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Intensive magnitude calculation?

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Surely the radius is extensive right? Yes but is it extending?
An extensive magnitude is a combinatorial contiguous form. Thus the extensive magnitude would be the dynamically changing one , not a balanced one. The extensive magnitudes naturally rely on calculus ideas to be expressed.

Norman on computation
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Ausdehnungslehre 1844

Induction

C. The exposition of the label of the Extending /Extensive  Magnitude doctrine

9!  The continuous  Becoming entity, in its moment laying hewn apart, appears as a continuous rooted and risen up entity with firm holding of the already reified entity.

Considered by the Extending magnitude thought pattern, the each time new rooted and risen up entity is set as a  Differing entity  : now hereby considering, we do not hold  fixedly  the eachtime reified entity , thus we reach  to the label of the Continuous Varying entity. What this Varying entity experiences we name the Created whole Element,
And the Created whole Element in a random one of the condition/ status markers, (which marker it adopts by considering its Varying entity,) we name "an element of the continuous thought pattern".

Thus hereconcording, the gathered system of all the elements is the extending magnitude thought pattern, in and  over which the Created whole Element travels by continuous Varying entity.

The label of the Continuous Varying entity  of The Element can Only by considering the Extensive / extending magnitude step hereforward.: considering the Intensive  magnitude:-by considering the on top piling of the eachtime reified entity, only the continuous alongside State description to the becoming as a "completed empty" entity would be remaining afterwards .

In the space doctrine , the point appears as the element ; the place varying, or kinematics as its continuous varying;  the Differing positions of the point in Space as its differing status/ condition markers.

Aufbung I have translated as "heaping on top", eigen I translate as " inward looking" in antonymony to augen which I perceive as " outward looking" and hence Augen or eyes.

Now Reihe as rank array is increasingly fertile. Firstly I recognise by it the work done by Cayley and Gauss and others in developing this notational language. Secondly I mean to imply the generalty of Hermanns conception upon which Cayley was able to construct " his" notational device.

Every concept of products in current use, derived from Haniltons works , especially on quaternions, have a deeper ocher nd more general conception in Hermanns work. Firstly Hamilton acknowledged this directly in 1853 upon reading the Ausdehnungslehre for the first time, and secondly Hermann is discussing this precisely in the essay that is the title of this thread, to which I will return after this long exodus into the related hinterlands.

Thus I preempt myself in this translation which is not true to Hermanns context, because I do not intend to explore the murky world that leads to the current Notation as exposited by Norman Wildberger, if I can help it.

To me the interesting thing about arrays is the fundamental positional structure. Thus I can start by positions in space, and thus by representative sequences. The mosaic or proportional aspect of the positioned quantities is maintained.

From this structural representation of sequence or array, Series of products are constructed by. Matrix producing.

Now tht is intriguing,

Ausdehnungslehre 1844
Induction

10.  The differing entity should develop itself according to a rule,  even if the created whole  result is a chosen one. The rule, by considering the simple thought pattern, should be the same for all moments of the becoming entity.

Thus the simple extended magnitude thought pattern is the thought pattern which roots and rises up through a varying of the creating whole element , a compliant varying  concording to the same rule.

The gathered totality of all the createable whole elements  concording to the rule we call  a System or a Field.

The differing quality was coming to be ; there, the  "from a Given" entity  can endlessly manifoldly be differing entities, running themselves everyway completely in the Incoherent differing quality,  If it  were not to submit to a fixed rule.

 Now, however, this rule, in the pure doctrine of the thought pattern is not chosen through some "random whichever  Content", rather, the label of the extensive magnitude is  chosen through the pure abstract idea of the rule governed Entity , and through which idea of "the same rule for all moments of the varying", the label of the Simple extending magnitude  is chosen .

Now, hereconcording, the simple extensive magnitude has the Attributes of condition and property , that even if  out of an elemental extensive magnitude  of the same simple extending magnitude a an other element b of the same simple extending magnitude hereforward goes,  through an act of varying , then out of b  through the same act of varying, a third element of the same simple extensive magnitude c  hereforward goes!

In the space doctrine is  the analogous Atttibute of Direction, the singular varyings  all embracing rule,
therefore
 the line segment in the space doctrine inter communicates to the Simple  extending magnitude,

the unending direct line intercommunicates to the complete System.

This simple rule every moment of the reifying entity must obey the sane rule, underpins the Mandelbrot fractal  , whose process   z =z² + C is the same rule applied repeatedly.

Commentary
If , like me , Peano meditatively translated Hetmanns doctrine into Italian, it is not surprising that he should utilise the Sigma summand notation to describe systems of elemental extensive magnitudes, of the simplest kind. However it is Levi and Riccis more elaborate notating of these Summand forms that lead o the Tensor notation. The concpt has not changed, just the notation, and thus the utility and the flexibility in labelling and thinking.

So it should not be a surprise that Cayley's matrix notation is in fact a variant notation for the same conception, here laid out generally by Hermann, but also set out specifically by Hamilton for Couples and Quaternions. The utility of Matrices over Tensors is their greater transparency to the underlying basis, and greater pecificity of that basis, especially in examples with finite elements.

Once the numbers become large or infinite, there is very little notational difference between matrices and Tensors. The concept of a " vector" therefore is confused, because it is a notational device that belongs to both notational systems.

In the space of 3dimensions we do not have this confusion if we stick to the suggested line segment exemplar. In this setting Hermann identifies an all encompassing rule that governs singular variations in direction for simple extensive magnitudes, that is line segments.

While it is common to present these attributes as falling out of some general law, yet here we Knis Hermann developed the general law by observing the specifics of line segments in their behaviours. The whole of the Ausdehnungs doctrine is thus concocted from an intensive and prolonged study of the behaviours of line segments in modelling or describing attributes and or phenomena observed in space. Because of this focus it is clear to me that Hetmann missed the significance of his generalising approach for curved line segments in general, and circular arc segments specifically. Thus when he makes hs simple correspondences for the simple extensive magnitude he misses a trick. By including the circular arc segment at this juncture and the circle as the corresponding system he would have provided a truly more general method ncompassing all forms of curve and curvature.

In addition, the category of intensive magnitude would not be modelled by a "step rise" but rather by a circular disk or a spherical ball! The advantage of this would be that the radius of curvature clearly then becomes an Inverse measure of intensity or density. Point like spaces would clearly be huge intensive magnitudes while circular disk regions would represent lower regions of intensity. The perimeters of such regions would have curvatures approximating to straight lines.

We could thus model straight line and flat plane geometry as occurring on the surface of a sphere or spheres whose near common centre is very far away or at " infinity" .

Immediately we have a polarisation into local and universal behaviours or attributes. Local attributes would tend to be more curvilinear while general or universal ones would tend to be flatter.

From the outset Galileos fractal for the solar system develops this local and far difference in absolute systems. Thus the greater system of the sun does not exclude the absolute powers of the local system of Jupiter, et al. Newton in formulating his Astronomical Principles was well aware of this feature of Galileos Fractal. The inverse square law or rule was thus a rule to be applied at all scales and all moments of the systems reification, that is as an eternal dynamic.

Is the inverse swuare law gravity? Clearly it is no such thing. It is an all encompassing general rule of Newtons system which applies to the simplest exemplars of the extensive magnitudes. To apply it to a galactic mass is therefore to invite error. It is not a rule for such a complex extensive magnitude!

What such a rule would be , or how we might develop such a rule is precisely what the doctrine of Extensive magnitudes would guide us in formulating! This is assuming the doctrine is correct and robust enough.

In any case it is clearly a system suited to the fractal nature of our experiential continuum, and still intriguingly relevant!

Commentary on §11

Ok, so now, as I should have guessed, further clarification of the ideas in §10 swim into view.

It is hard to simply express how this exposition is so well constructed. The necessary ideas are vaguely or generally described prior to when they achieve their maximal impact in specificity. So it is helpful to re-read the earlier definitions and labels, notions and concepts. The acceptability and applicability of notions presented by Hermann in your immediate awareness hang on these prior considerations that you have accepted or allowed. Like a game of chess Hegel and so Hermann locks you in to his Schema inductively.

Consequently it is important to state that these techniques and methods of persuasion constitute the Hegelian logic, an advance on the -Aristotelian logic system. The construction is hypnogogic, homiletic , designed to convict and convince.

From the outset Hermann and his brother  Robert have been working in the framework of the early 19th century view of philosophy, Religion and Mathematics, but with a view to purify and correct it from perceived and actual error. From a religious standpoint or even a high brow ethical one this is a Laudable goal  " advancing the cause of Truth" . However I do not personally subscribe to such motivations, and consequently view all such claims as skeptically as I can.

The Hegelian logic system makes this an impossible thing to do, because it includes my and any skepticism within its operating principles. I am thus reduced, as I must be, to my innate flawed opinion, and no better or worse than any other opinion of this content.

To say that Hermann concocted his generalities out of the specifics of his contemporary experience is hardly enlightening, since Hermann specifically describes in the Vorrede exactly that! And not only that in passing but in step by step detail.  So to make this remark is not to castigate Hermann, but myself for intuitively or subconsciously seeking some kind of divine Epiphany from these general statements and their specific expositions.

That is not to say that there are no epiphanies to be had, because there are. But each is personal to myself , revealing where I have made some unjustified or justified assumption; or where I have accepted some camel while straining out a gnat; or where I simply was ignorant of how what I was taught was originated.

In section 11 Hermann introduces the coordinate system of geometry (a la DesCartes), Its very general principles in a formal statement which is meaningless until exemplified in the space doctrine.  Because of this approach, it is almost automatic to ask is there another interpretation of this general formality? That truly is the " power" or absorbing interest of Hermanns method . Thus my earlier comments about using curved segments constitute an alternative .

But Hermann dissects how we arrive at the labels and notions we use in the coordinate system. In so doing he Maps out how we might apply the general formalism in an alternative setting. We see how he adjusts tetnimology or label to a specific exemplar, and learn that this is what we must perhaps do in any alternative System.

But further still, using direction as his general rule, he now can apply differing rules. Because he has not analysed direction itself I think he missed an even greater fundamental, but that in fact is a detail to which The Method he is inducing can be easily turned. It is the Method that one must pick up on as best one can.

Because of the method and earlier distinctions he points out that while it is impossible using mathematics to construct any space whatsoever, let alone 3 dimensional space, using his method the doctrine of extending magnitudes , one can rise up to greet it and indeed rise up beyond it into endless space!

The notion of climbing or enhancing to a System or level or stage is a crucial notion in his system, and in Hegels categorical logic and Dalectical system.

We see now why it was important to remove geometry from the formal expertises. It is " Geometry" that initially establishes the validity of his general statements.

But his general statements are validated in more ways than just geometry. However, remaining with Geometry one is bound to ask what is a 4 dimensional space and can geometry represent it? The answer is yes. What one has to let go of is orthogonality. Hermanns general formalism makes no mention of orthogonality, it relies solely on Independence, that is Differring at an elemental level.

Orthogonality is a unique property of the circle and sphere, it is not a general property. The belief that space is 3 dimensional is foisted upon us when we are very young with no critical apparatus to question what we are being taught. As a crystallographer one can appreciate the need for Space to be more than 3 dimensional. With this freedom the facets and crystal organisations can be more coherently and satisfactorily described.

What do we mean by independence in space? Very simply and very naturally any 2 orientations are independent! In Hermanns notion Directions are independent. Thus we have an endless crowd of independent directions to utilise, but we need the first ones to specify all the others.

Currently we are taught that the first 2,3 etc are a basis if they are independent, but all others concocted from them are not independent! This is a language confusion between dependent and dependant. The German is abhängich, and literally means "hangs off". No direction hangs off another. However we construct directions in reference to elementals which can be whimsically chosen. Any line can be referenced by an endless crowd of related pairs, etc. These pairs,etc are in the same class of multiples. These multiples are in an Equivlence relationship. It is this equivalence relationship that is confused with dependence .

Now some have sought to show that a maximal basis exists for 1,2,3 space. In fact Hermann denounces that widespread belief in announcing the 3rd stage System. Here he  goes on to show that this is an imposed constraint that hinders rather than Aids understanding spacelike objects and their behaviours relative to themselves and others. This is alluded to through an accounting Bill , the account tally. The extensive magnitude method keeps track of its elemental constructors. Thus as long as they are independent, systems of directions can be built endlessly.

However we cannot build them in the same orthogonal manner we have built the Cartesian reference frame. This is ironic, because DesCartes did not construct this commonly accepted frame, Wallis did! DesCartes frame was in fact very very general. Today this generality is wrongly ascribed to Lagrange in terms of origin. LaGrange indeed pioneered generalised coordinate systems as a development of DesCartes original prescription.

We can live intuitively in an n-dimensional space, if we recognise orthogonality as special cases not general rules.