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**Odessa Sabah**: *on* 1/5/17 at 13:17pm UTC, wrote That's great news and well deserved! :) Keep up the great work and I'm...

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**Alma Ionescu**: *on* 4/21/15 at 10:09am UTC, wrote Dear Rick, No need to apologize, I understand completely. It was my...

**Rick Searle**: *on* 4/21/15 at 1:42am UTC, wrote Dear Alma, Sincere apologies, but I have been very busy of late. I will...

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FQXi FORUM

October 21, 2019

CATEGORY:
Trick or Truth Essay Contest (2015)
[back]

TOPIC: Pythagoras versus the Mad Tailor: an essay on the unreasonable effectiveness of mathematics by Rick Searle [refresh]

TOPIC: Pythagoras versus the Mad Tailor: an essay on the unreasonable effectiveness of mathematics by Rick Searle [refresh]

The success of mathematics in the natural sciences, and especially in physics, suggest that mathematics is a real and deep feature of the natural world rather than a mere convention invented by human beings. Yet does this suggest the idea first imagined by Pythagoras and Plato that nature itself is fundamentally mathematical? This essay proposes a weak version of Max Tegmark’s Mathematical Universe Hypothesis that might allow us to use the past and future of physics as a benchmark for whether the universe is a mathematical structure.

Rick Searle is a writer and educator living in central Pennsylvania. He is an affiliate scholar with the Institute for Ethics and Emerging Technology whose upcoming anthology “Rethinking Machine Ethics in the Age of Ubiquitous Technology” explores the intersection between science, technology, and philosophy. He is one of the winners of last year’s FQXi essay contest “How Should Humanity Steer the Future” and blogs at utopiaordystopia.com.

Dear Rick Searle,

Pythagoras versus Mad tailor explains the subtle difference between the weak and strong MUH. You propose the weak version, “We live in a mathematical structure that is fully homeomorphic with a language of mathematics that retains …...Platonic features”. A still weaker version can be obtained by replacing 'homeomorphic' with 'dictated by'.

Regarding the question 'where the mathematical truths reside', I agree with your third 'option', “all mathematical truths, including undiscovered ones, can be said to be embedded in the range of possibilities that emerge one once defines some set of constraints”.

“Yet we’re still left with a big question; namely, what is the relationship between this mathematics and the world it so accurately describes?” I think the above referred third option is the answer. First we define the set of properties of matter. Mathematical laws will dictate the final emergent structure, the only structure possible. The reason: the physical world has no laws of its own, it has only some basic properties, the laws that it follows are mathematical. The same is applicable to chess: we decide the arbitrary properties of the chess-pieces; mathematics decides the emergent structures. I invite you to read my essay A physicalist interpretation of the relation between Physics and Mathematics.

Matter has only four basic properties: mass, volume, energy and force. We do not know why it is so. This four variables can lead to a single final structure - refer my site: finitenesstheory.com. Chess has six variables and it leads to a large number of final structures.

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Pythagoras versus Mad tailor explains the subtle difference between the weak and strong MUH. You propose the weak version, “We live in a mathematical structure that is fully homeomorphic with a language of mathematics that retains …...Platonic features”. A still weaker version can be obtained by replacing 'homeomorphic' with 'dictated by'.

Regarding the question 'where the mathematical truths reside', I agree with your third 'option', “all mathematical truths, including undiscovered ones, can be said to be embedded in the range of possibilities that emerge one once defines some set of constraints”.

“Yet we’re still left with a big question; namely, what is the relationship between this mathematics and the world it so accurately describes?” I think the above referred third option is the answer. First we define the set of properties of matter. Mathematical laws will dictate the final emergent structure, the only structure possible. The reason: the physical world has no laws of its own, it has only some basic properties, the laws that it follows are mathematical. The same is applicable to chess: we decide the arbitrary properties of the chess-pieces; mathematics decides the emergent structures. I invite you to read my essay A physicalist interpretation of the relation between Physics and Mathematics.

Matter has only four basic properties: mass, volume, energy and force. We do not know why it is so. This four variables can lead to a single final structure - refer my site: finitenesstheory.com. Chess has six variables and it leads to a large number of final structures.

report post as inappropriate

Dear Jose Koshy,

Thank you for reading and commenting on my essay. I really enjoyed your essay and I completely agree with you on this:

“Thus the question whether the connection between physics and mathematics is a trick or truth is very relevant at present.”

And that the whole issue can be traced back to Newton. I am still left with the question, though, of why Newton’s turn to mathematics was so successful, or why, if the mathematical approach is not the right one, these approaches have been able to give us both precision and sometimes predict unanticipated phenomenon we are only later able to confirm.

Best of luck in the contest!

Rick Searle

Thank you for reading and commenting on my essay. I really enjoyed your essay and I completely agree with you on this:

“Thus the question whether the connection between physics and mathematics is a trick or truth is very relevant at present.”

And that the whole issue can be traced back to Newton. I am still left with the question, though, of why Newton’s turn to mathematics was so successful, or why, if the mathematical approach is not the right one, these approaches have been able to give us both precision and sometimes predict unanticipated phenomenon we are only later able to confirm.

Best of luck in the contest!

Rick Searle

Dear Rick,

I would agree with you that much of the content of mathematics is discovered rather than invented. I think you maintain that what we ordinarily think of as the external world is a mathematical structure, as contrasted with merely corresponding to such a structure in some way. So, you accept Tegmark’s view on that. My question is about the location you assign to the mathematical structures which apparently have no application in the external world. Tegmark envisions all these other structures as more or less alongside one another in a very capacious logical space. You prefer to include all the non-evident structures inside nature, and perhaps even inside minds inside nature. The question, then, is what advantages does your treatment have over the way Tegmark handles the multiplicity of mathematical structures?

Larry Hitterdale

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I would agree with you that much of the content of mathematics is discovered rather than invented. I think you maintain that what we ordinarily think of as the external world is a mathematical structure, as contrasted with merely corresponding to such a structure in some way. So, you accept Tegmark’s view on that. My question is about the location you assign to the mathematical structures which apparently have no application in the external world. Tegmark envisions all these other structures as more or less alongside one another in a very capacious logical space. You prefer to include all the non-evident structures inside nature, and perhaps even inside minds inside nature. The question, then, is what advantages does your treatment have over the way Tegmark handles the multiplicity of mathematical structures?

Larry Hitterdale

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Dear Larry,

Thank you for reading my essay. I think the advantage over the way "Tegmark handles the multiplicity of mathematical structures" is that my weak MUH is not dependent on the existence of the multiverse, a hypothesis that may never be provable and for which even its inferred existence may at some point become suspect. I also think Tegamark's version of the MUH leaves the old questions regarding dualism in tact, for how is it that we have access to these mathematical structures that exist in other universes?

I am glad to see you are back for this contest, I loved your piece last time, and intend to read and vote on your new one sometime today or tomorrow. Please vote on my essay if you haven't dome so already.

Best of luck in the contest!

Rick

Thank you for reading my essay. I think the advantage over the way "Tegmark handles the multiplicity of mathematical structures" is that my weak MUH is not dependent on the existence of the multiverse, a hypothesis that may never be provable and for which even its inferred existence may at some point become suspect. I also think Tegamark's version of the MUH leaves the old questions regarding dualism in tact, for how is it that we have access to these mathematical structures that exist in other universes?

I am glad to see you are back for this contest, I loved your piece last time, and intend to read and vote on your new one sometime today or tomorrow. Please vote on my essay if you haven't dome so already.

Best of luck in the contest!

Rick

Dear Rick,

I understand what you are saying here. I think your approach might encounter another difficulty, but the approach does avoid the very serious epistemological problems that Tegmark’s MUH faces. At least in contest, the discussion of this topic is coming to a close. I do read your IEET blogs on a regular basis. I appreciate your choice of topics, by which I often encounter new items. Your presentations are always worth reading, and I tend to agree with you most of the time. Beginning in the next few months, I intend to participate in some of the discussions which you initiate, and I shall try to bring in new or at least less-familiar ideas as often as I can.

Best wishes,

Larry

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I understand what you are saying here. I think your approach might encounter another difficulty, but the approach does avoid the very serious epistemological problems that Tegmark’s MUH faces. At least in contest, the discussion of this topic is coming to a close. I do read your IEET blogs on a regular basis. I appreciate your choice of topics, by which I often encounter new items. Your presentations are always worth reading, and I tend to agree with you most of the time. Beginning in the next few months, I intend to participate in some of the discussions which you initiate, and I shall try to bring in new or at least less-familiar ideas as often as I can.

Best wishes,

Larry

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Rick -

You have written a delightful essay, and I have ranked it highly. I'm sorry it has not (yet) gotten the attention it deserves! I think you offer a useful critique of the strong MUH that so many seem to have rushed to embrace.

I also applaud your hopefulness that further developments in math/physics will provide a better sense of whether we will gain confidence or lose confidence in your weak MUH formulation. I am not so sanguine, as I feel we are seeing (and will continue to see) more of a divergence rather than a convergence. The explanations we all seek need to be transcendent!

Sincere regards - George Gantz

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You have written a delightful essay, and I have ranked it highly. I'm sorry it has not (yet) gotten the attention it deserves! I think you offer a useful critique of the strong MUH that so many seem to have rushed to embrace.

I also applaud your hopefulness that further developments in math/physics will provide a better sense of whether we will gain confidence or lose confidence in your weak MUH formulation. I am not so sanguine, as I feel we are seeing (and will continue to see) more of a divergence rather than a convergence. The explanations we all seek need to be transcendent!

Sincere regards - George Gantz

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Dear Rick,

Great essay! I liked very much your tailor analogy, and your well-written arguments for a weak version of the Mathematical Universe Hypothesis.

Good luck in the contest,

Mohammed

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Great essay! I liked very much your tailor analogy, and your well-written arguments for a weak version of the Mathematical Universe Hypothesis.

Good luck in the contest,

Mohammed

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Hi Rick--

I enjoyed your essay very much. Great title and concept. I had not heard the story of the honeybee. In the small world department, we both relied on science fiction authors to make points (you, Lem; me, Zelazny).

I was intrigued by your Weak MUH argument. I like the way that you set it out and, most important, detailed ways in which the hypothesis might fail. That's fantastic--the true sign of a first-class critical thinker!

Best regards and good luck,

Bill.

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I enjoyed your essay very much. Great title and concept. I had not heard the story of the honeybee. In the small world department, we both relied on science fiction authors to make points (you, Lem; me, Zelazny).

I was intrigued by your Weak MUH argument. I like the way that you set it out and, most important, detailed ways in which the hypothesis might fail. That's fantastic--the true sign of a first-class critical thinker!

Best regards and good luck,

Bill.

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Rick,

If we successfully model a Theory of Everything, does this mean a strong MUH and thus that the universe is a mathematical structure? Would we say the the universe is knowable then? Does a weak version of Max Tegmark's MUH suspend the belief that nature itself is fundamentally mathematical until one model is established?

I am not a mathematician, so I find such questions unfathomable.

You essay is a great discussion and prompts many questions.

Jim

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If we successfully model a Theory of Everything, does this mean a strong MUH and thus that the universe is a mathematical structure? Would we say the the universe is knowable then? Does a weak version of Max Tegmark's MUH suspend the belief that nature itself is fundamentally mathematical until one model is established?

I am not a mathematician, so I find such questions unfathomable.

You essay is a great discussion and prompts many questions.

Jim

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Hi Jim,

I am very glad to see another of your pieces in the contest again this year.

I will try to answer you question this way: based on the way I understand it the longer we go on without a clear path to a Theory of Everything, and the longer there are competing equally likely versions of a TOE in the running the less likely, in my view, that even a weak version of the MUH is true. Even if we are in a multiverse, only one mathematical structure should map onto our particular universe and if we can't find a way to complete this mapping then we should conclude that mathematics is not isomorphic with our world, but a very good tool for negotiating our way through it.

I intend to comment and vote for your essay tomorrow at the latest. I liked it very much. Please vote for my essay if you haven't done so already: I'm trying to get to that "magic number 10" and have been delayed in sitting down to give other essayists the attention they deserve.

Best of luck in the contest!

Rick

I am very glad to see another of your pieces in the contest again this year.

I will try to answer you question this way: based on the way I understand it the longer we go on without a clear path to a Theory of Everything, and the longer there are competing equally likely versions of a TOE in the running the less likely, in my view, that even a weak version of the MUH is true. Even if we are in a multiverse, only one mathematical structure should map onto our particular universe and if we can't find a way to complete this mapping then we should conclude that mathematics is not isomorphic with our world, but a very good tool for negotiating our way through it.

I intend to comment and vote for your essay tomorrow at the latest. I liked it very much. Please vote for my essay if you haven't done so already: I'm trying to get to that "magic number 10" and have been delayed in sitting down to give other essayists the attention they deserve.

Best of luck in the contest!

Rick

Rick,

I always enjoy your writing, and I think this is a top notch essay even though we have different interpretations of Tegmark's program.

I think weakening the MUH will kill it -- for the reason that the weaker version depends on the same "equally likely" hypothesis on which a probabilistically random world depends. Tegmark is quite straightforward in admitting that if the universe is random at foundation, MUH is refuted.

My argument for the MUH is based on classical probability. Given a binary choice, MUH is probably true.

Thanks for the good read, and best wishes in the competition. I hope you get a chance to drop by my forum.

Best,

Tom

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I always enjoy your writing, and I think this is a top notch essay even though we have different interpretations of Tegmark's program.

I think weakening the MUH will kill it -- for the reason that the weaker version depends on the same "equally likely" hypothesis on which a probabilistically random world depends. Tegmark is quite straightforward in admitting that if the universe is random at foundation, MUH is refuted.

My argument for the MUH is based on classical probability. Given a binary choice, MUH is probably true.

Thanks for the good read, and best wishes in the competition. I hope you get a chance to drop by my forum.

Best,

Tom

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Tom,

I just finished your essay which I thought was excellent. I'll comment on that in your forum, but I also realized that I hadn't answered your question regarding killing the MUH and randomness here.

I am not sure that "weakening the MUH will kill it" in that as I see it a weakened MUH isn't so much an issue of the existence or none existence of the "strong" MUH, as it is a both easier and more robust form of empirical evidence that something like the MUH is true.

As for randomness if in either or both a weak or string version of the MUH are true wouldn't randomness disappear once we abandon the notion of time? That is, if the probabilities have "already" been "decided" and our perception of randomness is merely subjective and based on our subjective experience of moving through time combined with our inability to see the structure as a whole.

I am hoping to comment and rank your essay this evening. Again, it was excellent.

Best of luck!

Rick

I just finished your essay which I thought was excellent. I'll comment on that in your forum, but I also realized that I hadn't answered your question regarding killing the MUH and randomness here.

I am not sure that "weakening the MUH will kill it" in that as I see it a weakened MUH isn't so much an issue of the existence or none existence of the "strong" MUH, as it is a both easier and more robust form of empirical evidence that something like the MUH is true.

As for randomness if in either or both a weak or string version of the MUH are true wouldn't randomness disappear once we abandon the notion of time? That is, if the probabilities have "already" been "decided" and our perception of randomness is merely subjective and based on our subjective experience of moving through time combined with our inability to see the structure as a whole.

I am hoping to comment and rank your essay this evening. Again, it was excellent.

Best of luck!

Rick

Hi Rick,

Thanks. I thought I had rated your essay when I commented -- apparently not (forgive my senior moment :-) ). Done now, with my best.

This is certainly worth commenting on: "As for randomness if in either or both a weak or (strong) version of the MUH are true wouldn't randomness disappear once we abandon the notion of time? That is, if the probabilities have 'already' been 'decided' and our perception of randomness is merely subjective and based on our subjective experience of moving through time combined with our inability to see the structure as a whole."(?)

In my view, while randomness would disappear, classical binary probability would not. That's one of my main points -- that decidability with a time parameter implies a pairwise stochastic function (See Hess-Philipp 2002). That is, past and future simultaneously correlated events imply that the MUH is true with a probability of unity. That's why I think it cannot be weakened, unless one abandons classical probability along with randomness, which would obviate the hypothesis altogether.

All best,

Tom

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Thanks. I thought I had rated your essay when I commented -- apparently not (forgive my senior moment :-) ). Done now, with my best.

This is certainly worth commenting on: "As for randomness if in either or both a weak or (strong) version of the MUH are true wouldn't randomness disappear once we abandon the notion of time? That is, if the probabilities have 'already' been 'decided' and our perception of randomness is merely subjective and based on our subjective experience of moving through time combined with our inability to see the structure as a whole."(?)

In my view, while randomness would disappear, classical binary probability would not. That's one of my main points -- that decidability with a time parameter implies a pairwise stochastic function (See Hess-Philipp 2002). That is, past and future simultaneously correlated events imply that the MUH is true with a probability of unity. That's why I think it cannot be weakened, unless one abandons classical probability along with randomness, which would obviate the hypothesis altogether.

All best,

Tom

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Dear Rick,

I really like this essay, it is both well-argued and well-written :) I (or Pragmatic Physicist respectively) also approve of the pragmatism to get something useful out of the mathematical universe.

-- Sophia

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I really like this essay, it is both well-argued and well-written :) I (or Pragmatic Physicist respectively) also approve of the pragmatism to get something useful out of the mathematical universe.

-- Sophia

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Dear Rick,

I enjoyed reading your essay. I like your weak MUH, that "We live in a mathematical structure that is fully homeomorphic with a language of mathematics that retains this mathematics’ Platonic features i.e. it is timeless and reversibility." And I also like that you made it testable, so that our confidence in it may decrease or increase under some conditions. Very well written and well argued!

Best wishes,

Cristi

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I enjoyed reading your essay. I like your weak MUH, that "We live in a mathematical structure that is fully homeomorphic with a language of mathematics that retains this mathematics’ Platonic features i.e. it is timeless and reversibility." And I also like that you made it testable, so that our confidence in it may decrease or increase under some conditions. Very well written and well argued!

Best wishes,

Cristi

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Hello Rick,

Your essay was able to capture very well the theme of this year's essay. And nice to read.

If I may take you up on some aspects...

"One way that has been proposed that would overcome this problem is simply to do away with the map/territory distinction entirely in favor of the ultimate reality of the map itself. This is the case with Max Tegmark and his...

view entire post

Your essay was able to capture very well the theme of this year's essay. And nice to read.

If I may take you up on some aspects...

"One way that has been proposed that would overcome this problem is simply to do away with the map/territory distinction entirely in favor of the ultimate reality of the map itself. This is the case with Max Tegmark and his...

view entire post

report post as inappropriate

Hi Akinbo,

Thank you for reading my essay and for your comments. At least mathematical objects as a subset of Plato's idea of the Forms would be both timeless and reversible. 2 + 3 = 5 is not only eternally true in that it has always been and always will be true, but, like all mathematical equations is reversible- you can solve it in either direction. The equations found in physics share this timeless and reversible aspect as well. Whether reality itself does is another question entirely.

I am looking forward to reading your essay sometime tonight or tomorrow.

Please give me your vote if you haven't done so already.

Best of luck in the contest!

Rick

Thank you for reading my essay and for your comments. At least mathematical objects as a subset of Plato's idea of the Forms would be both timeless and reversible. 2 + 3 = 5 is not only eternally true in that it has always been and always will be true, but, like all mathematical equations is reversible- you can solve it in either direction. The equations found in physics share this timeless and reversible aspect as well. Whether reality itself does is another question entirely.

I am looking forward to reading your essay sometime tonight or tomorrow.

Please give me your vote if you haven't done so already.

Best of luck in the contest!

Rick

Rick,

Time grows short, so I am revisiting essays I’ve read (3/23) to assure I’ve rated them. I find that I have not rated yours, so I will rectify. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Time grows short, so I am revisiting essays I’ve read (3/23) to assure I’ve rated them. I find that I have not rated yours, so I will rectify. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Hi James,

I finally got around to reading your essay. Your passion for mathematics shines through. Loved what you wrote regarding the Euler identity- and the graphics were great!

Best of luck!

Rick

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I finally got around to reading your essay. Your passion for mathematics shines through. Loved what you wrote regarding the Euler identity- and the graphics were great!

Best of luck!

Rick

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Dear Rick,

I share your argumentation about the objectivity of the Platonic world and your conclusion that the

"actual history of physics itself should put to rest the argument that mathematics is a mere invention of the human mind which we impose upon nature. Mathematics is truth not trick".

In this respect, you may see in our essay the arguments similar to yours. I think though that answering the question

"how, among such a huge number of mathematical structures are we able to find the one that is actually ours?"

you lost an important fact that the laws of nature are expressed by rather simple equations. This fact is used in our refutation of Tegmark's "mathematical democracy".

Regards,

Alexey.

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I share your argumentation about the objectivity of the Platonic world and your conclusion that the

"actual history of physics itself should put to rest the argument that mathematics is a mere invention of the human mind which we impose upon nature. Mathematics is truth not trick".

In this respect, you may see in our essay the arguments similar to yours. I think though that answering the question

"how, among such a huge number of mathematical structures are we able to find the one that is actually ours?"

you lost an important fact that the laws of nature are expressed by rather simple equations. This fact is used in our refutation of Tegmark's "mathematical democracy".

Regards,

Alexey.

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Dear Rick,

Thank you for your kind comment and your interest in my thought. I don't know about an interview for I have never given any before. KQID is based on Xuan Yuan's DAO concept that it is the substance that creates and distributes everything that is. Dao as Giving first Taking later (Love) has been unfolding itself from its first emergent out of Non-existence by its own free will by itself and for itself. As I mentioned in my blog, this Philosoy has its theory, equations and numbers like that can be falsified or verified in scientific experiments or by everyday life experiences. As mentioned here, KQId is the only terry right now that can calculate with numbers precisely the size of our baby universe at birth, its temperature and its speed.how c evolve from the creation to the present light speed in the vacuum within 5 thousand years after the Bit Bang.

I looked at your essay and your Mad Taylor story is so wonderful. I like it very much. KQID is more agreeable with Pythagoras that "all things are numbers". In KQID, these numbers are Ψ(iτLx,y,z, T), 4 vector Einstein complex coordinates.

Wonderful essay and rated it accordingly.

Let us continue our discussion and I believe in strongly the symphony of ideas,

Sincerely yours,

Leo KoGuan

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Thank you for your kind comment and your interest in my thought. I don't know about an interview for I have never given any before. KQID is based on Xuan Yuan's DAO concept that it is the substance that creates and distributes everything that is. Dao as Giving first Taking later (Love) has been unfolding itself from its first emergent out of Non-existence by its own free will by itself and for itself. As I mentioned in my blog, this Philosoy has its theory, equations and numbers like that can be falsified or verified in scientific experiments or by everyday life experiences. As mentioned here, KQId is the only terry right now that can calculate with numbers precisely the size of our baby universe at birth, its temperature and its speed.how c evolve from the creation to the present light speed in the vacuum within 5 thousand years after the Bit Bang.

I looked at your essay and your Mad Taylor story is so wonderful. I like it very much. KQID is more agreeable with Pythagoras that "all things are numbers". In KQID, these numbers are Ψ(iτLx,y,z, T), 4 vector Einstein complex coordinates.

Wonderful essay and rated it accordingly.

Let us continue our discussion and I believe in strongly the symphony of ideas,

Sincerely yours,

Leo KoGuan

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Dear Leo,

Sorry it took me so long to respond: I have been traveling and very busy with work.

I am very glad you liked my essay. Should you change your mind on the issue of a short interview you can always email me here:

rsearle.searle@gmail.com

My only hope would be to provide some insight into your ideas for an English speaking audience interested in such issues.

All the best,

Rick Searle

Sorry it took me so long to respond: I have been traveling and very busy with work.

I am very glad you liked my essay. Should you change your mind on the issue of a short interview you can always email me here:

rsearle.searle@gmail.com

My only hope would be to provide some insight into your ideas for an English speaking audience interested in such issues.

All the best,

Rick Searle

Dear Rick,

I enjoyed the way you used the Mad Taylor metaphor to describe the modus operandi of mathematics regarding the characteristic of the natural world. Your proposition of a weak MUH is well thought out and very balanced especially now, when we seem to have circumstantial and somewhat conflicting evidence about the possibilities of unified descriptions of the universe. As you say "Our confidence in a weak MUH should decrease should there be notable theoretical and practical progress in other scientific fields that have embraced alternative or ad hoc mathematical models[..]". I think this is well written and well argued and I will rate it accordingly. Wish you best of luck in the contest! If you have the necessary time, please read my essay and let me know your thoughts in a comment.

Warm regards,

Alma

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I enjoyed the way you used the Mad Taylor metaphor to describe the modus operandi of mathematics regarding the characteristic of the natural world. Your proposition of a weak MUH is well thought out and very balanced especially now, when we seem to have circumstantial and somewhat conflicting evidence about the possibilities of unified descriptions of the universe. As you say "Our confidence in a weak MUH should decrease should there be notable theoretical and practical progress in other scientific fields that have embraced alternative or ad hoc mathematical models[..]". I think this is well written and well argued and I will rate it accordingly. Wish you best of luck in the contest! If you have the necessary time, please read my essay and let me know your thoughts in a comment.

Warm regards,

Alma

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Dear Alma,

Sincere apologies, but I have been very busy of late. I will try to get to your essay by tomorrow evening at the latest.

Rick

Sincere apologies, but I have been very busy of late. I will try to get to your essay by tomorrow evening at the latest.

Rick

Dear Rick,

No need to apologize, I understand completely. It was my pleasure to read your essay. I wish you a very nice day and a great week!

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No need to apologize, I understand completely. It was my pleasure to read your essay. I wish you a very nice day and a great week!

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Hi Alma,

I wanted to let you know that I thought your essay was wonderful and that I am glad you seem to have done well in the contest.

All the best,

Rick

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I wanted to let you know that I thought your essay was wonderful and that I am glad you seem to have done well in the contest.

All the best,

Rick

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I pronounce Pythagoras the winner..

While I didn't know Mad Max was a tailor, I suspected it all along, and you did an excellent job of showing people the difference between an approach that yields a Mad Tailor and a true Platonic model - where Math is an enduring ideal. High marks from me for an excellent paper. I'll say more, when there is time.

All the Best,

Jonathan

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While I didn't know Mad Max was a tailor, I suspected it all along, and you did an excellent job of showing people the difference between an approach that yields a Mad Tailor and a true Platonic model - where Math is an enduring ideal. High marks from me for an excellent paper. I'll say more, when there is time.

All the Best,

Jonathan

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That's great news and well deserved! :) Keep up the great work and I'm looking forward to the new and improved 192.168.1.1 192.168.1.1 192.168.1.1 192.168.1.1

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