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

[jehovajah]

The Logos Response compares and distinguishes Ratios, but the ratios are all related to one another, so focus is necessary to concentrate the attention on a particular region. But this means that the region focused on is in a ratio with the whole ratio data set from the logos response. Thus the ratios i may divine from a region of attention are in fact ratios within a ratio itself.

The Golden ratio is perhaps a formal recognition of this fractal state of affairs.

[jehovajah]

The Auricle has other structures which function in the logos response for vertebrates; resonance chambers, pressure amplifiers through lever actions,pressure equalisers and head tilt sensors,gyroscopic motion sensors, amplitude and frequency sensors.

Although i began my exploration of the sensory system with the eyes and discovered the logos response in that system, it i clear to me that the auricle system is fundamental in a way the visual is not. The sense of rotation lies in the auricle system and the sense of orientation lies in this system also at least the sense of head orientation vis a vis( or rather ratioed against) the proprioceptive sense of orientation. The auricle and kinesthetic senses of orientation are more fundamental than the visual reference of that orientation.

Thus in the auricle and propriceptive systems i find the fundamental notions of orientation and rotational movement relative to the organisms form and structure. If the visual sensors are combined i find that rotational movement and orientation can be referenced against a visual map which includes a map of the organisms form and structure.

Thus i note the systems fundamentally provide information /ratios about orientation and rotational movement  Where then does Extensional movement reside and the notion of extensional direction (from which and upon which we base the notion of Axis) ?

This is of interest to me because despite seeming fundamental Axis is not at all represented in the sensory system, and therefore seems to arise from the processing algorithms that form the basis of extension.

As far as i can tell the notion of extension arises because of the interference patterns produced by the binaural, stereoscopic, dual gustatory and rich proprioceptive web-maps of the inter-reacting sensory systems. The point here is that extension is a computational output that is at a different computational level to rotation and orientation. Thus extension will have more computational artefacts than orientation and rotational movement.

What this all means is that "Perspective" (as a relevant example is a computational effect) and the vanishing point are a computational effect. Of course the structural form of the eye contributes to this, and the ratio of "interference  pattern source" signal processing also contributes. Thus if i focus attention on the interference pattern at a particular region in the visual data/ratio streams the perspective and the detail and even the image size output changes to reflect that. The output to memory thus can significantly differ from the raw signal output from the sensors themselves.   However it is unlikely that the raw signal output will be interpretable without the processing that occurs to "make sense" of it so to put my point another way: the data that is currently being processed in your processor to produce this screen you are currently looking at will be unintelligible without the visual data processing algorithms that re-translate them to a visual image.
The perspectives in this visual image are not reflected in the raw data that is being processed as it may take only a few bits of machine code to describe the colour and area of the screen but masses of data points to modulate each pixel to give the correct amplitude and relative colour and persistence, etc. And of that vast ocean of data only the selected part is output to screen, and only in the window assigned to it, at the resolution assigned to it . Thus these very words are the focus of your attention, but the processor and arrays have present on the whole screen more information than you are actually focused on right now. Your perspective of he screen has thus been altered, so that your experience highlights these words and their references and not the physical display.

Their is no vanishing point on the physical display so every selected  thing that is currently being processed is flat in front of you, but by focusing you have made various parts of the screen vanish from attention ( this is now called attentional blindness). But what if the algorithms put a vanishing point on screen to reflect the computational ratio of the data being processed? Thus the processes with smaller cpu clock cycles  would be represented on the screen with smaller windows, and the larger cpu clock cycles with larger windows. The resulting quilt map would mono-scopically reference a ratio field. If these windows were arranged on a spiral we would observe a natural spiral vanishing point that would immediately make visual sense to us.

The point then is that the notion of extension is as much a computational effect as a "real" spatial arrangement in notFS, and the notion of Axis derives from this computational output. The cone effect of the vanishing point is likely to be a computational artefact of the arrangement of processing loads on the CNS distributive parallel computational system.

Thus despite the seeming fundamental nature of axis it is not as important as relativity, and the relational arrangement of certain structures within the organism are sufficient for rotational movement and orientation to make sense.

Now orientation as a spaciometric attribute relies, as it must, on an individual organism defining it and demonstrating it. It is a real thing only in this sense, and makes no sense as a line on a piece of paper.

However, if the defining organism can spaciometrically reference these marks and lines  they may have a significance as an orientation or set of orientations to another organism. You may have heard of the dance of the bumble bee? This is such a non human example of the general system i refer to, and exemplifies the basis underlying our graphical representation of orientation. When extension data/ ratios are combined with that orientation definition we have the basis for the notion of Axis.

It is worth noting that in none of this has it been necessary to define a surface called a plane. A plane therefore is even more of a computational artefact than an axisIf i spaciometrically rotate and keep the relative ratios within my organism the same i might visually reference a surface which i can define as a plane, but equally i might reference a surface which is a cone, or edges or surfaces of any form that encloses me. What i will not reference obviously is the spiral or vorticular nature of all my observations, in all sensory systems.

[jehovajah]

This is a 2d notion of a 3d DNA wrap.

[bib]

Hi jeovajah,

I don't mean to disturb you and intervene in your blog (which is a little bit too complex for me I must admit). Let me know if it's OK or I will remove this post. I'm not a mathematician, and not a biologist, but I just wanted to share some thoughts here, in case you find them interesting.

When I see your comment about the DNA I can't help trying to find similarities between fractals and life. First I find it completely amazing that DNA is the unique genetic coding system for ALL species on Earth. Has it won the competition several billions years ago against other codes? If not, would it be the same code if we eventually find life on another planet? I think these are questions no one can answer (yet?)

What is even more amazing with DNA is that it was the first molecule complex enough to have this incredible power of self-replication (due to a very simple chemical concept : the fact that the ATCG bases work in pair). That's what we call "life", but it's just a step in the chemical complexity of our universe, it's not magical nor supernatural.

And I think we could draw a parallel between this concept of self-replication and the notion of iterations, so of fractals. Each time a cell splits in 2, it's just like an iteration. So all living creature are built thanks to an iteration process. So they're really fractal in essence, meaning that all the "program" to build them is included in the "seed", the DNA, and if you take any living cell (like if you zoom in a fractal), you could potentially extract its DNA and rebuild the full creature with it (like you can see minibrots everywhere in the Mandelbrot set).

Iterative process, self-similarity, "seed containing the program" (=fractal formula) : are there any other conditions to verify in order to validate that life is fractal ??

My 2 cents...

cheers
bib

[jehovajah]

Hey Bib , Welcome!

I thought some evil genius had stuck a sign on my Thread saying "mathematicians only" ! No way man! It is a thread after all and not a blog 

So i welcome everyone .

First let me apologise for my complexity. Although i accept the criticism/comment  i cannot promise to change much as the thoughts come to me as is, literally. I am more interested in capturing the thought than in making it intelligible to the reader, including myself!

Anyway this is fun for me and a reference notepad of insights. Anybody can contribute , challenge correct, question (what happened to the alliteration?) , but i hope someone will want to collaborate. I do not have the answers and i do not intend to judge others contributions. You know the pedagogues of my day called teachers of math did that to so many that they have become timid and bereft of their mathematical heritage. Its a power trip man. Pure and simple! And while i am on a roll i have a beef with Gauss for his treatment of Riemann!
  Anyway now i got that off my chest i feel a lot better!

I think the more i learn about Feynman and Dirac the more i see their Autism, but Feynman was able to use his Aspberger"s in a field where it was valued and he could mature in a positive framework. Dirac had to overcome much more difficult circumstances and a different level of Autism. So now what about Stephen Hawkings?

Mathematical physicists from Newton to Feynman are my favourite maths teachers, but of course they can be self righteous too, a trait i think to be avoided.

So welcome, Welcome Welcome ! All are Welcome.

[jehovajah]

I want to post a note about the iterative structure of the rod and cone sensors and how they are formed for their function and provide stepwise variation in signal output which is akin to digital sampling. Thus the notion of analogue and digital evaporates as a biological sensor can be shown to demonstrate digital sampling like all of the camera sensors which are in use nowadays. Plus how a drop of oil provides an interferometer /diffraction system that isolates "colour" frequencies and enables a colour signal output from a cone, while a rod provides a contrast ratio  signal output.

It's all iteration, its all biological and it's all there, man!

The rod cell
  .The cone cell

Image from The Internet Encyclopedia of Science.

The rod and the cone cell develop  the rod and the cone from a  cilium. This hair like structure is associated with the mitochondrial cell elements and are usually spiral or helical in form and function. Thus a cylindrical helix and a conical helix form are the precise descriptions of these cell segments.

The helical form enwraps regions which are here called flat and parallel unit membranes but which in fact are helical "slices". These slices contain the distribution of the photosensitive pigment; in the cone they contain 3 pigment types and in the helix just one.

This spiral arrangement of pigment provides the analogue to digital sampler framework. The heliical forms both the cone and the helix act as collectors of light signals (photons) and the light enters into them in the human eye from the base of the image. Immediately the role of the oil drop in the cone becomes apparrent as does the cone shape. The oil diffracts the light into its constituent colours and refracts it to different  parts of the cone. Thus each part of the cone registers different colours and has a different photosensitive pigment concentration. It also digitises the analogue signal to that part of the cone.

So the helical sensor due to its size and shape is capable of measuring intensity of light to a very fine contrast ratio and the analogue signal is digitised by the amount of pigment that gets changed increasing the electron transfer to the helical wrap spaciometrically. Thus the duration for a helical element to transmit its charge to the mitochondrial elements in the rod cell is also measured via the "spiral length".

Intense lights therefore convert more pigment and take longer to fade the signal output. The mitochondrial capacitors are slower at discharging because there is more of them, but the ones in the conical helix structure discharge more quickly because there is less of them. Consequently the conical system is faster and able to deal with finer detail as well as colour but not as sensitive to intensity or contrast as the helical structure because the "raw light" is impinging on the helical unit layers while the diffracted and thereby diminished light is impinging on only a part of a conical unit layer.

The complex functioning that produces The Logos Response i have discussed elsewhere, but suffice it to say that despite the sophistication of the sensor it is down to the processing and wiring as to how the signal is managed; and at every stage we can identify a reduction in informaton content as the signal is transduced to make its way to the CNS.

Thus whether we liike it or not the setFS is only a poor model of the set notFS, and human beings who claim extra perception may in fact be able to demonstrate this in their wiring diagrams, processing algorithms and perception schemas, to name a few.

My joy however is to point out the presence of spiral and vorticular structures in the biological solution to signal processing and to suggest that these forms may be useful in modern electronics.

[jehovajah]

"Imagine as a/ the disc starts with 1/4 of the area of the circle in the box; a 1/4 turn brings in 1/2 the circle area, a 1/2 turn brings the area up to 3/4 and a 3/4 turn brings in the whole area and the final turn brings in another 1/4 area. This manoeuvre actually covers twice the area of a circle as Theodorus realised, thus the area of a circle was  1/2*r*c"

This maneuver is not unique to the circle, but in fact relates the perimeter of a convex 2d shape to its area. By finding the geometric centre of a shape and measuring the perpendicular distance to one of it sides that gives the height of the rectangle measure the shape will rotate into along its perimeter. The area of the rectangle is twice the area of the shape.

[hermann]

Hallo Jehovajah,

thanks for all the commends on my work here in this forum.
When I read this thread I get a lot of new Idears cause some things are very fundamental but I have a lot of Problems to followe your train of thoughts.

So let me ask you at the begining some simple Questions:
What does the abreviation FS mean? Fractal System?

For me a set is something like this:
{1, 2, 3, 4, 5, ...}
But what is a universal set?

For me a fractal system is something like:
z_n+1 = z_n² + c;

When I look on the set {1, 2, 3, 4, 5, ...}
I have the Problem with the ... which means Infinit, which is something I canot measure and what is not computable.
So I would prefere to have something like:
{1, 2, 3, 4, 5, ... n}
From this kind of set it is easy to come to the set:
{1, 2, 3, 4, 4, ... n+1}
So I will be possible to define a lot of Maths in a recursive way like a fractal.

Is this the direction in which your idears go?

[jehovajah]

Hello Hermann

I apologise frequently to you and others about the difficulty in following my thread, or particularly my train of thought. In part as i have explained before this is due to the intuitive nature of the writing, and the declarative style that i seem to have. I have also to point out that for me this is a journal, a noting of random thoughts and intuitions that drive me to write before the notion fades back into the vast computation that surrounds me, and of which i partake but a few morsels, crumbs of consolation for my tardiness and human frailties!

Also, as for a long time no one has been moved to interact with the thread i have not needed to clarify anything much except when i am moved to do further work on an idea. Sometimes i find myself inspired to write a full description of an idea, a truly satisfying or sometimes driven experience, only to find i have written the idea down somewhere else already! Some of the ideas i admit would be better spun off into their own threads, but i do not propose to impose on our host by multiplying topics endlessly.

Because this thread purports to be about Axioms i have had to spin off the axioms just to provide a more straightforward access to them, but as you will notice my housekeeping is not the best, and i need to tidy up some of the numbering system. I will get to it, just as i eventually get to my typos, which are frequent and occasionally interesting: for example Axiom!   i have never changed it because it says something i cannot define!

So when i started it was not out of the blue but after years of self reflection and working through practical philosophical issues about the nature of knowing: Epistemology and religion and language and communication etc.

Man i had a lot of issues! But any way the epistemology had to start with me, because i had no idea if anything else existed independently of me. But of course this is not satisfactory because "me" is a construct developed over "time", etc and "space" is a construct i develop over time.

I had got as far as the concepts of infinite possibility space condensing into probability space instancing in a particular statistical reality space,from which and in which i have my experiential continuum. This was driven by a common religious concept of "ALL in ALL", and "in whom we have our being". Also the infinite pattern formations were inescapable, but the epistemology of pattern recognition or perception was not clear.

Then i came upon Fractals, and immediately found a resonance. Space then became infinite fractal possibility space, etc.. I literally woke as if from a dream and began to look at this thing called fractal. Who should pop up bur Benoit Mandelbrot and then Arthue C Clarke. Arthue i could understand having read his science fiction from a child. Benoit i had come across earlier in life, seen the clouds and the mountains , been put off by the mathematics and put it back in the library as crazy abstract mathematicians stuff! (it seemed arch and Facile and it still does, but at least i have the resources now to get at the basis of it- i intend to explore dimensionality along with parametrisation).

I had a long term beef about numbers and why "i" could not possibly be a number imaginary or otherwise. Thus i started my exploration into FractalSpace, and the foundations of maths, seeing that "Fractals" were fundamental to the order of all things.

I start off all Axiom this and corollary that, quite the mathematician , darling! A bit up myself?  Not really, just Autistic and not sure where i was going who i would meet along the way, how out of date my ruminations may be etc. I had assumed that the artists here would all be jobbing mathematicians. How ironic and special to find that they were players and playful! Just what i needed.

The thoughts range over my areas of insight and interest, and some do not apparently appear mathematical. They probably "ain't" in the old sense, but since i discovered, uncovered The Logos Response all things are mathematical or strictly "ratios".

My current aims are to fully explore the Logos Response and its impact on Epistemology, to continue to review the basic "mathematical" structures of which Spaciometry is primary as a fundamental output of the Logos Response, and in conjunction with spaciometry algebraic thinking. Algebraic thinking is how i formally construct and apply and deploy axioms, definitions, transformations, products of transformations, theorems, notation and rules of computation or manipulation,mapping, iteration, boundarisation, enumeration and transformation rules.(Some of these overlap).

 from these Algebras i derive specific arithmetics which are the fundamental computational tools i use in everyday calculations of one sort or another. Arithmetics can be very intense and algebraic at times, so i welcome the tools of computation now available, especially symbolic logic applications.

Calculus is usually marked out on its own but is a type of arithmetic for dealing with change, so modern computing platforms that can animate those changes are a fantastic aid.

I have found out a few things of great interest so far, and i need to note something in more detail about the change from ratios to Fractions, and thus the whole misleading conception of number leading to the complex numbers so called.

A lot of the "genius" of early mathematicians is due to the fact that they thought in and were fluent with ratios and proportions. For example i recently found that the greeks were not squaring the circle to find its area they were triangulising it (if such a word exist!). This lead me to observe a curious fact between rotating the perimeter of a convex 2d shape into a rectangular area, and the area of the shape. I am currently wondering if a similar relationship exists between surface area and volume, or is it just rotation around a given axis.

Hope you find this helpful Hermann. Essentially the whole of my thoughts are to explore as rigorously as i can the fundamental function of iteration in defining and developing our mathematical ideas, and as a consequence the inevitable fractal nature of everything, but this time not in Benoit"s sense but the modern sense of self referencing, self similar patterns at all scales.

[jehovajah]

Hallo Jehovajah,

thanks for all the commends on my work here in this forum.
When I read this thread I get a lot of new Idears cause some things are very fundamental but I have a lot of Problems to followe your train of thoughts.

So let me ask you at the begining some simple Questions:
What does the abreviation FS mean? Fractal System?

For me a set is something like this:
{1, 2, 3, 4, 5, ...}
But what is a universal set?

For me a fractal system is something like:
z_n+1 = z_n² + c;

When I look on the set {1, 2, 3, 4, 5, ...}
I have the Problem with the ... which means Infinit, which is something I canot measure and what is not computable.
So I would prefere to have something like:
{1, 2, 3, 4, 5, ... n}
From this kind of set it is easy to come to the set:
{1, 2, 3, 4, 4, ... n+1}
So I will be possible to define a lot of Maths in a recursive way like a fractal.

Is this the direction in which your idears go?

Not really. Set theoretic descriptions are a language algebra with operators in the notation and rules of the set. The set you denote    (see how the language has to change to become rigorous?) is not a real set except by definition. You won't find it at the bottom of your garden for example. You won't even find it among communities of religious monks dedicated to chanting and counting! So what is it you are denoting? My contention is that this is not a set but a cultural iteration called by me +1. But now perception comes into it. What do you perceive you are looking at? I do not know, and any way is it the same as what i perceive or intended to communicate? Again i do not know.

Only yesterday i was struck by the enormous assumption we are taught to make through number bonds: 20 *20 = 400. Is it really? What the hell have i just written? some marks on your screen, some buttons i have just pushed, and all of us reading that agree that it is correct. How did i do that?  Did you do in your head what i did? i did 2*2 =4 put 00 on the end you get 400, but i could have done 2+2=4 ,10*10=100, these are 4 hundreds, that gives 400! i might even have done 2+2=4 put 00 on the end that gives 400. Why do these numbers do that? Why is one wrong and the other right, but they both give me the right answer?

So i never could understand Russels Problem, until i realised it stemmed from the flip flop of undecidability. Undecidability exists in the real world and we solve it by making a real decision. The real decision is a bit like Archimedes principle, we avoid these "unreal" ,not everyday abstractions and deal with what is quantifiable. This is what you have done. But hey when you make that decision you set up consequences that shape your world, a bit like Schroedinger's cat!

The only set of all things that can include itself is the set of all things. And as soon as you realise that you realise the iterative nature of all things and how one condition drops out a universe of consequences from that set which partition that set of all things. Thus as soon as consciousness or perception is allowed the set of all things is partitioned by that perception. This essentially and abstractly is the effect of the Logos Response: the set of all things is ratioed, that is partitioned, proportioned and related irreducibly in some way to a portion of the whole.

I find now that i am comfortable with iteration being foundational to every perception and notation, and thus the set with the ellipsis means an iterative process is being evoked to describe what goes in between the brackets. Your alternative definition is equally valid and in fact i would define away the issue by formally making them equivalent, much in the same way that 0.99999999999999....... is formally equivalent to 1.

My whole thesis if you like is that mathematics is a recursive system, that is all of mathematics is iterative, and finite systems are only stages in the overall scheme of things.The setFS was initially going to be a Universal set, but then i realised that it was really a model of another set that is universal: notFS. SetFS then has really become my mental model of reality if you like, and the place where all paradoxes live! Thus it represents discovered knowledge particularly overtly iterative.

There is a lot more that can be done to make this obvious, and a lot of overlap with computability. I am happy and grown up enough to accept maths as a subfield of computing, despite the obvious irony. In "truth" maths has always been at the behest of one patron after another, like music,so where is the problem?

[jehovajah]

I can begin to address the issue of the concept number. At the heart of it must be spaciometry. I have abstractly refered to the tensor spaciometry as quantity and "number". The relational ratios in a tensor encapsulaing Quantity and the boundary of the tensor encpsulating unity or one.

Numbering is a simple naming activity, but the naming activity can be made complex or rhythmic or repetitious or systematic. It is these intuitions that cultures bring to their numbering that inform their concept of number. Value is also attributed to their cultural counting /naming iteration, and that value is proprtioned throughout the whole process of numbering so that each number may hold an ordinal value or a cardinal value or both, and a rank according to the proportion of value. The map between value and number is spaciometric and thus provides a circular or tautological basis to value. The source of value lies within our own neurology and is culturally maintained, defined and standardised and enforced, at least in weights and measures and SI units etc.

Thus number becomes a name that identifies a stage in a cultural iteration onto which a culture encrusts many meanings, all of which reflect a spaciometric attribute of the many tensors in space.

Given this description i venture to add that the attempt to tie number mathematically to one abstract tensor, a linear fractal called a line distorted the concept of number and confused those who had cultural attachments to number. Newtons tutor John Wallis was principal in achieving this and despite the neatness of it the underlying fractal has come to the fore when mathematicians were not ready for them in general. Thus Cantor, Julia, sierpinski, Peano, all gutturally felt these fractals as monsters and horrors eating away at the basis of reality and of course mathematics.

The area of solving arithmetic problems using algorithms led to the development of Babylonian binomial equations to trinomial and quartic and eventually quintic. The increase in the number of terms in the equation reflected the effect of iteration on these algorithms as they described relational aspects in spaciometry,and the systematic relations that underlie manipulations. This "attacking" of a problem by "manipulations" is a very militaristic paradigm, and underlies all the notions f combination and permutation, issues that would very much concern the militaristic mind through the ages, but also the commercial or merchant mind would consider these aspects of the spaciometric tensors under its hand.

This rich appreciation of number and value is what the number line threatened, and that is why it was a tool for mathematicians per se. Fractions and the numberline are where mathematicians withdrew contact with the general culture and began to distinguish mathematics as a specialist field of study with certain enforced tools.

Fortunately for us the iterative nature of reality put a cold hand of dread on them and hopefully will prevent mathematicians from disappering up heir own anus!

So in the times of the great Taxonomists the subject of mathematics came under taxanomic scrutiny, and among other things the taxonomy of equations was updated to "polynomials". Mathematical reference for the body of knowledge to do with algorithmic solutions to quadratic, cubic, quartic and quintic equations became subsumed under the heading polynomial of rank or order 2,3,4,5 etc. The term binomial had existed prior to this for a while , so this represented a tidying up of the taxonomy for ontological purposes.

Early on in the development of the solution to the equations surds had been encountered as solutions. Surds are purely geometrical values, in that they naturally arise in euclidean geometry of the right angled triangle. The very name of the equations quadratic and cubic testify to the geometrical basis of these algorithms. Going beyond the cubic meant that no geometry informed the solution, Thus it made solution harder and less intuitive and relied much upon "abstract" relationships and symbolic manipulations,and analogy of form. Essentially try to view the quartic and quintic equation as some kind of quadratic or cubic one. That is simply to utilise the spaciometry of the day to intuit the solution.

Without formally recognising the difference mathematicians had come across a type of value in solving their equations which were geometrical, ratioed and measured, not counted. Thus they were not numbers, nor were they the ratio of any common or archimedian numbers. They were thus called surds and meant "geometrical measurements".

In the course of this feverish activity mathematicians came across a curious surd √-1. As mathematicians new general surds had a value they did not reject this as meaningless, but as some geometical measurement they did not yet understand. They were necessary for many solutions of quartic and cubic equations and so had an algorithmic value.

It was not until Argand that their geometric meaning was hinted at, and by then the number line had queered the pitch and the surds had become irrational Numbers, rather than geometrical measurements. it could not be seen for a long while that √-1 was a geometrical measurement of rotation. It is still not appreciated as that even today.

Due to abstract and symbolic manipulations some mathematicians had developed algorithms that gave solutions to the quadratic and cubics both as numbers and surds, particularly when the negative number rules had become well established. The negative numbers were another geometrical value, but because they were defined  in terms of a balance, and from that the commercial bookkeepers financial sheets/ tablets, their geometric meaning  was obscured. Their geometric meaning is in fact a rotation through π radians, if one accepts the number line.

The chinese and the indian mathematicians had a good understanding of them in the commercial context, and in the context of quadratic equations, but they were not easy to accept just as √-1 was not easy to accept.

It is of great importance to realise the spaciometric origin of these quantites and how they do not exist without the awareness of a mathematician and his/her paradigm. The concept of number is a cultural totem, based on identifiable tensors in space. The relational ratios in a tensor quantity are key to the distinction i am about to make:measurement and distinction .

The Logos Response provides me with measurements of ratios. These ratios are a field effect in my experiential continuum, and i respond to them by processes within my CNS and Peripheral NS with an action that boundarises regions in that field, based on comparison of the relativistic motion attributes within those regions. Thus the field of ratios from the Logos Response is a Motion Field. Although i cannot say much more about that yet i am working on it in the thread on the Axioms of setFS. Nevertheless the point is that Measurements of ratios not Counting is the fundamental response to the motion field in the set notFS.

The distinctions we make by boundarisation are the source of our language response. Thus our language response holds the bounded distinctions in and among  the tensors. One aspect of our language response is the identification of plurality, which means the recognition of more than one and the recognition of repetiton of bounded regions: identical, similar, or none similar. At the same time i recognise the relativistic relationships between these regions in this plurality. Thus the spatial arrangement is inherent within this notion of plurality. Thus to sum it up almost the first notion that arises through the logs response is a measurable spaciometry; the second notion is a languaged spaciometry and the third notion is a countable spaciometry, in that order.

All Founding mathematician exclusively engaged with the spaciometry in doing and thinking about their mathematics. Thus while a region is real when it is in front of one, it is also a real memory that can be in front of the mind at the same time , Recognising this the indian mathematician in particular were able to conceive of debt as an absent region, a re-balancing of scales or the filling in of a hole, or the removal or changing of a colour. The chinese used coloured rods to represent a region that was removed from the direct view of the mathematician, but was important to account for the regions in view. Spaciometrically the red rods were removed from the relationships under purview, but needed to be accounted for. Each red rod thus told a story, and the story might be one of debt, loss, investment, advance or retreat, whatever the mathematician wanted to account for over a sequence of events. a set of relativistic motions.

With these spaciometric tools and memory tools in mind Indian mathematicians were able to give rules of manipulation which became the rules we use for signs today. How they arrived at -*-=+ i have yet to uncover, but our mathematics is the way it is because of this rule. We now can explore different "sign" rules and see what mathematics they produce, but the one we have resonates in so many ways with the natural order that it is unlikely to be replaced.

Thus the geometric/ spaciometric underpinnings of number are clear but mathematicians began to confuse measurement and a number concept. The geometric measuring scalar fractal was and is different to the number line concept, although of course the number line concept is an analogous system that principally John Wallis used to great effect. However Wallis used it as a geometric measuring line, later mathematicians like Cauchy and Dedekind ripped it away from these geometrical roots and created the number concept we use today. This number concept is a aggregation of number, memory tools like number, surds and Fractions, and infinitesimal like limit values including continued fractions and e and π cognates.

It seemed crazy to extend numbers by fractions, even crazier by negative numbers, but to then attempt to add √-1 was a step too far. Mathematicians have resolved the conflict by the invention of vector mathematics, but few recognise the work of Bombelli 

1572 book-keeping was highly
developed in northern Italy,
but even "simple" negative numbers were just introduced
(and the + and - signs unknown). So the hydraulic ingenieur
Bombelli  wrote a poem about "piu" and "meno" to teach
calculating these.  In his book "L Algebra" he didn't try to
solve x²+1=0 any longer; instead he recognized the "necessarissimi"
existence of squareroots of negative numbers and introduced
sign-rules,f.e.:


Just insert numbers and you can calculate every combination.
So he introduces them as new members to the family of numbers,
or, more precise, of quantities, lets say of a different branch (we
express this by the word adjungate or adjoin). In modern words
he is calculating vectors. For him, it was not an abstract construction.
Solving equations were done with geometric constructions and
Bombelli used L-shaped rulers for this:

The aftermath of this age long construction has been a confusion in the concept of number, carried on today through the use of number when referring to geometric values and operations, and relationships. The concept of a tensor has the power to resolve this issue of geometrical measurements in a vector type relationship, or a matrix or indeed a relational database called a tensor. This allows quantities and measurements to be separated from number and number to be returned to its cultural role in naming stages in the counting iteration.

Tensors, by which i mean weights and measures and dimensional units are the geometrical heirs of the "number line" concept, modified now to a geometrical vector.

[hermann]

Hello jehovajah,

thank you for the survey you have written for me.
But I thing it is bejond my time, energy, an knowledge of mathematics to understand your writing.

It is very inspirational reading your writing, even if I find more questions and answers.
Maybe I can elaborate some of my idears and post them here.

[jehovajah]

For sure, Hermann, 2 heads are better than one they say

Welcome aboard!

[jehovajah]

http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/60/

In the light of further research i appreciate that the rotational and extensional aspect of z^2+c is in fact codefying spiral orbits, as a rotation while extending is precisely a form of Spiral. Thus the mandelbrot set is what remains after vectors c  trace spiral paths through the vector field with a sculpting effect whenever |c|≥2.

This i think is what Vector is exploring, but the spiral orbits exist in z^n+c where n>1 regardless of changing power or logarithm.

Julia is a spiral orbit traced by the z vector, which is the stepped vector added to the same c vector every time after squaring, thus spiraling and translating.

[hermann]

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.