END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

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.

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 :

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.

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

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.

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.

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 .

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!

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.