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

[jehovajah]

On further reflection i have a modification to the root of unity distinguisher.

To make it less clumsy and to link back to some earlier exploration i did in another thread i wondered about using signal instead of distinguisher?  

However that loses the fact that the sign does pick out or identify the specific root of unity so what about a root of unity identity?

To avoid confusion with r or R for radius what about and ?

In line, perhaps, with the name identity i think it is more useful to recombine the distinguisher with the unit magnitude thus making

= = unity in magnitude or

|| =| | = |1|
an unsigned magnitude

with , being "signed" or signal magnitudes which i will name in full as root unity identies. Does that work?

Got any suggestions to improve?

Extending it to the unit sphere looks not to bad now:
|| =| | = |1|
an unsigned magnitude, and are the radian arc measures on the surface of the unit sphere which use the great circle, circle of latitude scheme common in spherical trig, and spherical geometry.

So the root unity identies can now be assigned by the flip algorithm, and the
action () can be called rotation and distinguished as it must be from multiplication.

I think the rotation rues would look like
action() =|1|²

But i am not sure yet.

Any body help out here?

So for n rotation actions i would expect a logarithmic effect on the magnitude but an arithmetic effect on the root unit identity.

This is to be expected in spherical trig, and is the basis of Napier inventing his Logos Arithmoi.

Incidentally Napier had no word for it then, hence logarithms, but as has been pointed out this would directly equate logos to geometric series and arithmoi to arithmetic series and the tables being function tables enabling a look up of related terms to aid in calculation. Logarithms truely are a wonder of the world.

So i have chosen and to enable a relativity measure, which consists of at least two unit spheres touching each other , where is the root identity in , and is to be considered the basis sphere and is the root identity in , and is to be considered the relative sphere.

From these i can adopt a minimum 2 tuple as a tensor description of relativity.
(, )

And i will just need to specify the aggregation rules.

If this in any way produces geodesic curves and links them to 3d trochoids i will be happy!

Any one want to check this?

After all, this is not a blog.

[jehovajah]

From Brahmagupta to Sophus Lie there is a fascinating and wonderfully exhilarating set of connections to the study of Shunaya.
 Today Lisi ,Garret Lisi is continuing in that vein of exploration, studying the fundamental unity of Shunaya [1,0)!

Many Great scientists have drawn from this rich vein, but my favourites are Napier,Bombelli, Newton,Cotes, De Moivre, Hamilton and Dirac and Feynman. I guess Einstein if his wife and Ricci, and  Levi are included with him.

[jehovajah]

Boy am i going to love reading this!

[jehovajah]

Shunayasutras are dynamic forms of magnitude from shunaya. The arithmoi are a special group of shunayasutras with rectilinear form.

So standing in a supermarket brought home the existence of shunayasutras to me. Every stacked shelf was full of them. But they were in static equilibrium. The only ones in dynamic equilibrium were the people and trolleys, and tru to form they had direction magnitude and rotation. Then I realised that this rotation comes out of shunaya with the form thus an object has motion and maintains motion because of dynamic equilibrium.

I then noticed that shunaya coming put of the unit circle expand  while objects moving into the unit circle contract. Depending on the rate of rotation is the size of the objects that come out and expand. The slow rotations produce bigger shunayasutras and the forms break up as they rotate faster, and the continue to expand as they move away from the source of origin.

Shunayasutras of all shapes and sizes explode out of the unit sphere  and expand until their spin breaks them up into smaller pieces.

Then I thought about the yoked sunayasutras have, no mater how far they spin away from each other the yoke cannot be broken but only annihilated when they spin across each other again.

Then I thought a root 3 unit yoke cannot be annihilated by a root 2 unit yoke, but they will still interact! Thus the root unit yokes  mean that bonding takes place between shunayasutras  and we have all forms of attractions from gravity to nuclear forces to spin!

I then thought gravity would be a root unit 2 yoke, which in terms of bonding means opposites attract and nothing is repelled or spun away? All othe root units spin other root units in various ways only root unit 2 attracts all in some way.

I enjoyed that because it seemed as if I had a full description of the behaviours of all shunayasutras in my reality.

[jehovajah]

Although not surprising, the idea of a polynomial developed in the west quickly once Tagliaterri, Cardano Viele and Bombelli published and disseminated their works, Although the precise time when "poly" was chosen is a matter for research, certainl by Newtons time it was accepted to refer to the terms as Nomials after the bi nomial expression. In this heady time after Descartes and Bombelli, the discovery of the compound interest formula in he west and the introduction of repeated fractions from indian influenced arabic sources, with Newton's infinfite binomial series! which he invented lead to the fashion of infinity and multi. So De Moivre was able to deliver a paper to the Royal Society headed".... infinite roots for multinomial equations".

Around this time also is when analytical methods started to undermine the older synthetic geometrical demonstrations, and the gentleman's agreement thai if it had been shown so plainly that it was obvious by inspection, then that was proof enough. Now arguments were flaring up about who was right, who was prior , and who actually had demonstrated something!

This is why besides his autism Newton was loathe to publish anything that would leave him open to argument which could also lead to duels! Galois is a famous example of a "light" unnecessarily lost in this way.

Many people questioned established "sages" and so schools of defenders sprung up. Newton's happened to be in part Cotes and De Moivre; and research shows that alliances were about more than philosophical argument, they extended to politics and religion too.

[jehovajah]

Anybody know where sequences and series come from?

[jehovajah]

Now that i have identifed the flip algorithm, it puts periodicity into perspective, and i can now experience another measurement called "duration" or some say "time". But i experience duration as a "change in state of tension". it is associated with motion, or observation. Its a kind of "bated breath" and "expectant state" a waiting- ready to react when something happens!

It is very wearing, but it seems to be a proprioceptive response to measurement, like  balancing on a rope.

This "duration" is event dependent, and a sense of quickness or hyersensitivity to detail accompanies it. Therefore there is a sense in which the events seem to take longer to happen. or pass by very quickly. Tus duration is subjective and not suitable as a standard such as time is. This is why duration is allied to periodicity to form the full conception of "time".

Time therefore is completely dependent on the motion of objects, but without "duration"we would not sense periodicity, and know when to apply the flip algorithm.

Of course you might think we would notice when the sun rises! But that does not seem to be a time marker! It is the single most visible phase change which catalyses and coordinates many chemical change reactions, and thus acts as a kickstarter to activity, but it is the change of state , the increase in tension that signals duration. Thus we may sleep with no sense of time passing, relying on periodic  chemical reactions some reliant on the sun to wake us up to activity and duration. We use the sun and any other periodic motion to atach our sense of duration to, or to attribute a sense of duration to and so measure time by motion.

This leads to strange experiences of time, which can be explained in this way. So when you appear to be craling allong at 30 when previously travelling at 60, this is due to a hypersensitive adjustment of the sense of "duration" bought about by a habituation to travelling at speeds requiring fast reaction times!

So in my view time travel backwards is not a realistic possibility, but different rates of time are to be expected, because time is based on motion and a reference periodic motion.

Does duration extend with a slowing down of time? Only if each and every aspect of space is affected by motion, thus slowing down chemical and nuclear events. If chemical and nuclear events are slowed down by motion, it follows that every aspect of space is in relative motion, and thus space is a motion field.

the Lorentz transformation is therefore a key measurement of the relativity in a a motion field.

[jehovajah]

One unfortunate result of the Lorentz transformation, at least according to the current view of it, is the result that not only does time slow down but molecular interaction and quantum interaction. What this means is that the closed the speed gets to light speed the more fragile the bonds within any molecular conglomeration become. Eventually ceasing to be at light speed. A molecular body would thus disintegrate if it even went one zillionth of a billionth of a quintillionth faster than light speed!

Very unstable!

However there is the symmetrical consideration of the Lorentz Transform to consider. Whatever happens to observer A should by relativity happen to traveller B.

To explain what i am thinking, and it does not originate with me, i am going to explore symmetry primarily and centre of relativity == centre of gravity as a consequence.

Anybody know what symmetry is fundamentally?

[jehovajah]

It seems clear to me that either Wallis, or Newton, or Cotes calculated the Logarithms to the base e and introduced the term Naperian logarithms to distinguish them from Briggs base 10 logarthm. Either of the three were certainly capable of computing the value, but of the 3 Wallis and Cotes were more  to naturally inclined do it, just for fun.

It is also clear that Wallis in dealing with the squaring of the area of a circle was able to introduce a more algebraic formulation of the problem based on the solving of rational roots of equations. This would inevitably mean that Wallis was probably first or among the first to link the trig functions to the unit circle, before Bernoulli or Euler or Leibniz.

This gave Newton, and subsequently De Moivre and Cotes a tremendous insight into the analytical power of Wallis, and trigonometric functions in general. Newton stopped short of what became known as the De Moivre Cotes Equation, and certainly did no more than intimate the Cotes-Euler Equation. At this time Newton was busy in other spheres developing the calculus of Fluxions.

Cotes was able also to describe the roots of unity as depending on "ratios and angles in a progressive series" based on his share insight of Newton's work with De Moivre.

There is no doubt that these 4 gentlemen, in secret, formed and defended the foundations of Modern Mathematical science.

[jehovajah]

It seems fair to point out that Newton Like Jakob Steiner was a synthetic geometer, and most if not all his formulaions are derived through that means, using the mathesis of proportions. Whereas He initially balked at Euclids geometry, finding it alien and confusing, he upon his return took it up with ease finding its conlusions rather obvious, and its method straight forward. From that time forward he did always employ the method of the greeks in his expositions.

Wheras Wallis looked forward to a greater algebraisation of geometry after Harriot, as he disputed Descartes primacy, Newton actually looked back to the Greeks for inspiration and mathesis. Thus cerftain things are rather obvious to himas a geometer as they were to the greeks, and the dynamic nature of units , and the need for proportion and their resultant scalars was ever present to his mind.

Though he wrote in Latin, he understood the greek well enough to know that plethoration was fundamental to measurement and is the basis of calculus. Thus to Newton Algebra was servant to geometry not its master as Wallis would have it.

Wallis was an avid cryptographer, a breaker of codes; thus an algebraic form would be as intriguing as breaking a code to him. Most of us are, however, mystified by codes! This is why Newton saw what no one else saw very often, and derived whar no one else derived frequently, he refused to codify it beyond the greek mathesis.In this way he was very much like Steiner.  Algebra, i am afraid obscures by its generality. It hides detail by a layer of abstraction, and it lead to the false notion of number over measure.

Within measure , and within geometry in general, orientation, magnitude and circular rotation, neusis and relativity, parallel and orthogonality plus trigonometric ratio have always to be accounted for. And the missing element in my understanding, usually i call the affine transformations, but in reality the affine symmetries: the description of plethoration in action!

Plethoration requires production or reproduction of units, and as may be observed in cell mitosis this requires the affine transformations or symmetries.

Because of the fractal nature of all things this motile behaviour is repeated at all scales. This is the basis of Newton's understanding of dynamic magnitudes, which he called fluxions, and i call shunayasutras.

The fact that indian mathematics was known in Europe is also attested to in Newtons papers, and by Napiers comments on navigation. Thus the notion of shunaya was communicated to Newton by all means available to him, but it was because he was "aware" that he picked up on it more than any man of his time or since Brahmagupta.

Had Bombelli not developed his operators i think Viele might have, but in a different way which would have held back the algebra of the time, and possibly not given Newton the perspectives he needed to distinguish his thoughts from Wallis and Descartes and the general analytical trend subverting geometry, the artisan and the Artists of renown.

It is a joyous discovery to find that many upon Napier's revealing of logarithms, immediately applied it to the scales and rhythms of music, and thereby affected rhythms of dance and movement with it!

But for me the Reasoning behind spaciometry is justified by no less a seer than Newton, whose mathesis was geometrical through and through.

"... Now, the geometry is the thing...!"

So symmetry is really that fundamental and is related to shunaya fundamentally.

[jehovajah]

Videos of three nice lectures by Robert Devaney on the dynamics of complex maps are available here:

http://www.globalpres.com/mediasite/Catalog/pages/catalog.aspx?catalogId=00282ad2-f5b5-4991-a75e-d4ac72b6196e&folderId=48820f4c-94dd-4827-b33d-84d3b5fcee1c

The first lecture is on the Mandelbrot set, the second is on iterated exponential maps, and the third is on iterated rational maps.

These lectures help to explain the dynamic magnitudes i have been mentioning and why a revisison of the foundations of mathematics is not just an alternative, but a necessity.

This also visualises shunayasutra for me as bulbs on the mandelbrot, and why the search for the holy grail is looking for a simila r symmetry.

The lathed mandelbrot is that symmetry, so we have found it and it is not very interesting!

We want another symmetry that probably exists in the hextonians that is generalised coordinates with 12 yoked axes,6 positive with 6 opposite negatives.

My guess.

[jehovajah]

I awoke thinking about Wallis and his use of Viete's law of cosines, deeply,deeply impressed by him and his influence on Newton, and in awe of Newton's geometrical facility and faculties, the same i suspect as that of Wallis.

In Newton as De Moivre recognised there is a richness and subtlety of vision, relation, action and co-mensuration of proportion, and comparison of ratio, with distinction of kind.

The construction from his analyses of these relations and apprehension of these distinctions are magical. And from them he easily derives solutions and forms and curves and areas and dynamic relations.

At the heart of this wonderful facility is a powerful and endlessly applicable set of relations and notions known as Trigonometry, which is supported by an axiomatic base of theory that is known as Euclidean geometry.

This is not the Euclidean geometry that i was fed as a student, nor i suspect a vast majority of us who were and are educated in mathematical knowledge.

I have never read Euclids geometry. I daresay i know theorems from it but i have no conception of the plan of the book, the discussions the lemmas and dilemmas and the historical glosses it contains.

Firstly i am not a classical scholar, and so the perceptions and distinctions would be beyond my quick reading ability. Secondly i fall into the category of being bainwashed by my tutors into thinking all that is important and necessary they have taught me. And thirdly i have little time to spread upon a lot of things.

But this i note Newton read it from cover to cover and found it of great simplicity, and Wallis read it and promoted it to all and had great facility in invention of solutions and arrangements.

Newton is a geometer through and through, and such a geometer as to make the mind boggle!

In Newton the unit circle diagram is used to define the trig relations, but then to define new relations and conceptions and ratios. This he undoubtedly learned from Wallis.

But Wallis did one other thing with enjoyment, he explored new relations and from the diameter of the unit circle he derive the law of Cosines.

But shortly after that he derived equatioms for the circle the ellipse and all the conic section curves! This is from the diameter of the unit circle and the observation that by varying the relations between the dynamically changing sides of any triangle he could by trigonometry measure the position of any point on any curve!

Further using his trigonometric measure he could posit relations that drew new curves!

This immensely powerful trigonometric tool the unit circle with inscribed right triangle was relaed directly to neusis, the greeks . Bombelli, and calculus! Newton focused on the unit circle with the tangential right angled triangle to develop the calculus, and Steiner focussed on the escribed right triangle to develop his synthetic geometrical insights.

Geometry with trigonometry therefore has always been the most erudite analytical tool known to western man!

Without hesitation i say that these 4 men, Wallis, Newton, Cotes and De Moivre had a geometric understanding of i the √-1 that has no equal until wessler!

And it was not just a tentative experimentation, it was a mathesis of the imaginary magnitudes that Hamilton had to defend when he took up his algebraic studies!

This goes deeper, as Cotes equation shows: the Napierian method of logarithms is nothing if it is not an addition to the set of fundamental trigonometric relations, and can be based on the unit circle. Thus Newton and Cotes could easily calculate logarithms for the tangents, and the sines and relate those simply to lengths of arc on the unit circle.

They could also calculate e from this same tool, and Cotes actually did.
This to me makes the 5 men the biggest innovators in geometry and trigonometry since the greeks and the foundation of the modern mathematical sciences

The polynomial has its roots in these 5, and starts from the quintic of Bombelli to the Multi nomials of Newton and De Moivre.

Napiers Logarithms highlights something easily forgotten: Relations, proportions and detailed exploration of relations,when tabulated are the basis of the notions of symmetry, similarity, congruence, equality. We tend to skew our use of geometry toward notions of Equality, and so cut our minds off from the relations and proportions and ratios underlying that notion. We miss a lot by rushing through.

We ought to play a little, like Newton. on the beach of a vast ocean.

[jehovajah]

Relations, proportions and detailed exploration of relations,when tabulated are the basis of the notions of symmetry, similarity, congruence, equality..

For me relations means  contiguous connections in space and orientations in space relative to those contiguous connections, and thereby by  a chain of contiguous connections (with their respective orientations) a relation between non contiguous objects/ forms. Thus for action at a distance to occur space itself must have some contiguous connection between dense objects.

I am not a proposer of Aether, but i am of shunaya.

Proportion is the comparing of dimensioned magnitudes be they in static or dynamic equilibrium. I believe that Newton understood Einsteinian relativity, but unfortunately lost Cotes too soon, and was too distracted to pursue the matter to the depths of Einstein. His religious convictions and his autism restricted him to a less revolutionary conception of "reality", and his memory was fading for the finer sutleties of his earlier explorations. Quite often Decades would go by before someone would ask him what he meant by such and such! Can you imagine being questioned on what you did yesterday in detail, let alone 10 years ago?

However his "less revolutionary" is my turning the whole universe upside down!

So proportion in conjunction with relation leads t a legion of distinct arrangements of forms in their magnitudes.

Now proportioning is the very heart of rational thinking! There is no other word to describe the fundamental link between proportioning and reasoning, Certainly in greek the fundamental concept is described by only one word: Logos. Hence my dicovery of the Logs response. There are other words in greek for thinking and measuring, and even minding, but the fundamental nature of Logos is such that all other things stem from it. It is the symmetrical heart of all reason and rational thought and measurement and management and manipulation of and description of and distinction of the world of the greek mind, our mind.

So all our arithmoi stem from our logos response, which is our response to the measurements and comparisons in our experience, This response is to distinguish, to make distinctions.
So my shunayasutras stem from my logos response, and each one of us uniquely has our own logos response that contains measurement and reasoning of the reality we experience.

Hold on to what you have, and do not give it up because somebody says there's is better! You can do amazing things with your Logos response that no one else can do!

So our "relations" with others is at a different or distinct level of supporting sublevels, to our relations with ourselves, or if you prefer within yourself. These distinctions become platforms for other distinctions and so on in a never ending fractal pattern of learned reponses. thus we have so called subjectivity and so called objectivity, and through these we form relations and alliances of relations to develop a common language and a gateway of communication between our unique perspectives on "reality". But the communication is not perfect, it is fraught with wrong associations and relations and is as obscuring as it is revealing.

It is like traveling into the mist  on the mountains to find who knows what and who knows where. Sometimes the mist clears reveling vistas that are either breathtaking or perilous! This in my view is he nature of communication.

Some useful attributes of these relations that we have distinguished are symmetry, similarity congruency and equality, and this is through the application of proportioning to everything we do or create or think. The notion of relativity therefore arises from the more general notion of symmetry.

[jehovajah]

The state of play this morning:

Man is a magnitude measuring entity who responds to magnitude through thr Logos Response.
Man makes measuring tools as often as he draws breath!
Man shapes his experience of his reality around his culturally accepted measuring tools.
Mans greatest measuring tool besides himself is the automatic energy driven arrangement called a computer.
I have called all magnitudinal forms and media shunayasutra, out of respect to the BFG that is BG or Brahamagupta.
The greeks Called a particular set of shunayasutra Arithmoi
If i were to give an english name to shunayasutra it would be "convoluted magnitudes" and i would use very curly C to denote it with C being contained within it as the complex magnitudes, with being contained in them as real magnitudes  etc. All these magnitudes will be aggregated structures based on unities of choice.

By such a magnitude measurement system we can not only measure but apprehend and construct or reconstruct our relaity.
Does man become godlike? This certainly was a Greek goal, and may be being fulfilled as one of many teleos for the system.

[jehovajah]

Reciprocals and computing

It is a stunning realisation that all my measuring is founded on the scalars of the ratios of the sides of a right angled triangle, the orientation of that right angle and the aggregation rules of Brahmagupta-Bombelli, and the assigment of an orientation by a flip algorithm.

One wonders if it could be otherwise arranged?

If not we have really come to as Cotes put it the Harmonium Mensuraram.