On 1/10/2012 12:03, Bruno Marchal wrote:
On 09 Jan 2012, at 19:36, acw wrote:
On 1/9/2012 19:54, Craig Weinberg wrote:
On Jan 9, 12:00 pm, Bruno Marchal<marc...@ulb.ac.be> wrote:
On 09 Jan 2012, at 14:50, Craig Weinberg wrote:
On Jan 9, 6:06 am, Bruno Marchal<marc...@ulb.ac.be> wrote:
I agree with your general reply to Craig, but I disagree that
computations are physical. That's the revisionist conception of
computation, defended by Deustch, Landauer, etc. Computations have
been discovered by mathematicians when trying to expalin some
foundational difficulties in pure mathematics.
Mathematicians aren't physical? Computations are discovered through a
living nervous system, one that has been highly developed and
conditioned specifically for that purpose.
Computation and mechanism have been discovered by many people since
humans are there. It is related to the understanding of the difference
between "finite" and "infinite". The modern notion has been discovered
independently by many mathematicians, notably Emil Post, Alan Turing,
Alonzo Church, Andrzei Markov, etc.
With the comp. hyp., this is easily explainable, given that we are
somehow "made of" (in some not completely Aristotelian sense to be
sure) computations.
They are making those discoveries by using their physical brain
though.
Sure, but that requires one to better understand what a physical brain
is. In the case of COMP(given some basic assumptions), matter is
explained as appearing from simpler abstract mathematical relations,
in which case, a brain would be an inevitable consequence of such
relations.
We can implement
computation in the physical worlds, but that means only that the
physical reality is (at least) Turing universal. Theoretical computer
science is a branch of pure mathematics, even completely embeddable
in
arithmetical truth.
And pure mathematics is a branch of anthropology.
I thought you already agreed that the arithmetical truth are
independent of the existence of humans, from old posts you write.
Explain me, please, how the truth or falsity of the Riemann
hypothesis, or of Goldbach conjecture depend(s) on anthropology.
Please, explain me how the convergence or divergence of phi_(j)
depends on the existence of humans (with phi_i = the ith computable
function in an enumeration based on some universal system).
The whole idea of truth or falsity in the first place depends on
humans capacities to interpret experiences in those terms. We can read
this quality of truth or falsity into many aspects of our direct and
indirect experience, but that doesn't mean that the quality itself is
external to us. If you look at a starfish, you can see it has five
arms, but the starfish doesn't necessarily know it had five arms.
Yet that the fact the starfish has 5 arms is a fact, regardless of the
starfish's awareness of it. It will have many consequences with
regards of how the starfish interacts with the rest of the world or
how any other system perceives it.
If you see something colored red, you will know that you saw red and
that is 'true', and that it will be false that you didn't see 'red',
assuming you recognize 'red' the same as everyone else and that your
nervous system isn't wired too strangely or if your sensory systems
aren't defective or function differently than average.
Consequences of mathematical truths will be everywhere, regardless if
you understand them or not. A circle's length will depend on its
radius regardless if you understand the relation or not.
Any system, be they human, computer or alien, regardless of the laws
of physics in play should also be able to compute (Church-Turing
Thesis shows that computation comes very cheap and all it takes is
ability of some simple abstract finite rules being followed and always
yielding the same result, although specific proofs for showing
Turing-universality would depend on each system - some may be too
simple to have such a property, but then, it's questionable if they
would be powerful enough to support intelligence or even more trivial
behavior such as life/replicators or evolution), and if they can, they
will always get the same results if they asked the same computational
or mathematical question (in this case, mathematical truths, or even
yet unknown truths such as Riemann hypothesis, Goldbach conjecture,
and so on). Most physics should support computation, and I conjecture
that any physics that isn't strong enough to at least support
computation isn't strong enough to support intelligence or
consciousness (and computation comes very cheap!). Support computation
and you get any mathematical truth that humans can reach/talk about.
Don't support it, and you probably won't have any intelligence in it.
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. 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). 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. 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...
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) - would it even retain CT's current "immunity"
from diagonalization?
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).
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.
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? 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. 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.
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.
If any intelligent system capable of interpreting the same idea will
always reach the same conclusions about it (if they followed the same
steps), I'd call that an external truth, it's about as external as it
can get. If your consciousness or physics were a direct result of such
abstract relations, it would also be both an internal and external truth.
What about arabic numerals? Seeing how popular their spread has been
on Earth after humans, shouldn't we ask why those numerals, given an
arithmetic universal primitive, are not present in nature
independently of literate humans? If indeed all qualia, feeling,
color, sounds, etc are a consequence of arithmetic, why not the
numerals themselves? Why should they be limited to human minds and
writings?
I think you're confusing numerals with numbers. Numeral systems are
just an encoding we have for talking about numbers. Numeral encodings
are a matter of history, which is a matter of physics, and in case of
COMP, is a matter of arithmetic (or any other universal computational
system - they're all equivalent by the Church Turing Thesis). In that
sense, numeral systems(encodings) are a consequence of arithmetic.
The encoding itself is irrelevant, you could use tally notation (such
as ||| + || = |||||) and it wouldn't matter. Nor is the choice of the
universal system - all that matters is the ability of following simple
finite rules and getting the same result each time you do.
Us finding about the CTT or any other mathematical truth is also such
a consequence of arithmetic. In a less serious way, you could say:
"It's turtles all the way down!". In a more serious way, you could
think of quines and Kleene's recursion theorems about fixed points.
Craig
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