Cognitive Theoretic Model of the Universe

[Quentin Anciaux-2]

Well as FORTRAN is a turing complete language, then you can.

As long as the programming language is universal/turing complete you can.

http://en.wikipedia.org/wiki/Turing_completeness

Regards,
Quentin

2009/6/5 ronaldheld <RonaldHeld@...>:

>
> Bruno:
>  I understand a little better. is there a citition for a version of
> Church Thesis that all algorithm can be written in
> FORTRAN?
>                         Ronald
>
>
> On Jun 4, 10:49 am, Bruno Marchal <marc...@...> wrote:
>> Hi Ronald,
>>
>> On 02 Jun 2009, at 16:45, ronaldheld wrote:
>>
>>
>>
>> > Bruno:
>> >   Since I program in Fortran, I am uncertain how to interpret things.
>>
>> I was alluding to old, and less old, disputes again programmers, about
>> which programming language to prefer.
>> It is a version of Church Thesis that all algorithm can be written in
>> FORTRAN. But this does not mean that it is relevant to define an
>> algorithm by a fortran program. I thought this was obvious, and I was
>> using that "known" confusion to point on a similar confusion in Set
>> Theory, like Langan can be said to perform.
>>
>> In Set Theorist, we still find often the error consisting in defining
>> a mathematical object by a set. I have done that error in my youth.
>> What you can do, indeed, is to *represent* (almost all) mathematical
>> objects by sets. Langan seems to make that mistake.
>>
>> The point is just that we have to distinguish a mathematical object
>> and the representation of that object in some mathematical theory.
>>
>> I will have the opportunity to give a precise example in the 7th
>> thread later.
>>
>> In usual mathematical practice, this mistake is really not important,
>> yet, in logic it is more important to take into account that
>> distinction, and then in cognitive science it is *very* important.
>> Crucial, I would say. The error consisting in identifying
>> consciousness and brain state belongs to that family, for example. To
>> confuse a person and its body belongs to that family of error too.
>>
>> All such error are of the form of the confusion between the Moon and
>> the finger which point to the moon, or the confusion between a map and
>> the territory.
>>
>> I have nothing against the use of FORTRAN. On the contrary I have a
>> big respect for that old venerable high level programming language :)
>>
>> Bruno
>>
>> http://iridia.ulb.ac.be/~marchal/
> >
>

--
All those moments will be lost in time, like tears in rain.

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[Bruno Marchal]

On 04 Jun 2009, at 21:23, Brent Meeker wrote:

>
> Bruno Marchal wrote:
>> ...
>>> Bruno Marchal wrote:
>>
>> The whole point of logic is to consider the "Peano's axioms" as a
>> mathematical object itself, which is studied mathematically in the
>> usual informal (yet rigorous and typically mathematica) way.
>>
>> PA, and PA+GOLDBACH are different mathematical objects. They are
>> different theories, or different machines.
>>
>> Now if GOLDBACH is provable by PA, then PA and PA+GOLDBACH shed the
>> same light on the same arithmetical truth. In that case I will
>> identify PA and PA+GOLDBACH, in many contexts, because most of the
>> time I identify a theory with its set of theorems. Like I identify a
>> person with its set of (possible) beliefs.
>>
>> If GOLDBACH is *true, but not provable* by PA, then PA and PA
>> +GOLDBACH
>> still talk on the same reality, but PA+GOLDBACH will shed more light
>> on it, by proving more theorems on the numbers and numbers relations
>> than PA. I do no more identify them, and they have different set of
>> theorems.
>>
>> If GOLDBACH is false. Well GOLBACH is PI_1, that is, its negation is
>> SIGMA_1, that is, it has the shape "it exist a number such that it
>> verify this decidable property". Indeed the negation of Goldbach
>> conjecture is "it exists a number bigger than 2 which is not the sum
>> of two primes". This, if true, is verifiable already by the much
>> weaker RA (Robinson arithmetic). So, if GOLDBACH is false PA +
>> GOLDBACH is inconsistent. That is a mathematical object quite
>> different from PA!
>
> So what then is the status of the natural numbers?  Are there many
> different objects in Platonia which we loosely refer to as "the  
> natural
> numbers" or is there only one such object and the Goldbach  
> conjecture is
> either true of false of this object?

Nobody can answer this question in your place.
But if you believe that the principle of excluded middle can be  
applied to closed arithmetical sentences, like 99,999% of the  
mathematician, then you have to believe that the Goldbach conjecture  
is either true or false.
Even intuitionist will admit that Goldabch conjecture is true or  
false, given its Sigma_1 character. This means that, about the (true-
or-false) nature of GOLDBACH is doubtable only for an ultrafinitist.
BTW, Goldbach conjecture asserts that all female (even) numbers can be  
written as a sum of two primes, except the number two. (I forget the  
word "even" in my enunciation above!).





>
>>
>> Here, you would have taken the twin primes conjecture, and things
>> would have been different, and more complex.
>
> Because, even if it is false, it cannot be proven false by  
> exhibiting an
> example?

Yes. And this entails that both PA+TPC and PA + (~TPC) could be  
consistent, yet one of those theory has to be unsound, or if you  
prefer has to enunciate false arithmetical statements (yet consistent  
with PA).
"Sound" is relative to the usual understanding of the natural numbers  
which is presupposed in any work in mathematical logic or computer  
science, like it is presupposed in any part of any physical theory.  
That usual meaning is taught in primary school without any trouble.
In model theory, this notion of soundness can be made more precise,  
through the notion of standard model of PA for example, but this  
presupposes, in the meta-theory, an understanding of that usual notion  
of numbers.
Nobody doubts the consistency and soundness of the theories like RA  
and PA. (Even Torgny, who fakes that he doubts them for a  
philosophical purpose unrelated to our discussion, like he fakes to be  
a faking zombie, etc. This is clear from older post by Torgny).

>
>
>>
>> Note that a theory of set like ZF shed even much more large light on
>> arithmetical truth, (and is still incomplete on arithmetic, by  
>> Gödel ...).
>> Incidentally it can be shown that ZF and ZFC, although they shed
>> different light on the mathematical truth in general, does shed
>> exactly the same light on arithmetical truth. They prove the same
>> arithmetical theorems. On the numbers, the axiom of choice add
>> nothing. This is quite unlike the ladder of infinity axioms.
>>
>> I would say it is and will be particularly important to distinguish
>> chatting beings like RA, PA, ZF, ZFC, etc... and what those beings  
>> are
>> talking about.
>>
>> Bruno
>
> Do you mean PA talks about the natural numbers but PA+theorems is a
> different mathematical object than N?

I am not sure I understand what you mean. PA is an (immaterial)  
machine, or a program if you want. I guess that, by PA+theorems, you  
mean the set of theorems of PA. In some context we can identify PA and  
PA+theorems, because the context makes things unambiguous. But  
strictly speaking those are different mathematical object: PA is  
finite (well, as I defined it usually), But PA+theorems is infinite.  
Both talk about N, and both are different of N. Indeed PA is a finite  
(or infinite in the usual first order presentation) set of axioms and  
rules, PA+theorems is an infinite set of formula, and N is an infinite  
set of numbers. That is very different. Of course both PA and PA
+theorems (your wording) talk really about the structure (N, +, x),  
that is the set of numbers N together with its additive and  
multiplicative structure, as studied in school.

It is important to distinguish a theory or a machine (usually a finite  
object), with the set of statements proved by that theory or machine  
(usually an infinite set).
And it is important to distinguish both of them with the semantical  
content of those statements produce by that theory or machine. In  
metamathematics (or mathematical logic) that "semantical content" will  
itself be represented by a mathematical object (a model) in some other  
theory (usually set theory, or category theory, or model theory).

With respect to the current thread on the seven step, this is of  
course sort of advanced remarks. But mathematical logic is not an easy  
subject. Many things which are not distinguished in the usual practice  
of mathematics or physics are distinguished by logicians.

Bruno

http://iridia.ulb.ac.be/~marchal/




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[Bruno Marchal]

On 05 Jun 2009, at 14:23, ronaldheld wrote:

>
> Bruno:
> I understand a little better. is there a citition for a version of
> Church Thesis that all algorithm can be written in
> FORTRAN?


The original Church Thesis, (also due to Post, Turing, Markov, Kleene,  
and others independently)

is this:

A function is computable if and only if it is programmable in LAMDA  
CALCULUS.

Then it is an easy but tedious exercise of programing to show that you  
can simulate LAMDA CALCULUS with FORTRAN, and that you can simulate  
FORTRAN with LAMBDA CALCULUS. So they compute the same functions.

And the same is true with LISP, or JAVA, or ALGOL, or C++, etc... in  
the place of FORTRAN.

A thorough introduction to Church thesis, and I would say one far  
deeper than usual, is integrally part of the seventh step of UDA. So  
we will come back on this soon or later. Church thesis is really the  
key and the motor of both UDA and AUDA. I have discovered that it is  
rarely well understood, even by many "experts". Like Gödel's theorem,  
Church's thesis is often deformed or misused.

Bruno


http://iridia.ulb.ac.be/~marchal/




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