Re: Combinators 1 (Introduction)

[Jason Resch]

On Sun, Aug 5, 2018 at 2:05 PM Bruno Marchal  wrote:

> Hi Jason,
>
>
> On 5 Aug 2018, at 05:24, Jason Resch  wrote:
>
>
>
> On Sat, Jul 28, 2018 at 2:19 PM Bruno Marchal  wrote:
>
>> Hi Jason, people,
>>
>>
> Hi Bruno,
>
> Thank you for this. I've been trying to digest it over the past few days.
>
>
> No problem.  It was hard to begin with , and I was about sending few easy
> exercise to help for the notation. But you did very well.
>
>
Thank you. :-)


>
>
>
>
>>
>> I will send my post on the Church-Turing thesis and incompleteness later.
>> It is too long.
>>
>> So, let us proceed with the combinators.
>>
>> Two seconds of historical motivation. During the crisis in set theory,
>> Moses Schoenfinkel publishes, in 1924, an attempt to found mathematics on
>> only functions. But he did not consider the functions as defined by their
>> behaviour (or input-output) but more as rules to follow.
>>
>> He considered also only functions of one variable, and wrote (f x)
>> instead of the usual f(x).
>>
>> The idea is that a binary function like (x + y) when given the input 4,
>> say, and other inputs, will just remains patient, instead of insulting the
>> user, and so to compute 4+5 you just give 5 (+ 4), that is you compute
>>  ((+ 4) 5). (+ 4) will be an object computing the function 4 + x.
>>
>>
>> The composition of f and g on x is thus written  (f (g x)), and a
>> combinator should be some function B able on f, g and x to give (f (g x)).
>>
>> Bfgx = f(gx), for example.
>>
>
> So am I correct to say a combinator "B" is a function taking a single
> input "fgx”,
>
>
> Three inputs. B on f will first gives (B f), written Bf, then when B will
> get its second input that will give ((B f) g), written Bfg, which is a new
> function which on x, will now trigger the definition above and give the
> combinator (f (g x)), written f(gx) and which would compute f(g(x)) written
> with the usual schoolboy notation.
>
>
Okay, I see.


>
>
>
> but is itself capable of parsing the inputs and evaluating them as
> functions?
>
>
> It just recombine its inputs, the functions will evaluate by themselves.
> Don’t worry, you will see clearly the how and why.
>
> B is called an applicator, because given f, g and h has arguments, Bfgh,
> it gives f(gh). I have used f and g and h has symbol, but I can use x and y
> and z instead. Those variables are put for combinators. Bxyz = x(yz).
> Formally B only introduce those right parenthesis. With full parentheses we
> should write:
>
> (((Bx)y)z) = (x(yz)). But we suppress all leftmost parentheses: Bxyz
> =x(yz).
>
> The interesting question is: does B exist? Which here means —is there a
> combinator (named B) which applied on x, then y, then z, gives x(yz).
>

I believe B exists, given combinators are universal, but I don't know what
it is.


>
> Later I will provide an algorithm solving the task of finding a
> combinators doing some given combination like that.  But here I just answer
> the question: YES!
>
> Theorem B = S(KS)K, i.e. Bxyz = S(KS)Kxyz = x(yz)
>

Ahh here it is!


>
> Proof: it is enough to compute S(KS)Kxyz and to see if we get x(yz)
>
> Let us compute, and of course I remind you the two fundamental laws used
> in that computation:
>
> Kxy = x
> Sxyz = xz(yz)
>
> S(KS)Kxyz =
>
> OK let see in detail that is the combinator S, which got a first argument,
> the combinator (KS) this gives (S (K S)) written S(KS), which remains
> stable ("not enough argument”), then S(KS) get the argument K which gives
> S(KS)K, which remains stable (indeed it is supposed to be the code of B)
> and indeed S has still got only his first two argument and so we can’t
> apply any laws to proceed, but now, S get its third argument x so
>
> we are at S(KS)Kx, that is S (KS) K x, and here S has three arguments and
> so match the left part of the second law S x y z, with x = KS, y = K and z
> = x.
>

Okay, I follow so far.  The idea of waiting until having enough arguments
before activating is helpful and I think I was missing that before.


>
> Now the second law is triggered, so to speak, and we get xz(yz) with with
> x = KS, y = K and z = x, and that is gives (KS)x(Kx) = KSx(Kx). OK?
>

It isn't fully clear to me how the "eliminate left parenthesis" rule works.
Is it that you only cancel the left-most parenthesis until you hot a
non-left-parenthesis element, or is it more complex than this?


>
> You always add the left parentheses, or some of them to be sure what we
> have obtained. KSx(Kx) = ((KSx) (Kx)), but “KSx” is a redex, as it match
> Kxy, with x = S and y = x, and so get “reduces” into S, so we get S(Kx)
> (starting from S(KS)Kx, which is Bx, waiting now for y and then z.
>
> We are at Bxy = S(KS)Kxy = (we just computed) S(Kx)y, which is S with “not
> enough argument” so we give the remains z and get
>
> S(Kx)yz
>
> Which triggers again the second law to give (x = (Kx), y = y, z = z)
>
> (Kx)z(yz) = Kxz(yz)
>
> And again, Kxz gives x (by the first law) so we get
>
> x(yz).
>
> OK?
>

It is 

Re: The codical-material universe

[Brent Meeker]

On 9/16/2018 11:07 AM, Bruno Marchal wrote:
No problem. I can explain why you will need a non computationalist 
theory of mind.


Given that there is no evidence at all for primary matter, nor for a 
non computationalist theory of mind, that seems very speculative to me.


I am aware that many materialist use (explicitly or implicitly) a 
mechanist theory of mind, but I can explain why materialism (or 
physicalism) is (are) incompatible with Indexical Digital Mechanism. 
(The doctrine that the brain function is Turing emulable at *some* 
level of description).


But it seems to me that a subconscious is incompatible with your theory 
of mind.  So far as I can see you only propose to explain conscious 
thought as computation, i.e. some computations instantiate conscious 
thoughts and some don't.  But it appears that many, if not most, of our 
thinking is subconscious.  Where in your theory is this distinction encoded?


Brent

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

Re: The codical-material universe

[Philip Thrift]

On Sunday, September 16, 2018 at 1:07:34 PM UTC-5, Bruno Marchal wrote:
>
>
> On 15 Sep 2018, at 22:56, Philip Thrift > 
> wrote:
>
>
>
> On Saturday, September 15, 2018 at 10:27:07 AM UTC-5, John Clark wrote:
>>
>> On Fri, Sep 14, 2018 at 12:22 PM Bruno Marchal  wrote:
>>  
>>
>>> >>> you claim nonphysical calculations can be performed, but you also 
 say space and time and even spacetime are physical, so how can you have 
 change without them and how can you have calculations without change?
>>>
>>>  
>>
>> >> *The changes are digital, discrete.*
>>>
>>  
>> Changes in what? You say electrons, atoms, photons, neutrinos, 
>> gravitational waves, time, space and even spacetime are all matter, so the 
>> change can't be with respect to any of them; so what's changing and with 
>> respect to what?  And how can you make a calculation without changing 
>> something?
>>
>> >>Electromagnetic waves of 650 nanometers is physical , the color red is 
 a qualia and can be produced without Electromagnetic waves of any sort.
>>>
>>>  
>>
>> >*One reason more to understand that the qualia can raise just with any 
>>> number relations mimicking the relevant brain parts relation in 
>>> arithmetic. *
>>>
>>
>> And yet you agree that "even qualia is matter". And please note that in 
>> a MRI scanner similar parts of the brain light up when one sees a red 
>> lights and when one just thinks about a red light. And also note that the 
>> brain is made of matter that obeys the laws of physics.
>>  
>>
>>> *>If some primary matter is at play for consciousness, can you explain 
>>> what role it has, and what it is?*
>>
>>
>> If matter is primary then obviously I can't say what its made of, if I 
>> could it wouldn't be primary. And if it's primary then it's needed to 
>> perform calculations.  And after saying consciousness is the way that data 
>> feels when it is being processed there is simply nothing more that can be 
>> said about consciousness. 
>>
>> >>the blueprints for a 747 airliner can fly me to London. 
>>>
>>>
>>> *>The analogy does not work. A machine is more than a blueprint.*
>>>
>>
>> A machine needs 2 things, data and matter. The blueprints supply the data 
>> but I can't fly to London on half a machine.
>>
>> >*But the step 3 can be done with robots instead of human, and if you 
>>> disbelieve in first person indeterminacy,*
>>>
>>
>> I don't believe in "first person indeterminacy" and I don't 
>> disbelieve in it either because there is nothing in it to believe or 
>> disbelieve. There is no there there. If you could specify who exactly what 
>> "the first person" is in a world that contains "the first person" 
>> duplicating machines I might perhaps be able to tell you what I think about 
>> it, but until then the idea is far worse than just being wrong, it is 
>> gibberish.   
>>  
>>
>>> > tell me how you program the robot so that he []
>>>
>>
>> As always in your thought "experiments" the gibberish kicks in 
>> immediately. Right at the start you postulate a world that contains both "
>> *THE*" and "*HE*" duplicating machines so your reference to "*the*" 
>> robot and "*he*" no longer refers to anything unique.  
>>  
>>
>>> >  i*s able to predict the results of opening the door after pushing on 
>>> the button.*
>>>
>>  
>> Well I predict that before the doors open there will only be one 
>> conscious mind regardless of how many brains have been duplicated. And I 
>> predict the moment the doors are opened revealing 2 different things there 
>> will be 2 conscious minds. And I predict the mind that observes Moscow will 
>> become the Moscow Man.  And I predict the mind that observes Washington 
>> will become the Washington Man. And there is nothing more to predict.   
>>  
>>
>>> *> The robot survive, by computationalism (or even without in this 
>>> case), and the robot can predict with certainty that he will not write in 
>>> his personal diary that he is seeing both city at once,*
>>
>>
>> No, that is incorrect, "*THE* robot" can't predict that and neither can 
>> anybody else because you haven't specified what or who the prediction is 
>> supposed to be about.
>>
>> > So, if you have an algorithm just show it to us.
>>>
>>
>> I will show you such a algorithm as soon as you show me a algorithm to 
>> determine what one and only one thing will happen to 1 banana after 1 
>> banana becomes 2 bananas. But even then you'd have no way of knowing if my 
>> prediction turned out to be correct or not because even after the 
>> "exparament" is long over you *STILL* couldn't say if the correct 
>> prediction turned out to be Moscow or Washington. And that my dear Bruno is 
>> why this isn't a exparament at all and it's not even a thought exparament, 
>> it is just a showcase displaying your massive confusion and the difficulty 
>> you have in handling philosophical matters.   
>>  
>>
>>> *>The question makes sense, as you have agreed that we survive such an 
>>> experience,*
>>
>>
>> I have 

Re: The codical-material universe

[Bruno Marchal]

> On 15 Sep 2018, at 22:56, Philip Thrift  wrote:
> 
> 
> 
> On Saturday, September 15, 2018 at 10:27:07 AM UTC-5, John Clark wrote:
> On Fri, Sep 14, 2018 at 12:22 PM Bruno Marchal  > wrote:
>  
> >>> you claim nonphysical calculations can be performed, but you also say 
> >>> space and time and even spacetime are physical, so how can you have 
> >>> change without them and how can you have calculations without change?
>  
> >> The changes are digital, discrete.
>  
> Changes in what? You say electrons, atoms, photons, neutrinos, gravitational 
> waves, time, space and even spacetime are all matter, so the change can't be 
> with respect to any of them; so what's changing and with respect to what?  
> And how can you make a calculation without changing something?
> 
> >>Electromagnetic waves of 650 nanometers is physical , the color red is a 
> >>qualia and can be produced without Electromagnetic waves of any sort.
>  
> >One reason more to understand that the qualia can raise just with any number 
> >relations mimicking the relevant brain parts relation in arithmetic. 
> 
> And yet you agree that "even qualia is matter". And please note that in a MRI 
> scanner similar parts of the brain light up when one sees a red lights and 
> when one just thinks about a red light. And also note that the brain is made 
> of matter that obeys the laws of physics.
>  
> >If some primary matter is at play for consciousness, can you explain what 
> >role it has, and what it is?
> 
> If matter is primary then obviously I can't say what its made of, if I could 
> it wouldn't be primary. And if it's primary then it's needed to perform 
> calculations.  And after saying consciousness is the way that data feels when 
> it is being processed there is simply nothing more that can be said about 
> consciousness. 
> 
> >>the blueprints for a 747 airliner can fly me to London.
> 
> >The analogy does not work. A machine is more than a blueprint.
> 
> A machine needs 2 things, data and matter. The blueprints supply the data but 
> I can't fly to London on half a machine.
> 
> >But the step 3 can be done with robots instead of human, and if you 
> >disbelieve in first person indeterminacy,
> 
> I don't believe in "first person indeterminacy" and I don't disbelieve in it 
> either because there is nothing in it to believe or disbelieve. There is no 
> there there. If you could specify who exactly what "the first person" is in a 
> world that contains "the first person" duplicating machines I might perhaps 
> be able to tell you what I think about it, but until then the idea is far 
> worse than just being wrong, it is gibberish.   
>  
> > tell me how you program the robot so that he []
> 
> As always in your thought "experiments" the gibberish kicks in immediately. 
> Right at the start you postulate a world that contains both "THE" and "HE" 
> duplicating machines so your reference to "the" robot and "he" no longer 
> refers to anything unique.  
>  
> >  is able to predict the results of opening the door after pushing on the 
> > button.
>  
> Well I predict that before the doors open there will only be one conscious 
> mind regardless of how many brains have been duplicated.  And I predict the 
> moment the doors are opened revealing 2 different things there will be 2 
> conscious minds. And I predict the mind that observes Moscow will become the 
> Moscow Man.  And I predict the mind that observes Washington will become the 
> Washington Man. And there is nothing more to predict.   
>  
> > The robot survive, by computationalism (or even without in this case), and 
> > the robot can predict with certainty that he will not write in his personal 
> > diary that he is seeing both city at once,
> 
> No, that is incorrect, "THE robot" can't predict that and neither can anybody 
> else because you haven't specified what or who the prediction is supposed to 
> be about.
> 
> > So, if you have an algorithm just show it to us.
> 
> I will show you such a algorithm as soon as you show me a algorithm to 
> determine what one and only one thing will happen to 1 banana after 1 banana 
> becomes 2 bananas. But even then you'd have no way of knowing if my 
> prediction turned out to be correct or not because even after the 
> "exparament" is long over you STILL couldn't say if the correct prediction 
> turned out to be Moscow or Washington. And that my dear Bruno is why this 
> isn't a exparament at all and it's not even a thought exparament, it is just 
> a showcase displaying your massive confusion and the difficulty you have in 
> handling philosophical matters.   
>  
> >The question makes sense, as you have agreed that we survive such an 
> >experience,
> 
> I have agreed that if survival today means being able to remember being the 
> Helsinki Man yesterday and somebody remembers today then the Helsinki Man has 
> survived today. And obviously if that's what the word means and if we have 
> Helsinki Man duplicating machines then the 

Re: The codical-material universe

[Bruno Marchal]

> On 15 Sep 2018, at 17:26, John Clark  wrote:
> 
> On Fri, Sep 14, 2018 at 12:22 PM Bruno Marchal  > wrote:
>  
> >>> you claim nonphysical calculations can be performed, but you also say 
> >>> space and time and even spacetime are physical, so how can you have 
> >>> change without them and how can you have calculations without change?
>  
> >> The changes are digital, discrete.
>  
> Changes in what? You say electrons, atoms, photons, neutrinos, gravitational 
> waves, time, space and even spacetime are all matter, so the change can't be 
> with respect to any of them; so what's changing and with respect to what? 


The state of a Turing machine with respect to the universal number which 
implements it, in arithmetic to fx the things.

Or equivalently, a computation can be defined by a sequence of reduction, i.e. 
an application of the two laws of combinator reduction. Again with respect to 
the universal number which implements it.

I define a universal machinery by what Rogers called “an acceptable 
enumeration”. That is a (recursively enumerable) sequence of the partial 
computable functions phi_i such that 

1) it exist a u such that phi_u() = phi_x(y)
2) it exist  computable function S such that for all i: phi_i() = 
phi_S(i,x) (y). (Curryfication on the phi_i, this is offered for free with the 
combinators).

Note that Kleene, using his famous “Kleene’s predicate” make clear how 
statement about the phi_i can be translated in purely arithmetical sense.



> And how can you make a calculation without changing something?


Are you assuming time to be fundamental? In GR, there is a notion of block 
universe, where time can be defined indexically. The same for the multi-dreams 
or multi-histories canonical interpretation by universal number in arithmetic 
occur. Computations needs a clock of some sort, and it is provided by the 
number of steps of the relative computation.

I say that x emulate y on z when phi_x(y, z) = phi_y(z).

Then we can associate to each computation of phi_i(j) a notion of number of 
step, made by i, or made by the universal u emulating i. 

At no moment we need to assume anything,ng more than some induction axioms, and 
the usual axioms I have given. More details in the combinator threads, or in 
the phi_i (Church’s thesis) thread.





> 
> >>Electromagnetic waves of 650 nanometers is physical , the color red is a 
> >>qualia and can be produced without Electromagnetic waves of any sort.
>  
> >One reason more to understand that the qualia can raise just with any number 
> >relations mimicking the relevant brain parts relation in arithmetic. 
> 
> And yet you agree that "even qualia is matter”.


It is intelligible matter. It is eventually defined by the logic of []p & <>t & 
p, with p a sigma_1 arithmetiical proposition, and this at the level of G*. The 
qualia are defined in (second order) arithmetic by qZ1*. See my papers for all 
details, or ask any question. 




> And please note that in a MRI scanner similar parts of the brain light up 
> when one sees a red lights and when one just thinks about a red light.


Yes, that is an argument in favour of Mechanism. 




> And also note that the brain is made of matter that obeys the laws of physics.

Yes, no problem with this. But that does not make the matter of our brain 
fundamental, nor do that entail the identity thesis between the brain and 
consciousness, as the thought experiences illustrates and the math proves 
(assuming Mechanism of course).




>  
> >If some primary matter is at play for consciousness, can you explain what 
> >role it has, and what it is?
> 
> If matter is primary then obviously I can't say what its made of,

That was not asked. The question is why do you assume this for saying that 
material brain can have consciousness but the immaterial one in arithmetic 
can’t.




> if I could it wouldn't be primary. And if it's primary then it's needed to 
> perform calculations.  And after saying consciousness is the way that data 
> feels when it is being processed there is simply nothing more that can be 
> said about consciousness. 

But infinitely many data are processed relatively in arithmetic. That leads to 
the problem of justifying the stability of the appearances by a statistic on 
all computations. 

You cannot invoke a God, nor any metaphysical ontological assumption, to select 
a computation in arithmetic. Or you give a role to that matter, which is not 
Turing emulbale, nor representable in arithmetic (which is far rocher than the 
Turing emulable).




> 
> >>the blueprints for a 747 airliner can fly me to London.
> 
> >The analogy does not work. A machine is more than a blueprint.
> 
> A machine needs 2 things, data and matter.

That is plain wrong.




> The blueprints supply the data but I can't fly to London on half a machine.
> 
> >But the step 3 can be done with robots instead of human, and if you 
> >disbelieve in first person indeterminacy,
> 
> I don't 

Re: Combinator 5 (Numbers)

[Bruno Marchal]

Hi, 

We will implement the numbers with the combinators. In particular we will 
emulate Robinson Arithmetic(*) with the combinators. 

Robinson arithmetic is classical logic +

1) 0 ≠ s(x) 0 is not a successor of any number x
2) s(x) = s(y) -> x = y Different numbers have different successors
3) x = 0 v Ey(x = s(y)) Each number has a predecessor
4) x+0 = x  Adding nothing keep a number invariant
5) x+s(y) = s(x+y)  Recursion equation of addition
6) x*0=0Multiplying by 0 gives 0
7) x*s(y)=(x*y)+x   Recursion equation of multiplication

On the right I have put the intended intuitive semantic.

Usually I use the "Church numerals”, but I will use those of Barendrecht. 
Smullyan’s book convinced me of their elegance, and it gives a nice opportunity 
to use our new recursion tools, as they strikes the eyes in line “5)” and “7)”. 
Isn’t it? “+” is defined from itself. “*” also. 


Barendrecht's Numerals

Definitions.

The number 0 is defined by I, that is SKK, but I will abbreviate it by I.
The successeur function is defined by the combinator Vf. V is the Vireo,
Vxyz = zxy. f is KI. We have seen that V = BTC (and that B = S(KS)K, …). I mean 
V is a combinator (a combination of K and S), and it does what it does (that 
circular permutation of its argument). Reread previous post if necessary.

f is the “logical” abbreviation for false, that is KI, i.e. K(SKK), as defined  
in the logical interlude.

Does it work? 

We have that 

0 = I
1 = VfI which remains stable, as V has not enough of its “arguments”.
2 = Vf(VfI) idem
3) = Vf(Vf(VfI) It looks everything is fine and (by induction) will remain fine.
Etc.

So a number n, which is defined by s(s(s(s(…(0))…), with n “s“ in, or by, 
Robinson Arithmetic is defined, or represented by 

Vf(Vf(Vf(Vf( …(I))…)with n “Vf”.

It will be handy to have a predecessor. 
You might try to find one by yourself before reading what follows, but there is 
no obligation. What is obligatory is to verify that it works. Barendrecht 
proposes Tf. (Where T is the “trush”: Txy = yx).

OK, but does it works? We need to verify this. So let us try it on 0, just to 
see!

TfI = If = f. 

Well, that is OK. All we need is that it does not give some number, and giving 
f is not so bad, almost an “error message” :)

I will abbreviate Vf(Vf(Vf(Vf( …(I))…) by n (hoping the underline will not 
disappear).

We have tested the predecessor Tf on 0, now we must test it on some successor, 
that is some Vfn 

Tf(Vfn)
Vfnf(Vfn)f  for the beginners
ffn f is KI, KIxy = y, revise the preceding posts if needed)
n

It works!

Now, to implement “x + y” we need an ability to distinguish between a null and 
a non null number, to decide between using axiom “4)” or “5)”.

So we need a combinator Z which answer truth, i.e t, that is K, when given 0, 
that is I,  and gives KI when given a non null n. 

We want Z0 = ZI = K, and Zn = KI in case n is different from 0.

Barendregt's solution: Z = Tt.

We have already met Tt. It played the role of the “OR” in logic, and works very 
well also to test if a number is null or not:

TtI = It = t
Tt(Vfn) = Vfnt = tfn = f.

Let me sum up:

 t = K
 f = KI

0 = I
s = Vf   (successor)
p = Tf(predecessor)

Z = testing “nullness” = Tt.

And I recall that, thanks to t = K and f = KI, we have that 

if A then B else C

becomes simply ABC, as tBC = B, and fBC = C.

===

ADDITION


Now, we have all we need to program addition. 

Addition is defined by its recursive equation (cf above)

4) x+0 = x  Adding nothing keep a number invariant
5) x+s(y) = s(x+y)  Recursion equation of addition

I replace “5)" by

5’) x+y = s(x + (py))

which is more suitable here.

So a (recursive) program, for adding x + y,  could be

If y = 0 then output x else output s applied to the addition of x and py.

I will use “a" for the combinator doing the addition. It is the one we are 
searching.

axy = if (Zy) then x else  (Vf(ax(py)))

And thus (cf: if A then B else C = ABC. I wrote (Zy)x = Zyx, as always.

axy = Zyx(Vf(ax(py)))

Do you see this?  You should read it:  If Zy then x else Vf(ax(py)).

Of course, we need to solve the recursion. You might need to revise the post on 
Recursion (Combinator 4).

1) renaming of the variables:

azy = Zzy(Vf(ay(pz)))

2) There is a combinator A such that (a is replaced by x)

Axyz = Zzy(Vf(xyz))

Indeed A = [x][y][z]Zzy(Vf(xyz))

Then, the adding combinator a is just the fixed point of A:

a = YA = Y[x][y][z]Zzy(Vf(xyz))

Done.

We know that the paradoxical combinator does well its job, so let us test the 
recursion directly, on 2+2=4, say:

a 2 2 = a(Vf(VfI))(Vf(VfI) 
= Z(Vf(VfI))(Vf(VfI))(Vf (a (Vf(VfI)) (p (Vf(VfI
= Z(Vf(VfI))(Vf(VfI))(Vf (a (Vf(VfI)) (VfI)))
= Vf(a