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CATEGORY:
Undecidability, Uncomputability, and Unpredictability Essay Contest (2019-2020)
[back]

TOPIC: Mathematics, Shaken but not Stirred by Kevin H Knuth [refresh]

TOPIC: Mathematics, Shaken but not Stirred by Kevin H Knuth [refresh]

The last century saw the breakdown of the dream of the mechanical universe where it was imagined that given sufficient intellect and effort all truths could be discovered and known. Gödel’s Incompleteness Theorems, which form the basis for the concept of Undecidability, revealed that there exist true statements that cannot be proven to be true. This undeniably shook the Mathematical Worldview, but was this Worldview shaken enough? When deduction is not possible, inductive inference can serve in its stead and form a different kind of base. What progress could be made in Mathematics if uncertainty was embraced and inductive inference employed to its maximum potential? Perhaps Mathematics should have been sufficiently stirred to follow the lead of the Physical Sciences and learn to embrace uncertainty when necessary.

Kevin Knuth is an Associate Professor in the Department of Physics at the University at Albany. He is Editor-in-Chief of the journal Entropy. He has more than 20 years of experience in applying Bayesian and maximum entropy methods to the design of machine learning algorithms for data analysis applied to the physical sciences. His current research interests include the foundations of physics, autonomous robotics, and searching for extrasolar planets.

Respected Professor Kevin H Knuth,

Wonderful essay about Godels theorem applicability to Physics and Quantum Mechanics.

I got a small question here, will this theorem be applicable to cosmology also?

I developed and working on**Dynamic Universe Model** for the last 40 years. I never come across such problem of non testable hypothesis.

Can you please clear this doubt by hopefully reading my essay " A properly deciding, Computing and Predicting new theory’s Philosophy "

Best Regards

=snp

report post as inappropriate

Wonderful essay about Godels theorem applicability to Physics and Quantum Mechanics.

I got a small question here, will this theorem be applicable to cosmology also?

I developed and working on

Can you please clear this doubt by hopefully reading my essay " A properly deciding, Computing and Predicting new theory’s Philosophy "

Best Regards

=snp

report post as inappropriate

Dear Satyavarapu,

Thank you for your kind comments and question.

Godel's theorem is generally applicable to mathematics. Whether there are implications for cosmology really comes down to whether the cosmological questions, that one wishes to prove, rely on mathematics that is undecidable.

In science, we only entertain testable hypotheses. These are hypotheses that make predictions. And it is by making predictions, using our theories, that allow us to test them against experimental data by applying the weakest of the logical syllogisms listed in my essay. This is the process of inductive inference, also known as Bayesian inference. And it is possible that one can get around Godel's theorem in a way by assigning probabilities to hypotheses rather than truth values. Of course, nothing will be known with certainty. But this is the situation that we are very familiar with in the physical sciences.

My essay claims that mathematicians threw up their hands and gave up, whereas they (especially now) could re-visit these issues and consider applying inductive inference rather than deduction.

As an amusing aside, I was watching the TV show, Sherlock, about the famous fictional detective Sherlock Holmes, and in the show, Sherlock claims that he arrived at his conclusions via deduction. However, this is not the case. Sherlock routinely uses inductive inference rather than logical deduction.

I think that mathematicians should give it a go and try it too!

Last, thank you for pointing me to your essay. I look forward to reading it.

Sincerely,

Kevin Knuth

Thank you for your kind comments and question.

Godel's theorem is generally applicable to mathematics. Whether there are implications for cosmology really comes down to whether the cosmological questions, that one wishes to prove, rely on mathematics that is undecidable.

In science, we only entertain testable hypotheses. These are hypotheses that make predictions. And it is by making predictions, using our theories, that allow us to test them against experimental data by applying the weakest of the logical syllogisms listed in my essay. This is the process of inductive inference, also known as Bayesian inference. And it is possible that one can get around Godel's theorem in a way by assigning probabilities to hypotheses rather than truth values. Of course, nothing will be known with certainty. But this is the situation that we are very familiar with in the physical sciences.

My essay claims that mathematicians threw up their hands and gave up, whereas they (especially now) could re-visit these issues and consider applying inductive inference rather than deduction.

As an amusing aside, I was watching the TV show, Sherlock, about the famous fictional detective Sherlock Holmes, and in the show, Sherlock claims that he arrived at his conclusions via deduction. However, this is not the case. Sherlock routinely uses inductive inference rather than logical deduction.

I think that mathematicians should give it a go and try it too!

Last, thank you for pointing me to your essay. I look forward to reading it.

Sincerely,

Kevin Knuth

Dear Prof. Kevin Knuth

Thank you for nice reply. Your excellent Knowledge cleared this doubt.

**You are the only person WHO CLEARED MY DOUBT!!! **

REQUEST YOU TO PLEASE LOOK AT MY ESSAY AND RATE IT...

Best

=snp

report post as inappropriate

Thank you for nice reply. Your excellent Knowledge cleared this doubt.

REQUEST YOU TO PLEASE LOOK AT MY ESSAY AND RATE IT...

Best

=snp

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Hi Kevin,

a very interesting essay I enjoyed a lot. Sorry, if I first want to mention some mistakes, I think, I have spotted:

Shouldn't in page 6 the syllogism SS-2 be: If A true, then B true. Learn B false, then A false.

And similarly WS-2: Learn A false, then probably B false.

Finally on page 8. If u is undecidable and u follows from x1 and x2 and x3, than at least...

view entire post

a very interesting essay I enjoyed a lot. Sorry, if I first want to mention some mistakes, I think, I have spotted:

Shouldn't in page 6 the syllogism SS-2 be: If A true, then B true. Learn B false, then A false.

And similarly WS-2: Learn A false, then probably B false.

Finally on page 8. If u is undecidable and u follows from x1 and x2 and x3, than at least...

view entire post

report post as inappropriate

Dear Luca

Thank you very much for your comments and kind words.

You note some potential typos or mistakes. Let me look at these first.

> Shouldn't in page 6 the syllogism SS-2 be: If A true, then B true. Learn B false, then A false.

You are correct. It should be:

Given: If A is true, then B is true.

Learn: B is false.

Deduce: A is...

view entire post

Thank you very much for your comments and kind words.

You note some potential typos or mistakes. Let me look at these first.

> Shouldn't in page 6 the syllogism SS-2 be: If A true, then B true. Learn B false, then A false.

You are correct. It should be:

Given: If A is true, then B is true.

Learn: B is false.

Deduce: A is...

view entire post

Hi Kevin,

you are right. In the context of lattice theory, atomic statements are of the kind: this particle is at location x and so on. Atomic statements are composed through disjunction.

I falsely have mistaken atomic statements as axioms in the context of undecidability. There theorems are derived from axioms by conjunction. If now a theorem is undecidable one of the axioms must be undecidable. In Gödel's case this would be the axiom non contradiction. Which is the second Gödel's theorem, that consistency is not decidable. But I don't know if my simplified argument applies here, since what is an axiom and a theorem is not uniquely defined. Contrary to atomic statements.

I certainly will look up your recent papers, as your approach to derive probability from the underlying symmetries and logic is really interesting.

Luca

report post as inappropriate

you are right. In the context of lattice theory, atomic statements are of the kind: this particle is at location x and so on. Atomic statements are composed through disjunction.

I falsely have mistaken atomic statements as axioms in the context of undecidability. There theorems are derived from axioms by conjunction. If now a theorem is undecidable one of the axioms must be undecidable. In Gödel's case this would be the axiom non contradiction. Which is the second Gödel's theorem, that consistency is not decidable. But I don't know if my simplified argument applies here, since what is an axiom and a theorem is not uniquely defined. Contrary to atomic statements.

I certainly will look up your recent papers, as your approach to derive probability from the underlying symmetries and logic is really interesting.

Luca

report post as inappropriate

Thank you, Luca.

It is not clear that the axioms are the atomic statements. So I do not think that one can use my arguments to say something, in general, about the axioms. In fact, it could very well be that the undecidable statement is an atomic statement.

Thank you for your kind words about my approach to deriving probability from symmetries. If you have any questions about that work, please feel free to email me.

Sincerely,

Kevin Knuth

It is not clear that the axioms are the atomic statements. So I do not think that one can use my arguments to say something, in general, about the axioms. In fact, it could very well be that the undecidable statement is an atomic statement.

Thank you for your kind words about my approach to deriving probability from symmetries. If you have any questions about that work, please feel free to email me.

Sincerely,

Kevin Knuth

Dear Kevin,

I'm happy to see you at this new contest with a new excellent essay! I enjoyed it very much. Especially the spirit of reopening cases considered closed for long time, and reconstructing them from other basic principles. In particular, the results you presented with the symmetrical foundations. Thanks for another beautiful and thought-provoking essay!

I wish you all the best in this contest!

Cheers,

Cristi

P.S. I wanted to point out some minor typos, but I see they are already mentioned in Luca's comment.

report post as inappropriate

I'm happy to see you at this new contest with a new excellent essay! I enjoyed it very much. Especially the spirit of reopening cases considered closed for long time, and reconstructing them from other basic principles. In particular, the results you presented with the symmetrical foundations. Thanks for another beautiful and thought-provoking essay!

I wish you all the best in this contest!

Cheers,

Cristi

P.S. I wanted to point out some minor typos, but I see they are already mentioned in Luca's comment.

report post as inappropriate

Dear Cristi,

Thank you for your kind words about my essay.

Luca did indeed find some typos. Specifically, I wrote the two syllogisms wrong. It was a transcription error that I did not catch. Very unfortunate. :(

I wish you the best in this contest as well!

I have printed out your essay and I am looking forward to reading it.

Thank you, again!

Kevin

Thank you for your kind words about my essay.

Luca did indeed find some typos. Specifically, I wrote the two syllogisms wrong. It was a transcription error that I did not catch. Very unfortunate. :(

I wish you the best in this contest as well!

I have printed out your essay and I am looking forward to reading it.

Thank you, again!

Kevin

Very interesting essay! I agree that it's important to interrogate the nature of certainty in mathematics, given that many of us feel that it is somehow the last bastion of 'absolute' certainty. The idea that math (or perhaps a branch of math) should embrace uncertainty reminds me of Horgan's famous "Death of Proof" article in Scientific American, wherein he discusses alternative standards of...

view entire post

view entire post

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Hi Kevin,

I really enjoyed reading your essay and heard your talk as well in Vaxjo.

What do you think about the non-local box in this context? This is in the general probability theory (GPT). In this context, is the GBT final to resolve the Godel incompleteness theorem?

On the context of the Turing machine, my essay was briefly discussed.

Best wishes,

Yutaka

report post as inappropriate

I really enjoyed reading your essay and heard your talk as well in Vaxjo.

What do you think about the non-local box in this context? This is in the general probability theory (GPT). In this context, is the GBT final to resolve the Godel incompleteness theorem?

On the context of the Turing machine, my essay was briefly discussed.

Best wishes,

Yutaka

report post as inappropriate

Dear Yutaka,

I am very glad to see you here in this essay contest.

It is a shame that the Vaxjo meeting had to be delayed this year. I do hope to go again next year.

I do not quite understand what you are asking:

>

Perhaps you can reply with a more detailed explanation of your question.

I wish you all the best in this essay contest and I hope to read your essay on the Turing machine.

Sincerely,

Kevin

I am very glad to see you here in this essay contest.

It is a shame that the Vaxjo meeting had to be delayed this year. I do hope to go again next year.

I do not quite understand what you are asking:

>

Perhaps you can reply with a more detailed explanation of your question.

I wish you all the best in this essay contest and I hope to read your essay on the Turing machine.

Sincerely,

Kevin

Dear Professor Kevin nuth. Had a great insight in your well crafted essay,you raise pertinent philosophical issues well blended into mathematics. in particular your statistical analysis diagram and simplification via the Boolean lattice.very impressive.Rated you well. How about you see something simple I have submitted on cognitive bias-https://fqxi.org/community/forum/topic/3525. Thanks and all the Best in the essay contest.

report post as inappropriate

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Dear Michael,

Thank you for your kind words.

I am very glad that you appreciated my approach, especially the collapse of the Boolean lattice under the truth-falsity equivalence relation.

I hope to get to reading your essay as well, especially since I am curious how you relate cognitive bias to undecidability and such.

I wish you all the best in this contest.

Sincerely,

Kevin Knuth

Thank you for your kind words.

I am very glad that you appreciated my approach, especially the collapse of the Boolean lattice under the truth-falsity equivalence relation.

I hope to get to reading your essay as well, especially since I am curious how you relate cognitive bias to undecidability and such.

I wish you all the best in this contest.

Sincerely,

Kevin Knuth

Greetings Professor Knuth,

I have downloaded and begun to read your essay. Looks interesting. In my essay; I take almost the polar opposite tack. Recently; Giulio Tiozzo proved some very broad connections between the Mandelbrot Set and entropy, completing some of the last work of Thurston (with whom he collaborated). I have been exploring some specific connections of M to Physics for more than 30 years now, and the location I focus on in my essay has relevance for entropic theories of gravity.

I'd appreciate your comments on my essay. I will be back once I have read your paper in its entirety.

Warm Regards,

Jonathan

report post as inappropriate

I have downloaded and begun to read your essay. Looks interesting. In my essay; I take almost the polar opposite tack. Recently; Giulio Tiozzo proved some very broad connections between the Mandelbrot Set and entropy, completing some of the last work of Thurston (with whom he collaborated). I have been exploring some specific connections of M to Physics for more than 30 years now, and the location I focus on in my essay has relevance for entropic theories of gravity.

I'd appreciate your comments on my essay. I will be back once I have read your paper in its entirety.

Warm Regards,

Jonathan

report post as inappropriate

Dear Jonathan,

Thank you for your comments. I look forward to your return and further comments.

I am not aware of these connections between the Mandelbrot set and entropy. I will have to look into this as you have piqued my interest. And entropic theories of gravity are interesting in their own right, so I am eager to look at your essay as well.

I wish you the best of luck in this essay contest.

Thank you again,

Kevin

Thank you for your comments. I look forward to your return and further comments.

I am not aware of these connections between the Mandelbrot set and entropy. I will have to look into this as you have piqued my interest. And entropic theories of gravity are interesting in their own right, so I am eager to look at your essay as well.

I wish you the best of luck in this essay contest.

Thank you again,

Kevin

Thsnks Kevin,

Tiozzo was a PhD student of C.T. McMullen at Harvard...

He was set to work on a problem by Tan Lei, to make general a result involving local symmetry at Misiurewicz points against global asymmetry, which I think applies broadly to Physics. This led to his work with Thurston beofre Bill's demise.

More later,

Jonathan

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Tiozzo was a PhD student of C.T. McMullen at Harvard...

He was set to work on a problem by Tan Lei, to make general a result involving local symmetry at Misiurewicz points against global asymmetry, which I think applies broadly to Physics. This led to his work with Thurston beofre Bill's demise.

More later,

Jonathan

report post as inappropriate

This essay is beautiful work Kevin...

I am not in full agreement however. Some statements are only absolutely true if you rule out the possibility of hyper-dimensional super-determinism - an avenue I have been exploring of late. I have a paper in peer-review on "Painting, Baking, and non-Associative Algebra" that talks about forced ordering in higher-order Maths. Things must be done in...

view entire post

I am not in full agreement however. Some statements are only absolutely true if you rule out the possibility of hyper-dimensional super-determinism - an avenue I have been exploring of late. I have a paper in peer-review on "Painting, Baking, and non-Associative Algebra" that talks about forced ordering in higher-order Maths. Things must be done in...

view entire post

report post as inappropriate

Hope I didn't gush too much Kevin...

But I really like that your more down to Earth approach yields similar answers as a more wildly open-ended version of reality might. I think I'll probably read this essay again for detail, even though I've already given my rating. Good luck in the contest.

Best,

JJD

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But I really like that your more down to Earth approach yields similar answers as a more wildly open-ended version of reality might. I think I'll probably read this essay again for detail, even though I've already given my rating. Good luck in the contest.

Best,

JJD

report post as inappropriate

Dear Jonathan,

Thank you so much for your kind words and the detail with which you discussed my essay.

> But I really like that your more down to Earth approach yields similar answers as a more wildly open-ended version of reality might.

I think that things are fundamentally simple. And that we add layers of complication, which is not always necessary.

Thank you again!

Kevin

Thank you so much for your kind words and the detail with which you discussed my essay.

> But I really like that your more down to Earth approach yields similar answers as a more wildly open-ended version of reality might.

I think that things are fundamentally simple. And that we add layers of complication, which is not always necessary.

Thank you again!

Kevin

Dear Prof Kevin H Knuth

This post is a discussion post after well knowledged reply given on May. 4, 2020 @ 07:55 GMT above ... Your nice words.........

Godel's theorem is generally applicable to mathematics. Whether there are implications for cosmology really comes down to whether the cosmological questions, that one wishes to prove, rely on mathematics that is...

view entire post

This post is a discussion post after well knowledged reply given on May. 4, 2020 @ 07:55 GMT above ... Your nice words.........

Godel's theorem is generally applicable to mathematics. Whether there are implications for cosmology really comes down to whether the cosmological questions, that one wishes to prove, rely on mathematics that is...

view entire post

report post as inappropriate

Dear Professor Kevin Knuth,

Your essay was a joy to read your essay!

Your focus on addivity and expounding on its centrality to the sciences made for a nuanced and intelligent argument. My submission (co-authored) has a similar focus, where we interpret mathematics as the exclusive language of natural science and try to work out the implications it has and how the 3 un's become relevant. We view science as a function mapping observations to numbers, which I think is consistent to your statement:

"In science, we quantify things so that we can rank them: quantity, mass, volume, voltage, probability.

To maintain such rankings, quantities must be assigned consistently, especially in situations in which

things are combined or partitioned to form other numbers of things."

And, thus, like you conclude ( though nowhere as rigorously as both authors are undergrad level students) that we will have statements that are undecidable.

Whereas we differ in our work , is that we do not think observation and its interpretation through probability theory can rescue us, and found your arguments very convincing!

Raiyan Reza

(PS: I have you find time to read our work : https://fqxi.org/community/forum/topic/3563 )

report post as inappropriate

Your essay was a joy to read your essay!

Your focus on addivity and expounding on its centrality to the sciences made for a nuanced and intelligent argument. My submission (co-authored) has a similar focus, where we interpret mathematics as the exclusive language of natural science and try to work out the implications it has and how the 3 un's become relevant. We view science as a function mapping observations to numbers, which I think is consistent to your statement:

"In science, we quantify things so that we can rank them: quantity, mass, volume, voltage, probability.

To maintain such rankings, quantities must be assigned consistently, especially in situations in which

things are combined or partitioned to form other numbers of things."

And, thus, like you conclude ( though nowhere as rigorously as both authors are undergrad level students) that we will have statements that are undecidable.

Whereas we differ in our work , is that we do not think observation and its interpretation through probability theory can rescue us, and found your arguments very convincing!

Raiyan Reza

(PS: I have you find time to read our work : https://fqxi.org/community/forum/topic/3563 )

report post as inappropriate

Nice essay Kevin. Perhaps not surprisingly, I am one of those people who actually *has* wondered about addition. Bertrand Russell actually talks a little bit about this in the Introduction to Mathematical Philosophy, but what I find most intriguing is how this becomes known to sentient beings. I use the term "sentient beings" simply because it is known that many species can actually perform simple addition. Strangely, there is a point at which children (when they are very young), confuse strict addition with spatial extent. That is, for a certain period of time as they develop their understanding of space (usually during the toddler years), they often will mistakenly think that fewer objects spread out over a greater extent, are greater than a larger number of objects packed close together. There was some pioneering work done on this in the late 60s. Anyway, interesting stuff and, as I said when you commented on my essay, we need to have a beer when this whole pandemic is over.

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Dear Professor Kevin Knuth,

Just now I gave the best rating for your wonderful essay, Earlier yours is 6.3 ( 16 ratings) now 6.5. Congratulations!!!

Hope you will get some time to look at mt essay TODAY and rate it. Tomorrow contest is closing is closing!!!

Best wishes to your essay!

=snp

report post as inappropriate

Just now I gave the best rating for your wonderful essay, Earlier yours is 6.3 ( 16 ratings) now 6.5. Congratulations!!!

Hope you will get some time to look at mt essay TODAY and rate it. Tomorrow contest is closing is closing!!!

Best wishes to your essay!

=snp

report post as inappropriate

Hello again Kevin,

I am curious what you will think if you read my essay. I hope you get the chance, while there is still time for ratings to count, because I have the sense you reward good scholarship. Even though we have near-opposite approaches in some areas, I think you will be rewarded for taking the time.

All the Best,

Jonathan

report post as inappropriate

I am curious what you will think if you read my essay. I hope you get the chance, while there is still time for ratings to count, because I have the sense you reward good scholarship. Even though we have near-opposite approaches in some areas, I think you will be rewarded for taking the time.

All the Best,

Jonathan

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Perhaps after the ratings wars have concluded...

I can contact you about an Entropy submission. I was invited to write a paper for a special issue once, but then the market crashed and I could not cover the publication fees, so I had to pull it. That work was on a common basis for non-locality and entropy, which I presented on at FFP10. But my current research is in a different direction. My essay will provide insights.

Regards,

Jonathan

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I can contact you about an Entropy submission. I was invited to write a paper for a special issue once, but then the market crashed and I could not cover the publication fees, so I had to pull it. That work was on a common basis for non-locality and entropy, which I presented on at FFP10. But my current research is in a different direction. My essay will provide insights.

Regards,

Jonathan

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It looks like we both made it by a hair...

But either way, I'm glad we got a chance to interact this round, professor Knuth. I'd be pleased if we can continue the conversation. I'm tempted to give a brief explanation of the entropy and non-locality connection now. But I'd better leave off here. I get up to Albany from time to time, if you would care to meet in person at some point.

All the Best,

Jonathan

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But either way, I'm glad we got a chance to interact this round, professor Knuth. I'd be pleased if we can continue the conversation. I'm tempted to give a brief explanation of the entropy and non-locality connection now. But I'd better leave off here. I get up to Albany from time to time, if you would care to meet in person at some point.

All the Best,

Jonathan

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Hi Kevin,

Pleased I got to your essay. As high quality in content and well written as usual.I loved your identification that;*"The effectiveness of mathematics is reasonable because mathematics is designed to work*. Yet it still has it's limits in approximating nature itself! Good score coming. I'm sure you'll also like mine, challenging some foundations to good effect, and hope you get to it.

Very best

Peter

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Pleased I got to your essay. As high quality in content and well written as usual.I loved your identification that;

Very best

Peter

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Kevin. Thanks for your post on mine. I copy me response below;

Kevin,

Thanks, but it clearly DOES "apply to objects"! Galaxies down to snowflakes, grains of sand & atoms, so not so easily dismissed! You must keep Booleing at higher orders! I also agree the 2nd 'Born rule' case, and please now sit down and prepare to be shocked; describe a physical interaction sequence that produces...

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Kevin,

Thanks, but it clearly DOES "apply to objects"! Galaxies down to snowflakes, grains of sand & atoms, so not so easily dismissed! You must keep Booleing at higher orders! I also agree the 2nd 'Born rule' case, and please now sit down and prepare to be shocked; describe a physical interaction sequence that produces...

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p.s. Great to see your essay seems to have sneaked in as one of the 40 finalists. It was up to 6.4 after my score. I see it slipped again, but just, deservedly, hung on. Something really must be done about the trolling with 1's. I didn't give out any but seemed to get around 6!

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