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

[jehovajah]

Axiom 6 has an interessting corollary. Energy and motion are by it recursive or iterative processes. This leads on reflection to the notion that a set wide iterative process may be a hypothesis worth making with regard to the notions of energy and motion in FS. This set wide process will only be worth making if the energy and motion laws that Einstein derived can be shown to be consistent by every measure with the axioms of the set FS. If this can be done then starting with a suitable fractal rule if recursive processes can be shown to generate einstein like motions and energy equivalents then a convincing case may be made for the recursive action of space. If space has this recursive action it may then be possible to relate each iteration to a notion of sequential statuses which may be similar to the notion of "time" in modern physics. I will be more rigorous in a following post, but essentially the elements of the set FS are many and varied but for it to be useful FS must explain the nature of space or rather have an equivalent definition of space to what is in notFS.

[jehovajah]

IN being rigorous the notions of recursive or iteration need to be explored and defined. The notion of iteration is the deja vu notion. The latin word iterum reflects the cry of a joyous child "again!" so "I" instinctively know the initial conditions and the process i followed to produce the outcome/ product. "Again" however includes a modified set of initial conditions, modified by the first outcome of the first process. So "I" instinctively sequence the initial states, process, and outcomes. Sequencing in this way is ordering, ranking and eventually arranging by some inate cultural or idiosyncratic schema. The schena itself is influenced by cultural norms and boundary conditions.
My definition of iteration is therefore an abstract from this notion and itself will inflence the notion.

Recursion is a mathematical conception of cyclng or running back on 'itself' to go through 'itself' again which influenced the much more general linguistic notion of recursion found in sanskrit grammar formulations and subsequent grammar formulations. The lingustic form arises as an artefact of attempting to define precisely without an inate understanding that language is a human enterprise that models rather than gives magical control over. The power of language, the alphabets the rules of spell casting etc are all intertwined at this initial phase. Later philosopher magicians grammarians were more able to accept the notion of self evident, that is one must experience it oneself before being able to have a referrent for a symbol or word or group of words acting as a description. I am put in mind of the wonderful creation of artistic  monstrosities which arose from attempted linguistic definitions of what we refer to as a giraffe.

The simpler notion of "again" lies within the more elaborate "running back", as formally running back carries the notion of an initial place to which one is returning from a place at which one turned back. So this place at which one turns back is not an end place so implies a continous action requiring one to race back to the start. But at the start the run back instruction is applied again propelling one toward the turning back place. This conception is so poorly defined that it begs the question what is it one is doing? For this reason alone recursion is presented as an abstruse difficult subject, when in fact it is just a poorly defined one.

Mathematical definitions building on this foundation suffer the same confusing trait. My definition of recursion will therefore be formally equivalent to my definition of iteratiion so that the two notions will be replaced by a single consistent one.

A proceedure is a sequence of actions that act either at the same position in the sequence or in following positions in the sequence or both.

A sequence is an inate organisational structure that i have which is appreciable by me through boundarisation and movement relative to boundaries and is one of a set of inate perceptions including shape angle orientation. extension etc. A sequence is generalised by me to include symbolic arrangements and through synesthesia to include sound patterns and touch and other sense sensation. My very memory structure of these sensations is a contributory factor to my perception of sequence. Culturally sequencing is intertwined with concepts called "time" and though useful and powerful these are not necessary or presuppositionally neutral, neither is "time" inate.
 a sequence may be as long as i like! however some sequences i may use may be truncations of infinite sequences.

Actions or interactions are perceivable changes in a bounded region or at a boundary or both as another perceivable change occurs in a referrence bounded region or boundary in contact in some way with the former.

An Iteration is then a bounded region acted upon by a finite or truncated procedure (the sequence is finite or truncated) whose outcome is then the bounded region that the procedure acts on again.

[jehovajah]

Axiom 6 requires some careful handling. Axioms 1 - 5 lay out an underpinning framework for 6 but do not define a set or set notation. This is in fact assumed to be the standard mathematical definition and usage. However, the axiom itself is attempting to draw together axioms 1 -5 under a mathematical notation system. Thus axiom 6 is a tautology expressing a symbolic representation (set FS with rule ) of axioms 1 - 5. Tautologies like this are indicative of the iterative nature of my consciousness, and the question arises if the generalised notion of iteration does not preclude me from coming to any other description.

The definition of iteration is clearly to specific to explain everyday usage as I recognise iteration not only on bounded regions but also on values and symbols. Turings machine for example is a symbolic iteration, and Newtons iteration is a value iteration. Then there are the iterations which can be seen in design modification, Editing, scientific inductive reasoning, the scientific method itself acting on a group behavioural process; cybernetic and feedback systems etc. Therefore for FS to represent axioms 1 -5 the rule of iteration must take a more general form which can not rigorously be defined and is subjective to my appreciation of iteration. This is not a new problem. I mention it to highlight the fact that my so called knowledge of set FS is more likely to be a knowledge of a partial or restricted subset of FS or subsets of FS which may or may not be cofactors of one another.

[jehovajah]

Establishing that the definition of the rule on FS is a weaker form of the definition of iteration clarifies a consistent activity of Mathematicians: They love to focus on specific simple cases which can be well defined or rigorously defined and generalise from those special cases. As they generalise they explore the rules which change to accommodate familiar operations and they define rules to enhance this familiarity. For example the iteration we call counting (addition /subtraction) is used as the design for the union operation in set theory. Therefore the iteration property is inate within this set operation, but more than that the mathematicians have created an iteration of the counting iteration! This is a fundamental example of symbolic iteration where the + has by a creative process been iterated to a U. This begs the question already posed: are iterations endemic to our understanding and perception because we are inately fractal? By fractal i mean the result of an iteration process. Axiom 2 is this systems answer to that question.

Because of the definition of iteration it is more likely that the product of an iteration will have a property called self similarity. Consequently we can use self similarity when it is defined as a measure of how well defined and rigorous the definition of iteration used to produce an object or result is.

[jehovajah]

Mathematicians study the workings of operators, their larger counterparts algorithms and their iterations. Axiom 2 declares that all results in mathematics are the products of iterations and thus fractals, by my definition. Therefore all mathematics will have the characteristics of fractal geometries and or fractal symbolic logics of values and Turing machines on symbols, to identify a few familiar lines of thought. Take for example the familiar 1 + 1, this is in fact a useful stage in the iteration of counting, or it is a symbolic logic statement. A Turing procedure is at the other end of specification and needs to be precisely specified.Operation of the procedure is fully determined by a finite set of elementary instructions such as "in state 42, if the symbol seen is 0, write a 1; if the symbol seen is 1, shift to the right, and change into state 17; in state 17, if the symbol seen is 0, write a 1 and change to state 6;"

The point of note is that these are only stages in the iterations which can continue for as many times as we need or wish or is determined by some other iteration rule. Fractals are the stuff of my conscious thought, and i as a mathematician specify the area of interest i wish to play with.

The concept of continuous and discrete needs to be viewed in the light of iteration. Graph construction for example is an iterative process and in fact produces discrete locations through which an abstract locus is deemed to move.The continuity therefore is imported into the graph by a property of the abstract locus

[jehovajah]

I have already remarked on multiplication and division as being repeated addition and subtraction of a fixed value. I will restate this as a form of truncated counting iteration. So the 5 times table is the iteration "+5". All counting iterations are infinite but i truncate them where they are of use to me. ``i can represent this iteration procedure and the truncation by using the notion of multiples or divisions. As a mathematician it is second nature to use a bit of symbolic logic and indeed most of my thinking is cluttered with these symbolic logic statements. What they refer to often i do not know or understand unless i can relate to a learning experience of doing the logical procedure. But even then the playfulness of the procedure is lost under memories of grimacing pedagogues whose scorn or whit drove home the rote learning of the task and not the thrill of exploration.

So 5 X 5 is immediately recognisable but not understood as a truncated iteration procedure expressed in symbolic logic. 5 : 5 curiously is not so well recognised because for me division and repeated subtraction were an unlinked notion until well into my 20's.
 Symbolic logic and the rich history of infighting among mathematicians is also to blame for competing and confusing symbology .

In all counting iterations the space that is being iterated is formally included as 0. For this we have to thank indian mysticism as the greek mystics apparently had no appreciation of the void. A simple but powerful description of the counting iteration is the description of bean counters. A space would be cleared to count the beans and then the iteration begins. The procedure modifies the space by adding 1 bean. This modified space then becomes the basis for the next iteration which is again modified by adding 1 bean. This new modified space is again modified by the iteration through adding 1 bean. These iterations continue ubtil all beans have been added to this modified space.

This procedure links names to each modification of the space, and each modified space is the referent or the value of the name. It takes a while for the name and the referent to become so conditionally linked that the name has value. The symbol of the name which i call a numeral then has a value attached to it. Modern numerals we can trace back to arabic influence but of course there are many earlier name and numeral links. The history of names is linked with a deep mysticism which i call number and numerology, but the symbolic logic of modern maths requires only the numerals. However i acknowledge that the mysticism of former times was and is still a driving motive for playing with the relationships between the values or referents. At the same time the mystics obscured the iterative nature of counting by downplaying or ignoring its fundamental , crucial role in the world around me. It is not a criticism to call iteration boring and so likely to be ignored except by a few extremely ocd monks or ascetics!

A snobbishness arose among mathematicians about elegance and style, which translated into a conceited anachronistic approach to calculators and computers. In addition the reverencing of Euclid in the west in particular led to a false notion of proof. I have to note that this is not a universal feeling among mathematicians but was particular to western european mathematics. Eastern maths has always welcomed any tool that aided the "number crunching".  It is fair to say that if i as a  mathematician pure or otherwise  ignore the tool of computing i am jeopardising the relevance of what i have to say because i can only do a few iterations of my elegant procedures.

The definition of iteration then is strongly represented in counting iterations, and anything directly based on counting iterations will be highly fractal.

[jehovajah]

Axiom 1 holds together some fairly hard to define but necessary and unavoidable notions. `My' abilities or attributes are only hinted at but implied. The notion of quantity is inate in this axiom as is the notion that i have this ability to quantify. Axiom 1 states explicitly that i can construct an experiential continuum but not in isolation, and that i can model notFS. These abilities are not to be tgnored, for counting iterations are not the basis of my appreciation of quantity. I continually appreciate quantity without being trained to name it in the form of a stage in a counting iteration. Today we have digitising devices that can sense the environment as a signal in myriads of sensors ,and the output from each individual sensor is almost instantaneously given a digital value or a count by a device or circuitry that converts the sensors output into a digital value. What i am alluding to by this is that i have within my perceptual function all the information and more which i am tediously naming and defining in the study of maths. Typically i learnt number bonds and numeral links up to the value of 12, but i could just as easily learnt them up to higher values. Some autistic individuals demonstrate this ability quite well. The thing to note is the sheer power of the iteration that is taking place in the perceptive faculty, and the modern sensor systems allied to the computing platforms give testimony to that.

[jehovajah]

The perception of boundary is a synesthesia of several sensors of which vision is the dominant one. It is not clear what boundary can be perceived by the ear say which does not immediately elicit a stored image of the same or similar. Similarly the sense of touch is so local that image perception is key to it being in any way indicative of boundary. Orientation extensibility etc all make more powerful sense in the visual sensors but nevertheless have contributory components in the other sensors. Shutting my eyes for example and trying to explore the orientation of my teeth in my mouth by touch of my tongue or finger reveal the intricate link to the image in my visual faculty. Therefore boundary is a perceptual iteration result with many sub systems to the visual contributing. Consequently the abstraction i call a line is an overlay of an image onto a perception of boundary. This image can be drawn from visual experiences of say silk or spider web or even a sharp pencil line. To abstract it further to nothing but extensibility is a faux pas that leads to certain inconsistencies.

The notion of straight and parallel derive from boundary segments that have certain inate properties. These properties are explored by euclid quite well and do not require abstract infinite lines. The notion of circle and spiral and oval derive from natural observations, but for me to observe them i must have some inate sense of curve, and an iteration process for that sense makes good sense of the recognition or perception procedure.

[jehovajah]

The context referred to in axiom ! is undefined and undefinable which is why i define my experiential continuum and formalise it in the set FS in axiom 6. However much of my inate abilities and functions which are in the set notFS  are gradually by iteration processes being found to be mapped / modeled in the set FS. The vision system within my symbiotic microbial colonic system has a counterpart in the ccd and cmos sensor systems used within cameras and electronic visidn systems. A study of the biological vision system : the eye the optic nerve and the visual cortex reveals that the retina is modeled by the pixel system the analogue/digital converter is modeled by modifier cels just behind the retina and the visual cortex models the digital/signal processors. Their is also a clear regular pattern of rod and cone arrangement in the retina which provides a grid like arrangement. This grid like arrangement is mimiced in the arrangement of decoding neurons in the visual cortex. The eye and the visual cortex developmentally (by iteration) are extensions of each other. These regular arrangements of cells, crystal lattices,packing of small objects, molecules etc are being studied under the heading of self assembling structures. Suffice it tosay that my inate sense of shape angle line boundary and orientation can be found in the patterns that this vision system is able to respond to in conjunction with the other sense sub systems.So for example a boundary arises when an arrangement of rods and or cones fires off at a particular action potential within a region on the retina and a different action potential on the "other " side of that region. The arrangement of rods and cones are the inate shape or angle or line which the processing cortex uses to engage in the perception iteration. Somewhere along that set of iterations i am able to make a connection with a stored model that gives rise to the recognition response. The iteration then proceeds but a "higher" level iteration now dominates and uses the recognition iteration in a verification iteration. IF the verification is not found then i may begin the recognition iteration again from a different perspective, whether that be a different angle or a closer(magnified) look until verification is achieved. This process is the basis of curiosity, and i may never achieve verification so i may always be curious about some experience. At another level of functioning this particular non verification may be used to detect a whole class of similar situations. 

[jehovajah]

Axiom 2 needs to be further explored as to the all inclusive nature of its statement. Clearly some iteration will have to be defined as null iterations if it remains in its present form. However how to distinguish between  a null iteration and a fixed result iteration may be a valuable thing to explore with regard to iterations that transfer energy into a fixed region, gravity for example.

[jehovajah]

I am setting down a few notes about the conception of time and space common in the west. Axiom 2 declares that these are fractals. It is important to distinguish time because this is a fractal used as a foundational scale like length but is based on movement in space. The fractals that i am looking at are therefore space and movement in space.

Like all conceptions there is an inate sense of the fractal we call time, but what i am sensing is the iteration process or procedure. Axiom 3 declares that i am able to have multiple orientations on this inate sense and my chief sense is how personal to me the iterations are. Then as i look out or sense other bounded regions i can experience a different iteration process happening there within or including the boundary region. I can also find similar iteration procedures going on at a larger scale and at larger scales than that. These larger scale iteration procedures imply that the iterations processes within smaller scale boundaries are not isolated from each other by the perceived boundaries but may in fact be entrained to the larger scale ones. This causative relationship is perceived but as "chaotic" or complex. To this list of descriptors i may now add fractal. But it is not just a descriptor it is a fundamental reorganisation of any notions of cause and effect i have formerly introduced.

Axiom 9 therefore i sketch out here as: large scale and small scale iteration procedures in FS are fractally entrained at any scale size i wish to examine.

When i take a large scale iterative process such as a solar cycle as a standard or the rotaton of the earth about its axis, i can then subdivide it into smaller and smaller segments and use it as a measure or metric. Measures or metrics are some of the most obvious fractals that i can create or design in FS, but as such they are abstracts. Each iteration process i use as standard has to include the sub iterations within its "orbit" to imply any useful iteration link. So to become so abstract that a metric is applied "outside" of its defining iteration procedure is likely to lead to problems of scale.

When i experience a fractal zoom it reminds me so strongly of the differences in structure which scale changes reveal and therefore it is a wisdom to me not to generalise in an assumption of a "smooth" continuous development beyond a certain iteration procedures  defining region of operation. Rather i should expect discontinuity and discrete regionalised developments. So for example the quantum physics and classical physics are fractally entrained by axiom 9 so they will operate on each other,but there will not be a smooth continuous link between them. However i can approach iteratively close to the "boundary" between them in a wada "point" sense. I could refer to this as asymptopic, but i am not assuming a smooth continuous progression.
 
So using the movement of space in space i have traditionally accepted an abstract conception of iteration procedures and caled this "time". What i have perceived within the iterations is movement, and in movement the speed or rate of movement. So using my own visual sensors and the inate notions of sequence position/location and my own internal iteration procedures i am able to develop a sense of speed or rate. Even within my own body if i take my heart beat as a standard i can develop a rate assessment of say how fast i am breathing. This is a useful example as it reveals that many iterative internal procedures are combined in developing this sense of rate of breathing besides heart beat, and again axiom 3 leads me to expect this.

Space and movement of space in space under an iterative procedure is what i will look at next post.

[jehovajah]

The following are candidates for Axioms of set FS but i have just considered them so i have yet to assess them.

Inertia, Equilibrium, Syntax, Parsing and Equivalence.

By Equivalence i mean that things `are not the same despite the objects being seemingly identical. Every thing is unique but some things are apparently the same, that is perceived similarity, or apparently identical, that is perceived congruency. The class of sets of "things" for which these relationships are a good description i am calling equivalence, and the specified sets within that class i am calling equivalences. The specification of a set in Equivalence is those aspects of a "thing" which are perceivable  as the same in all elements of that set. These sets are in general containing similar elements. A set  which is consisting of similar elememts which upon further iterative investigation are found to be the same for any arbitrary new rule imposed on the set is defined as a set of identical elements and called a congruence.

For a given set in Equivalence we can form  ratios of its elements specific properties such that for a property p in the set the ratio of p for element i [ p_i] to the same property p for element j [p_j]  is p_i  :  p_j  or alternatively . This ratio wherever it makes sense for elements i and j will be a constant value in the sense that the ratio will always be reducible to the smallest numerals by "dividing out" common multiples.

I can establish a system of ratios based on the same property for many elements in a specific equivalence and these allow me to represent  an operation or an action or a relation that applies to many elements at once. This also enables me to represent the many levels of a fractal system by one convenient ratio.

Between specific equivalences there may also be a property that allows a ratio to be formed. Such ratios between equivalences are in the Equivalence class and may represent relationships of one equivalence acting on another. The action of an equivalence on another may be indicated and a property linking the two equivalences be implied by the ratio between the equivalences being constant.

Thus the iterative process of enquiry may be characterised by the forming of Equivalences, the forming of ratios around a common property, and the seeking of properties to link found constant ratios.

Many things i take for granted are equivalences. For example mass is an equivalence. My mass is the equivalent mass of water at a certain temperature that has the same moment on a balance scale as i do. The specific property is the moment on a balance scale. By this property many equivalences can be put into ratio with each other. However these properties may not specify the set of the participating equivalences, rather they define a new set a new equivalence.

[jehovajah]

Before i continue i want to sketch out a possible universal iteration procedure. I am thinking of a relative vortex for each individual. So the entire experiential continuum i have constructed is based on a procedural vortex relative to myself. Each iteration applying the vortex results in motion within my experiential continuum. At each fractal level fractal entrainment across the boundary generate motion on the other side of the boundary and either side of the boundary. The vortex procedure moves a region at each iteration to a new position, and regions within regions to new positions within the regions. The universal application does not imply uniformity, it implies fractality at all scales, which is to say that the product of the iteration under the vortex procedure is a fractal with infinite levels, and these products would be vortex motions at all scales. Wile this may appear to one observer to be chaotic to myself it would appear fractal, and would generate a search for the self similarity ratios the boundaries of regionalisation and the evidence of fractal entrainment. As this vortex procedure is universal, all motion that results will be voticular to scale. Thus i would expect to find that all forms of motion from seemingly straight line motion to hyperbolic parabolic elliptic circular, cardioidal  and spiral and even brownian would be apparent in its region of operation, which is universal and thus at all scales.

Since we use elliptical and circular motion to define periodicity i would expect periodical forms of motion to be linked to the iteration cycle of the universal vortex iteration in some way.

I would also expect brownian motion to be linked to vorticular motion and fractal entrainment both ways across a boundary with a wada basin condition.

Whatever descriptions we have of vorticular procedures should have this fractal nature if this is a universal iteration, and boundary conditions will need to be generalised to reflect the wada nature of all boundaries in a fractal.

[jehovajah]

<a href="http://www.youtube.com/v/hPgq4AbLKyU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/hPgq4AbLKyU&rel=1&fs=1&hd=1</a>

The vortex procedure in the set FS will have properties demonstrated above. Thus regions around the vortex will be moved in a coherent way and at different scales. Within the layer generated at each scale by fractal entrainment, regions that are in sympathetic vibratory lock , that is resonance move as a whole with gradual sheer representing growth. Those regions not in resonance move in a way that dissipates them rather like evaporation. Evaporation and condensation can be expected within the layers and across the layer boundaries. Within the layers vortex motion will be evident again fractally entrained by the vortex procedure driving the layer system.

The coherent motion generated by the vortex procedure with these stationary boundary systems and a smooth and flexible boundary condition (at least on a normal everyday scale) exists everywhere even in the twist of the vortex. Only the sheer that cuts spirally down the eye wall of the vortex induces chaotic motion always and spirally on the wall of the eye, Elsewhere resonance induces coherent modes of motion some conformal some sheer, some growth some contraction. Dissonance generates evaporative destructive motion on a region assonance and resonance a conformal motion on a  region and euphony a condensing motion on a region. Notice this coherency in the following

<a href="http://www.youtube.com/v/4_72HbbiG6A&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/4_72HbbiG6A&rel=1&fs=1&hd=1</a>   

The coherency demonstrated here under these conditions allows me to posit that the coherent light produced in a laser or a coherent maser are evidence of vorticular  procedures within the generating crystals under the influence of fractal entrainment by a changing electromagnetic field which itself is vorticular in operation.

With this extreme level of coherency the wada basins for the vorticular motions should have a definite axial shape which demonstrates concentration of the motion along that direction. Also the boundary of a region as a wada basin may be significant in the propogation of fractal entrainment across a boundary at any scale. It may be that all boundaries are wada basins in some sense.

http:"//video.google.com/videoplay?docid=4532767129040787318&ei=NixUSveREYqUqQL1gMGqDw&hl=en#QL1gMGqDw&hl=en#

This tendentious video nevertheless illustrates the conceptual basics but much has to be sifted to get the insight. {copy and paste in the address bar removing the "}

The navier stokes equations for fluid flow of compressible and non compressible fluids have vortex solutions that describe the propogation of vortices in interrupted flows, but the point is not the flow of vortices but the vortex procedure which they describe. This procedure is one model for the universal procedure  in setFS and helps to understand the propogation of wave forms as vortices in this space. For example a vortex or vortex formation can establish a sinusoidal type vibration that is propogated radially or in a flow.

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


<a href="http://www.youtube.com/v/_AJgEa2dbJU&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/_AJgEa2dbJU&rel=1&fs=1&hd=1</a>

Complicated standing forms can be produced from vortex motion at specific resonances to the medium


http://www.climateaudit.org/?p=466

The calculated solution and the photographed image show not only compliance but other effects due to non resonance.

<a href="http://www.youtube.com/v/5_drmTOe424&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/5_drmTOe424&rel=1&fs=1&hd=1</a>


 

[kram1032]

I want such a string-device-thing