<p>Concerning my limited understanding of your last question and my interpretation of Alan Turing's outlook: the answer leans towards 'no'.</p>
<ul>
<li>Daylight, E. True Turing: A Bird’s-Eye View. Minds &amp; Machines 34 (Suppl 1), 29–49 (2024)</li>
<li><a href="https://link.springer.com/article/10.1007/s11023-023-09634-0" rel="nofollow noreferrer">https://link.springer.com/article/10.1007/s11023-023-09634-0</a></li>
</ul>
<p>More generally, we live in an uncountable physical world (according to many but not all of us) and there are only countably many Turing machines.</p>
<ul>
<li>Edward A. Lee's &quot;Plato and the Nerd&quot;</li>
<li><a href="https://ptolemy.berkeley.edu/%7Eeal/pnerd/PlatoAndTheNerd.pdf" rel="nofollow noreferrer">https://ptolemy.berkeley.edu/~eal/pnerd/PlatoAndTheNerd.pdf</a></li>
</ul>

CC BY-SA 4.0
2024-09-05T03:32:55.230Z
Hao Wang's book Reflections on Godel has discussed some aritcles and works that do some related researches. But those works and articles have not answered the questions.
XL _At_Here_There
CC BY-SA 4.0
2024-09-05T07:09:16.793Z
The inconvenient truth for some theorists is that not everything is a Turing machine. Suggesting that logic is limited w.r.t. the physical world, as you do in your questions, makes them vote down your contribution and follow-ups. They want to dictate the do's and don'ts of engineers and physicists without listening to counter-arguments coming from people versed in both logic and engineering. It so happens that Alan Turing would not have endorsed a "one size fits all" mentality in theoretical discourse.
Edgar Daylight
CC BY-SA 4.0
2024-09-05T20:09:37.473Z
Uncountability does not imply non-computability.
Andrej Bauer
CC BY-SA 4.0
2024-09-05T21:00:45.583Z
Please read the sources (which takes days, if not weeks) to avoid misquoting Edward A. Lee's book on uncountability and computability.
Edgar Daylight
CC BY-SA 4.0
2024-09-08T14:03:25.983Z
@EdgarDaylight: with all due respect, your statements about "theorists" and "inconvenient truth" display an off-putting degree of hubris. At the very least you should be aware that there is extensive research and literature by "theorists" on alternative models of computation, i.e., *not* Turing machines or anything equivalent to them.
Andrej Bauer
CC BY-SA 4.0
2024-09-08T14:10:12.210Z
I downvoted the answer because it cites literature that is irrelevant to the question posed (does a certain least-principle formulation imply Turing computability). The first reference is an essay about what Turing's philosophical outlook might have been. The second one does not say anything about the relationship between the least-action principles and computability. I admit to having had a cursory look only, so i would appreciate more specific references to relevant sections and pages that discuss the question, as I might have missed them.
Andrej Bauer
CC BY-SA 4.0
2024-09-08T16:04:10.127Z
Dear Mr. Bauer. Is it not the other way around? You downvote our contributions without providing reasons and *then* I provide my take on your actions. Only recently (today) are you *trying* to justify your dictatorial actions. You are acting like an emperor without clothes. And you have succeeded in kicking me out of StackExchange (I've been on it for less than a week) because I am now leaving for good. You are abusing your power. You are spreading myths about Turing's 1936 paper yourself, not realizing how unsophisticated you are. Please take some time to reflect.
Edgar Daylight
CC BY-SA 4.0
2024-09-08T22:11:32.993Z
I have no power here, I am not a moderator, but I do believe I have the right to vote. Ans since it was explicitly asked who down-voted, I did not hide behind the anonymity, but explained publicly my reasons for downvoting. Isn't that how this site functions? In any case, I would appreciate your pointing to the relevant passages in the references you provided, in case I missed them. However, if you're going to continue in this style, I am afraid a productive communication will be quite difficult.
Andrej Bauer
CC BY-SA 4.0
2024-09-04T19:51:38.847Z
2024-09-04T19:59:44.633Z
2024-09-04T19:59:44.633Z
Edgar Daylight
<p>The Hamilton action <span class="math-container">$S$</span> is defined as following:
<span class="math-container">$$S=\int^T_0 L(q,\dot{q})dt$$</span>
the integral along any actual or virtual (conceivable or trial) space-time trajectory q(t)
connects two specified space-time events, initial event t=0 and final event t=T ,
And the principle of least action is
<span class="math-container">$$ \delta S=0$$</span>
There are questions:</p>
<p>Can any procedure satisified by the principle be implemented in real physics world?</p>
<p>Is any process in real physics world able to be discribed with the principle?</p>
<p>Is any procedure satisified by the principle able to be simulated by Turing Machine ( computed by Turing Machine or appoximated numerically to arbitary precision by Turing Machine) ?</p>
<p>Any reference is appreciated</p>
<p><strong>UPDATE:</strong> Two articles that try to prove there are physics procedures which can nont simulated by Turing Machine / quantum computer.</p>
<p><a href="https://academic.oup.com/philmat/article-abstract/1/2/97/1482843?redirectedFrom=fulltext&amp;login=false" rel="nofollow noreferrer">https://academic.oup.com/philmat/article-abstract/1/2/97/1482843?redirectedFrom=fulltext&amp;login=false</a></p>
<p><a href="https://www.sciencedirect.com/science/article/pii/000187088390004X" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/pii/000187088390004X</a></p>

CC BY-SA 4.0
2024-09-04T15:39:46.247Z
The second question is out of scope, the first one is likely not in scope but it might be. The third question requires clarification as to what is your model of computability. In any case, you might be interested in https://worldscientific.com/worldscibooks/10.1142/8306#t=aboutBook
Andrej Bauer
CC BY-SA 4.0
2024-09-05T03:35:30.593Z
Yes  and thanks, those questions have to be posted on mathoverlfow, physics, but they will refuse or have refused to accept the post. :D
XL _At_Here_There
CC BY-SA 4.0
2024-09-08T11:40:48.417Z
Two articles that try to prove there are physics procedures which can nont simulated by Turing Machine / quantum computer.

https://academic.oup.com/philmat/article-abstract/1/2/97/1482843?redirectedFrom=fulltext&login=false

https://www.sciencedirect.com/science/article/pii/000187088390004X
XL _At_Here_There
CC BY-SA 4.0
2024-09-08T11:44:50.490Z
It seems that logicians and mathematician are less likely interested in the question: is  any procedure  in real physics world able to be simulated by Turing Machine ( computed by Turing Machine or appoximated numerically to arbitary precision by Turing Machine) ?
XL _At_Here_There
CC BY-SA 4.0
2024-09-08T12:01:13.293Z
Mathematicians and logicians are firstly interested in what you might mean by statements which are quite vague from a mathematical point of view.
Andrej Bauer
CC BY-SA 4.0
2024-09-08T12:10:58.227Z
The first paper is philosophy and I prefer to be gracious rather than to have a mathematical opinion about it. The second paper is a well-known one by Pour-El and Richards. Where to they say that there are physical procedures which cannot be simulated by Turing machines?
Andrej Bauer
CC BY-SA 4.0
2024-09-08T12:12:38.510Z
If you think that their statement "unbounded operators are not computable" says anything about physical procedures (because physicists use such operators), tthen I would disagree with that. What they say is "if your operator is unbounded then you're using the wrong space".
Andrej Bauer
CC BY-SA 4.0
2024-09-08T13:21:23.810Z
@AndrejBauer, I mean Theorem 9 in the second paper.
XL _At_Here_There
CC BY-SA 4.0
2024-09-08T14:32:47.590Z
Oh, I didn't know that theorem appears there too. I am aware of https://doi.org/10.1007/BF02650184 and https://doi.org/10.1016/0001-8708(81)90001-3, which might be of some interest to you. However, see the explanation given in  https://doi.org/10.1112/S0024611502013643 (section 5) by Weihrauch and Zhong as to why Pour-El and Richards cannot actually be physically realized.
Andrej Bauer
CC BY-SA 4.0
2024-09-08T14:34:59.870Z
BTW, I am very interested in this topic, but your question ist too vague. For starters, you never defined "procedure". You would have to (1) fix a model of computation such as the Kleene-Vesley topos, (2) ask about computability within that model of a specific and well-defined mathematical gadget, probably some sort of differential operators. However, just asking (2) might require you to first develop a sufficiently good theory, such as the one developed by Pour-Ell and Richards (questionable), or Weihrauch and Zhong (better). Which is why I down-voted, as this is a *research-level* site.
Andrej Bauer
CC BY-SA 4.0
2024-09-08T15:16:54.790Z
@AndrejBauer, thank you for your coments and your URL for reference, I will think about the questions carefully and throughly
XL _At_Here_There
<reference-request><computability><physics><computational-mathematics>
Is any procedure satisified by the principle of least action able to be simulated by Turing Machine?
XL _At_Here_There
# Is any procedure satisified by the principle of least action able to be simulated by Turing Machine?

(Asked by: username_0 on 2024-09-04)

The Hamilton action $S$ is defined as following:
$$S=\int^T\_0 L(q,\dot{q})dt$$
the integral along any actual or virtual (conceivable or trial) space-time trajectory q(t)
connects two specified space-time events, initial event t=0 and final event t=T ,
And the principle of least action is
$$ \delta S=0$$
There are questions:

Can any procedure satisified by the principle be implemented in real physics world?

Is any process in real physics world able to be discribed with the principle?

Is any procedure satisified by the principle able to be simulated by Turing Machine ( computed by Turing Machine or appoximated numerically to arbitary precision by Turing Machine) ?

Any reference is appreciated

**UPDATE:** Two articles that try to prove there are physics procedures which can nont simulated by Turing Machine / quantum computer.

<https://academic.oup.com/philmat/article-abstract/1/2/97/1482843?redirectedFrom=fulltext&login=false>

<https://www.sciencedirect.com/science/article/pii/000187088390004X>

**Tags:** reference-request, computability, physics, computational-mathematics

> **Comments:**
> **username_1**: The second question is out of scope, the first one is likely not in scope but it might be. The third question requires clarification as to what is your model of computability. In any case, you might be interested in https://worldscientific.com/worldscibooks/10.1142/8306#t=aboutBook  
> **username_0**: Yes  and thanks, those questions have to be posted on mathoverlfow, physics, but they will refuse or have refused to accept the post. :D  
> **username_0**: Two articles that try to prove there are physics procedures which can nont simulated by Turing Machine / quantum computer.  https://academic.oup.com/philmat/article-abstract/1/2/97/1482843?redirectedFrom=fulltext&login=false  https://www.sciencedirect.com/science/article/pii/000187088390004X  
> **username_0**: It seems that logicians and mathematician are less likely interested in the question: is  any procedure  in real physics world able to be simulated by Turing Machine ( computed by Turing Machine or appoximated numerically to arbitary precision by Turing Machine) ?  
> **username_1**: Mathematicians and logicians are firstly interested in what you might mean by statements which are quite vague from a mathematical point of view.  
> **username_1**: The first paper is philosophy and I prefer to be gracious rather than to have a mathematical opinion about it. The second paper is a well-known one by Pour-El and Richards. Where to they say that there are physical procedures which cannot be simulated by Turing machines?  
> **username_1**: If you think that their statement "unbounded operators are not computable" says anything about physical procedures (because physicists use such operators), tthen I would disagree with that. What they say is "if your operator is unbounded then you're using the wrong space".  
> **username_0**: @AndrejBauer, I mean Theorem 9 in the second paper.  
> **username_1**: Oh, I didn't know that theorem appears there too. I am aware of https://doi.org/10.1007/BF02650184 and https://doi.org/10.1016/0001-8708(81)90001-3, which might be of some interest to you. However, see the explanation given in  https://doi.org/10.1112/S0024611502013643 (section 5) by Weihrauch and Zhong as to why Pour-El and Richards cannot actually be physically realized.  
> **username_1**: BTW, I am very interested in this topic, but your question ist too vague. For starters, you never defined "procedure". You would have to (1) fix a model of computation such as the Kleene-Vesley topos, (2) ask about computability within that model of a specific and well-defined mathematical gadget, probably some sort of differential operators. However, just asking (2) might require you to first develop a sufficiently good theory, such as the one developed by Pour-Ell and Richards (questionable), or Weihrauch and Zhong (better). Which is why I down-voted, as this is a *research-level* site.  
> **username_0**: @AndrejBauer, thank you for your coments and your URL for reference, I will think about the questions carefully and throughly  

---

## 1 answers

### Answer by username_2 on 2024-09-04. Score: -1

Concerning my limited understanding of your last question and my interpretation of Alan Turing's outlook: the answer leans towards 'no'.

* Daylight, E. True Turing: A Bird’s-Eye View. Minds & Machines 34 (Suppl 1), 29–49 (2024)
* <https://link.springer.com/article/10.1007/s11023-023-09634-0>

More generally, we live in an uncountable physical world (according to many but not all of us) and there are only countably many Turing machines.

* Edward A. Lee's "Plato and the Nerd"
* [https://ptolemy.berkeley.edu/~eal/pnerd/PlatoAndTheNerd.pdf](https://ptolemy.berkeley.edu/%7Eeal/pnerd/PlatoAndTheNerd.pdf)

> **Comments:**
> **username_0**: Hao Wang's book Reflections on Godel has discussed some aritcles and works that do some related researches. But those works and articles have not answered the questions.  
> **username_2**: The inconvenient truth for some theorists is that not everything is a Turing machine. Suggesting that logic is limited w.r.t. the physical world, as you do in your questions, makes them vote down your contribution and follow-ups. They want to dictate the do's and don'ts of engineers and physicists without listening to counter-arguments coming from people versed in both logic and engineering. It so happens that Alan Turing would not have endorsed a "one size fits all" mentality in theoretical discourse.  
> **username_1**: Uncountability does not imply non-computability.  
> **username_2**: Please read the sources (which takes days, if not weeks) to avoid misquoting Edward A. Lee's book on uncountability and computability.  
> **username_1**: @EdgarDaylight: with all due respect, your statements about "theorists" and "inconvenient truth" display an off-putting degree of hubris. At the very least you should be aware that there is extensive research and literature by "theorists" on alternative models of computation, i.e., *not* Turing machines or anything equivalent to them.  
> **username_1**: I downvoted the answer because it cites literature that is irrelevant to the question posed (does a certain least-principle formulation imply Turing computability). The first reference is an essay about what Turing's philosophical outlook might have been. The second one does not say anything about the relationship between the least-action principles and computability. I admit to having had a cursory look only, so i would appreciate more specific references to relevant sections and pages that discuss the question, as I might have missed them.  
> **username_2**: Dear Mr. Bauer. Is it not the other way around? You downvote our contributions without providing reasons and *then* I provide my take on your actions. Only recently (today) are you *trying* to justify your dictatorial actions. You are acting like an emperor without clothes. And you have succeeded in kicking me out of StackExchange (I've been on it for less than a week) because I am now leaving for good. You are abusing your power. You are spreading myths about Turing's 1936 paper yourself, not realizing how unsophisticated you are. Please take some time to reflect.  
> **username_1**: I have no power here, I am not a moderator, but I do believe I have the right to vote. Ans since it was explicitly asked who down-voted, I did not hide behind the anonymity, but explained publicly my reasons for downvoting. Isn't that how this site functions? In any case, I would appreciate your pointing to the relevant passages in the references you provided, in case I missed them. However, if you're going to continue in this style, I am afraid a productive communication will be quite difficult.  

---

