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Author Topic: The Ausdehnungslehre of Hermann Grassmann 1844 reprinted in 1877  (Read 3863 times)
Description: Hopefully a steady translation of the whole 1844 version over time.
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« on: March 04, 2015, 12:03:01 PM »

This translation has begun in parts in several threads which I will link below.

But first let me say. The forum is not a blog. I know several members and guests have been following along . To those I say: Thankyou for your  support. However I am not a native German speaker and I am not an advanced student of German , I just like languages and learned German doing my " O" levels! So some of you could usefully critique my quirky etymological translations!

Please do so!

I only stipulate that criticising my translation should be accompanied by your own translation. That way we all learn and I will become better educated in German history and culture, both then and now.

Please contribute to the thread even if it is just to express an opinion.

The opening pages of the book translated here with much commentary and useful links
http://www.fractalforums.com/complex-numbers/the-theory-of-stretchy-thingys/
 
The induction and the general doctrine are translated here in the context of researching the basis of Hermanns later work

http://www.fractalforums.com/complex-numbers/der-ort-der-hamilton'schen-quaternionen-in-der-ausdehnungslehre/.

If you want to post a picture, a relevant article or video or even( please) your own translation you are more than welcome.
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« Reply #1 on: March 04, 2015, 02:20:16 PM »

From time to time I publish the commentary before the post, usually because I am reworking the translation.

If you feel inspired to share your insights from reading the translation, or even working on your own translation please do so, because then we all learn!

In the first section , the extending magnitude and the first chapter Hermann takes the reader in the stadium position through his adventure to find the well spring of Mathematical creativity and expression. In this manner he hopes to give the reader the maximum freedom and mastery over his experience of Hermanns researches! Hermanns researches by reason of rigour, and by the nature of his necessary abstract approach is unlikely to suit many of his intended audience, and very likely to cause established mathematicians to recoil in Dread! Thus he elects to do the hardworking on his own offering up the results to his audience for scrutiny as propositions, demonstrations and the like in an easily accessible form.

Hermann in setting about his work in this way hoped not to tax or tire the reader unduly by pages of abstract concepts thoughts and reasoning, and in particular makes a point to use long familiar concepts. In fact even his phrasing and adjective use was designed to be as accessible as possible. Thus we often get long, wordy adjectival phrases attempting to communicate important distinctions.

But the most important aspect of the 3 writings plus this first chapter is the gradual revelation of the expertise as it is put into practice! Here we find the clearest summary of it: that the worst possible approximation to a solution or description is taken as a starting point and then by a process of reasoning and experimentation a process called a dialectic , that situation is resolved into a better and better solution! In the course of do doing certain labels play a vital role as keepers. These keepers enable the refining process to tautologically anchor to a label and gradually transform from bad to good to better and eventually the best representation of the situation circumstance or process.

The labels are varied, from letters to every typeface in the printers block. Some labels are signs some are names some are symbols, and all combine to express the idea or resolution sought or under discussion.

The use of other relevant doctrines particularly the Combinatorial doctrine is mployed for method,notstion, terminology and Analogy and inspiration, and by these means and methods an expertise is established based on concrete developed and developing resultd.

The nature of his material is that it is dynamic and ever changing, but yet certain invariant patterns inevitably appear. It is these that form the fundamental basis of his application of his expertise, developing and as it develops, to the dynamism of real space and time .

Certain concepts of creating, being and existing , coming into being and dissolving from being are deeply established so that the reader knows that what is demonstrated is fully considered and not half baked.

Because of this approach, the work is an introductory philosophical text, which enlightens any reader who engages with it. The important attitude is to believe that it is intended to be crystal clear, and consequently the sudden realisations one obtains while reading are likely to be cathartic of many years of wrong headed thinking and teaching!

At the end of the day everything he says is empirically testable by the simplest of Geometries. Thus the works of Euclid, Apollonius, Archimedes etc are brought into light as examples of clear empirically based, dynamically testable thinking. This type of dialectic differs immensely from the linguistic grammar based logic which we are presented with through our education system.
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« Reply #2 on: March 06, 2015, 06:40:50 AM »

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction  of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§13 The pure expertise like way, to treat the Doctrine of the extending magnitude, the pure expertise would be, that we concord with the artform, how it in the Induction is pursued from the labels out., which labels,    by this expertise, lay to ground ( as a basis) all individually developed  entities.

Alone,

to not tire out the reader through continued forth abstractions , and at the same time therethrough  

to set him   in the ( stadium) Stands ,

to move himself with greater freedom and self standing quality ,

that we tie up besides familiar entities,

I tie up over everything else, by considering the deriving of new labels alongsides the Geometry;

  our  expertise builds a representation of the Geometry Basis

Therefore in the entity I lay the abstract label to ground (as a basis), every time,  by considering the deriving of the enduring  qualities, which qualities build a representation of the content of this expertise ,   without myself, to shore it up, thereby considering each upon some random one in "the geometry demonstrating" truth ;

 thusly I still  hold out  the  expertise concording to its content, completely pure and independent from the Geometry.•)

Therehere   I tie up besides the Creating whole enity of the line to gain something in the vicinity of the extending magnitude.

Here it  is a creating whole Point,  which point takes aside ( to scrutinise) differing positions in continuous succession ; and the totality of the points , over which the creating whole point is going  by scrutinising this everyway varying ( in the totality), builds a representation , the line.

The points of a line appear thuslywith,
essentially as differing entities, and become also as such besigned ( with differing printers block types);

 therefore, how the Like is  adhered alongside to the differing entity always at the same moment in time  ( imagine wholly over in a subordinate  sense) , thus  also here the differing points appear as differing positions of one and the same creating whole point.

Upon like manner we reach our expertise to the extending ( magnitude), only  if we  set here the "intercommunicating label-like" entities in place of the there "inter-stepping space-like" Relatings  .

Initially in place of "the point", that brands, "of the special place", we set here "the element", whereunder we everyway stand the special entity badly way off apprehended as  differing from the other special entity. And indeed we lay by considering "the element" in the abstract expertise no half baked other content.

Therehere, It, the conversation here, can not be half baked therefrom  , what conversation were usually set for a special entity, because this entity is central-like –

 because it is plainly   badly way off the special entity,  without all real content—,

or in which relating the one from the other is differing –

because it is plainly in a bad way appointed as a differing entity, without that one random real content , in relating onto which it is a differing entity,

By This label of "the element" is our expertise of a common design with the Doctrine of the combination ( combinatorial theory), and therehere also the besigning scheme of the elements ( through differing printers block type) are both of a common design with the combinatorial doctrine.

The differing elements could now at the same time become apprehended as Differing Condition/Status markers of the same created whole element  , and this abstract Differing quality of the condition/ status markers is it, which of the differing place quality inter communicates. The overgoing of the creating element out of a condition/status marker in one other we name "a varying"/ "transforming" of the same , and this abstract transforming of the element inter communicates therefore to the "place transforming " or the kinematic moving of the point in the Geometry.

Now, how in the geometry, through the forward kinematic moving of a point, immediately nearby a line roots and rises up,

and for the first time ,
in which entity one under throws/ subjugates the achieved representation onto the new representation due to the kinematic moving ,
space-like representations of higher steps/ ranks/stages can root and rise up,

 thusly roots and rise up also in our expertise through continuous transforming of the creating whole element immediately nearby the extending representation of  former/ prior step/ stage/ rank.

The result of the until here developing Summarily Grabbing together report, the definition, we can   set down :
Quote
" Under an extending representation of former / prior step/ stage / rank, in which a creating whole element by continuous Varying over travels, we everyway stand the totality of the elements, "

And in particular we name the creating whole element in its former condition/ status marker the Beginning element, in its latter condition marker the End element.

Out of this label output results itself thuslike, that to each extending representation a "running into against set "entity relates, which entity the same elements enholds( holds within) but in turned around rooting and rising up manner/ cognisance, thus therefore,that  name- like becomes  the Beginning element of one the End element of the other. Or ,  more concordingly expressed, if through a varying  out of a   b comes to be, thus is the running into agains set varying  the varying, through which out of  b  a comes to be, and the representation relating to an extending representation "running into against set" entity  is the indicated entity, which through the running into against set varyings in turned around succession goes henceforward, wherein at the same time lies, that the running into against set being is a side changing entity.


Footnotes

•In the Induction (Nr.16) I have shown, how by considering the presentation of  each an expertise   and in particular the presentation of the mathematical one, two developing rank arrays grip in one another ; from which the one  delivers the Material, that brands, "the complete rank array  of the truths", which builds a representation of the central- like content of the Expertise; while the other to the reader  should give the mastery over the material . That former developing rank array now is it, which I have  to hold out completely iindependent of the Geometry  , while I  have placed the greatest Freedom before me by considering the latter array commensurate to my goal .  
•• the difference lies only in the artform, how in both expertises  the thought patterns come to be achieved out of the element: in the Doctrine of the combination specifically through direct knitting together, therefore discrete, but here  through continuous creating whole.
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« Reply #3 on: March 06, 2015, 10:27:50 AM »

Commentary on §13

Once again I can say I am profoundly surprised by this section.

I have some redacting to do  but I hope it is clear that Hermann has a different take on interacting with space. Firstly it is Newtonian, and that particular conception of Newtons called Fluxionic. This is not surprising in the least as LaPlace, LaGrange and indeed Euler were " students" of Newton. However, Berkley attacked the Fluxionic method on the basis that it was as unfounded as religion, that is if one is accepted by the scientific, and to him atheistic community, then so ought the other!

However, as piercing as his argumentation was it was nevertheless based on misreadings, mis-teachings and misunderstandings of Newton's Astrological Principles. Berkeleys criticisms apparently goaded Mathematicians into  wrong headed number concepts .

It was said that Newton had a dream, as of God, in which the Fluxions were revealed. In that dream a glowing point moved to trace out a line, and that line itself moved to trace out a surface; and that surface itself  moved to generate a solid, and that solid itself moved to trace the locii of a moving object.

The story related, that from then onwards Newton pursued these Fluxions to derive his theory of the calculus, that is the Fluxions.

It is quite clear that this dream influence LaGrane and LaPlace and Euler and was the basis of their concpt of Mechanics.

I on the other hand was fed the Leibnizian view of Mechanics in which motion replaces Fluxions and hard billiard balls and thn elastic billiard balls replace the Dynmics of Fluxions!. The differential Calculus thereof was made up of discrete minutiae called infinitesimals.. They were arrived at by analytical abstraction whereas Newton was given a synthetical Geometry to found the theory of Fluxions and Fluents upon.

The analytical Leibnizian approach actually was initiated in he School founded by Cartesian scholars, and developed by the Brilliant work of Leibniz, whose notation by and large is the standard notation in integral and differential Calculus. On the other hand the Synthetical approach of Fluxions was adopted only by British scholars who found in it a national pride of Newtonian genius . However the French, Kant and Euler also like De Moivre found Newtons conception to be to their liking and thus much of the Mathematics of the Ecole is based on Newtonian principles thoroughly revised and extended by French Scholars and engineers.

Hermann was aware of this and he called these Approaches , or rather I have interpreted Weise as Cognisance! Of course I could not justifiably maintain a constant translation for so psychological a word, and I have ranged across the vocabulary appropriate to that concept in context. Thus the cognisance is exhibited in a certain style or manner of proceeding, thinking, speaking snd notating..

Thus the 2 cognisances resulted in 2 competing versions of calculus. It is this historical opposition and competition that dialectically has given us the modern Calculus. It is also a fascinating case study of how dialectical reasoning is naturally embedded in historical movements both progressive and regressive. This was a constant Hegelian theme, which he had much to say about. Hegel advised that we identify and choose the progressive processes of the dialectical process.

There is much to comment on in this section and I will do so in several posts , but Hermann here builds on the work of his 3 writings in an ever more fractal and detailed way which he is at pains to make absolutely clear to the reader in the Stands!
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« Reply #4 on: March 07, 2015, 03:29:59 PM »

Commentary on §13 continued

The fractal structure of thought is well documented by Hegel's writings. This carries over into Hermanns structuring of his work.. The fractal is based on a repeated pattern of 3 at several levels . The general pattern is : comparison, contrast and conclude.

Of course you need two items to compare and Hermann chooses the GegenSatz format. To make a comparison one must have 2 statements to compare, and Hegel starts his discourse with a historical narrative. . Hermann in the Vorrede does the same.

Then Hermann gives a formal induction covering the whole topic once again, but differently , in fact in a philosophical format establishing propositions. The third concluding treatment is the general Doctrine of the thought pattern. On e again he covers the whole topc but this time he focuses on elements that can go forward into the later works . Readers, from their vantage point are conducted backwards to previousvandvother ideas in order to draw out the resolving conclusions from the comparison and contrasting.in the discussions.

We see clearly how by carefully revisiting each aspect of the case in focus how the proposition is carefully revised redacted and updated : it is gradually resolved into the best expression of the conception.

A fractal is a broken of piece ( fractus) or a pattern of such pieces. It is usually formed by recursion or more simply reiteration, repeating the process over and over on the same material , but with a slight difference every time. It is that slight difference which takes " almost" self I liar patterns and transforms them into a fractal synthesis or analysis.or some combination.of both.

This revisiting of the same material oncept or definition in order to slightly adjust its meaning in use is called Tautology. It is a fundamental aspect of reasoning apprehension and dialectical development. Yet some frown upon it as sophistry or bad practice! You need to understand tautology as part of self reflexive thinking, this is easier for languages that retain self reflexive verbs..

With that fractal process in mind we can move into the Details of §13
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« Reply #5 on: March 08, 2015, 06:24:59 AM »

Commentary§13 continued

The Induction is a piece of work!

I recommend reading and retreading it meditatively. It is a true hypnotic induction piece of writing so do not be surprised if you find ideas popping into your mind from just about everywhere!

So given its " general" approach( rather mind blowing nd mind bending presentation) we have to touch base somewhere.. We have been on a wild horse ride across un familiar terrain, or a white water rapid ride through echoing canyons and now it's time to " tie up" the horses reins , or "tie up" the boat to some moorings and lay down some firm footings to the ground, plant our feet firmly to ground and set up a base camp, as a basis for more specific examination of the Locle or local terrain.

I hope you appreciate the similes or metaphors because Hermann not only uses thrm explicitly and directly, but his whole writing style and word choice is suffused with them, along with word puns and other literary devices. Many times even his grammatical constructions and sentence structure reflect or convey the relationship he is discussing, or the " connection" he is making or expressing.

The expression is the thing. It not only conveys a semantic meaning it also conveys a grammatical design. That grammatical design often and deliberately becomes the mathematical synthetic and product design.

You and I need to ponder that deeply.

I have rather deliberately translated the word Begriffe as Label ( or sometimes handle) wherever it has occurred. This was because it is absolutely foundational to his approach, and his Förderung, or promoted mindset. The word itself etymologically derives from griefen that is "to grip ". The participle " Be" underpins " bei" which sounds like " by" but I translate as "by considering". This translation serves 2 purposes: one it deals with the inner space of the reader, it implies the outer spatial relationship of the reader and author to the subject under consideration.

Thus Begriffe I like to think of as the handle I use to grasp something, some notion, some concept, some idea.. Thus it intimates the notion or idea so labelled, but sometimes it refers simply to the handle that will be used to attach to some yet nebulous concept. It is thus areal thing that refers and relates to many abstract immaterial or none concrete ideas and concepts. The real thing by which we grasp the un realities of thought!

So here Hermann states that the pure way is to work from these labels out wards!
Pure because he values the highest objectivity in his thinking and results. The subjective and objective processing conundrum I have dealt with in the Fractal Foundations threads, but here, in a romantic age, it was believed and hoped one could really divorce the Two. Surprisingly Hegel confounded this conception and thus for Hegel, the pure way involved not deluding oneself any longer about these " absolute " distinctions which are not so. Hermann is of the same mind, as he demonstrates in the Induction. So pure here refers to the purity of the dialectical process he has expressed prior to this section.

As a consequence he refers us back in the footnote to the last and quite difficult section in the Induction, where he describes the 2 rank arrays. I have to admit I do not fully grasp this description, and here he succinctly clarifies it a bit more. So I will be revisiting and reviewing that section with a view to clarifying the translation.

However, I do grasp that these 2 arrays are a piece, they "grip " in one another. That essential nd central notion is the real message of these arrays: they will not be separated, like Siamese twins! How these arrays are realised is the whole "point" of the Work! The Transformation or Varying or Developing of one into the other is what the labelling is all about. It is the labels that we expertly use to ground real truths of space and time , set out in or populating an array, and it is the freedom we then have to vary that array toward some appropriate goal that is the power and the purpose of this whole entire work.
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« Reply #6 on: March 08, 2015, 07:35:09 AM »

In a commentary on the induction translations I dealt with the use of the labels " algebraic" and " combinatorial".

It is worth pointing out here that Algrbra has had a number of meaning changes over the period since Hetmann wrote the Ausdehnungslehre. However the meaning I shall be using in this thread is simply nd straightforwardly symbolic arithmetic.

We need a cold shower sometimes to wake up and realise how we get hoodwinked by others misconceptions. Symbolic arithmetic is nothing different to ordinary arithmetic. We need to wake up and smell the BS! The Arabic numerals are symbols.

We could and indeed have used many symbols in the past to perform so called arithmetic. The roman and Greek numerals are a case in point. The Arabic numerals were mo different to these.

As a child you are taught to believe that you could not do sums easily with these symbols and thus Arabic ones are the best! In fact this is far from true. The ancient Chinese used a stick symbol in various patterns which  are in fact a base 10 system. The Ancient Sumerians used a base 60 system, all of them are advanced civilisations, capable of Astrological thought and apprehension,

The true benefit of the Indian system is their absolute mastery of long intense calculations, often done entirely within their heads apparently. This so impressed other merchants that they abandoned their traditional systems to gain the bartering advantage of the Indian superlative system. This was not well received by the Greek loving West who opposed its creeping insurgence vehemently, as a bishop to India found to his cost!

Although we now adopt the Indian system of numerals we never fully adopted the Vedic Ganitas and Sutras tHat underpin it. In particular we disdained the absolutely fundamental secret of the Indians ability to calculate so well! They used their gingers and thumbs and toes!

Even now the west cannot accept the plain fact that neurological programming is the fundamental consequence of using your digits in this way. People who count on their gingers are laughed at in the west, whereas they are all trained to begin there in the Vedic school systems. Contrast that with say a Chinese savant who can calculate at blinding speeds on an abacus!

We are hypocritical . The abacus is no better or worse than our fingers and toes, and an adept, or savant can indeed out perform any mentalist calculator using his fingers and toes to reorder and retain the out put result of the calculation. This is neurological programming at its finest.

There are many other advantages to the Indian 10 fingered Shunya is everything system , which the Vedic schools exploit and explore, promoting confident , swift, accurate and able computations on the spot!

Thus we should be aware that symbolic arithmetic is not some abstract weird thinking, but instead it is the foundation on which all numeral systems are designed and based. These basic properties by which we implement a design derive as much from the behaviours of real objects as from aesthetic design patterns we might like to promote. In contrast to the symbols , counting is our true vocal and emotional response to these dynamic patterns and pattern transformations in space. We can go further and incorporate dance position and poses, and indeed many movements within Indian dance styles encode such counting patterns.

The point is that counting will occur any and everywhere in our response to and expression of dynamic geometries, but it should not obscure or define the dynamic Geometries per se. The dynamic Spaciometry should  promote the design of numeral systems in keeping with the synthetic and product design constraints and protocols established by empirical evaluations and experimentation.
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« Reply #7 on: March 08, 2015, 07:05:19 PM »

Commentary on§ 13 continued.

In 1844 when Hermann finished the Manuscript he sent it to Gauss on the advice of Möbius who found the work beyond his capabilities .  After a while a busy Gauss returned the manuscript advising a rewrite using more familiar less original terms. He had not been able to devote timee to learning Hermanns new terminology .

Later in 1853 Gauss directed Riemann to speak on the topic of the hypothetical basis of Geometty in his 1853 Habilitations speech. This is the topic of the next few sentences of §13 in which Hermanns explains his contribution to the growing debate in 1844.
Riemann
Quote
To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics.

There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an n-dimensional space and ended up giving a definition of what today we call a Riemannian space. Freudenthal writes in [1]:-

It possesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras' theorem.

In fact the main point of this part of Riemann's lecture was the definition of the curvature tensor. The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was too far ahead of its time to be appreciated by most scientists of that time. Monastyrsky writes in [6]:-

Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts. ... The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented.

It was not fully understood until sixty years later. Freudenthal writes in [1]:-

The general theory of relativity splendidly justified his work. In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data.

So this brilliant work entitled Riemann to begin to lecture. However [6]:-

Not long before, in September, he read a report "On the Laws of the Distribution of Static Electricity" at a session of the Göttingen Society of Scientific researchers and Physicians. In a letter to his father, Riemann recalled, among other things, "the fact that I spoke at a scientific meeting was useful for my lectures". In October he set to work on his lectures on partial differential equations. Riemann's letters to his dearly-loved father were full of recollections about the difficulties he encountered. Although only eight students attended the lectures, Riemann was completely happy. Gradually he overcame his natural shyness and established a rapport with his audience.

Gauss's chair at Göttingen was filled by Dirichlet in 1855. At this time there was an attempt to get Riemann a personal chair but this failed. Two years later, however, he was appointed as professor and in the same year, 1857, another of his masterpieces was published. The paper Theory of abelian functions was the result of work carried out over several years and contained in a lecture course he gave to three people in 1855-56. One of the three was Dedekind who was able to make the beauty of Riemann's lectures available by publishing the material after Riemann's early death.

Whatever Riemann presented in his lecture was hardly more subtle and far reaching than Hermanns conception. And yer Gauss raved about his protegé's lecture some 10 years later after reading the Susdehnungslehre!

To be fair Geometry was in a very bad plight and was being thoroughly analysed for its weaknesses by many great and disappointed philosophers and Geometricians, Justus Grassmann amongst them as a lowly cleric and district School Inspectir. But I still find it a curious tale that Gauss ignored one and promotes the other without any compunction!

Be that as it may the labelling and ideas are related but clearly different. Hermann singlehandedly created the Lineal Algebra  from his conception, while Riemann as brilliant as he was lauded to be merely posed the questions..

I have redacted the section of the translation to hopefully make this point clearer from the text..
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« Reply #8 on: March 10, 2015, 08:54:06 AM »

Commentary on §13 continued.

This is even more genius than I could imagine !

The point is first addressed.
There is much I have written about the point but nothing like this!

Hermann does not start with Geometry or even Astrology . He starts with thought patterns represented by labels.

But to ground these thought patterns he binds them to a well known geometrical label.

However the geometrical label has no influence on the thought pattern!

Well of course it does, but not in the expected or traditional way or cognisance!

In a way it is precisely like saying: I am going to change the referrent of your most fundamental labels before your eyes, so that when you speak in your everyday language you will be saying something with an entirely new semantic!mor another analogy would be : you speak your language in such a way that there is a direct intercommunicating , a 1 to 1 transliteration into another language, for example griffen is 1 to 1 mapped ont gripping.

But this is also a serviceable " translation" going beyond the direct transliteration!

Because Hermann has separated geometry as a real , concrete space-like experience, his labels, and indeed all labels are forml, independent and thought- like experiences that relate to our subjective modelling or copying of the real entities. But how well we label and how good our modelling is depend on our skill sets and life experiences. This he sums up in the term Cognisance. Allied to our subjective cognisance is our expertise . It is this combination of Cognisance snd expertise that makes it possible to build weird and wonderful mental representations of real behaviours:Subjective processes representing real objective processes.

In an analogous way or by simile we may anthropomorphise certain natural processes, that is we may speak of water as if it had legs and could voluntarily run down hill to the sea!

This is th kind of place Hermann has brought us to, where what we label here has an intercommunicating equivalent there. What is loosed here on earth is loosed also in heaven where God oversees everything.

So we start with creating elements. Our thought patterns perceive or we perceive through thinking that entities come into ontological existence and then may exist or pass out of existence, leaving only a memory as a built representation within us. In that process Hermann has identified nd thus labelled a class of entities called Creating Elemnts. As you study the Induction you experience how these elements tke shape and receive their formal labelling.

Without contradiction then Hermann can only start his process on the a priori foundational primitives that bring everything into real existence and thus apprehendable status! But in so doing he cannot define such primitives within his system. He has to relate to a prior system of definitions , or a larger system beyond the one he is specifying now.

Thus he cn specify the line segment, but not the point in his real system model. His model thus is founded on the line segment.

We will see how this enables him to model the most general transformational processes by the permutation processes explored in the Doctine of Combinations; how permutations can be directly linked to translation, rotation, and anti translation and contra rotation in space. We will also see how permutations can have no real objective space like nehaviour yet be subjectively useful as stages in our mental journeying to space-like solutions. In this category I specify the notion of reflection in a mirror.
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« Reply #9 on: March 11, 2015, 01:25:18 AM »

Commentary on §13 continued.

The remarkable process that Hermann now goes through is important for its honesty and clarity.
The creating whole entity will initially be called " the point" and the context will be the line.
Its creative role consists in taking to one side and scrutinising each point in space.. Thus the creating point behaves precisely like a Human judge or a computer programme that tests every point.!

The inescapable involvement of human or computer agency is acknowledged from the outset. As a consequence of this activity, by going over all the points , the line appears. This line is not necessarily a straight line! The line that appears is dependent on this process of scrutiny and what constraints or conditions are placed on the output result.

Thus the dynamic motion in Newtons dream is replaced by a continuous process of input, scrutiny output as a point in the line or not, making plain that any model is engendered  by human / computer activity.

However what follows after this derivation of the label point, which is independent of a geometrical description of the points of space (The line so constructed is also free of any specific geometry, except that it is labelled as a line due to it being differing to a point) is a typical design or construction model Hermsnn uses time and again.

But now, this becomes the " Mould" for creating all manner of representations of continuously extending magnitudes.

It is vital for his method that labels are " badly fitting" abstract terms or labels. Thereby much inane chatter is avoided. The focus is then on how to achieve the output result by synthesis.

The process involves this individual scrutiny in a continuous , successive manner, of every point in some totality of points. Thus I may constrain or place a condition on the totality of points  which the process may scrutinise.

The process of scrutiny itself implies that some condition or status is bing tested for, and so once again conditions and constraints are vital to achieving an output result that is a line, or an arc, or a spiral or in some cases a "random " scatter - like appearance. These are all possible outputs of a fractal generator app!


A duality or dual cognisance always exists for a process output: the output may be considered as a collection of differing  results or positions that fulfill a system of constraints, or they may be considered as the successive ( and thus differing) results or positions of the creating whole entity. This duality pervades all our thought patterns and Hermann acknowledges and utilises it.

Many of us are taught to remove all duality, all tautology all analogy in our pursuit of some thing called "the Truth " ; whereas the Truth is : duality and Tautology and fractal structure are all we can ultimately derive!

Now , surprisingly, the bad fit of the abstract contentless label makes it a prime example of a combinatorial label!

This surprising connection is not explicitly stated, even in modern Group theoretic discourses and discussions. And yet now that it is here blatantly expressed it is obviously " true!".

Every course on group theory it has been my misfortune to endure has obscured this combinatorial simplicity in implicit examples of the " way it could go" . Terms like Homologous and Homotopy obscure this simple combinatorial fact: we interact with space in fundamental sequence patterns, and we can do no other, no matter how random the output result may appear to others.

Thus ultimately all our calculations are constrained by the principles of Combinatorial Doctrine! Justus Grassmann, Abel and Galois were perhaps amongst some of the pioneers exploring this early Ring Theoretical behaviour.

Because of the confusion about multiplication, the combinatorial properties and principles were obscured by multiplication tables . However Cayley and others gradually clarified these tables as fundamental product primitives and started the Matrix notation.

Sequences, permutations of sequences and combinations of labels within sequences are fundamental to our synthetical structures  we call Algebra, Arithmetic , and Geometry.

So now realise that Newton pioneered all of this in his work on infinite series and the Binomial series expansion. In a very real sense all our calculation schemes are based on finite ring theoretical structures and their attendant group theories.
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« Reply #10 on: March 11, 2015, 09:11:45 AM »

Commentary on§13 continued
Lancelot Hogben intouced me to permutations when I was an adolescent determined to become a mathematician of renown!  Of course he useda pack of gaming cards to expound on the subject, and forever made cards an enduring interest!  Later I was introduced to the permutation tree method , which I alo taught with as little understanding as I was taught it . Later my own research with letter symbols rather than cards established a direct link to the tabular form for multiplication and bracket expansion.

Despite these experiences these observations were in no way flagged up as fundamental. They were of interest but that was it.

Oh I did think it rather wonderful that God had so ordained the behaviours of the world or Cosmos that they were first possible, then measurably probable, and finally statistically inevitable( stochastic processes). The concept of Random being equally likely made no sense to me as a child, and still makes no sense, which is why I soon realised it was a human designed artifice!

At university I was introduced to the Axiomatic approach, in Analysis, and then the Group theoretic approach in Algebra. Neither of which were particularly attractive, but the analytical treatment took pole position because I encountered it first. Group theory I dismissed as so much insane babbling!

So where were the lectures on Combinatorics? Confusingly buried in a semi computational subject called number theory which seemed to be attempting to ape its more respected academic counterpart Analysis.

I went to university to study " mathematics" and ended up as a casualty in a war zone  engendered by the subject boundary wars! Fortunately for me I found David Hilberts book on the foundation of Mathematics and read that over the course of 3 years, so my university education was not a total waste of time and effort!

Oh yes and I learned how to programme a computer in an elective course!

Combinatorics thus figured little in any formal way as fundamental. Even Hilbert in his section on Galois did not elaborate on the ring structure he was investigating for the quadratics, not that I would have grasped it back then.

Lancelot Hogben stands out as one of my most informative sources with Hilbert Next. But I have to say that Hermann and Norman have both been vital to my putting everything in some sensibly constructed order.

in all my years of exploring combinatorics I have never thought of the combinatorics of a sequence of oriented line segments, even when I was precisely using the notion to construct polynomial rotations and the Newtonian Triples, as well as to critique Hamiltons Quaternion 8 group!

There has to be something wrong with mathematics if this fundamental observation is not taught at an advanced level even when it really belongs in a primary level curriculum .
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« Reply #11 on: March 11, 2015, 09:47:49 AM »

I have a copy of Justus Grassmanns Verbindungslehre which I will read and review in this thread  :smiley

http://books.googleusercontent.com/books/content?req=AKW5Qafo2jMc6tsVPa9jRqpjEj3V4iELc7pgmhpYRh7Q8332YYh3pHgU-e6hbYwElgLofevs1XbWCANmf0q7Ci5QKat9N4Vk8EERDLbsZnCnG8DtF-46vAVoXytwkdgT0pQEsRxnLzXJd29gO6JOEQjH0SnHetwuKhEiWpNVFg1Zk27SyrXUverjGOzJdTbx9CGW8oUI-ze3AxtoZTP0l0lwtlBVS8TH8sghJeYP6iP2aAx26wVR52ImmSYI3GcxvqgL2EZMcwQdCK1Xna7TCstHWrwu5gHtpNj-4RZ-bE90PEf7vyACx5E
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« Reply #12 on: March 12, 2015, 06:41:19 AM »

Commentary on §13 continued.

I have skirted around the many ground breaking propositions or promoted viewpoints in this section, mainly because they are so deep that I would end up writing extensively on topics hich Hermann fully plans to extend on in the rest of the work. However I cannot stress enough that Hermann is not reiterating what one is commonly taught in mathmatics even today.

Pay attention to the labels and symbols and how they are used Rhetorically. Have in mind that subjective and objective experiences and viewpoints are constantly employed explored and contrasted . Realise that the labels are a formal abstract that brands, "virtually empty" entity waiting to be populated with true empirical data which comes from the real empirically observed and geoetrical space we call reality. Understnd that every label is held independent from the geometrical Doctrine and is derived by a scientific scrutiny of empirical behaviours snd properties of space-like objects that are related to a geometrical label .

That last point is crucial. One may start with the geometrical point, which thus makes that thing a real entity to be explained, or one may start with a label " point"  and describe what it is Designed or defined or constrained to do, or how it is to behave. However once we go down this route we must specify precisely what it can and can not do , what it can or can not be, and what it's principal functions and interactions are.

If you have done any objective programming, that is a design coding system called object oriented, you will recognise the task of specifying how the " object" may or may not interact or relate to other bits of code within the grater code body of a programme or application.. This is the kind of discipline and constraint Hermann worked under..

But now, as coders know, once you have designed and implemented how that piece of cide will run,it now can be used for any analogous operation! In fact, as the actual elements in the hardware are bistable floppy circuits, what you designed to behave like a point will actually accept any data encoded within that precise data structure. So if a code a square function/ procedure to square any double length byte, it will square any data encoded in that format, whether it is a numeral or a chRacter!

Thus Hermann states: if we go down to the level of the bistable circuit , that is the fundamental element of the hardware we can make that whatever we want/ physically can and the exact same process will run it and produce the analogous output.

This is a Functional description or expression of behaviour. The function is fixed, the input and output are free, up to certain conditions or constraints of the function.

So the creating whole element labelled "the point" will out put a line, labelled "a line" it will out put a surface, labelled "a surface " it will output a space.... All of course subject to the constraints. And since the process involves scrutinising every "point ", line etc it naturally is related to every permutation of every Zustände or method of recording the condition and status of a point, line etc, within the totality of these condition / status markers.

You can feel why the labels must be so Abstract, that is devoid of real content! While this initially feels so weird,eventually, as in programming or coding, you realise that you can code for a general behaviour by coding for a specific behaviour and then adjusting to give it the greater generality. This is the greatest Freedom Hermann is referring to, and the way two arrays grip in one another, one specific the other extremely general based on that specific one..

As a consequence of linking to the combinatorial Doctrine Hermann has to have a beginning and end " point" or as it is branded " a special or specific position".

The final property he discusses in the context now of beginnings and endings is the contra processes that reverse the out put relative to the original process. Again there is this duality of opposingly running through each other processes which nevertheless use precisely the same element in reverse order.

Remember now, as Hermann points out, any permutation of symbols or labels can be interpreted as a variation of position or a transformation of position. Thus the permutations now encode translation, rotation, reflection etc ,in these extended magnitude structure creating processes.

These transformations are viewed as Sidechanging ones!
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« Reply #13 on: March 24, 2015, 12:07:50 PM »

The translation of §14 is again throwing up deeper pedagogical issues ! It does not at all go as I initially anticipated.

When I was first introduced to " algebra" at primary school level, it was called geometry. Consequently after learning some basic geometrical formulae and demonstrations at primary level I somehow grasped the adhoc jumbled and confusing introduction of Algrbra at secondary level.

At that stage it was all a,b, apples , oranges,x,y and let x be thus and such!!

To the question 'what is x, what does it "stand  for"(Zustand)?' several replies were given! Initially " an unknown" number, later quantity; a variable(?); an empty "box " in which we can place anything. To a 12 year old this was not apprehensible! I picked the empty box idea.

Later when we were introduced to equations, and the term "an expression", I already knew how to duck the brain frying effect of this confusion of terms!. Oh yes the word " term" crept in there too! I had an English language grammatical apprehension of that word.

My point is, my elementary teachers and later I found my university teachers knew how to use the symbols, per se, but we're incapable of explaining or expounding on the labels or names which were attached to these symbols!

One of the causes for this was the originators of these terms were French and German. The French were very fluid and expressive thinkers, proud of their achievements and their mindset on the world, society and historical movement. They were the originators of Revolutionary thinking,Liberty, equality and Fraternity or all!  Thus their philosophy and mathematical expression reflects that.

The Prussians were more rigidly structures, and feared the revolutionary chaos on their borders! But the younger Prusdians were liberated by it, drawn to its exciting possibilities. Most were educated by French thinkers. However the Prusdian expression was deeply explorative and connected, based on homiletics and liturgical exposition of biblical texts. Philosophers generally left no sentiment unexplored and so their expositions were weighty, tautological and full of case by case arguments. The same text or symbol thus could generate multiple meanings related by the same textual form. Thus Prussian philosophers used symbols as a kind of variable expression whose meaning has to be determined by and which is constrained by context.

The difference between equating( equalité ) and likening( gleichung) is sometimes profound. The elevation of an equation to a symbolic expression in France , liberated it from any common meaning to a specific " mathematique", whereas in Prussia , the writing of an expression symbolically was still a rhetorical shorthand. Consequently the symbolic form were exegetically treatable: they were mired into common sensibilities and circumstance, case by case.

Hermann therefore takes this second route. He treats the label of the creating whole element as an exegetical exercise, in which the reader is brought to an intimate apprehension of the feelings and sensibilities of that element!

What is it like for a creating whole element to vary? According to Hermann it is as agonising as giving birth!

With that analogy in mind, and the general background of these differing cultural conceptions one may grasp how the variable concept permeates modern physical theory, while the abstract symbolic expression dominates Mathematicl thinking. Mathematicians do not suffer through variables as Theoretical physicists do. In fact they only have a hard time when they come to explain what they are doing ! Unfortunately we have lowed them to hide behind the mantle of God, asserting that they are s
Raking in Gods tongue!

That is why to most ordinary people mathematicians speak gibberish.

Translating §14 has brought this pointedly to the fore. Hermann strives to be crystal clear, but our mindsets make us confused about what his exegesis is about. It is about what we are fundamentally thinking when we describe a formal meta geometry or a subjective Spaciometry that models empirical behaviours of partial objects as they transform. What we really do is cherry pick from an infinite set of possible outcomes( output results). Whatever we cherry pick is determined somewhat by constraints.

Constraints are supposed to be objective: but everything we do or select is ultimately subjective! The discipline therefore is to be autisticlly rigorous! If we choose a whimsical set of constraints, and we design the product knitting and the synthetic and analytic knitting of these elemental magnitudes, then we must watch ourselves strictly so we do not fudge the results! We have to let the rules play out as they were designed and implanted!

The confusion that many Mathmaticians felt when an automatic system comes up eith an unexpected result is profound. I realised this when computer programming in 1973 using Algol60. I spent many weeks being feverishly ill trying to get my programmes to work as expected! I eventually embraced the programming paradigms and let go of the subjective mathematical intuitions I had built thus far. I could admit to the sloppiness of the way I had been allowed to think!

Rigour was a new and cleansing baptism of fire back then, and it has been a foundation of my meditation ever since.

Proceed slowly to gain the maximum benefit from Hermanns exegesis.
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« Reply #14 on: March 24, 2015, 01:39:10 PM »

I have had a problem with Normans promotion of Quadrance since I understood wht it was!

I suddenly understood where my difficulty lies. Normn bases his development on some abstract thought pattern he calls a rational type or a rational number. Both I and Hermann base our undertaking on magnitude, a general experience of space.

In particular I have learned by Hermann about the intenive and extensive magnitude experiences. I was thereby willing to accept the bounded line segment as the best symbolic expression of these aspects of magnitude in both it's continuous and discrete expression.

But the line segment is only an aspect of a higher level/ step representation of a extensive/ extending magnitude. . In particular the flat or planar extending magnitude is the primary magnitude we experience as magnitude both extensive and intensive, do why pick the line segmnt as a primitive?

The reason is subjective, we can draw a line segment, either as an arc or a straight(ish) line. This has meaning for us, connected with motor muscle activity and thus with proprioceptive functions that occur within our deeper unconscious processing. This is why the point as a physical object and a line are indissolubly connected. Thus a line can be associated with 2 points or it can be associated with planar figures. It is the most generative symbol we apprehend.

Length and orientation and direction are also attributable to the properties of a line, but do not define a line! Rather a line segment defines these properties!
Thus a line segment is the foundational symbol allied to both a point and a plane .

Here Norman gives his clearest explanation thus far

<a href="http://www.youtube.com/v/YfY7RyAzHPE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YfY7RyAzHPE&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/YfY7RyAzHPE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YfY7RyAzHPE&rel=1&fs=1&hd=1</a>

The use of quadratic versions of quadratic forms(!) cannot be apprehended without the " functional" role that the line takes in ordering magnitudes in space. The line segment thus becomes an arbitrary Metron or Monas.

Realising this the difficulty with roots or surds is not a problem. The number theoretical difficulty is not Pythagorean in origin. The incommensurability of the units as Metrons is not only embraced but utilised in the distinction of Protos Arithmos. The protoi Arithmoi are the first or base Arithmoi for an accounting system . Thus within a square, the perimeter can be accounted by one system, the diagonals have to be accounted by another .

In attempting to place all the Arithmoi on a number line the difficulty of reducing all to one fundamental unit system becomes dramatically illustrated. To say that the Pythagoreans were upset by this is one of those perennial myths . It was the Pythagoreans that distinguished the difference  between monads.

The difficulty for the number line disappears if the Pythagorean theorem is used as the fundamental organising principle. Thus the use of "squares" is the secret of the Pythagorean "unit". This means surfaces are the fundamental form in geometry. The line segment however is a " secret" encoding system the Pythagoreans used to record important ratios.

The surds form the ratio segments that encode a spiral form called the spiral of Theodorus.

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