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

[jehovajah]

Experiment in proprioception 2

Pick something you know how to do well. Do this as fast as you can. Repeat and attempt to go faster. If you can repeat faster still.

Your consciousness of doing this task is altered. You will not know how each movement flows into the next, because your attentional focus is limited.

You routinely do more through unconscious proprioception than you can distinguish.

@ lycium  Hey man if you want weed, you can smoke weed, but if you want what i am on you need to spin out, dude--right to the edge and beyond! 

[jehovajah]

So now the spiral gives up another secret.

The circle and the spiral are in competition in my mind. Well not really as i think that the spiral is prior to the circle, and in fact i have gone a long way to generalising the boundary properties for generating spirals. So it is really a bit of penile envy. Why is the circle so damn easy to use to explain spiral motion in general?

The circle is a very very special spiral form, and i have yet to rigorously determine it as an intersection form for the general spiral boundary. It can't be because the general spiral form is chaotic, so why do circles introduce an order to that chaos? Why are they so facile as a starting point for anything spiral?

A sphere or a circle can enclose any shape, and then the boundary of the shape shrunk to, thus providing an increasingly accurate map based on the radial adjustments to a circle or a sphere. This means i can develop a theory of interactions based on circles or spheres and then apply it to real objects by using a radial transform map.

Is the theory based on the circle or sphere more general than the one based on the actual shape of things? Yes, but that makes it an approximation which has to be made more applicable by minute adjustments.

Well i have heard of this type of limit process before, and it in fact underlies the notions of differential geometries. So even in our most general thinking the circle plays a powerful role, and there is no obvious sign of the ubiquitous spiral or torus forms, In fact we tend to explore these form using the circle!

And i have done the same.

I was thinking about trochoids and cardioids in their relation to the Lai Zhide Torus based on the I Ching algorithm. I wondered what the path of points on an unrolling spiral would look like. It turns out to be trivial in one sense but non trivia in another as the programmes that can generate an "unrolling carpet" animation are non trivial.

It is trivial because the answer is that they form a set of trochoids originating on the line along which the spiral is rolling. It is non trivial in that the trochoids dimensions are radially decreasing to zero for unrolling, and increasing for rolling.

Thus i can see the straight line and the trochoids are geared to the spiral and the circle is inextricably enmeshed. I can of course transform to a different boundary say a spirangle but that only emphasises the general applicability of the circle in solving these types of problems.

A general geometry for the trochoids is found here with some illustrations here

The usefulness of these curves is probably an industrial open secret but even toys inspire this deep use.

However if only for the Quote at the end i like this exploration into thr problems of locomotion on roads.

I always enjoy a good blow in 3 dimensions! but for me the simple bouncing of a ball before it comes to rest hopefully before smashing that window is illustration enough of the trochoidal nature of movement and motion and the interlinking of spiral and circle, of vortex and torus.

The circle is a very special form of spiral as the torus is a very special form of vortex. I do not think spheres actually are real objects,but ovoids are which are a form of torus approaching zero central radius.

The power of the circle or sphere lies in its unique abstraction from every vorticular form, and that is the sense in which i think it is an intersection of vorticular or spiral forms unique to animate consciousness.

[jehovajah]

Ah yes, i forgot to add that i think the mandelbrot set is intimately related to the trochoids of an unravelling system of spirals in the polar coordinate plane, and consequently the mandelbulb is intimately related to the three dimensional trochoid shells unravelling from a system of tori (toruses!) cut off, and ultimately shaped by the circular or spherical boundary condition.

[jehovajah]

Proprioception experiment 3:

Go into a familiar room and stand in a familiar space. Now reach behind you to grab, or make contact with something.

Notice how proprioception is tied in with visualisation, and how you use the visualisation and the audiometric senses to locate and position your hand and body to ensure maximum success.

If you are struggling, try to smell out the object or surface or use your Whiskers (small hairs on your body) to sense how close you are. Maybe a temperature feedback might help, but these are all exterioceptors, so realise how exterioception assists and coordinates with proprioception.

And you thought geometry was just Euclid!

[jehovajah]

i have commented on the notion of dimension. in trems of the characteristic 3 d and 2d nomenclature. So essentially the idea is that a 4th orthogonal dimension exists, etc. Of course a 3 d consciousness cannot perceive a 4d spatiality, so the rubric goes.

Well i will be clear and perhaps reactionary in saying i do not think an mutually  orthogonal direction beyond 3 exists.

I think that there are lots of dimensions, but orthogonality is a unique characteristic to 3d. That is to say to flatlanders that they would not perceive orthogonality in their flat world, only us 3d types can see it in their world and in ours.
 
Now flatlanders is a "teaching myth", and thus it is not an analogy that i would care to continue beyond 3d,because in 3d we have all the dimensions we wish.

So why is orthogonality important? Orthogonality is in fact a sensor driven notion, and we have an orthogonal sensor within the vestibular system . That for me locates it in what we call 3d. Now if we had a 4d orthogonal sensor i would concur, but we do not.

What we do have is a proprioceptive mesh network which drives our motion of geometry,and our imagination. Thus proprioceptively we can imagine another dimension, but in the sense of a "degree of freedom" . We do not like to be stuck and 3d is so often defined in that sense of a limit to our motion. Of course mechanics and engineers know that there are degrees of freedom and the rigidity implied by 3d is a difficult thing to achieve without taking these into account.

So while it is a nice story and all that the 4d fiction is actually masking what is going on. We are actually dimensioning in terms of degrees of freedom.

Thus if i build a so called hypercube "shadow" i am actually building a shape that with an extra degree of freedom providing it is an elastic or compressive degree of freedom can deform while preserving local relationships. As a consequence like any of my real world clothes seen as mesh networks i can turn them inside out continually by rotating in the correct way. I can also twist them and deform them in other ways while maintaining relative relationships.

Some of the more complex web like structures can still do this but in a complex internal radiating in/out pattern which in fact may be good models for growth and development within a 3d cellular structure.

So are my clothes shadows of 4d objects? You know when you get that feeling you are not lone....

[jehovajah]

Just a note to myself. Trochoids and torii are related in some way. Maybe that torus is a 3d trochoid mesh, and this links to helical wraps of torii, maybe they are types of trochoid in 3d. Now i have seen one description of the lie group as the symmetry of <a href="http://www.youtube.com/v/pCDxsBmaXU0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/pCDxsBmaXU0&rel=1&fs=1&hd=1</a> arrange in an orthogonal twisted like pattern so maybe trochoids relate to Lie groups in some fundamental way.

[jehovajah]

Benfords Law, Eudoxus and Archimedes are related i think. We typically do not use the full range of any aggregate number system, preferring to rescale. This kind of hides the infinite fractal nature of things, but also relates to computational boundaries being a scale issue. That means that my PNS and CNS based massively parallel distributed computing system computes surfaces based on scale information. Therefore a smooth surface at one scale becomes a rocky mountainous terrain at a high magnification because the scale information falls within the computational accuracy limits.
In one sense computational error is what i live with and in everyday, so notFS may be a scale issue as well as a sensor issue. Who knows what i am missing because of scale or not having the right sensors?!

It's a bit scary really, cos i am blind and deaf and insensate in so many areas! Or its exciting news- plenty to explore and find out for everyone!

[jehovajah]

I want to extend the conic section curves to include loxodromes and trochoids. The way i would do it is by projecting the poxodromes onto the cone that is enclosing andtangential to the sphere. Similarly i would project the trochoids onto the cone.

All these additional conic section curves would then be the defining set of curved motion for gravity.

Additionally i would project thr trochoids and loxodromes on a torus onto the cone. The torus would again be enclosed and tangential to the cone.

With this thought in mind i would project the curves of a lie group onto a cone and search for matches with the conic section curves. For the lie group to model gravity it would have to match precisely with all the conic curves including the extensions. 

[jehovajah]

Topology, i say, is the analysis of the the properties of the space that inheres within a form. And when i say propeties i mean attributes to that space.

So i attribute a property to a space and maintain that attribute by some means and explore the consequences of that attribution to that space by any specific or general action.

This is as general as i want to get because i feel that the mathematical definitions of topology are way to abstract to be graspable. I have used form property and agent to define topology so that i can locate the subject in reality. One of the reasons why i escaped to spaciometry was so that i could get clear of all the abstractions and words. It seems that mathematical definitions are more about words than anything tangible, whereas i know every mathematical idea comes from a tangible source, but some would want to make out otherwise.

Topology naturally includes any geometry and is an interface between geometries in the abstract and the material arts and sciences.

So a concrete example: an elastic band inheres a space to which i attribute elasticity as a property. I manitain that property by playing with loads of elastic bands and referring to a cultural database on elastic bands, which informs my exploration and apprehension f the property of elasticity.
One of my explorations for example may be what happens to to marks on the elastic band as i act on it by stretching twisting folding. A question arises? what happens when the band breaks? This is of course explored in material sciences and thus the tw studies mutually inform each other.

So topology is not an abstract set of words and notations but an exploration of real spaces.

With this in mind i look at the property of "tangential". In reality this does not exist.

A surface of one form grazing another surface does exist. In practice we deal with tangentiality by using a variable area of contact that can be made arbitrarily small.

In fact it is a variable volume of contact, and the arbitrary measure of closeness has to be suitably defined. The tangent is then defined as the limit of a process that makes this volume tend to zero. Although this is defined as a point in practice it is never achieved as the materials breaks free of each other before this singleton point is ever reached. The reason is quite simple: a singleton point is a fiction that does not exist. The consequence of this is that tangents do not exist as defined normally.

This leads to a point about points in Euclidean geometries. They do not exist. Instead intersection of lines is and always has been used. The introduction of the notion of point into Euclidean geometry is a later addittion to try to axiomatize the system. Since it has no practical geometrical purpose that intersection does nor fulfill, point has not been analysed that much. However i say that it does not bear scrutiny and its legacy in mathematics has always been troublesome.

[jehovajah]

Manipume‘.

That's my first idea for a new name for mathematics based on man and me the indo european roots and manipulate which itself has an interesting etymology.  The idea is :think, manipulate, measure.

I really like the quote below so i have attached it.

[jehovajah]

Rolling a carpet produces radial order from linear order while maintaining linear order.

Rolling a carpet enables a linear sequence to connect to sequences that appear radially in a unique way for each radial sequence.

Rolling a carpet takes a linear, sequenced order of regions and connects them radially to each other in a radial linear sequenced order, but the motion to connect them is  trochoidal and the combined motion of these  spiral trochoids may appear chaotic.

Rolling is one way of ordering chaotic motion.

Rolling a ball of wool produces radial order from linear sequence order in 3d while maintaining linear sequence order.

The motion of the regions in the linear sequence order may define a 3d trochoidal type.

Trochoids are possibly the curves of creation, by which i mean the curves involved in every motion that creates order in equilibrium. They may also be the curves of destruction, that is the motion curves of the breaking up of an ordered system through dis-equilibrium.

Trochoids especially spiral trochoids i think are involved in torus formation, which means in the formation of general forms from bubbles to clouds, and in the formation of plants like trees from root to branch from bud to leaf.

[jehovajah]

Quite generally mathematical proof in the west changed after the introduction of calculus.

The main contributions to this were an introduction of fractions particularly continuous fractions, the argument by Berkely about Newton's Fluxions, the development of the number line concept and the notions of Dedekind, the formalism of the notion of function in russian and european mathematics, Boolean Algebra as a symbolic reasoning tool connecting philisophical argumentation (ie Logic) with a mathematical equation form; Cantors set theoretic work revolutionising the categorisation and taxonomy of just about everything, but particularly the abstract entities of line points and curves of geometry and number in arithmetic.

The Taxonomic movement of the late 17th century onwards lead to a hope of categorising and consolidating everything in neat boxes, which highlighted the consistency of a subject and introduced boundaries of categorisation. This was felt to be epistemologically helpful and academically a justifiable thing to do. Many research grants flowed from this effort and in a very real way it mirrored the Islamic effort centuries earlier to coordinate all knowledge in the empire, which lead to the significant mathematical categorisation of geometry algebra and arithmetic, and in the sciences Alchemy which is the foundation of chemistry.

It was Bertrand Russel who was the principal catalyst for the change in the notion of proof, as a focal point for many mathematicians and philosophers who were calling for greater rigour greater certainty in the foundational notions and methods and theorems of mathematics. David Hilbert is no less a champion for a more rigorous and defensible form of proof in mathematics. His aim was to vindicate the long held belief that mathematics was the only eternal verity among the knowledge of man besides religious formulations. Russel's aim was  to firmly embed mathematics in philosophical logic, as a branch of philosophy.

He embarked on this with his collaborator A. N. Whitehead with great vigour and dedication. using the most exacting tools of logic which we hardly ever use even today.He was inspired by Boolean algebra and boolean logic.

Despite the fact that he failed what he did was to change the course and mindset of academic  mathematicians to this day. He was the Father of increasing abstraction in mathematics, especially notation.

The Hilbertian school of thought managed to reveral that mathematics can never be proved to be consistent of itself, it needs to be embedded within a larger system to prove its consistency within that. However the same thing applies to the larger system and so on. The fractal nature of consistency and proof was thus revealed and the futility of Russel's main goal Highlighted

Mathematics benefits from rigour in its demonstration, but the abstraction cripples it even as it seemingly unites it.

Today it is more reasonable to ally mathematics with computer science, but the real font of mathematics is The Logos Response. This gives risse to both measurement and Language to distinguish the measurements and the framework to perceive and explore a geometry through proprioception, which underlies the manipulations and associations and syntax and procedures of spaeaking , describing and thinking and thus logic. Logic arises out of the proprioceptive geometry of The Logos Response as does mathematics in general, but measurement precedes these as does computational processing.

So one "saw" based on logic 101 made the claim that mathematics being based on all statement "A = A"   was inherently trivial.

Of course this is one falsehood placed on a truism, and is not trivial to explain.

A =A is the formalism of mathematics, but in fact it is not exclusive to maths. It is used every day in criminal justice systems in the sciences apart from calculations, in virtually every description we make using language. It starts off in general as B = A but finishes with A = A. These are not only procedural steps , they are state or stage notation and evaluation of a process, be it an argument, or an investigation, or a journalistic description.
Although this is an equality. it is used to denote an equivalence relation until the equality is accepted as an identity. Identity involves similarity and congruency, and even exactness, which are all subjectively determined by inspection procedures of one form or another.

As usual communication and notation are all subject to personal interpretation, but the marks are out there for personal inspection and by making the same mark using the same notation some measure of coherency may be established. Nevertheless these symbols have more than one use and meaning even in the most rigorous mathematical demonstration. This is why Russel was at pains to demonstrate that his system of notation was necessary to establish even the most "trivial" of definitions 1+1=2.

He showed that maintaining the exact meaning of a notation necessitated a circuitous method of proof, but he justified the ellipsis and subtle shifts of meaning in rigorous formalism as not being detrimental to the conclusion, and therefore rigorous formalism was a safe way to produce certain and logically sound answers.

That being said mathematicians use the formalism as a way of notating a discovery arrived at by different means and in a different order frequently. The procedural formalism thus becomes a test of the veracity or logical consistency of the relations in the notion, and in this sense they test or "prove" in the old fashioned sense the relations and statements with these relations being put forward.

Now for the reader they hold a different function: they are a meditative statement identifying a stage and a stopping point for meditation and reflection. The statement starts as B = A and B begs the Question which is answered by A, This is why it is called the answer. However this is not the full answer because the relationships may not thave been fully demonstrated. Therefore the next = means i have a further answer to the preceding questions. This answer is usually some transformation of form or manipulation of form that upon meditation and reflection reveal some other relation. And thus the presentation proceeds from equality to equality, from equivalence to equivalence.

Occasionally the meditation is halted while some related notion is reminded and brought into play, and then the presentation precedes to its conclusion.

Now this meditation is formalised in statements A = A, but the action involves manipulation of symbols or thoughts or geometrical objects or real physical objects, the inspection of relationships or measurement of quantites, the perception of forms, and relations and contexts and links and equivalences, and the motions of wholes or parts relative to frames of reference. The imagination is excited and the geometry is perceived in a new or more fundamental way.

Of course abstraction hinders this by making the referencing tortuous and in some cases tautological. I hope that by giving maths a new name Manipume‘ and building it from the ground up again that this fun, meditative and playful pastime might recommend itself to more ordinary mathematicians in the real world.

[jehovajah]

Hairy balls and cowlicks!   sounds a bit saucy  i know, but these i think are indicators of the fundamental reference frame of reality, certainly my experiential continuum. Along with the advice from my friends the spiders i have decided that the reference frame i have been seeking is not spiral but radial.

Thus the reference frame has a large set of radials radiating out from any "point" and these radials are orientations from any chosen radial. A particular set of radials is chosen as the special case and these radials ar chosen as straight lines intersecting at a unique region, which i will later define as a point.

Although radials are usually thought of in this way in general a radial set may be any set of curves that intersect in a unique "point" with no intersection at any other "point". This allows each radial to be part of a unique reference for another point.

The limiting set for such radials will be a set of trochoids without the loops, so from a straight line to the cardioid.

I feel like a gambler because these curves are now considered as roulettes

So maybe this is my lucky day and i have spiralled onto the right track! I am betting on Red  .

Sexual innuendo and gambling! Whatever next!

[jehovajah]

Proprioception experiment 3
Warning: mind your fingers!

Its breakfast. so placing it before you shut your eyes. Now carefully find the breakfast and implements and start to feed your mouth,
Notice how you use other exterioceptors to locate where the food is relative to you and your mouth. Notice the visual map of your near surroundings and the additional details invoked by the other exterioceptors. It is almost like "seeing", but this is a generated image and map based on your other sensors and illustrates the density of the mesh of sensors involved in the internal re presentation.

The proprioceptive sensory mesh input can be distinguished by the geometrical grounding of the other sensory information. The kinesthetic whole links the proprioceptors to the exterioceptor input and is then represented as an output in all sensory systems visual, auditory, gustatory and kinesthetic( specific feel and touch systems within the overall kinesthetic).

Because of this kinesthetic coordination of proprioceptive signals and the other sensory signals, dynamic referencing is possible by using one or more of the sensory systems to hold or store a computed output. Thus i can compute an orientation in terms of sensory system output and turn to a new orientation while the process computes the changes necessary in the output system to maintain that orientation. Thus my muscle tension referencing the initial orientation "rotates" and twists around my proprioceptive mesh as i turn. However many of these tension sets i can compute and hold is how sophisticated  can be at spaciometrically analysing my position and orientation in the place around me.

Typically these tension sets are easily disturb and so my orientation is easily confused, but practice makes this skill more reliable.

Hence feeding with your eyes shut  is usually quite safe, but over a prolonged period i find i go out of synch and unless i keep refreshing the map i can go way off course. Hence the warning .

When you open your eyes the kinesthetic map becomes dominated and washed over by the visual signal input. The dominant signal input is isual because of the amount of processing required to produce a visual output, so the other signals interfere with a more dominant or greater amplitude signal and produce a carrier wave effect on that signal.

However each individual has a choice represented by the filters hardwired into the neurology and the selection by preference, habit, or design of the set and arrangement of those filters. These filters emphasise for each individual the main or preferred learning and referencing style. Thus a person may prefer to attend to the weaker but for them more significant proprioceptive signal or the auditory signal etc. Thus attention represents and consists in a choice of signal filters and a style of using those filters and an intention to use those filters.

Spaciometry makes no sense therefore without this set of fundamentals and the intention of the Logos Response : to measure, compare and distinguish.

In case one is thinking of measuring in the logos response as an active agent, this is not the case. Measurement is inate within the system of sensors: that is the purpose of a sensor i to measure and output its measurement as a graduated signal. The sensor mesh itself compares  and passes on a comparison as a distinction. Thus the fact that at the level of a conscious organism i do these things is of no surprise because this is the ineluctable structural function of the sensory system.

[jehovajah]

In my reality spirals and vortices are about rolling and twisting and curving and bending all of which are dynamic spaciometric rotations.

Rotations of what? Well space essentially and what inheres space. Thus carpets and scrolls and rolls and ropes and twists and coils and weaves and knits and knots and motions and matter whatever phase state and fields of motion whatever type.

It is interesting to think of fields of motion such as electromagnetohydrodynamical ones to gravitionucleardynamic ones as rolling and twisting and curving and bending. Many QCD and classical attributed properties of space which are apparently problematical will be explainable in a dynamic explanation of these fields as opposed to an axiomatic stationary one.