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RECENT POSTS IN THIS TOPIC

Mohammed Khalil: on 4/24/15 at 11:01am UTC, wrote Dear Steven, Thank you for your kind words on our essay. I am glad you...

Steven Sax: on 4/23/15 at 1:06am UTC, wrote Dear Mohammed, Thank you for your kind remarks on my essay and I enjoyed...

Mohammed Khalil: on 4/20/15 at 22:59pm UTC, wrote Dear Armin, Thank you very much for your interesting comments. I respect...

Mohammed Khalil: on 4/20/15 at 22:51pm UTC, wrote Dear William, Thank you very much for your kind and encouraging words. I...

Mohammed Khalil: on 4/20/15 at 22:47pm UTC, wrote Dear Alma, Thank you very much for your kind and encouraging words. I...

Armin Nikkhah Shirazi: on 4/20/15 at 22:29pm UTC, wrote Dear Basem and Mohammed, Though I disagree with your instrumentalist...

William Parsons: on 4/20/15 at 19:39pm UTC, wrote Hi Mohammed and Basem-- An absolutely brilliant essay. I concur fully...

Alma Ionescu: on 4/19/15 at 19:23pm UTC, wrote Dear Mohammed, Basem, It is a well established fact that people will...


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FQXi FORUM
October 24, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: How Accurately Can Mathematics Describe Nature? by Basem Galal and Mohammed M. Khalil [refresh]
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Author Mohammed M. Khalil wrote on Mar. 10, 2015 @ 15:41 GMT
Essay Abstract

Mathematics and physics are different, yet they are closely connected. The effectiveness of mathematics in physics is unparalleled in any other branch of knowledge. In this essay, we try to explain the reason for this effectiveness based on the view that mathematics is invented. We also question the accuracy of mathematics in describing nature, and argue that mathematics does not provide us with the truth about how nature works, but with models that enable us to make predictions about the outcome of observations and experiments.

Author Bio

Basem and Mohammed are undergraduate students at Alexandria University, Egypt. Basem is interested in doing research in machine learning (theory and application), especially deep learning. Mohammed is interested in theoretical physics. He won third prize in the previous FQXi essay contest, and coauthored 7 research papers http://inspirehep.net/author/profile/M.M.Khalil.1

Download Essay PDF File

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John C Hodge wrote on Mar. 11, 2015 @ 02:21 GMT
Great essay.

I think math is discovered. I think you present a very good analysis of the comparison of discovered vs. invented. I’ll spend more time thinking about your `invented’ arguments.

The following addresses your objections to `discovered’ (``Math is a part of nature…”.)

A There is a need to discover the truth of how nature works to advance our survival....

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Author Mohammed M. Khalil replied on Mar. 11, 2015 @ 16:59 GMT
Dear Mr. Hodge,

Thank you for your comments. I looked at your essay, and I appreciate your arguments for mathematics as a characteristic of the universe.

I will read your essay in more detail and write my comments on it, but here I would like to address your comments about the discovery of mathematics.

A) Different formulations are equivalent, but they tell us...

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Alan M. Kadin wrote on Mar. 11, 2015 @ 02:33 GMT
Dear Mr. Galal and Mr. Khalil:

Your essay is very well written and insightful. Are you really both undergraduate students?

Given that your essay focuses on the role of mathematics in providing models for physical systems, you might be interested in my essay, "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory". I argue that contrary to universal...

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Author Mohammed M. Khalil replied on Mar. 11, 2015 @ 17:03 GMT
Dear Dr. Kadin,

Thank you for your interesting post. I am glad you liked our essay. I agree with you that mathematics provides models for nature. Some models might accelerate the progress of physics, while others might hinder it.

Your model for quantum mechanics seems interesting. I will read it in more detail soon.

Best,

Mohammed

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James Lee Hoover wrote on Mar. 11, 2015 @ 22:11 GMT
Basem and Mohammed,

I salute you for your cogent and straightforward argument that math is invented.

I also believe that it is an effective invention to represent and model the natural world, providing all the advantages you mention. My views are similar in my essay, "Connections: math, physics and the mind"

Regards,

Jim

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Author Mohammed M. Khalil replied on Mar. 12, 2015 @ 20:22 GMT
Dear Jim,

Thank you for your kind comments. I am glad we agree that mathematics is an effective invention. I looked at your essay and I also salute your well-written arguments. I enjoyed your analogy with the Euler’s identity and the connection between math, physics, and the brain.

Best regards,

Mohammed

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Ed Unverricht wrote on Mar. 12, 2015 @ 01:04 GMT
Dear Basem Galal and Mohammed M. Khalil,

Very interesting and thought provoking essay. Your initial premise that mathematics is invented leading to its effectiveness, is followed up by a very solid argument "we argue that the usefulness of mathematics in discovering new theories is limited, and that it does not provide us with a real picture of the world, but with models useful as calculational tools for making predictions.".

Your statement "Our theories are models of nature. Some models are more useful than others." is the lead in to the geometric models in my essay here of the particles of the standard model. Hope you get a chance to read and comment on it.

Great essay, best of luck in the contest.

Regards, Ed Unverricht

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Author Mohammed M. Khalil replied on Mar. 12, 2015 @ 20:31 GMT
Dear Ed Unverricht,

Thank you for the kind and interesting comments. I looked at your essay, and I am glad we agree that mathematics provides useful models for nature. I enjoyed very much your beautiful images, and I could only wonder at the amazing usefulness of mathematical models.

Good luck to you too.

Best regards,

Mohammed

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Edward Michael MacKinnon wrote on Mar. 12, 2015 @ 04:30 GMT
I think that this is a remarkable essay for two undergraduates. It should signal a great future for both. Since my contribution takes a very similar position, but develops it historically, I really have no adverse comments. However, I find the claim that nature is symmetric a bit ambiguous. We use symmetry principles in developing theories especially the standard model of particle physics. Then we have to add qualifications because what we find are broken symmetries.

Ed MacKinnon

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Author Mohammed M. Khalil replied on Mar. 12, 2015 @ 20:46 GMT
Dear Ed MacKinnon,

Thank you for your kind comments. Your essay is interesting, and you gave well-written arguments for the coevolution of physics and mathematics. I am glad we agree on some points.

By "nature is symmetric", we mean that the laws of physics are invariant under specific transformations, such as translation in space and motion at constant velocity. If not for this property, describing nature would be very difficult indeed.

Best regards,

Mohammed

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Joe Fisher wrote on Mar. 17, 2015 @ 16:42 GMT
Dear Mssrs. Galal and Khalil,

You wrote: “A theory can have support from its theoretical foundation and solid mathematics,

but until it can produce predictions that can be compared with experiment, we must not evaluate it as the only truth. We should always consider alternative approaches even if they are less developed.”

Please behold my alternative approach: This is my...

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Author Mohammed M. Khalil replied on Mar. 17, 2015 @ 18:35 GMT
Dear Mr. Fisher,

Thank you for your comment, and for sharing your views.

Best regards,

Mohammed

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adel sadeq wrote on Mar. 22, 2015 @ 17:23 GMT
Hi Mohammed,

Thank you for reading my essay. You have written a very well essay although you probably know that our philosophies are different.

I have seen that you have co authored with Das and Faraj, that is very impressive. How did you do that? Is it possible to show my idea to Faraj and also get Basem to run the simple simulation(at the end of each section written "program link") to confirm the results.

Thanks and good luck.

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Author Mohammed M. Khalil replied on Mar. 22, 2015 @ 21:59 GMT
Dear Adel,

Thank you for your kind comments. I suggest you send Dr. Farag an email yourself, because you will be able to explain the idea better than me, and I am sure you will find him very cooperative.

Best regards,

Mohammed

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Member Rick Searle wrote on Mar. 22, 2015 @ 21:05 GMT
Dear Basem Galal and Mohammed M. Khalil,

This is by far the best essay to argue that mathematics is invented which I have read, and, believe me, in researching for this contest, I read a lot of them!

Please take the time to check out and vote on my own essay:

http://fqxi.org/community/forum/topic/2391

Best of luck in the contest!

Rick Searle

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Author Mohammed M. Khalil replied on Mar. 22, 2015 @ 21:58 GMT
Dear Rick,

Thank you for your kind comments. I really liked your essay; I rated it and wrote you a comment there.

Best regards,

Mohammed

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Thomas Howard Ray wrote on Mar. 23, 2015 @ 14:46 GMT
Mohammed (& Basem),

Thanks for commenting in my forum. I am at a loss, however, to know why you think our ideas are opposed -- I found your excellent essay to reflect an entirely rationalist view of science, as does mine.

I want to point out something to you: You quote Einstein on mathematics as a human invention:

“How is it possible that mathematics, a product of human...

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Author Mohammed M. Khalil replied on Mar. 23, 2015 @ 18:27 GMT
Dear Tom,

Thank you for your comments. I really like that quote of Einstein; his closed watch analogy agrees well with our essays.

I admit that our essays have some similar views. However, I think that mathematics is invented, and hence, there is no preexisting correspondence between mathematics and the physical world. Mathematics is effective because it was invented to describe patterns and regularities in nature. Mathematics provides models that describe nature, and most of those models are not exact, i.e. they do not correspond exactly to the phenomena they describe. This view is different from that of the mathematical universe hypothesis which your essay supports.

All the best,

Mohammed

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Thomas Howard Ray replied on Mar. 24, 2015 @ 00:54 GMT
Mohammed, I'm afraid you still miss the point. I don't support the mathematical universe hypothesis a priori. The philosophical question of whether mathematics is invented or discovered has *nothing* to do with the correspondence of mathematics to physics, i.e., the corresponding truth content of their respective models.

You wouldn't say that natural language has truth content independent of physics, would you? In other words, the string of symbols C-A-T is true if, and only if, there is a physical counterpart to the symbols. That is what Einstein was saying -- e.g., he favored the introduction of extra dimensional models, even in his day, " ... if there exist good physical reasons to do so."

The MUH is based on physical probability, not mathematical philosophy.

Best,

Tom

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Author Mohammed M. Khalil replied on Mar. 25, 2015 @ 14:27 GMT
Dear Tom,

Thank you for your comment and for the explanation. I think I see your point now.

Best,

Mohammed

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Jose P. Koshy wrote on Mar. 24, 2015 @ 06:21 GMT
Dear Galal & Khalil,

You have tried to give a logical picture how mathematics has become effective in physics, and suggest the need for new mathematical inventions to solve the hitherto unsolved problems. You say correctly, “A great mystery about nature is that we can describe the same phenomenon with different mathematical formulations”. Have you thought of the reverse possibility? A...

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Author Mohammed M. Khalil replied on Mar. 25, 2015 @ 14:33 GMT
Dear Jose,

Thank you for your comment. I think mathematics is invented in the sense that we define/invent a set of rules (axioms) and then discover certain relations based on them (theorems).

Best,

Mohammed

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Janko Kokosar wrote on Apr. 4, 2015 @ 13:09 GMT
Dear Basem and Mohammed

I agree with your that mathematics is not enough to describe physics. Thus, that many mathematical theories predicted something in mathematics, but predictions were wrong. My opinion is that math is only an abstract language, which tell more simple what happening in physics. (Torsten Asselmeyer-Maluga used the best words: ''' abstraction is necessary concept for...

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Author Mohammed M. Khalil replied on Apr. 7, 2015 @ 16:35 GMT
Dear Janko,

Thank you for your interesting comments. We seem to agree about the limitations of mathematics but disagree about the accuracy of theories in describing nature.

Best regards,

Mohammed

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Cristinel Stoica wrote on Apr. 5, 2015 @ 17:07 GMT
Dear Basem and Mohammed,

I liked reading your essay. It is very well written, and you explain very well the role of mathematical models, and how they can turn out to be inadequate to describe the physical world. I also like that you let open the possibility that a mathematical theory well suited to describe the universe may exist, although we will never be sure it is the true one.

Best wishes,

Cristi

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Author Mohammed M. Khalil replied on Apr. 7, 2015 @ 16:14 GMT
Dear Cristi,

Thank you for reading our essay and for your kind and encouraging comments.

Best regards,

Mohammed

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Jacek Safuta wrote on Apr. 10, 2015 @ 10:06 GMT
Dear Basem and Mohammed,

You have presented one of the best essays in the contest. Very clear, modest and not intrusive like many others. You deserve very high rating what you will observe in a minute. However I want to address some issues.

You present important objections to the view No. “2. Mathematics is discovered because it is part of nature just like physics.” I agree with...

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Author Mohammed M. Khalil replied on Apr. 10, 2015 @ 21:46 GMT
Dear Jacek,

Thank you very much for your kind comments and for the rating.

I am glad you agree with our objections to the discovery of mathematics if we define math as an abstract language of equations. However, I also think that geometry is invented not discovered. In the real world there are no straight lines extending to infinity, or perfect circles that exactly lead to pi when...

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Jacek Safuta replied on Apr. 11, 2015 @ 07:35 GMT
Dear Mohammed,

The principle of minimum energy (really the second law of thermodynamics) states that for a closed system, with constant external parameters and entropy, the internal energy will decrease and approach a minimum value at equilibrium. It means that every object/structure shall deform to the shape that minimizes the total potential energy. That shape is an ideal shape. But the system cannot approach that ideal shape.

I am not a Platonist. Obviously, in nature we cannot find ideal shapes. As I have mentioned, the geometry is about shapes that we perceive in real world and not equations. However it is not practical or possible to make calculus on real shapes. To make predictions we need calculus. That is the reason we need idealizations (approximations) of real complexity. In my opinion, the lack of ideal shapes in nature is not an intrinsic feature of real objects (the second law) but the outcome of complexity , interactions and dynamics of interacting objects.

Best regards

Jacek

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Vladimir F. Tamari wrote on Apr. 14, 2015 @ 01:19 GMT
Dear Basem and Mohammed

Congratulations on an exceptionally thought-out and well-written essay. It helped me appreciate it that I agree in my own essay with many (but not all) of the points you made. For example you mention symmetry and universality as explanation of the effectiveness of mathematics. I go much further and speculate that at the deepest level mind, mathematics and nature...

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Author Mohammed M. Khalil replied on Apr. 14, 2015 @ 22:44 GMT
Dear Vladimir,

Thank you for your kind comments, and good luck in the contest.

Mohammed

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Torsten Asselmeyer-Maluga wrote on Apr. 15, 2015 @ 08:14 GMT
Dear Mohammed,

thanks for reading my essay. As I see we are agreeing in many points. But more importantly, I also think that math is an invention. Thanks for bringing your essay to my attention.I rate your essay high.

Best

Torsten

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Author Mohammed M. Khalil replied on Apr. 15, 2015 @ 17:52 GMT
Dear Torsten,

Thank you very much for the kind comment and for the rating.

Best,

Mohammed

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Michel Planat wrote on Apr. 15, 2015 @ 09:05 GMT
Dear Basem and Mohammed,

Your point is very well argued and reasonable, very close to the one by the great mathematician, physicist and thinker Henri Poincaré in Science and Hypothesis

http://www.gutenberg.org/files/37157/37157-pdf.pdf


"Rôle of Hypothesis.—Every generalisation is a hypothesis. Hypothesis therefore plays a necessary rôle, which no one has ever...

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Author Mohammed M. Khalil replied on Apr. 15, 2015 @ 17:49 GMT
Dear Michel,

Thank you for your kind comments. I wasn't aware of Poincaré's book, but it seems very interesting, and I am glad our essay agrees with his ideas. I think that currently the main problem with theoretical physics is the wide gap between hypothesis and verification.

I have read your essay and I find it very interesting.

Best regards,

Mohammed

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Michel Planat replied on Apr. 15, 2015 @ 18:01 GMT
Dear Mohammed,

Congratulations, you are on the right way already gussing what matters, soon being involved in a great chapter of science.

All the best,

Michel

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Member Noson S. Yanofsky wrote on Apr. 17, 2015 @ 15:54 GMT
Hi,

You wrote me that you looked at my paper and I am very appreciative. I read your paper and I like it.

I wish you would work out some more of your idea that simplicity and beauty in physical theories can be understood from the computational complexity point of view. Has anyone else talked about this?

Thanks again!

All the best,

Noson Yanofsky

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Basem Galal Ahmed replied on Apr. 18, 2015 @ 13:35 GMT
Dear Noson,

Thank you for reading our essay

In our article we suggest that we can use computational complexity as a measure of simplicity because we use computers today for almost all physical computations and simulations .Hence , it's reasonable to choose measure of simplicity relative to computers, I am not sure if anyone else talked about that.

All the best,

Basem

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Alma Ionescu wrote on Apr. 19, 2015 @ 19:23 GMT
Dear Mohammed, Basem,

It is a well established fact that people will mostly speak of things that impress them, either good or bad. I couldn't agree more with your point that Wigner's speech is not shedding light on theories that have not worked as hoped. For every successful theory like relativity or quantum mechanics, there is a bunch of other theories that we know are wrong, like N=4 super Yang Mills or your SU(5) example. I like your idea of quantifying the elegance of a theory through the number of dependent variables and the relations between variables. It sounds like the most compact network model possible. Your arguments do a very good job getting the point across as well as your clear and enjoyable writing style.

Warm regards,

Alma

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Author Mohammed M. Khalil replied on Apr. 20, 2015 @ 22:47 GMT
Dear Alma,

Thank you very much for your kind and encouraging words. I agree with you about wrong theories like N=4 super Yang Mills. We suggested using computational complexity as a measure of simplicity because we use computers today for almost all physical computations and simulations. Hence, it's reasonable to choose measure of simplicity relative to computations.

Kind regards,

Mohammed

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William T. Parsons wrote on Apr. 20, 2015 @ 19:39 GMT
Hi Mohammed and Basem--

An absolutely brilliant essay. I concur fully with your analysis. In fact, your essay is so good that I'm glad I didn't write on your precise topic. You would have put me to shame. And, needless-to-say, I certainly agree with your last bullet on page 3.

I can't believe you two are only undergraduates. Your professors are lucky to have you. I predict shining futures for both of you. Keep up the good work!

Best regards and best of luck in the contest,

Bill.

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Author Mohammed M. Khalil replied on Apr. 20, 2015 @ 22:51 GMT
Dear William,

Thank you very much for your kind and encouraging words. I am glad you liked our essay. From your essay, I am sure you would have done a better job if you wrote about that topic.

All the best,

Mohammed

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Armin Nikkhah Shirazi wrote on Apr. 20, 2015 @ 22:29 GMT
Dear Basem and Mohammed,

Though I disagree with your instrumentalist perspective on physics and your antiplatonist view of mathematics, I am genuinely impressed by your eloquent defense of your positions. It is all the more impressive considering that, if I infer correctly, you are not even native English Speakers.

Despite our philosophical differences, there is much in your essay that I do concur with, especially the notion that observation and experiment trumps any other consideration in science.

Your ideas about quantifying aesthetic notions like beauty are interesting and deserve more detailed treatment.Finally, I could not agree more with your last paragraph as my own research effort is geared precisely to developing new mathematics that helps us model and especially understand fundamental aspects of reality.

Overall, you did a great job

Best wishes,

Armin

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Author Mohammed M. Khalil replied on Apr. 20, 2015 @ 22:59 GMT
Dear Armin,

Thank you very much for your interesting comments. I respect your opinion about the Platonist view, and I am glad you concur with other points.

Indeed the importance of experiments and observations in science cannot be overstressed. I think that currently the main problem with theoretical physics is the wide gap between theory and experiment.

Finally, I am glad to...

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Steven P Sax wrote on Apr. 23, 2015 @ 01:06 GMT
Dear Mohammed,

Thank you for your kind remarks on my essay and I enjoyed reading yours as well. As you know, we agree on many points. It's very true that many theories which were considered beautiful or simple ended up failing to explain observations and experimental data. I liked your discussion on simplicity and computational complexity, and on how the same phenomenon can be described with different mathematical formulations. You present throughout a very interesting essay, and I rate it highly.

Kind regards,

Steve Sax

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Author Mohammed M. Khalil replied on Apr. 24, 2015 @ 11:01 GMT
Dear Steven,

Thank you for your kind words on our essay. I am glad you enjoyed reading it, and that you agree with us in many points.

Best regards,

Mohammed

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