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

[jehovajah]

So the geometrical /spaciometric concept of measure in the process of measurement is my foundation for aggregating "values" of various measures and forming a measure-line concept which includes the natural number names plus many more for example π, e. The point here is the measure line concept is a convenient organisation of spaciometric measures which uses numerals in a namespace to identify the measures specifically.

Remembering the fundamental process involves ratios was brought home to me by the following project.

I want o construct a protractor that measures in radians. A simple enough idea as most mathematicians measure in radians not degrees. To do so i came up with the notion of rolling a circle of circumference R around a circle of Radius R and marking off the cardioid points. Clearly this will lead to procession around the circle radius R, but i only want a simple radian measure at this stage.

The construction cannot be done by compass by the way, and the ratio is r : R = 1 : 2π. So i can use some euclidean properties to construct the measurement r given R. I need r to draw the circle to roll around the bigger one.

Firstly i choose an r and draw a circle. I roll the disc carefully to avoid slippage to mark out R. Job done. I construct the circle using this new measured portion.

But what is this measurement 2π ? By constructing a equilateral triangle base R and marking of the distance r we have a ratio comparison equivalent to division ( it is the geometrical basis of division). From this i can see that 2π is slightly more than 6*r but a good approximation is given by 3 : 19. I obtain this ratio by carefully "dividing" the small difference, and marking it off carefully on the other side of the triangle.

This process of approximation of the measure for the ratio can be repeated for further accuracy, thus highlighting the measure as the value of the division process, not a number!

Of course the numeral namespace has been constructed to be able to name these measurements.

Thus i think the measure line concept along with ratios provides a foundational basis for ordering and evaluating the world and with a namespace we can describe that evaluation.

In addition the measure line is a flexible bur inelastic measure and so can measure curves, depending on the degrees of freedom given between the fractal regional boundaries as the iterations are increased. Thus a tailor's tape is an adequate measure line for most discussions.

[jehovajah]

When I read this thread a lot of idears come to my minds. So reading and understanding nothing makes me very creative.
To think about the fundations of mathematics a first question comes to my mind:

Are this to numbers equal?
1 = 1

This should be less a question than a starting point of an essay I have in mind and would like to write down.
I think I can not do this in a short time. But I can come back later and can change this post.
For this I have to write down my understanding of a matched filter in digital signal processing.
Then go to artifical neural systems and then one has to understand the behavior of real neural systems and our knowledge of the human brain.

Big a program. I will surly fail cause of the lack of time and energy.

But I can post some reverences to books I like:

The first one I used at University. It gave me a deep view in digital signal processing.
For me this book was a starting point to develop my idears in a scientific context.
Digital Filtering and Signal Processing
from Donald Childers and Allen Durling
the book is from 1975 an I think it is no longer available from the book shops.
It is more my personal begining of digital signal processing and not a starting point for learning digital signal processing in the year 2010.

A great inspiration for me was the reading of the Chapter 3 the design of digital filters.
I was very impressed by a recursive digital filter.
Where the output of the filter was again feed back to the input.
Building an infinit loop that can produce infinit patterns.
(May be equivalent to Stephen Wolframs rule 30 when setting the parameters right. (So I have discovered it first!))

The next book is:
Introduction to "Artificial Neural Systems" by
Jacek M.Zurada
This book lead my idears further in the direction of how information processing and system control can be done in biological systems.

The last book is:
The Priniples of Neural Science
from
Eric R.Kandel
James H.Schwartz
Thomas M.Jessell

I bought this book because I was very much impressed by the lecture that Prof. Kandel gave in the Iconic Turn lecture serial at the LMU in Munic.
=6&cHash=98c6b2bafc]http://lectures.iconic-turn.de/iconicturn/programm/video/?tx_aicommhbslectures_pi1[showUid]=6&cHash=98c6b2bafc

I have not read the book complete but I am always impressed how recursive processes are implemented in biological systems when only looking at the pictures.

Cool! May the muse inspire you!

[jehovajah]

So i fell to wondering how the proto-Akkadians as early Sumerians and Babylonians were able to divide the circle into 360.


I remember Euclidean constructions and mistakenly thought that this was the sum of their technique back then. Thus the issue was trisecting and quntisecting an angle. This i thought must have been easy. In fact it is not that hard for a practical person. This is when i found out about Galois theory and the greek "game" or agonia of trying to do this with an unmarked ruler and a pair of compasses!

so i fell under that spell for a while and diverted much of my thought and time to impossible pursuits! Why would you?

This is precisely where Geometry and thus a whole swathe of mathematics becomes obscure and agonising! The greek love of the Agonia!

Like all things the game is a diversion that produces amazing and interesting responses, but ultimately is not a model of "reality", by which i mean many of us have had our world view twisted by this type of practice and constraint. It is also why i decided to coin the term spaciometry to reveal in myself these imposed twists by adopting a different viewpoint.

So enjoy the pursuit as much as anything else but the fundamental connection does not require one to undergo "agonia", or any other "initiation" rite into the secret society of the pythagoreans etc.

Now this is out in the open, my appreciation of the vortex and its surface manifestation as a spiral has been enhanced, and the contention that the circle is a special spiral has an increased confidence value. The fundamental motions in the spaciometric motion field are spaciometric rotation and spaciometric extension. These mean that the fundamental form is the vortex/spiral (torus-vortex in 3d). If i have a condition that fixes the extension to a constant then the special form of a spiral circle or sphere is the result. Thus the extension quantity and its ratio to the rotation quantity determine all forms from the so called singularity to the infinite "straight" edge.

The proto-Akkadians are linked by trade and via the Dravidians to the Chinese ancestors. There is also a land bridge between them admittedly over difficult terrain. The Chinese have a construction of the i Ching based on 24 divisions of the 6 concentric circles. This tool enables the practical division of any circle into 360 parts, by qunitisection, and bisection and trisection.

[jehovajah]

Lai Zhide is credited with introducing the Taijitsu called the Yin Yang, and in particular contributing to Chinese "analogical thinking" or philosophy . One must not get hung up on the apparent absurdity of Chinese astrology, as it is in fact an ancient and well respected body of knowledge which is common in the west and indeed Astrology  gave birth to western Astronomy and Cosmology!

It is a thing of note that the Babylonian magi began to demonstrate philosophical thinking, dealing with abstractions and principles in their cuneiform tablet records early in our common history. Chinese Astrology is nothing if it is not an exact example of this ancient "scientific" analysis of the cosmos.

Analogical thinking requires a form to make the analogy with. In chinese this is called the Yi. Thus the five elements are just 5 Yi by which chinese philosophy assembled the chemical and physical properties of the cosmos. They represent in one analogy a kind of periodic table, and in another a type of electromagneto hydrodynamic spectrum. The simplicity of analogical thinking is that it can be ratioed to just about anything.

In this current western absolutist, abstract phase we find it hard to conceive of the usefulness of this conception in terms of dealing with information overload. In terms of creativity, it is not at all restrictive or stifling, nor does it preclude innovation. Chinese scientists are as objective or subjective as i and i require an organising principle or set of principles to store knowledge!

Lai Zhide i think would have been at home with Carl Sagan, Richard Feynman even Einstein and Newton in his philosophical reduction of the cosmos to the i ching and his consumate Yi-ology.

[jehovajah]

There is a recurring relationship between spirals vortices and the measure 3. this in fact could be π 0r e but this certainly has a spaciometic form which is irregular angular and cone or pyramid-like. This has a bearing on close packing in the sense that a spiral based around a 3 measure centre may provide the best bundling spaciometrically. Certainly i notice the measure 3 in a lot of natural plant forms and structures.

[jehovajah]

I thought it might be worth exploring the notion of extension spaciometrically, as the notion of straight and right are cultural paradigms, and somewhat idealised. I will consider this more fully after i have completed a survey of kinesis and kinesthesia with its subfield proprioception.

The Logos Response is informed and grounded in my sensor systems and processing, and thus extension is not as simple as it is made out to be.

I rather suspect that spiral motion will be he outcome and that orthogonality will be important along with orientation and feedforward feedback cybernetic systems.

It is of interest that orthogonality is a distinct sensor system of itself, irrespective of any spaciometric definition. Orthogonality founds nd contributes to spaciometry, and in a very real sense spaciometry could not exist without it.

By the way i feel a clear distinction between humans and other animates is spaciometry as an axiomatic system. Other animates i am sure have spaciometries and even codify and pass them on to the next generation, but it seems only humans seek principles on the basis of analogy, and reconstruct there experiential continuum accordingly.

That is not to say that other animates do not develop erroneous concepts of the environment, cos clearly they do, but there information and processing is empirical not axiomatic. I guess in just about everything else we are similar including inductive and deductive reasoning!

The notion of relative ratioed radial expansion suggests itself as the fundamental motion with relative ratioed radial extension being the specific or abstracted orientation. Thus extension and orientation are the exact same notion. The ratio is specific to the form and or region of focus, but the relativity relates to the quantifying of expansion by ratioed extension in a standard orientation.

If the radius is special, that is inelastic it produces a circular expansion and the notion of perfect or universal applicability of the measure: a universal metric. However if the radius is "elastic" or deformable in some way then there is no universal metric and all measurement of extension is local and dependent on orientation. This would be the case if a spiral reference framework is utilised.

Vorticular motion fields may indicate that the underlying metric is based on a ratioed radial expansion that is "elastic" relative to orientation.
Dilative rotation

[jehovajah]

Polar coordinates, spherical coordinates are the most natural relativistic reference framework currently in use, but it is still made out to be special, whereas it is due to historical primacy that Cartesian coordinates are still presented first. The weight of mathematical description is still presented as Cartesian, thus Special Relativity seems inaccessible and strange, and the "complex plane" can pass as a number system rather than the coordinate system which it is.

Spiral coordinates will be a truely strange system, but one i am laying the groundwork to explore.

[jehovajah]

Plato was one of the first to discuss the problems of perspective. "Thus (through perspective) every sort of confusion is revealed within us; and this is that weakness of the human mind on which the art of conjuring and of deceiving by light and shadow and other ingenious devices imposes, having an effect upon us like magic... And the arts of measuring and numbering and weighing come to the rescue of the human understanding—there is the beauty of them—and the apparent greater or less, or more or heavier, no longer have the mastery over us, but give way before calculation and measure and weight?"[15]

from Perspective (graphical)
From Wikipedia, the free encyclopedia

To which i add the art of languaging, without which the full context of the other three is missed and the logos response- the calculation-takes abnormal preeminence.

By the way only the ascetics among humans describe the human mind in these pejorative terms: a confused "mind" is no weaker or stronger than a seemingly understanding "mind" it is merely confused. The actions proceeding from a confused "mind" may appear weak and ineffectual but appearances can be deceiving, as outomes determine the relevant value of states of mind. And in any case many confused 'minds" have taken decisive and bold action without understanding, just as understanding "minds" have taken weak and indecisive action.

[jehovajah]

We have no real sense of "time" that is not spiral, thus our modern abstract notions of "Time" are simply deceptive, hiding the creeping advance of precessional motion and indeed all vorticular motion in the void. Instead of shadows we replace our connection to our universe with something more insubstantial: the notion of time.

[jehovajah]

The eyes have it for more go here

The important point is  it is not the technology it is the cybernetic system, based on iterative processing called here convolution.

[M Benesi]

It's nice you're keeping this thread going.  It's enjoyable to wander among your thoughts occasionally. 

[jehovajah]

It seems that projective geometry, in particular mapping projections like mercator et al may be a way to enscribe vorticular spiral wraps onto any object, thus providing a transform from its usual geometrical representation to a sonic wrap. This spaciometric transform means thqat in essence i would be able to represent all objects by these wraps and deal with the motion laws for these wraps for a theory of everything.

[jehovajah]

Somewhere nice and spaciometric to play

[jehovajah]

It strikes me this morning that the measuring instrument that solves all my problems is the pedometer. No not all! my problems,just the curious reversal tht has taken place between the spiral and the circle.

To whit, using a pedometer where the circumference of the wheel is the unit of length allow everything to be measured in radians. In particular every curve from a straight line to a circle can be measured simply and easily by pedometer. Screw the greek straight rule without any marks i say! .

So then the straight line and the circle become two special bounding curves for every other curve, in that every other curve is spiral and tending toward a straight line or a circle, no matter how many twists and turns or knots and winds it gets itself into.
 I could go further and define each curve as a unique spaciometry, to be explored, and thus categorize spaciometries by the fundamental curve used in their constructions. Thus a circular spaciometry and a straight line spaciometry would be 2 special spaciometries, and more general spaciometries would be spiral.

Releasing the measurement method from the spaciometry, or rather restricting it to the spaciometry that can accommodate all others allows me to apply one single fractal pattern to all measurements and inherently simplifies things.

This does not require a new aggregation base as it is a scalar of the old . so it is not (mod π^n) based but π*mod(10^n).

In practice then if i measure a length 3π it is up to me to specify the curve that i am measuring along. The curve itself informs me whether it is a rotation or a straight line or a combination. Angle measure becomes plainly a tool utilised to describe an aspect of a form and may be generalised by the measuring curve used in the tool. thus angle measure does not have to be exclusively a circle anymore, but it does have to be a specified and constructible boundary curve element.

[jehovajah]

using the pedometer measure it is straightforward to mark off a radian: it is the distance the pedometer rolls equal to a radius of the pedometer. So using my pizza cutter for measuring radians is quite fun. It is clear that if you roll along a straight line that the distance in radians is simply how many radii divide the length in normal units, but for a curve the distance in radians has to be read off from the pedometer. This makes it clear that for a curve a judgement of the lIne is taking place, and in fact in general the curve will be measured by an approximation to its shape.

This makes every curve measure potentially a bezier polynomial, and polynomials in general the family of curves for approximating a general curve and a general curve length. Therefore polynomial interpolation and extrapolation remain the fundamental basis for measurement, and the nature of calculus is preserved and revealed as polynomial approximation to curve shape, curve length and curvature itself, based on a spaciometric notion of dividing off a standard curve of best fit for each division of the focus curve.

In a straightforward approach this leads to the binomial series as newton discovered, but in general to the Taylor series and the special case the maclaurin series.

As much as these are presented as results of differentiation they are no different to dividing off the curve and seeing which polynomial best fits. The use of the binomial theorem and the binomial coefficients is one of those wonderful things that occur in investigating the set notFS,the universal application of an algorithm.

The recipe for finding the best fit is the same as the recipe for organizing patterns of medic rhyme and the organizing of finite sets of things. So if we want to choose from the possible permutations of a finite or ordered set it is not unreasonable to try and organise it by the binomial expansion. What is sup rising is how well it works apparently.

I say apparently because it is an algorithm, which essentially means an iterative set of instructions on how to tackle the problem in general. A valid algorithm will produce valid results. Two valid algorithms may not produce the same looking results by form but by a common measure the assigned value usually converge.

However sometimes the results of valid algorithms diverge which begs an interesting question? How is validity assigned?

The exploration of this strikes at the common notion of proof, and in fact in general I adopt an evidential basis for a theory not a so called "proof" basis. Falsifiability as Karl popper states is in fact the only occurs razor we have. I use it all the time to review my ideas and welcome a falsification by evidence as a new avenue of exploration and play