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. 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.
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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