> On 15 Nov 2018, at 18:13, Philip Thrift <[email protected]> wrote:
>
>
>
> On Thursday, November 15, 2018 at 5:15:39 AM UTC-6, Bruno Marchal wrote:
>
>> On 13 Nov 2018, at 11:06, Philip Thrift <[email protected] <javascript:>>
>> wrote:
>>
>>
>>
>> On Monday, November 12, 2018 at 8:35:23 PM UTC-6, Bruno Marchal wrote:
>>
>> A model is a model of a theory. The notion of model of a model can make
>> sense, by considering non axiomatisable theory, but that can lead to
>> confusion, so it is better to avoid this. When a model is seen as a theory,
>> if it contains arithmetic, the theory cannot be axiomatised, proofs cannot
>> be checked, the set of theorems is not recursively enumerable, etc.
>>
>>
>> Bruno
>>
>>
>>
>> This is why some have mathematical theories (alternatives to ZF) that have
>> finite (i.e. Only a finite number of numbers needed!) models (e.g. Jan
>> Mycielski, "Locally Finite Theories" [https://www.jstor.org/stable/2273942
>> <https://www.jstor.org/stable/2273942> ]). In this approach quantifiers are
>> effectively replaced by typed quantifiers, where the type says "this
>> quantifier ranges over some finite set".
>>
>> Another approach is to nominalize physical theories theories (Hartry Field,
>> Science Without Numbers, summary [
>> http://www.nyu.edu/projects/dorr/teaching/objectivity/Handout.5.10.pdf
>> <http://www.nyu.edu/projects/dorr/teaching/objectivity/Handout.5.10.pdf> ]).
>> In this approach the model of the theory is a finite set of (references to)
>> physical objects.
>>
>> This is the best point-of-view to have: The set of natural numbers simply
>> doesn't exist!
>
>
> I agree. It is actually a consequence of mechanism. The set of natural
> numbers does not exist, nor any infinite set. But that does not make a
> physical universe into something existing. Analysis, physics, sets, … belongs
> to the numbers “dreams” (a highly structured set, which has no ontology, but
> a rich and complex phenomenological accounts).
>
> I gave my axioms (Arithmetic, or Kxy = x, Sxyz = xz(yz)). As you can see,
> there is no axiom of infinity.
>
> Bruno
>
> PS Sorry for the delay.
>
>
>
>
> The "highest" programming may be higher-type (or higher-order) programming:
>
> http://www.cs.bham.ac.uk/~mhe/papers/introduction-to-higher-order-computation-NLS-2017.pdf
> examples @ http://www.cs.bham.ac.uk/~mhe/
>
>
> "Higher-order [programming involves] infinite objects, such as infinite
> strings, real numbers, and even functions themselves, etc. [which themselves]
> are computable. And, more importantly, how to compute them. In practice,
> computation with infinite objects often takes place in languages such as ML,
> Haskell, Agda etc. In theory, some canonical systems are Godel’s system T,
> Platek-Scott-Plotkin PCF, Martin-Lof’s dependent type theory, among many
> others. But how can we (or a computer) compute with infinite objects, given
> that we have a finite amount of time and a finite amount of memory and a
> finite amount of any resource? Topology comes to the rescue [revolving]
> around the [finite vs. infinite dichotomy], mediated by topology. We can say
> that topology is precisely about the relation between finiteness and
> infiniteness that is relevant to computation."
>
>
>
> But there is a new biochemical programming language:
>
> CRN++: Molecular Programming Language
> (Submitted on 19 Sep 2018)
> https://arxiv.org/abs/1809.07430
> "We present its syntax and semantics, and build a compiler translating CRN++
> programs into chemical reactions...laying the foundation of a comprehensive
> framework for molecular programming."
>
> A programming language whose purpose is to create bugs!
>
> So the question becomes: Is bioprogramming > programming? (if biomatter has
> experiential qualities in addition to informational quantities)
Assuming some primary matter, and some non mechanist theory, why not. That
seems to quite speculative, though, and adding difficulties to a subject which
is already difficult when assuming the “simplifying” assumption of Mechanism.
With mechanism, the mind-body problem reduced into justifying the existence of
a canonical measure on all computations “seen from inside” (which admits a
number of modes, imposed by incompleteness). In case the physics in the head of
the universal machine/number departs from observation, we get the mean to make
sense of some non-mechanism, and this might show you right. So let us continue
the testing/comparison.
What do you think your biomatter do which would be non Turing emulable, nor
“first person measurable(*) and in what sense would that be relevant with
respect of consciousness?
I have no doubt chemical computation is a wonderful subject, but with
“Indexical Digital Mechanism”, the theology and the physics is independent of
the language and the basic theories as far as they are Turing complete(*), the
physical appearance, needs to be justified in term of a relative measure
state/computations "seen from inside” (Incompleteness makes the usual standard
definition getting sense in those “enough rich” Turing complete(**) theories.
Bruno
(*) This provides some “free oracle”, like the random oracle and the halting
oracle, due to the limiting behaviour of the first person indeterminacy).
(**) Turing complete means that for all p sigma_1 (shape ExA(x, y), A
decidable) we have, with “[]” Gödel’s arithmetical provability predicate,
p -> []p
is true.
Löbian (sufficiently rich) means that for all such p,"p -> []p" is not only
true, but provable. Put it in another way, this means that
[](p -> []p)
is true. (This makes the machine obeying to G and G* and their intensional
variants).
(See all definitions in the second part of sane04, I recall them in most of my
papers).
>
> - pt
>
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