Re: Combinators 1 (Introduction)

[Bruno Marchal]

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.



>  
> 
> 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.




> 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).

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)

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.

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?

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?

How could we have found that B was computed by the combinators S(KS)K?

We can do this by guessing and computing in reverse, introducing K or other 
combinators so that we can reverse the fundamental laws. So in x(yz), we can 
replace x by (Kxz) that is ((Kx)z) so that we can apply axiom 2 to x(yz) = 
(Kxz)(yz) = S(Kx)yz, then, well the “yz” are already in the good place, but the 
x is still in a parenthesis which has to be removed: we just do the same trick 
and replace S(Kx) by (KSx)(Kx), and so we get S(KS)Kx and we are done: B = 
S(KS)K.

Don’t worry too much, I will soon or a bit later provide an algorithm which 
from a specification Xxyzt = xt(ytxzx) 

Re: The Ilusion of Branching and the MWI

['scerir' via Everything List]

> Il 4 agosto 2018 alle 23.32 agrayson2...@gmail.com ha scritto:
> 
> AFAIK, no one has ever observed a probability wave, from which I conclude 
> the wave function has only epistemic content. So I have embraced the "shut up 
> and calculate" interpretation of the wave function. I also see a connection 
> between the True Believers of the MWI, and Trump sycophants; they seem immune 
> to simple facts, such as the foolishness of thinking copies of observers can 
> occur, or be created, willy-nilly. AG
> 

Frankly I cannot understand, from the following famous page, whether 
Schroedinger thinks the wavefunction as ontic or epistermic or both!

Erwin Schroedinger - § 7. The psi-Function as a Catalogue of Expectations.

Continuing with the exposition of the official teaching, let us turn to the 
psi-function
mentioned above (§ 5). It is now the instrument for predicting the probability 
of
measurement outcomes. It embodies the totality of theoretical future 
expectations, as laid
down in a catalogue. It is, at any moment in time, the bridge of relations and 
restrictions
between different measurements, as were in the classical theory the model and 
its state at
any given time. The psi-function has also otherwise much in common with this 
classical
state. In principle, it is also uniquely determined by a finite number of 
suitably chosen
measurements on the object, though half as many as in the classical theory. 
Thus is the
catalogue of expectations laid down initially. From then on, it changes with 
time, as in
the classical theory, in a well-defined and deterministic ("causal") way - the 
development
of the psi-function is governed by a partial differential equation (of first 
order in the time
variable, and resolved for dy/dt). This corresponds to the undisturbed motion 
of the
model in the classical theory. But that lasts only so long until another 
measurement is
undertaken. After every measurement, one has to attribute to the psi-function a 
curious,
somewhat sudden adaptation, which depends on the measurement result and is 
therefore
unpredictable. This alone already shows that this second type of change of the 
psi-function
has nothing to do with the regular development between two measurements. The 
sudden
change due to measurement is closely connected with the discussion in § 5, and 
we will
consider it in depth in the following. It is the most interesting aspect of the 
whole theory,
and it is precisely this aspect that requires a breach with naive realism. For 
this reason,
the psi-function cannot immediately replace the model or the real thing. And 
this is not
because a real thing or a model could not in principle undergo sudden 
unpredictable
changes, but because from a realistic point of view, measurements are natural 
phenomena
like any other, and should not by themselves cause a sudden interruption of the 
regular
evolution in Nature.

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Re: The Ilusion of Branching and the MWI

[agrayson2000]

On Sunday, August 5, 2018 at 4:43:21 PM UTC, Bruno Marchal wrote:
>
>
> On 4 Aug 2018, at 23:32, agrays...@gmail.com  wrote:
>
> AFAIK, no one has ever observed a probability wave, from which I conclude 
> the wave function has only epistemic content.
>
>
>
> Then you need to explain how that epistemic content interfere in nature. 
> Your idea might make sense, and indeed if we believe in a collapse (as you 
> have to do if you believe in QM and that the superposition does not apply 
> to us) the idea that consciousness collapse the wave is perhaps the less 
> ridiculous idea. That idea has indeed be defended by von Neumann, Wigner, 
> and some others. But has been shown to lead to many difficulties when taken 
> seriously by Abner Shimony, as well guessed by Wigner itself. Obviously 
> that idea would be inconsistent with Mechanism.
>

*Easy to show that consciousness doesn't collapse the wf. Just do repeated 
trials and don't look at the screen until the experiment is finished. I 
forget; what is mechanism? AG *

>
> There is no probability waves. There is only an amplitude of probability 
> wave, and the weirdness is that we have strong indirect evidence that the 
> amplitude of that wave is as physically real as the particles that we can 
> observe, because the particle location is determined by that wave having 
> interfered like wave usually do. In particular, even if send one by one, 
> the particles will never been found where the wave interfere destructively, 
> and the pattern on the screen will reflect the number of holes, and their 
> disposition. 
>
> It is OK to say that probability comes from ignorance, and that the wave 
> describe that ignorance, the extraordinary thing is then that  this 
> ignorance interfere independently of you.
>
>
>
>
>
> So I have embraced the "shut up and calculate" interpretation of the wave 
> function.
>
>
>
> That can be wise. Nobody can enforce the search of the truth. It is 
> frustrating because we can’t be sure if we progress toward it or the 
> contrary, and it is shocking because truth always beat fictions.
>
>
>
> I also see a connection between the True Believers of the MWI, and Trump 
> sycophants; they seem immune to simple facts, such as the foolishness of 
> thinking copies of observers can occur, or be created, willy-nilly. AG
>
>
> That remark deserves your point and diminish your credibility. It also 
> suggests that you are a “True Believer” in something.
>
> Assuming Mechanism in cognitive science, you don’t need quantum mechanics 
> to understand that there are infinitely many relative computational states 
> corresponding to you here and now emulated by infinitely many universal 
> machines. Even without mechanism this is a theorem of arithmetic using only 
> Church thesis. With mechanism, we have to derive the “guessable wave" from 
> a statistics on those computations, and so we can test Mechanism if it 
> leads to more, or less extravaganza than Nature. It fits up to now. So with 
> Mechanism, we get the *appearance* of many interfering “worlds”, and this 
> without any worlds, from just the natural numbers and the laws of addition 
> and multiplication. I will show that with the combinators as it is much 
> shorter (but still long) than showing this with the numbers. This is known 
> by logicians since the 1930s (I mean that a universal Turing machine is an 
> arithmetical object). Computationalism, or Indexical Digital Mechanism 
> imposes a Many-Dreams internal interpretation of Arithmetic (or combinator 
> theory, or game-of-life theory, … we have to assume only one universal 
> machinery).
>
> Bruno
>
>
>
>
>
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Re: The Ilusion of Branching and the MWI

[Bruno Marchal]

> On 4 Aug 2018, at 23:32, agrayson2...@gmail.com wrote:
> 
> AFAIK, no one has ever observed a probability wave, from which I conclude the 
> wave function has only epistemic content.


Then you need to explain how that epistemic content interfere in nature. Your 
idea might make sense, and indeed if we believe in a collapse (as you have to 
do if you believe in QM and that the superposition does not apply to us) the 
idea that consciousness collapse the wave is perhaps the less ridiculous idea. 
That idea has indeed be defended by von Neumann, Wigner, and some others. But 
has been shown to lead to many difficulties when taken seriously by Abner 
Shimony, as well guessed by Wigner itself. Obviously that idea would be 
inconsistent with Mechanism.

There is no probability waves. There is only an amplitude of probability wave, 
and the weirdness is that we have strong indirect evidence that the amplitude 
of that wave is as physically real as the particles that we can observe, 
because the particle location is determined by that wave having interfered like 
wave usually do. In particular, even if send one by one, the particles will 
never been found where the wave interfere destructively, and the pattern on the 
screen will reflect the number of holes, and their disposition. 

It is OK to say that probability comes from ignorance, and that the wave 
describe that ignorance, the extraordinary thing is then that  this ignorance 
interfere independently of you.





> So I have embraced the "shut up and calculate" interpretation of the wave 
> function.


That can be wise. Nobody can enforce the search of the truth. It is frustrating 
because we can’t be sure if we progress toward it or the contrary, and it is 
shocking because truth always beat fictions.



> I also see a connection between the True Believers of the MWI, and Trump 
> sycophants; they seem immune to simple facts, such as the foolishness of 
> thinking copies of observers can occur, or be created, willy-nilly. AG

That remark deserves your point and diminish your credibility. It also suggests 
that you are a “True Believer” in something.

Assuming Mechanism in cognitive science, you don’t need quantum mechanics to 
understand that there are infinitely many relative computational states 
corresponding to you here and now emulated by infinitely many universal 
machines. Even without mechanism this is a theorem of arithmetic using only 
Church thesis. With mechanism, we have to derive the “guessable wave" from a 
statistics on those computations, and so we can test Mechanism if it leads to 
more, or less extravaganza than Nature. It fits up to now. So with Mechanism, 
we get the *appearance* of many interfering “worlds”, and this without any 
worlds, from just the natural numbers and the laws of addition and 
multiplication. I will show that with the combinators as it is much shorter 
(but still long) than showing this with the numbers. This is known by logicians 
since the 1930s (I mean that a universal Turing machine is an arithmetical 
object). Computationalism, or Indexical Digital Mechanism imposes a Many-Dreams 
internal interpretation of Arithmetic (or combinator theory, or game-of-life 
theory, … we have to assume only one universal machinery).

Bruno




> 
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