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Axiom !: The Experience "I" have is solely my own and is constructed by "me" within a context that i can only model using cultural "forms".
Note: The structure of "my" experience i have constructed using the paradigm of a continuum between poles that are indefinitely located in a location that has at least an inner region separated by boundarisation processes from an outer region. It is consequential of the construction process that the outer region has no boundary that cannot be enclosed by another constructed boundary.

Axiom 2: All processes within my experiential continuum are iterative or recursive.

One important corollary of Axiom1 is that Everything i construct is necessarily relative to me in "my" model. Hence:
Axiom 3:- i can construct any given number of other reference points within "my" model by iterative processes.

This among other things allows me to rotate {in fact all affine transforms} my model relative to myself so i can gain insights from different vantage points within my model/ experiential continuum. The number of reference points is unbounded. The iterative processes are the basis of "trial and error" within my experiential continuum as i construct the re configured model from my initial assumption of cultural paradigms.

Axiom 4: The "context" in Axiom 1 is not constructed or definable within my model but is perceivable by "me" by a iterative process of negation of all elements within my model.
Basically i can't say what the context is but i can say what it is not by recursive means.

Axiom 5: I stabilise "my" model by an iterative process of "acceptance".

"My" experiential continuum changes with what i "accept" as a basis for the iterative processes of perception and recognition. the cultural forms which i accept from conception are numerous and pervasive and as i alter these my perceptions change as does my experiential continuum. "not altering" then is a nascent notion of acceptance.

Axiom 6: The Set FS is the universal set within which my model/ experiential continuum is defined and has a rule:- all processes on its elements are iterative /recursive and all its elements are determined by iterarive processes. notFS will be the recursive definition of the context in axiom 1. However there is a mapping from notFS onto FS such that FS is a model of notFS.

Axiom 7: Iterative/recursive processes operating on notFS are perceivable.

These processes will be compared with enegetic transfomations within FS.

Axiom 8:All perceived boundaries involve an iterative process or processes.

Definition: Infinfite is unbounded and large Infinitesimal is unbounded and small.

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.

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.



   

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.

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.

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.

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.


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.

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.

So to follow on from a vortex process operating in an iterative way in set FS leads me to re emphasise newtons laws of motion in FS. A body continues in a state of vorticular motion iteration by iteration unless impressed upon by a force. A body impressed upon by a force changes its motion in proportion to the force and along the vorticular path of the force. And finally the impressed or drawn force is opposed by an equal and opposite force acting on the bodies involved  .

Newton accepted the states of motion and rest, but in set FS the iteration of the vortex is the source of all motion by fractal entrainment. A body is only at rest to an observer with the same vorticular motion, and i will discuss this more when i consider equilibrium and inertia.

A body at rest is in a state of force equilibrium which is to say that all forces acting on the body cancel out. Only when this condition fails does a net force impress upon or draw upon a body in a "right" direction in the newtonian sense. This right direction is not a straight line but a perpendicular direction to the place of contact of the force. In a vortex field this approximates to a straight line as a first order approximation, but it is more accurately a logarithmic motion.

That iteration is the fractal entrainment for motion will be a basic axiom of the set FS and will be a development of axiom 2.

Axiom 10:
 The fundamental geometry of the set FS is non Euclidian and as a first statement i will align it with Riemann.

I have yet to study Riemann's geometry and those of others in order to make a finer determination.

However this begs the question : What is "fractal geometry"? I shall endeavour to set some distinctions to give referents to the name.

I will tidy up the numbering system for axioms in due course.

In the set FS the fundamental Energy notion is motion. The word energy is not only synonymous with motion but exactly congruent with it in the set FS. The various forms of motion therefore are different exhibitions of the former conception of energy. Displacement is a quantized exhibition of energy while uniform motion is a continuous energy event occurring at a uniform rate . Acceleration then becomes an event exhibiting the growth and development of energy. The relativistic attribute of motion thus applies to the former concept of energy with potential energy being a relativistic notion. That there are stable courses of motion indicates stable relativistic energy differentials. more later.

These stable courses of motion i will say are in relativistic motion equilibrium, but as a vector or directional tensor entity.

I have to quickly note  several notions developing in my head. Relative motion, the patterns of relative motion which i will call relativistic motion, relativistic motion equilibrium and relativistic motion transfer. the idea that relativistic motion is the inherent ground of all attributes of space and the sole property of space is constant fractal vorticular/ trochoidal motion. The basis of this notion is testable in that it would not require any special material to reproduce all the known attributes of my experiential contiuum merely patterns of relative motion.

The concept of relativistic motion therefore requires a more complete exposition. Quick examples are stationary solid objects where the solidity is a function of the relative motion and the relativistic equilibrium of the parts in the general reference frame on the planets surface. The symphony of motion i call a tree with leaves, where the relative motions of each part from the atoms to the cells to the structures of leaf and bough and trunk illustrate the complexity of relativistic motion, and dynamic relativistic equilibrium on a comprehensible scale. The clouds in the sky which illustrate the relativistic motion of gaseousness but also the impression of solidity and electromagnetic variation. The notion of light as being a relativistic motion that has general and widespread effect or resultant on all other relativistic motions it "contacts" with and the notion of relativistic motion transfer on an enormous scale differential, from the smallest to the largest.

The notion thaty relativistic motion transfer is related to the mysterious attribute called momentum and represented by  

And the formulae 

and

or partial derivative forms on a Riemann surface or more hopefully a surface of a fractal vorticular nature with conic, helical and loxodromic attributes.

In such a conception "time" is replaced by a periodic length and a standard is used to establish a fractal scale, but it is clear that if a given conic surface is used to describe the periodic motion used in the standard that the periodic length will vary with the plane of intersection of the conic. Thus periodic length will be dependent on conic sectional properties but how numerous the counting or clocking of the standard periodic length will depend on the "swiftness" of motion used to define the periodic length. And the swiftness of motion is a perception of the former energy notion. So higher "energy" would give more standard periodic lengths which would mean that my perceptions would either speed up or slow down "time" depending on how i want to describe it. 

Some research i am doing into diffraction gratings gives me the notion of a relativistic structure for the set FS of a spaceometric manifold. It will need some work but i note here the concept and the directions i want to develop it in, as a fuller descrption of the universal iterative procedure in set FS for entraining motion.

http://books.google.com/books?id=bFcL0bbsDP4C&lpg=PA83&ots=f4exT7ZzPy&dq=3%20dimensional%20grating%20diffraction&pg=PA83#v=onepage&q=3%20dimensional%20grating%20diffraction&f=false (http://books.google.com/books?id=bFcL0bbsDP4C&lpg=PA83&ots=f4exT7ZzPy&dq=3%20dimensional%20grating%20diffraction&pg=PA83#v=onepage&q=3%20dimensional%20grating%20diffraction&f=false)
And of course
http://en.wikipedia.org/wiki/Diffraction (http://en.wikipedia.org/wiki/Diffraction).

There are various arrangements of diffraction gratings and various designs of aperture or/and spatial relativity of the "slits" on the gratings. I would be interested in the spiral relativity arrangement(cd disc) and the vortexian crystal like arrangement for the Bragg distribution. All these structures would be in motion and thus the arrangements would be relativistic.

One of the limitations i feel of the current QCM theory is that it is ultimately based on harmonically oscillating lattice structures, and i feel a description based on relativistic elements ie elements in relative motion to one another would be possible and of course more fluid. The motion field equations would be some forms such as

http://en.wikipedia.org/wiki/Field_equation (http://en.wikipedia.org/wiki/Field_equation)
and
http://en.wikipedia.org/wiki/Electric_field_integral_equation (http://en.wikipedia.org/wiki/Electric_field_integral_equation)
and
http://en.wikipedia.org/wiki/Fluid_solution (http://en.wikipedia.org/wiki/Fluid_solution)

and
http://en.wikipedia.org/wiki/Dust_solution (http://en.wikipedia.org/wiki/Dust_solution)

and

http://en.wikipedia.org/wiki/Schrödinger_field (http://en.wikipedia.org/wiki/Schrödinger_field)


and it is always good to note ideas do not develop in isolation http://en.wikipedia.org/wiki/Relativity_priority_dispute (http://en.wikipedia.org/wiki/Relativity_priority_dispute)

Other analogues

http://en.wikipedia.org/wiki/Gravitomagnetism (http://en.wikipedia.org/wiki/Gravitomagnetism)

http://en.wikipedia.org/wiki/Stationary_spacetime (http://en.wikipedia.org/wiki/Stationary_spacetime)

http://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation (http://en.wikipedia.org/wiki/Nordström's_theory_of_gravitation)

http://en.wikipedia.org/wiki/Brans–Dicke_theory (http://en.wikipedia.org/wiki/Brans–Dicke_theory)

http://en.wikipedia.org/wiki/Tensor (http://en.wikipedia.org/wiki/Tensor)

http://en.wikipedia.org/wiki/Fluid_mechanics (http://en.wikipedia.org/wiki/Fluid_mechanics)

http://en.wikipedia.org/wiki/Free_field (http://en.wikipedia.org/wiki/Free_field)

http://en.wikipedia.org/wiki/Magnetohydrodynamics (http://en.wikipedia.org/wiki/Magnetohydrodynamics)

http://en.wikipedia.org/wiki/Bernoulli's_principle (http://en.wikipedia.org/wiki/Bernoulli's_principle)

http://en.wikipedia.org/wiki/Metafluid_dynamics (http://en.wikipedia.org/wiki/Metafluid_dynamics)


So i see the ideas have legs.


What caught my attention is that the general description of these lattice like arrangements is as triply periodic functions of position (at least in the vector form) and that all physical properties of crystals can be represented by functions of this type.

This is a clear relativistic description of attributes of a region of space, and as the function is a function of position it will admit to a differential form which may allow me to explore a relativistic description based on motion.

Relativistic motion when localised to a small spaciometric volume is meaured as a "cavitation energy". Cavitation energy is thus a relativistic motion transfer event which transfers potential relativistic motion to a small spaciometric volume giving it kinetic relativistic motion as a region.

Small volume relativistic motion is proportional to the spaciometric density of the volume which is itself proportional to the mechanical density of a small volume. Spaciometric density is a numeral expressing the count/measure and and relativity/ratio of structures, forms and surfaces and distinctions within a region in space. I will revise this definition in spaciometry to recognise the tensor nature of this definition, thus the numeral becomes precisely a scalar to a unit quantity of spaciometric volume which will be a tensor or a differential tensor element.

This is just a note and i will need to work on clarifying the notion of a spaciometric tensor, and the link to a mechanical tensor of density.

Nevertheless , cavitation relativistic motion transfer is proportional to the mechanical tensor for density in which the density is a dynamic function of pressure and temperature, allowing a local gas law to obtain in a super phase or plasma state. Such plasmas will have electromagnetic properties as well as chemical ionic properties. The permeability of the spaciometric structures around these localised events will be part of a full tensor description of the volume, it seems to me if the resultant transport behaviour of the superphase is to be described and the electromagnetic phenomena are to be described and understood.

Pressure and temperature are normally understood as mechanical strain, so i am applying a ratio between Boyles law and mechanical strain. with elasticity and deformation characteristics being phase change phenomena with associated plasma properties entangled in the permeable structure of the spaciometric volume.

Fracture occurs as a latent energy function for the whole spaciometric volume , but for the tensor each differential element will have phase change boundary quantity  which will determine when it becomes relativistically non cohesive, ie relativistic motion exceeds the bond motion quanta, that is relativistic motion transfer imparts relativistic motion in excess of the escape motion for coherent relativistic motion. The spaciometric object shears along planes and boundaries where these escape motions are proportionally weakened by cavitation relativistic motion transfer, so a clean break is not to be expected, and close examination of the fracture plane should reveal plasma state phenomena with phase change phenomena leading to "condensation " events and relativistic motions of these condensing regions in a type of gas behaviour, as well as a distinct electromagnetic pulse behaviour.

Draft To be corrected.

Axiom: the natural motion of any region at any scale within the relativistic motion field is vorticular/trochoidal. The size and strength of this motion is proportional to scale, speed, and radial distance from the vorticular/trochoidal axis centre, in fact inversely proportional to the radial distance. The limit motion at the vorticular/trochoidal centre is an "angular", that is spaciometric rotational action called twist or spin which is a spiral/trochoidal reference framework value representing the distance traveled along the vortex/trochoid locus to the centre.

Axiom the spiral/trochoidal referennce framework value is quantized, and this is achieved by either the growth in the value by quantized amounts or the decrease in the value by quantized amounts.

Thus the spaciometric centre of the rotation will increase in spaciometric density or decrease in spaciometric density by a quantized amount. These quantized changes represent spaciometric phase change boundaries for the absorption or emission of other spaciometric forms.

Thus applying this to a recognisable example: the interaction of an electron orbital with a photon may lead to a spaciometric change in the electron orbital value on absorption, which subsequently returns to its previous value with the emission of a photon orbital value.

Axiom : a "straight line" motion from a vortex centre is in fact a helical vortex emission of a quantized spiral reference value. The motion appears straight because the helical motion is of such a great rapidity. This rapidity of helical motion i will call spaciometric frequency and it represents a comparison between a standard period length and the period length of the observed spaciometric rotation.

Axiom: The motion field in set FS has as a consequence of vorticular/trochoidal motion at all scales and in all direction an entraining effect throughout a spaciometric region which i will call spaciometric gravity.

Axiom: the boundary of a region completely determines the spaciometry within a region.

As an example Euclidean geometry exists only within abstract boundaries, whose abstract  axiomatic components are line angle and point, arc of a circle and of course, the ubiquitous plane.

Riemannian geometry allows for a general curve component and a general surface component. Although parallelism is also an important distinction it does derive from the other components in Euclidean  geometry, but is falsifiable in Riemannian geometry.

Axiom a region moving n a straight line across an observers cone of orientation has a spaciometric rotation.

a spaciometric rotation is an activty so either the oserver is active or the region is active or both. Whichever is active has the spaciometric rotation.

Axiom a spiral locus is the only locuss that results in acceleration or deceleration of rotation infinitely.

Ratios produced by proportioning of one change to another change sequentially  is precisely logarithmic and the anti ratio of a changing proportion is precisely exponential. That is the  changing differential portions  ratioed change logarithmically but the  differential changes summed  increase exponentially.

So 1:2:3:4:5:6:7:8:9:10
 The sums are given by s= n( n+1 )/2 for n=0,1,2,3...

The ratios are given by n: n+ 1 for n=1,2,3...
writing the ratios as fractions n/ n+1 reveals the logarithmic sequential change.

It is of interest that until this form of writing ratios was widely adopted and treated as an enumeration system, that is as a value assigning protocol, which in simpler words merely means a number system like the reals and decimal system today, this insight could only be intuited not notated.

To my mind this is the first time a number system was created by an algebraist, the first constructed number system that facilitated an interaction with a fractal universe (notFS). The invention combined the fundamental logos response into an iterative and self similar fractal pattern that has never been beaten only utilized more extensively in the extensions to the number line.

 The rational comparisons are where we as a culture will always find our most intuitive response to notFS , and where polynomials and calculus spring from as well as the essential ratio methods that so inform our apprehension of the changes in notFS. All of this is now hidden, implicit within the notion of rational numbers, and this is bad or our creative response to not FS and our Logos Response. Ratio and proportion keep us real and fractions and rational numbers keep us grounded in the iterative nature of "reality"

Axiom: motion flows in a structure or in a chaotic trochoidal form. The structure of motion flow that stores motion is a vortex and the anti vortex consequently dissipates motion. The vortex-torus structure is  fundamental motion store/dissipation structure that is stable and enduring.

Axiom: There are an infinite number of clocks cycles of the Ed Lorenz equation type, and all these "clocks" are regional and relativistic. There is thus no universal time frame, just relativistic ones.

Clock cycles therefore are great for regional ordering, sequencing, synchtonising and entraining, but do not represent any fundamental memory store of information that can be accessed by a "time machine". The chaotic nature of these quantum clocks makes that a non starter.

The effect of these clocks on motion:extension and rotation, not as simple as they seem. In addition motion ratios can be proportioned from any of the attributes that vary in a motion, not only the sequencing of extension ratio changes. Finding a clock cycle enables synchronicity of these attributes; as for example the information stored on a video flash drive.

If such a universal flash drive existed that stored holographically all information in all regions, then time travel would be a possibility.

Because the logos response is able to perceive and deploy ratios speed ratios and acceleration ratios and control ratios and then system ratios or cybernetic ratios and finally computational output ratios are all perceived by an individual. To describe some of the more complex ratios as chaotic and unpredictable is to shirk the task of apprehending them, leaving the way open for mystical or mythical explanations of them.

I have to run but i will detail an axiomatic corollary of axioms 1 to 5 under the notion of proprioception.

Kinesthesis (http://dictionary.reference.com/browse/kinesthesis)/Proprioception (http://dictionary.reference.com/browse/proprioception) is essentially the biological/physiological source of all reference frameworks for geometrical and spaciometric conceptions, and is self referencing and defining of the structures of the Logos Response as it applies to the conception of Spaciometry/geometry.

I cannot have spaciometry without innately having proprioception to give rise to spaciometric notions through the Logos Response, and kinesthesis to act out that response.

The fundamental processes of The Logos response is the measurement through analogue to digital sampling of sensors, the comparison of the measurement signals within the neural network structures of the CNS distributed processing systems and the distinctions assigned to given stable signal patterns.

It is clear that distinguishing is a response to the stimulus of the signal inputs. Thus the Logos Response entails a communication network to effect that response (http://www.nature.com/news/2006/060320/full/news060320-5.html).Up until now the response has only been thought of as a naming or associating response without exploring any mechanism or structure for this: in other words an intuitive description of a faculty i demonstrate without really being able to give a referent for it. (http://news.bbc.co.uk/1/hi/health/4861268.stm)

I think the proprioception (http://www.daviddarling.info/encyclopedia/C/cranial_nerve.html) faculty or faculties provide a basis for exploring this further.  tbc

The Logos Response provides an epiphenomenal route map from the sensory signal input through the PNS and CNS parallel, widely distributed processing network to an objective observable, testable and demonstrable explanation and exhibition of so called consciousness.

Part of consciousness has always been the Skinner stimulus-response arc which is the groundbreaking model that Skinner developed to objectively quantify all behaviours of complex systems.

There is no absolute objectivity or subjectivity in my view, but only greater or lesser degrees of detachment and observer position displacement in any descriptive or observation based assessment and apprehension of a subject of study. The reason why these particular sentences describing this activity are so long and torturous, is because of the Heisenberg uncertainty principle, which applies to all observations not just physics!

So any complex behaviour will be part of some stimulus response arc and The logos response is no exception.  The Logos response Measures, Compares and Contrasts. In other word the Logos Responses measures and compares and provides a response as an output that Contrasts, by Ratio, and distinction which at its basis is a boundarising process. The contrast therefore is the major part of the reponse experienced in so called consciousness and is the source of all our nascent notions of the Logos Response experience.

Thus my spaciometry arises out of the contrast response of the Logos response and so do all my particular and specific a well as general and expansive notions

Specifically my basic notions of spaciometry and kinesiology and proprioception arise from this contrast.  All other motions( and by relativistic motion attributes all other notions) are derived (at least i hope to show this) from spaciometric rotation and spaciometric extension. The Logos response provides the impulse to attribute dimension, and to specify parameters by which i am or will dimension, As usual the notions are distinguished by two different languages Latin and Greek. This testifies to the general education in the classics that was the curriculum for most of ou advanced western thinkers at the time,

Languages are Logos Response products, thus we can understand all languages once we know their spaciometric references, which are usually prepositional in form even if they are placed anywhere ib=n relation to the noun in the sentence!, or injunctive!!

Thus dis means a small space between, as does para, but para additionally orientates the observer listener to one "side" of the object or activity. Both use the small separation to convey boundary or distinction, but the greek para uses that boundary to orient the observer listener. The greek spaeker thinker would naturally ask Which orientation of the cultural orientations allowed or usually referred to : side is not specific on its own.

Thus i come to our use of the word dimension on the way to describing kinesiology and proprioception. Culturally and of recent origin we have begun to expect other worldly connotations (http://www.miqel.com/fractals_math_patterns/visual-math-hyper-dimensional.html) of the word dimension. While this is
 entertaining and of great facility in science fiction writing it has led to a slight confusion in popularising scientific and mathematical conceptions, slight but profound.

You see i am allowed to dimension a subject in any way i wish, and because these regions are tensors i can have as many dimensions for the properties and structures as i wish and need. When it comes to measuring these dimensions i use some fractal scale meausre alongside the dimension, thus it is called a parameter, and the readings from these scalar fractals are called parametric readings or generally "parameters".

in general there is a natural order to the presentation or perception of dimensions. First visually is area then some conception of volume and then perception of boundary. These require differing amounts of perceptual processing to identify and in the process other dimensions and their parameters are invoked.

 Auditorilly the first dimension is possibly amplitude followed by pitch and then resonance harmony etc. A musician probably can make more accurate lists of the primacy of the dimensions. The parameters may be derived or natural language ones like loudness etc.based on a common unit such as a group of people.

The identification of parameters and the derivation and use of parameters is fundamentally based on the notion of standard and it's quite recent cognate unit.

By means of this continuing impulse and manipulation of space in order to parametrise, through a process I may call mensuration or the establishment of weights and measures, our apprehension of space has developed and refined into the commensurate description we have of the physical world today.
 

Axiom: Any void is a trochoidal motion field.




The tentative and potential postulates of a trochoidal motion field are:

A: the conic sectional curve motions are precisely gravity;

B: the high frequency helical curve motions are precisely electro magnetism;

C: the  spaciometrically dense conic helical motions are the strong nuclear force;

D: the spheric helical motion and/or  loxodromic motions are the weak nuclear force.


Polarity is accounted for by relative "direction" of spaciometric rotation and is quantized.


Spaciometric rotation is a store for all vorticular motions and both aggregates and dis-aggregates vorticular motion in a quantized way. This may be an artefact of the observer under the Heisenberg uncertainty principle and the sampling constraints of the CN- PN widely distributed massively parallel processing System.

Periodicity and rhythm are statistically averaged and standardized polarity switch signals in an Ed Lorenz spiral system and are the basis of all clock cycles.

Loxodromic motion behaviour describes the boundary of a region, and is the chief contributer of mechanical properties ascribed to a region.

The basic motions of a motion field are radial expansion and rotational movement around a radial centre. This is very Keplerian, but in this case the statement refer to all motions in general not simply to a specified plane surface.

The Basic measuring tool is thus for a plane surface a "protractor" marked off in degrees  or radians and in a set of concentric boundaries whatever curve, or a radial 3d version of this in a set of concentric boundary surfaces or surface elements around each intersection with a radial .

The tool thus covers dimensions, orientations and scalar magnification for any general form and provides a reference frame for locii, that is paths of movement. This is essentially a spherical coordinate system, if the boundaries chosen are a sphere, but clearly is ellipsoidal or cubical of those boundaries are chosen.

Thus the fundamental reference frame ideally consists of radials and concentric surfaces intersecting those radials.

God i love those cascade events! You know when something simple happens and a huge series of events is triggered off.

Everything is in motion, but i generally do not perceive this because i have such a vast array of equilibrium positions!

Cascade events tend to occur when something changes from one equilibrium position to another! On its way an object may disturb a set of equilibria so that we see a series of equilibrium adjustments, rearrangements spreading out from an initial "cause", but more accurately -entrainer.

 Here (http://www.exploratorium.edu/snacks/proprioception/index.html) is something to do, while i tell you a story.

Are you sitting comfortably? Then, a person sat at a computer, reading and deeply engrossed in what is on the screen, hears something over to the side. Looking briefly over, hardly turning the head a clearer sound is picked up. `it is late, and time is passing more quickly than expected and there is something else to do. Being called reminds, and slightly annoys, but motivates standing up and moving toward a door, passing a mirror on the wall. Looking deep into the mirror the reflection reveals tired eyes, a slightly disheveled appearance, and a harried look.

Straightening up and breathing in, while moving toward the door, somehow seems to clarify the mind and the handle feels familiar as it turns, and the door opens onto a raging, foaming torrent! The wind whips flecks of foam onto the face and in the soft light fall someone is waving, out in the midst of the boiling deep! My god, they are drowning! Drowning and crying for help.

Moving out into the icy, swirling waters, tugging at legs and taking the breath away as it reaches waist height. the person seems to be slipping away. Urgency motivates a strong determined swim towards the person and the gap closes and the coldness of the current tearing the person from the ground abates as a firm grip on their wet clothes reassures every fibre that they are safe!

Turning back with the person in tow an island appears nearby with a sandy beach and an anxious group of rescuers coming out into the blue waters to help bring all to safety, a warm fire and a blanket.

Sat on a rock dripping , a pain in the left hand causes a jump! The beach is gone , the computer is hissing, the hand is stuck fast to the mouse as if in a death grip and dribble is running softly onto the chest.....

Now that is proprioception (http://en.wikipedia.org/wiki/Proprioception)

Some of the proprioceptors are discussed here (http://courses.washington.edu/conj/bess/spindle/proprioceptors.html), but this is quite basic.

Here (http://docs.google.com/viewer?a=v&q=cache:Ma0WWnf8RW0J:www.janushead.org/9-2/colemontero.pdf+proprioception&hl=en&pid=bl&srcid=ADGEESgYBsvSEVbP7dNzDBQ8WyAlIywluurAPMbH7fVW9WZa3XBs9Dg2ObyRYhVpPoiSKMwKarGtvdyH0umTsWzv_AIrSM0HsiB7cgoQwbWo1eYETXLreYRijYUXSK_AgNngtZqF1kdT&sig=AHIEtbQ7GRb5cRIzMYHsiHNetYV4Z9IqXA) a more emotional aspect of proprioception is discussed in the context of the CNS and PNS.

The neural network basis of proprioception is discussed here  (http://physrev.physiology.org/cgi/content/abstract/81/2/539) And a exploration of the Geometry of proprioception is detailed here (http://docs.google.com/viewer?a=v&q=cache:FVvECvrBMPoJ:citeseerx.ist.psu.edu/viewdoc/download%3Fdoi%3D10.1.1.43.4375%26rep%3Drep1%26type%3Dpdf+proprioception&hl=en&pid=bl&srcid=ADGEESi4nvuaEAx83cBwhq798Ln05NyaiXJLBWLsH2_gKom38qRmnXCy_JRbAnCEvx-5H4ig_fw3xV6pXhz2EgMq9c-ERWIK-vflZ2Asc5pS8SzHr4XXgevZVZc35oza9Q_GnCzMiLmD&sig=AHIEtbSde9BueAL_wP4iYMX9nT_dC0IkGg).

The illness aspect or disease of lack of proprioception , while interesting is not foundational in explaining its function in the Logos Response.

Having recently viewed a detailed image of the vestibular systems so called semi circular canals, i am clear that they are spiral in nature and only 2 are orthogonal, the third is at an angle to the other two, not quite orthogonal. This seems to provide an arrangement for detecting rotation.

Proprioceptors (http://jn.physiology.org/cgi/content/full/96/4/1789) and their action (http://stason.org/TULARC/sports/stretching/1-6-1-Proprioceptors-Stretching.html) and the differentiation (http://www.britannica.com/EBchecked/topic/371976/mechanoreception/64732/Chordotonal-proprioceptors).

Some factors that inform equilibrium (http://www.livestrong.com/article/49331-factors-body-equilibrium-balance/) and general navigation functions (http://www.livestrong.com/proprioceptors/)

Finally a respectful system (http://docs.google.com/viewer?a=v&q=cache:YkcEVFyYpBQJ:www.grayinstitute.com/media/3.8_Proprioceptors_Manual.pdf+proprioceptors&hl=en&pid=bl&srcid=ADGEESi_oE5aJg0WnqsUj_onJBANcu8kQWLDV7gt3KQqRuVVeawUn_-rkqIw1x8bCiU5claqi1hLdDeXtdIuTJc3BE9pRPP3wftQ1Y004TT1oJgc10St318lAFw8iLez0S89YUDk96Zx&sig=AHIEtbS5natKGuVLRkIEBSumRmjYr6VCNw) that emphasises proprioception.

In any re-definitiopn of consciousness the sensors have a defining role, and the sensory systems they mesh into, but amongst those sensors the proprioceptors are the most fundamental for engaginng with and apprehending notFS. And whenever an inate or intuitive notion arises it has its origin in these fundamental meshes of sensors, and their local data stores and processing processes. Kinesthesia is the summation of the proprioceptive information to an "inward" audience, while kinesis is the sometimes mysterious and involuntary but proportionately voluntary motion of a system or its parts to an outward audience.

The Harmony rhythm and flexibility of this kinesis can be joy, can bring joy, and can hold and store joy as a physical manifestation, or state memory/storage system. Thus each system state is not only a spatial position, but also a data storage position and allows access for reading and writing this data as in a hard drive paradigm.

Fibromyalgia (http://www.fibrofactpage.com/definitions20.html) and a general slideshow (http://homepages.uel.ac.uk/R.Carpenter/SR/sld001.htm) on sensory receptors, the body mesh from which The Logos Response derives

in spaciometry periodicity is derived from motion on closed boundaries and on self similar open boundaries. Thus peano and hilbert curves as well as sinuous curves are suitable for deriving periodicity. Periodicity requires a mark for the period and a parameter that fulfill this marking function will do as well as a direct parameter usually called a dimension. The parameter enables non closed self similar curves to be viewed as periodic .

It occurs to me that having found out Descartes and Spinoza were about the same exploration of reality that i can now review and revise the axioms in comparison to their deductive, absolutist schema.
But before that i thought it might be fun to learn the connection between all the words based on -ducere.

abduce (http://www.thefreedictionary.com/abduce)
 
adduce (http://www.thefreedictionary.com/adduce)

circumduce (http://www.thefreedictionary.com/Circumduce)

conduce (http://www.thefreedictionary.com/conduce)
co produce (http://www.thefreedictionary.com/co-produce)

Deduce (http://www.thefreedictionary.com/deduce),
duce (http://www.thefreedictionary.com/duce)

induce (http://www.thefreedictionary.com/induce)

educe (http://www.thefreedictionary.com/educe)

interduce (http://www.thefreedictionary.com/Interduce)
introduce (http://www.thefreedictionary.com/introduce)
obduce (http://www.thefreedictionary.com/Obduce)
reduce (http://www.thefreedictionary.com/reduce)
re-introduce (http://www.thefreedictionary.com/re-introduce)
reproduce (http://www.thefreedictionary.com/reproduce)
subduce (http://www.thefreedictionary.com/Subduce)
subinduce (http://www.thefreedictionary.com/Subinduce)
superinduce (http://www.thefreedictionary.com/superinduce)
traduce (http://www.thefreedictionary.com/traduce)

transduce (http://www.thefreedictionary.com/transduce)
produce (http://www.thefreedictionary.com/produce)

seduce (http://www.thefreedictionary.com/seduce)

A motion field is a velocity field because motion is of course "coursive" that is it has a locus or a course or is curvaceous  We might call it a vector ,or attribute the properties of a vector to it, but it would perhaps be a nonstandard vector with partialderivative components with respect to "time".

Such a velocity field would of course seem a conceptual nightmare because of the mathematics, and of course would have a mix of attributes like commutativity in certain motions and at certain scales but non commutativity generally., that is things only seem to go one way generally but in certain situation they may be reversible locally theoretically.

Such a velocity field would have to have relativistic motion, motion transfer and motion equilibrium in a non einsteinian sense. Their is such a field that has been applied to the motion field concept, the velocity field, and it is called Hamiltonian Quaternions. (http://www.quaternions.com/),

The principle vector there is thought of as a velocity and is called a 4velocity! This is different frpm an einsteinian 4velocity

The quaternion in the motion field is of the form ø(x,y,z)+X(ø)*i+Y(ø)*j+Z(ø)*k

Where ø(x,y,z) is a periodic function and X,Y,Z are position functions dependent on ø.

There is an iterative version ø(X(ø),Y(ø),Z(ø))+X(ø)*i+Y(ø)*j+Z(ø)*k

Hamilton always linked x with i y with j and z with k. This made the scalar suitable for a kind of clock function. Hamilton called it time, cos he was interested in time. I do not believe in time in this way and generalise it to a periodic function of motion, so Huygens pendulum period equation (http://en.wikipedia.org/wiki/Christiaan_Huygens) might be one example.

Because dimensionally they are equivalent it seems tautological, but that is the nature of the modern concept of time. What i am drawing attention to is the variable acceleration that is observed but factored out in the definition of time, Therefore the scalar in the Hamilton for me represents several things, chosen according to what one wants to examine: measurement of space deformation, measurement of reference speed, measurement of reference acceleration, iteration control, which in the line of thought thus presented would be a propensity to accelerate, and then i nest that in propensities  thus: a propensity toward a propensity to accelerate etc.

So having found what i was looking for it is necessary to make explicit an implicit notion in the set FS : unity.

It seems that all our science is and has been based on this notion, which has been presented as simplistic and trivial, but which in fact is not. Having numbed my brain on platonic dialectic, and on Vedic subhatras i think i can safely say that it is not trivial! It is a load of drivel, but then that is a different matter, and my opinion borne of the labour they have put me too!

Axiom, space has and is replete with infinite possibility. Of the possibilities i may individually select a collection and designate it as a standard for measure. The rules of the activity of measurement will be, endless iteration of the collection to be placed against or within a form in space untill the space is "covered". This aggregation of the basic collection will be called the measurement, and measuring is the act of iterating and placing this collection together.

If the basic collection can do this then the basic collection will be named a unit, and the collection called the measurement will be a scalar of the unit and may be given a name by some scheme of counting or comparing.

Because this aggregation is arrived at by iteration the aggregation will also be recognised as a Fractal.

The unit is the basic measurement and no smaller measurement can be made with that unit. However i may extend the fractal by proportion either by magnifying or shrinking this basic unit to form a proportionally smaller basic unit. By this scheme i may measure smaller and smaller collections in space.

Whichever unit i choose all my measurement must be done in that unit. If i wish to scale up or down i must do that at the end or the beginning of any,measurement method.

Finally there is no fundamental basis to unity, but each unity is nested within every other unity as a scale fractal.

Shunaya

I have drunk of the waters of shunaya, and am drunken

I have swum in the oceans of shunaya, the warm trochoidal currents have spun me round

I have played on the shores of shunaya, a vast and ceaseless ocean of currents and waves and whirlpools and becalmed waters

I have bathed in the milk of shunaya, and nourished my "atoma"

And, like some replete and burgeoning grub, grown fat on the royal jelly of Shunaya

The motions of shunaya are 3 but the secret of shunaya is the chinese yoke

All who run hither and thither , all who whirl like dervishes and all who are poised like yogi

Do so by the chinese yoke of shunaya

The balance of shunaya is weak yet eternally strong

it is light yet infinitely heavy

it is void yet infinitely dense

From shunaya and self is self reflexive shunaya

And from self and self reflexive shunaya shunaya stands alone

I have seen the map of shunaya and it has many faces without number

Faces that we may ceaselessly number, and never come to the core of shunaya.

Like some maggot i came upon the core and have lodged there growing fat

I will, yes we will emerge as butterflies before we return.

As if given birth to by a swollen womb we will emerge twins into the pregnant void,


And through trochoidal paths dance and play and sing until crossing we return to shunaya

Sines cosines and spherical trigonometry and logarithms are the fundamental relations of all measurement and the yokes of the roots of unity which we derive from shunaya. And the fundamental algorithm of directed ratios of dynamic magnitudes has to be the flip algorithm. This flip i hoped would arise naturally from the trigonometric relations and i believe wihout proof at the moment tnat it does if all measurements are allowed to have opposites, including the radial arm in a spherical coordinate system.

A fundamental insight for me in studying fractal generation is the universal nature of the relative motion field. Thus by a few simple rules and relations a computation can sculpt and place various regions in relation to each other that we perceive as connected statically or dynamically. These relative motions may be interpreted as other things, but what they in fact are, are computationally placed and highlighted and coloured regions of space fulfilling a simple motion rule such as Z_n^2+c =Z_n+1, and sculpted by |Z_n|< x^2

What i am saying is that it is possible to think of chemical, nuclear and electromagnetic phenomena as being the result not of inherent attributes, but of a universe wide fractal generation algorithm.

This at once means that action at a distance is not by gravity as described normally, but by computational iteration as it works out region by region.

Those of you who are familiar with zooming into the mandelbrot will be aware of the time lag in sharpness of image or even presentation of image as the computer manfully struggles to compute the next region!

We often think that some things are still, whereas in fact this is a perspective issue. Nothing it appears is static, All is constantly in motion, even if it is too slow for us to perceive in our lifetime.

I have actually revisited these axioms in many different guises in my blog, but have not set them out in a list fashion.

As i have, i feel, come to a natural conclusion in a theory of space, i feel inclined to slowly and carefully sift and revise the axioms.

It is more inviting, and it makes more sense to start afresh in a new thread, and that i propose to do. However, it is likely to be very terse and a bit like a contents table, hence my reluctance to do it. Therefore i will retain this thread as a commentary on the progress.

I have to say, that the axiomatic approach has been a kind of philosophers stone. As a springboard it has been useful but in reality i have written as the mood and insight took me. Therefore i may find inconsistencies and contradictions as i do this, and that in itself will be interesting!

I have now selected 3 guides to help me in my revision: Euclid, Hermann Grassmann, and sir William Rowan Hamilton.

I apologise to any one who has been following my ramblings, as i got the preeminency of Hamilton over Grassmann slightly wrong. The correct relations are discussed in my blog. (http://my.opera.com/jehovajah/blog/)

As usual, further research leads to revision of concepts. I have come upon the source of all mathematics and science in the western world as is named mathematics and Philosophical science; that is Pythagoras. Consequently  have added his work to the body of advisors i shall need to review my axioms.

What are axioms anyway? When i started 3 years ago i was under the impression they were greek in origin and established by Euclid. But i am reading Euclid in the greek and have yet to find the word. I will ponder mor e on this as i shape the matrial.

As luck would have it, my axioms have been very essential to the unraveling of this whole fractal structure, and so i am encouraged to pursue a more accurate version.

I have posted the revision to axiom1 or axiom! as i like it. It is hard and meant to be rigorous and obscure! This is not deliberate except in the sense that it invites any one to deliberate the axiom. The terms and notions are fully derived, as far as possible in my blog (http://my.opera.com/jehovajah/blog/2011/10/11/the-logos-kairos-sumbola-sunthemata-summetria-theurgigical-response). You will have to search the archives there but that is not a bad thing  :embarrass:.

Hey, do not be afraid to reply guys and gals, Damen und Herren, Mesdames et Messieurs. Take a bite. I promise i won't bite back, but those who have followed my meanderings know that already. However, the moderators may moderate any vitriol :crazyeyes: so be considerate :yes:

The second axiom is quite detailed, but gives a potential dynamic process that underpins the first axiom. At some stage i want to support the notion of self and then i as a fractal product of iterative processes, and i think i need axiom 2 and 1 before i can posit that.

Axiom by the way is derived from the same root as axis, and that is a kind of direction to travel in.

Axiom 3 is a draft. "What a mighty work is man...... in intellect how like a god!", and all that jazz..
Pretty cool guy,Shakespeare.

"Time is not a mensurable quantity" I think Lagrange may have said that

Axiom 3 in fact applies to all self aware animates and biological entities from microbe to mammoth, from bacteria to Blue whale, which evolve over time.

The virus is not excluded but is a special cse of horizontal gene transfer as opposed to vertical gene transfer, Without the sensorially dense boundary only simple interactions can take place with a virus. A bacteria however with a more sensorially dense boundary can exhibit more behaviours and interactions over time.

Axiom 4 is more related to the old axiom 3.

The philosophical axioms i originally proposed have served me well, but new insights have clarified things in my mind. I know the new axioms are even more weird and wonderful, but so are fractals! Hang in there and see where they take you!

Less is more on many an occasion. Although i have conducted research into the basis of the axiomatic system, and know it to be  a Euclidean teaching style, based on the best Rhetorical practices of the day, that same research indicates that Kairos, or proportionality is required. Thus, to over complicate the presentation is in fact to go against the rhetorical rules established millennia ago and championed by Pythagoras.

The Axiomatic approach of course has a history, but the Euclidean and indeed the Newtonian examples of it are models of appropriateness and accessibility. One cannot help being the self that the neural interaction has established, but one can review and modify expressions to make them plainer or more accessible.

The seemingly pernicious goal of mathematical abstruseness is a difficult habit to eradicate, seeing as all so called mathematicians are spoon fed it as pabulum. However when one matures it is hoped that one will put aside childish things in favour of portehouse, that is good convivial communication.

Quite often, when this is done the triteness of the topic is revealed, and indeed we may fall prey to thinking in terms of trivialities. However Fractal geometry comes to the rescue, for it is from such trivialities that the most beautiful and profound effects are constructed via iteration. Thus a true democracy ensues, because anyone can contribute, and watch the unfolding fractal patterns that iteration produces.

So 2300 years old and still packing a punch! While reading the greek of Euclids Stoikeioon i found out that The so called postulations are in fact demands! The word postulate does in fact mean to demand or petition, and not as i thought to propose! I guess the clue is in the use of 2 different words :embarrass:

In any case the master of axiomatic style is no more than a petulant teacher demanding of us that we accept certain things as so!
Thus my lofty ingrained ideals of the merit of such an approach are brought crashing down into the mud of human frailty!

I do not demand of you that you accept these axioms, i beg you to critique them.

I have no confidence in the systematic logic  per se, because the conclusions depend entirely on the quality of the premises.

Therefore, Axiom this or axiom that is not to be taken in its highbrow meaning in this thread. It is merely the ramblings of the fevered mind of some old geezer in the uk!

However, Euclid's ramblings have had quite a powerful effect on modern intellectual thought, due entirely to their Pythagorean content. So make of it what you will, but my intention is not to browbeat anyone, but to welcome all  ;D

I have recovered slightly from my shock, and have researched the terms axiom and axis in my blog. Plus also, Plato is the kingpin here in the west. He redacted Pythagoras and employed Euclid in his Academy. Thus his demands are not petulant, but driven by a need to establish academic standards of construction and rigour. Later pedagogues may have acted petulantly.

The so called axiomatic system is also a late 19th century invention. Euclid and Plato relied upon Duality and veinal systematics, not axiom. There are no axioms in Euclid, only Aitema, that is "Items demanded". and Ennoia, that is "common notions".
The axiomatic system i will thus modify accordingly, retaining what is useful.

I am a recent (FF) member and sparse contributor.  Until about the time you relocated some of your writing, I was reading with interest.  I was not comprehending all of it, and cannot personally contribute to you lines of reasoning.
While I was musing around this morning I did have a curious thought and I wondered what you might think about it.

You are interested, I think, in general, of "other" sets.  I get the notion you are presenting is that the recognized sets in mathematics are incomplete.

What do you think of the concept of "impossible sets".  heheh...  Something besides "infinite sets", or "complex sets".  I don't do well at those classifications.  Actually I sort of thought that "impossible sets" were what you might be pointing toward with your descriptions and analysis of the setFS.
For me as a human being, impossible sets are valve to escape compulsive trying to solve for the construction of things constructed.  
We aren't really using the "sets" we know of now very well as a species.
Or something like that.

Could I also intrude upon your interest in fractals to introduce my main interest, which is a script I (continue to) write that animates fractal flame parameters.  (I promise I will be impolite only this one time intruding into the development of your thought here.  But I would like you to see. please. I don't have time to be here at FF myself: [edit 120717: I didn't fully realize the link would put the actual animation here, which I think is inappropriate.  So here's the link separated a bit to keep the anim from loading:] http:/ /www.youtube .com/watch?v= uAJ-WASvX30  :))

I hope I am able to get back into reading your setFS and otherwise interesting postings.

Fred's my name.  and on the web of course, others a few. :)
pleased to virtually meet you, if I may.

Hiya Fred!

Am i allowed to say OMG? Well i have said it anyway!
I am Sam, Fred and pleased you took time out to just chat a little.

Set Theory was developed particularly by Cantor and Venn, and i forget who else, though Boole made a significant contribution. It was particularly Bertrand Russel who highlighted the paradoxes of the theory as written, ie the predicate paradoxes. In so doing he motivated a whole generation to a form of rigourous mathematics that has turned into rigour mortis! It is so abstract it just about kills the subject for most people!

It just about killed it for me too, until i came to FF as you call it, and got caught by the wonderful surge toward the mandelbulb.
When i first came here i assumed everyone would be mathematicians!. Lickily for me they were Artists and artisans in the main. Lucky in 2 ways: the mathematics forum was under used; my meandering and sometimes obtuse style was not vigorously challenged, so i did not have to spend time in argumentation etc. I could philosophise.

Impossible became a word i rarely used because i thought we had a handle on infinity(which we don't by the way),so the strongest i could commit to would be improbable(because i thought we had a handle on probability, which again we kind of do not).

So set theory as a means of describing a collection of anything is  a bit of "mathematical window dressing". i picked up the lingo when at school and it was established in me at university, but what it means other than a Summetria i have not got a clue. It sounds mathematical though. ;D

I shall give your impossible sets some thinking time, and look at your interest too.

Hi Sam!

It's good to be able to use a shorter name. Thank you.  Although I've seen your user name in several similar forms - I like your spelling choice.
Using or saying OMG is not something I find offensive.
There are so many I'd like to chat with more substantially.  Oh the time...

I have so enjoyed your survey of earlier mathematicians.

I was seen to be good in math when I was in my teens and twenties.  But I didn't want to be a teacher.  What was left to interest me?
Hahaha... I didn't know, but it would be fractals that rekindled my interest.  Oh the time...

I follow your obtuse writing ok, I think, except for its mathematic depth.
I've read considerable material from the 19th century and lacking media writers of the era tending to compose very long sentences.
It's good really... reading long complex sentences helps hone one's concentration - at least in some cases including mine.

I think I presented you with some kind of paradox really.
By impossible I more or less meant "humanly unreachable" - or for those who take such "other realms" to be existences of fact, I meant these "impossible sets" to also be unreachable by or through any beings capable of consciousness.
So anyway, an impossible set is impossible of definition. 
I suppose in that, I've left mathematics entirely and ventured into philosophy an area which I try not to tread over.
I'm a nitwit when it comes to philosophy! ;-/

So... just some bits more of my fractal flame interest.  The most difficult math I use is spline interpolation.  Not much really, but once I learned that splines were used once to frame boat hulls - to make the necessary smooth curves - I began to see how widespread their use is in our modern world.  Some of the math I encountered was developed by a person working for "Peugeot".  The math was so developed so that engineers had an easier way to describe the smooth curves of a car's body and I suppose other features.
When I was a kid, the shapes of cars were very square. Now cars are all bubbles.  I think the car mfgrs got somewhat carried away with their curves.  :-(
But the curves are everywhere.  I think nearly everything that's made in some kind of mold uses some type of spline-like interpolation math to design and manufacture the shapes.
As Artie Shaw used to say:  "Interesting, but not funny".
I mean, a 1957 Ford Thunderbird is a very interesting car to look at.  About any late model car I look at today *might* be a late model thunderbird...

But wait... there's more.  The windows on cars at one time were flat.  One would only see the reflection of the sun bouncing off those windows if one were in some particular spot.  Now, it doesn't matter where a person might be.  ALL cars on a sunny day reflect light at a person as long as there is not obstruction between the person and the rays from the cars.
My feeling is that all of these inescapable rays of sunlight on streets where there are many cars are very very bad for our eyes.   

Popping on spline-designed, darkly shaded sun-glasses is not a very good solution. 

Fred

Fractal "splines", Fred are related to trochoids and Roulettes, so you got my interest there. Trochoids are what i once called the manufacturers dirty little secret, because these smooth polished curved surfaces rely on them for manufacture and finish. and of course, trochoids are at the heart of the Mandelbrot set.
http://en.wikipedia.org/wiki/Spline_(mathematics)

Thanks very much for that excerpt Sam.
I've been walking and looking at some of the "Victorian" vintage architecture lately.

I've been wondering how some parts of some of the structures were built.
I cannot quickly find a good example, but I believe some similar technique would be used in the construction shown here:
http://leadingleadworkservices.com/ESW/Images/Cupola.jpg?xcache=2874

Haven't time to look further now, but I will try to snap a photo of some very particularly interesting "domes" I've seen to show you. 

Ummm... is all of this ok in the setFS thread here?

I so appreciate your considerate reply, once again!
Fred

Hi Sam

I haven't figured out yet how to use a digital camera I recently borrowed.  But I will try to get a picture of that architecture I mentioned in my last reply when I understand using the camera.

Some clarity about the way the notion of an "Impossible Set" occurred to me has returned.
I was simply thinking about the way "imaginary numbers" are in fact, not "imaginary" at all.

So my notion was that an "Impossible Set" was some set not genuinely "impossible".

Maybe related somehow was a "koan" I've been wrestling with lately.  I suggested in another topic to those who have pondered the idea "I think therefore I am" to do the thought experiment: "I am not".
Or in other words, what is the experience of "I think 'I am not'"? 
And what about those who can be "I am" with a stilled mind, absent of the arising and fading away of thought constructions?
(Maybe such yogis and monks would understand "impossible sets".)

The connections between these two notions are still in a formative state in my mind.

Fred

Only just recently learned the term koan in some research i was doing on the pythagorean scholarly material. Pythagoras may have been mythologised who knows, but some of his harder sayings seem to fit the description of a koan.

While you may think you are off topic in this thread, you are not. The axioms are meant to be the most tautologically obvious statement of an attribute. So imagine my shock when i find that axioms are not even from Euclid, but a late 19th century school of thought about logic and logical systems. Aitema and Ennoia and Oroi are all that Euclid apparently laid out as the entry requirements to his course on Philosophy. The rest he builds as he goes along using these foundational principles.

Now splines may be foundational principles that need to be added, particularly if we want to describe human engineered products within the set. Trochoids for me are fundamental in a way i can not fully explain. I derive much benefit from meditating on them, especially as Laz Plath presents them.

Your particular koan i have tackled in the other thread on the Axioms, which is slowly being developed and refined.

I have an orientation inclined to the East (but have lived on the West coast of the USA about 4/5 of my life).  It is so interesting to me Sam, with your wide range of knowledge of historic mathematicians (East and West) that you should be also interested (it seems) in the mystical(?) representations you've mentioned.

I pulled myself back from "metaphysical" speculations some time ago.  I had a terrible time when I overindulged my belief in some of the presentations.
Once all that outer shell was removed, I started understanding (I think) some of the firmer bases of practices and the thinking behind them that I've read in scriptures of all the religions. 
I must say though, that I recognize only superficially, many of the names you note.  And many names I recognize not at all, whether the names represent ideas or people.

Din't someone, at sometime, say something like "East is East and West is West, and never the Twain shall meet"?
(I cheated and looked: it was Rudyard Kipling.)

Perhaps both the author (or speaker) of those original words and myself both need some further study time!

Cheers!
Fred

I am not sure if i wrote this in my blog or here, but one of the things i found out when i got older, was just how many "fairy" tales i was told as a kid to keep me interested! Now i have come to see how human our knowledge is, and thus dependent on individuals and their belief systems.

We arrogantly were taught to pooh pooh those belief systems, while we used the results of that way of thinking. Very pragmatic, but ultimately creatively sterile.

I do not say that one has to believe a belief system to use a method derived from it, but i do say the belief system deserves equal merit of mention when expositing a method, otherwise the heuristics is lost, and creativity is harmed. This is why i think empiricism is so important. By being empirical we can judge one method against another as regards its underlying principles, and determine which principles are empirically more "congruent".

Unfortunately you do hurt peoples"feelings" in doing this, but the individual's beliefs, though judged, are not hopefully otherwise denigrated. And every belief system i know of has that big area called "the mysteries" from which devotees may garner further and more compelling insights.

Howdy,

I mentioned around reply 44 that I would try to snap a photo of a structure that I believe was constructed with the help of some kind of "spline" technique.  It's the top piece of the cylindrical section of the building pictured in the attached photo.
If you turn the photo upside down you can see that the profile of that "dome" has some resemblance to the profile of a ship's hull.
I know that some kind of spline mechanisms were used in constructing ships but I don't have a very good grasp of how that was done.  I think the method used to construct the dome must have been similar.

The building is a relatively modest structure compared to others in this area which I think may have been built around the same time.
But this one's dome shape is more unusual than the others I've seen that have similarly curved or rounded structural features.  Most of the others have simpler shapes like cones or multi-gons.
(My architectural vocabulary is miniscule.)

As far as I know there was no mathematical spline at the time these building were constructed.
Even blueprints would have been drawn with the assistance of stiff metal guides that were bent and kept in place by pins.

Fred

The house looks modern to me, judging by the facade. Is it really that old?
The mathematics of splines developed from the artisans pragmatic approach. Many pagodas and other Chinese roofs will fit a spline model in some aspect.

A lot of architects perhaps forget the considerable wisdoms in basket weaving, beadwork and knitting. These do not look like hi tec solutions, but in fact they are way beyond hi tec in terms of material technologies. When an earth process crumples a layer of the ocean sea floor like a piece of cloth, one should look beyond just the analogy to the scales of forces involved.

Fractals as a paradigm provide some vey intuitive insights and expectations.

Since I discovered that axioms are not what I was taught to believe they were, I have not been so inclined to axiomatise the thread. Instead, I am thinking carefully about indexing it under common or fundamental ideas/forms.

I cannot deny that the axioms I started with, no matter how much they were not what I thought they were, or what they aspired to be, have been pretty crucial to holding my meandering thought processes together!

I do not know how useful this thread is as a resource , whatever, but I do know that if I had not moved to my blog site, it would have been about ten times longer! So some indexing is in order!

Currently i am working around the development of the axiomatic system in a direct Historical line from Euclid to Wallis to Newton to Kant to Schelling and Ficht and to Justus Grassmann. This is revealing the historical dynamic to the axiomatic school of thought and of course many fundamental mistakes and misdirections and misinterpretations! Its a wonder we get anything right at all!! :rotfl:

http://www.youtube.com/watch?v=ct_uGOSPtok
Norman gives this lovely insight into the algebraic constructions of splines, and indeed they are fundamental!

Recently certain insights have transpired to lluminate my path Ahead. I have come to see that my research is close enough to that o Normans for me to usefully spark off his research and motives.

I just want to recommend his universal Hyperbolic geometry as the best set of "Axioms" you could currently have from a geometrical point of view.

The Axioms i have historically laid out here are more `philosophical viewpoints in that regard, a bit like Newton's `principles of Philosophy. The newer Thread on the Axioms Exemplifies this and i will tackle them on that basis rather than as so called axioms.

If i have not said it before i am saying it now:

The dynamics of the sphere underpin all of our Astrological, so called mathematical, notions , and the spaciometry of that sphere, its so called geometry, derives from Mechanical Philosophy.

The Stoikeia of Euclid are not mathematical textbooks but a 2 or 3 year introductory course into Platonic Socratic Theory of Form/ Ideas in which the Pythagorean School of Thought is most purely transmitted, with Aristotle's version being a variation which introduces some logical inconsistencies, due to his incomplete graduation.

The Euclidean outpost of the Academy recognised the Graduated status of Eudoxus as mathematikos, and passed on his introductory teachings on the Logos Analogos method of Rhetorical justification and measurement.

Beyond this introductory course in Philosophy there were many other rhetorical works dealing with Mechanical , Astrological and Social, metaphysics which the Seniors of the Platonic Academy engaged in Discourse.

That Newton, particularly was sensitive to the profound teachings and investigations of the Pythagorean schools, and penetrated their heart, goes without question, and that he laid out such a groundwork for philosophical reasoning that we might always conclude aright, is commended by many if not all.

To  this i might add the thoughts of Hermann Grassmann and Sir William Rowan hamilton as able commentators and developers on the Newtonian Principles derived from the Pythagorean schools of great Antiquity.