Hi Jonathan ,

no worries, it certainly does meet with approval. Your enthusiasm shines through even if you don't feel your presentation is as polished as you would like.I enjoyed your descriptions of mathematics in the universe itself. I have been thinking of it as mathematics 'in the wild' controlling what can and does happen, being the relationships at the heart of everything. I also like the idea that you mentioned of lots of kinds of mathematics co-operating. Bringing an ecosystem to my mind- Organisms doing different things but somehow all working together as an entirety.

I'm not sure it is necessary for the mathematics, the relationships, to be distilled from reality and placed in an abstract theoretical space or to assume that the mathematics precedes the concrete universe. It can be distilled into pure maths but that seems to me a bit like distilling the psychology of a man and speculating that it somehow exists separately from the body of the man and the environment in which the man finds himself. (Sensory experiences, thoughts and knowledge can be considered a different facet of reality but they are still within the minds within the concrete reality influenced and shaped by it.)

Thank you for sharing your work with us, I do appreciate it. Kind regards, Georgina

Dear Jonathan,

Thank you for the reference to Tiozzo's thesis.

It is interesting that what he defines as a quadratic interval looks the same as what we measure in the "superheterodyne calendar" of my paper [see Fig. 2 and eq. (17)] from continued fractions

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zet

a/planat6.pdf

At least the starting point and the connection to Thurston's "quadratic minor lamination" is encouraging. I would also like to recognize a possible link to the f Farey fractions of hyperbolic polygons (tesselations of the upper-half plane, or of the conformally equivalent unit disk) that I mention in Sec. 3 of my essay.

We know what we have to work on. Thanks.

Michel

    Jonathan,

    Perhaps I don't understand what you mean by "pure Mathematics." I think math does reveal many elements of structure and form utilized by nature because scientists model them that way. I agree that using math to model physical systems is a very effective way to do physics.

    Perhaps my view is simplistic in seeing a integral connection between math, the mind, and physics. What are your thoughts here?

    Jim

      Jonathan, the mandelbrot set arises in chaotic systems as a natural and universal structure so it is related nicely to my work too. It seems to be a common feature of universal structures that they have a fractal self similar structure as the mandelbrot set does when you zoom in. This is close to what I see happening when iterated quantisation is used to form a universal structure emerging in mathematics. Quantisation itself is then the transformation under which the laws are self similar like a fractal structure. It would not surprise me if the Mandelbrot set turns up in physics at the deepest level in this way just as you suggest.

      Your butterfly idea adds an extra level of originality to the use of fractals. Thanks for the mention too.

        Jonathan,

        I am a little late reading your essay, though I had been anticipating it. I was not surprised that you addressed the Contest Topic in terms of the Topic! While many entrants in any FQXi Essay Contest quite naturally piggy-back their own special interest on any given topic, the harness generally needs an "evener", which is a common accoutrement in Amish Country where I come from. A team of horses is seldom matched in all proportions so a clever piece is built to couple the harnesses and even the load, and spare the less powerful animal being subjected to a greater part of the pull. No such devise is necessary for your superb essay, it IS your specialty!

        "The question of whether ideal forms predate their physical representations, and to what extent all physical forms are a representation of their mathematical ideal, remains open." And well it should.

        Perhaps there is some thing we might call geometry where the finite value obtainable for rectilinear space is also obtainable for curvilinear space, and we only lack a better way of trying to compute 'pi' that returns a finite ratio. But what we observe physically argues well that we can trust our foundational maths as being inherent to reality. The only difference between space and time might be in that dichotomy of finite vs. infinite geometries. and the origin of energy in a continuum of creation. If 'pi' reaches a finite limit, would energy cease to exist? Is it physically possible to devise a mathematic form that finds easement to directly equate flat geometry with curved geometry? If it were, wouldn't all be symmetric and the universe a singularity?

        Your fascination with fractals and the asymmetry at scales is communicated well in your essay, and it should be appreciated by all that you invite the totality of mathematics to the party. Well done, Jonathan! Best wishes, jrc

          Thank you my friend..

          I appreciate your reference to Linas Vepstas, and his work with modular groups and the Mandelbrot Set. This is a significant thread to follow. As I mentioned on your essay page, this connection is expanded in recent work by Giulio Tiozzo and somewhat in the earlier work by Tao Li.

          I also find the Monster fascinating, and I am certain that it plays a part in Physics, as it is one of those archetypal structures shaping whole broad areas of Math. If the totality of Math concept is more than a theory, the Monster group must be essential to understanding some areas of Physics.

          All the Best,

          Jonathan

          Gracious thanks Sir,

          I appreciate your good regard Michael, especially after reading your excellent essay. I know you are one of the people participating who truly understands why certain fundaments of Math must find their expression in Physics. Just as with some of the most basic concepts in geometry and topology, the application of the normed division algebras in Physics is merely an expression of the natural order, and their universality reflects that order.

          All the Best,

          Jonathan

          Thanks George,

          I appreciate your comments and have read your wonderful essay. I have no problems with certain varieties of self-referential systems, and I think on some level they are essential to life, but Gödel is another matter; his contribution was brilliant, but only limits a subset of the available options, or is somewhat over-applied as a general limiter on what can be known. Sometimes a priori knowledge, in combination with what can be ascertained through reasoning, provides insights that greatly exceed what formal systems alone can reveal.

          All the Best,

          Jonathan

          Thanks Ed,

          You give me quite a lot to think about. I think the biggest determiner of what fundaments find expression in Physics is that structures must be consistent both internally or externally, both globally and locally. That is; a form must agree with itself, and also with the space or universe it inhabits, including any fields the space or its forms might contain.

          I see self-agreement of this type and the self-similarity in fractals to be harmonious concepts. There is an internal symmetry to the star-like sunburst shapes, for example, but they conform at the periphery to the surrounding space. This reflects a similar sensibility to your comments, as what is observed from the macro scale is always an inexact symmetry, but asymptotic to an exact and ideal symmetry at the core.

          All the Best,

          Jonathan

          Thanks greatly Colin,

          I'm glad you enjoyed my essay, and also that you found my Mandelbrot Butterfly video. I figured that since I've already outed myself in the YouTube videos, I might as well engage this topic with my research into this topic highlighted. The 'Hitchhiker's Guide' is a story I love too, and I'm glad to hear you make that connection plain.

          Who knows? It may turn out that we live in a neighborhood of the Mandelbrot Set where there are precisely 42 spokes on the wheel of the nearest spiral.

          All the Best,

          Jonathan

          You are generous Leo KoGuan,

          A unit that describes its bigger self can explain how the smallest possible and largest possible systems are alike. And this decidedly is true of the self-similarity seen in fractal forms. In nature; there are ferns whose 'leaves' resemble the fronds, and whose fronds resemble the whole fern - to such a degree that we have exquisite self-similarity there too.

          Perhaps; it's not information itself, but the thing we can call knowledge, and is obtained by awareness or observation, that connects the levels of scale, or the abstract and objective universe. The utilization and processing of information is what makes it matter, and so the entity that makes this happen is the key - or explains why we are here. If the Qbit is that kind of unit, as your work would suggest, it is significant.

          All the Best,

          Jonathan

          It is my pleasure Michel,

          I find it exciting, that some of these connections between different areas of Math are being seen at this time. I am glad this is happening in my lifetime, and I hope to contribute to this process, as I learn more about those connections myself. I also see that Tiozzo has some papers in the pipeline, which offer additional information on some of the relevant topics we have touched on. So the process continues, and the subject evolves.

          All the Best,

          Jonathan

          Thanks very much Georgina..

          I am happy that my offering meets your approval, even if it was a bit rushed, or feels incomplete to me. The idea of an ecosystem for various activities is one that resonates with me, as well, and I find it is an important consideration for all kinds of endeavor. But as to the core message; I feel like only the messenger, where my discovery forced me to accept something akin to the mathematical universe hypothesis - and I am only now coming to grips with the implications of what I have learned.

          All the Best,

          Jonathan

          Thanks Jim,

          I think that where we agree is that Math and Physics resemble each other because they both partake of the same nature of the process by which things arise. I think that developing a universal measurement protocol is a good way to generate the entire body of Math, for example. And the specific form of the Mandelbrot algorithm speaks to that ideal, where something of great complexity arises from core principles that are very simple - and relate to measurement.

          All the Best,

          Jonathan

          Excellent thoughts Phil!

          The universality of the Mandelbrot Set is definitely worthy of note. To see that it shows up in other settings - apart from the familiar algorithmic generator or equation - is surprising but relevant. I should go back to Peitgen and Richter, and include some examples of this in my upcoming paper.

          If nothing else, the Mandelbrot Set is full of delightful examples showing self-similar structures that can also be seen elsewhere. And by training eyes and minds to discern such features, we are more likely to see the principles of universality at work in nature's laws.

          All the Best,

          JOnathan

          Thanks for coming by John,

          No apology is needed, because with the sheer number of essays there is plenty to read before you get to mine. I agree though; this topic is tailor-made as a forum for me to introduce ideas I've nurtured for quite some time, to the FQXi community. Honestly; I've spent most of my time in these contests being on the fence, because I had my reservations about both choice A and B. But in this contest; they have given me an incentive to focus on something I am emphatic about, so no fence sitting this time.

          Still, I realize that without decisive proof air-tight arguments, I need to be humble and leave room for other possibilities. But after considering the pros and cons for years now; I think I've found a fair number of reasons why the universe must borrow some structures found in Math, as they are the right tool to get the job of creating a universe done. I look forward to reading your essay in the near future.

          Warm Regards,

          Jonathan

          Dear Jonathan,

          I greatly enjoyed you beautifully written essay. It gives me confidence that someone such as yourself who has spent so much time pondering the question has reached conclusions similar to my own.

          Please take the time to read, comment on and vote on my essay.

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

          Best of luck in the contest!

          Rick Searle

            Thanks Rick,

            I appreciate your comment, and the discovery that you are a kindred spirit. I shall likely get to your essay later today, and I look forward to reading it.

            All the Best,

            Jonathan

            Jonathan,

            I was so pleased to hear from you, don't spend too much time looking for an essay from me I can get in enough trouble just commenting.

            In the post you made on the T or T page 3/22/15 @ 20:41; you explained, "The Mandelbrot formulae involves multiplying a complex number by itself, then adding the result back to the original number. That is; you square the starting value, then add the initial value."

            Question; Doesn't squaring a complex value result in a real number, so that then adding the initial complex value reduces the real value of the final result? This appears suspiciously akin to the Lorentz Factor, as well as the exponential function and the inverse square law. Am I dreaming or just profoundly deficient in math tools?

            Fondly, jrc

            Good question John!

            Squaring a pure imaginary number gets you a negative real number, which then gets added to the original pure imaginary number, and this becomes a complex number - as it has both real and imaginary components. If we start out with a complex number, it will almost always give us a complex result, although sometimes summing makes terms cancel out - and we end up with a pure real, a pure imaginary, or a null result (0,0i).

            If we take out the addition step, and just iterate the squaring function, we find that any initial value whose distance is greater than 1 from the origin will grow unendingly, and for initial values whose distance is less than 1 from the origin, the successive values approach the origin or shrink monotonically. Of course; for a value or distance in C of exactly 1, the function remains at the boundary forever, neither growing nor shrinking.

            However; when I tried to use a similar shortcut for the Mandelbrot algorithm, by looking for a result that shrinks over 3 successive iterations; I didn't get the Mandelbrot Set at all, and what I found was the Mandelbrot Butterfly instead. Pretty weird, huh?

            Regards,

            Jonathan