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

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## Sequencing

Given any scatter of points we define 2 kinds of sequencing.
The first is sequencing by observer orientation;
The second is sequencing by observer translation.
Given points a,b,c,d.... The observer either positions itself on dot a or establishes itself as an extra dot O. Then by Turns (!) the observer includes each of the dots in a sequence. It is this " by turns " that establishes the sequence of orientations that maps the sequence created from the points.
If a and b are any 2 points then a+b is the orientation of an observer at a looking directly at b . It is clear that a +b is not the same as b + a, but they can be defined as contras of each other

Thus a + b + c +... Represents a sequence of orientation changes which are defined as a rotation. This is thus a rotation sequence.

Now let the observer orient himself a + b, then define ab to be the observer looking directly at b from a precisely when the observer is a + b.

Then ab means a projection of a onto b, a direction drawn from a to b by the observer translating in a straight line onto b. it could also be thr system of points that define a straight line at each point between a and b that are precisely on a straight line.

They could equally be defined as the straight line H generated by ab.
If ab is a line then abc cannot be a line. It has to be a projection of a point onto a line or a line onto a point.

To translate around a sequence of points I have to do ab + bc +' cd..............in which the observer must be oriented correctly at each pont for the form ab ,bc... etc. to make sense..
Although rotation seems to be a magnitude, it is a special kind of magnitude: a self referetial one. We could define rotation as the fundamentl ratio Logos in which any 2 magnitudes of the same kind have to be copared. it is a matter of fact that unless this minimal condition is fulfilled we cannot subjectively determine rotation, let alone objectively. The 2 magnitudes used are the radius and the arc on a sphere with one being fixed either objectively or subjectively or both. Any change is compared against the fixed magnitude, usually the radius or diameter of the sphere.

If the arc is fixed, then the centre of the circle moves away or closer but the arc curls or straightens as it does so. This behaviour is used to define curvature.

This behaviour of the circle is perhaps the clearest example of quadratic proportionality or perhaps mor generally hyperbolic proportionality. It intimates that some magnitudes we experience do not vary in a simple logos Analogos relationship like projective geometrical ones with fixed constraints. Our experience of variation in sequence is more complex than that.

## Motion

In the System i propose there are only two fundamental motions: closed and open rotation relative to a subjective processing centre.

In relation to any motion there is subjectively and objectively a contramotion. Thus subjectively all motions are paired to contramotions, and consequently basic analysis of anything must acknowledge the contra description subjectively. The balancing of the contra motions means that a third motion is universal and it is called null motion. Null motion is subjective and relativistic and is the essence of invariance, motion without apparent motion, difference without change, analogy without exact reproduction unless it is in process, in which case it becomes identity:one and the same;indistinguishable. And yet identity does not have to mean indistinctness. It may be that the distinction is subjective, and thus only accessible to that processing centre.

In other words, there is no objective difference,but there is a subjective difference that the subjective processing centre may exploit to advantage or disadvantage. What may be subjectively determined is thus always potential to objectivity, but what is objectively determined is offered up for acceptance by all.

Within this consensus of agreement lies many compromises and diversions. Thus it should always be remembered that the subjective experience for an individual is king, and not to be down graded to the consensus, but rather the consensus is a model for agreed sequential action which may require revision and renewal in the light of experience and some more utilitarian subjective approach of one or many.

As an individual, value your subjective experiential continuum, and do not be disavowed of it by bullies or social pressures. Learn to maintain and switch between your notions and the consensus without umbrage or aggression. Be proportionate in all things good or bad, who knows what will turn to your fortune or misfortune?

Contra will be given a vector orientation interpretation not a sign or process interpretation. From the vector orientation analogies will be drawn wit regard to process direction, or pole status. Thus + and - will be freed for a specific use defined in terms of vectored motion. Laz Plath's Circa. Brilliant app.

This picture represents rotational motion contra and progressive relative to 4 polar centres, with the finished path being relative to the subjective processing centre.

We clearly need the notion of semeia in motion and objective centres of rotation,but more fundamentally the compass multivector network is what the subjective experience consists in. We experience it ,thus it becomes invisible as it is the medium of the experience. However without it we cannot define plane or rotation or direction etc.. The compass multivector is thus the informational basis of our experience,and as a tool is so pervasive that we often do not realise that it is in constant use by a subjective processing centre.

http://wsdt.office-on-the.net/resources/DTonCD1/school/motion.html

Perpetual motion is found everywhere in nature, an atom with the electrons spinning in orbit around the nucleus, the earth spinning around it's star (Our sun), and so forth. As newton put it, an object in motion, tends to stay in motion unless acted upon by an outside source. So in this case a motor has all sorts of losses, resistive losses in the coil resistance, which is inevitable, air friction, gravity friction, and friction in the bearings and then of course bemf, which in this case has been turned around to act as an advantage, not a hinderance. These frictions are very small losses and there is more than enough energy to compensate for the losses, therefore boosting the energy well over 100% and into overunity ranges also known as free energy.

It is impossible to generate more energy in a system than what it consumes. If this were true then the device would be running in overunity mode and would be over 100% efficient which is impossible. However it is completely possible to generate more energy than what the device consumes if the local enviroment is involved or some other energy that is outside of the system which can be drawn or attracted into the system. Namely the the power of magnets and how to unlock it's hidden power and potential will be discussed.

A system may only appear as perpetual motion and may appear as if it's charging it's own battery or running itself if the observer doesn't understand the "how" or "why" it is doing so or where the extra energy is coming from. This extra energy maybe hidden from the view of the observer because it is outside of the system, not from within it and not well understood. The device in question is really actually drawing in the extra energy into the system from outside of the system. I would like to emphasize that it is important that there is energy outside of the system that is available to be drawn into the system. In this way we can get a system to self sustain once the energy outside of the system is realized and understood.

In the case that is currently being spoken of, the source of energy is very simplistic, and that is the magnetic field of a magnet. This can be considered a natural source of energy from the earth in the form of magnetite, but it is sometimes difficult to understand how to unlock the hidden power of a magnet and make it do real work and produce useable amounts of current and power, even above the amount used to power the whole thing.

Often the terms zero point energy, radiant energy, aether, cosmic energy, dark energy and antigravity are spoken of in the free energy field. These all represent basically the same concept. It is often said that we are surrounded in a vast sea of seething energy and that we can tap into this universal power that surrounds us. However it is somewhat difficult to understand exactly what that means and how to create a device that taps such a field.

So I will explain it right here in simplistic terms that a layman can understand. To make it easy, Einstein said that mass is equal to energy, that means we are mass therefore we are energy! The fact that we are surrounded by a reality that mainly consists of mass that we can physically see, can only mean we are surrounded by energy! I believe this is what is meant by the phrase that has been mentioned before "We are surrounded by a vast sea of seething energy"!

http://eternalmotor.t15.org/

## Rotation

Rotation is the most surprising action, especially when decentralized.

Within rotation is paralleism and parallax and synchronism and origami. The notion of vectors is encapsulated by rotation. Some experimentation i am doing is showing how rotation has generalised "number line" concepts into spatial vector concepts, and preserved the notion of combination.

Euclid when defining "multiplication" by arithmos used a gnomon, the essence of rotation,to place the combining monads. Thus 2+3 in vectors rotates, generating area volume and form with form relationships as it does.

Any 2 centres of rotation can rotate in "parallel", "synchronous", equiangular rate of spin, etc providing a dynamic basis fro relationships which are periodic and stable; providing the notion of reference frame; providiing the notion of transformational geometries, and finally the notion of sympathetic vibration and resonamce.

In the previous post i set out some formulae for rotation vectors. The consequence of iterating the yin yang formula is quite dramatic in Quasz. It also highlights the point that sign , although a useful shorthand is very misleading spatially. Vectors completely replace the need for sign, and vectors are of course combinations of the trig ratios /functions. Combinatorics needs to be emphasised over addition and subtraction which are specific forms of combination or aggregation, as i have called it while searching for the meaning of it. And this meaning is to be found in Kombinationslehre.
In any case the combination of 2 and 3 has to be specified by object and by vector, and the resultant is a spatial arrangement, or rather a selection of spatial arrangements, from which we choose as we need.

The compass multivector network thus specifies an arrangement of objects in space, and if these objects are contiguous or continuous then a single larger object id described, but if not then an arrangement of objects is described which may represent anything from a dust cloud to a solar system and beyond.

The rigidity of the relationships in the compass multivector network describes the additional attributes of the larger body with regard to flexibility fluidity and phase state etc.

The conservation of energy and mass principles may be described in terms of compass multivector networks, and the growth of a leaf and a child may similarly be described.

The data required to process these changes is huge, but computational tools not only make it possible, but feasible, and it is regularly achieved through modern computing platforms.

How do we teach this stuff? Surprisingly Euclid set out a method 2300 years ago, based on a platonic redaction of Pythagorean philosophy which is more than equal to the task. It has been called Stoikeioon ever since. Roughly translated it means "Data set out in an orderly fashion for teaching". The Data is about space, that is our subjective interaction with space..

Many geniuses have learned a thing or two by studying it.

The path notion arises in the contex of semeia as a dynamic referenc to a moving semeion. As a consequence it relates also to moving subjective vectors and thus a path can be described by a set of subjective vector p + p(). The set of subjective vectors can be seen as a set of dynamic forms p+p()+∂(p+p()), and a path may in some circumstances be describable by a function of subjectiv vectors Ω(s) where S is the subjective processing centre, which is also the hub for all subjective compass vector networks, which would be used in describing a path.

A path Ω = ∫∂(p + p()) which is a continuous or contiguous combination of all the transformations in the subjective vectors from p + p() to p' + p'().

The path notion is very similar to the trace ofa moving semeion, or a gramme, but a path is a subjective experience.

The circular path or the path around a closed object are special forms of a path, as are straight lines

## Compass Multivector `Network

The first notion is a compass vector

z= e z

or z=f(z)

Z is a vector that means it has a subjective orientation relative to the subjective processing centres frames of reference. Additional notions of magnitude may be attached to this orientation vector as per the arithmoi gematria or Hamilton's algebraic arithmetic, a modern exposition of the gematria of the arithmoi.

The recursive definition is because z is dynamic and iterative.
an explicit definition would reqire a dynamic form using the differential calculus such as

z=f(v)+∂f(v) where v is a vector, f a vector function, and the change in the vector function f describes the changing vector z. This is the calculus of variations set up.

I could also approach it from z= v+∂v where ∂v is a change in the vector v not the scalar multiple so ∂v can be any resultant vector so it is in effect and nature a vector function and thus should be exposed as such. Thus z=v+∂f(v) making it clear that this is special form of the calculus of variation set up.

This may seem hard but all i am doing is describing using a notation for brevity that my subjective vector changes relative to my initial set up. My iniial set up is therefore my reference frame for any and every change.

I have not specified a sequence against which the changes will be observed and "measured", but it is crucial to note that i decide on the sequence on which the changes will be dependent. Thus i cannot take any sequence and compare the changes with that sequence, Unless that sequence was chosen to determine my sequence of observations and measurement. This requirement is called dependency but it really means simultanaeity, synchronicity, co temporality,in step, in rhythm, confluent,in harmony, in proportion etc. It is fundamental notion of comparison that is subjectively apprehended and then objectified.

A periodic motion for example is appreciated as periodic by subjective apprehension. Thus the initial compass vector description of a sequence of motions is stored in memory, and the processing of further motions compares for differences and updates. In this way cyclical or periodic behaviour becomes apparent as an epiphenomenon of the measuring process, ie one realises or perceives the motion is repeating!

A pendulum or any rotating motion has been the sequence motion of choice for measurement because of this,

Now a pendulum has curious behaviour under gravity that Huygens formulated. The period of the swing is defined! as the same in order for the clear decrease in motion quickness to be measurable. Subjectively we see the speed of the bob slow down, and we experience it take "longer" to complete its period route. But its route actually also gets "shorter". The proportional behaviour thus is apparent and the only determination we can make of such proportional behaviour is that a constant relationship exist. This is an "invariant" relationship and so we can use it to define other relationships. In order to do that Huygens defined his formula as unit so that the variation in pendulum length could be used as a scalar for period or duration.

The formula looks a little complicated, as they all do sometimes, but it obscures the simple subjective act of setting up the period "metron" by defining it for unit values of the factors in the formula

http://en.wikipedia.org/wiki/Pendulum

When i first learned of this i thought of it as a tautology, a mind bending definition? Now further and fuller investigations, particularly using Lagrangian techniques has shown this invariance is only approximate. Thus the definition of period is totally subjective and culturally enforced. "Time" if it is compared against some even more stable system will vary, but that is not an issue wih "time" it is an issue with the tautologies that we hav to use to define anything. In truth, what we accepts defines our apprehension of the experiential continuum.

Making change sequence dependent enhances the ability to describe change. but it also gives a quality to change. Tf the sequence is not continuous or contiguous, the change cannot be described at each intervening interval of the sequence. Thus what happens between sequence intervals may be so different as to fundamentally ater our apprehension of the change.
As we have used faster and faster film/video frame rate speeds we have been able to measure the actual changes with startling revelations.

The rotating vector therefore is important on describing these changes and relativity of each vector is essential to describe fully a changing scene

A network is built up by combining a scatter set of rotating networks with a relativity factor.

## The compass-Vector Network

The gematria of the semeia being a subjective experience of objective locations in space, both relatively static and dynamic, highlight the subjectivity of the notions of point, encapsulating it within certain indicator activities which derive from subjective motivations,and experiences.

Nevertheless we still can and do build a representational, recording and on some way countable or mensurable and definitely paradigmatic set of "standard experiences. These are subjective objective links,by which we may apprehend space ,somewhat objectively or through objective means which nevertheless are entirely dependent on subjective processing.

One structure that arises is the compass vector model of a semeion or a semeiotic experience. In this model both changes in orientation and orientation are encompassed as a whole experience. Changes in orientation are otherwise known as spaciometric rotation.. There is also a subjective a priori notion of distance, and consequently the notion of vector attaches to this change in focus and parallax.

The full definition of a compass vector involves a network of vectors relative to one or more compasses, and in itself forms a basis for all fundamental subjective spatial notions including perspective.

It is therefore possible to asses the notion of vector in terms of compass vectors that associate to relative perspectives ,and any dynamic within space corresponds to a dynamic change in compass vector network perspectives.

This more general notion of a compass vector network as encompassing associated perspectives and parallaxes , meand that a compass vector network is sufficiently complex to describe any motion in space subjectively without recourse to special spaces such as planes and angles. Thus every motion is describable by a compass vector network with perspectives and parallax and the associated changes of the same.

From these basic compass vector networks, i may Then develop the notions of planes reference frames and fixed centres of rotation and relative motions and relativity itself.

The compass vector network with perspective and parallax enables any motion to be described by an iterative sequencr of compass vector networks with relative parallax and perspective changes, and thus the constiute for me a definite notion of the motion sequent, introduced in an earlier blog.

The compass-vector model however does not naturally include the twisting and bending proprioceptive senses that underpin rotation and change of orientation and the fixing of orientation. Thus a compass vector is a resultant of a subjective process of relative strain statuses within my main body. These relative strains not only inform orientation but pitch and roll and yaw. This revels that any vector carries a lot more subjective information than just rotating to orientation. In fact at least 2 centres of rotation are necessary to describe any orientation. In practice if i pick any semeion it is located by at least 2 subjective processes: one involving the eyes the other involving the body and ears.

## The shunya Field

This is simply a rebranding of my concept of space, but highlighting a substructure called a subfield. There are 3 empirical subfields , density variation(gravity) electric variation, and magnetic variation.

These subfields are aspects of the shunya field, like a diamond with many facets, remove a facet and the diamond is destroyed. So space as a 3 dimensional entity cannot exist unless these 3 subfields interact exactly as they do. Removing one subfield may create a subspace never known before.

The subfields have the same characteristics but different actions. They are characterised by multipolar interactions, and are in fact multipolar fields. Their actions are different at the boundary of a rotating surface in the shunya field. The shunya field condenses or rsrefies surfaces around it multiple poles. The subfields have their own unique multipole distribution. The electric multipoles are where the subfield concentrates and rarefy electric action, and the magnetic subfield does the same. What is not admitted is that the gravity or density subfield concentrates and rarefies density action around its multiple poles.

The subfields do not have to coincide in terms of their multipoles, but when they do there is a special coincidence of the actions.

The subfields are like elastic bands. Some of the poles cluster together under the field action while others are pushed apart. The difference in action leads to the differences in motion. Fields and subfields, whatever they are are made objective to us by the motion they engender on a substantive and dense region of the shunya field. Thus fields are causal as far as motion is concerned. However the presence of multipoles in the field is an unexplained reality.

The action of the empirical fields generates rotational motion in the Shunya field, but it is possible that many other fields may yet be found empirically that contribute.

I have not mentioned till now the aether concept. Although i do not work with that concept personally, i do not denigrate anyone who does. However the philosophy of those who accept an aether is different to those who use a Pythagorean philosophy.

http://farside.ph.utexas.edu/teaching/jk1/lectures/node6.html in which the section on multipoles is of interest, but does not precisely match what i mean by multipolar.

http://galileoandeinstein.physics.virginia.edu/more_stuff/Maxwell_Eq.html
http://www.space-mixing-theory.com/
http://buphy.bu.edu/~duffy/PY106/Electricfield.html

http://pw1.netcom.com/~sbyers11/#10.2

http://library.thinkquest.org/10796/ch12/ch12.htm#s1

Multipoles for me are the centres of rotation and condensing and rarefaction within a field. Their distribution is not uniform and neither is the condensation or rarefaction around a pole of the multipolar subfield.

http://world.std.com/~sweetser/quaternions/qindex/qindex.html

## The vector gnomon

Vector addition is based around the trigonometric attributes around the triangle, thus the gnomon in the unit circle is very instructive.

The gnomon in the unit circle ties together the three sides of a triangle and the associated arcs. In vector terms any two vectors when added rotate and extend or diminish each other . Thus addition is effectively a rotation transformation.

The summation of vectors is defined geometrically around a form and it is an action. The action is to get from A to B. Why aggregate around a form? The behaviour of space and the motion in space respects form or deforms form. Respecting form means we have to aggregate distances and going around a perimeter does that. Deforming space means the lengths or edges have to be elastic and defining a unit vector with a scalar represents that. So noting the form produced y ana ction records all these conditions. So why highlight one dimension as the resultant when the geometric form contains the information?

When i perform the action of measuring or traveling my interest is the direct measuremnt, but i do not want to lose the other information. so the resultant vector is not always the goal.The systematic application of operations produces a record, which can reveal combinatoric information and derive formulae. Vectors begin to highlight effective organisation of the permutation and combination of actions in space.

Suppose we concentrate on the resultant, then addition traces an extension and then a combined rotation and extension. This describes well the effect of non parallel forces acting on the same object. Thus the form we trace out by drawing motion holds true for the motion of larger objects. The trace records motion under the same type of force distribution scaled down. Thus the vector trace also records model forces, and suggests applicability. The geometry in this case is secondary to the force model used to trace the geometrical figure.

Using this analysis we can define a rotation vector that represents a drawing force that rotates a centre. Again the motion can be characterised as an extension followed by a extension around the circumference of a circle. In this translation there is no further radial extension. Not shown usually in drawing the circle is the vector triangle that is in a pair of compasses, or the tension vector in a taught string anchored at the origin. or the action and rection vectors as one draws around a circular form, Every one of these vectors needs to be recorded to describe the actual situation. When that is done then the proper vector description can be noted and the proper geometrical form results which gives the desired results. We can also se how a rotational unit vector is interrelated to other vectors in the situation.

In one case vector addition is the same as just an extension, when the vectors are parallel or contiguous, but as i have said before position of a vector has also to be recorded otherwise the missing positional information can lead o a distorted result.

When i look at the analogical frame of references that encapsulate the analogies vectors are linked to i see that vectos as extensions of forms are used to model motion But we have no direct way of using motion as an analogical basis in our current systems. The tautology always starts with extensions of space or form, and we represent motion by a trace which is an extension of form called a trace line or a locus.

The only pure way to capture motion is in sequents and these seqoents have to be ordere and displayed exactly as in a film sequence or a series of frames. Motion sequents are the only analogical base that uses motion not extension, abd i will continue to explore its impact on describing reality.

The research question is using the definition of a motion sequent as an image frame in a fim reel how do i devise a notation system to describe what is happening in sequence of frames. How do i generalise this to 3d, and how do i directly relate this to electrnic storage of film sequences? Do the different forms of recording alter the nascent algebra ? Does the media editing industry already have a model i can use and generalise? What is the significance of the computing paradigm and the influence of computer science? Has computer science and programming languages already developed an algebra in flow charting? Does iteration and convolution have a role in such an algebra? Does fractal geometry< and the work of Benoit Mandelbrot?

The possible answer is out there. I can feel it in my flowing water!

## Euclid and Hamilton

A point is that which hath no part

A point is that which has no part.

Euclid and his school set out a developmental, systematic and thus logical exposition of the subject they were engaged in.

Why start at a point. I have discussed before the difficulties around the notion of point.

It is unlikely that Euclid started at a point, and more likely this is a redaction introduced over time as the system grew in influence and popularity.

If this observation is accurately translated then we have no problem, for it clearly states a point has no part. Thus the problem lies with the interpreters, and for me the most useful interpretation is that a point is an abstracted notion derived from the sharpened corners, hooks and vertices of objects, and serves as a reference marker to hook, position, or measure from.

The common, observational nature of these definitions is not to be overlooked, for Euclid wrote for those who were artisans, not just abstract thinkers. All his audience therefore knew how to abstract from a real thing and were never confused about there role in the elevation of everyday palpable thing to eternal verities.

In every definition Euclid appeals to their perceptions, there experiences , and their judgements of form, shape and surface to establish a common denotation, a common jargon that they will use to shorten communication, and to increase precision, but never to obscure or mystify for the purpose of aggrandizement. His 23RD DEFIFNITION is a very practical explanation of what he will denote by the term parallel.

Some have taken these definitions in the sense of properties, but i take them in the sense of attributional denotations, after all what is a property if it is not attributed?

Thus we may see Euclid's practical use of the sensory system to establish a common language, and to promote a common observation.

Without this kind of acceptance of common terms, no agreement can be reached .

Much is different between Euclid and modern terminology . Definition 8 for example defines angle as "angle using boundary lines", and the inclination of those boundary lines. This is very interesting, because an angle to euclid is a measure of inclination and is dynamic. Therefore if a boundary line is a circle the inclination to a straight line is constantly changing as one progresses along the boundary: inclination therefore can only be a measure of rotation relative to an orientation.

Euclid then denotes a special type of rotation one that is fixed and unmoving, in terms not of rotation, but motion along a direction traced out by a boundary line. Rectilinear motion produces fixed inclinations , non rectilinear motion produces variable inclinations/rotations.

Euclid uses this notion to demonstrate that a circle boundary is a greater rotation than any acute rectilinear rotation that is clockwise or less than any that is anticlockwise depending on which boundary you are rotating from as you move along the curve boundary.

The notion of angle therefore is precisely linked to rotational motion and comparison with articulated jointing.

`in exploring this in one of his theorems Euclid appeals to contradiction of established rules, as a explanation, by demonstration, why he asserts certain things. But the support in this manner is given by contradictions of simple real life situations, so for example a length cannot be both longer than it must be, when the length it must be is determined precisely.

The definition of plane angle, definition 8 is i suspect not couched using the modern notion of angle , because Euclid appears ti define form in terms of boundaries not angle. Thus we have trilateral, quadrilateral and multilateral rectilinear forms ,with these other curvilateral shapes.

The concept of angle is a very important concept for all of Greek geometry. Many of the propositions require angles even for their statements.

The two lines are meant to emanate from the same point; two intersecting lines will actually make four angles.

The concept is also a difficult one, and, surprisingly, broader than our modern concept of angle.
As can be seen from the next definition of rectilinear angle, angles do not have to have straight sides; they can have curves as sides. The size of the angle does not depend on the length of the sides, but is determined only by how the two sides meet. In the Elements nearly all the angles are rectilinear, but angles with curved sides appear in proposition III.16. In that proposition, a so-called horn angle CAE is described as the angle between a circle and a straight tangent line and is shown to be smaller than any rectilinear angle FAE. Even though the curved side of the horn angle extends beyond any rectilinear angle, it is considered to be smaller since near the vertex A of the angle, the curvilinear angle CAE is entirely contained in the rectilinear angle FAE. Thus, an angle doesn't have an extent.

As you may see , the interpreter has the problem of what is meant by the terms. I believe the notion of angle or hook is an foreign one to the greek notion of inclination/rotation as a dynamic constituent of motion along a "curve". For a rectilinear"curve": that is a straight line there is no rotation from a fixed line or orientation.

Greek geometry is dynamic and full of motion and actual sensate experience from which deductions and abstraction, definitions/fixed starting points are drawn. to not observe that that is the case is to create unusual problems of your own devising.

Euclid could easily consider a rotation relative to a curved reference trace, but it had been found that the most communicable reference frames were rectilinear trilateral forms with perpendiculars, and rotation relative to the boundaries of the space between the perpendicular could be most easily exposited.

To think that the greeks were not aware of relativity and the need to establish and use common reference frames is to be disingenuous

Note that there is no requirement that the angle be rectilinear, indeed, the horn angles mentioned before are not rectilinear, but they are less than right angles, and so are acute (notwithstanding Proclus' remarks to the contrary).

With these definition, we see another aspect of magnitudes, namely, two magnitudes of the same kind, such as two angles, can be compared for size. Euclid frequently uses what is known as the law of trichotomy: given two magnitudes F and G of the same kind, exactly one of the following three situations hold, F is less than G, F equals G, or F is greater than G. See the comments after the Common Notions for more discussion on magnitudes and the law of trichotomy.

Trichotomy is so evident in the very fabric ans structure of Hamilton's Exposition of conjugate functions or couples, that it is hard not to cry out Euclid/Eudoxus when reading this paper. Therefore the soundness of Hamiltons conceptions depend entirely on he soundness of the greek style of reasoning. Experiencing Hamilton's thorough exposition is a modern example of an understanding of greek deduction, unfettered by translation. Hamilton emulates the style and dependencies of the greek style and thus communicates an activity Also encoded in the greek, but not written in greek: Style.

In scripted form, in dance moves and in spoken or song words, the nonverbal communication is as important and relevant as the denotation of a grapheme or a phoneme. Thus, though Hamilton clearly limits his subject to "time", he , as he states, uses geometry analogically. Therefore, in reality, there is nothing that is not touched or informed or shaped by the greek science of geometry. I refer to it as spaciometry partly to free myself from modern misconceptions of dynamic greek geometry, and partly to emphasise that it is about the measurement of space by comparison of unit spaces in motion.

The revealing of the foundation of ars, that is motion sequents, though fundamental, and involving experiential,sensate perception, sensory mesh computational outputs, is more detailed rather than new , Greek dynamic geometry would always arrive at Newton' s Fluxions, Hamilton's couples and quaternions, my moment sequents, but only as the ideas of Brahmagupta, Bombelli, Descartes,and DE Fermat, Laplace,Lagrange all contributed to a common rhetoric and notation for referring to it.

Amongst the greek ideas, those who were students to the philosophers had understanding, those of us who have learned from translations of their work have conjecture and opinion. It is only as the utility of the ideas are vrified that we gain insight and understanding of what the philosophers undoubtedly knew and meant.

Do i mean by this that we have not advanced beyond our father? Indeed i do. And by what little we have gained in advancement, the more we have appreciated the state of their wisdom and knowledge and conjecture.However i am not saying that we should look back, or "go back", but rather we should go forward, bringing all wisdom forward with us so we might properly understand it by experience.

I am reminded that it is but 200 years since it was scientific dictum that life arose spontaneously from common everyday materials like wheat and rags! You may laugh, but it is still scientific dictum that life arose from common elements in chemical reactions under very violent conditions: the same idea but with more detail!

When i develop a wave theoretic description, i fundamentally utilise a dimension called a radial. Of course no such radial line exists, as it is entirely an abstract attribute that i make to distinguish rotation around a circumference or on a spherical shape.

In point of fact the radial line is an integral part of a right or orthogonal triangle. Therefore, inorder to establish a "wave" theory i require some actual distinctions in spinning space to represent these relationships.

Firstly i note that the sphere is a perfect abstract concept. What i actually perceive are what may be termed spheroidal spinning spaces or even more distinctly orticular spinning spaces. Both these visual types of spinning space have features which serve as anchor point to attach the abstract concept of a right /orthogonal triangle. thus it is the variation in these perceptible distinctions which enable a wave theory to be constructed.

Now for a given spinning space the distinctions may have a fixed relationship to one another, and so facilitate the apprehension of the space spinning relative to the background. On the other hand they may not have a fixed relation and therefore make it possible only to determine a local self referential motion or a convoluted motion relative to the background motion. These motions define relativity, and display the reference frame anchoring freedom that exists in perception.

To each centre of rotation perception may attach a relative local reference frame, so that i as a perceiver may adopt any of these frames as frames of reference from which to perceive all the others. However these reference frames are not possiblr if the spinning spaces have no identifiable feature or features to anchor the right triangles to.

Therefore spheroids and vortices will be fundamental to any constructed wave theory, and the wave theory itself will be an alternative description to the rotational behaviour of spinning spaces. As a description it introduces, by way of exposition and explanation redundancy of description of the rotation, but detail of the orthogonak direction and generation of the rotation.

## "IT's Hamilton time!..... Can't touch that!.......can't touch that!

Hamilton begins his development by sytematically laying out all the common relationships between moments of time. Hamilton is able to draw on a consensus of these "time" relationships and therefore is able to "perceive" these things as being in time. However this is a "mis-perception" as these relations are a refined subset of spatial relations used to construct the notion of time.

It was an interesting find of mine when reading some presentation on the arguments of Leibniz against Descartes philosophical notions of motion. The lecturer noted in passing that it was not until Galileo that our forefathers had any modern notion of time. Acceleration for example was as easily related t distance traveled as "time" 'traveled', and some examples were given of ow ancients described time in terms of travel distance or distance in terms of days traveled.

So it is not surprising that such a "close" linkage exists between time and space, as one is in fact a refined suset of the other!

It is not enough to say that they are distinguishable, which by instrumentation they are. One must recognise the tautological construction of time from spatial concepts and notions. In fact, the concept of ratio is a powerful enough matrix to comprehend the notion of "time" in all its forms. We may easily conclude that "time" i the motion of space in space relative to a standard periodic spatial motion. that we tend to use rotation is not strange either, as all motions are examples of rotational motion, or/and rotational expansion/contraction motion /and/or reflected motion in the "centre" of rotation between one region and its π radian opposite. This last is a special motion that i will write again: reflection occurs only in a "point" that is a centre of rotation between a region and its 'mirror" opposite. Thus any region within any subsuming region will have a reflection pi radians from it around a centre of rotation. AS we investigate subregion by subregion we will note that the subsuming region is reflected in an AXIS made up of these subregion centres of rotation. The behaviour of a mirror axes reflection is entirely different to the behaviour of a reflection through a centre of rotation for an infinitesimal subregion.

In fact if we reflect all the subregions through a common point we get a different behaviour to reflecting a subregion through its unique point connecting it with its rotation/reflection by π radians.

I have only recently elucidated this relationship, and so am still exploring it. Nevertheless, these 3 motions are necessary and sufficient to describe all motions, and therefore are to be found in any utilitarian construction of the "time" ratio.

Concisely, then, Hamilton is expounding on dynamic Spaciometry by developing his "pure Time" conception, beginning with the systematic survey of geometrical relationships.

Now some may confuse his systematic approach with Logic, and they would be justified because Logic is precisely a systematic survey of the relationships between all forms of expression and there attendant referents. In addition it provides on the basis f these studied relationships rules of conduct to "win" an argument, including"ad hominem".

While many do not regard its full field of study, Logic has become a weapon in the hand of able and articulate rhetoricians, while not in the least validating anything at all as "true" except in the specific sense of "consistent".

While consistency may e a prize worth having, it is salutary to realise that truth at the last evades us as it is an abstract notion of little substance, but major consequence.I myself prefer the empirical notion of "true": congruent,exact, in line with, coherent etc.

December 2013
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