On 06 Jan 2014, at 20:05, Telmo Menezes wrote:

On Mon, Jan 6, 2014 at 6:31 PM, Bruno Marchal <marc...@ulb.ac.be> wrote:
Dear Stephen,


On 03 Jan 2014, at 20:21, Stephen Paul King wrote:

Dear Bruno,

 I do not understand something.



OK. (good!)



Your idea


It is not an idea, but a result in an hypothetical context (or theoretical
context).



seems to me to be a very sophisticated and yat sneaky way of reintroducing Newton/Laplacean absolute time and/or Leibnitz' Pre-established Harmony.


It is only a remind of elementary arithmetic. The music 0, s0, ss0, sss0,
ssss0, sssss0, ssssss0, sssssss0, ...
You can see it as an elementary block digital time. If you want. And then
all other times are relative indexicals, including the physical and
subjective times.

Bruno,

I think I (perhaps naively) understand what you mean. My understanding
is that, if comp is true, then the relationship between comp and the
physical laws we observe is not a simple one. Even QM would be at a
high level of abstraction in relation to raw reality. In this case,
the recursive definition of integers would be the simplest possible
expression of a fundamental building block that is responsible for
time -- although the time we experience is a much more complex
phenomena.

It makes sense to me that time is strongly related to recursivity
(maybe because of a CS background). I imagine moments being "copied
forward" and changed in some fashion.

Would you agree with these intuitions?

Yes. But recursivity relies on the ordering 0, s0, ss0, ... which is admitted in the axioms, and so is a notion of time more primitive than the recursive definition by themselves. Another notion of time, which is still rather primitive is the time step of the computations implemented by the UD (or arithmetic) like the phi_344(76)^1, phi_344(76)^2, phi_344(76)^3, phi_344(76)^4, phi_344(76)^5, phi_344(76)^6, phi_344(76)^7, ... (with phi_i(j)-n = the nth step of the computation of program i on input j. It is different from the UD steps, because the UD dovetails, and can execute billions of steps between phi_344(76)^6 and phi_344(76)^7 for example. But those times have no direct link with the "observed time" from inside, which emerges from the logic of the first person points of view. The logic of Bp & p, and Bp & Dt & p, have some canonical temporal significations. Your time is eventually defined by the set of your continuation. It can happen that phi_i(j)^n is lived by you statistically after some phi_i(j)^m with m < n, a priori.

Bruno











I recall reading how much Einstein himself loved the idea and was loath to give it up, thus motivating his quest for a classical grand unified field
theory. Physics has moved on...


After Aristotle Physics has also moved on ... I think Einstein was right on
QM, and wrong on GR, in the sense that GR has to be justified by the
quantum, before, perhaps justifying the quantum by the "digital seen from
inside".





You recently wrote:

"The only "time" needed for the notion of computation is the successor relation on the non negative integers. It is not a physical time, as it is
only the standard ordering of the natural numbers: 0, 1, 2, 3, etc.

So, the 3p "outer structure" is very simple, conceptually, as it is given by the standard structure, known to be very complex, mathematically, of the additive/multiplicative (and hybrids of course) structure of the numbers (or
any object-of-talk of a universal numbers).

That is indeed a quite "static" structure (and usually we don't attribute consciousness to that type of thing, but salvia makes some (1p alas) point
against this)."


Let me try to clarify how I am confused by this claim.


OK.



How many different versions of the integers "exist"?

AFAIK, there can be only One and it is this *One* that acts as the "time"
(maybe) in your argument for all other "strings" of integers.


?

I have no clue what you are talking about. I am talking about the usual, standard, finite and non negative integers, also known as natural numbers. I am not doing philosophy, so any problem you might have with this might comes from unecessary over-interpretation you make, over what you have been
supposed to have learned in high school.





Are the "strings" distorted and/or incomplete "shadows" of the One?

Are we permitted to use the allegory of the cave here? :-)



Yes, but you need to do the work to understand the "real thing". We start
from arithmetic, that is:

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

or even just

Kxy = x
Sxyz = xz(yz)

((K x) y) = x
(((S x) y) z) = ((x z) (y z))

And we stay in that theory.

In that theory we define the observer by a believer in the axioms:


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

together with the infinity of beliefs in the following induction axioms
(with F any formula in logic + {0, s, +, *}):

(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x)

Just that is already very long to do, but that is done in the literature and
is basically the "known" arithmetization of meta-arithmetic.

Then incompleteness entails the nuances between proof and truth, and
consistency, and the double completeness theorem of Solovay provides the 8
hypostases, and we see that the classical introspecting machines can
understand by herself that what she observe might be only the shadow of the
truth.
Indeed.





How many "shadows" are there and how are they "distinguished" from each
other such that the notion of a computation is not lost?


By the study of the degrees of unsolvability. Notably.




In my work I have found that theoreticians in computer science completely take for granted that a computation is a process that can only occur in the
absence of randomness.


That is well studied. It is computability relativized to oracles.
Computability on random oracle has been studied.





Imagine if the atoms making up the CPU of your computer where to suddenly start changing their positions and states due to outside interactions in a
random/uncontrolled way?


That happens when I smoke a psychotropic plant, if not when I breath the
polluted air.



  No computation would occur!


Let us not exaggerate. No need to smoke the grass of Fukushima.




In fact, this is the situation that we find when, for instance, the cooler
fan fails and the CPU overheats.



Yes. The hypostases might be used to study the 1p associated to such extreme
events. Would this give a NDE? Difficult questions, which needs some
technical progresses.




My point here is that the string of states that is a von Neumann computation


von Neumann, Babbage, Turing, Church, Conway, Post, McCarthy, etc. OK.



is something that has to be separable and/or isolated to be able to be said
to "occur" or -to use the Platonic metaphor- "exist".


We start from the "E" interpreted in the usual way, like in "16 has a
successor".
And gives 8 different notion of existences, in the eight hypostases (which are each a mathematics with an intensional arithmetical interpretation).

You get physics when you restrict the arithmetical interpretation on the
sigma_1 sentences, on the material hypostases.




So, what exactly is separating the "strings of integers" from each other and
the One, such that we can coherently discuss them as actually being
computations and not just "representations of computations"?


The trueness of their relative association, together with their
redundancies. At the bottom, what do the separation are the additions and multiplication, they separate the computations which halt from those who
does not halt, the first person views do the rest.

Hope this helps.

Best,

Bruno




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Kindest Regards,

Stephen Paul King



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