On 18 Feb 2016, at 17:53, Brent Meeker wrote:
On 2/18/2016 1:36 AM, Bruno Marchal wrote:
On 17 Feb 2016, at 21:50, Brent Meeker wrote:
On 2/17/2016 12:37 AM, Bruno Marchal wrote:
What would be the property of the generic that is not Turing
emulable. Existence? That would beg the question
How would that be begging the question and more than asserting
that RA exists?
RA's existence is a theorem of elementary arithmetic, which you
need to define what is a computation.
But definitions and existence follow from many axioms. We don't
suppose dragons exist just because we can define them.
The theory of dragon is not that useful. We can prove that dragons
spit fire, but as there is no dragon alive on this planet that has not
much uses.
Nor can the theory of dragon explain the why and how of bosons and
fermions, for which we have more evidence.
Attempts to use the theory of dragon in number theory has failed
miserably. Some believed that if the Riemann hypothesis was false
dragon might exist, but the argument lacked of rigor and the notion of
dragon was a bit fuzzy for the member of the jury.
I suggest we don't waste our time in the Dragon theory.
And I hope you agree with elementary arithmetic, including the idea
that a closed arithmetical sentence is either true or false.
An arithmetical realist is someone who does not write a letter of
protestation to the director of the school when his/her daughter came
back from school with the statements in their school diary: 7 - 2 = 5,
8 + 1 = 9, 2 + 2 = 4, 3 * 8 = 24, 10 * 10 = 100, etc.
But to use some primary matter to avoid logical consequence of a
theory is like to invent ether to refute special relativity.
To say that matter exists (whether primary or not) is simple
empiricism. It doesn't beg any question.
To say that matter exists is simple empiricism. Hmm....OK.
To say that primary matter exists is not at all simple empiricism. It
is an ontological commitment. It is an assumption of a fundamental
reality or a god. We can do that, but not for hiding a problem or
stopping the questioning, which is where I were to... Its
epistemological variants: the physical science is the fundamental
theory of reality is equivalently as much a strong assumption. It is a
truism if we define reality by physical reality, but it is a big
theological assumption in theology.
Then I have argued that Primary Matter God has been shown to be unable
to do its job, that is connecting the first person experience with God/
Reality.
But apparently, the much smaller conceptual God provided by Mechanism
(any Turing universal system, like Robinson Arithmetic, and some model/
truth notion) seems to be able to do the job, and explains why nature
looks like a self-interfering sheaf of infinitely many computations.
Also, all self-referentially correct machine clever enough to know
that she is universal (like PA) can't miss the point when reasoning
and thinking about herself.
True or not it is a beautiful theory, which can be used as a rigorous
etalon to compare other theories. It works for a huge class of machine
(sigma_1 complete sets) and gods (pi_i complete, or even analytical
gods like pi_1^1).
I define a god by a non Recursively Enumerable (RE) set of numbers/
beliefs.
Unlike some theology, but like some other, between the machine and the
big ONE, there is an infinity of gods. G and G* remains sound and
complete for most of them, but Gods exist for which G and G* are no
more complete, although still sound. Solovay found the axiom needed to
add to get the completeness.
This shows only that by adding infinite possibilities we hardly escape
the standard logic of classical self-reference.
The divine is the range of the non constructive, not computable, not
nameable, according to computer science. The universal machine, alias
the creative set of Emil Post, are related (complementary) to a
productive set, which is not just not-RE, but is constructively not-
RE, that is, for any w_i in its complement, the RE creative set can
generate an element of the complement, not in the w_i. That can be
repeated along the transfinite ordinals. So some Gods of the machine,
like the Inner God, can be approximated, and have interesting
topological models. Computationalism needs to invest in computer
science of course. Machines are confronted with infinities all the
time, but with the mechanist assumption, infinities and those gods can
be put in the epistemological, as objective complexity measure. The
set of codes of total programs is a more powerful God than the set of
codes of halting programs, for example (the "halting Oracle").
But I know that your background is in physics, you need to study one
good book in mathematical logic, like the Mendelson, or the Boolos and
Jeffrey.
Bruno
Brent
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