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

[jehovajah]

So this morning it becomes clear that the tools and customs we use are so embedded as to make it seem fanciful to use any others. However as far as rotation is concerned i may use any specified tool . Thus degrees are merely a fractal pattern marked on a circular tool(protractor), and radians are a different fractal pattern marked usually on a unit circle. The measure line concept is a fractal pattern marked on a line (rule) and the line  is said to be "straight", but this is an undefined concept. However we all agree more or less what "straight" is, but again this is a cultural definition. Some attempts to tautologically define straight are usually made .

Thus it is clear that to be rigorous one needs to define the fractal pattern and the "curve" used in the measuring tool, and the application of the tool for its measuring purpose. This in essence is a simple algebra, or rather the axioms of the algebra of the measuring tool. Changing the measuring tool changes the algebra, but in most cases only by a scale factor. the measuring tool can be exotic in curve and shape, so a spiral form may seem a little tame after a while!

Weights and "forces" represent a different measure principle, as a fractal pattern is used only after the motion has been transduced to a form that can be measured by a fractal "Scale". This scale is arrived at by dimensioning the motion and then parameterising the dimension. A parameter that has a tractable or useable uniformity is the preferred choice for a scale, but again this is a cultural decision. The transducing of the motion by dimensioning and parameterisation is the attributing of measurable properties to the motion, and the establishing of a relativistic reference frame. Relativistic motion and relativistic motion transfer follow from this transduction, but independent of the reference frame we observe motion and motion interactions.

It is to be noted that dimensioning is governed by the measuring tool. Thus we dimension on a cartesion coordinate basis because we tend to mark the boundary of objects by straight lines. With a minimum of 3 distinctions we can record the boundary of an object if it is regular. Irregular objects require many more dimensions. The number of dimensions also varies with the meassuring tool. A tool that fits the form typically requires fewer dimensions.

[jehovajah]

Spiders are a friendly folk imparting wisdom as they go, spinning yarns of interminable length and spiraling in the wind.

So i noticed hat a spider does not build a spiral on circular elements, but on triangular and radiating elements.

The spider seems to typically lay down or attach a main anchor line. This is so fine and uner tension so it is the nearest thing to a straight line unaffected by gravity, and interacting with the aerodynamic forces that counteract any gravitational ones. The spider then drops at least 2 more lines to provide a stable platform to anchor the radiating spokes of its web. Thus it uses radiating lines a the basis of its spiral . Radiating lines are interesting.

I probably first came across radiating lines as a pattern due to sunlight passing through my eye lashes, and then spokes in all sorts of situations from basketwork, sowing patterns , wheel spokes, spider webs.
Radiation is an attribute of straight lines originating from a single point and spaciometrically rotating a boundary. It is a structure that has spaciometric mass and spaciometric density which varies with relative radial expansion from the origin.  

Spiders typically lay out between 28 qnd 34 of these radiating lines before spiralling. Thus a circle does not seem to preoccupy a spider's mind, simply a centre point from which to radiate to any given boundary, preferably stable.

Radial expansion then is a spaciometric concept based on spaciometric orientation and spaciometric rotation of a boundary from a point of origin or "centre". This is a concept built on iterative processes, and my perception of radiating lines is an iterative process.

Relative radial expansion therefore is a process that describes the general radial motion in motion space and spaciometric rotation is a process that describes the general motion in the same space that is not solely radial.

Spaciometrically i would want to reduce motion to these 2  fundamental types: radial and rotational. All motion are then some combination of these fundamentals.

I suspect my spider friends have more to teach me about spirals, especially spiral wraps.

[jehovajah]

As i have recently added to the Axioms, it occurs that relativistic motion attributes of solidity, liquidity and gaseousness relate to the density and vigorousness of spiral motion and the direction of spin. Thus a hot gas has a higher spaciometric density of helical vortices all spinning in the same way which would tend to make hem energetically bounce off each other and spread apart in all directions. A cooler boundary condition would tend to absorb this spin and start a counterclockwise spin. This would tend to keep the helical vortex motions fluidly linked. Finally a cold region would tend to have very slow rotation and therefore the regions would tend to stay together in viscous lattice forms, with slow thermodynamically generated spins. Nevertheless the electromagneto hydrodynamic spin would then take precedence giving rise to electro mechanical phase attributes.

[jehovajah]

Often the philosophy of the scientist is squeezed out of the description, but it is all important at the level of observation. What one thinks determines what one perceives

[jehovajah]

I am thinking about the foundations of mathematics and what i have explored so far. So I feel i will move toward a tidying up and consolidation of where i have got to soon. Fractions i feel are the only outstanding thing to explore and Abstract Geometry.

I am not surprised to find that mathematics is not solely a human animal affair, and that aspects of it exist from viral through microbial to zoological forms of life. The simple description is everything is mathematics and nothing is mathematical!

I was privileged to see my spider friends constructing their spirals in the early morning sunlight!

As Geometers go, they ain't half bad! They measure, they locate, they mark and they construct, and they use their whole body to do all of this because it is a survival technique that engages them in the panoply of life.

Even the spider shows the fundamental importance of radials and rotations. If i have observed anything of significance in this exploration it is the Logos Response and that the fundamental reference framework is Radials, and the fundamental motion is Rotation and Radial Expansion and the fundamental form is Vortex-Torus, and this is fundamentally because their is no constant but Constant Change.

That's life, as they say, and mathematics has been an organisms response to life from inception, and will continue to grow and develop with organisms as they respond to constant change. Mathematics therefore is inate and intrinsically simple and intuitive, and anyone or anything that makes it not so is perhaps not really a mathematician.

In fact, get rid! I do not want mathematicians anymore i want craftsmen, artists, artisans, builders, potters, weavers, jewelers beaders, shapers who have a deep love of what they do and who study and play with what they love to make it more utilitarian, more enjoyable for themselves. That is where mathematics is born, nurtured and thrives.

[jehovajah]

The i ching algorithm and the Lai Zhide Taijitu or yin Yang symbol are a result of a common principle which applies to rotating reference frames. Essentially Radials and rotation will produce similarity in form    irrespective of the topic: mathematics, science art astronomy astrology....

[jehovajah]

Brownian motion and Bachelier theory applied to stock markets get heat from Benoit! Very stochastic!

<a href="http://www.youtube.com/v/vxbxXBrOPS8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/vxbxXBrOPS8&rel=1&fs=1&hd=1</a>

What interests me is the non linear time scale. Very iterative,non intuitive and fractal self similarity.

[jehovajah]

Fractions and fractals  before breakfast! 2 impossible things, eh?

[jehovajah]

Some foundational words and where they come from.
Thank 'h¶h for the Brahmins! Fractions would be a piece of cake otherwise!
Romans of course weighed out their portions. Give them an uncia and they would take a Roman Legion!

So lets throw in some polynomial thinking and see what we get. Continuous fractions!

Now we can see the Egyptians in a new light, with some nice hieroglyphs.

Oh the Humanity! Oh the irrationality of it all! The magnitude of these ratios and incommensurables is a significant clue to the measure line concept.

By this means of education and cross fertilisation and promulgation of conceptions Fractions morphed from portions, and ratios into magnitudes and numbers: and i propose that number morph to the concept of measure as a spaciometric entity of dimension, with wholeness linked to boundary and fractal pattern self similarity.

Fractions are "always" presented bereft of their ratio origins. And even as Rational numbers the ratio geometrical thinking of the earlier mathematicians is downplayed. Ratios and there cognates proportion and portion are crucial to an intuitive and spaciometric understanding of measure and quantity and rank and enumeration etc. The notion of Tensor is accessible through these fundamental notions of ratio and proportion and portion. So good on yer! Eudoxus, and any other ancient mathematicians who valued proportion. And Archimedian properties are attributable to Eudoxus it seems. Archimedes always was a good student!

Enjoy the reading. It's Intense!

[jehovajah]

The google sketchup tool/ app is a sufficiently complex and sophisticated Representational system for me to use it as an instance of what i mean in part by spaciometry.

Despite its complexity it is intuitive and easy to learn and engaging.

With a tool like this i can re-explain notions of parameter, dimension, radials and spaciometric rotation, spaciometric mass, density and surface form and boundarisation. I can also emphasise the tool nature of many fundamental geometrical "givens" and the abstract nature of Euclidian, Greek Geometry. I can illustrate Riemannian geometry and how restrictive Euclidean geometry is compared to it, but how even Riemannian geometry has to give way to a more general spaciometry.

This fundamental tool is not new, but new use can be made of it in redefining the foundations of mathematics.

[jehovajah]

   Mathematics derives from the french mathematique which derives back to the indo European roots through Latin and Greek and the Sanskrit: man.

So in every sense mathematics is essential to the defining of man or what it means to be a human. This does not imply arrogance ut rather a time when animals became self aware enough to describe there learning and thinking ability as "man". From this one distinction flows every human notion of difference to other animals.

However the Root of interest for the Logos Response is: me. Now this distinction is what our CNS and PNS do prior to and during man (that is learning and thinking) and that is MEAsuring and comparing.
Thus me and man are possibly the 2 oldest "human" notions, but are not unique to humans as all animals and even plants and fungi do this. It is a fundamental response of life (living things).

From this root analysis no human is excluded from mathematics as it is part of ones definition of being human!  And every human not only thinks and learns but also measures and compares. Thus the problem is the distinctions that are drawn, or posited and adopted. These are what set up and establish barriers as well as boundaries between the myriad human activities that would otherwise qualify to be called mathematical!
So be proud and loud you craftsmen and craftswomen, for everything your hand produces from your mind is mathematical through and through, especially if done with love:

Mind \Mind\ (m[imac]nd), n. [AS. mynd, gemynd; akin to OHG.
   minna memory, love, G. minne love, Dan. minde mind, memory,
   remembrance, consent, vote, Sw. minne memory, Icel. minni,
   Goth. gamunds, L. mens, mentis, mind, Gr. me`nos, Skr. manas
   mind, man to think. [root]104, 278. Cf. Comment, Man,
   Mean, v., 3d Mental, Mignonette, Minion, Mnemonic,
   Money.]

From Free dictionary.

From this it is but a short step to........play!

The establishment of distinctions is not only a language response, but also drives communication, explanation, education and differentiation. It is a powerful aspect of the Logos Response which accounts for many communication motivations and road blocks.

However just as thinking and learning measuring and comparing are at the heart of being human, so my friends is doing math, or rather playing with the things around one and thinking and learning about what that reveals.

[jehovajah]

Given the derivation roots man and me can anyone suggest a more friendly name for mathematics?

<a href="http://www.youtube.com/v/BQRIIeMwfR0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/BQRIIeMwfR0&rel=1&fs=1&hd=1</a>

On the derivation of measure, and play.
I just found it.

[jehovajah]

Experiment in proprioception 1
Slowly, very slowly turn, or twist with eyes closed.

There is a quantum size to sensors action potential. So i know i am moving by 1 intention, 2 lateral differences between proprioceptors, 3 mesh of proprioceptors self referencing, 4 exterioceptors (open eyes) audiometric and gustatory differentials 5, memory comparisons that is  perception comparisons.

Each one in this list requires an increased potential to activate.

[lycium]

Experiment in proprioception 1
Slowly, very slowly turn, or twist with eyes closed.

There is a quantum size to sensors action potential. So i know i am moving by 1 intention, 2 lateral differences between proprioceptors, 3 mesh of proprioceptors self referencing, 4 exterioceptors (open eyes) audiometric and gustatory differentials 5, memory comparisons that is  perception comparisons.

Each one in this list requires an increased potential to activate.

can i get some of what you're smoking? 

vorticose pulsations of extravagance!

[jehovajah]

When i generalise the notion of a coordinate system to have n ordinates/ axis not necessarily mutually orthogonal, which in any case is a non sequitur beyond 3, i find myself tracing a path through all the ordinates ending on the point i am referencing. Thus it seems to me that a generalisation of a coordinate bracket would be a path bracket, based on a sequence of visiting the ordinates.


Thus it seems fair to think of a peano curve as the basis of a sequence for visiting each ordinate on an extremely dense ordinate system.