On 10 Jan 2012, at 12:58, acw wrote:
On 1/10/2012 12:03, Bruno Marchal wrote:
On 09 Jan 2012, at 19:36, acw wrote:
To put it more simply: if Church Turing Thesis(CTT) is correct,
mathematics is the same for any system or being you can imagine.
I am not sure why. "Sigma_1 arithmetic" would be the same; but higher
mathematics (set theory, analysis) might still be different.
If it's wrong, maybe stuff like concrete infinities,
hypercomputation
and infinite minds could exist and that would falsify COMP, however
there is zero evidence for any of that being possible.
Not sure, if CT is wrong, there would be finite machines, working in
finite time, with well defined instructions, which would be NOT
Turing
emulable. Hypercomputation and infinite (human) minds would
contradict
comp, not CT. On the contrary, they need CT to claim that they
compute
more than any programmable machines. CT is part of comp, but comp
is not
part of CT.
Beyond this, I agree with your reply to Craig.
In that response I was using CT in the more unrestricted form: all
effectively computable functions are Turing-computable.
I understand, but that is confusing. David Deutsch and many physicists
are a bit responsible of that confusion, by attempting to have a
notion of "effectivity" relying on physics. The original statement of
Church, Turing, Markov, Post, ... concerns only the intuitively human
computable functions, or the functions computable by finitary means.
It asserts that the class of such intuitively computable functions is
the same as the class of functions computable by some Turing machine
(or by the unique universal Turing machine). Such a notion is a priori
completely independent of the notion of computable by physical means.
It might be a bit stronger than the usual equivalency proofs between
a very wide range of models of computation (Turing machines, Abacus/
PA machines, (primitive) recursive functions (+minimization), all
kinds of more "current" models of computation, languages and so on).
Yes. I even suspect that CT makes the class of functions computable by
physics greater than the class of Church.
If hypercomputation was actually possible that would mean that
strong variant of CT would be false, because there would be
something effectively computable that wasn't computable by a Turing
machine.
OK.
In a way, that strong form of CT might already be false with comp,
only in the 1p sense as you get a free random oracle as well as
always staying consistent(and 'alive'), but it's not false in the 3p
view...
Yes. Comp makes physics a first person plural reality, and a priori we
might be able to exploit the first plural indeterminacy to compute
more function, like we know already that we have more "processes",
like that free random oracle. The empirical fact that quantum computer
does not violate CT can make us doubt about this.
Also, I do wonder if the same universality that is present in the
current CT would be present in hypercomputation (if one were to
assume it would be possible)
Yes, at least for many type of hypercomputation, notably of the form
of computability with some oracle.
- would it even retain CT's current "immunity" from diagonalization?
Yes. Actually the immunity of the class of computable functions
entails the immunity of the class of computable functions with oracle.
So the consistency of CT entails the consistency of some super-CT for
larger class. But I doubt that there is a super-CT for the class of
functions computable by physical means. I am a bit agnostic on that.
As for the mathematical truth part, I mostly meant that from the
perspective of a computable machine talking about axiomatic systems
- as it is computable, the same machine (theorem prover) would
always yield the same results in all possible worlds(or shared
dreams).
I see here why you have some problem with AUDA (and logic). CT = the
notion of computability is absolute. But provability is not absolute
at all. Even with CT, different machine talking or using different
axiomatic system will obtain different theorems.
In fact this is even an easy (one diagonalization) consequence of CT,
although Gödel's original proof does not use CT. provability, nor
definability is not immune for diagonalization. Different machines
proves different theorems.
Although with my incomplete understanding of the AUDA, and I may be
wrong about this, it appeared to me that it might be possible for a
machine to get more and more of the truth given the consistency
constraint.
That's right both PA + con(PA) and PA + ~con(PA) proves more true
arithmetical theorems than PA.
And PA + con(PA + con(PA + con (PA + con PA)) will proves even more
theorems. The same with the negation of those consistency.
Note that the theory PA* = PA* + con(PA*), which can be defined
finitely by the use of the Kleene recursion fixed point proves ALL the
true propositions of arithmetic!!! Unfortunately it proves also all
false propositions of arithmetic. This follows easily by the second
theorem of Gödel, because such a theory can prove its own consistency
given that (con "itself") is an axiom, and by Gödel II, it is
inconsistent.
But, yes, once you have a consistent machine, you can extend its
provability ability on the whole constructive transfinite.
As for higher math, such as set theories: do they have a model and
are they consistent? (that's an open question) If some forms close
to Cantor's paradise are accepted in the ontology, wouldn't that
risk potentially falsifying COMP?
Trivially. You can refute comp in the theory ZF + ~comp (accepting
some formalization of comp in set theory).
Remember that, like consistency, (~ probable comp) is consistent with
comp. If comp is true, like consistency, it is not provable, and so
you can add (~ provable comp), or (con ~comp) to the comp theory of
everything (arithmetic) without getting an inconsistency with comp. Of
course you should not add ~comp to comp at the same (meta)level. You
will get a contradiction.
Likewise, as above, the theory PA + (PA is inconsistent) is
consistent. You can prove, in it, that the false is provable, but you
cannot prove, in it, the false. Bf -> f is not a theorem (Bf -> f) =
~Bf = consistency.
I can see many reasons why a particular machine/system would want to
talk about such higher math, but I'm not sure how it could end up
with different discourses/truths if the machine('s body) is
computable.
Here there is a difficulty, and many people get it wrong. There is a
frequent error in logic which mirrors very well Searles error in his
chinese room argument. With comp I can certainly simulate Einstein's
brain, but that fact does not transform me into Einstein. If someone
asks me a complex question about relativity, I might be able to answer
by simulating what Einstein would respond, but I might still not have
a clue about what the meaning of Einstein's answer. In fact I would
just make it possible for Einstein to answer the question. Not me.
Like wise, a quasi debilitating arithmetical theorem system like
Robinson Arithmetic, which cannot prove x+y = y+x, for example, is
still Turing universal, and as such can imitate PA perfectly through
its provability abilities. That is, RA is able to prove that PA can
prove x+y=y+x. But RA has not the power to be convinced by that PA's
proof, like I can simulate Einstein without having the gift to
understand any words by him.
RA can simulate PA and ZF, and even ZF+k (which can prove the
consistency of ZF), but this does not give to RA the *provability*
power of PA, ZF or ZF+k.
PA can prove that ZF can prove the consistency of PA, but PA can still
not prove the consistency of PA.
I can see it discovering the independence of certain axioms (for
example the axiom of choice or the continuum hypothesis), but
wouldn't all the math that it can /talk/ about be the same? The
machine would have to assume some axioms and reason from there.
Yes. And with different axioms you get different provability
aptitudes. Once a machine can prove all true arithmetical sigma_1
sentences (with the shape ExP(x) with P decidable) she is universal,
with respect to *computability. You can add as many axioms you want,
the machine will not *compute* more functions. But adding axioms will
always lead the machine into proving more *theorems*.
In AUDA, "belief" is modeled by provability (not computability), and
then knowledge is defined in the usual classical (Theaetetus) ways.
All beliefs of the correct machines will obeys to the same self-
referential logic, but all belief-content will differ from a machine
to a different machine. PA and ZF have the same self-referential
logics, but they have quite different belief, even restricted on the
numbers.
For example ZF proves more arithmetical truth than PA. ZFC and ZF+(~C)
proves exactly the same theorem in arithmetic, despite they proves
quite different theorem about sets (so arithmetic is deeply
independent of the axiom of choice). ZF+k (= ZF + it exists an
inaccessible cardinal) proves *much more* arithmetical theorem than ZF.
To sum up:
Computability is an absolute notion.
Provability is a relative notion.
BTW, acw, you might try to write a shorter and clearer version of
your
joining post argument. It is hard to follow. If not, I might take
much
more time.
Bruno
I think I talked about too many different things in that post, not
all directly relevant to the argument (although relevant when trying
to consider as many consequences as possible of that experiment). If
some parts are unclear, feel free to ask in that thread. The general
outline of what I talked about in the part you have yet to comment
on: a generalized form of the experiment the main character from
"Permutation City" novel performed is described in detail(assuming
COMP), a possible explanation for why it might not actually be
useless to perform such an experiment and why it might be a good
practical test for verifying the consequences described in the UDA,
various variations/factors/practicalities of that experiment are
discussed (with goals such as reducing white rabbits, among a few
others), some not directly relevant to the argument and at the end I
tried to see if the notion of observer can be better defined and
tried to show that the notion of "generalized brain" might not
always be an appropriate way to talk about an observer. That post
was mostly meant to be exploratory and I hoped the ensuing
discussion would lead to 2 things: 1) assessing the viability of
that experiment if COMP is assumed AND 2) reaching a better
definition of the notion of observer.
OK. This I think I understood this, but your style is not easy, and it
might be useful, even within your goal, to work on a clearer and
shorter version, with shorter sentences, without any digression, with
clear section and subsection, so as to invite most people (including
me) to grasp it, or to find a flaw, and this in reasonable time. In
particular I fail to see the point of discussing the use of different
universal systems like you did with the Cellular Automata (CA).
Bruno
http://iridia.ulb.ac.be/~marchal/
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