On 14 May 2015, at 22:04, John Clark wrote:
On Wed, May 13, 2015 Bruno Marchal <[email protected]> wrote:
> Why would Turing machine obeys the laws of physics?
Because a Turing Machine like all machines involves change.
The change are injection in N.
A clockwork must read a cell on a tape made of matter
Buy the Davis Book 1964, in the cheap paperback from Dover.
and determine if it is white or black, and a clockwork must
determine if it should change the color of that cell or not, and a
clockwork must determine if it should move the tape one space to the
right or one space to the left or just stop. And nobody knows how to
make clockwork without using matter that obeys the laws of physics.
Nobody, absolutely nobody.
Oh, so you do assume primitive matter.
But you are just wrong on what is a Turing machine. Turing makes it
look material for reason of pedagogy.
> You can implement Turing machine in Lambda calculus
No you can not!
Then not only Turing and Church were wrong, as they will both proves
this.
You can find the proofs, or similar, in any textbooks in computer
science.
All known universal systems have been proved to implement all known
universal systems. And with CT you can suppressed the "known".
The word "implement" means to put a plan into effect
Not in computer science. It means you can write a translator of one
universal system in another one, or you can write an interpreter of
one language in another one.
and Lambda calculus or any other type of ink on paper can not do that.
Lambda calculus, like number theory, has no relationship with ink and
paper.
You can find books about Lambda calculus that describe how Turing
Machines operate but it's just a description,
No. It is either compilation or interpretation, as universal entities
can do. You confuse computer science and physical computer science.
Those are different, and the second use the basic definition of the
first, up to now.
to actually make a Turing Machine as opposed to just talking about
one, you'll need matter and the laws of physics. A book about Lambda
calculus or about anything else can't calculate diddly squat.
> You can implement them in Fortran, in Algol,
Not unless you have a computer made of matter that obeys the laws of
physics to run those Fortran or Algol programs on.
No, if you agree that 2+2=4, and if you use the standard definition,
then you can prove that a tiny part of the standard model of
arithmetic run all computations.
>> nearly all numbers are non computable
> I told you that by numbers I mean integers, what you call number
here are non computable functions.
And what you call non computable functions Turing himself called non
computable numbers in the very 1936 paper that introduced the
concept that would later be called a "Turing Machine".
OK, Turing made two pedagogical mistakes, relatively to the question
treated here.
Read any of his other paper, in fact it hows the relative vice-versa
implementation of lambda and his "machine" formalism in that basic
paper.
Note that his definition of computable real numbers is wrong (as he
himself realized, and changed later. There is no Church-thesis for the
notion of real number computable.
> If we are machine, reality is not a machine, and with comp physics
is an important part of that reality
> If by mathematics you mean tha arithmetical truth, then
mathematics knows the arithmetical truth.
Nothing can divide all arithmetical truth from all arithmetical
falsehoods. Nothing can do it including arithmetic.
Sorry Arithmetical truth does it, trivially. Then that reality can
have his complexity measured, and their are degrees of unsolvability.
You just dismiss a whole branch of math.
> At this stage, a plea for intuitionism is inadequate. It implies
non-comp (strictly speaking).
I don't care, I'm not interested in "comp".
But you are a comp1 believer, and "comp" is comp1. Then it implies
comp2, which you fake to not understand, or you just play the
advocate's devil.
>> Ink on paper is in those textbooks, there is no evidence that any
book has ever been able to calculate anything, not even 1+1. You
want to fly across the Pacific Ocean on the blueprints of a 747 and
it just doesn't work.
> Grave confusion of level.
Maybe on some level our entire universe is just a simulation program
written in Fortran, but if it is as far as we know that program is
running on a computer made of matter that obeys the laws of physics,
How do you know that?
Especially knowing than the the sigma_1 tiny part of the arithmetical
truth realizes all computations, even all quantum computations.
We don't know that, and have no evidence, and even clues that the
physical universe might be the border of something else.
>> In other words those computer textbooks provide simplified and
approximated descriptions of how real computers operate.
> They described fundamental mathematical object which have been
discovered by mathematician working in the foundation of mathematics
Yes exactly, those textbooks DESCRIBE a Turing Machine,
They do both/ they define it, and then it will happens that such being
admits many description, but you need to distinguish between the
things and their description.
and the blueprints of a 747 DESCRIBE that airplane, but you can't
fly to Tokyo on a description.
But the numbers themselves, when implemented by certain type of
universal numbers can do that.
You just show that you are unaware of some fundamental discoveries in
logic and math.
Or you invoke a God for which we have no evidence.
> physical computation is defined by the ability by nature to
emulate (approximatively) those mathematical objects.
Mathematical computational "objects" are defined by their ability to
approximate physical computational objects.
Amen (you are a good Aristotelian Theologian). But, this is an
assumption.
>>> But the physical reality is used only for that relative
manifestation,
>> If so then physics can do something mathematics can not, make a
calculation that has a relationship with our world. Physics must
have some secret sauce that mathematics does not.
> Only if computationalism is false,
Sounding rather theological you just said that physical reality is
needed for some manifestations, so physics must have something
mathematics does not. QED.
Of course physics has something different from mathematics, like the
even numbers have something different from the set of all natural
numbers. Physics is the product of the machine's notion of
observability, but you need to go up to step 7 to grasp this.
>> Godel said there are an infinite number of statements that are
true (so you can never find a counterexample to prove it wrong)
> ?
!
>> If Goldbach's conjecture is in that second category (and if it
isn't there are an infinite number of similar statements that are)
then mathematicians could spend eternity looking (unsuccessfully)
for a way to prove that Goldbach's conjecture is true, and spend an
infinite number of years building ever faster computers looking
(unsuccessfully) for an even integer that is not the sum of two
prime numbers to prove that Goldbach's conjecture is false. So
after an infinite amount of work you'd be no wiser about the truth
or falsehood of Goldbach's Conjecture than you are right now.
> But I am pretty sure, to tell you my opinion, that the conjecture
is either true or false.
I accept that Goldbach is either true or false, Godel and Turing did
too, but the question is does anything, ANYTHING, know if it is true
or not.
Today we don't know, although most would bet it is provable in ZF.
Godel and Turing say not necessarily, and even if Goldbach is
provable or unprovable there are an infinite number of similar
statements that are not.
Yes, and they are all true of false. So the set of true sentences is
well defined, even if non constructively so. We can defined it in
second order logic, or analysis. Actually, we can defined it is much
restricted theories. There is lot of literature on this.
> mathematicians have studied the difference between truth and proofs,
The difference is not all true statements have a proof,
Proof is relative.
it less misleading to talk in term of those theories or machines. It
is that for all theories and machines, they will be true statement
(true in the standard model of arithmetic) that such machine cannot
prove. Usually they can find it, and change themselves accordingly.
There is a theory of autonomous progression.
and in general there is no way to determine which statements it is
possible to prove to be right or wrong and which statements you can
not.
Gödel and Post provided a constructive way to do exactly this.
Chaitin and Post provided also a non constructive way.
> comp uses only the arithmetical realism,
I don't care, I'm not interested in "comp".
You say that like a Mantra, but you would not participate to a
discusioon on comp for so long if you were not interested, especially
that you are a comp practioner.
>>> You confuse again the mathematical reality and the
mathematical theories.
>> Godel and Turing proved that there is no way even in theory to
totally separate mathematical reality from mathematical non-reality,
> They don't talk about reality.
But you just did.
Yes, but they did not.
I do in step 8. We have the time, I think.
Bruno
> with comp, Gödel's theorem is an argument in favor of arithmetical
realism.
I don't care, I'm not interested in "comp".
John K Clark
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