Re: A Church-Turing-Thesis for Infinitary Computations

[Lawrence Crowell]

On Thursday, March 19, 2020 at 11:48:00 PM UTC-5, Philip Thrift wrote:
>
>
>
> On Thursday, March 19, 2020 at 7:13:10 PM UTC-5, Lawrence Crowell wrote:
>>
>>
>>
>> On Thursday, March 19, 2020 at 9:54:58 AM UTC-5, Bruno Marchal wrote:
>>>
>>>
>>> On 18 Mar 2020, at 14:42, Lawrence Crowell  
>>> wrote:
>>>
>>> On Wednesday, March 18, 2020 at 4:13:36 AM UTC-5, Bruno Marchal wrote:


 On 17 Mar 2020, at 16:14, Lawrence Crowell  
 wrote:

 I pretty seriously doubt these things will enter into physics. 
 Computations with Cantor's aleph hierarchy of transfinite numbers seems 
 pretty far removed from anything really physical.



 OK. It is just abstract recursion theory, has been with Turing, and it 
 concerns divine creatures, and the goal is to show that even those 
 “divine” 
 (infinite being) cannot effectively recover the arithmetical 
 truth/reality. 
 I doubt too that such machine can be brought to Earth, and they do play 
 some role in the internal phenomenology of the “terrestrial machine”, a 
 bit 
 like infinite sums can play a role in physics, like the zeta 
 renormalisation in superstring theory.
 Note that most of those divine (infinite) machine are still obeying to 
 the same self-reference logics (G and G*), and would not change much in 
 the 
 mathematical derivation of physics from arithmetic. Note also that the 
 first person indeterminacy is connected to the machine + random oracle, so 
 that the “sigma_1” predicate is really sigma_1 in all oracles, and this 
 makes it possible to use some strong set axiom (like “projective 
 determinacy”) to assure the existence of the measure, and probably of the 
 needed generalisation of Feynman integral in arithmetic. (This needs 
 indeed 
 a generalisation of the Church’s thesis, called the hyperarihmetical 
 church’s thesis in the classical  textbook by Rogers).

 Bruno



>>> When we get into subtleties of ZF set theory we get into the application 
>>> of axioms, eg the axiom of replacement, axiom of infinity, axiom of choice 
>>> etc, that have a limited bearing on most standard mathematics. 
>>>
>>>
>>>
>>> I am not sure why you say this. Most mathematician would say that ZF is 
>>> just the usual intuition of sets made precise. Most people understand the 
>>> need (and the power) of the infinity axiom, so that we can talk about 
>>> infinite set like N, Q, R, C etc., and the functions in between those sets 
>>> of numbers, matrices, etc. The axiom of choice is just obvious, and the 
>>> fact that it cannot be proved is no more astonishing than the fact that we 
>>> cannot prove any axiom of Robinson Arithmetic from any of its other axioms.
>>>
>>>
>>>
>> In the field of physics mathematics is regarded as a tool or language. I 
>> think that Feynman put it best with his talk about Greek mathematics vs 
>> Babylonian mathematics. Greek mathematics with its precise ε - δ 
>> theorem-proof system is "hard" in that you know within any system its 
>> results are absolute. However, the Babylonians tended to do practical math, 
>> what sometimes is called "maths," which has a certain quick utility to it. 
>> I tend to have a foot in both camps, though I have to admit more weight is 
>> placed on the Babylonian side. For this reason few physicists give much 
>> consideration of ZF axiomatic models. I have to admit I generally give 
>> these things much consideration.
>>
>> LC
>>  
>>
>
>
I guess I am not sure why you say there is actual damage done. It see the 
two approaches as worthy in their own right. What might be called Greek 
mathematics has lead us into a field where proofs of the great unknown 
problems are huge affairs with hundreds of pages. Wiles' result with the 
Fermat Theorem is 200 some pages of detailed work with algebraic varieties. 
Perelman's result on the Poincare conjecture for homotopy of the 3-sphere 
relies on the Hamilton equation of Ricci evolution. In both of these cases 
I read these and followed the key results and ideas, though there were a 
lot of details I tended to run over. Proofs seem likely to press into ever 
more complex and lengthy things. A more empirical stance on mathematics has 
taken off, where many people just do not have the patience for working out 
massive proofs. 

I can imagine GH Hardy, *A Mathematician's Apology*, is rolling in his 
grave.

LC
 

>
> *Opinion 174: The "Greek" (Euclidean, Axiomatic, Deductive. ...) approach 
> to mathematics did much more damage than just ruin mathematics, as is clear 
> from Amir Alexander's fascinating new Book "Proof!". It is high time that 
> we move over to the much more democratic and egalitarian (and far less 
> boring!) "Babylonian" (Algorithmic, Inductive, Experimental,...) approach 
> to mathematics*
>
> By Doron Zeilberger
> Written: Oct. 20, 2019
>
> Way back in the mid 1960s, when I was in ninth grade, we still

Re: It's totally under control

[Lawrence Crowell]

On Friday, March 20, 2020 at 6:30:06 AM UTC-5, John Clark wrote:
>
>
> I've done a Great Job 10 out of 10 
> 
>
> John K Clark
>

[image: cartoon presidential quotes.jpg]
 

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Re: A Church-Turing-Thesis for Infinitary Computations

[Philip Thrift]

On Friday, March 20, 2020 at 4:03:36 AM UTC-5, Bruno Marchal wrote:
>
>
> On 19 Mar 2020, at 20:05, Philip Thrift > 
> wrote:
>
>
>
> On Thursday, March 19, 2020 at 9:34:36 AM UTC-5, Bruno Marchal wrote:
>>
>>
>> On 18 Mar 2020, at 11:38, Philip Thrift  wrote:
>>
>>
>>
>> It is a contradiction for a physicist to say, literally,
>>
>>  *nothing real is infinite*
>>
>>
>> http://backreaction.blogspot.com/2020/03/unpredictability-undecidability-and.html
>>
>> yet their physics extensively uses infinitary mathematics
>>
>>
>> https://physics.stackexchange.com/questions/149786/why-do-we-need-infinite-dimensional-hilbert-spaces-in-physics
>>
>> (To be consistent: All theoretical physics should be based on *discrete, 
>> finite mathematics*. with *no violations permitted*. That, or the 
>> premise itself should be voided.)
>>
>>
>>
>> The problem is that any finite mathematics, rich enough to define what it 
>> is a computer, will have a extremely complex semantics, full of infinities.
>>
>> Before Gödel, we though we could protect the use of the infinities by 
>> handling their finite descriptions only.
>> After Gödel, we know that we cannot even protect the finitely realm by 
>> itself, and that we have to use infinities to manage even very partial 
>> “protection”.
>>
>> We can no more separate the finite from the infinite. In fact, we cannot 
>> even really get a precise (first-order) analysis of what finite means, even 
>> by using strong powerful theory like ZFC. Some same object can be finite in 
>> a model, and infinite in another model.
>>
>> What we can do, and what I have done with Mechanism, is to limite the 
>> ontology on the finite things, in their intuitive usual sense, and put all 
>> the infinities (including the physical) in the phenomenologies associated 
>> with the observers.
>>
>> Bruno
>>
>>
>>
> From the *P-L-T-O-S (program-language-translator-object-semantics) 
> *framework, 
> the only models are only those that exist in the universe - the 
> physical-material universe, or PMU, to use the vernacular of the science 
> types. 
>
>
> If you *assume* a primitive physical universe, then you have to abandon 
> the digital mechanist thesis. (That is the first half of my contribution, 
> and if you have not grasp this I can explain).
> No ontological commitment can select a computation, as seen from the first 
> person perspective, and make it more real than those emulated in the 
> arithmetical reality, because this would provide a non Turing emulable role 
> of that physical reality with respect to the mind/consciousness.
> Also, my goal is to explain matter, or its conscious appearrances, so I 
> prefer to be neutral on this at the start.
>
>
>
>
> So whatever can be mapped to some subset of the PMU are the only models 
> there can be.
>
> Now it is possible that 
>
> *a computer operating in a Malament–Hogarth spacetime or in orbit around a 
> rotating black hole could theoretically perform non-Turing computations for 
> an observer inside the black hole*
>
>
> In principle, but quantum mechanics makes this quite unlikely to happen.
>
>
>
>
> if the PMU permits it.
>
> (Scientific theories are completely irrelevant to whether this phenomenon 
> can happen. If it can't happen, it can't happen.)
>
>
> ?
>
> It can happen with GR. But it cannot happen with QM. Eventually we need a 
> theory making QM and GR consistent. We are not there yet, and with 
> mechanism, we have to extract it from the relative measure on all 
> computations.
>
>
>
> In any case, statements like* there are an infinite number of integers* 
> may be false (which would automatically let the air out of the 
> hypercomputation balloon).
>
>
> Frankly, I don’t see how “there are an infinite number of integers” can be 
> false, no matter which metaphysics principle we accept. But this is not 
> relevant, as the ontological part of the “theory of everything” does not 
> use any infinity axiom, and Robinson arithmetic is consistent with the 
> existence of a greatest integer.
>
>
>
>
>
> In any case: 
>
> *Mathematics (at least Applied Math, and its twin sister, Computing) have 
> only to do with what one can do, not with what is true.*
>
>
> Theoretical computer science has a lot to do with the things that we, the 
> machines, cannot do, indeed, even with what we cannot do despite the use of 
> powerful oracle. Recursion theory is the study of the degree of 
> unsolvability, and the while truth has something to say for theology and 
> the origin of the physical laws.
>
> Bruno
>
>
>
What a theory (a linguistic expression we make up and hope turns out to me 
useful) permits or does not permit is irrelevant to whether nature permits 
it or not.


Max Tegmark's dictum:

"Not only do we lack evidence for the infinite but *we don’t need the 
infinite to do physics*. Our best computer simulations, accurately 
describing everything from the formation of galaxies to tomorrow’s weather 
to the masses of elementary particl

It's totally under control

[John Clark]

I've done a Great Job 10 out of 10


John K Clark

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Re: A Church-Turing-Thesis for Infinitary Computations

[Bruno Marchal]

> On 20 Mar 2020, at 01:13, Lawrence Crowell  
> wrote:
> 
> 
> 
> On Thursday, March 19, 2020 at 9:54:58 AM UTC-5, Bruno Marchal wrote:
> 
>> On 18 Mar 2020, at 14:42, Lawrence Crowell > > wrote:
>> 
>> On Wednesday, March 18, 2020 at 4:13:36 AM UTC-5, Bruno Marchal wrote:
>> 
>>> On 17 Mar 2020, at 16:14, Lawrence Crowell > 
>>> wrote:
>>> 
>>> I pretty seriously doubt these things will enter into physics. Computations 
>>> with Cantor's aleph hierarchy of transfinite numbers seems pretty far 
>>> removed from anything really physical.
>> 
>> 
>> OK. It is just abstract recursion theory, has been with Turing, and it 
>> concerns divine creatures, and the goal is to show that even those “divine” 
>> (infinite being) cannot effectively recover the arithmetical truth/reality. 
>> I doubt too that such machine can be brought to Earth, and they do play some 
>> role in the internal phenomenology of the “terrestrial machine”, a bit like 
>> infinite sums can play a role in physics, like the zeta renormalisation in 
>> superstring theory.
>> Note that most of those divine (infinite) machine are still obeying to the 
>> same self-reference logics (G and G*), and would not change much in the 
>> mathematical derivation of physics from arithmetic. Note also that the first 
>> person indeterminacy is connected to the machine + random oracle, so that 
>> the “sigma_1” predicate is really sigma_1 in all oracles, and this makes it 
>> possible to use some strong set axiom (like “projective determinacy”) to 
>> assure the existence of the measure, and probably of the needed 
>> generalisation of Feynman integral in arithmetic. (This needs indeed a 
>> generalisation of the Church’s thesis, called the hyperarihmetical church’s 
>> thesis in the classical  textbook by Rogers).
>> 
>> Bruno
>> 
>> 
>> 
>> When we get into subtleties of ZF set theory we get into the application of 
>> axioms, eg the axiom of replacement, axiom of infinity, axiom of choice etc, 
>> that have a limited bearing on most standard mathematics.
> 
> 
> I am not sure why you say this. Most mathematician would say that ZF is just 
> the usual intuition of sets made precise. Most people understand the need 
> (and the power) of the infinity axiom, so that we can talk about infinite set 
> like N, Q, R, C etc., and the functions in between those sets of numbers, 
> matrices, etc. The axiom of choice is just obvious, and the fact that it 
> cannot be proved is no more astonishing than the fact that we cannot prove 
> any axiom of Robinson Arithmetic from any of its other axioms.
> 
> 
> 
> In the field of physics mathematics is regarded as a tool or language.


That is reasonable in physics,.

But it is the Aristotle axiom in metaphysics/theology, and I have shown that it 
is incompatible with Mechanism, where the reality is the elementary 
arithmetical truth, which is beyond all language and theories, as we know since 
Gödel and Tarski.

The number of triangle with an integer perimeter equal to its area does not 
depend on the choice of any language, and can be computed (it is 5). 
The fact that the nth program in an enumeration of all programs P_n stop or 
not, on some input, is a truth which is independent of us.




> I think that Feynman put it best with his talk about Greek mathematics vs 
> Babylonian mathematics. Greek mathematics with its precise ε - δ 
> theorem-proof system is "hard" in that you know within any system its results 
> are absolute. However, the Babylonians tended to do practical math, what 
> sometimes is called "maths," which has a certain quick utility to it. I tend 
> to have a foot in both camps, though I have to admit more weight is placed on 
> the Babylonian side. For this reason few physicists give much consideration 
> of ZF axiomatic models. I have to admit I generally give these things much 
> consideration.


We must distinguish clearly the truth or the intended model of a general 
theory, and the (human) theories which are just little lantern to put some 
light on the realm explored with those theories. A mathematical theory is a 
tool, but that makes not the mathematical reality dependent on us.

This is even more important when we postulate mechanism, as the physical 
reality becomes a computation-as-seen-from-the-1p-perspective statistic 
emergent appearance. 

Bruno




> 
> LC
>  
> 
> 
>> This means physics is even further removed.
> 
> 
> I am not sure. Physicists works in a tiny fragment of ZFC, but are very happy 
> of C (the axiom of choice), which makes every linear spaces and Helbert 
> spaces having a base, and making mathematics much easier. 
> 
> 
> 
> 
>> Transfinite numbers and the question of א_0 ≤ c ≤ 2^א_0 or the continuum 
>> hypothesis has a role with Robinson’s numbers.
> 
> 
> You mean Robinson non standard analysis. This shows something 
> “sociologically” interesting, as it refutes many old authors, like Carnot for 
> example, or Berkeley, who were convinced that Lebiniz and Newton’s no

Re: A Church-Turing-Thesis for Infinitary Computations

[Bruno Marchal]

> On 19 Mar 2020, at 20:05, Philip Thrift  wrote:
> 
> 
> 
> On Thursday, March 19, 2020 at 9:34:36 AM UTC-5, Bruno Marchal wrote:
> 
>> On 18 Mar 2020, at 11:38, Philip Thrift > 
>> wrote:
>> 
>> 
>> 
>> It is a contradiction for a physicist to say, literally,
>> 
>>  nothing real is infinite
>> 
>> http://backreaction.blogspot.com/2020/03/unpredictability-undecidability-and.html
>>  
>> 
>> 
>> yet their physics extensively uses infinitary mathematics
>> 
>> https://physics.stackexchange.com/questions/149786/why-do-we-need-infinite-dimensional-hilbert-spaces-in-physics
>>  
>> 
>> 
>> (To be consistent: All theoretical physics should be based on discrete, 
>> finite mathematics. with no violations permitted. That, or the premise 
>> itself should be voided.)
> 
> 
> The problem is that any finite mathematics, rich enough to define what it is 
> a computer, will have a extremely complex semantics, full of infinities.
> 
> Before Gödel, we though we could protect the use of the infinities by 
> handling their finite descriptions only.
> After Gödel, we know that we cannot even protect the finitely realm by 
> itself, and that we have to use infinities to manage even very partial 
> “protection”.
> 
> We can no more separate the finite from the infinite. In fact, we cannot even 
> really get a precise (first-order) analysis of what finite means, even by 
> using strong powerful theory like ZFC. Some same object can be finite in a 
> model, and infinite in another model.
> 
> What we can do, and what I have done with Mechanism, is to limite the 
> ontology on the finite things, in their intuitive usual sense, and put all 
> the infinities (including the physical) in the phenomenologies associated 
> with the observers.
> 
> Bruno
> 
> 
> 
> From the P-L-T-O-S (program-language-translator-object-semantics) framework, 
> the only models are only those that exist in the universe - the 
> physical-material universe, or PMU, to use the vernacular of the science 
> types. 

If you *assume* a primitive physical universe, then you have to abandon the 
digital mechanist thesis. (That is the first half of my contribution, and if 
you have not grasp this I can explain).
No ontological commitment can select a computation, as seen from the first 
person perspective, and make it more real than those emulated in the 
arithmetical reality, because this would provide a non Turing emulable role of 
that physical reality with respect to the mind/consciousness.
Also, my goal is to explain matter, or its conscious appearrances, so I prefer 
to be neutral on this at the start.



> 
> So whatever can be mapped to some subset of the PMU are the only models there 
> can be.
> 
> Now it is possible that 
> 
> a computer operating in a Malament–Hogarth spacetime or in orbit around a 
> rotating black hole could theoretically perform non-Turing computations for 
> an observer inside the black hole

In principle, but quantum mechanics makes this quite unlikely to happen.



> 
> if the PMU permits it.
> 
> (Scientific theories are completely irrelevant to whether this phenomenon can 
> happen. If it can't happen, it can't happen.)

?

It can happen with GR. But it cannot happen with QM. Eventually we need a 
theory making QM and GR consistent. We are not there yet, and with mechanism, 
we have to extract it from the relative measure on all computations.


> 
> In any case, statements like there are an infinite number of integers may be 
> false (which would automatically let the air out of the hypercomputation 
> balloon).

Frankly, I don’t see how “there are an infinite number of integers” can be 
false, no matter which metaphysics principle we accept. But this is not 
relevant, as the ontological part of the “theory of everything” does not use 
any infinity axiom, and Robinson arithmetic is consistent with the existence of 
a greatest integer.




> 
> In any case: 
> 
> Mathematics (at least Applied Math, and its twin sister, Computing) have only 
> to do with what one can do, not with what is true.

Theoretical computer science has a lot to do with the things that we, the 
machines, cannot do, indeed, even with what we cannot do despite the use of 
powerful oracle. Recursion theory is the study of the degree of unsolvability, 
and the while truth has something to say for theology and the origin of the 
physical laws.

Bruno




> 
> @philipthrift
> 
> 
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