very recently Stefan O’Rear has reduced Aaronson' s 7918 number so
now we know that BB(1919) is not computable.

So we know that:
• BB(1)=1
• BB(2)=6
• BB(3)=21
• BB(4)=107

and that's all we know for sure, but we do know some lower bounds:

• BB(5) ≥ 47,176,870
• BB(6) ≥ 7.4 *10^36534
• BB(7) >10^((10^10)^(10^10)^7)

> *BB(n) is not computable means that there is no algorithm, which given n,
> will give BB(n).*
>

Yes, so what are we arguing about?


> > *what Aaronson has shown, is that above 7918, we loss any hope to find
> it by using the theory ZF. But may be someone will find it by using ZF +
> kappa, which is much more powerful that ZF,*
>

It's easy to find a system of axioms more powerful than ZF, the problem is
it may be so powerful it can even prove things that aren't true. Would you
really trust a system that claimed to be able to solve the Halting Problem?
I certainly wouldn't! And if you can't solve the Halting Problem then there
is absolutely no way to calculate BB(7918) or BB(1919) and I wouldn't be
surprised if even BB(5) is out of reach.

> There are only 2 possibilities, a program will halt after a finite number
>> of steps or it won’t.
>
>
> > Yes. But the program which computes BB(n) always stop.
>

if it stops then it is successful but if n is 1919 then it never stops so
BB(1919) is never computed.


>>I would maintain that you haven't solved a problem if you can't give the
>> right answer more often than random guessing would.
>
>
> *> You are restricting computability to a string notion of intutionistic
> computability. *
>

What grade would you give a student who couldn't answer your questions
better than you'd expect from random guessing?

John K Clark

>
>

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Re: Are there real numbers that cannot be defined?

[Philip Thrift]

On Monday, March 4, 2019 at 12:00:05 PM UTC-6, John Clark wrote:
>
>
>
> And proof is not truth.
>
... 

>  
>
John K Clark
>
>
>
>>
Of course *truth == proof *in the land of radical 
intuitionists-constructivists.

(And what is proof anyway?)


From: Doren Zeilberger
To: Scott Aaronson
[ http://sites.math.rutgers.edu/~zeilberg/Opinion155.html ]

...

As I have said before, there is a quick dictionary to turn all this 
undecidability babble and the obsession with related problems, like the *"busy 
beaver"*, into purely meaningful, albeit uninteresting, statements. Every 
statement that involves quantifies over "infinite" sets, even such a 
"trivial" statement like

n+1=1+n , for EVERY natural number n   ,

(tacitly assuming that you have an "infinite" supply of them) is a priori 
meaningless, but many of them (including the above, and the statement that 
"for all" integers x,y,z > 0 and n > 2 , xn+ yn -zn < > 0) can be made a 
posteriori meaningful, by proving them for symbolic n (and x,y,z). So the 
right dictionary (for statements that involve quantifies over "infinite" 
sets)

Provable : a priori meaningless (taken literally), but a posteriori 
meaningful, when interpreted correctly (for symbolic n)

Undecidable: not even a posteriori meaningful, impossible to make sense of 
it symbolically
So, like the proof that the square-root of two is irrational, Gödel and 
Turing did prove something seminal, but it was a negative result, that they 
(and you, and unfortunately so many, otherwise smart, people), in their 
naive platonism, interpret in a wrong way. So the initial "paradox" was 
very interesting, but all the subsequent "busy beaver" bells and whistles, 
is just a meaningless game.

I am not saying that you are not brilliant, you sure are (and you are also 
a brilliant speaker, as I found out from your stimulating and engaging talk 
at AviFest last week), but you are wasting your talent on uninteresting 
research. Perhaps even worse than "undecidability" is your main research 
area on "quantum computing", that once again is a challenging intellectual 
mathematical game, but with empty content. The history of science and 
mathematics is full of people who had superstitious beliefs: Kepler 
believed in Astrology, Newton in Alchemy, but they did many other things 
besides. The great debunker, Gil Kalai, (who debunked the Bible Code, along 
with co-debunkers Dror Bar-Natan and Brendan McKay), has recently pointed 
out (unfortunately in his understated, gentle, way) the shortcomings of 
research in "quantum computing", and my impression is that he is right. It 
is indeed amazing how in our current "enlightened" age, that (allegedly) 
abhors superstition, such superstitious people as you (and many other, e.g. 
MIT cosmologist, Max Tegmark, another admittedly brilliant, but 
nevertheless superstitious, scientist) can be full professors at MIT.

But then again, it supplies some comic relief, and some of us still enjoy 
Mythology and Theology, but it is not nice to be dismissive of people who 
do not share your superstitions.


- pt

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Re: When Did Consciousness Begin?

[Brent Meeker]

On 3/4/2019 3:45 AM, Bruno Marchal wrote:

  Unconsciousness is an illusion of consciousness … It should be obvious that 
“being unconscious” cannot be a first person experience, for logical reason. To 
die is not a personal event. That happens only to the others.

I agree.  Except I don't suppose that all events are personal.

Brent

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Re: When Did Consciousness Begin?

[Brent Meeker]

On 3/4/2019 3:45 AM, Bruno Marchal wrote:

   I have had two relatives die of Alzheimers and they lost their identity 
gradually as they lost memory.

They lost they memory. Not their identity, but the apprehension of their 
identity. If not, when you ask where they are in the hospital, the nurse would 
say “what are you talking about”. Even a corpse has an identity.

At last you recognize the importance of the material.

Brent

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Re: When Did Consciousness Begin?

[Brent Meeker]

On 3/4/2019 3:32 AM, Bruno Marchal wrote:


On 3 Mar 2019, at 20:15, Brent Meeker > wrote:




On 3/3/2019 3:34 AM, Bruno Marchal wrote:


On 1 Mar 2019, at 21:36, Brent Meeker > wrote:




On 3/1/2019 7:08 AM, Bruno Marchal wrote:


On 28 Feb 2019, at 22:47, Brent Meeker > wrote:




On 2/28/2019 1:17 PM, Philip Thrift wrote:



The best current philosopher of (and writer about) consciousness 
is *Galen Strawson*.


https://en.wikipedia.org/wiki/Galen_Strawson
https://sites.google.com/site/galenstrawson/
https://liberalarts.utexas.edu/philosophy/faculty/profile.php?id=gs24429

There is a lot of his material (PDFs, articles, videos, etc.) 
freely available online.


The main word that is synonymous with /consciousness /is 
/experience/.


Which is something bacteria and plants and my thermostat 
have...and ability to detect and react to the environment based 
on internal states.


What the thermostat lacks, that the bacteria and plants do not 
lack, is Turing universality. That gives the mind, and even the 
free-will.


A bacterium doesn't have Turing universiality, only bacteria in the 
abstract of a potentially infinite set of evolving bacteria 
interacting with their environment.  But if a consider a 
potentially infinite set of thermostats interacting with their 
environment of furnaces and rooms, it will be Turing universal too. 
Turing universality is cheap.


Yes, it is cheap, like consistency, and plausibly consciousness.

But it is more cheap you might think, because even one bacteria is 
fully Turing universal. The genome of Escherichia Coli can be 
“programmed” to run a Turing universal set of quadruplet. Of course, 
the bacteria’s “tape” is quite limited, and they can exploit their 
universality only by cooperation in the long run, and so no 
individual bacteria can be self-conscious or Löbian.


I think that's what I said.  Except I also noted that all this 
requires an environment within which the bacteria can metabolize.


That is contingent with respect of the bacteria “mental life”. All 
programs needs a code, and an environment which run it, but it can be 
arithmetic. Then a physical reality emerges as a means on all 
accessible computations-continuations.


Mentioning the environment can be misleading. If a material 
environment is needed, matter would play some role, and there is no 
more reason to say accept a digital, even if physical, brain.


I didn't say anything about the environment being "material".    But 
your objection seems to reduce to, "But that's contrary to my theory."  
It's no good saying your theory is testable when you only test it within 
the assumptions you used to derive it.


In a dream, we create more clearly the environment by ourself, and 
that is enough for being conscious, or even self-conscious, like in a 
lucid dream, or a sophisticated virtual environment.


The dream is realized by the brain and it is about elements of our real 
environment.






So to say bacteria have Turing universality is like saying water is 
lemonade...if you add lemons and sugar.


It means that with the 4 letters, you can program any partial 
recursive function. Of course you need the decoding apparatus, but 
that is entirely in the bacteria. It means that you can simulate any 
other computer, with a basic set of DNA-enzyme molecular interaction. 
A universal machine is just a number u such that for all x and y 
phi_u(x, y) = phi_x(y) *in principle. You can implement all control 
structure. The operon illustrates a “if-then-else”, and the regulation 
apparatus is enough to get universality. René Thomas, in Brussels, has 
succeeded to make a loop, with a plasmid (little circular gene) 
entering in the bacterium, and then going out, repetitively. It is 
even a “fuzzy computer”. Some product are regulated in a continuum, 
depending on the concentration of the metabolites. When I was young, I 
have made e project for a massively parallel computers which was a 
solution of bacteria (E. coli) and bacteriophage. One drop of it could 
process billions of instructions in a second. But the read and write 
was demanding highly sophisticated molecular biology. I think that 
such ideas have more success today. After all, molecular biology 
studies “natural nanotechnology”.


You're wrong.  The environment is essential.  The fact that DNA can 
encode functions means nothing without the ability to read and execute 
the code.  RNA, proteins, krebs cycle, and proton pumps are all 
necessary for that.


Brent

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Re: Are there real numbers that cannot be defined?

[John Clark]

On Mon, Mar 4, 2019 at 7:56 AM Bruno Marchal  wrote:


> *> BB(7918) is just a number, say k, and the program “print k” will do.*
>

Unless God wrote the program the output would just be "k" or perhaps
"BB(7918)".
And as I said before unlike most Real Numbers Busy Beaver Numbers can be
named but just like most Real Numbers they can not be computed.

> I* doubt Scaott Aaronson makes such a big mistake. B(7918), is
> computable.*
>

And I'm flabbergasted you don't understand that you can't compute BB(7918)
unless you know how to solve the Halting Problem. And I don't know how to
do that and you don't either.


> *> they show is that B(7918) = k might be unprovable in ZFC. **That is
> not equivalent with B(7918) is not computable.*
>

You confuse provability with soundness. Any proof is only as good as the
axioms it is based on and almost all of mathematics is based on ZFC because
although incomplete it is believed to be consistent and most mathematicians
believe any alteration in ZFC would reduce its "Soundness", that is to say
it's ability to preserve the truth. It would be easy to prove the Goldbach
Conjecture is true if you just changed the ZFC axioms; but what do you do
after that if a computer cranks out a huge even number that is not the sum
of 2 primes?  Without soundness doing mathematics would be no more profound
than doing a crossword puzzle.


> *>  It is a natural number. All constant function are computable. *
>

Let's look at something simpler, the 5th busy beaver number.  We know for
certain there is one 5 state Turing Machine that stops after 47,176,870
operations and there are five other machines that are still going past that
point. If one of those five machines eventually halts then we'd know 47,176,870
is *NOT* BB(5) but we still wouldn't know what BB(5) is unless all five
machines halt. And if one of them never halts then Turing proved we can't
know it won't halt so we'll just be sitting there waiting for it to stop
and we'll be waiting forever and never know what BB(5) is even though the
number exists is finite and is well defined.

>> The 8000th Busy Beaver Number is the largest number of FINITE operations
>> a 8000 state Turing Machine will make before it halts. Some programs we can
>> observe halting and with others it's easy to prove will never halt, that's
>> why we know the first 4 Busy Beaver Numbers, but Turing Proved you can't do
>> that in general and  Aaronson proved you can't do that for the 7918th; and
>> you probably can't even do it for the 5th.
>
>
> *> In ZF, we cannot prove that. *
>

In ZF we *CAN *prove we can never know BB(8000), and Turing proved over 80
years ago that BB(n) is unknowable for any value greater than some integer
n. Nobody knows what n is, all we is it's greater than 4 and less than 7918.


> > *But provable is not computable.*
>

And proof is not truth. If a even number that is not the sum of 2 primes
can be computed and at the same time a proof can be found that the Goldbach
Conjecture is true then we'd know that ZF is not sound. I do not expect
this to happen.


> *> Of course BB(n) is not feasible for large n.*
>

It's not a question of feasibility, technological limitations have nothing
to do with it. Even though the number exists and is finite the largest
computer in the world couldn't find BB(7918) even if you gave it infinite
time to do so, and it probably couldn't even find BB(5).

>> It is entirely possible that the 5th Busy Beaver number is  47,176,870
>> because a 5 state Turing Machine has been found that halts after 47,176,870
>> operations,  the problem is there are still 5 different 5 state turing
>> machines that are well past 47,176,870 and they have not halted. If none of
>> those 5 machines ever halts then 47,176,870 really and truly is the 5th
>> Busy Beaver Number, but if that is the case we will never know that is
>> the case because we'll never know that none of those 5 machines ever halts.
>
>
> *> No problem with this. But the never know is made precise in term of
> “provable in ZF, or not”, and that is what Aaronson wrote about.*
>

You're saying if there were an omnipotent God that was more powerful than
the laws of logic and was able to endure self contradiction and thus solve
the Halting Problem He could simply type out BB(7918) = 94722568219 or
whatever the gargantuan number is, then a program could just say "print
BB(7918)" and it would print out the correct answer provided there was
enough paper and ink in the observable universe to do so, which there
wouldn't be.

I'm saying forget about ZFC and proof, a program will either halt or it
won't and if it won't then, provided we do *NOT* have access to an
omnipotent God more powerful than the laws of logic and  able to endure
self contradiction and thus solve the Halting Problem, there is no way for
us to know if the program will halt or not and thus we have no way to
calculate BB(7918).


> > *BB(n), as function of n, is not computable, but all BB(n) is trivially
> c

Re: Recommend this article, Even just for the Wheeler quote near the end

[Bruno Marchal]

> On 4 Mar 2019, at 15:13, Philip Thrift  wrote:
> 
> 
> 
> On Monday, March 4, 2019 at 5:54:24 AM UTC-6, Bruno Marchal wrote:
> 
>> On 3 Mar 2019, at 20:43, Brent Meeker > 
>> wrote:
>> 
>> 
>> 
>> On 3/3/2019 4:52 AM, Philip Thrift wrote:
>>> 
>>> 
>>> Here's an example David Wallace presents (as an "outlandish" possibility): 
>>> Suppose in pi (which is computable, so has a program (a spigot one, in 
>>> fact) that produces its digits. Suppose somewhere in that stream of digits 
>>> is the Standard Model Equation
>>> 
>>> (say written in LaTeX/Math but rendered here)
>>>  
>>> https://www.sciencealert.com/images/Screen_Shot_2016-08-03_at_3.20.12_pm.png
>>>  
>>> 
>>> 
>>> So what could this mean? (He sort of leaves it hanging.)
>>> 
>> 
>> Nothing.  Given a suitable mapping the SM Lagrangian can be found in any 
>> sequence of symbols.  It's just a special case of the rock that computes 
>> everything.
> 
> Even if rock would exist in some primitive sense, which I doubt, they do not 
> compute anything, except in a trivial sense the quantum state of the rock. A 
> rock is not even a definable digital object. If someone want to convince me 
> that a rock can compute everything, I will ask them to write a complier of 
> the combinators, say, in the rock. I will ask an algorithm generating the 
> phi_i associated to the rock.
> 
> Bruno
> 
> 
> 
> There are "smart" rocks that do signal processing.
> 
> http://science.sciencemag.org/content/350/6258/289.4
> 

Pebbles have a very small of natural computational power, in that sense. They 
memorise events, like photon memorise his encounter with an election, by some 
change of direction. Then, if you add that pebbles interact with water, and can 
cooperates, yes, in that sense rocks can be “smart”.

In that sense, I have already demonstrate that pebbles are very intelligent, 
and wise. They never asserts stupidities, and nobody has heard a pebble lying 
for its personal benefits, nor do they hung their fellows. So much more 
intelligent than the humans. But pebbles do not compute in those simple task, 
except what I mentioned, they can bounce, and remember in that way, which makes 
a lot of pebbles capable of accomplishing some task. 

Here with have the problem to delineate the computability of the environment, 
which eventually emulates the pebble, and the computability of the pebble. When 
I say that a rock does not compute, it is a reference to Putnam’s suggestion 
that all the movement of the rocks part (the electron in the atoms) can be 
combined to obtained all computations. But that makes no sense, because the 
computation is in the logical relations, incarnate in a very special way, so 
that the computational state are related by some relation (the arithmetical 
one, and the physical one if the computation is entangled to you). That is true 
in a trivial sense, because with quantum mechanics, even the vacuum is already 
Turing universal. This is predicted by Mechanism, the closer you look at the 
possible environment, the closer you look to a universal dovetailing, or to 
*all* computations, even "the aberrant one” (those inconsistent relatively to 
your normal histories). Mechanism predict the “many-world appearance” but also 
that there is a lot of things at the bottom, virtual things of course, but 
everything is virtual with Mechanism.

It is like the question, can apes solve trigonometrical equation? You can say 
no. But some will show you that the brain of the visual cortex of the apes do 
solve trigonometrical equation. But of course, the apes is not its brain. With 
such a confusion, I could say that I have solved the protein enfolding problem, 
indeed about a billions times everyday, after they emerges from my ribosome. I 
do that /// naturally! Lol.

When talking about subject and object, there is always some ambiguity possible, 
and caution to take.

Bruno





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Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 4 Mar 2019, at 14:24, John Clark  wrote:
> 
> On Sun, Mar 3, 2019 at 6:38 PM Russell Standish  > wrote:
> 
> > ISTM that the 8000th BB number is unknowable, rather than uncomputable.
> 
> I don't follow you. If the 8000th BB number is unknowable then it is 
> certainly uncomputable


That is not true. All natural number n are computable. The program is “output 
n”.

4 is computable, by the program "output 4”.

45643900765502223 is computable, by the program “output 45643900765502223”.

Even “990801733966657343489 … 678907”, with “…” for a row of 10^(10^100) 
digits, is still computable, by the program “output “990801733966657343489 … 
678907”.

BB(n) is not computable means that there is no algorithm, which given n, will 
give BB(n).

The number BB(n), unlike the function  [n] BB(n), is computable, despite we 
cannot find constructively, mechanically, its programs.

We might say that you are using “computable” in the intuitionist sense. The 
program computing BB(n) is just a constant number, and it belongs somewhere in 
the phi_i. We cannot find it, and what Aaronson has shown, is that above 7918, 
we loss any hope to find it by using the theory ZF. But may be someone will 
find it by using ZF + kappa, which is much more powerful that ZF, and might be 
able to solve much more halting question, so that the correspond limit for ZF + 
kappa would be something like 6758918. It is sure that it will be bigger than 
7918. 
For PA, the limit can be expected to be much smaller than 7918. 



> but if it's uncomputable then it's only *probably* unknowable because you 
> could get lucky and correctly guess the truth.

BB(n), the number, is computable, because there is a problem which computes it, 
independently of me guessing it or not. The program exists, if you accept 
classical logic. Even if you will never guess it correctly, or even if you 
guess an incorrect number. That will not change the fact that the number exists 
and can, like all natural number, be computed.

That is the reason, except for Turing fundamental paper, all the theory of 
computability is based on the natural numbers, or any precise finite entities. 
It makes the primitive objects all trivially computable, and the non 
computability is only in the functions and relations.

When using the real numbers, a confusion is forced to occur, notably between 
intuitionist real numbers and classical real numbers. Then, also, a real 
numbers can be seen as a total computable function (on N, in N). But then, the 
set of such computable numbers is no more recursively enumerable, and we miss 
the partial computable functions, which are capital for the whole theory, or 
worst, we need to add a notion of real numbers having some undefined digits.

Today, we have progressed, and there are very nice notions of computable real 
numbers. Some do take them as primitive (like with an implicit oracle). That 
gives the Blum, Small and Shub theory of computable real numbers (and functions 
on them). Other just take the classical theory of real numbers, restricted to 
intuitionist mathematics, and here, some use Browuer’s intuitionism (all 
functions are computable, simply) or any of the many variant of intuitionism. 
There is no Church-Turing thesis for them, and are treated in recursion theory 
by using the analytical hierarchy, where second order logic is used (variable 
represents both number and functions).
In all reasonable theory of the reals, the computable reals are not 
mechanically enumerable, like the total functions from N to N, and so cannot be 
used for a general theory of computations. That was just not clear at Turing’s 
time, even if Turing is the guy who will benefits a lot to the development of 
computability theory. 




> So unknowable covers more ground than uncomputable
>  
> > As Bruno said, there is a program that outputs the 8000th BB number, but we 
> > can never know that this program is the correct one.
> 
> There are only 2 possibilities, a program will halt after a finite number of 
> steps or it won’t.

Yes. But the program which computes BB(n) always stop. It is a program like 

“output “990801733966657343489 … 678907”. 

Now, you cannot recognise that is the correct program. But that is frequent. By 
Rice theorem, you cannot recognise if an arbitrary program computes the 
factorial function or not. Most attributes of program are non computable, and 
most relations between programs are undecidable, in all effective theories. (A 
theory is effective if we can test mechanically the proofs). Usually theories 
are effective, but logicians have enlarged very much the notion of theory, and 
they admit non effective theory. They consider for example the set of true 
sentences of arithmetic to be a theory, but that does not concern us).



> So can you solve the halting problem by just flipping a coin?

Certainly not.


> I would maintain that you haven't solved a problem if you can't give the 
>

Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 4 Mar 2019, at 11:35, Lawrence Crowell  
> wrote:
> 
> On Saturday, March 2, 2019 at 8:28:01 PM UTC-6, John Clark wrote:
> 
> On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell  > wrote:
> 
> > There are numbers that have no description in a practical sense. The 
> > numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a vast number of 
> > numbers that have no description with any information theoretic sense.
> 
> The 8000th Busy Beaver Number can be named but not calculated even 
> theoretically, but most Real Numbers can't even be uniquely named with ASCII 
> characters, not even with an infinite number of them.  
> 
> John K Clark
> 
> There exists an uncountably infinite number of reals in the interval (0, 1), 
> and they exhaust all possible information theoretic description. Some 
> mathematicians have argued this means they do not in some ways exist. Most 
> mathematicians disagree with that by arguing computational tractability is 
> not equivalent to mathematical existence. 


Yes. Most mathematicians accept the laws of the excluded middle principle, 
which makes them tolerant to the idea that something can exist, even if we are 
unable to build it “concretely". They accept to prove a disjunction, or an 
existence (which is a sort of infinite disjunction) by reductio at absurdo. 

We can prove in three lines that

ExEy (x is irrational & y is irrational & x^y is rational).

But the three-lines proof proves only that

  (x = sqrt(2)^sqrt(2) and y = sqrt(2)) is a solution, or (x = sqrt(2) and 
y = sqrt(2)) is a solution.

Without being able to say which one(*).

Now, in this case, a constructive solution exists, but it is far more complex, 
as it uses elliptic function, modular form, and is much longer!
In many branches of math, but especially in theoretical computer science, some 
proof of existence can be proved to be necessarily non constructive. We can 
know that some machine exist, yet without ever know if we have or will met her.

An intuitionist will restricted existence to constructive proof. But it does 
not matter if the proof is tractable or not, and they share the same computable 
functiosn as the classical mathematician. Basically, they share the same 
arithmetic.

Only ultrafinitist, and perhaps engineers when they work, would disqualify the 
existence of a *computable* object if it is not tractable (the computation is 
not feasible or the proof of the existence is beyond all physical range of 
expression.

Bruno


(*) I give the proof here, as it beautiful, and so you see the importance of 
the secluded middle principle: It is used at the start, when we suppose that 
either a number is rational, or it is irrational (not rational).

Suppose sqrt(2) ^sqrt(2) is rational. Then we are done. And x = sqrt(2) and y = 
sqrt(2) is the solution.
Or sqrt(2) ^sqrt(2) is not rational, i.e. irrational. But then , in that case

sqrt(2)^sqrt(2) is irrational, and (sqrt(2)^sqrt(2))^sqrt(2) gives sqrt(2)^2 = 
2, which is rational! So we get also a solution:  x = sqrt(2)^sqrt(2) and y = 
sqrt(2).

A remark. A non constructive proof can be useful. If you are in war, and your 
secret service tells you there is a big chance that the enemy will send an 
atomic bomb on your country, and that their missile will be sent from place A 
or place B, with a non constructive or, it makes sense, to survive, to drop 
some bombs on both place A and place B. Nature made choice like that. A lion 
can kill all young lions in its neighbourhood, knowing that one of them might 
become a threat. There are certainly better example.





> 
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Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 4 Mar 2019, at 00:37, Russell Standish  wrote:
> 
> ISTM that the 8000th BB number is unknowable, rather than
> uncomputable. As Bruno said, there is a program that outputs the
> 8000th BB number, but we can never know that this program is the
> correct one.

Yes. Clark mention an interesting paper making it possible to add information 
on this, where "not knowable" becomes “not provable in ZF”. It is quite amazing 
that 7918 would be the smallest number from which the non halting becomes 
undecidable.

It should be much smaller if we use “provable in PA”. I would like to know 
which one.
It should be a very little number  for theories like RA or the SK-combinator. 
As they lack induction,  They cannot prove any non halting machine type of 
truth.

Cheers,

Bruno




> 
> 
> 
> On Sun, Mar 03, 2019 at 12:28:54PM -0500, John Clark wrote:
>> 
>> 
>> On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal  wrote:
>> 
>> 
>> 
 The 8000th Busy Beaver Number can be named but not calculated
>>even theoretically,
>> 
>>> The busy beaver function is not computable, but on each individual n, it
>>is computable theoretically,
>> 
>> 
>> No it is not, not if n= 7918, to compute that the program would have to solve
>> the Halting Problem. The first 4 Busy beaver numbers have been computed and
>> Scott Aaronson proved that the 7918th Busy Beaver Number is not computable,
>> most people think n=5 is not computable either but that has not been proved.
>> 
>> The 7918th Busy Beaver Number
>> 
>> 
>>> The 8000h BB number is well defined,
>> 
>> 
>> Yes.
>>  
>> 
>>> so it is a (finite) number,
>> 
>> 
>> Yes,
>> 
>> 
>>> and so you there exist a finite program computing it
>> 
>> 
>> No. The 8000th Busy Beaver Number is the largest number of FINITE operations 
>> a
>> 8000 state Turing Machine will make before it halts. Some programs we can
>> observe halting and with others it's easy to prove will never halt, that's 
>> why
>> we know the first 4 Busy Beaver Numbers, but Turing Proved you can't do that 
>> in
>> general and  Aaronson proved you can't do that for the 7918th; and you 
>> probably
>> can't even do it for the 5th.
>> 
>> It is entirely possible that the 5th Busy Beaver number is  47,176,870 
>> because
>> a 5 state Turing Machine has been found that halts after 47,176,870
>> operations,  the problem is there are still 5 different 5 state turing
>> machines that are well past 47,176,870 and they have not halted. If none of
>> those 5 machines ever halts then 47,176,870 really and truly is the 5th Busy 
>> Beaver Number, but if that is the case we will never know that is the case
>> because we'll never know that none of those 5 machines ever halts. 
>> 
>> John K Clark
>> 
>> 
>> 
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Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 3 Mar 2019, at 22:58, Brent Meeker  wrote:
> 
> 
> 
> On 3/3/2019 1:37 PM, John Clark wrote:
>> On Sun, Mar 3, 2019 at 2:52 PM Philip Thrift > > wrote:
>> 
>> > If a program "represents" a real number (e.g. in the spigot sense), then 
>> > that could be said to "define" it.
>> 
>> But for most Real Numbers there is no such program. 
>> 
>>  > But what does it mean for a real number to be "defined"?
>> 
>> If you can point to a property that a Real Number has that no other Real 
>> number does then it is defined;
> 
> The smallest real number that is not defined.

That is a version of Berry Paradox. Chaitin used it to give a proof of its 
incompleteness theorem (proved a long time before by Post).

It means that we do not have an all encompassing notion of “definable”, a bit 
like Truth. We can, nevertheless, define it for large class of objects.

So, here, “the smallest non definable real number” does not define a number.


Bruno



> 
> Brent
> 
>> a number like PI can be defined and you can approximated it with a 
>> calculation, a Busy Beaver Number can be defined but not approximated, most 
>> Real Numbers can not be defined or approximated and thus most Real Numbers 
>> can not be named.
>> 
>> John K Clark
>> 
>> 
>>  
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Re: Recommend this article, Even just for the Wheeler quote near the end

[Philip Thrift]

On Monday, March 4, 2019 at 5:54:24 AM UTC-6, Bruno Marchal wrote:
>
>
> On 3 Mar 2019, at 20:43, Brent Meeker > 
> wrote:
>
>
>
> On 3/3/2019 4:52 AM, Philip Thrift wrote:
>
>
>>
> Here's an example David Wallace presents (as an "outlandish" possibility): 
> Suppose in *pi *(which is computable, so has a *program* (a spigot one, 
> in fact) that produces its digits. Suppose somewhere in that stream of 
> digits is the Standard Model Equation
>
> (say written in LaTeX/Math but rendered here)
>  
> https://www.sciencealert.com/images/Screen_Shot_2016-08-03_at_3.20.12_pm.png
>
> So what could this mean? (He sort of leaves it hanging.)
>
>
> Nothing.  Given a suitable mapping the SM Lagrangian can be found in any 
> sequence of symbols.  It's just a special case of the rock that computes 
> everything.
>
>
> Even if rock would exist in some primitive sense, which I doubt, they do 
> not compute anything, except in a trivial sense the quantum state of the 
> rock. A rock is not even a definable digital object. If someone want to 
> convince me that a rock can compute everything, I will ask them to write a 
> complier of the combinators, say, in the rock. I will ask an algorithm 
> generating the phi_i associated to the rock.
>
> Bruno
>
>
>
There are "smart" rocks that do signal processing.

http://science.sciencemag.org/content/350/6258/289.4

- pt 

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Re: Are there real numbers that cannot be defined?

[John Clark]

On Sun, Mar 3, 2019 at 6:38 PM Russell Standish 
wrote:

*> ISTM that the 8000th BB number is unknowable, rather than uncomputable.*


I don't follow you. If the 8000th BB number is unknowable then it is
certainly uncomputable but if it's uncomputable then it's only *probably*
unknowable because you could get lucky and correctly guess the truth. So
unknowable covers more ground than uncomputable


> * > As Bruno said, there is a program that outputs the 8000th BB number,
> but we can never know that this program is the correct one.*
>

There are only 2 possibilities, a program will halt after a finite number
of steps or it won't. So can you solve the halting problem by just flipping
a coin? I would maintain that you haven't solved a problem if you can't
give the right answer more often than random guessing would.

John K Clark

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Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 3 Mar 2019, at 20:52, Philip Thrift  wrote:
> 
> 
> 
> On Sunday, March 3, 2019 at 11:29:32 AM UTC-6, John Clark wrote:
> 
> 
> On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal > 
> wrote:
>> 
>> >> The 8000th Busy Beaver Number can be named but not calculated even 
>> >> theoretically,
> > The busy beaver function is not computable, but on each individual n, it is 
> > computable theoretically, 
> 
> No it is not, not if n= 7918, to compute that the program would have to solve 
> the Halting Problem. The first 4 Busy beaver numbers have been computed and 
> Scott Aaronson proved that the 7918th Busy Beaver Number is not computable, 
> most people think n=5 is not computable either but that has not been proved.
> 
> The 7918th Busy Beaver Number 
> 
> > The 8000h BB number is well defined,
> 
> Yes.
>  
> > so it is a (finite) number,
> 
> Yes,
> 
> > and so you there exist a finite program computing it
> 
> No. The 8000th Busy Beaver Number is the largest number of FINITE operations 
> a 8000 state Turing Machine will make before it halts. Some programs we can 
> observe halting and with others it's easy to prove will never halt, that's 
> why we know the first 4 Busy Beaver Numbers, but Turing Proved you can't do 
> that in general and  Aaronson proved you can't do that for the 7918th; and 
> you probably can't even do it for the 5th.
> 
> It is entirely possible that the 5th Busy Beaver number is  47,176,870 
> because a 5 state Turing Machine has been found that halts after 47,176,870 
> operations,  the problem is there are still 5 different 5 state turing 
> machines that are well past 47,176,870 and they have not halted. If none of 
> those 5 machines ever halts then 47,176,870 really and truly is the 5th Busy 
> Beaver Number, but if that is the case we will never know that is the case 
> because we'll never know that none of those 5 machines ever halts. 
> 
> John K Clark
> 
> 
> 
> The original issue is what real numbers can be described or defined.
> 
>   https://twitter.com/JDHamkins/status/1100090709527408640 
> 
> 
> Must there be numbers we cannot describe or define? Definability in 
> mathematics and the Math Tea argument 
> 
> 
> 
> If a program "represents" a real number (e.g. in the spigot sense), then that 
> could be said to "define" it.
> 
> But what noes it mean for a real number to be "defined”?


It means that we have some meaning for the digit. 

I have proved more or less recently that the function TOT, such that

TOT(i) = 0 if phi_i is a total partial recursive function,   (total = 
everywhere defined on N)
= 1, if not.

Is not computable. Yet, a partial recursive function is either total or 
strictly partial (not total), so that function, despite being not computable, 
is well defined.

Now, you can construct a similarly well defined real number, which is not 
computable, nor appoximable: it is:

0, a_1 a_2 a_3 … with a_i = TOT(i).

That is well defined real number, that, provably, nobody can compute (unless 
the Church-Turing thesis is false).
This one can be shown to be non computable even with the Oracle for the halting 
Problem. It is P2_complete, that is, more uncomputable than the halting 
problem. TOT belong to a higher degree of unsolvability than HALT.


Bruno




> 
> - pt
> 
> 
> 
>  
> 
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Re: Are there real numbers that cannot be defined?

[Bruno Marchal]

> On 3 Mar 2019, at 18:28, John Clark  wrote:
> 
> 
> 
> On Sun, Mar 3, 2019 at 9:26 AM Bruno Marchal  > wrote:
>> 
>> >> The 8000th Busy Beaver Number can be named but not calculated even 
>> >> theoretically,
> > The busy beaver function is not computable, but on each individual n, it is 
> > computable theoretically, 
> 
> No it is not, not if n= 7918, to compute that the program would have to solve 
> the Halting Problem.


Why?


You confuse computable with “provably computable”. BB is not computable, but 
BB+ is computable.

BB(7918) is just a number, say k, and the program “print k” will do.

Similarly, the function F defined by

F (x) = 0 if the twin prime conjecture is true, and equal 1 if not.

Is a computable function; because it is the same function as y=0, or the same 
function as y = 1. Both are computable. 

F is computable, despite its definition is a not a code for computing it.



> The first 4 Busy beaver numbers have been computed and Scott Aaronson proved 
> that the 7918th Busy Beaver Number is not computable, most people think n=5 
> is not computable either but that has not been proved.


I doubt Scaott Aaronson makes such a big mistake. B(7918), is computable. It is 
BB which is not computable. It means that there is no algorithmic to compute BB 
on each numbers. But on each number, BB, and all total functions, are 
computable. 


> 
> The 7918th Busy Beaver Number 
> 


Let me see. Indeed, what they show is that B(7918) = k might be unprovable in 
ZFC. That is not equivalent with B(7918) is not computable. It is a natural 
number. All constant function are computable. 




> > The 8000h BB number is well defined,
> 
> Yes.
>  
> > so it is a (finite) number,
> 
> Yes,
> 
> > and so you there exist a finite program computing it
> 
> No. The 8000th Busy Beaver Number is the largest number of FINITE operations 
> a 8000 state Turing Machine will make before it halts. Some programs we can 
> observe halting and with others it's easy to prove will never halt, that's 
> why we know the first 4 Busy Beaver Numbers, but Turing Proved you can't do 
> that in general and  Aaronson proved you can't do that for the 7918th; and 
> you probably can't even do it for the 5th.

In ZF, we cannot prove that. But provable is not computable. In lambda notation 
[n]BB(n) is not computable, but all BB(n), for each individual n is trivially 
computable. Aaronson takes a problem of provability and independence, not of 
computability. Of course BB(n) is not feasible for large n.




> 
> It is entirely possible that the 5th Busy Beaver number is  47,176,870 
> because a 5 state Turing Machine has been found that halts after 47,176,870 
> operations,  the problem is there are still 5 different 5 state turing 
> machines that are well past 47,176,870 and they have not halted. If none of 
> those 5 machines ever halts then 47,176,870 really and truly is the 5th Busy 
> Beaver Number, but if that is the case we will never know that is the case 
> because we'll never know that none of those 5 machines ever halts. 

No problem with this. But the never know is made precise in term of “provable 
in ZF, or not”, and that is what Aaronson wrote about. BB(n), as function of n, 
is not computable, but all BB(n) is trivially computable for each n, as it is 
only a number. We just don’t know the algorithm, but we know that algorithm 
exists, as BB(n) is just a number, and the algorithm will just be “print that 
numbers”. 

Bruno



> 
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> 
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Re: Recommend this article, Even just for the Wheeler quote near the end

[Bruno Marchal]

> On 3 Mar 2019, at 20:49, Lawrence Crowell  
> wrote:
> 
> On Sunday, March 3, 2019 at 7:58:01 AM UTC-6, Philip Thrift wrote:
> 
> 
> On Sunday, March 3, 2019 at 7:32:00 AM UTC-6, Lawrence Crowell wrote:
> 
> Bringing Gödel into physics is treading on a mine field as it is. Believe me, 
> most physicists react in horror at the mere suggestion of this. I have this 
> suspicion however that quantum measurement is a a sort of Gödel 
> self-reference with quantum information or qubits. This may, at least within 
> how we describe quantum mechanics if it should turn out to be not how the 
> quantum world actually is, be one reason why we have this growing pantheon of 
> quantum interpretations and no apparent way to decide which is definitively 
> correct. 
>  
> 
>  
> I still think it's Darwin, not Gödel,  that has anything to do with  "quantum 
> measurement".
> 
> But physicists recoil in horror from that.
> 
> - pt
> 
> Darwinian logic did put down the Aristotelian-Cartesian hierarchical 
> structure with respect to biology.

OK. Darwin use both mechanism (quasi-explicitly), and is understood usually in 
the materialist frame, but Darwin just do not address that question.



> Aristotle and Plato are the two most known Hellenic philosophers because 
> their systems of thought were wrapped into the New Testament Bible. Plato had 
> this idea of there being a hierarchy of being, which was taken up by St Paul, 
> carried further by Augustine, Aquinas and eventually encoded by Descartes. 
> Descartes had this hierarchy of structure over function, design over material 
> form etc, which was carried into science during the 17th and 18th century. In 
> some ways Newtonian mechanics was seen as a confirmation of Descartes' 
> metaphysics.

That is true. Today we know that Newtonian Mechanics is highly not computable. 
But Newton saw that, and indeed, distrusted his Mechanics, and saw it as an 
approximation. 



> Darwin struck a fatal blow to this with respect to biology.

He struck the wrong view on Descartes and Mechanism, but his own Mechanism is a 
foreseen of digital mechanism, and its confirmation by molecular genetics, and 
the genetical code.



> 
> Darwin did away with Aristotle and Descartes with biology. Gödel had an 
> impact on Plato, though it is not clear to me how. Gödel saw himself as a 
> Platonist and that his incompleteness theorem demonstrated how mathematical 
> truth is independent of knowing it. I tend to see this in terms of Turing 
> machines, which would say that certain problems are not computable and as 
> such no information can be derived.

… can be derived mechanically. But the truth can be guessed and experience by 
non algorithmic, mechanical, means, even by a machine. Gödel’s theorem is 
already proved by machine, which can even prove their own Gödel’s theorem, and 
enforces them to be mystical, that is, to believe that there is something more 
than their own consciousness.



> Whether there is a self-referential truth that is not enumerated is less 
> important. The real number line has a continuum of elements and there is not 
> enough information, even if that is infinite, to encode it all. We might say 
> in some sense that these numbers exist as if being in Plato's cave we can 
> imagine the existence of things by looking at shadows.


Yes. For a set-theoretical realist, there are aleph_0 computable functions, and 
thus 2^aleph_0 non computable functions.

Now, in many toposes, all functions are computable, and all real-functions are 
continuous. That is the case for the effective topos of Highland, based on 
Kleene’s notion of realisability. Mechanism ask for arithmetical realism, just 
to define what is a machine, but it does not asks for set-theoretical realism, 
or analytical realism.



> 
> Kant proposed a metaphysics that is somewhat parallel to quantum mechanics. 
> The noumena of what "actually is," that Bohm insisted we could come to really 
> know, is the unknown of QM. We probably can't know if the quanta is epistemic 
> or ontic.

If you *assume* Mechanism (the indexical weak version I present) then we 
already know that the quanta are epistemic.





> The phenomena are the measurements and predictions. This has a certain 
> Platonic character to it, but within the physical domain. The noumena are 
> similar to Plato's pure forms and the phenomena are similar to the physical 
> forms. The employment of Gödel with physics might be compared to shifting 
> from Plato to Kant. However, if quantum interpretations are Gödel 
> self-referential physical axioms


The whole of the appearance of the physical reality is brought by 
incompleteness, for anyone rational and saying “yes” to the doctor. To assume a 
physical reality is like assuming that a car is driven by invisible horse. It 
add nothing to the thermodynamics, and add only new question, like what are the 
invisible horse made of, and in what sense thermodynamics is not enough. It is 
like adding a new go

Re: Recommend this article, Even just for the Wheeler quote near the end

[Bruno Marchal]

> On 3 Mar 2019, at 20:43, Brent Meeker  wrote:
> 
> 
> 
> On 3/3/2019 4:52 AM, Philip Thrift wrote:
>> 
>> 
>> Here's an example David Wallace presents (as an "outlandish" possibility): 
>> Suppose in pi (which is computable, so has a program (a spigot one, in fact) 
>> that produces its digits. Suppose somewhere in that stream of digits is the 
>> Standard Model Equation
>> 
>> (say written in LaTeX/Math but rendered here)
>>  
>> https://www.sciencealert.com/images/Screen_Shot_2016-08-03_at_3.20.12_pm.png 
>> 
>> 
>> So what could this mean? (He sort of leaves it hanging.)
>> 
> 
> Nothing.  Given a suitable mapping the SM Lagrangian can be found in any 
> sequence of symbols.  It's just a special case of the rock that computes 
> everything.

Even if rock would exist in some primitive sense, which I doubt, they do not 
compute anything, except in a trivial sense the quantum state of the rock. A 
rock is not even a definable digital object. If someone want to convince me 
that a rock can compute everything, I will ask them to write a complier of the 
combinators, say, in the rock. I will ask an algorithm generating the phi_i 
associated to the rock.

Bruno



> 
> Brent
> 
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Re: Recommend this article, Even just for the Wheeler quote near the end

[Bruno Marchal]

> On 3 Mar 2019, at 16:46, Philip Thrift  wrote:
> 
> 
> 
> On Sunday, March 3, 2019 at 8:54:42 AM UTC-6, Bruno Marchal wrote:
> 
>> On 3 Mar 2019, at 15:32, Philip Thrift > 
>> wrote:
>> 
>> 
>> 
>> On Sunday, March 3, 2019 at 6:52:41 AM UTC-6, Philip Thrift wrote:
>> 
>> 
>> On Sunday, March 3, 2019 at 5:58:17 AM UTC-6, Bruno Marchal wrote:
>> 
>>> On 1 Mar 2019, at 19:32, Philip Thrift > wrote:
 
 
 Reading all the above in the context of 
 
 Naturalness and Emergence
 David Wallace
 February 20, 2019
 http://philsci-archive.pitt.edu/15757/1/naturalness_emergence.pdf 
 
 
 leads to the conclusion that our current language(s) of physics is(are) 
 most likely wrong.
>>> 
>>> 
>>> A proposition can be wrong. I am not sure what you or Wallace would mean by 
>>> a language being wrong. Perhaps Wallace meant that our metaphysics (most of 
>>> the time the materialist one) is wrong, which makes more sense. Perhaps he 
>>> does not dare to say so. It is not well seen in some circles.
>>> 
>>> Bruno
>>> 
>>> 
>>> 
>>> 
>>> By 'language' in the above paper he means 'mathematical language' and he 
>>> means precisely the language in which QFT an GR are actually written in 
>>> (seen when you look at them on paper or on a screen): Sentences are made of 
>>> mathematical symbols and variables, but the basics begin with a selection 
>>> of sentences (axioms) from which a theory is made.
>> 
>> OK. Thanks. That makes more sense.
>> 
>> 
>> 
>>> So he is really saying the axioms are likely wrong, and even new primitives 
>>> (mathematical symbols) may have to be invented.
>> 
>> Of course, I don’t think so. It is phenomenologically true, but for the 
>> ontology, i.e. the minimal amount of things which needs to be assumed, s, 0, 
>> + and x are enough (added to logic). In fact, S and K, with “(“ and “)”, 
>> plus “=“ are enough, even without logic. (I always assume Mechanism, by 
>> default, to be sure).
>> 
>> Bruno
>> 
>> 
>> 
>> 
>> 
>> Here's an example David Wallace presents (as an "outlandish" possibility): 
>> Suppose in pi (which is computable, so has a program (a spigot one, in fact) 
>> that produces its digits. Suppose somewhere in that stream of digits is the 
>> Standard Model Equation
>> 
>> (say written in LaTeX/Math [ Unicode ] but rendered here)
>>  
>> https://www.sciencealert.com/images/Screen_Shot_2016-08-03_at_3.20.12_pm.png 
>> 
>> 
>> So what could this mean? (He sort of leaves it hanging.)
>> 
>> - pt
>> 
>> 
>> Apropos Dilbert cartoon:
>> 
>> https://dilbert.com/strip/2019-03-03 
>> 
> 
> 
> Poor Dilbert will have an infinite task to fail its simulated creature. He 
> will have to revise its limit an infinity of times. If the simulation is 
> “physically correct”, there is no sense to localise the mind of the creature 
> in the simulation, as its supervene on infinity of computations. If the 
> simulation is physically incorrect, the creature will see it, by comparing 
> the arithmetic physics with their observation, unless Dilbert intervenes each 
> time to make them dumb.
> 
> We can experimentally test if the empirical world is fundamental or not, and 
> the results obtained today is that it is very plausibly only a symptom of a 
> deeper, and simpler, non material reality. That’s why we have to come back to 
> Plato, and take distance from Aristotle, at least if we are willing to bet 
> the brain is a (material/natural) digitalizable machine.
> 
> Bruno 
> 
> 
> 
> 
> Paralleling the Donald Rumsfeld quote:
> 
> You go to do science [ or engineering ] with the matter you have, not the 
> matter you might want to have. 

Einstein said that he needed just a pen and a bit of papers. 

With mechanism, pen and papers is offered freely by the generous arithmetical 
reality, in the limits of infinitely many first person experiences. 

Bruno



> 
> - pt
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Re: When Did Consciousness Begin?

[Bruno Marchal]

> On 3 Mar 2019, at 20:23, Brent Meeker  wrote:
> 
> 
> 
> On 3/3/2019 3:45 AM, Bruno Marchal wrote:
>>> On 1 Mar 2019, at 23:21, Brent Meeker  wrote:
>>> 
>>> 
>>> 
>>> On 3/1/2019 7:04 AM, Bruno Marchal wrote:
 The “minimal” consciousness require only a weak notion of self. It does 
 not require memory, nor any sense. It is a highly dissociated state of 
 consciousness. It is quite different from the usual mundane consciousness 
 of the everyday life.
 
 
>>> How can there be a notion of self...something that persists through 
>>> time..without memory?
>> I am aware this is highly counter-intuitive, but consciousness does not seem 
>> to require time, nor even any notion of identity. Identity is already a sort 
>> of global illusion, a construct of the universal machine, even if it looks 
>> “persistent” in the terrestrial (effective) plane.
> 
> I agree that consciousness, at the level of my thermostat, does not require a 
> notion of identity. 

I don’t think so. You need only a personal body. And you thermostat has one. If 
not, there is no more un thermostat.



> But identity requires memory.

Consciousness of one’s identity requires memory. But that is close to 
self-consciousness. For consciousness, any (universal) body will do. It will be 
a highly dissociative state of consciousness. To get a life from it will 
require memory, but consciousness is more general, and will use the memory only 
for the self-differentiating purpose.



>   I have had two relatives die of Alzheimers and they lost their identity 
> gradually as they lost memory. 

They lost they memory. Not their identity, but the apprehension of their 
identity. If not, when you ask where they are in the hospital, the nurse would 
say “what are you talking about”. Even a corpse has an identity. 




> Eventually they were only reactive, living in the moment...like my 
> thermostat, but more complicated.

In slow sleep, out of the dream REM sleep period, some onirophysiolog 
(neuroscientist specialised in dreams) begin to acknowledge that we might be 
conscious, but the consciousness does not imprint new memories, and we forget 
it at each instant, giving us a feeling that we have been unconscious. With 
practice, we can learn to get micro-moment of awareness helping us to notice 
it. Unconsciousness is an illusion of consciousness … It should be obvious that 
“being unconscious” cannot be a first person experience, for logical reason. To 
die is not a personal event. That happens only to the others. 

Bruno




> 
> Brent
>> 
>> I don’t use this in my derivation of physics from arithmetical theology, but 
>> many evidences accumulate for this, but they are so close the te logical 
>> trap that I have no way to convey this without feeling uneasy.
>> 
>> Memory and persistence through space and time is required only for 
>> self-consciousness? It is something build from consciousness and 
>> self-consciousness, eventually, but as subject, we see those logical 
>> causality in reverse. The theory explains why we cannot understand this, or 
>> justify it rationally.
>> 
>> Bruno
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Re: When Did Consciousness Begin?

[Bruno Marchal]

> On 3 Mar 2019, at 20:15, Brent Meeker  wrote:
> 
> 
> 
> On 3/3/2019 3:34 AM, Bruno Marchal wrote:
>> 
>>> On 1 Mar 2019, at 21:36, Brent Meeker >> > wrote:
>>> 
>>> 
>>> 
>>> On 3/1/2019 7:08 AM, Bruno Marchal wrote:
 
> On 28 Feb 2019, at 22:47, Brent Meeker  > wrote:
> 
> 
> 
> On 2/28/2019 1:17 PM, Philip Thrift wrote:
>> 
>> 
>> The best current philosopher of (and writer about) consciousness is 
>> Galen Strawson.
>> 
>> https://en.wikipedia.org/wiki/Galen_Strawson 
>> 
>> https://sites.google.com/site/galenstrawson/ 
>> 
>> https://liberalarts.utexas.edu/philosophy/faculty/profile.php?id=gs24429 
>> 
>> 
>> There is a lot of his material (PDFs, articles, videos, etc.) freely 
>> available online.
>> 
>> The main word that is synonymous with consciousness is experience.
> 
> Which is something bacteria and plants and my thermostat have...and 
> ability to detect and react to the environment based on internal states.
 
 What the thermostat lacks, that the bacteria and plants do not lack, is 
 Turing universality. That gives the mind, and even the free-will. 
>>> 
>>> A bacterium doesn't have Turing universiality, only bacteria in the 
>>> abstract of a potentially infinite set of evolving bacteria interacting 
>>> with their environment.  But if a consider a potentially infinite set of 
>>> thermostats interacting with their environment of furnaces and rooms, it 
>>> will be Turing universal too.  Turing universality is cheap.
>> 
>> Yes, it is cheap, like consistency, and plausibly consciousness.
>> 
>> But it is more cheap you might think, because even one bacteria is fully 
>> Turing universal. The genome of Escherichia Coli can be “programmed” to run 
>> a Turing universal set of quadruplet. Of course, the bacteria’s “tape” is 
>> quite limited, and they can exploit their universality only by cooperation 
>> in the long run, and so no individual bacteria can be self-conscious or 
>> Löbian.
> 
> I think that's what I said.  Except I also noted that all this requires an 
> environment within which the bacteria can metabolize. 

That is contingent with respect of the bacteria “mental life”. All programs 
needs a code, and an environment which run it, but it can be arithmetic. Then a 
physical reality emerges as a means on all accessible 
computations-continuations.

Mentioning the environment can be misleading. If a material environment is 
needed, matter would play some role, and there is no more reason to say accept 
a digital, even if physical, brain. In a dream, we create more clearly the 
environment by ourself, and that is enough for being conscious, or even 
self-conscious, like in a lucid dream, or a sophisticated virtual environment.



> So to say bacteria have Turing universality is like saying water is 
> lemonade...if you add lemons and sugar.

It means that with the 4 letters, you can program any partial recursive 
function. Of course you need the decoding apparatus, but that is entirely in 
the bacteria. It means that you can simulate any other computer, with a basic 
set of DNA-enzyme molecular interaction. A universal machine is just a number u 
such that for all x and y phi_u(x, y) = phi_x(y) *in principle. You can 
implement all control structure. The operon illustrates a “if-then-else”, and 
the regulation apparatus is enough to get universality. René Thomas, in 
Brussels, has succeeded to make a loop, with a plasmid (little circular gene) 
entering in the bacterium, and then going out, repetitively. It is even a 
“fuzzy computer”. Some product are regulated in a continuum, depending on the 
concentration of the metabolites. When I was young, I have made e project for a 
massively parallel computers which was a solution of bacteria (E. coli) and 
bacteriophage. One drop of it could process billions of instructions in a 
second. But the read and write was demanding highly sophisticated molecular 
biology. I think that such ideas have more success today. After all, molecular 
biology studies “natural nanotechnology”.

Bruno





> 
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Re: Are there real numbers that cannot be defined?

[Lawrence Crowell]

On Saturday, March 2, 2019 at 8:28:01 PM UTC-6, John Clark wrote:
>
>
> On Fri, Mar 1, 2019 at 4:23 PM Lawrence Crowell  > wrote:
>
> > There are numbers that have no description in a practical sense. The 
>> numbers 10^{10^{10^{10}}} and 10^{10^{10^{10^{10 have a vast number of 
>> numbers that have no description with any information theoretic sense.
>>
>
> The 8000th Busy Beaver Number can be named but not calculated even 
> theoretically, but most Real Numbers can't even be uniquely named with 
> ASCII characters, not even with an infinite number of them.   
>
> John K Clark
>

There exists an uncountably infinite number of reals in the interval (0, 
1), and they exhaust all possible information theoretic description. Some 
mathematicians have argued this means they do not in some ways exist. Most 
mathematicians disagree with that by arguing computational tractability is 
not equivalent to mathematical existence. 

LC 

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