Le 13-août-07, à 17:37, David Nyman a écrit :
> On 11/08/07, Bruno Marchal <marchal.domain.name.hidden> wrote:
>
>> That the 'comp reality' is founded on the number realm, is almost
>> trivial. What is not trivial at all, and this is what the UDA shows,
>> is
>> that, once you say "yes" to the digital doctor, for some level of
>> substitution, then your immateriality (somehow like the quantum
>> superpositions) is contagious on your most probable neighborhoods. So
>> that physics has to be shown to emerge statistically from the
>> "measure"
>> on the UD accessible relative states.
>
> This perhaps needs to be explained more slowly. What - exactly - do
> you mean by "immateriality", and "contagious on your most probable
> neighbourhoods"?
Nothing magical. For example the game of bridge is immaterial, despite
you need some local matter to implement some particular bridge game.
All abstract concepts are immaterial. You don't have the need to
explain what bosons and fermions are to explain what is a number.
Like Stathis argued a lot, if you identify yourself with your
history/personality there is a sense to be 50 years old, but if you
identify yourself with your matter, you disappear a bit by eating and
shitting (is this correct? polite?) and respiring, etc. and never
really exist for more than seven years, etc... In the first case you
admit somehow your immateriality despite contingent and local
appearance of being composed in some substancial way.
What comp (by UDA+FILMED-GRAPH) shows, is that, once the digitalness of
your local relative description is taken seriously, you can no more
distinguish the comp stories existing below your comp substitution
level. Eventually the laws of physics will be the law of what remains
or emerges as observable in all computations. From inside this has to
interfere statistically (by UDA).
> Also, precisely what do you intend to be included in
> - or excluded from - the notion of 'physics' in this context? Is it
> to be equated with what is observable, and if so, how and by whom?
By lobian machine. By lobian machine, as they are emulated by a weaker
machine (the universal dovetailer, alias the arithmetical Sigma1
sentences).
How? I would say by self-measurement relatively to their most probable
(or credible ...) comp histories. There is always an infinity of them.
I said in Siena, and already in this list, that for Plato, what *you*
see (observe, measure) is the border of what *you* don't see. In the
universal machine context this can lead to a recursive but solvable
equation where physical reality is a sort of border of the comp-
indeterminacy or the comp intrinsical ignorance.
>
>> To be sure, it is needed,
>> however, for the understanding that with comp, we *have to* derive the
>> physics from "intensional numbers prevailing discourses". With comp,
>> postulating a physical world cannot be used as an explanation relating
>> mind and appearance of matter (memory-stable observations).
>> It is not that (aristotelian primary )substance does not exist, but
>> that such primary substance is provably (with the comp hyp) void of
>> explanation power.
>
> It strikes me, reading the above, that it might be a good idea to find
> a way to limit ourselves -
Yes. (but hard!)
> at this deliberately elementary stage - to
> an agreed set of terms with which to designate each of your key ideas,
> for example with respect to physics deriving from "intensional number
> prevailing discourses". Perhaps what we need is not so much
> grandmother-version, but a kindergarten-level introduction to the key
> terms and concepts, which we can then use slowly and clearly to build
> up the argument.
I have to explain enough notions so as to be able to explain Solovay
theorem, which links precisely the modal logic G and G* with the
discourse of the ideally correct lobian machines. "Machine theology" is
really captured by the study of G*, G* \ G, and their intensional
(modal) variants (but this is again an anticipation).
Most key terms and concepts are from mathematical logic and computer
science key terms and concepts, at least for the arithmetical part.
You can see my thesis either as a the showing that comp necessitates to
generalize Everett embedding of the subject into the physical world.
(Cf also Rossler endophysiocs). Indeed comp forces us to embed the
arithmetician (or any memory machine) in numberland (something for
which we will never have a complete unification).
But you can also see the technical part of the thesis as a
reconstruction of Lucas-Penrose godelian argument. This reconstruction
has a long history which, paradoxically enough precedes both Lucas
(1960) or Penrose (Much latter). The line is: Post, Benecerraf, Wang,
Reinhardt, (myself), Webb, etc.
In a nutshell, you cannot use Godel incompleteness to show that we are
not machine (or that we are not lobian), but you can use Godel
incompleteness to argue that IF we are sound lobian machine then we
cannot know which machine we are, still less which computations support
us. It gives the arithmetical origin of the first person comp
indetermincacy, which you are supposed to have already intuitively
swallow from the UDA, OK?
> At each stage, perhaps you could refer to the
> appropriate points in the UDA, or other key papers, that could then be
> consulted for comparison and further elucidation. Would this work?
OK. Don't hesitate to remind me.
>
>> The best book is without
>> doubt the one by Cutland:
>>
>> CUTLAND N. J., 1980, Computability An introduction to recursive
>> function theory,
>> Cambridge University Press.
>
> Thanks
I think this one is the best. But the subject is so beautiful that you
can find other very nice book (I think of the classical bible of
recursion theory, the book by Rogers, Tourlakis comes to my mind.
Calude also did wrote a cute technical textbook, Saloma too, aargh
there are too much good books! OK, keep on on Cutland and Franzen.
(A chef d'oeuvre which is hard to find is the preprint by Machtey and
Young, Cutland acknowledge being inspired by it).
>
>> OK. I will begin by saying two words on the language we will use when
>> discussing with the machine. I can already explain the difference
>> between the layman (or grandmother) and the logician. This is not just
>> for you (I guess you know what I will say) but for those who just
>> abandon logic for reason of notation.
>>
>> The main difference is that where a layman says "Alfred is serious",
>> the logician says serious(Alfred).
>>
>> Where the layman will say there is a ferocious dog, a logician will
>> say
>> that it exists something such that that something is a dog and is
>> ferocious. Because of laziness he will write Ex(dog(x) &
>> ferocious(x)).
>> For saying that all dogs are ferocious, he will say that for all dogs
>> (i.e. choose any thing that is a dog) that things will be ferocious:
>> and he will write Ax (dog(x) -> ferocious(x)).
>>
>> Of course, there is perhaps no effective test to see if a dog is
>> ferocious or not, perhaps the notion is not well defined, but we have
>> to live with things like that: even in the pure realm of numbers we
>> will encounter some unexpected (I guess) complexity.
>
> Thanks, this is useful.
So, where the layman says "17 is prime", the logician writes
"prime(17)".
But let me add two words on function. Actually you can see "prime(17)
as a function, defined on numbers and having {true, false}, or {t, f}
as range: for example prime(17) = t, and prime(14) = f. OK?
Where a layman says: the temperature in Toulouse is 34.5, the logician
says: temperature(Toulouse) = 17. And, here, you can see
"temperature" as a function from location (city) in the real numbers. A
function is just such an association. The key idea is that a function
will always have zero or one value. There is no sense to say that the
temperature in toulouse is 34.5 and 23.7.
Childhood operation on numbers are usually function, even binary
functions (functions with two arguments, inputs, ...). Where the layman
will say: the sum of 3 and 4 gives 7, the logician will write: +(3,4) =
7. They put the name of the function in front, and they put the
arguments after. So an arbitrary function from n-tuple to number will
be denote by f(x_1, x_2, ..., x_n). Exactly like in the definition of
an arbitrary derivative in calculus: the limit, if it exists, for h
going near zero of the quotient f(x + h) - f(x) with h. OK?
>
>> By the way, David, do you know what is called "classical propositional
>> calculus", the truth table method? Do you need some refreshing?
>> Some refreshing is in Smullyan's FU, but I can do it, or focus on some
>> difficulty (classical propositional calculus is not so simple indeed,
>> even if simpler than most other logics).
>
> I can always use wikipedia - which I've looked at - or other sources
> online, but anything you would also be prepared to do here would be
> most helpful.
OK. The road is the following:
I propose to go, from the Cantor non-enumerability of the reals (or
things equivalent) to Kleene non recursive enumerability of the
recursive reals, by Church thesis. Comp, both in the UDA, and in the
arithmetical UDA, is mainly Church thesis. I want to show you how
strong and deep that thesis is. OK?
>
>> I don't think Church thesis can be grasped
>> conceptually without the understanding that the class of programmable
>> functions is closed for the diagonalization procedure.
>
> Please explain 'programmable functions' and 'closed for the
> diagonalisation procedure'.
Of course, today, there is a simple way to say what is a programmable
function: it is a function that you can program on a computer. But wait
I explain Church thesis, for really making sense of this.
I can explain to you what I mean by: a set is closed for this or that
function/operation/procedure. A set is closed for some operation
applicable to its elements when such application does not lead outside
the set. Examples:
the set of natural numbers N = {0, 1, 2, 3, 4, 5, ...} is closed for
addition
the set of natural numbers N = {0, 1, 2, 3, 4, 5, ...} is NOT closed
for subtraction
the set of natural numbers N = {0, 1, 2, 3, 4, 5, ...} is closed for
multiplication
the set R of real numbers is closed for addition, multiplication,
division, subtraction, but ...
is not closed for the square roots
the set of humans is closed for sexual reproduction
Etc ... OK?
Now diagonalization will appear to be a sort of "transcendental
operation". Its main use is for going outside some set, and I would
like to convey why the fact that the set "programmable functions" is
closed for diagonalization is truly a miracle! (to borrow Godel's
expression). It is really that miracle which makes the set of
programmable or computable function fitting so well the search for
universal everything theory.
>
>> Do everyone
>> (interested) know how to prove the non enumerability of the subset of
>> N
>> by diagonalization?
>
> Which subset do you mean? I've encountered the
> diagonalisation/enumerability argument, assuming it's the one I
> referred to above.
Well, ok, sorry. Instead of "the non enumerability of the subset of N",
read "the non enumerability of the set of subsets of N".
Have you take a look on my old diagonalization post which I send to
Lennart?
>
>> Let us go slow and deep so that everybody can understand, once and for
>> all. OK?
>
> Definitely OK.
Did this post helped? I want you to understand Church thesis, before
the description of some formal language. This will economize work, and
help you disentangle the rigorous from the formal. In our setting,
"formal" will always mean "output by a machine". Don't believe that
formal = rigor. That would be equivalent to believe that all machines
are correct (a nonsense). OK?
Bruno
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Aug 15 2007 - 07:36:40 PDT