Fractal Geometry

I’m posting this article by Ed Kellogg, who has generously contributed many interesting ideas and links to our local Ashland, Oregon geometers group (a.k.a. ‘Hedronists’). Enjoy! :-)

Bruce Rawles

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Mathematical Universe Hypothesis Emails (Recently sent out to a local group of people interested in Sacred Geometry from Ed Kellogg, Ph.D)

Hi all  –

Some important points in respect to the role of mathematics in the universe came up in the last group meeting that I see as important enough to follow up on.

Someone stated something to the effect that “the Universe is not, could not, be mathematical” because, first of all, mathematics seems a human invention, and second of all, because mathematics can never, will never, ever (adequately) describe what goes on in Nature, even in something as commonplace as a tree, or even a leaf on a tree, as it varies in time.

In a sense I agreed with him – in that mathematics as commonly understood consists of a map, and a style of mapmaking invented by humans, that describes the universe, but is not the universe – i.e. “the map is not the territory. The idea that the universe exists as an expression of a “Finite Human Mathematics,” (FHM) no matter how far we might develop it, does indeed seem absurd.

However, although I agreed with him that while FHM most likely could never absolutely describe what goes in a leaf or a tree, let alone what goes on in Nature as a whole, I disagreed with him when he asserted that it could never, ever, adequately do so.

FHM up to a 100 years ago repeatedly proved itself grossly inadequate to describe the natural world – the usual geometric shapes, triangles, squares, cubes and spheres applied at best poorly to the complex shapes of the natural world – with leaves and trees as great examples of non-conforming shapes and processes. However, then Mandelbrot came along and developed a new field of human mathematics – fractals – and suddenly human mathematics became several orders of magnitude more adequate in modeling “what goes on” in Nature. Chaos theory has had a similar effect.

Given that the potential field of mathematical development seems infinite, FHM as best seems in its infancy.  If you look at a map of different mathematical areas of research, and their relationships to each other ( http://www.math.niu.edu/~rusin/known-math/index/mathmap.html ) this point becomes obvious, as many of these fields did not even exist until recently. And as one recent article in Science News put it, computer power “is now enabling researchers to discover hidden corners of the mathematical universe, many of which earlier researchers had never even dreamed existed.”

Given the history and potential of FHM, it does not seem hard for me to believe that given time, eventually more breakthrough’s like Mandelbrot with allow FHM to adequately describe what goes on in a leaf or a tree (or in Nature as whole), to whatever degree of adequacy one requires. Perhaps a mathematics based on Holophasec will provide another major breakthrough. <g>

However, I feel happy to admit that my belief that FHM will eventually have the ability to adequately describe what goes on in a leaf or a tree does indeed, at this point in time, seems just that – a belief.

However, I also posit that the belief that FHM will never have the ability to adequately describe what goes on in a leaf or a tree also simply seems a belief.

As far as the plausibility of Tegmark’s Mathematical Universe theory goes, I reread his paper ( http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.0646v2.pdf )

and still find it compelling. Please note that his theory sharply distinguishes “physical reality as mathematics,” given that “mathematics” seems a limited human construct, from “physical reality as mathematical structure.”

In the abstract: “I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure.”

And later he writes: “Let us clarify some nomenclature. Whereas the customary terminology in physics textbooks is that the external reality is described by mathematics, the MUH states that it is mathematics (more specifically, a mathematical structure). This corresponds to the “ontic” version of universal structural realism in the philosophical terminology of [14, 22]. If a future physics textbook contains the TOE, then its equations are the complete description of the mathematical structure that is the external physical reality. We write is rather than corresponds to here, because if two structures are isomorphic, then there is no meaningful sense in which they are not one and the same [19]. From the definition of a mathematical structure (see Appendix A), it follows that if there is an isomorphism between a mathematical structure and another structure (a one-to-one correspondence between the two that respects the relations), then they are one and the same. If our external physical reality is isomorphic to a mathematical structure, it therefore fits the definition of being a mathematical structure.”

As people in this group know, scientists routinely find that even the most esoteric discoveries in mathematics often relate to phenomena in the physical world, with no a priori reason that they should do so. This curious fact supports the belief that a mathematical reality underlies or interpenetrates physical reality in some way not easily explained. Of course, although natural languages suffice for practical communications, their idiosyncratic and often false to facts structure limits their usefulness in describing the world. Mathematical language on the other hand provides a coherent, adaptable, and potentially infinite set of structures that allows mathematicians and scientists to “describe the indescribable.”

Does Nature seem beyond any conceivable mathematical structure, or at least beyond the capability of any possible FHM to adequately describe? Perhaps, but given the lack of data, especially with respect to the ultimate characteristics of reality as such – comprised of consciousness?, matter?, consciousness and matter?, real? illusion? simulation? this question at present seems theological, and one’s answer to it a matter of personal belief.

However, to those who might dismiss the Mathematical Universe Hypothesis out of hand, Tegmark wrote: “It is arguably worthwhile to study implications of the MUH even if one subscribes to an alternative viewpoint, as it forms a logical extreme in a broad spectrum of philosophical interpretations of physics. It is arguably extreme in the sense of being maximally offensive to human vanity. Since our earliest ancestors admired the stars, our human egos have suffered a series of blows. For starters, we are smaller than we thought. ( . . . ) The MUH brings this human demotion to its logical extreme: not only is the Level IV Multiverse larger still, but even the languages, the notions and the common cultural heritage that we have evolved is dismissed as “baggage”, stripped of any fundamental status for describing the ultimate reality. The most compelling argument against the MUH hinges on such emotional issues: it arguably feels counterintuitive and disturbing. [emphasis added] On the other hand, placing humility over vanity has proven a more fruitful approach to physics, as emphasized by Copernicus, Galileo and Darwin. Moreover, if the MUH is true, then it constitutes great news for science, allowing the possibility that an elegant unification of physics, mathematics and computer science will one day allow us humans to understand our reality even more deeply than many dreamed would be possible.”

As for myself, I believe that we live in an information universe, where at the deepest level, we input the universe primarily as code, as an information pattern, a code that we learn to habitually translate and then experience in terms of sight, sound, touch, etc. The movie The Matrix illustrates this idea in an entertaining way, where the characters live in a virtual reality experienced and only perceived as physical, but which at its root consists entirely of a mathematical code. And articles like those of Whitworth “Exploring the virtual reality conjecture” http://brianwhitworth.com/FQXiWhitworth.pdf makes even far-out (for most people) speculative theories such as these worth considering seriously. And if we do live in a virtual reality, Tegmark’s MUH becomes a slam dunk.

My point of view.

Ed

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(A follow-up email.)

Hi all –

A few more comments on the Mathematical Universe Hypothesis. First, if true, it means that one would necessarily expect that Sacred Geometry, in some form, plays an essential and inevitable role in all “physical processes”, including healing processes. So one would expect that sciences based on sacred geometry/sacred mathematics, could have extraordinary effects. The more closely they tune into and congruently understand the actual essential mathematical structuring of the Universe, the more powerful the effects.

On the other hand, if false, if the Universe does not exist as a mathematical structure, if it really seems something OTHER, to which we as human beings mistakenly ascribe a mathematical structure, in a crude attempt to approximate and understand it, then Sacred Geometry does not really seem so much sacred as artificial. And if it appears to have effects in some instances, for example for healing, it does through through happenstance, happy accident, or placebo effect rather than through a resonance effect based on having a profound correlation with the Universe as such.

Of course, throughout history many very intelligent individuals, have declared their belief in something very much like the MUH. A few of my favorite quotes:

“Number rules the Universe.” and “Geometry is knowledge of the eternally existent.” Pythagoras

“Philosophy is written in this grand book – the universe – which stands continuously open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these one is wandering about in a dark labyrinth.” – Galileo Galilei, 1623

“One can not escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them?” Heinrich Hertz, German physicist, who proved that electricity can be transmitted in electromagnetic waves, and travel at the speed of light and which possess many other properties of light.

” . . . the Platonistic view is the only one tenable. Thereby I mean the view that mathematics describes a non-sensual reality, which exists independently both of the acts and the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind.” and, “Mathematical objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions.” Kurt Gödel, the most influential mathematical logician of the 20th century, who revolutionized mathematical theory with the discovery the Incompleteness Theorem.

“You conceive of my mathematical reality as a part of the external physical world. . . . For me it’s just the opposite: external physical reality is a part of archaic mathematical reality. . . . To me, our ability to comprehend the external physical world implies that there exists an archaic mathematical reality, a reality that exists on the same footing as external physical reality. . . . From my point of view, it would be nearer the truth to say that it’s the physical universe that’s inside archaic mathematical reality.” Alain Connes, Winner of the 1982 Fields Medal.

Last week one of the group members  used one of the main arguments against the MUH, when he argued that it does not seem possible that something as complex as nature, or even a living leaf on a living tree, could ever get reduced to a set of mathematical equations and relationships.

But Stephen Wolfram pretty much invalidated that argument in principle, in his work with cellular automata, by clearly demonstrating that simple rules can give rise to incredibly complex behaviors.

“Wolfram gives the following examples as typical rules of each class.^[1]

• Class 1: Cellular automata which rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 250.
• Class 2: Cellular automata which rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 232.
• Class 3: Cellular automata which appear to remain in a random state. Examples are rules 22, 30, 126, 150, 182.
• Class 4: Cellular automata which form areas of repetitive or stable states, but also form structures that interact with each other in complicated ways. An example is rule 110. Rule 110 has been shown to be capable of universal computation.^[2]“

http://en.wikipedia.org/wiki/Cellular_automaton#Elementary_cellular_automata

Rule 30 for example generates a product that looking from the outside in, with the best analysis methods available, tests out as entirely random, yet this “random” pattern seems entirely deterministic and reproducible through the application of a simple rule. The fact that the processes of a living leaf seems bewilderingly complex to us as human beings looking from the outside in does not mean a priori that no mathematics can explain it, or that one would need a mathematics as complex as the leaf processes itself to do so.

I should note that Wolfram himself apparently does not believe that a cellular automaton model will work in describing the Universe, as it fails to account for the quantum and relativistic properties of nature. However, he has demonstrated that in principle at least, complex phenomena can derive from very simple rules. He has also had some apparent success in finding other kinds of simple model systems – not CAs – that do allow for quantum and relativistic effects.

Now I do not agree with Wolfram in many respects – some fundamental. To me he seems a “reductionist-immaterialist”,” who apparently regards consciousness seems an epiphenomenon, but not of physical matter, but of immaterial code. I see consciousness/beingness as primary, and the Matrix code as secondary in respect to the roles they play in the Universe.

Still, I believe he has had brilliant insights and has done some important foundation experimental work in a number of new areas, making many intriguing discoveries relevant to the subject of Sacred Geometry.  I really recommend this outstanding lecture of his I found online yesterday, that he did a year after the publication of his book, A New Kind of Science. He does a great job of clearly summing up some of the most important and fascinating points in his book – a book that weighs in at well over 1,000 pages – in the first twenty five minutes!

http://www.youtube.com/watch?v=_eC14GonZnU&feature=related

Two highlights to fast forward to if you wish, after watching the first 25 minutes:

29:03 – on the shapes of leaves, where he summarize features of leaves in a parameter space set that turns out to correspond to simpler, linear analogue of the Mandelbrot set.

59:00 Maps out different systems of mathematics based of different axioms. Human logic appears 50,000th or so from the beginning of his computer generated list of possibilities, with those regions explored by human mathematics comprising a very small subset of the possibilities, something I commented on in my last email.

Yours towards greater lucidity,

Ed