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

[jehovajah]

Hamilton sucessfully developed the algebra of couples and that of quadruples called quaternions. The algebra of triples will be found in the geometry of triples as per the logos response.

Clearly the geometry of triples has been a well worked field in terms of the Euclidian geometry, and this may be why it has appeared so elusive, in that it lies obscured in everyday commonplaces. For example trigonometry must play a significant part in such an algebra, and to the extent Hamilton used trigonometry in his algebra of couples is the possible obscuring of an algebra of triples.

[kram1032]

(60)+ mod (60)+ mod(24)+ mod (7 )+mod(29,30,31)+ mod( 12)

shouldn't it be mod((28,29),30,31)?

[jehovajah]

Hiya Kram1032, of course it should, and thanks for contributing.

I am going to suggest a relationship in the primes resembling the spiral threads of a phylotaxis. Now i have seen one and been intrigued but i do not know what it relates to beyond naturally occurring biological growth patterns.

To understand a prime i think i have to steer clear of the notions of number and stay with the spaciometric precursors. Thus the issues relate to a phylotaxis, and how to multiply, or bundle efficiently to fill a region,

Of course from observation this seems to be a spiral, and that is the spiral of the phylotaxis.

One spiral thread cannot cover or fill every part of a region and so there is a space every so often to start another thread. I think that these spaces will exactly match the prime number patterns, and may be got at by using an exponential aggregation clock arithmetic based on the known primes, and extended by every new prime found.

That is a mod(p(ø)^n) clock arithmetic aggregation where p(1)is the first prime and p(r) is the rth prime and the aggregation is sequenced by the logarithm of the clock arithmetic and the prime indicator ø. That is a 2 parameter  sequencing which spaciometrically should cover any given surface area in a kind of spiral pattern. Thus i would look for a prime on a spiral linked spaciometrically to a phylotaxis.

In my previous post i loosel used+ as a sequencer and a connective, not at all as an operation or even a set addition property.

The rigorous notion that i am using is in fact subset inclusion, but with a rule that retains exclusivity in set operations . The rule is precisely the mod(n) clock arithmetic.

So my 60 second bundle is a subset of a 60 minute bundle, but the enclosing set can only operate on complete 60 second bundles. In this way the enclosing set has a mod(60^2) clock arithmetic, but its operations are restricted to a mod (60) arithmetic on the mod (60) bundles. similarly the mod (24) clock arithmetic is actionable only on mod (60^2) bundles  included within it etc.

The interesting thing to me is that if i draw a graph of these relationships or a set diagram, i spaciometrically have an image of a shattered or disintegrating whole! This of course takes me right back to the notion of disintegration and a resonance with fractions .

It is immediately clear that whereas disintegration breaks a whole spaciometric mass into chaotic pieces in general, aggregation in general puts these chaotic piecees back together into a whole spaciometric mass, and the sequence i do that in determines my aggregate clock arithmetic system, or is fixed by imposing such a system. Pn a similar way the way a spaciometric mass disintegrates can inform me as to an appropriate aggregate clock arithmetic of fractions to describe it with, or the imposed aggregate system determines how i describe the way a thing is disintegrating.

The less i impose  and the more i follow the lead of the spaciometric form i play with the more intuitive is my understanding of the behaviours of the spaciometric forms under aggregation and disintegration.

The polynomial aggregate mod(e^n) clock arithmetic by implication should be more in tune with spaciometric behaviours than the decimal system. It would use 0,1,2,e as clock arithmetic elements where e would be the mod divisor/ signal to go to the enclosing set and i would in fact generalise to the rationals as clock arithmetic elements and would find that surds naturally would be necessary. Surds are the basis of the polynomial numerals we call complex and quaternion and indeed hypercomplex, they are also the basis of the irrationals: yes mathematicians are and were slowly going Mad!  

However spaciometrically these ratios are very real for mod(e^n) clock arithmetic aggregates, and will include that favourite the golden mean ratio or ø.

i hope to see and show how √-1 is a departure from extension in spaciometry to spaciometric rotation, eventually. However i need a scaffolding to support the restructuring i am exploring and one that mathematicians have used in the past intuitively, and i think this scaffolding is the general polynomial form. Hence, why i refer to them as polynomial numerals

[jehovajah]

In terms of representing a generalised mod(e^n) the sin and cos ratios on the unit circle are going to be useful, whereas the tangent ratio may have some bearing on the spaciometric arrangement of the prime bundle threads as ratio threads.

And i have just developed this linking rule for the ingredients u,x,y in my parametrise recipe for a helical spiral cake, Its not finished yet

x=u*cos(.5)*sin(u), y=u*sin(.5)*cos(u) : cartesian

ø=u*cos(.5)*sin(u),r=u*sin(.5)*cos(u)  :polar.  < This one is interesting because the finished image has the yin yang in it, but   
                                                                    this is a perception as the path traced is a cardioid,

u is the parameter for the length along the spiral coil, and the trig ratios model the pitch and the periodicity.

[jehovajah]

So i was thinking: Hausdorf and his partially ordered sets  . Spaciometrically an aggregation is either ordered by imposition or it has its own relational order based on contiguity. In general i cannot perceive contiguous relational order sequentially, it is just perceived. Auditorially i can experience this relational order as a chord for example, which is a interference pattern relationship with phase distinctions as well. This is still sequentially experienced in a sampling mode that picks up the variations and assigns additional values such as distance to source location and speed and pressure etc. this visual relational order assigns relational structure distances, spaciometric rotations. and spaciometric mass and density. The kinesthesia adds more in terms of mechanical relationships.

So partial ordering means what exactly? Is it a partially imposed order? It can not really be anything else can it as a relational order is a perception that is nascent . t b c

[jehovajah]

Yesterday was a good day. Spiralman pointed me to  this book by Ginzburg, Terry W.Gintz RSK gallery is freakin awesome and i found this 3d calculator on my ipad apps site.

I can honestly say i have never heard of Ginzburg before now, but his ideas are not new just popularized and worked out in detail. He is in fact heir to much eastern european thinking about an aether, or a substratum of a fluidic type analogy. I do not propose to read his books yet as i am happy that i do not have to do the math so i can play around with the notions i have set out and have fun. Any way i have no one to convince but myself, and that is how it should be. I mean i do not know if what i am playing with is going to harm anyone because i do not know what i am playing with exactly. That is why it is fun. One day like Oppenheimer and Feynman i might be coraled into making a bomb or something! Yh¶h forbid!

Try as i might i find no relationship between Vitaly and Vladimir both new to me. However vitaly is a nobel prize winning physicist and vladimir is a metallurgist who has studied theoretical physics as a hobby, not unlike myself really.

Vitaly i am drawn to by the mathematics, Vladimir i will have to look at in the cold light of day to see if he is an armchair pundit, like me!

[jehovajah]

I am looking at packing and stacking, and more general arranging and heaping under the notion of bundling, in a motion field which has the conic sectional curve motion properties/ attributes (including helical) as a gravitational description, or law.

I have also been looking at spaciometric restrictions on bundling in cases of relational arrangements like plant branching analogies.

The bundling notion is in fact  a nascent multiplication notion, but it applies to an aggregation notion as well as to a disintegration notion. Disintegration is a special form of disaggregation, the "reverse" of aggregation.

Aggregation is a nominalisation  of a process: to aggregate, with clear links to : to collect so to distinguish 'an aggregation'(collection) from aggregation i will use the word collection or a collection.

Clearly the speed and force of these processes has not been a consideration at this stage but these qualities are quantative measures of the motion field, and are present in some mechanical and biomechanical attributes in the description,for example :stability and growth.

Aggregation consequently can be imposed by a biological mechanism( including human interaction) or arrived at by dynamical mechanisms in the environment including electromagnetic and thermal dynamic processes .

I therefore distinguish organic aggregation and disintegration, and inorganic aggregation and disintegration. This is done primarily spaciometrically so tht by inspection and exploration i can make these categorizations. The spaciometry of the inorganic and organic categories is significantly different and therefore will lead to different algebras and thus different maths. It is therefore significant that there is a deep self similarity with regard to the construction of the numeral systems. This may simply lie in the cybernetics of the observer and its sensor system arrangements.

[jehovajah]

Soo ! i was thinking about my Instruction anaology for mathematical functions and procedures, and my ingredients were parameter elements? so what did i mean by that? Does that make a parameter a set of elements?

A parameter is a dimensional fractal quantifier, that is a fractal employed as a standard reference for measurement or quantification purposes. Spaciometrically things are quantifiable but the measure is totally relativistic, that is dependent on the person, the standard, the temperature and pressure and location relative to other regions etc.

So the parameter is the ingredient and it is a fractal withe named distinctions. The named distinctions are not elements as in set theory, although there is a one to one correspondence with the elements in an abstract set like the set .

The abstract sets of number theory treat the elements as reified mathematical objects while procedurally instructing their use according to the circumstance, for example as marks along a number line, as a counted out quantifying and or ordering process (the cultural iteration +1) and often as both or flitting between the two. Culturally the rule is our only visual referent and the count our only auditory referent. This is why mathematics and music seem so intimately linked.

So what do i mean by dimensional? Merely that each use of the parameter is strictly related to a measurement of a dimension of a spaciometric form I have opined before on the confusion between this use of the word dimension as in dimensional analysis of physical quantities and the more science fiction use of the term! Dimensions derive from the practice of categorizing process of measurement and distinguishing it from a different process of measurement. These differences can be as simple as orientation differences to spaciometric form and mass and density differences. Thus the spaciometry and the cone of orientation and my own sensory sampling systems and procedures are what underlie a dimension. I therefore find it hard to subscribe to "other " dimensional objects except in the sense that the parametrisation of these spaciometric forms requires more than the standard 2/3 dimension description.

It is often hard to see that xyz are as much parameters as r and ø,as are any other parameter in maths. For example the angle is a parameter of the ratios sin cos tan. i can call them an Instruction using the ingredient "the angle" but it has to be in a right angled triangle. So the specific ingredients in my sin cos tan cake would be a whole bunch of juicy right angled triangles. mmmmm! Delicious!

[jehovajah]

i want to introduce two opposing characters into the plot line. The first is Thermal, and he always applies the laws of thermo dynamics while the second is Electral and she always applies the laws of elecromagnetic hydrodynamics.

Electral is always struggling to develop and maintain order while Thermal is just happy to create chaos! He is the proverbial chaos monster!

Electral uses all in her power from the mighty electro magnetic gamma ray to the tiniest electron spin to create wonderful and delicate arrangements of order with space as her medium.Thermal however takes space and kicks it about all over the place, giving it terrific velocites and vortices of fantastic rotational velocities. When they clash-sparks fly, thunder rolls and heat and light radiate everywhere, and space warps wondrously!

The plot thickens!!!.

[jehovajah]

I have learned today that the whirlwind has a spaciometry with a beginning a middle and an end. So the vortex is itself part of an inclusive system which is dynamically balanced. This means that vortices can exist as distinct objects within this spaciometry or they can combine their spaciometry to achieve a greater or lesser vortex spaciometry. Thus fractal arrangements of variable interactive complexity can exist within a super vortex spaciometry.

Such an inclusive spaciometry is applicable to all scales.

It is interesting that the low pressure core is a helical vortex updraft, as an exploration i did based on assumptions that motion is not in a straight line unless that line represents a high frequency helical transport, predicted that "heat" plumes would "rise" in a high frequency coherent column vortex due to high frequency "electron" emission not due to bouyancy effects, that is lighter density rises while denser material rushes in to fill the so called vacuum .

These helical vortex emissions would interact with less motile space and establish a radiative wave transport which would eventually dissipate the motion. However if the helical coherent motion is at the resonant frequency for the surrounding systems then extended and conical vorticular flow should develop protecting the helical core. Thus the helical core would dissipate but be replaced by an updraft fed by inflowing vorticular air which when it reaches the eye of the vortex will have a fixed rotational motion transfer value, which will be dynamically balanced by a downward back pressure vortex. The updraft air would then rise under shear tension and dissipate when the equilibrium could not be maintained with a twist out at the top of the column.

[jehovajah]

Vladimir B Ginzburg is a heavy metal dude! He has produced some serious stuff.

His tables speak of the underlying spaciometric symmetry in the universe, but this time using a spiral form as a tool. Unfortunately for me he has not gone on to develop a spiral reference frame so i do not see in his work the expected simplification in notation. So i am still living in hopes.

See attachments of his serious metal, dudes.

Helicola a new drink that makes you see the world with fresh eyes. It also makes you pee your pants!

[jehovajah]

So the spaciometric form of a vortex iwas thinking: obviously the torus where the boundary conditions are the vortex itself. This has been called a minkowski bubble, but it is very like a bead used in beadwork constructions (can you see the where that may take me with a bit of thread?!) But i also thought of marbles-- you know how they get that unique swirl in the centre?

And finally for you boy scouts knots and knotworks. The not is interesting because it links the spiral filament trail to a general toroidal form which is not obviously a donut shape and extend the toroid to klein models for arranging vortex interactions.

[jehovajah]

Just got time to state this:

By iterating a motion statement on the complex plane  so called, and now motion statements in the Hamilton Clifford Void(for want of a better term for hypercomplex space) we have precisely discoverd the spaciometry of the vortex-torus form.

The vortex torus form is the spaciometric free form for a torus vortex arrangement without any boundary conditions other than itself. Thus the "filaments" of the vortex in this form lie on concentric torri and the centre of a torus is a spiral vortex.

If the form has boundary conditions imposed it behaves like a vortex in a fluid flow at a certain scale but like a bubble at a small scale compared to the fluid current flow.

Thus this form is a spaciometric form that will be fractaly ubiquitous under an iteration process with self referrential boundary conditions, or boundary conditions that approximate a torus.

The algebra that describes this form or rather generates this form is hypercomplex but this may change when a spiral reference frame is developed.

[jehovajah]

Spin a dollar and watch it fall. Why does it vibrate like that? The pitch increases doesn't it? but does it cut off or go ultrasonic?

Ginzburg V and B and Haramein and Rauscher they all get you thinking. But i find out from Tombe that Maxwell was a vortex theoretician. The trouble seems to be the aether concept. Einstein followed it for a while then abandoned it. `but not because it was wrong. I think that Einsein realised that  he did not need an aether, And rather than spending valuable time and resources on proving that an aether existed he could simply proceed "mathematically".

I take issue with this substantiation of mathematical equationing! Feynman typically relied on the procedural and syntactical and symmetrical notions inherent in the mathematical developments of his day without ever being able to "Know" what he was calculating . That is not to say that he did not have insights, but rather he took a philosophical viewpoint that it did not matter what the referent was as long as it was consistently and accurately referred to! Hence mathematical rigour was crucial.

Dirac took the view that the electron was real enough to base his equationing on and that Einstein had neglected the negative solutions to his equations. For a while he was strictly censured, but now is vindicated. There is anti matter! Einstein to be fair was not a brilliant mathematician like Dirac or Levi, and often was helped by his wife to do the calculations. He really thought that only the positive answers made physical sense, and thus used the mathematics as a tool rather than a model. Schroedinger and Dirac set out to model the statisitical and probabilistic behaviour of an electron as a real entity. So their mathematics was a descriptive , ballistic model of a particle.

Feynman was heir to that strain of thinking but did not feel the need to come down on one side of the fence about the existence of these particles/waves, especially when that meant one might be accused of supporting an "aether" hypothesis.

Nowadays some scientists in the west openly propose an aether of sorts as the standard model is so inadequate at fully explaining everything. This seems to raise the ire of some who do not seem to realise as Einstein did that life is too short to engage in this kind of debate.

Religious and mystical people want to connect the aether to their deity, but they make a false premise in assuming that space is outside of their deity! And if that is  false premise then it means that space is their deity and we can proceed as Einstein did without recourse to an aether: space itself warps.

If however space is outside their deity then what is space and where did their deity come from to inhabit space. and more importantly who else is in that space?

Fractal geometry or rather fractal spaciometry is the only geometry that adequately deals with this question, whichever premise you accept, and one can still proceed with recourse to an aether exactly as one would without an aether, except that the aetherists will feel a certain "gnosis" about why things happen a they do.

Since the mathematics is the same but the interpretation is not, proceeding mathematically seems like an attractive option. It is not.

Mathematics by the Logos Response derives feom a spaciometry, and if one posits angels in the spaciometry angel will appear in the mathematics! The Logos response is key to eventual mathematical inspiration and insight, and one cannot be divorced from the other. Feynman's spaciometry influenced his calculations just as much as Newtons did his, and Newton was a very religious man of his time.

Newtons spaciometry was an absolute space of perfection inhabited by god in which all reference frameworks were true to god. Everyone elses reference frameworks were relative to that absolute one. The only change that Einstein made at this level was that all reference frames were equally valid. This was like saying god does not have any better handle on this universe than you or I!(if s/he exists.)

Einsteins spaciometry was the emerging geometries of Riemann in particular but other non Euclidean Geometries. This of course revealed algebras which influenced his mathemaical description of physical relationships. When Tensor maths was developed by Levi And Ricci Einstein struggled to learn this description of relational reference frameworks because it simplified is mathematical description of the quantities he was relating, and it was invariant under transformation. So he hoped he could develop a theory of everything at every scale due to scale invariance.

As it turned out tensors are only invariant under affine transformations, and Levi had a fondness for cartesian tensors which meant that the properties of polar coordinate tensors were not properly perceived until recently, and are not even fully investigated now, because complex and hypercomplex tensors are more readily accessible(clifford-Hamilton operators).

Today our spaciometry is not so easily defined, but it has to be at least hypercomplex, and many are reaching forward to a fractal geometry.

When i started to explore spaciometry it was to free my mind to look with fresh eyes over  a ploughed field. In doing so i find that the field has not been ploughed but hacked at and trampled over, with a few walled gardens of exquisite beauty.

As for my own musings i find the simplicity in spaciometric rotation and extension. Two motions in a motion field which characterise every other motion. As a result of these fundamental motions my spaciometry is vorticular.

[jehovajah]

I mentioned the spaciometry of the torus-vortex as a complete spaciometric form. This form being fractal and at all scales is found in the spinning electron, the solenoidal magnetic flux field, the electromagnetic radiation wave around a dipole aerial, the magnetic flux field around planets with active cores and stars, the gravitiational shaping of galactic masses from single galaxies to galactic clusters, and of course the ubiquitous black holes.

My definition of fractal by the way is the product of an iteration process.

It is so strange how the cultural legacy i have inherited puts blinkers on the senses. By simply changing the tool which i use to reference a direct sensory experience i can avoid the blinkers of using defined terms. Thus the appearanve of the torus-vortex forms in all these examples really does indicate an underlying unity. And this unity is plain to see and experience especially here in fractalforums, when 3d fractal generators can routinely sculpt these basic forms without hard mathematics getting in the way.

Spaciometric rotation and extension are the basics for fractal images as well as the experience of notFS that i have through the setFS