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**Jens Koeplinger**: *on* 11/4/12 at 22:23pm UTC, wrote Dear Eckard, thank you for leaving your note, and for referring to your...

**Eckard Blumschein**: *on* 11/4/12 at 16:37pm UTC, wrote Sorry for my typo.

**Eckard Blumschein**: *on* 11/4/12 at 16:35pm UTC, wrote Dear authors, While my conclusions are rather contrary to yours, I agree...

**Jens Koeplinger**: *on* 10/8/12 at 15:12pm UTC, wrote Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I...

**Benjamin Dribus**: *on* 10/5/12 at 4:41am UTC, wrote Dear Jens and John, What a lot of profound topics you weave into your...

**Jens Koeplinger**: *on* 10/5/12 at 2:25am UTC, wrote Hello Sergey - yeah, the ratings ... with almost 300 submissions I estimate...

**Sergey Fedosin**: *on* 10/4/12 at 5:56am UTC, wrote If you do not understand why your rating dropped down. As I found ratings...

**Jens Koeplinger**: *on* 9/26/12 at 17:41pm UTC, wrote Dear Caohoàng, Thank you for leaving your thought provoking note. You...

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FQXi FORUM

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CATEGORY:
Questioning the Foundations Essay Contest (2012)
[back]

TOPIC: "1 + 1 = 2" A Step in the Wrong Direction? by Jens Koeplinger and John Shuster [refresh]

TOPIC: "1 + 1 = 2" A Step in the Wrong Direction? by Jens Koeplinger and John Shuster [refresh]

Fundamental questions in physics can be asked anytime, anywhere. Often they arise at the interface of physics, mathematics, and philosophy – where scrapping conversation turns into testable hypothesis. This essay explores the idea that the primitive act of counting "1, 2, 3 ..." makes an implicit assumption that ultimately causes some of the challenges faced in quantum mechanics today. A hypothesis for what could be done differently is developed during a humorous, yet serious, conversation among a physics student, a math student, an ex-philosophy student, and a city councilor. Beginning with a physics student's ill-fated attempt at bargaining for a lower price, the essay touches upon beauty in numbers and nature; repetition, inversion, and algebraic closure in mathematics; and observability in quantum mechanics. A surprising property of the complex numbers will be shown to indicate incompleteness or inadequacy in regard to resolving certain questions in quantum mechanics. A new kind of number and arithmetic may be needed, and a proposal for such is sketched using the E8 lattice.

Jens Koeplinger received a "Diplom" (M. Sci.) in physics at the University of Heidelberg, Germany, in 1999. When not exploring possibilities in physical mathematics, he is working daytime hours as IT Systems Analyst for AT&T, and off-hours developing mobile apps at "Dirty Little Cyborg". John A Shuster earned an A.B. in math and economics (physics minor) from Kenyon College (OH) in 1971, then did graduate work in operations research at the University of Rochester (NY). He is a retired Systems Analyst who enjoys grandchildren, travel, writing, and exploring new math systems.

Dear Messrs Koeplinger and Shuster,

Despite the dismal fact that I do not know an awful lot about mathematics, I loved reading your exceptionally well written, humorous, yet perceptively cogent essay, and I do hope it garners one of the prizes. I hesitate to mention what my picayune quibble with the essay might be lest you might question my motive for bringing it up, but I will risk elaborating on it if you do not mind. In my essay Sequence Consequence, I concentrate on reality. I believe that one real Universe having one real appearance can be perpetually occurring in a real here for a real now in one real dimension once. Real stuff has always to be in one real dimension. I think that if there were three abstract spatial dimensions, it would be difficult if not impossible to determine how abstract stuff was distributed. Would heavy abstract stuff helpingly remain in dimension A, moderate abstract stuff stay in Dimension B, and light abstract stuff linger in dimension C. I prefer to think that only 1 of anything could only ever exist once. Unfortunately, the most confounding illogical code seems to be the numeric representation of numbers. For instance, a single line is used to depict each of the numbers from zero to nine. The number 0 could visibly equal the number1if only the number of lines used to construct both numbers was considered. Does the space inside of the 0 have a value? Is that spatial value greater, equal, or less than the space between the 0 and the 1? There has been a standardization of the measured speed of light. Why has there never been a set standard for the presentation of numbers?

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Despite the dismal fact that I do not know an awful lot about mathematics, I loved reading your exceptionally well written, humorous, yet perceptively cogent essay, and I do hope it garners one of the prizes. I hesitate to mention what my picayune quibble with the essay might be lest you might question my motive for bringing it up, but I will risk elaborating on it if you do not mind. In my essay Sequence Consequence, I concentrate on reality. I believe that one real Universe having one real appearance can be perpetually occurring in a real here for a real now in one real dimension once. Real stuff has always to be in one real dimension. I think that if there were three abstract spatial dimensions, it would be difficult if not impossible to determine how abstract stuff was distributed. Would heavy abstract stuff helpingly remain in dimension A, moderate abstract stuff stay in Dimension B, and light abstract stuff linger in dimension C. I prefer to think that only 1 of anything could only ever exist once. Unfortunately, the most confounding illogical code seems to be the numeric representation of numbers. For instance, a single line is used to depict each of the numbers from zero to nine. The number 0 could visibly equal the number1if only the number of lines used to construct both numbers was considered. Does the space inside of the 0 have a value? Is that spatial value greater, equal, or less than the space between the 0 and the 1? There has been a standardization of the measured speed of light. Why has there never been a set standard for the presentation of numbers?

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Dear Joe,

Thank you for your kind words! I'm glad you enjoyed the essay, we sure had a lot of fun writing it. Relating nature's observed geometric dimensionality and magnitudes to abstract defined algebraic dimensionality and numbers is one of the big riddles to be solved, we feel as well. Thanks for sharing your thoughts towards the numbers 0 and 1, and for referring to your essay.

Jens

Thank you for your kind words! I'm glad you enjoyed the essay, we sure had a lot of fun writing it. Relating nature's observed geometric dimensionality and magnitudes to abstract defined algebraic dimensionality and numbers is one of the big riddles to be solved, we feel as well. Thanks for sharing your thoughts towards the numbers 0 and 1, and for referring to your essay.

Jens

Jens,

Your essay reminds me of the many times over the past decade that I have been trying to communicate with "System International (SI) loyalists" that their base units are not suitable for "scientific units of measure." One of my communicants even stated "they are totally anthropocentric, arbitrary, and non-natural base units, from the POV of physical law."

My paper had not been...

view entire post

Your essay reminds me of the many times over the past decade that I have been trying to communicate with "System International (SI) loyalists" that their base units are not suitable for "scientific units of measure." One of my communicants even stated "they are totally anthropocentric, arbitrary, and non-natural base units, from the POV of physical law."

My paper had not been...

view entire post

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Hello Frank, thank you for pointing out your research. Aside from formal publication, do you have a reference to freely available material that would give the reader here an overview of your thoughts? You must admit that accusing the IEEE as having an "anthropocentric" bias is a bit odd, given that the customers of an Engineering society are humans after all. It's like accusing Barnes & Noble of selling books. Re "E8 based QM", I wish I knew what that is ... Best wishes, Jens

Jens, John,

I did not state the IEEE has an anthropocentric bias. I don't know if the person I quoted is an IEEE member, that used the term anthropocentric, but I do know he is an electrical engineer.

The IEEE publication I cited was the culmination of over ten years of trying to get the concept published. It was rejected by several publications before I submitted to an IEEE publication, and it was rejected. I rewrote the introduction and submitted the paper to another IEEE publication and it was accepted.

Presenting a physical law in the form of two right triangles is not taught in the text books.

The methodology in the IEEE paper disposes of anthropocentric bias in how base units of measure should be derived. Physicists are trying to derive physical laws that govern the characteristics of the universe, and the use of man-defined base units does not help. My topic, 1294, discusses the multi-century assumption that SI units are suitable for scientific units. Even the BIPM admits they are not based upon fundamental physical constants; they don't know how to correct it. My emails to various BIPM officials have never been answered.

The BIPM is a bureaucracy that exists for the purpose of preserving artifacts that represent purely anthropocentric base units of measure. SI units are fine for commerce. Bureaucracies do not take any action that will eliminate their existence.

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I did not state the IEEE has an anthropocentric bias. I don't know if the person I quoted is an IEEE member, that used the term anthropocentric, but I do know he is an electrical engineer.

The IEEE publication I cited was the culmination of over ten years of trying to get the concept published. It was rejected by several publications before I submitted to an IEEE publication, and it was rejected. I rewrote the introduction and submitted the paper to another IEEE publication and it was accepted.

Presenting a physical law in the form of two right triangles is not taught in the text books.

The methodology in the IEEE paper disposes of anthropocentric bias in how base units of measure should be derived. Physicists are trying to derive physical laws that govern the characteristics of the universe, and the use of man-defined base units does not help. My topic, 1294, discusses the multi-century assumption that SI units are suitable for scientific units. Even the BIPM admits they are not based upon fundamental physical constants; they don't know how to correct it. My emails to various BIPM officials have never been answered.

The BIPM is a bureaucracy that exists for the purpose of preserving artifacts that represent purely anthropocentric base units of measure. SI units are fine for commerce. Bureaucracies do not take any action that will eliminate their existence.

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Ok - thank you for clarifying. It would still be good to have a public overview of your work somewhere. It could be as simple as a personal web page or so. I'd be glad to have a look. Thanks, Jens

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Another essay on considering the context of numbers as opposed to just considering their value as a number is at

http://fqxi.org/community/forum/topic/1375

Thanks.

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http://fqxi.org/community/forum/topic/1375

Thanks.

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Another dream where we wake up just before the answer is given.

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:) Well then I suppose we better get to work and find the answer while awake!

Seriously, going beyond discussing a physical assumptions that may be wrong, to actually provide a fully working answer, that would be truly amazing. We decided against writing about published ideas that are developed further. With that, the essay sides in favor of inspiration, but at the expense of presenting a working model. Those familiar with my line of research know of course where the work with John is heading: From the 4D Euclidean quantum gravity model that needed a geometry, to the octonionic background geometry that needed a quantum theory, to the nonassociative quantum theory in one dimension that needs math yet to be determined in order to go higher-dimensional. "Just" about a year ago did I learn about a technique that lets us do exponentiation and differentiation on one-dimensional lattices. In a very optimistic estimate, this essay is a half-way point for our work of making this a reality on the E8 lattice. Next to formal publications ( http://www.jenskoeplinger.com/P ) we're working in a glass house ( http://groups.yahoo.com/group/hypercomplex/ ). Open-source research, so to speak - contributors welcome, to the least we appreciate if you post to our group if you've done related work, or work inspired by us.

Hope this helps describing where we come from in a bit more detail! Thanks, Jens

Seriously, going beyond discussing a physical assumptions that may be wrong, to actually provide a fully working answer, that would be truly amazing. We decided against writing about published ideas that are developed further. With that, the essay sides in favor of inspiration, but at the expense of presenting a working model. Those familiar with my line of research know of course where the work with John is heading: From the 4D Euclidean quantum gravity model that needed a geometry, to the octonionic background geometry that needed a quantum theory, to the nonassociative quantum theory in one dimension that needs math yet to be determined in order to go higher-dimensional. "Just" about a year ago did I learn about a technique that lets us do exponentiation and differentiation on one-dimensional lattices. In a very optimistic estimate, this essay is a half-way point for our work of making this a reality on the E8 lattice. Next to formal publications ( http://www.jenskoeplinger.com/P ) we're working in a glass house ( http://groups.yahoo.com/group/hypercomplex/ ). Open-source research, so to speak - contributors welcome, to the least we appreciate if you post to our group if you've done related work, or work inspired by us.

Hope this helps describing where we come from in a bit more detail! Thanks, Jens

Hi Jens,

I liked your essay, but it did leave me wanting more explanation. I think due to the length limitations the story line took up too much space. Knowing you a bit, I am quite sure you could have provided more content. I do appreciate your intent using the method you did, and had mused myself about presenting my essay within a story line with the theme Crazy Uncle O’s Magical Mystery Tour of Physical Reality. The wife talked me out of it. Just as well, without the prop I had to leave out quite a bit of content I wanted to put in to be able to shoe horn it into 9 pages.

I look forward to announcements on your blog http://groups.yahoo.com/group/hypercomplex/ about further developments. Keep up the good work.

Regards,

Rick

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I liked your essay, but it did leave me wanting more explanation. I think due to the length limitations the story line took up too much space. Knowing you a bit, I am quite sure you could have provided more content. I do appreciate your intent using the method you did, and had mused myself about presenting my essay within a story line with the theme Crazy Uncle O’s Magical Mystery Tour of Physical Reality. The wife talked me out of it. Just as well, without the prop I had to leave out quite a bit of content I wanted to put in to be able to shoe horn it into 9 pages.

I look forward to announcements on your blog http://groups.yahoo.com/group/hypercomplex/ about further developments. Keep up the good work.

Regards,

Rick

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Hello Rick - thank you for leaving your note. John and I noticed your essay as well. You understand that I am disappointed about seeing no mention, favorable or otherwise, of my analysis of your work ( arxiv:1103.4748 ). It is of course your choice on what to write about, and what to ignore. Jens

Jens,

Trust me when I say I did not intentionally fail to mention your paper, there just was no space available. I chose to discus algebraic invariance in terms of what I called Iso(). I was remiss not to include your paper in the Reference section, and have posted such in my essay blog. This also was not an intentional act, I just tried to give references related to the essay content. It is easy to leave things out of the references, like your omission of this in the very same arxiv paper. No worries, it is all good.

Rick

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Trust me when I say I did not intentionally fail to mention your paper, there just was no space available. I chose to discus algebraic invariance in terms of what I called Iso(). I was remiss not to include your paper in the Reference section, and have posted such in my essay blog. This also was not an intentional act, I just tried to give references related to the essay content. It is easy to leave things out of the references, like your omission of this in the very same arxiv paper. No worries, it is all good.

Rick

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Ok, I appreciate your note. You wrote at some point about your vision: "Algebra, analysis, topology and groups are interlocking parts. The most fundamental is the algebra, for it sets the tone for the remainder." Your 'octonion variance sieve' can indeed be expressed elegantly using derivation algebras. Those exhibit properties similar to what one would expect from arithmetic. There are a couple of formal bugs in my paper on this part of your work, in its current version on the arXiv at least; but since it has attracted no feedback whatsoever I'm somewhat demotivated towards fixing them. Maybe that explains my negativity.... I do believe that your octonion variance sieve works, and that - for differential equations - there are solution spaces that don't simply collapse into the quaternion case. Best wishes, Jens

Hello Jens,

Please do not be "demotivated" towards fixing the bugs in your arXiv paper. It has been more comprehensible to me than Rick's own writings and explanations (because of my own limitations as a physicist rather than a mathematician, and because of my associative, Clifford-algebraic perspective derived from the works of Hestenes and Lounesto). So, please, do revise your arXiv paper if necessary because it has been useful at least to me. In particular, I would be interested in understanding how the solution spaces for some differential equations do not collapse into the quaternion case. This is not what I would expect from my topological perspective of the octonionic 7-sphere.

Best,

Joy

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Please do not be "demotivated" towards fixing the bugs in your arXiv paper. It has been more comprehensible to me than Rick's own writings and explanations (because of my own limitations as a physicist rather than a mathematician, and because of my associative, Clifford-algebraic perspective derived from the works of Hestenes and Lounesto). So, please, do revise your arXiv paper if necessary because it has been useful at least to me. In particular, I would be interested in understanding how the solution spaces for some differential equations do not collapse into the quaternion case. This is not what I would expect from my topological perspective of the octonionic 7-sphere.

Best,

Joy

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Hello Joy - thank you so much for posting!

Regarding the bugs, it's good to see your interest. I'll try to get to them soon. Essentially the problem is that I'm treating polynomial functions and algebras as if they were the same thing. E.g., on the right-hand side of (5.7), a set of functions { f[N], u, v } is of course not contained in the quaternions. Oops - that doesn't make sense. What I meant to write was that the multiplication rules used in the polynomials f are quaternion, therefore making the f[N] quaternionic polynomial functions.

Then regarding where the approach collapses into the quaternions, I admit that my work is incomplete in that I only state in (5.9) that

der( Df ) contained in H

does not necessarily require that the polynomial f is quaternionic. In order to be complete it needs to be shown exactly where f may be octonionic and whether there exist any interesting differential operators D such that (5.9) still holds *and* Df is not already quaternion. Rick is proposing such a construct for his recovery of the Maxwell equations; and I've checked his multiplication rules by hand and found no error. But that doesn't make it formal proof, of course ...

Thank you again for your interest!

Jens

Regarding the bugs, it's good to see your interest. I'll try to get to them soon. Essentially the problem is that I'm treating polynomial functions and algebras as if they were the same thing. E.g., on the right-hand side of (5.7), a set of functions { f[N], u, v } is of course not contained in the quaternions. Oops - that doesn't make sense. What I meant to write was that the multiplication rules used in the polynomials f are quaternion, therefore making the f[N] quaternionic polynomial functions.

Then regarding where the approach collapses into the quaternions, I admit that my work is incomplete in that I only state in (5.9) that

der( Df ) contained in H

does not necessarily require that the polynomial f is quaternionic. In order to be complete it needs to be shown exactly where f may be octonionic and whether there exist any interesting differential operators D such that (5.9) still holds *and* Df is not already quaternion. Rick is proposing such a construct for his recovery of the Maxwell equations; and I've checked his multiplication rules by hand and found no error. But that doesn't make it formal proof, of course ...

Thank you again for your interest!

Jens

Hi Jens

After reading your essay i would like to send my observation

http://vixra.org/abs/0907.0014

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After reading your essay i would like to send my observation

http://vixra.org/abs/0907.0014

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Hi Jens,

I wish I could find some young students who really do have a dialogue like that. :-) Maybe I shop in the wrong stores.

Delightful essay! Going to spark a lot of conversation here, I predict -- and certainly most relevant to how physics and mathematics intersect. I hope to have something of substance to say later; a little pressed right now.

Best wishes in the contest -- hope you get a chance to visit my own essay site.

Tom

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I wish I could find some young students who really do have a dialogue like that. :-) Maybe I shop in the wrong stores.

Delightful essay! Going to spark a lot of conversation here, I predict -- and certainly most relevant to how physics and mathematics intersect. I hope to have something of substance to say later; a little pressed right now.

Best wishes in the contest -- hope you get a chance to visit my own essay site.

Tom

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Dear Jens Koeplinger and John Shuster

Very strange!like a fable.

If two you can give such a direction, why not boldly gives a new theory such is sketched using the E8 lattice (whether a expected)?

It looks like you a little lack of confidence in yourself?

Kind Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Very strange!like a fable.

If two you can give such a direction, why not boldly gives a new theory such is sketched using the E8 lattice (whether a expected)?

It looks like you a little lack of confidence in yourself?

Kind Regards !

Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY

August 23, 2012 - 11:51 GMT on this essay contest.

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Dear Caohoàng,

Thank you for leaving your thought provoking note. You suggest that it might be lack of confidence that caused us not to boldly sketch a new theory on the E8 lattice. To me it is more a reality check that assures me that most all suggestions for new physics that we can come up with, at any time, are wrong. This assurance comes from the mere number of possibilities out there. In order to keep this essay both entertaining for a wide audience, but also to make a strong point nevertheless, we focused on what it most important to us: To sketch a sense of naturalness, beauty, and simplicity as motivating some of our current and future work elsewhere. We did that at the expense of actually proposing a model, granted, but isn't it amazing with how few and simple assumptions you necessarily arrive at the E8 lattice? Repeat, invert, closure; and self-duality of the space under addition and multiplication. Lattices in 1, 2, 4, and 8 dimensions satisfy these very simple assumptions; and only in these dimensions. The E8 lattice is of course an immensely complicated construct if you attempt to understand its properties and automorphisms - something I will never succeed in fully. But conceptually it is *that* simple. Pure math at its finest, we feel, and that's what we want to convey to the reader.

Best wishes, Jens

Thank you for leaving your thought provoking note. You suggest that it might be lack of confidence that caused us not to boldly sketch a new theory on the E8 lattice. To me it is more a reality check that assures me that most all suggestions for new physics that we can come up with, at any time, are wrong. This assurance comes from the mere number of possibilities out there. In order to keep this essay both entertaining for a wide audience, but also to make a strong point nevertheless, we focused on what it most important to us: To sketch a sense of naturalness, beauty, and simplicity as motivating some of our current and future work elsewhere. We did that at the expense of actually proposing a model, granted, but isn't it amazing with how few and simple assumptions you necessarily arrive at the E8 lattice? Repeat, invert, closure; and self-duality of the space under addition and multiplication. Lattices in 1, 2, 4, and 8 dimensions satisfy these very simple assumptions; and only in these dimensions. The E8 lattice is of course an immensely complicated construct if you attempt to understand its properties and automorphisms - something I will never succeed in fully. But conceptually it is *that* simple. Pure math at its finest, we feel, and that's what we want to convey to the reader.

Best wishes, Jens

If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is and was the quantity of people which gave you ratings. Then you have of points. After it anyone give you of points so you have of points and is the common quantity of the people which gave you ratings. At the same time you will have of points. From here, if you want to be R2 > R1 there must be: or or In other words if you want to increase rating of anyone you must give him more points then the participant`s rating was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.

Sergey Fedosin

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Sergey Fedosin

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Hello Sergey - yeah, the ratings ... with almost 300 submissions I estimate that any community vote will favor known authors or known topics, or both. This is not out of boredom or disinterest or dishonesty of the voters; but stems from mere statistics when overwhelming a decision engine with too many choices to evaluate. Forced to employ some kind of efficiency scheme, the likely pattern of an interested reader (and voter) would be to review topics that sound the most interesting - which in turn include a disproportional amount of known authors and topics. I projected that if I were to ring the advertising bell really loud here and attempt to create more visibility, people would still not really have the time to read and evaluate the essay - instead, we would receive a well-meaning "7" at best, which would give us no chance of reaching the top 12% (to arrive in the first 35 essays that are planned to be considered for an ordinary prize). But all of that is OK - for one there's always the off-chance for a special commendation prize. But much more so, we are very satisfied to have communicated our research vision to the few people who we wanted to reach. Small group work has been my preference always, which makes this essay contest so valuable: For one we reached the handful of people who expressed interest in our work; and for the other we reach the other handful who might be interested but isn't quite keen to show face yet. Imagine you're working on something as remote as we are, and there are 10 people who actually care! To me, that is a big achievement. Best wishes, Jens

Dear Jens and John,

What a lot of profound topics you weave into your story! The following thoughts come to mind:

1. I guess the “loss of information” involved in addition is a very general aspect of “noninvertible morphisms;” for example, maps that aren’t injective (one-to-one). It’s interesting to regard this as a foundational problem and a viewpoint I hadn’t considered in this explicit way! After all, the superposition principle is an example of this, and superposition occurs even for classical waves. But your analysis goes much deeper than this…

2. For the logarithm function (and other similar functions), the usual way of dealing with this in complex analysis is of course to use Riemann surfaces; this was one of the ways in which such objects were first introduced. These play a striking role already in quantum information theory, but this seems like a new possible application.

3. The suggestion of introducing internal structure (in this case for purposes of distinguishability) is embodied in a cutting-edge area of abstract algebra that hasn’t yet been properly applied to physics. This is the theory of “categorification,” in which elements are elevated to objects; for instance, lattices. I have written about this near the end of my essay here; it might interest you.

4. This specific use of root systems of exceptional Lie groups is something I have not seen before. It’s a good idea, regardless of its ultimate scope of applicability.

Yours is one of the few submissions that earns a solid “10” from me. Thanks for the great read! Take care,

Ben Dribus

P.S., Regarding your previous comment, I'm sorry you didn't "evangelize" more actively... your title didn't stand out to me, and I read your essay only because I read them all. Hence, you nearly missed out on a thoroughly deserved top rating. Yours is about the 245th I've read, and it's one of the best in the contest.

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What a lot of profound topics you weave into your story! The following thoughts come to mind:

1. I guess the “loss of information” involved in addition is a very general aspect of “noninvertible morphisms;” for example, maps that aren’t injective (one-to-one). It’s interesting to regard this as a foundational problem and a viewpoint I hadn’t considered in this explicit way! After all, the superposition principle is an example of this, and superposition occurs even for classical waves. But your analysis goes much deeper than this…

2. For the logarithm function (and other similar functions), the usual way of dealing with this in complex analysis is of course to use Riemann surfaces; this was one of the ways in which such objects were first introduced. These play a striking role already in quantum information theory, but this seems like a new possible application.

3. The suggestion of introducing internal structure (in this case for purposes of distinguishability) is embodied in a cutting-edge area of abstract algebra that hasn’t yet been properly applied to physics. This is the theory of “categorification,” in which elements are elevated to objects; for instance, lattices. I have written about this near the end of my essay here; it might interest you.

4. This specific use of root systems of exceptional Lie groups is something I have not seen before. It’s a good idea, regardless of its ultimate scope of applicability.

Yours is one of the few submissions that earns a solid “10” from me. Thanks for the great read! Take care,

Ben Dribus

P.S., Regarding your previous comment, I'm sorry you didn't "evangelize" more actively... your title didn't stand out to me, and I read your essay only because I read them all. Hence, you nearly missed out on a thoroughly deserved top rating. Yours is about the 245th I've read, and it's one of the best in the contest.

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Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I was out of town with my family. Just to let you know, I came across your essay on 30 August, loved it as well, told John about it, and gave it the top rating as well - whew :) You should have good chances of winning a prize, and hopefully you earn FQXi membership! I do have a question regarding your causal metric...

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Dear authors,

While my conclusions are rather contrary to yours, I agree with you on that some fundamentals of mathematics may play a crucial role in physics. Please feel challenged to factually object to my arguments.

The style of your essay reminds me of a book by Detlef Spalt: "Vom Mythos der Mathematischen Vernunft" Wiss. Buchgemeinschaft: Darmstadt 1987.

May I ask you to comment on Spalt's opinions too?

Eckard

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While my conclusions are rather contrary to yours, I agree with you on that some fundamentals of mathematics may play a crucial role in physics. Please feel challenged to factually object to my arguments.

The style of your essay reminds me of a book by Detlef Spalt: "Vom Mythos der Mathematischen Vernunft" Wiss. Buchgemeinschaft: Darmstadt 1987.

May I ask you to comment on Spalt's opinions too?

Eckard

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