Fractal Foundations of mathematics: Axioms notions and the set FS as a model

[jehovajah]

I was going mad while doing this research, and my blog shows the extent of that madness. Twice i came upon stories of geniuses who had gone to an early grave through too much thinking! And the chilling ability of De Moivre to predict his own death indicated the intense regulation these men had over their own systems in order to explore these issues. Fortunately for me the incessant computation can be handed off to computers.

Still, it has taken some time and my wifes intervention to return me to some sense of normalcy and normal interaction. As much as i am able i shall refrain from this deep research and enjoy the beautiful fractal patterns of each passing moment. Maybe i shall attain to great age and senility in a happy frame of mind!

[jehovajah]

I am still at it!, but hopefully in a more philosophical relaxed way. No hurry, no rush.. time to stand and stare.

[jehovajah]

Still at it!
http://jehovajah.wordpress.com/jehovajah/blog/2012/07/05/newtons-third-law-and-torque
Enjoy yourselves, cos i am!

[jehovajah]

In my latest blog i give manipume some structure, which of course i was not able to do earlier, so i gave a poetic experience of it and of  Shunya.

http://jehovajah.wordpress.com/jehovajah/blog/2012/07/13/maipume

Manipume you may recall is a word i constructed from the Proto Indo Eurpean roots "ma, man, me" and the Old french Latin Manipule which is a military detachment used to strike a blow, militarily.
By the way, you are welcome to comment here or anywhere in the forum, because this is not my blog!

[jehovajah]

It is always satisfying to see "mathematical" developments coming from experimentation, rather than abstraction, and so from outside the 'precious' boundaries which are no largely irrelevant. In fact Grassmann said they were irrelevant back on 1844, but who listens to Grassmann?

To be fair, an increasing number of scientists and philosophers nowadays, but their understanding has been "poisoned" by Gibbs reworking of his Ideas. Clfford gave them a good reworking as well and may also have skewed the formalism on which they are based.

Ok, so what does it matter as long as the formalism is consistent. And there is the issue, the underlying formalism is not consistent , where both Hamilton and Grassmann had consistent formalisms, they were not the same. Grassmann comments on this in his critique of Quaternions.

Any way the reason for the post is this link:
http://www.newswise.com/articles/mathematicians-develop-new-method-for-describing-extremely-complicated-shapes

And the full online article:
http://jmp.aip.org/resource/1/jmapaq/v53/i7/p073516_s1?view=fulltext&bypassSSO=1

and of course my blog:
http://jehovajah.wordppress.com/jehovajah/blog/2012/07/29/the-formalism-of-the-unit-sphere-is-a-2

Enjoy.

[jehovajah]

So i get to combine process algebra with my research into Grassmann's Ausdehnongslehre. The two really are the same thing in a kind of action and reaction way. I read Grassmann in German, not my native tongue, so i have to read meditatively, looking up certain surprisingly small words that make all the difference, in the meantime my unconscious mind has already got it and is processing the insights while the conscious is struggling to put two words together!

Thus iget all these new insights and they appear to come from me, but i know better: they come from engaging with Grassmann's own text!

I may be hampered research wise, but who is truely hampered in these days of the internet? But in any case whaat is research for- to clog up the arteries or to ispire? do i like to keep it light, get just enough to race ahead, and then come back to check if i got it righ. sometimes i do sometimes i do't, but on many occasion i turn over a forgotten leaf, a hissen treasue, a buried secret. I find that my teachers have either knowingly or ignorantly suppressed original thought and lines of investigation for very human, mundane reasons. the whole glossy image of Human Progress becomes just a bit of faded old newsprin you find under the carpet when you turn it over to find out what that awful smell is!


What's upwith Academia? well at one time it was portrayed as a band of happy travellers, venturing into the mysteries and secrets of the universe, a bit like NASA today. It was the dream of a questing young and wealthy pioneer  called Aristocles but better known as Giant Haystacks, or Wide Load. Yes you guessed it Plato!

As a young adventurous and curious man he travelled widely to take in the Hellenistic world, having found local politics to be too bloody and too "pragmatic". He was young, and looking for "Ideals" and some "utopia" in life, a real way of living in this world which would be for the good of all. Hang on though, this was no dwmocracy as we know it, for each wa to be assigned to his own level or place, butto be happy in that societal status.

In all his travelling, none impressed him more han the Italian Pythagoreans, and it is down to them that Plato established the style and the tenor f his Academy as he did.

The utopianism of the Pythagoreans and the closed community of scholars deriving inspration from the Muses was exactly what Plato was looking for. The significance of the shrines to the Muses only came apparent to me a few days ago while reviewing the notion of Arithmos. I had just completed some work on the Manifold concept and how, as i believe, Gauss and Riemann had decided to silence Grassmann, because he had a full working model of the concept while they only had a sketch of the idea. By this i mean that Grassmann a nobody had presented to Gauss, a somebody, the fundamental notion about the substrate of reality. Gauss knew exactly what it was, because, as usual, he had been "thinking along the same lines". So now the story should have been all sweetness and light, right?

What we do know is that Grassmann "acquired " a reputation for being obscure, and the most notable proponent of that was Gauss. We also know that Grassmann endured at least 10 years of little recognition that was almost soul destroying, and then a "mathematician" came to see him. This mathematician told him to rewrite his book as a mathematical text, as it would be a shame for these ideas which He (Riemann) had been pondering to lay idle!. The bitter irony does not escape Grassmann who writes of the incident obscurely in his Vorrede to the 1862 version of Ausdehnunglehre.

Gauss had clearly held Grassmann back so he could prepare Riemann to take on his role of lead Mathematician in the Academy. Gauss had clearly understood Grassmann's work, but Riemann had not. Riemann was tasked by Gauss to deliver a Habiltation on the concepts of Geometry for the future, and Gauss knew who Riemann had to learn it from, that is Grassmann.

So the deal seems to have been, Riemann gets the glory, Grassmann gets to write the first text book! Riemann delivered his speech on the underpinnings of Geometry in 1853/1854, and it was a great success for him. meanwhile it took Grassmann 8 years to rewrite his Ausdehnungslehre in such an alien mathematical form that he hardly could recognise it. The book of course then began to receive moderate recognition, but more recognition after Riemann died.

So beware Academia, for once where Pythagoras demanded that the Muses should be given space to inhabit, and their very natures worshiped , thereby rewarding obeisance with insights and music and poetry and dance; that place has become tha abode of dark souls, and a den of foxes and every sly and slothsome thing!
http://my.opera.com/jehovajah/blog/2012/08/12/aristotles-blunder
http://my.opera.com/jehovajah/blog/2012/08/10/from-sequences-and-series-to-grassmann-algebra
http://my.opera.com/jehovajah/blog/2012/08/07/the-notation-for-sequencing-sequence-construction-processes-by-combinatorial-cho
http://my.opera.com/jehovajah/blog/2012/08/14/the-matrix-mother-of-all-sequences

The technical details about the manifold i will get to at some other time.

[jehovajah]

The work i did here in exploring reference frames and systems has been fundamental to me grasping what quaternions really are. As Children of Shunya they represent the fundamental building Spheres of all vector systems for referencing space.

 Of course, as children of Shunya we can go further and further back in the gene pool without ever coming to an end, so we may as well stick with quaternions as our fundamental reference frame. Then i can look at Grassmann's work in its true perspective as the fundamental analytical tool for building manifolds, including the Hamiltonian Quaternions.

http://jehovajah.wordppress.com/jehovajah/blog/2012/08/27/sign
http://jehovajah.wordppress.com/jehovajah/blog/2012/08/28/hamiltons-bubbles

[jehovajah]

Well, as usual, there is one more step below the last one!
In my most recent series of blog posts, which result from working on the Quaternion 8 group, I realised that there was a fundamental subjective process underpinning all of this feverish proportioning. This led to some bold statements on I and pi, but it also lead to the revisiting of work I did on the roots of unity.
From their I was able to analyse Hamilton's quaternions and posit a reason for them being non commutative.
In the course of doing this analysis I realised the importance of sequence structures in delivering combinatorial certainties, and levels of arrays.

I left that analysis to pursue Hamilton's triples, work which he largely left unpublished, concentrating his efforts on the Quaternions. I did this to fulfil a wish to establish a "proper" triplex algebra, which I glimpsed while developing polynomial rotations. Several insights I had while struggling with that work have now been vindicated.

To say that I needed Hamilton's quaternions to proceed to Hamilton's triples is a fair and revealing comment on the profundity of Hamilton's work. Evenso I have chosen to call the work Newtonian triples, because both Hamilton and I know thst it rests solidly on the work of 4 men, of whom I have identified the last 3: Wallis, Newton, De Moivre, Cotes.

Nevertheless, I am drawn back to the simple facts, that our most complex theoretical constructs rest on the notions of spherical geometry and trigonometry, as well as the notion of differential forms as quantities.

I will not here multiply words, or aggregate arguments in the expositing of the said assertion, but will divide your attention between the forum and my blog, in order to subtract the maximum obscurity from your minds!

[jehovajah]

As a struggling undergraduate I recoursed to libraries to lose myself in a line of research questions on any topic that pressed hard upon my mind.

There were many questions, but one that pursued me was "what is mathematics?"

I thought as a young school pupil I had worked it out as a bunch of methods. I felt that the word mathematics was probably Latin for "methods", and I was comforted in that certainty. However, my mathematical experiences changed, introducing set theoretical notation, trigonometric notation. Polar coordinates and the imaginary numbers. This briefest of experiences happened when I was in germany and before I was plunged back into a traditional maths curriculum.

Traditional maths at university level is such a meagre and insubstantial thing! The culture shock of having to create a method and not just parrot it off was profound and unsettling. I can safely say that I was brought face to face with my many shortcomings, demons and groundless assumptions at this stage in my life. Very probably I was going mad! Nothing made sense anymore, nothing was meaningful anymore! My natural instinct was to tear it all down to the ground and build it up again in a way I could actually comprehend and trust.

Both religiously and philosophically as well as mathematically I was into fundamental "Truth".

How I found Hilberts book I do not know, but of all the books I thought I should read, and never, this was one I read and could not put down. It was on the foundations of mathematics. Thus, despite barely scraping through the course I had received the 2 most valuable aspects of my university education: the skill to programme a computer, and a knowledge that mathematics was a philosophical construct, developed on tramlines called axioms, and never therefore able to represent any profound or actual truth about anything, if I did not first place that truth in the axioms!

By nature I am a polymath, but a lazy one. When I was asked to describe my interests I usually put every subject down! But there is only so much the febrile young brain can take! I suffered several mental breakdowns after that, but fortunately I was too autistic to know they were not part of normal growth and development! Maybe they are, as current research into the plasticity of the young brain has shown the dramatic pruning that tales place in adolescence.

The cradle of my religion provided me not just with solace, comfort and hope, but also much needed socialisation skills. My mother told me that "the humble child sucks the most grace". Not that I knew what she meant back then. All I knew was I could read, but I could not talk! Therefore I became the most assiduous listener! Apparently this trait distinguishes me in ways I would never have comprehended in my youth. I just thought of myself as the "Lone Wolf", or the man of few words! Back then, these antiheroes were our heroes, whether they were cops or cowboys!

Suffice it to say, that a long period of religious and philosophical reflection with growing pragmatic social responsibilities prepared me for an adult life of high stress activities!. Eventually I had a proper diagnosed nervous breakdown which enabled me to take stock of everything and build up my life on a new basis.

Religious differences of opinion and greater self assurance enabled me to take a truely independent stance. If there is one thing I would say to any person, child or adult it is "Trust no one! Not even yourself!"

This enigmatic dictum has been a moral compass ever since I adopted it, and it rescues me from many hard and difficult situations, morally and physically. It also allows me to change and grow naturally!

Let me just say, that after philosophically determining that I had degrees of freedom in choices, often expressed as free will, I decided to accept reality as a continuum of experiences both internal and external. On that basis I needed to determine the "axioms" of my realities, and soon found thst the social cultural ones were of considerable value. I elected to confirm my Jewish faith, although I am a radical Jew in all aspects. And I searched for YHWH in my experiences as an empirical fact, according to a longstanding Jewish tradition.

Consequently I had philosophically structured my experiential realty into pattern formation and probability patterns, and used the structure of possibility/ probability to express it. I therefore accepted as fundamental that the possibilities were infinite, but probability clouds condensed out of this infinite spatial possibility. Within these probability clouds statistical clouds were realiseable  that is actualised/reeified, and these I experienced as reality.

In this line of thought I was introduced by a biological lecture on colonic microbial structures to the fractal paradigm. A couple of days of intensive googling brought me to Fractaforums.com. After a brief interaction with Enzedblue over fractally generating virus capsids I decided to make it my home fractal forum. The rest is history recorded in this thread.

I can truly say that fractals, and fractalforums.com have changed my life entirely.

[jehovajah]

So what is "Mathematics?"
It turns out that it is good , old fashioned Astrology!
The word itself I can trace back to the Pythagorean School in Platos time. Of course it's etymology goes back to the Dravidian era, bu no one before the 15th century AD would have a clue what you were talking about if you mentioned you were interested in Mathematics! The Mathematikos of Plato and Aristotles time were graduate Philosopher/Astrologers. Prior to that philosophers were not distinguished, because reputedly Pythagoras coined the term!
Sages, wisemen, Magi, Shamans were all terms used to distinguish what we now freely call "scientists", and oftentimes geeks, oddballs and even witches and warlocks! Of these name, Astrologer is the most translatable back through history.

We can trace back even to cave paintings using this notion, for many cave paintings are seasonal maps of the stars!

Few realise that Astrologers still function today in their official capacity, and that Newton was officially an Astrologer. The term became distinguished in its pejorative sense only in the late 18th century. When rationalists, seeking to distinguish themselves declared Astronomy as the title they would use rather than the " irrational" astrological connotations within the term Astrology.
http://Http://en.wikipedia.org/wiki/Astrology_and_astronomy

[jehovajah]

Currently, I am working on the nature of polynomials in my blog, a kind of corrected version of polynomial rotations. The notion of conjugacy and adjugacy is combined with the actions of sequencing and bracketing to explain the fundamental role polynomials play as the treasure chest of Kombinationlehre.

I am writing this ahead of delving into Justus Grassmann's fundamental contribution to German/ Prussian mathematics, and in fact how that has influenced the whole of western mathematics and physics through his two sons Hermmann and Robert.

German speakers have reason to be proud!

[jehovajah]

http://jehovajah.wordppress.com/jehovajah/blog/2012/12/15/in-the-image-of-si-d
Merry Christmas!

I want an ABHA coil for mine!
<a href="http://www.youtube.com/v/WhFMcuU82Fo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/WhFMcuU82Fo&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/rjTiWM2fJJA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/rjTiWM2fJJA&rel=1&fs=1&hd=1</a>

[jehovajah]

My latest blog is a reworking of the Shunya Field Theory of Everything in the light od the Electric Universe Theory. You can track the development of this theory from this thread and the axiom thread and the subsequent blog posts.

There is a fundamental spaciometry which is more complex than the sacred geometry. I find this is fundamental to all measurement schemes of astrology and astronomy and the basis of Newton's Electromagnetic Principles of an Astrologist (Mathematikos), from which he derives as a first order approximation the laws of "Gravity".

The complex mathematics develops the laws into electro magnetic fields.

[jehovajah]

It is always delightful to find out something new about which you thought you knew. The new insights enthusiastically outweigh any feeling of correction. However when the correction is major it is not so delightful!

Nevertheless one must correct mistaken notions , sins, youthful errors and the like if one wishes to progress on to the Elysium fields!
http://my.opera.com/jehovajah/blog/2013/03/05/heat-theory
I set out here some corrections to my apprehension of the relationship twixt Wallis, Newton and Barrow, which provide a clearer historical perspective on the course and causality of events in Newton's long and productive life.

Newton may not seem as important a figure as he really is, in my view. He, with Steiner, they together are perhaps the last great synthetic geometers in a long line of Astrologers. After Newton, algebraisation took over and blinded those who once saw clearly.

It is of great interest also that academic Geometry may perhaps trace its roots back to Plato's  Academy, but in fact Astrology is the one consistent term regarding all of these arts now called mathematics. Running alongside Astrology, but thought too low to be academic was the Pragmatic Techne subject later termed Mechanics. Much of what is Mechanical overlaps and is the source of those principles called geometrical, and sometimes distinguished as Gematria a corruption of the word geometry into another language..

The attempt to link Geometry to Euclid's Stoikeioon is a historical confusion probably stemming from the Renaissance thinkers who had little access to original documents. Arabian mechanical and Alchemical treatises must have seemed echoes of this revered work.

It bears repeating: the Stoikeioon is not a work of geometry, but an introductory course in Philosophy of the Socratic and Platonic Theory of Forms/Ideas.

If you meditate on that, you will come to realise that Newton indeed wrote a philosophy of Quantity as it is stated in the Preface to the Method of Fluents. Modeling it on the Stoikeioon hopefully ensures its longevity.

[jehovajah]

My re-education continues, happily. Below is a copy of an email sent to Norman.

I am checking that last claim out in your videos,and sorry about the German which i have corrected..

I defined a notion called Spaciometry years ago. At that time i thought Riemann was ahead of the game. I was about to change the fundamental definition to Grasmmanian, but now i am definitely changing it to Universal Hyperbolic spaciometry a Grassmannian Wildberger Projective spaciometry.

The inter relationship between Barrow, Wallis, Newton, Desargues, Leibniz, Kant, Schiller,Lagrange, Moebius, Poncelet and the Grassmanns , Hamilton and early ring and group theorists including Euler  i have yet to work through, but it is clear they drew upon Euclid's Optics, Appolonius' works and critical analysis of The introduction possible through the Stoikeioon, also by Euclid. expounded by Pappus. In a real sense Appolonius is the father of modern critical geometries, but all derive from the Pythagorean School of Thought and training in Astrology, and Aesthetics by consorting with the Musai.

------- Forwarded message -------
From: "sam gray" <jehovajah@ntlworld.com>
To: "Norman Wildberger" <n.wildberger@unsw.edu.au>
Cc:
Subject: Fwd: Suggestion of Perpendular for Perpendicular
Date: Mon, 27 May 2013 10:37:20 +0100


By a comment you just made and applying Pascals theorem points that are
perpendular will lie on a line that is perpendular, but that line may also
be a conic!
Astounding.

------- Forwarded message -------
From: "sam gray" <jehovajah@ntlworld.com>
To: "Norman Wildberger" <n.wildberger@unsw.edu.au>
Cc:
Subject: Suggestion of Perpendular for Perpendicular
Date: Mon, 27 May 2013 10:22:08 +0100

While Perpendicular is clearly important in Euclidean forms of the
pythagoras theorem, the actual Greek is ortho or ribht. Grassmann uses the
Austro Prussian senkrecht. Both of these are physical terms not
translating ortho but rather gravitational downward pull on a pendulum.
Thus i suggest the term perpendular as a more natural description of the
dual principle.

Sam


--
Using Opera's revolutionary e-mail client: http://www.opera.com/mail/

Here is the video referred to

http://web.maths.unsw.edu.au/~norman/YouTube.htm#WildTrig

<a href="http://www.youtube.com/v/AjVM5Q-pvjw&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/AjVM5Q-pvjw&rel=1&fs=1&hd=1</a>