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

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Subjective Groups Leading to the Origin of the Inner and Outer Grassmann Product

A point is that which has no part

Seemeioon estin, ou meros outhen

The Grassmann concept of a Strecken is like an object oriented class definition. The class line has three properties: direction; length; points that fulfill some function. We might attribute some colour to these points to visually identify them.
However a point has no parts by Euclids definition.

We cannot write a list of observables for meaurables or orientations for a point. But we can and do write a list of subjective experiences and descriptions of a point. A point has the subjective parts ( properties): meaning; significance, hen we communicate about a particular point we communicate about its meaning and significance, that is wholly subjectively. Thus we give points a subjective reference frame which we carry about with us internally and use to subjectively identify experiences including experiences of topos or place.

Colouring a point is just that, giving a distinguishing experience of a topos with its meaning and ignificance hooked onto that experience like a coat on a coat hanger.. These are subjective structures, internal models and maps of external experiences.

Even though a Metron is deemphasised in a Grassmann Algrbra, it is still one of the properties of a line. To maintain tht property Grassmann uses the interior Algrbra of points to define a line as a product of points A,B. the usefulness of this is that these points mark off a Metron in the extensive Algrbra, by which coefficients are derived. Thus in this form of lineal algebra there is an implicit Metron, and this guides the use of any explicit Metron.

The notion of a vector has this metrical implication implicitly, and so is a good instance of a Strecken. Where a vector concept is sometimes confusing is where it is suggested to be somehow implicitly free of all these relationships implicit in a Strecken. The mixture of implicit and explicit use of properties is why the algebra is so subtle. . Very often, Grassmann draws on the intuitive implicit properties without explicitly stating the fact. This he inherited from his Fathers struggles with rigour.

Hermann corrected mistakes his father Judtus had made without sacrificing too much of the elegance in this way of thinking. Later researchers, for rigours sake, attempted to split subtle points into 2 ignoramus concepts only to find they lead to other ifficulties.

The blend of what you fudge and what you expose is demonstrable in any system, axiomatic or not. Axiomatic systems tend to set the fudges out at the beginning, but they still inhere in he system!

We have to live pragmatically, and that is why the pragmatic seemeia on is so important. It's a fudge, but it makes the whole ytem work usefully., we can hide all our fudges behind the seemeia on! That means subjectively we knowingly or unwittingly delude ourselves in order to get a pragmatic result.

The Schwerpunkt developed from the observation that a point exists in a topos, that is a place. This place is not explicitly referenced, it is subjectively referenced. However, the practice developed by Descartes, DeFermat and organised by Wallis to set up a reference frame called a fixed axial sytm.mthe Measuring line was used to model these axes which were set orthogonally to each other in a standardised format. What this meant was a point could be referenced by two " numbers". .
This is a misconception of the reference frame, and it has persisted to its detriment.

Those who wanted to break free from Cartesian coordinates could not put their finger on the problem. Grassmann did. The point has 2 properties in a reference frame: position and magnitude. We generally ignore the position and focus on the magnitude. Grassmann realised tht this type of point was different to a Euclidean point which has no parts, except subjective ones. The point had a position and a magnitude on the axes. This is then used to project onto a third point in space by parallel lines to the axes.mthis point does not have a magnitude in the reference plane it has only a position specified by coordinates.

However it could be given a magnitude using Pythagoras theorem, and so a Schwerpunkt could describe a conic section point!

If I switch to a polar coordinate frame thn every point in the plane has a position and a magnitude. Thevschwerpunkt deals with a major inconsistency in traditional reference frame theories.

Grassmann uses this understanding to define the inner and outer products of Strecken under "parallelogram multiplication".

What is a Strecke? The simplest and noblest notion is " a construction line". It is a subjective notion of our intention and application to construct. We conceive it before we even draw it, and its meaning grows as we construct. Once its job is done, it fades into the background

The angle between the Strecken becomes crucial. Up until this point it had not been considered, but his work on the ebb and flow of tides advanced his conception of the algebra. In the case of the lineal algebra the angle has to be included in the analysis, and that means the trig functions and surprisingly the exponential logarithmic functions.

His concept of parallelogram multiplication meant naturally that 2 Strecken in the same line and in the same direction would produce a zero parallelogram. Also two Strecken in the same line but directed contra would do the same( gleichgerichtet). He called this behaviour the "Aussere produkt". This seems to be because there is no projection line involved in this conception of the parallelogram. The Two Strecken form the outer perimeter of the parallelogram, and both flow out of each other ( auseinander tretenden)

However there was another case when the Strecken produced a zero parallelogram: if one Strecke was projected onto the other Strecke this designated a shortened Strecke. If two shortened Strecken lie against each other then their product will be zero. This perpendicular projection involves the cosine function( arithmeticsche produkt but now called the dot product) and as these Strecken fall within the given Strecken the parallelogram constructed by these Strecken is an inner product!

The Grassmann Outer product is about the strecken spreading out from one another as you step away the strecken like clock hands, the Inner Product is about the nearness( Annaherung) of the Strecken in this same process, but for the Inner Product it was important that the projection was perpendicular onto each other. In this way the strecken have a reciprocal value applicable .This actually makes the Grassmann inner product

AB* cos^2¢*sin¢ if ¢ is the angle between them.

The outer product therefore represented a construction based on parallel lines, while the inner product is based on perpendicular projection and then a parallel line construction. Although this is not the work up for covariant and contra variant vectors, it is the source of that technology. Grassmann specified a vertical projection( perpendicular ) for his inner product, but the Euclidean inner product works slightly differently in where it projects the Strecken to.

Now Grassmann was keen to put his results and discoveries in a second " Volume". Especially as he believed he had found out how to represent undulatory motion and angle in his algebra. He was so excited that he wrote this in his first Vorrede as an overview of good things to come, in case pressure ( of circumstance) delayed the publication of the second volume. How true that fear turned out to be, and then some. The uptake of his first volume was minimal! Yet it alerted Peano and Hamilton to a great genius. I have written what I have written about Gauss and Riemann, with some corrections I might add, but the plot is the same.

The inner and outer product are crucial to representing angle and undulation. The use of the exponential is also novel, but well founded.

The inner product never exceeds the pi angle! This is due to the insistence on drawing perpendiculars onto the other Strecke. As the angle between the 2 Strecken alters the outer product goes from an acute to an obtuse parallelogram in its outer product. The two Strecken must step out from each other, that is emerge from a common join or point. This product actually goes negative when the angle exceeds pi. However many ignore this formalism in geometrically constructing the product, something Grassmann warns against, we must observe all the conventions! Thus as Grassmann points out, you can represent every outer product by accounting for and interchanging the signed designation. To keep equality when designations for Strecken change you must change the sign of the whole system.

Grassmann has contrasted Strecken that were connected to each other by a join, whose directions or orientations were ticking apart like the hands of a clock with the vertically projected Strecken which got closer the more the projecting Strecken got further apart to each other , that is in a partition sense they were reciprocal to one another. What he meant by that I think is that the Strecken rotated apart the shadows they cast vertically on each other drew closer to each other, not in orientation but in "nearness". For the inner product this gave "geltenden Werthe", that is applicable values by parting in a reciprocal manner to the angle spread. This must be a reference to a table of values namely the Sine table. The dot product makes use of the cosine tables, but the Grassmnn inner product is a more complex combination.

Spend some time just appreciating how the different internal angle changes the sign of the product! This is for the exterior product.

The interior product is a bit more involved. Drop vertical /per pendicular lines onto each Strecke from the other Strecke.. That means for points ABC and Strecken AB, BC drop vertically onto BC from A and vertically onto AB from C. The two Strecken from B to the perpendiculars are used to form the inner product.. It can only form in an acute or obtuse angle so it never becomes negative. The standard or Gibbs vector inner product ( dot product or Euclidean product) does go negative unless a restriction is set on the angles.

Grassmann insist only on the construction. This means that as the Strecken pass the pi/2 boundaries they have to be prolonged backward to perform the construction.. The Strecken marked off are now in the contra directions of the outer product Strecken so produce a positive parallelogram product. Strictly speaking, these constructions produce strecken outside of the initial Strecken but in the same line with them ( gleichgerichtet ), so the inner product is distinct from the outer product.

The construction constrains us to use the angle between these projected Strecken to construct the parallelogram. This is not the same angle used by the outer product, so because Grassmanns construction of the parallelogram involves 2 same signed cosines( cos^2¢) and never uses the reflexive angles the result is always positive for the inner product. The construction of the outer product involves only the sine of the angle, so the resultant parallelogram switches sign when the angle becomes reflexive..

Grassmann noted that the inner product did not change sign when the Strecken designation was changed, so for the inner product he had commutativity
AB = BA

Grassmann utilised this fact in his Ebb and Flow of tides paper, to establish an identity between the angle , the trig and the exponential functions of the angle, he appears to have expressed it in degrees, but the point is it is an IDENTITY. This means that we do not evaluate the numbers we switch between the two to get a facility for visualising what is being modelled. We could say that it is a map onto the Grassmann product planes if that helps!

Grassmann then goes on to formally deduce the Eulerian form from his algebraic representation in inner and outer product form.

His point, briefly highlighted in this Vorrede was that his analytical method was as general if not more so than Eulers!

How come it produces an analogous identity? This is simply a consequence of Grassmann writing a linear combination of the outer and inner products and applying the combinatorial rules, it is to be observed that the outer and inner products coincide in the same manner as the i and the numeral products, but in zero rather than in 1. Using this as the angle measure and the exponential function he models a sine and cosine identity. Clearly an evaluation of the inner product is required akin to but not at all the same as radians, because they are an arc diameter ratio, the inner product is to all intents and purposes an area of a variable parallelogram.

https://en.wikipedia.org/wiki/Bivector

http://www.amazon.com/wiki/Exterior_algebra

http://brickisland.net/cs177/?cat=5
http://ia700209.us.archive.org/7/items/collmathpapers11caylrich/collmathpapers11caylrich.pdf

The Theory Of Cartesian Extension Grassmann,Euclid and Eudoxus Style!

Platonic Socratic theory of Form/Idea as taught by Euclid is an analysis of the sphere, and indeed the relationships between spherical forms as they overlap, their spherical surface from total inclusion to total exclusion. In short, how bubbles connect and disconnect.

The notion of a sphere is analysed down to a seemeioon, and from the seemeioon several special forms and relationships are distinguished as a useful superstructure for synthesis of any form using distinguished seemeioon.
Here the distinction between the pragmatic seemeioon and the formal or analytical one can be made.
The theory of form/ idea requires a set of tools and procedures used in the analysis and synthesis. To formally explain these tools requires that the theory of Form/ Idea be tautological or self reflexive. To hide, obscure or avoid considering this fact theoreticians ether hand wave some convenient axioms into place as givens,( that is axles around which their demonstration will swivel!), or they simply demand or require you to assent to certain understandings, that is they postulate or itemise as a requirement.

It is perhaps hard to see how thy got away with it for so long, but these behaviours are enshrined in traditions, and are overlaid with some traditional benefits, not the least of which is societal acceptability! Indeed so jealous are the guardians of their society of their societal cohesiveness that they will require some sunthemata of their postulants( those begging to join) and seal the agreement with some sumbola.

The Academic system today still follows this ancient Aristotelian tradition, extolling entry requirements to any course as a way to maintain standards.

Regardless, Euclid makes pedagogical demands on his students , which of course results in a social norm being imposed by ability. The demands are constructional : you must be able to draw a straight line between points; you must be able to extend that straight line as far as necessary; you must be able to extend straight lines so they cross one another or meet , however far away that is: you must be able to draw a circle( in a plain surface set out for drawing) as big as is needed.

The final requirement is a assent to the notion that all orthogonal " knees" are identical . This in fact can be proved, but the Stoikeioon is an introductory course, and Euclid left the proof to a later stage or level of the course. It is in fact a theorem of Thales, which he reputedly brought back from Egypt .

The motion of identity or identicalness ( iso) is used to support common understandings or judgements about formal combinatorial relationships, but the final procedure invoked is a physical, pragmatic lifting and rotating of the rigid form and placing it on top of the putative identical form.

This physicality, which provides visual and kinaesthetic confirmation of "fit", that is contiguity of boundary, from which flows without definition collinearity of arbitrary lines , and also the notion of " Artios" and it's conjugate "perisos" which are fully introduced in Book 7 with the mosaic concept otherwise called Arithmos.
Arithmos allows us to interact with form by processes of factorisation, counting, fractionalisation, and synthesis of multiforms. This along with Book 6 represents Euclids teaching on the Eudoxian theory of proportionality, which Eudoxus derived analytically from his deep study of the sphere., the goal of the Stoikeioon.

The pragmatic seemeioon represents this formal setting in everyday affairs, thus attesting to Playo's theory of form through the intriguing question? Which is real: the pragmatic or the formal? Each man must decide for himself.

The pragmatic seemeioon is construct able where the formal one is not. Every formal seemeioon is in fact represented by a pragmatic one. Thus the Grassmann's in tackling this issue , struggling to make a university level course accessible to primary school kids, as well as to justify their resolutions gave such a fine reworking of Euclid's Stoikeioon that we have a full algebraic representation of it. The groundwork of "rigour" laid by Justus, enabled Herrmann and Robert to develop a consistent Formenlehre. But it was Herrmann who cracked the formal difficulties and inconsistencies of his Fathers approach. This he did by a rigorous re analysis of the problem based on his fathers framework..

If a Vedic scholar would recognise the Ausdehnungslehre as a philosophical Guna, and the lineal algebra as a Ganitas Algrbra, they would have a deep appreciation of Hermann Grassmanns intuitive thinking!

It is quite beautiful to read their thinking, as Robert saw an opportunity to build his own ideas on his Brothers. Roberts ideas are principally laid out in Herrmanns 1862 reworking of his 1864 Ausdehnungslehre. The surviving principle seems to be Keine Abweichung! That is invariance of result no matter what the form.robert published several more books on his version of the Ausdehnungslehre, in honour of his fathers pioneering work in what later became group and ring theory.

There were others working in this field at the time, most notably Hamilton, and to an unknown secret extent Gauss, but Hamilton acknowledged the superiority of Hermanns work, not realising perhaps that it was virtually ignored in Prussia!.

The reason I wrote this post is because Hermmann's lineal algebraic rules use the notion of parallel exclusively to define the lineal algebra. This constitutes a second decomposition of space into parallel and intersecting planes, contiguous with Euclid/Eudoxus decomposition into concentric and MULTI centred spheres. The order of priority is, for a given set of tools, points or seemeioons are required to synthesise spheres and spheres are require to synthesise circular planes and circular planes are required to synthesise straight lines, and all of these forms are constructed from "iso" " points" that is dual seemeioon. By intersecting spherical surfaces precisely in these dual points/ seemeioon.

All thes decompositions require 2 arbitrary seemeioon as the starting point for synthesis using the given tools.

Once one accepts these 2 kinds of decomposition, Archimedes realised there are a whole host of other decomposition based on the dynamic combination of these which lead to the varying forms of vortices, vortex surfaces like the cone and spirals, all of which I have called the Shunyasutras. In a more western terminology I have also called them vorticular space. Dynamic situations require shunyasutras to describe them, and this is what Brahmagupta was describing, not a perpetual mobile!. Brahmagupta had actually isolated what we call the roots of unity in the sphere, and what I have called the children of Shunya.

The choice, by Hermann to found the Algebra of the seemeioon on the bisection construction, and so the Schwerpunkt as a point midway between the 2 summe point, has intended or unintended consequences when we move up to the lineal algebra. The lineal algebra requires the parallelogram to underpin the process of combination called producting,
'multiplication", factorisation etc. As a consequence the midpoint theorem for triancles surfaces as the lineal equivalent to the addition of points, using parallel lines in addition to bisection to establish the schwerpunkt relations in lineal algebras..

The importance of this is that a complete decomposition of the plane is possible using parallel line constructions. In fact Norman"s course on Wildlinear algebra demonstrates this fully.

I must confess that i do balk at the use of the word multiplication anf producting when describing these group and ring theoretical actions. While i can except them in the course of the Grassmann struggling to reconfigure the logical base of geometry along the lines of Arithmetic, a pythagorean initiative if ever there was one, i find them retarding and confusing i the light of the full concept of group and ring theory we have today.

These are activities or sequential processes of synthesis that we engage in to construct a resultant form/idea. Consequently, since we construct from "elements" sequential combinatorial aspects play a vital role. Thus in general i prefer to rename these processes as combinatorial processes of synthesis.

Thus Hermann noted that one combinatorial process of constructing a triangle from 3 points enshrined the law of 2 Strecken. That a + sign is used in that law disitinguishes it from the law of 3 Strecken in the construction of a parallelogram , where he found that he could enshrine a distributive rule for parallelograms on the same base using the x or , to indicate a different construction process to the triangle which has the + associated with it. Thus the + symbolises a triangle construction process and the x or . symbolises a parallelogram construction process.

It took a while for Hermann to realise that he had in fact stumbled on a greater algebra, a process algebra in which the construction or combinatorial process between the elements and the resultant is key. He describes his exhilaration at realising this in his 1844 Vorrede.

Today Group and Ring theory carry forward his vision, often without refeering to the work of the Grassmanns and particularly Hermann. But in context, it is the result of the Euclidean standard of pedagogy, apprehended somewhat similarly by Leibniz and Newton, that all philosophical musings should be structured veinally, that is usig Isos, and even religiously using Isis. This seems to have been misinterpreted as an axiomatic development. a dream held since the 1500's, that everything can be explained using a few principles. I say misinterpreted because often scholars get their postulates mixed up with their propositions and do not distinguish between Aithema, ennoia and oroi as they should. If they did hey would have to admit that it takes ore than a few principles to explain everything! a lot of defining has to take place and a lot of careful systematic demonstration, proportioning and dialectic{logical consenting} has to occur!

Many others are creditied with the development of the field of ring and group theory, but i think that the Grassmann and Hamilton are fundamental to the present shape of the subject boundary.
Grassmann starts with the point.
Given 2 points the natural combination is a point midway between the 2. That sounds strange, but it is applying the rule of closure. It says that objects that combine must be the same "thing" or like each other and consequently result in the same "thing". Thus combining points must result in a point.
In general algebra the rule is gather like things together, and then form a combination of the unlike collections. Simplify the like collections by writing them as a multiple form, so your resultant is a combination of multiple forms.

Points are" like things" but they are also spatially different. The combination of two points should be just that a combination of 2 points. However, by choosing a third point on the straight line joining the 2 points Grassmann enforces a closure rule for the combination.

Now he can define the product of 2 points to be the line between them. The line between 2 points is usually said to be a straight line. As a collection of points it is clearly multiple. The notion of a multiple form is what Grassmann wants associated with placing symbols together. In defining this he follows Euclid who simply states that a segmented good line of two parts will be used to represent a rectilinear form! This is simply because the sphere or circle is Euclids primitive constructional process. Using a circle in the plane allows Euclid to transform a segmented straight line into a rectilinear form.

Rectilinear forms can be divided into multiple forms by selection of a suitable monad/Metron . Thus any rectilinear form can be made into a mosaic of smaller forms. The Greek word for this mosaic is related to epipedos and it is Arithmos.
It is from these processes that we derived our notion of Factorisation and then as a corollary multiplication. As you have just read division is a natural process underpinning factorisation.
So Grassmann chose the line as the multiple form of points. Precisely he factorised the line into its monads which he defined as points?

The combination of points therefore naturally should produce lines, by this reasoning. But of course which lines,? Restricting the definition to straight lines provides an answer, but of course restricts the kind of lines discussed. Grassmann planned to move to the circular line and who knows here next, but he did not imagine he would be able to do this on his own.. He hoped others would be stirred to action and take forward the research. I understand his hope now in terms of the great Humboldt reforms initiated in Prussia. Of course his primary school credentials meant that few read his book., and Gauss definitely assimilated it and critiqued it, but all were busy with the social and political turmoil of the times. Only Peano seemed to act on it.

So why define the Schwerpunkt?

The algebra of his father,Justus, resulting from his strict analysis of the underpinnings of what was then thought of as the 3 pillars of mathematics: algebra arithmetic and geometry/ logic considered addition and multiplication as key to distinguishing sound calculus from any other form of thought. Justus held with Schilling that arithmetic was the quintessential calculus, with algebra generalising it and geometry hanging about doing very little logically, but providing a kind of intuitive subtext. The difficulty was in defining multiplication in synthetic terms, because it always requires a logic to make sense of it. Unlike addition which seems to be innate in things. This was the strict constructivist viewpoint. To explain multiplication one has to have a prior logic in which to base it. Geometry provides it,mbut it is mute. It requires intuition to apprehend. Thus to claim that mathematic can be constructed falls at this point. Intuition was not believed to be constructed, even if it could be encouraged honed and sharpened by construction experience.

Herrmann disagreed with his father on this. His experiences seemed to show clearly that geometry is constructed from the point upwards! Hermann's inight was the bisection construction. Given any 2 points a third could be constructed between it. However to effect this the line had to be defined , and tht simply was the product of the two points. . Now there does not seem to have bern a definition of a point as two circular arcs crossing. It seems to have been a primitive notion . Similarly a circular arc is a primitive notion. In Euclid all these are carefully defined and one has to assume Hermann and Justus accepted these as not requiring questioning. Thus we can magically construct points given 2 points, a tight edge and a compass of sorts.

This is how it seems to have been started. It is somewhat unsatisfactory compared with Euclid, from which it derives it's logical primitives? It misses much that is of synthetic and analytical concern, but hurries onto secure ground if construction is the criteria using the tools mentioned.
So now I can produce a line AB from 2 points A,B. I can construct 2 other " unique" points using the line AB as a radius centred at A and then at B , thes points I call C,D. I can now produce the line CD, and this line meets AB in its mid point S. this point is the Schwerpunkt and is labelled A+B/2.

This label is designed to reflect the process of finding the midpoint of AB given A and B.

Because the sign+ is used, and because it is a single point, despite the elaborate construction Grassmann defined it as the summation of the 2 points.

Is it naturally the summation of 2 points? Well we seem naturally to want to take a point in between 2 points as somehow summing those 2 points. But where should it be? If we have 3 or more points it seems increasingly natural to choose a point that is somehow equally distant from all the points. This is not always possible, so the next best thing is a process that puts all points on concentric circles.

To take a scatter of points and find a point that places them in concentric circles in relation to it is a remarkable thought. It is of course useful for many things including finding centres of gravity etc. The technical issue of what constitutes a point etc is suddenly dupersceded by a utilitarian prospect: a notation that my consistently describe a process that produces concrete and constructed results!

This is where I leave it. What has emerged, drawing on the Stoikeioon is a procedural algebra, encoding clearly and precisely the elements in a construction but allowing the outcome to be constrained or entrained in the definitions. While looking like a symbolic notation, and drawing on that facility, nevertheless it is a mnemonic of a pencil and pare construction, and eventually an n dimensional construction.

Unlike Descartes Geometrie, the symbols do not stand for quantities, they stand for magnitudes. On these magnitudes we may impress whatever quantities we desire, as long as we remain rigid in tht impressing.. The manipulation of the symbols thn specifies corresponding spatial manipulations from which we can then derive surprising perspectives on the form we have imposed on the symbolic structure and construction.

Much has been investigated, refined, modified clarified since then, but this is in line with Grassmnn's on methodology. The point is that his Analytical method has proved to be robust and crucial to the development of modern Physics in particular.

When Grassmann defined the product of 2 Strecken to be the constructed parallelogram, again using the arc constructed point he did not then fully realise how easily this would generalise, because he did not then fully apprehend the role of the Stoikeioon in the development of human ideas/ forms over the millennia. For example Newton simply used it to define all his measures. Thus all measures are fundamentally based on these so called geometrical forms. If you have ever wondered what the velocity squared is it is a square form whose form is used to represent a quantity. If we have a mass moving then we can use that as the third side of a form which represents a quantity of energy. If you look at the energy box you see that one Sid of it is a surface now defined as momentum. You can see there are 2 sets of parallel faces called momentum and obe set of parallel faces which are squared velocity. At the moment it has no other name.

The box has no physical referent. It is a mental construct or idea of a prescribed set of relationships. It is just an Idea/form illustrating the real purpose of the Stoikeioon.
Why Euclid elements are not a library on geometry.
http://www.mpiwg-berlin.mpg.de/en/resources/index.html

http://www.mpiwg-berlin.mpg.de/en/research/projects/DeptII_McNamee_PicturingNumber/index_html

Picturing Number: Visualizing Quadrivial Concepts in the Central Middle Ages

Megan McNamee

Orlèans, Médiathèque d’Orlèans, MS 306, p. 10 (detail).

Education changed in the central middle ages. While the arts of grammar, rhetoric, and dialectic continued to be taught as the foundation of all learning, the quadrivium, the four arts of number—arithmetic, music, geometry, and astronomy—received new emphasis. Some of the greatest minds of the era, among them Gerbert of Aurillac (later Pope Sylvester II; c. 940–1003) and Abbo of Fleury (c. 944–1004) gained renown for their numeracy. Teachers both, they educated many of Europe's future abbots and bishops, emperors and administrators, and authored new works on calculation, time-reckoning, geometry, and music. Gerbert and Abbo drew their knowledge of number from late antique and early medieval tracts by such figures as Calcidius, Boethius, Martianus Capella, Macrobius, and Bede. Copied and recopied, the texts of these earlier authors changed little, but the pictures underwent alterations that suggest a significant shift in use and an upsurge in interest. Tenth- and eleventh-century makers of manuscripts provided pictures to elucidate textual passages that made no mention of them; where pictures were an organic part of the tract, they often added more. They experimented with placement, scale, color, and contours in ways that suggest a keen awareness of contemporary notions of materiality, sight, and the limits of representation that were themselves the product of explorations in the domains of mathematics and science.

Nowhere was the quadrivium more vigorously pursued than at the cathedral school of Reims and the monastic school of Saint-Benoît-sur-Loire (Fleury), where Gerbert and Abbo were masters. Picturing Number focuses on the rich material legacy of these major schools, which acted as epicenters of an extensive, pan-European network of exchange that linked monastic, episcopal, and lay institutions. It is grounded in the quadrivial manuscripts that members of these communities made, copied, and used at the turn of the first millennium. Over 600 extant manuscripts have been attributed to the Fleury scriptorium; more than sixty contain quadrivial material. Relatively few manuscripts can be traced with certainty to Reims. Of these, many can be tied to Gerbert, who became tutor to Emperor Otto III in 996 and dispatched a number of deluxe manuscripts—several on quadrivial topics—for the edification of his royal charge. These are today held as a group in the library of Bamberg. Gerbert’s epistles are filled with frequent demands and requests for manuscripts. Scholars have identified many of the exemplars, most now scattered across Europe in small municipal collections, from which Gerbert commissioned copies. Pictures abound in these manuscripts. They appear in the margins adjacent to the text and in breaks within the textblock, and are sometimes woven into the very syntax of a sentence. Elsewhere, they appear apart from the text, spanning entire pages, openings, even quires.

Art historians in other fields have looked to scientific paradigms to explain representational trends, notably the use of perspective in the Renaissance or the desire for verisimilitude in seventeenth-century Dutch painting. Their studies have colored present notions of how scientific thinking did or might manifest itself visually in the past. It is perhaps unsurprising that the characteristics of the Year 1000 styles and the Romanesque—the flattening of forms and space, a taste for un-modeled and patterned surfaces, a-systematic proportion, and the intermingling of textual and graphic elements—have not encouraged similar investigation. Yet, if we turn to quadrivial manuscripts, sources of scientific knowledge during this period, we find these same graphic tendencies in pictures constructed to convey quantitative concepts. Such pictures offer evidence that we are, perhaps, too quick to limit our notion of the visual “scientific” and its impact.

Picturing Number contributes to the study of medieval image theory. Visual and textual material drawn from quadrivial manuscripts permits investigation of period-specific notions on a range of integrated issues: the relations and tensions between word and image, the nature of cognition, and modes of representing both the sensible world and that which was considered to be beyond the reach of the senses.

http://www.mpiwg-berlin.mpg.de/en/staff/members/mnamee

Picturing as Practice: Placing a Square above a Square in the Central Middle Ages

Megan McNamee

This essay examines the many pictures of solids added to the margins of Macrobius's fifth-century Commentary on Cicero's Dream of Scipio in light of contemporary geometric practice around the turn of the first millennium. At this time, geometry was less a body of axioms and precepts to be demonstrated and memorized and more a tool that flexed and sharpened the mind, thus heightening the ability to comprehend worldly and divine things. This ability was not considered a native talent; it required teaching and practice. The definitions of geometry's so called elements—points, lines, planes, and solids—in the Commentary and elsewhere, provided opportunities for such practice, that is, for exercising the mind’s eye. Given this, the pictorial annotations added to these definitions are perhaps best understood as goads to the intellect and traces of an otherwise ephemeral activity. Picturing as Practice considers the graphic strategies adopted by annotators along side the didactic tactics and tools employed by medieval masters in the classroom seeking to cultivate the intellectual eye.

The Stoikeioon was repackaged for thes kinds of curricula, but Euclid's original Teaching goal may be surmised from the Title, and its place in the academic institution founded by Plato, supporting Platonic Ideals/Forms
On a Theory of Space
We may start with Space as the archetypal Shunya, whether i wan to divide it into multipolar vorticular fields of force equilibria, or motion field equilbria of a vorticular nature or not( where a pole in fact refers to that unreal entity called also a centre or a point of centrality, rather than a distinguished straight line relationship between two such centres!)

The concepts of materiality of space start with a conjugation into a boundary between my inner experience and my outer experience. Such a boundary is immaterial but serves to define a material experience in an otherwise indistinguishable continuum. It allows me to differentiate continuaa, and to perceive a notion related to continuaa which is extension.

Extension as Descartes eventually came to use it is a subtle concept enabling perceptible things to be in their positions in space according to my perception and distinguishing of them, and indeed my naming of them by a process of comparison. It is my ability to establish or define an immaterial boundary that gives extension its subtlety. Wherever i define a boundary to some perceeption of some quality or essence or experience of magnitude, that discretization of my continuaa allows extension to populate space with entities that are bounded and thus quantifiable.

The discretization of extensive experiences and the boundaries that entail this discretization enable contiguity, continuity, and discretenes to be identifiable concepts. Armed with these i may proceed to a more complex concept which at the last i may call materiality, but which is a convenience to mask the complexity and indeed tautology of experiences, extensivenessesm magnitudesm and quantities that now are laid out before me as Ideas or Forms in the Socratic and Platonic sense.

/the distinction of materiality, ill defined as it is , leads swiftly to another equally ill defined conception of immateriality, which latterly has come to be called synonymously "Spirit" or Essence. Proceeding by the man made "rules" of "logic" or grammar, such a description may not be allowed to mix in any way. Thus in Descartes time and beyond, much time was wasted in defending this "silly " rule and in maintaining gods independence from his creation, if you so will.

However, different cultures did not pin their very lives to such arcane rules, but allowed all things to be mixed as willed according to a Yin Yang polar continuum structure which neve achieved absoluteness, as in western Philosophical/theosophical debate. Thus allowing a more calm approach to the foibles of Natural "Law" behaviours.

The Euclidean approach to Philosophy of Socrates and {lato, The Newtonian Principles of Astrological metaphysics and his co commitant Philosophy of quantity as expressed in the Methid of Fluents, plus his Praxis of Natural Philosophy, all combined to produce a guideline for wesern industrial revolutionary Thinkers and inventors which gave them extraordinary technical abilities but misalligned them to Natural Law. It requires Ed Lorenz Aperiodic behavioural Theories to bring Western Metaphysical theory into line with the Easern Yin Yang theoretical Structures.

In the meantime, materiality and Spirituality have been divorced where they ought not to be in the west, whereas in the east They are one and the same in the polar description of existential experience.

Where Newton mislead the west is in dismissing fluids from consideration in his definition of the quantity of matter. This decision which i now call a mistake, was forced upon him by the medieval beliefs of the church at the time, but was not the case in enlightened Arabia. Where Alchemical 'Lore" was open.
Consequently, though electric and magnetic phenomenon were discussed as fluididc behaviours, they were divorced from matter in Newtons philosophy of Quantity and the quantities of motion.
Newton did discuss fluid mechanics, but only as some adjunct to his corpuscular definition of matter, which is extremely ill formed, but accepted everywhere as axiomatic.

We can address the matter today in some detail and with some confidence of improving the descriptions of matter.

The first metaphysical principle must be(by your leave):

the only goal of this endeavour is to construct a model that faithfully represents what is empirical and observable.

To what use that model may be put discoursively or in rhetorical discourse is not my concern, but misrepresentations of the model are my business to correct.

Should someone construct a similar model, identicality must be determined by comparison to empirical data. If both faithfully represent empirical data as it is known then both models(or all models which fulfill this condition) are deemed identical.The inner workings are of little account to the effocacy of the model, but individual users of the models may express a preference.

Now supposing the inner workings to be the model of the workings of space is not allowed, unless and until observable and empirical data of such inner workings may be presentable.

Thus Hypothesis, based on the models is not allowed, but hypothesis based on phenomenom and empirical data is ,
Recasting of te models, no matter how inconvenient is always allowed, with the aim of obsoleting the former model by the latter.

The duty then of the model maker is to defend the model against misrepresentation, but not necessarily to promote the new model beyond postulation to such as think they may be authorities. Should they reject the new wine, saying the old is better, this should come as no surprise, but whatever advantage the new model gives its creator should be exploited in the presence of the young, that they may be given access to the better way.

Should the old guard seek to destroy such advances and opportunities the creator of the new, improved model is advised to seek employment and living elsewhere, where his " magic " may be a[[reciated. In any case, the lesson of history is that empires are required to change the status quo, or many satisfied customers of your service. Secrets are manytimes necessary to intrigue.

The proposed model of space is simple and observable:
Space will consist in extensive magnitudes that are motile and perceivable as motile, bounded, extensive rigid and fluid. Each of these attributes will be tautologically dependent on the other as far as the processes of the perceiver are concerned, and the perceiver is aware of a fractal relationship of levels and scales and conjugations between itself and that which is not itself, the whole being termed in synthesis and in discourse as Shunya.

Accordingly, when Newton excludes fluids from his model of matter he set himself upon a different course to the one which is proposed here!
Keeping it simple
By conjugation Shunya is percieved as "me" and everything not "me". Me is a primitive undefined concept at present, but its meaning is subject to the reader.
Shunya is now going to be conjugated in an alternative way that cuts across and convolutes with the initial conjugation, but which is wholly dependent on it, that is "I" conjugate Shunya again into rigid and fluid.
I and me are identified as having the same referrent in these 2 cojugations.
Now i may through the Logos, Kairos, Sunthemata Sumbola Processes begin to develop and assign attributes and characterisitics by these convoluted conjugation processes.

My second conjugation thereby acquires the notion of extension, boundarisation, relative motions, centres of rotationasl motion, relative kinematic disitinctions, Relative intensities, and relative sensory representations, etc..

Conjugating rigid and fluid again with these additional distinctions i develop a fractal, scale free attribution of propeties and behaviours etc, a ontiguous and causal attribution of motion transformation, etc, and a sensory mesh distinction and representation of continual transformations. At the last i may adopt and adapt "Panta Rhei", the conception that everything flows.

This concept is important in my subsequent and dependent conjugation processes as i establish my scale metrons to quantify all thes qualitive experiences of magnitude. Later, i will deploy magnitude to cover the internal "m3" and so substantially change its apparent and undefined meaning for the reader.

The purpose of my study is as Newton put it , to by reason and experiment exposit the behaviours and workings of my experiential continuum, and the best model to do this with is with a fluid model.

The almost ideal introductory fluid is Water, because the notions of fluid dynamics can be developed in this medium. In particular, its phase transformations admirably indicate that Everything flows "forwards" as well as "Backwards"

Now , observing water i can make 2 conjugations : one is that water flows as a "body" which i will define as a voluminous stream.
The second is that water flows as an oscillator or undulator, which i will define as a consequence of coupled boundary conditions.

Now conjugating just to the notion of water or fluid motion i have to state fluid motion is an adjugation of bodily motion in a "stream" AND undulatory motion within a bounded condition.

This simple statement is the fundamental synthesisi of Fluid motion

fluid motion is an adjugation of bodily motion in a "stream" AND undulatory motion within a bounded condition.

One cannot fully describe fluid motoions without the 2 "components" being presented, whatever additional properties of fluids are attributed.

i may now complexify the analysis of fluid flow Phenomenon using the Newtonian Method of Fluents and the Principles of Astrology(Mathematica!)
By concentrating on flow elements called streamlines i can track differential bodily fluid flows. BUT, and this is where the simple synthesis principle above highlights a shortfall in experimental practice, i have to also add the boundary conditions of each streamline to describe its undulatory characterisitics!

Thus a complex flow should be analysed by some tool or method that quantifies both these aspects. bodily stream motion And undulatory stream boundary motion.

I am going to point out that the streamlines sysnthesise "extensively", that is within a given volume you combine all the streams for the volume description. but the boundary conditions on each stream synthesise 'intensively" that is by a process of constructive and destructive combination the internal flow characteristics of an internal "test" volume are described.

The combined , superposed results of both processes hopefully model closely the actual flow phenomenon on which they are based.

I also want to draw attention to in passing, that the boundary conditions of the streamline are sufficient , and in effect the complexities of motion internal to the stream line do nt need to be known in detail.

Issues of what may be happening in a streamline at one scale are addressed by using an iterative model , a fractal modeling concept that changes the scale of measurement in a related , almost self similar way.

Also in passing, The use of this fluid dynamic model in modelling Electromagnetic phenomenon must transfer fully, so that electromagnetism is not a "wave" Phenomenon but a Streamline AND a wave Phenomenon. The ignoring of the bodily motion of space in Electromagnetism is the cause of its present difficulties. Historically this was due , in part to the collapse in confidence in an " aether" medium.

While an Aether medium may lack certain experimental evidence, it nevertheless does not negate its role in the modeling process, which does not require the reader to decide on what is real or not, merely on whether the model fulfill its analytical and synthetical purpose of accurately expositing the phenomenon. In addition, if an analogical mechanism is found to function similarly to the observed phenomenon, this validates the modelling method, but dues not make any comment on "reality". What is "Real" at the end of the day is a personal subjective decision.
Before Dirac, none considered Instantaneous Action definitionally. Instantaneous action was always a ratio with time, and to avoid infinite values the units of time were changed, uniformity was posited and empirical measurements were used rather than theoretical. Good deal of common sense was exercised to keep mathematical formalisms from introducing non pragmatic solutions.

That all changed when new philosophers, spoon fed on the belief that mathematics can speak to the attentive about reality in a way no other reasoning can, started to believe their equations and identities and formulations were reality, not mere models of human experience of reality.
When one is brought up to suppose a uniform development of theoretical ideas, that the average is good enough . it becomes difficult to deal with actual data from sensitive measuring tools. Statistical methods were developed and applied for dealimg with large volumes of data. consequently no simple direct relationships and formulae could be derived withtheoretical hypothesis.
Eventually theoretical considerations became overwhelmed by the massive uncertainty in interpreting the data. The solution was to go probabilistic with the statisitical data. Now np one could be certain about anything!
The rise of computing machines able to cope with masses of data and apply the new probabilisitic methods restored some control to a nervous and jittery scientific community. But the answer was not to go probabilistic, but to go fractal, and to go into Aperiodic behaviours, notably called "chaos". And the one trick that was missed, explpoted by Dirac was to use the envelope around space filling curves to define the value of whar is inside the envelope. The boundary condition became essential to describe behaviours within a boundary.

The issues that come together to support the modern structure of mathematical physics are wide ranging, but not always apt. De Moivre himself developed probability to a high degree in competition with Strenger, nd based on refinements of Cardsnos work in combinatorial sequences and structures. De Moivre saw a connection with the sines because producing accurate Sine tables was the major work of the times, both for commercial navigation and astronomical navigation.

Few realise that all the work on polynomials and probabilities ultimately have one root, the unit circle. De Moivre had a considerable advantage due to his mentor and the development of the Cotes De Moivre Theorems, which evenso Cotes appreciated the importance of more than he.

Thus ultimately probability is defined on closed conditions, and to extend it to open conditions is an artifice of infinity!

De Moivre was Netom's turntable. Thus , like Newton he was Archimedean. He did not accept unending quantities. Thus he expected and utilised approximations , finite quantifications. His argument was simple, but based on Euclids algorithm., shoudl a process be perisos, that is approximate, exhaustion of the process is acceptable if handled correctly. Certain elements of the procedure can be left off. These parts do not vanish, but combinatorially they are too small to make much difference, so they are simply not combined.

We can then consider the perisos result as the approximate unit for " measurement.. This unit Is used in all subsequent synthesis, and the shortfall is rounded away. It is the failure to remember that the model is approximate that generates spurious small scale effects!

The calculus of continuous and infinite processes can also generate spurious effects. It is precisely when a " model" is mistaken for reality when this has it's most devastating effect.

The rhetorical paraphernalia, or terminology , is often mistaken for procedural combinatorics.mthus he mnemonic value of the notation is mistaken as an evaluation procedure, many such mnemonics are not evaluateable. Instead, some jiggery pokery is done, and an identification is made that is evaluative, and that is used instead of the terminology. Dirac's function is a simple example of this.
Drac's delta function.

The algebraisation of astrological combinatorics does not arise , as we are told from generalisation, or going from the particular to the general. It arises from rhetorical style, in which spaciometric forms/ideas and relations are described terminologically, symbolically or translatably into another language, as in a code. Any method is already general, and not a particular instance. When I apply a method of combining forms in space that method is as general as it will ever actually get!

Now the habit of applying numerical mnemonics to these methods is not in fact a habit of giving a particular instance. It is simply reiterating the general relationship using a set of sequenced symbols. This set of sequences whether" numerical" or " alphabetic" provides inversion, to be sure, but it give illustration to the already general method of combination, and disguises, encodes this method in a format that may or may not represent it. To call its representation a particular is misleading, unless such a reference be fully put as a " particular encoding" using a " particular" encoding sequence.

Further, should one" decode "the sequence, there is no meaningful information contained I thin it, because it is not an encoding of information but an application of a method that is already general.

These combinatorial methods or procedures are called algorithms, and of themselves encode no information. They are instructions in the sense of mnemonics of actual behaviours the recipient is expected to do. As uch, they are rhetorical, and may be rewritten in any rhetorical style as art, sculpture, dance, speech etc. in biological systems of procedures they may be "written" as pure sequences of actions, resulting from mechanical/ helical/ electromagnetical interactions.

Thus the use of these rhetorical forms is a programming instructional language which we have now developed extensively into omputer programming languages with the miraculous effect of creating interactive technologies from the elements of our experiential continuum.

Let me no longer confuse " mathematics" as being anything other than an ancient omputer programming language by which detailed process instructions are conveyed to an operator to perform.

The general method of quantification exposited explicitly by Newton, but implicitly utilised by all natural philosophers especially astrological and mechanical, is to use Spaciometry to represent experiences. Dynamic spatial events are represented by dynamic spatial models, and based upon the dynamic, metronomical response we often call counting. Static spatial events are based on static spaciometries, but the real observable precursor is that these are dynamic equilibria!

Thus nothing is truly, essentially static, all is dynamic and a consequence of an interplay, an interaction of pressures and forces inducing and directing dynamically all motion.

Newton's Principia acknowledges this, because Newton wrote it in the light of his methods of fluents, a dynamical Spaciometry. Many indeed try to locate a Geometry as a prior Art, but in fact Newton only gives Mechanics as a prior art. Geometry, as Justus Grassmann, Schilling, and others found, was an entirely made up subject, drawing on mechanical principles and attempting to adhere them, unsuccessfully to Euclid's Stoikeioon. It bears repeating: Euclids Stoikeioon is not a work of geometry, but an introductory course in Platonic philosophy and the " Theory of Ideas/Forms" that Plato and Socrates put forward as a metaphysical foundation to all their philosophising.

So, as Herakleitos opined, Patna rhei, and motion, music, rhythm and dance poetry rhyme,etc are the essential rhetorical styles needed to acquire the wisdom of the Musai. Why mathematics should assume this role is a perverse set of historical circumstances which shall not detain me here( read my blog posts).

Using Spaciometry, dynamic Spaciometry in this way enables one at once to record spatial dynamics graphically. But Leibniz wanted to record it using any kind of symbol, that s algebraically. It was Hermann Grassmann who provided the Rosetta stone to translate between graphical Spaciometry nd algebraic Spaciometry. Not many people would now thank him! However it truly is one of the most remarkable nalytical methods to date.
Plano recognised this as a young man, and translated Die Ausdehnungslehre1844 into his own, redacted form in Italian. This lead to an unexpected development. While Prussian society under Gauss retarded Grassmann, the educational reforms similarly hampered any progress with his work. The Prussian empire had a major task on its hands to equip its infrastructure with the new insights and technologies and homegrown ingenuity! The Grassmann's felt this was a major priority.
It was only some years later that Robert, hermann's brother, prevailed on him to revamp and republish through his own ( Roberts) printing company. As editor Robert infused the new version with his own ideas! (1862). It took Hermann by surprise and elicited some dismay, but it did popularise his ideas and lead to a renaissance of interest in his 1844 self published work.
Although the 2 are named similarly, they represent different ideals. Robert wanted to promote his father's work through his work and that of the family name. Hermann wrote out of sheer genius and passion, and tore up the ground! Consequently Robert knew it was too far out of line to be accepted, and sought to tone it down to a more acceptable mathematical form.

I can recommend reading the 1844 version in German. All translations of it are not faithful to Herrmann, but rather are more swayed by Robert. Hermann's ideas/forms as Peano realised, are truly radical, and are the basis of Peano space filling curves , n dimensional spaces etc.

The early group and ring theoretical ideas of Justus Grassmann are carried through, but corrected by Herrmann. It is this group theoretical structuring, ring theoretical and field theoretical structure which takes the place of early combinatorial theory in Hermann's works.

We find David Hilbert, Klein, Cayley all exploring and writing on the same theme. In Britain, An Whitehead and Russell were influenced heavily, I'd not converts.the impact of the Grassmann analyses have been far reaching and profound and rival the impact of their contemporary Gauss. What I have come to realise is that the two camps were writing for opposite ends of the Audience: gauss for Academia, Grassmann for primary efucation. In that regard only Hermann's work could have made the crossover.

The Shocking "Truth" about Euclid's Stoikeioon!

The breaking news is that the attack on Euclid between about 1780 and 1880 in the Prussian holy Roman empire was motivated by misconceived notions of the Stoikeioon. Few in the Prussian intellectual elite had access to the actual text, and therefore they were reduced to relying on redacted and theologically purged excerpts of what the text actually said and what it's intentions were.

The widely held and taught view that the Stoikeioon was a book on geometry derives from religious teachers recounting the impact of the material on roman emperors, who in being educated to befit their status often expressed the view that it's tedium should be communicated in a more interesting way! To which the reply was there is no royal road to learning Geometry!

The emperors and their children were not great scholars and therefore were not always able to grasp what they had access to. Thus no one informed the ignorant that geometry was but one small application of a system of knowledge and wisdom used to calculate the Astrologers art, and to divine the celestials Mechanics!

Thus the misguided attack on the Stoikioon stems from ignorance in the highest ranks of Prussian imperial society, and consequently reached down to its loyal subjects as gospel truth. It is therefore of no great wonder that those tasked with educating the nations young should find the emperor to have no clothes on!

The attack on Euclid starts in the entirely Euclidean debate about the parallel demand. Those that engaged in this debate in the Arabic Empire were in fact well versed in the actual texts and in Spherical trigonometry and Astrology besides. The debate was not about proving the demand, but about what relations could be explored where such a demand could not be used, such as on the surface of a spherical earth!. The importance of this question lay in a religious duty to always give the faithful Muslim, the correct direction or orientation to Mecca. The problem was solved in the 5 th century AD in the Persian part of the empire, so it is with some humour that the Prussian empire arising out of the dark ages of the medieval period, and struggling to catch up with thr Rennaisance should suddenly begin to tackle this question again, but with far fewer intellectual resources. The Russian empire also found itself in a similar catch up mode with Lobachewsky and Bolyai valiantly attempting to breal this " new" ground in ignorance of its prior thorough ploughing.

The Humboldt reforms in the Prussian empire opened up this debate on Geometry to the wider educational reform movement, which due to having to teach children provided the first real fundamental critical analysis of the didactic form of geometry in the higher learning institutes, and a much more direct system to engender the same insights and self actuating research into children. In doing so it provided a resounding critique of the Prussian concepts of Geometry which resounded all around Europe, only stopping at the door of the Arab empire who perhaps could not believe the ignorance portrayed by this critique?
http://www.1902encyclopedia.com/E/EUC/euclid-mathematician.html

The first six and, less frequently, the eleventh and twelfth books are the only parts of the Elements which are now read in the schools or universities of the United Kingdom ; and, within recent years, strenuous endeavours have been made by the Association for the Improvement of Geometrical Teaching to supersede even these. On the Continent, Euclid has for many years been abandoned, and his place supplied by numerous treatises, certainly not models of geometrical rigour and arrangement. The fact that for twenty centuries the Elements, or parts of them, have held their ground as an introduction to geometry is a

are, speaking generally, not too difficult for novices in the science ; the demonstrations are rigorous, ingenious, and often elegant; the mixture of problems and theorems gives perhaps some variety, and makes their study less monotonous; and, if regard be had merely to the metri-cal properties of space as distinguished from the graphical, hardly any cardinal geometrical truths are omitted. With these excellences are combined a good many defects, some of them inevitable to a system based on a very few axioms and postulates. Thus the arrangement of his propositions seems arbitrary; associated theorems and problems are not grouped together; the classification, in short, is imperfect. That is the main objection to the retention of Euclid as a school-book. Other objections, not to mention minor blemishes, are the prolixity of his style, arising partly from a defective nomenclature, his treatment of parallels depend-ing on an axiom which is not axiomatic, and his sparing use of superposition as a method of proof. A text-book of geometry which shall be free from Euclid's faults, and not contain others of a graver character, and which shall at the same time be better adapted to purposes of elementary in-struction, is much to be desired, and remains'yet to be written.

http://en.wikipedia.org/wiki/Euclid%27s_Elements
http://mathsisgoodforyou.com/topicsPages/geometry/euclideanandnoneuclidean.htm

Nevertheless, from this fundamental revision, several important elements of the Stoikeioon were brought to the fore in a new synthesis intended for children, but extended by Hermann Grassmann to the level of the highest institutes of learning. However, at the time, the reactionary groups including Gauss resisted these innovations until they could manage them in their own way to their own advantage.

Justus Grassmann began a family effort to implement the Humboldt reforms in Stetin which resulted in a revised text book for geometry being developed and promoted in their region. This then led on to reforms in the arithmetic and algebraic curricula. Due to Robert Grassmann publishing business he managed to promote the families work to a wider Audience, but his success has to be contextualised. There were many educational regions and many educational reformers tasked with delivering the reforms effectively. The Grassmanns work at the primary level of Prussian education was just one of many. If it were not for Hermann's unique contribution the Grassmann reforms would have been a lost treasure in Stetin, doing its job and being superseded by each new reform and eventually lost to history. It is the Ausdehnungslehre of Hermann and Robert Grassmann that rescues them from a Gauss imposed obscurity. It was Gauss intention to secure his contribution to Prussian advancement in the geopolitical arena, and this he hoped to do through his Student Riemann. The Grassmanns were a buzzing fly to be swatted down, dangerous innovators at worst, primary educators at best. They certainly were not in Gauss league and he felt Hermann could be ignored and his ideas given to Riemann under Gauss tutelage. I doubt if Gauss ever took time to seriously critique Die Ausdehnungslehre, rather he recognised in it his own ideas and got Riemann to do further research in that vein, giving him Hermann's book as a jumping off point. It is only because Robert was a mathematics lecturer that he recognised what Riemann had done in 1854 and he collaborated ith Hermann to rewrite his book to get as much advantage out of the situation as possible. Although Robert completely dominates the 1862 version, it doers allow Hermann to re introduce his 1844 version to a more appreciative audience who could not fail to recognise his priority over Riemann.
Unfortunately the Academic board again refused to give Grassmann the recognition he deserved due to his unorthodox education, but they had to recognise the international impact of his ideas. Klein explains that the religious fervour of the Grassmanians and the Hamiltonians was to be discouraged as not the right image for international mathematics!

As pointed out earlier, the Prussian idealistic movement filtered out into the rest of Europe, where it met with mixed response. Many malcontents attempted to use its fresh approach to eduction to destabilise the status quo, others simply appropriated what they thought they could get away with if dressed up in national esteem terms. Particularly the turmoil in France allowed many French intellectuals to plagiarise Prussian ideas and dress them up as their own. However Laplace and Lgrange and Fourrier were among the few that responded to the times with genuinely innovative thinking giving credit where it was due, or engaging in genuine new untouched areas of research.

In Britain A N Whitehead and his student Bertrand Russell attempted to engender a reform in the british educational system inspired by the Prussian reforms and the work of the Prussian intellectuals and the Grassmanns.

In America the scientific community there was also influenced by emiigrés who brought across Prussian reform ideas and developed them as there own. A case in point is Gibbs who undermined Hamilton's contribution by utilising his version of the Grassmann analysis.

Meanwhile, in all this turmoil and advantage taking, the reputation of Euclid was besmirched, and this was used as an excuse to sidestep the then notions of Geometry, allowing new and fresh approaches to be developed for the technical demands of the industrial and scientific and technological revolution. Although now one can look back over a period of reinventing the wheel, which always takes the naive response: look how clever the ancient Greeks were, they had a form of technology similar to ours!– in fact the agenda in Prussia was to develop self actuating research and understanding. The philosophers wanted to harness the resource of Prussian innovation, homegrown with that unique Prussian cultural background. For too long knowledge and wisdom had been imported or bought- in by wealthy patrons leaving the native talent impoverished and under resourced to do their own research and development. Thus this spirit of selbsttätigkeit also travelled with the Prussian reforms and juiced the reworking of received didactic information in ones own terms.

Even though the early reform educators in Prussia got the author of Geometry wrong, because Euclid was a professor of Philosophy not geometry, they would still have had to critique the genuine authorities works in the same way to demonstrate self actuation. The point is they were willing to do the work, tackle any so called authority and recommend reforms that would better prepare their pupils. This gave Euclid the unexpected chance to be looked at again with fresh questioning eyes, and much of hat he actually taught to be brought to the table. What Euclid actually taught was entry level undergraduate philosophy in the Platonic Academy. He developed a course in the Platonic theory of Ideas/Form which transcended its Astrological purpose and became a favourite of the Tekne, the skilled artisans for who philosophy would become a fast way to senior positions in the artisans guilds!

Today expected skill goes along with the title Doctor of Philosphy, and the same applied in Euclid's day. An artisan became a master of his trade if he also had a recognised training in philosophy. Of course the Stoikeioon was at just the right level for most Tekne who wanted this type of recognition. The Stoikeioon however would only have been the first stage in the training of an Astrologer.

The Synthesis of the Goddess Isis

The goddess Isis has a mythological story that reveals one aspect of her worship. She synthesised Osiris. While this is only a small part of her story it is a crucial one for understanding Euclid and the Greek use of the word isos.
http://en.wikipedia.org/wiki/Isis
http://www.akhet.co.uk/isisosir.htm
The etymology appears to be a Hellenization of the Egyptian Aset meaning throne. But Iso in Greek is usually taken to mean "even". However it's etymology seems to mean "vein or filament like", establishing a link with equality through splitting into 2. The mixing of these meanings occurs only in Greece as the loan word for Aset is absorbed into the earlier notion of splitting or dividing into veins. We're it not for the myth of Isis restoring the pieces of Osiris, the mathematical connection would have been lost and the notions distinct.

The adoption of Isis as a goddess by all walks of life reflects the accessibility she denotes mythologically. This accessibility extended to Astrologers for whom the title queen of the Heavens becomes highly significant. For Euclid, the secret magic enshrined in her was evoked in the power she gave him to analyse the world like Set, knowing the goddess Isis would restore it to its proper condition as she did with Osiris. This theme runs all the way through the Stoikeioon, but particularly as the notion of Isos, the filaments from the goddess isis which bind all things together.

The pattern runs deeper, because not just filaments, but also veins and skeletal bones are envisaged in the reconstructions and synthesis of Isis, as prepared by Euclid. The secret relationship appears to be the binary relationship of items or sites, as they join, ennoia in the magical faculty of the consciousness, that gonu or rotating corner of judgement we call intellect. The intimations are scant and febrile, and yet they cannot be dispelled. The analogies suggestive and potent and fecund. From this intense religious devotion Euclid fabricates the basis of the astrologers philosophy assenting to Plato's similarly tantalising analogies about form or ideas.

Isis was a real religious phenomenon prior to Christ, and many motifs of her worship are reflected in Christian worship of the Madonna and child. The promise of magical restoration on an individual scale was also appealing. Thus every element of the Christian salvation myths were evident in the Isis osiris Horus myths.

The Stoikeioon, the beginning philosophy of how the world is constructed, is an enduring presentation of a mix of beliefs expectations, hopes and insights which also was pragmatically accessible. The further works of Euclid in the progression of the course material did not enjoy the same popularity as the Stoikeioon. This has distorted the apprehension of the Stoikeioon and obscured its true nature as an introductory course in philosophy 101!

The philosophy of Isis is the pragmatic application of the content of her mysteries, focussing on the reconstruction and synthesis of reality after it has been torn apart by analysis! That we still use this synthesis method 3000 years after it was developed and 2000 years after it was written down testifies to an essential true synthesis method found for spatial objects. It also implies an essential analytical process that underpins it.

Both is and os derive from a common idea: an observation of the analytical structure of form . Form is analysable into fibre and this fibre is flexible (is) or rigid ( os) , and it is also connected. Thus analysis or breaking a form reveals its substructures (is and os) but also how they are synthesised. The notion iso relates to this connected fibrous structure and connotes how one becomes many and many one.

Closer inspection revealed a statistical process: one mainly split into 2 equal or nearly equal fibres. This is not always the case, but it is a common enough observation to form the notion of ISP:- dividing into 2. The notion of duality is not distinct here, iso covers all kinds of splitting into 2 parts.mit is only in rhetorical descriptions of pragmatic activities that it's meaning has to be pinned down as duality a precursor to equality.

A philosophy of quantity as standard measures has to define these things, but culturally the terms have a context based use and interpretation loosely around the idea of splitting into related parts. The connection of the Goddess Isis synthesising the related parts of Osiris is too obvious to ignore. The religious or prayerful act of combining pieces of knowledge and definitions was part of Euclid's devotional act of meditation as well as the material of his course in basic philosophy. Much of what follows is based on his apprehension of the mystery of synthesis and the inner understanding of isos–Isis and Osiris.

The methods of exhaustion rely on these immutable principles, and the old saw proving pi to be equals to 4 is an excellent example of this and how little we understand the mystery of duality, Isos.

It is a fascinating exercise to work out the relationship between a circular arc and the diameter. It is a quintessential relationship which gives rise to the ratio i, the ratio of units pi, the function call sqrt(–), the apprehension of orientation, and the specialness of the unit constant spiral we call a circle.

The motion of symmetry is founded on it , and every form derived from it by some extension. Despite all this, it is still a mysterious relationship defeating even the analytical tools of the calculus of differentials and integrals.

Newton made great use of it in developing his philosophy of quantity, and developing his secure methods of synthesis and soundness of analysis. Pythagoras was able to defeat his rebellious followers by it, maintaining the theorem of areas in a plane inviolate despite the challenge by contradiction. This form was the mysterious shape of the never ending process.

Newton, by it determined that 2 forces or motives must exist and counterbalance: the centripetal and the centrifugal. Many have rejected this exposition, only to find themselves enveloped in darkness, dark matter and dark energy! The synthesis of the reality is not in question. Our synthesis is! How can we synthesise reality from our analytical parts? This is why for Euclid the godess Isis was so important, revealing to him, so he thought, the secret of her magic in putting forms together. This was a philosophy that made sense of the world of forms/ideas.

Labels

The rhetorical style initiated the use of labels to specify particular points in the discussion. The use of these labels is ubiquitous in such texts, identifying noy only equations, but paragraphs, pages, particular citations, different referrents and different terms. We tend to ignore this wider use of labelling, and so it comes as a shock when Hamilton shows in hi Theory of Couples, that a label can be a complex notational device!

Until it dawned on me while studying the Quaternion 8 group, I hunted for the operator that formulated "rotation" so precisely in the complex systems. I eventually saw that it was a double take on my part. i was a " number", better appreciated as an operator, but really a constant magnitude, which in the end turned out to be a procedural call, a conjugate function!
Whaaaat!

Of course it drives you crazy when nobody can tell you what is going on, when you are asked to just accept it. Nobody thst is except Euclid,Hamilton and Grassmann.

The rhetorical style of algebra allows one to name whole sentences, paragraphs, chapters and books by a simple label. This experience is only ever repeated in the programming classes I took in high level programming languages. However it is most pressingly "felt" when programming in machine code or a low level language. Every carefully coded hole or punched card becomes precious. The strictures of the electronic substrates we call processors were such that a successful code was not to be messed with. So a pointer to it was always preferable. This pointer was labelling in its essential machine code, and it allowed programmers to call and route the clock cycles through the same bit of code as sub routines.

These labels were carefully distinguished in programming rules, just as syntax, string length, alphanumeric characters were all carefully distinguished. These structural distinctions are drawn from the inner workings of our own subjective processing, and necessarily at a fundamental level to be able to be general and generalised.. The syntax rules allowed a complex procedure to be called via a single character or a name. Each line of code could be labelled.mthe self referencing tautological possibilities were such that it was possible to create an infinite loop!

While some opined that this should not be allowed, the goto statement should be banned etc, etc. the enthusiasts had a field day creating tautological nightmares tht worked! But when they gave unexpected results it became a nightmare to find out why. Old fears of creating Frankensteins monster began to surface . How could we ensure Asimov's 3 rules for robots?

We do so by rooting out rogue programmers and enforcing "safe" traceable coding! However, never forget, these goto statements still exist and some function calls are little more than goto statements. Labelling and the routing command for a pointer really capture something essential about human subjective processing.

Imagine therefore, when programming came to a sufficient level of conformity, the surprise of programmers at how similar it was to Grassmann's analytical method for encoding interaction ith space! The surprise was cautious, but founded. In reality we can trace a route from grassmann into the fundamentals of programming language syntax, through symbolic language and algebra studies. Nevertheless, it is grassmann who posited it in a form that is still modern even today.

What this all means is that using labels to identify parts of space, the rhetoric could be shown to encapsulate space. Using the segmented string machinery the rhetoric could be shown to orient space and dynamic motion in space. To do this it became necessary to investigate the relationships between dynamic string segments. The first concerted effort was on rotation and 2 solutions were found. Both were trigonometric, but one was written in mysterious labels called imaginary quantities. These mysterious labels in fact I have determined are procedural calls.

What I did not apprehend was how the machinery of combinatorics enabled these labels to generate a "rotation". In fact they don't. They generate a symmetry, and that is used to encode and decode a rotation. Because it is a symmetry process, the labels have to do the work, they have to be associated with particular labels as a group product activity. For this to work, the group action has to be closed. With that condition symmetry cn be guaranteed. However, the division algorithm for a group action can oftentimes fail, or not be apparent.

I would and have queried why we are reiterating fundamental combinatorics actions in terms of derived functions of them. This is not only tautologically difficult, but screws with the derived function as well. We need to proceed on the general basics of combinatorics, and this is the reason why I have discussed the segmented string of Euclid. Both division and subtraction and multiple forms derive from the action of comparison of the factors of a form. These factors themselves come about through a conjugation process. Thst is where I start from.

Euclid's String Theory: notational devices

The notation introduced in Book 2 has had a generally powerful shaping effexpct on algebraic combinatorics, especially through the work of the master Geometers Descartes,De Fermat and Vieta. Wallis made no mean contribution to notational synonymy, matching labelling to the actual physical structures of the object. So Newton had a great convention or consensus of how to label Euclid's forms algebraically to draw on.

However, Euclid's distinctions were not always observed, and factorisation is a case in point. Napier's logarithms observe the distinction in the Euclidean notation , nd is the reason why I have chosen logarithms to represent the process of iteratively conjugating an object, using a factor that is commensurate as a metron.

http://www.yesiknowthat.com/the-word-zero-in-differnet-languages/

The notion or idea of combining a whole universe of things into a flowing, lengthening twist of string is a powerful one. It invests twisting threads with magical, symbolic power. It invests a Seemeioon as a indicator with infinite possibility, and a gramme as a type of string with incomprehensible potential, and possibility. It ravels up space and unravels the potential of Shunya. It binds, it bonds, it bounds all relationships in space. It covers and it captures and it emulates all forms.

When I choose a focus region in Shunya I conjugate Shunya into 2 labelled adjugates. Suppose one of those conjugates/adjugates is a piece of string. Why is a focus region that is a piece of string already complex? Because I am already complex! I want to see the whole as combined adjugates, a synthesis, bu t I want to analyse the whole into conjugates, a binary analysis. Then I want to compare and contrast, this factorises the pieces or parts or factors, and begins an algorithmic , iterative process of factorisation, that determines commensurability. In carrying out this process I fractalise the whole iteratively and methodically, and I create or resynthesise the whole as a multiple form of commensurate factors, if possible. In this sense possible means pragmatic, which means able to be completed. What is pragmatic changes with the tools we invent, also through iterative processes. Computers now make many processes that were pragmatically infinite now do-able.

So, my innate iterative conjugation processes fractalise my experience of reality, factorise reality and make reality commensurate with some chosen factor of it as a metron/Monas. And then all these fractal regions become adjugates that I can aggregate into the whole by a creative process of synthesis. The simplest aggregation structure, synthesised by twisting threads together is a piece of string.

String, by it's nature symbolises an unending process of twisting threads into strings which in turn become threads for cords, which in turn become threads fore ropes, and so on, until galaxies are constructed from huge threads of gas, and the whole cosmos from threads of dark matter.

Taking from a piece of string a drawn line, conjugating it to give 2 segments, Euclid defines, in conjunction with the spherical relationships of parallelism the notion of a representation of a rectilinear form. Further, each segment takes on a symbolic role for the factors of this rectilinear form, but also another role as a symbol of a count. Thus the segmented line, used as a symbol of a rectilinear form loses its status as 2 conjugates/adjugates and becomes one conjugate/adjugate with its count. It's count arises from the comparison algorithm, and ultimately the factor algorithm

In making this notational synonym, Euclid provides mnemonic help in a complex process. It is these tricks and observations that aid the fluid mind in safely navigating the complexities of detail. But they also obscure what is actually happening to the uninitiated. When Napier devised his " logos arithmos" it was precisely by doing the same notational trick. He too took a segmented line and let the segment that was the factor stand as the logos, and the other segment stood for the count or arithmos. The complexity of his system was thus notation ally simplified, but the iterative algorithm was as sequential and iterative as the Euclidean factor algorithm. The difference was that the metron did not remain constant, the factor was itself conjugated at each stage, producing smaller and smaller conjugated factors. These factors were pair wise ratios, but any scheme of ratios is known as a proportion, again after Euclid . This scheme of proportions were linked to a greater or increasing count, and this in essence represents the scaling process that we innately perform as we concentrate on smaller regions of focus.

Our scaling is logarithmic, and all scalars are logarithmic. We hide this fact behind the p-adic number system the Indians created out of polynomial forms. What we traditionally call a number, is in fact one unending polynomial to a fixed logarithmic base. As I have just intimated, this logarithmic base is structured by Euclid's segmented tring notation from book 2

Finally one other use of this segmented string notation was derived by Hermann Grassmann in his Ausdehnungslehre. He directly uses segmented string/ lines to represent the radial , dynamic cords of circles, and thus also spheres. These segments became symbols fo vectors in Hamilton's mind, but in Grassmann's they remained true to Euclid, and he always called them Strecken.

Using Euclid's notational device the segmented string, long ith it's parallel process lines that together synthesised an arbitrary parallelogram, Frassmann revealed a facility vailable to the Greeks, but hidden away from the modern mind by the artifices of Descartes. Descartes was not one to particularly give credit to any predecessor, making out that his meditational praxis gave him superior insights. Certainly he gave small credit tp Al Kwharzimi, whose rhetorical algebra underpins all algebra, and no credit to Bombelli whose book on algebra inspired so many engineers and geometers for so long. Rather he set his sights on Vieta, hoping to demonstrate French superiority over the Italian bombast. In so doing he neglected to mention hs teacher Harriot, something Wallis with meticulous research brought to the attention of his students in Cambridge.

Wallis too was a Euclidean scholar, and an enemy of plagiarism, something that survives today in all academic institutions. Thus when Descartes introduced the literal notation for lines and points, it was not of his own invention, neither was the use of ordinate and abscissae, or coordinates. These were well founded Greek ideas but written rhetorically. Descartes was not really the inventor of his Cartesian coordinates, or even a great promoter of them. These refinements belong to De Fermat, Herriot and Wallis, and the general pragmatics of the printing block!
Nevertheless he was credited with it due to hs fame in French intellectual circles, and the importance of his philosophical ideas about space and god.

Wallis determined, that if the coordinate method was going to be useful, then the co ordinate needed to be measured against a fixed axle. That is to say, the general geometer knew well enough that for any line and any point off that line any number of arcs intersecting the line from the point could be drawn, but there was only one perpendicular line that went through that given point. Thus if the position of a moving object relative to that line is RePresented by points, that is its locus is presented as this collection of points, then perpendiculars from each point could be drawn to the line.

These perpendiculars and their relationship to one fixed point called the focus was the way that a particular locus called a parabola were defined. Geometrically this produced a lovely fan like diagram and a couple of defining ratios, related to each of these perpendiculars. Descartes did show thst if the point was represented by 2 labels, x and y then a considerable simplification occured. Wallis showed that if you used a pair of fixed axles, called axes, then this simplification leads to a Characteristic equation.

The difference is profound. What wallis showed is that merely by looking at the form of this characteristic equation you could sketch or imagine what the curve would look like against these axles(axes in later translations). Wallis went on to defoine the conic equations algebraically using his system. Of course he could not claim originality for the axle framework, and neither could Descartes. Nevertheless it was given to Descartes by default, despite Wallis's evidence to the contrary,

In fact we owe the system to Euclid, who in book one lays out all it's principles and in book 2 demonstrated its application. Do not be fooled by scholars who like to claim that Euclid did not have a symbolic presentation of the subject. The symbols were more apt than any yet devised. They were the segments of a string.

The sophistication of Euclids notation escaped students in the west for centuries. Many commentators on Euclid puzzled over its meaning. It was not until grassmann that it became clear that it's meaning was hidden in plain sight! The so called Cartesian system was one derived application of it, mathematical notation was another directly derived notion from it and finally grassmann realised thst the line segments (Strecken) were themselves the symbol of what thy represented. They tautologically represented themselves.

The notion of a vector immediately strikes one as a tautology. By changing the name, even the language one can fool ones processing into downplaying the tautology. This is a modern predilection. In Greek , the Greek mind seemed to revel in tautology!

The nearest English equivalent I know are the definitions of Newton in the Principia. The English translations are rich in tautology.

Now I naturally balk at this, because my English language avoids self reflexive language as much as possible. Thus my very language culture makes me uneasy with tautology. However, several European languages are very much inhere in reflexive or tautological language structures. Thus grasping this aspect of Euclid ought to have been self evident to some, but Euclid was always presented in translation in these times, and thus all were denied access to the originals. In fact some Greek versions are translations back from the Latin into the Greek, doubly confounding he notions!

It was not until original Greek texts were found that any intimation of the mental warping hat had occurred was picked up. Nevertheless, the genius of Newton is he saw through the translators treachery to the authors intent, and the same understanding cme to grassmann by his linguistic abilities.

When one reads an original text rather than a translation, the mind grip is quite unique! What such an experience communicates is often extra to the words sounded, or the marks viewed on the page. Literally it can be like tuning the brain to a particular frequency which then pulls these resonant ideas out of the surrounding space!

Grassmann realised at an early stage that not all lines are the same! Some lines are powerful symbols around wich ideas form or in which information is encoded. So it was thst he got the Euclidean sense of string! Strecken were like stretched string, which could bend around points to form segmented lines which were parts of parallelograms. The conjugated string therefore was a powerful notation for a parallelogram. Using any parallelogram. Any point in space could be uniquely referenced.

The beauty of Grassmann's realisation is that it starts with the triangle,mand expands stepwise to the rectilinear figures and beyond, and then, by fixing the rectilinear as unique forms a nother line out of the plane takes him to die Raume, where all things become possible. At each stage grassmann adds another line, another dimension, but creating another facet rather thn some science fictional hyperdimensional space. Grassmann only ever creates faceted objects in der Raum

Using Euclid's segmented string to create nested triangles and parallelograms is how Euclid established methods of drawing parallel lines in any orientation and collections of vertices which lie on a line in any direction. This is fundamental to our understanding of space.

The direct use of nested parallelograms and nested triangles was as a precise reference frame. It is important to note that the segmented string is used only to act as a reference for orientstion( confused often with direction, or rather stacked underneath the notion of direction). Thus the segmented string and its whole paraphenalia of parallel lines takes on a new interpretation: it is a description of the extension of our experience in and interacting with space.

This use of the segmented string to draw out regions in and of space, and to fractalise the segments further by an iterative conjugation and compare and contrast procedure, provided the ancient Greeks with the methods of trigonometry!

It is a moot point if the Greek Hypatchus " invented" it or whether he revealed the temple knowledge of the priests of ancient Harappian and Mesopotamian priests, along with ancient Afro Egyptian civilisations. Nevertheless Thales is credited with the fundamental theorem of trigonometry, which reveals the ancient knowledge of spherical trigonometry.

The reference frame of Euclid's day was more sophisticated than the so called Cartesian coordinates. There is a case for the simplification the axle system brought, but there is also a case against the teachers and pedagogues who buried the sophisticates trigonometric knowledge , and the rhetorical algebraic representation under mounds of turgid, turdlike symbols.

It is apparent that the schematics of solids and the models of segmented line relationships easily generalise to the sphere by iterative processes. And from the sphere one can quickly reduce to many solid forms. This kind of apprehension of forms in space requires a particular use of the segmented string: the string must become multiple from the centre of nested spheres. The string is not only " split" from this centre, combining at the centre to produce this single cord which is the fundamental orientation; but they must also be labelled to distinguish them. The use of these labels is the basis of so called complex valued algebras.
May 2013
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