Hi Brent, and colleagues,
Wow, many posts. I read them in the chronological order, but will try
to limit the number of answers by starting from the most recent one
(with the other in minds). Some answer has been given by Telmo and
Brent, and I might just clarify some points.
On 12 Jun 2016, at 02:30, Brent Meeker wrote:
On 6/11/2016 3:50 PM, John Clark wrote:
On Sat, Jun 11, 2016 at 6:14 PM, Brent Meeker
<[email protected]> wrote:
>> As I've said 6.02*10^23 times it's irrelevant if
matter is primary or not, matter is still necessary to make
calculations or perform intelligent behavior or produce
consciousness.
> I think Bruno agrees with that
That's news to me. If so Bruno should have said that several
years ago and a great many electrons wouldn't
have had to give up their lives.
>> And even if matter isn't primary that doesn't necessarily
mean mathematics is.
> The question is can one be derived from the other?
I think so, but neither may be primary.
> William S. Cooper, "The Origin of Reason" makes an argument
that mathematics is a way of brains thinking about things that was
found by evolution, just like mobility, metabolism,
reproduction,...and a lot of other functions.
I agree with that, but evolution works according to the laws
of physics so a animal who thought 1+1=0 would have fewer
offspring than one who believed 1+1=2.
But those aren't laws of physics; at least according to Bruno.
1+1=2 is a necessary truth that is independent of physics and hence
"more fundamental". So evolution merely leads to (approximate)
beliefs in something more basic than physics and which we use to
describe physics.
So we'd agree with ET's mathematics because it's the language of
physics.
That's more Cooper's viewpoint. But taking mathematics (i.e.
computation/arithmetic) as basic, Bruno uses modal logic to derive
some interesting categorization of beliefs.
OK. Just to be sure that the relation between the modal logic I am
using are given by precise, and not easy to prove, mathematical
theorem. G and G* were known to be sound since Gödel and Löb with the
help of Bernays and Hilbert, but Solovay's double (G *and* G*)
arithmetical completeness theorem came later (1976).
It is important to keep in mind the difference between a machine's set
of belief (which is recursively enumerable, at least in some
tangential sense, as real person are related to sequence of machines),
and and the semantic of those belief, that is the notion of truth. In
arithmetic, truth is given by the standard model, which, by
incompleteness escapes all machines beliefs or all axiomatic theories.
Even a theory as rich as set theory + large cardinals (like ZF + Kappa
exists) can only scratch the Arithmetical truth, and all machines and
theories, with respect to the arithmetical reality should be seen as
tiny candle in an obscure infinite cave.
The set of (machine's, or instantaneous machine's) beliefs are sigma_1.
Arithmetical truth can be seen as the union of all sigma_i and pi_i
truth, for all i. (Riemann hypothesis is Pi_1). Above sigma_1 we are
no more in the computable realm. Most attributes and properties of
numbers and machines are described by non computable predicate. Being
a machine does not made you escaping the non-computable realm (even
without the FPI). On the contrary it directly confront the machines.
The russian showed that the quantified version of G and G* are as
undecidable as they can possibly be. G with quantifier is Pi_2
complete, and G* with quantifiers is Pi_1+ the whole arithmetical
truth as oracle (!) complete. Even God needs to do an infinite task to
get the machine "noùs" ! The "worlds of ideas" is bigger than God
(Plato would have appreciate this, I think, but Plotinus and the
neoplatonist would not necessarily have been happy, but they are under
the prejudice that nothing can make something more complex than
itself, which is refuted in computer science and arithmetic.
Bruno
Brent
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