On Friday, November 16, 2018 at 11:05:51 AM UTC-6, Bruno Marchal wrote:
>
>
> On 15 Nov 2018, at 18:13, Philip Thrift <cloud...@gmail.com <javascript:>> 
> 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 <cloud...@gmail.com> 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 ]). 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 
>> ]). 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).
>
>
>
*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?*

One analogy I came up with (will see how this goes): Think of a Turing 
computing that doesn't manipulate (only) symbols (information, or numbers), 
but manipulates (also) emojis [ https://emojipedia.org/ ]! Now emojis 
themselves are symbols of course, but suppose that they "embody" real 
elements of *experience* that are ontologically separate from *information* 
(or numbers).

(One could call this *e-Turing* computing non-Turing or not, depending on 
whether how one defines unconventional computing.)


- pt
 

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