On 12 May 2015, at 18:47, John Clark wrote:
On Tue, May 12, 2015 Bruno Marchal <[email protected]> wrote:
>> No, what they proved is that physical reality can emulate
arithmetic;
> False. Just read the original paper of Church, Post, Turing,
Kleene, please. They don't mention physics at all.
Please explain how to build a Turing Machine, or a machine of any
sort, without using matter that obeys the laws of physics.
Why would Turing machine obeys the laws of physics?
Turing machine, like numbers and combinators, does not obeys to laws
of physics, but to mathematics.
The Turing machine is a mathematical notion which mimics a
mathematician doing a computation, of some function from N to N, or
NXN to N, etc, with pen and paper.
You can implement Turing machine in Lambda calculus, and in that case,
you saty in the mathematical.
You can implement them in Fortran, in Algol, ... which are still
mathematical objects, immaterial, and duplicable.
And, you can implement them in the physical reality, apparently, but
in that case, and only in that case, you have to take into account the
physical laws.
Don't ask me to change water in wine.
> BTW, they prove that the arithmetical reality is NOT emulable by
anything. the computable is only a quite tiny part of arithmetic.
I know, nearly all numbers are non computable
I told you that by numbers I mean integers, what you call number here
are non computable functions. It is preferable to stick on the
function from N to N, to use the simple tools available there.
so physics doesn't know what they are,
You don't know that. If we are machine, reality is not a machine, and
with comp physics is an important part of that reality, doubtfully
completely computable (like the position of an electron going in the
slit, note).
but mathematics doesn't know what they are either,
If by mathematics you mean tha arithmetical truth, then mathematics
knows the arithmetical truth.
Let me ask you do you believe that the following proposition: either
there are positive integers x and y such that x^2 = 991y^2 + 1, or
there are none.
they aren't the solution to any polynomial equation and no function
can produce them, an infinite series can't even approximate one.
At this stage, a plea for intuitionism is inadequate. It implies non-
comp (strictly speaking).
>> and one sort of physical reality, like a electronic computer, can
emulate another sort of physical reality, like a galaxy, but we have
no evidence that arithmetic can emulate anything.
> The proof is in all textbooks
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.
> The sigma_1 arithmetical reality is Turing universal. Robinson
arithmetic is Turing universal.
Then I suggest you start the Sigma_1 Arithmetical Reality Computer
Company with a Robinson Arithmetic subdivision and become the
world's first trillionaire.
<sigh>
> No, the reason is that they want buy physical computer, to enacted
the computation relatively to our physical reality.
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,
bfore we build computers (except for Babbage). Formally, an important
set of those objects (functions) appears in Gödel 1931.
On the contrary, still today, physical computation is defined by the
ability by nature to emulate (approximatively) those mathematical
objects.
> 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, as physics has just to be redefined
by Plato-Aristotle "bastard calculus".
Logical mathematical tools gives already the logic of observable for
"reasonable machine".
>>> (once you agree that 2+2=4 is a simpole truth on which we can
agree on).
>> We may agree on that but Godel and Turing tell us that there are
an infinite number of mathematical statements we will NEVER agree on,
> They don't say that. They say that for all consistent machine
there are statement that they cannot prove.
Godel said there are an infinite number of statements that are true
(so you can never find a counterexample to prove it wrong)
?
You confuse with Chaitin. And Gödel took pain to not use the concept
of truth, which was unclear at that time. So I can't see to which
theorem you allude too. You would have the slighest understanding of
Gödel's theorem, you would know for long that a tiny part of
arithmetic emulates the whole range of the computable.
but that have no proof (so you can't demonstrate its truth in a
finite number of steps).
Actually the unprovable statements by machine M are provable by the
machines which can prove M to be consistent.
If there were a way to put statements into two categories, the
statements that can be proven true or false into one category and
the statements that are either false or true but have no proof into
another category we could concentrate our efforts on the first
category and just ignore the second, but Turing proved that in
general you can't even do that.
yes, it means the boundary is a bit like the boundary of the
Mandelbrot set. It does not mean one is fuzzy or not.
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.
Note that here one of the alternative will halt. It is still Pi_1,
like Rieman Hypothesis. For the Syracuse conjecture (all numbers go to
1 when iterating the rules: if n is odd divided by two else multiply
by 3 and add 1.This one is sigma_2, it involves soling a proposition
like ExAyP(x,y), thinking quickly. For such proposition the serach of
the negation and the affirmation can be both infinite, but this does
mean that it is not provable in PA, or in ZF, or in ZF+kappa, but as
Erdos said, it is far beyond what we can solve today. yet, we do have
the intuition here two, that each individual will either go to one or
will not.
Nowhere than in arithmetic mathematicians have studied the difference
between truth and proofs, and provided convincing mathematical
definition of both.
>> mathematical statements that even mathematics doesn't know if
they are true or not.
> By definition, math "knows" the truth of the statement.
90 years ago every good mathematician would have agreed with you,
even Godel would have agreed with you, he was as surprised as anyone
with what he discovered in his 1930 proof, but today no good
mathematician would agree with you.
I disbelieve this completely. Only a minority, and only at the pause
café. In science we don't mock the assumption of people, as long s the
theory is clear, the reasoning valid, etc.
True, some people does not understand the impact of Church thesis, and
know virtulaly nothing on Gödel, which indeed was set theoretical
realist.
But again, comp uses only the arithmetical realism, which is shared by
anyone succeeding the exam of primary school.
> 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 in term of interpretation and model.
Yes, machine can't separate effectively true arithmetical proposition
from false one, but that is the way such a truth kicks back, and with
comp, Gödel's theorem is an argument in favor of arithmetical realism.
nothing can do it, not physics and not even mathematics itself can
The oracle consisting of the set of Gödel numbers of the true
propositions (true in the standard model (N,+,*)) can do it. That set
is not computable, but this does not prevent it to be highly
structured, and that we can't prove many thing about.
Read the book by Torkel Franzen, "Inexhaustibility", which treats
rather well the notion of arithmetical truth.
Bruno
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