On 21 Jun 2016, at 18:29, John Clark wrote:
On Mon, Jun 20, 2016 at 9:44 PM, Jason Resch <jasonre...@gmail.com>
wrote:
> Bruno has shown that arithmetic is a viable candidate for
explaining physics:
Bruno wasn't the first to discover that, people
have known for 400 years that mathematics is the
best language for describing physics, but the point is mathematics
is a language and physics isn't, physics just is.
I give an example, with arithmetic.
You have a language, that is, symbols and grammar. The symbols are the
parenthesis, "=", the logical symbols (&, V, ->, <->, ~, E, A), and
the non logical symbols, proper to arithmetic: 0, s, +, *.
Then the grammar tells you which expression using only those symbols
is grammatically correct or not, for example s(0) + 0 = s(s(0)) is a
grammatically correct formula.
Then you have the semantics, given by the usual structure (N, 0, +,
*), with the usual abuse of language, denoting the same way some well
known function from N to N, or NXN to N, with the symbols of the
arithmetical language. With such a model, we have a notion of truth.
For example "(Ax((x = 0) V (~(x = 0))) is true if for each natural
number (object of N) we have that either that number or not.
Then you have the theories, which are like lanterns to look at what is
true in all or some model(s) satisfying some formula.
I gave two very different theories for arithmetic.
One is Robinson Arithmetic (RA, Q):
It is Classical Logic +
0 ≠ s(x)
s(x) = s(y) -> x = y
x = 0 v Ey(x = s(y))
x+0 = x
x+s(y) = s(x+y)
x*0=0
x*s(y)=(x*y)+x
That is a very weak theory. It cannot prove that 0 + x = x, for
example, nor that (x + y) = (y + x), and is very poor in
generalisation. Yet, it can already prove for all digital machine x
that it stops, in all case x stops. It can represents in some strong
sense all partial computable function; it is sigma_1 complete, that is
Turing universal.
The other is PA. It is RA + the infinitely many following axioms,
where F is for any arithmetical formula written in the language
described above:
(F(0) & Ax(F(x) -> F(s(x))) -> AxF(x)
That gives PA a strong generalization ability, and it makes PA Löbian
(its propositional logic of provability obey G and G*, which main
axiom is Löb's formula []([]p -> p) -> []p. Exercise: derive Gödel's
second incompleteness from L. Hint: replace p by the constant
proposition false (f).
And even if it turns out that I'm wrong and that in some sense
mathematics is more fundamental than physics it wouldn't change
the status of what this list is unhealthily (in my humble
opinion) obsessed with, consciousness.
Thanks for precising it is your humble opinion. But consciousness is
what you hope to be preserved when they will give you a digital brain.
And we are not obsessed. We might be tired of its being pushed under
the rug.
Whatever consciousness is one thing is very clear, it can't be
produced entirely from the stuff at the fundamental level of
reality,
Ah! Glad you saw this.
and being more fundamental is not the same as being more important.
Atoms are more fundamental than molecules but molecules have
properties than atoms don't have, and molecules are more fundamental
than life but life has properties that molecules don't have; in the
same way consciousness needs intelligent behavior and intelligent
behavior needs computation and computation needs physics.
No. The notion of computation belongs to arithmetic. Only a physical
implementation of a computation needs physical assumptions.
The physical remains possibly (and necessarily with digital mechanism)
an aspect of Arithmetic when seen from inside, and taking the First
Person Indeterminacy into account.
We just take into account that no universal machine can know by
introspection which computations realize her among all computations
which exist, in the sense of the theory above.
Digital mechanism predicts that when we look at ourselves below our
substitution level, we must witness the many-"realities/computations",
and that explains the startling aspect of quantum physics.
That does not mean computationalism will not been refuted tomorrow,
but until now, it is the only theory, it seems to me, which explains
both mind and matter in a coherent way, without discarding the
complexity of the 1p and 3p linked complex relation.
I don't think it is so much harder to swallow than QM without collapse.
Bruno
John K Clark
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