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Norman calls the roles of the x,y,z axes  "distinguished" but Hermann calls them elementals. The fact that we choose a mutually orthogonal set is of little consequence for referencing all and any other line segments, but it fors give us the anomalous cross product. To extend into higher ranks we have to loosen the grip of this Anomally.

Because hermanns fundamental systems are based on elements that conform rigidly to constraints and conditions and are coupled the fundamental planar form is the parallelogram. This form enables all the desired properties of 3d space and continues on into the nth-rank spaces. It is therefore distinguishable in systems of equations , renamed nth- grade systems by Hermann.

This form however is only elemental in the 3rd rank, because for Hermann the elements of a space are of the rank below. Thus for a 0 th- rank system we would expect that to be points. In fact for Hermann the fundamental geometrical element is the line segment with its dual point-line segment nature, thus the 0th-rank is occupied by tally marks. These form a calculative axis by which all elements can be fractally dismembered or divided into Metrons/ monads.

Thus 1 is a tally mark, and only has extension when applied to a first rank system. Tally marks are thus wholly formal, and "out" of any system of geometry that Hermann can devise. We as observers therefore can apply them anywhere at will, but once attached to an extensive magnitude they must remain fixed for the duration of the exposition. By attaching them to the first rank system we obtain the first of many types of Arithmoi or Zahlengröße.

These Arithmoi are the elementals of the next rank system. Thus a parallelepiped(Spath) is a fundamental element of a 4th ranks system. Rotations and other transformations are essentially mappings by transforms from one oriented, projected or curvilinear parallelepiped to another . Transformation of basis is therefore a fundamental method of translating, rotating, reflecting and projecting spatial objects.

The general Doctrine of Thought Patterns( Ausdehnungslehre 1844)

§1. We everyway stand the rank arrays, those from the truths, under the general doctrine of thought patterns, which arrays situate themselves on top of like cognisance over all branches of the mathematic , and therehere  only the general labels of the Like quality and the Differing quality, of  the Binding and the Loosing (Separating /Sundering) are pre set out ( prior to anything else)

Therehere it must, the general doctrine of the Thought Patterns, go on ahead into all speciality branches of the mathematic, for-there there  that common branch is still not as such to hand how we require of it for our expertise
and we  still are not  permitting it to travel over, without everyway wiggling to us in unutilised endlessly running   qualities ,  so nothing remains left over  to us , as the same things here are thus far developing .

 In order to firmly set the label for the Like Quality and the Differing Quality It is here initially. There the Like entity necessarily steps here out , also therewith is already the dual quality  , and the Differing entity also as like Entity must appear, only in Differing view from afar;
 thus it appears necessary by considering over-layering tracking, to place on top( of each other) differing relationships of Like quality and Differing quality ;

Thus would  be able to  become declared ,( to the considering game),  by considering everyway equating ( likening) of two bounded lines:
 the Like quality of
the Direction
or the Length,
or the Direction And the Length,
or the direction and the laid Position ,
and so further ;
and by considering other things to everyway  equateable constrained Things   other relationships of Like quality would again come to step henceforward.

Therefore that already these relationships become other relationships , each according to the Attributed quality of the "to everyway equateable" constrained Things, delivering the demonstrable   proof for  there , that these relationships are not next of kin related to the label of the Like quality itself, rather by the contents ( lists) , onto which the same label of the Like quality becomes applied,

In Practice from 2 "equal length" line segments, (to the by considering game), we cannot say that they compared besides themselves are equal, rather only , that their length  be equal, and this length then stands plainly also in the complete relationship of the Like quality.

Thuswith  we have to the label of the Like quality its simple quality returned,

and we can by the same quality therein assign that   "the  entity” is like:-
that one from which one can continuously the same entity declare,   is "like";
or the general one, what in each common judgement themselves compared side by side  can be substituted,  is "like"

How by the same quality herein immediate nearby lies declared:
that even if 2 thought patterns are like a third, they Also themselves are like one another ,
and that the   entity which out of the Like entity  was reifying  on this same cognisance ,  once again is " like",
by the same quality lies in the daylight !

Footnote page1

•see induction No.13
•• plainly the No.5

The second in Normans series on Axiomatics

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Just beware the assumption that Euclid used the concept of Axiom! The Greek language does not sustain such a notion and the nearest Greek concept is that of axle or axis, both referring to the central rotating region of a disc..

The history of Axiomatics seems to start in Kant, and represents a powerful redaction of earlier notions, including the muddled thinking of some Islamic scholars who mistook Euclids Stoikeia as a version of Aristotelian logic, being great admirers of Aristotles works..

They seemed not to differentiate between the Pythagoreans and the Platonists, and so did not realise that Euclid was a Pythagorean Mathematikos supporting Plato's promotion of Pythagoreanism, and actually achieving the recognised status of Mathematikos among the Pythagorean scholars who were still based in monasteries( Monas - fundamental Pythgorean ideal) in southen Italy and Mediterranean Aftica.

I feel I must reiterate: we have moved beyond the dual rank array of LaPlace and particularly LaGrange , although you would hardly know that from mathematicians. Computer scientists use multiple rank arrays to encapsulate reality. Quantum mechanics uses multiple  probabilistic rank arrays to portray their data set findings and models, but perhaps the most accessible example is the no humble video camera which is installed in every cheap phone! Each video frame in each short video is a rank array that encodes empirical data sets, the Codec enables those rank arrays to be pushed to and from input/ output media which we can see as a moving display.

The fact that we do not make these connections is the real shame. It allows the majority of us to be hoodwinked by the few.

The humble fractal generator is every bit as capable and tooled up for basic research as any Cray supercomputer, it just does it more slowly! Many of the artistic rendering members are achieving are of direct empirical relevance to our understanding of how our models of reality work.

The general Doctrine of the Thought Pattern (Ausdehnungslehre 1844)

§2 the second comparative statement, which we in view have pulled  to here, is of the entity of the every way knotting and sundering( binding and loosing)
Even If two magnitudes or thought patterns( which Name we prior pulled  as the general one. See Induction .3) under themselves are everyway knotted, thusly they brand them: 
limbs of the everyway knotting( knitting), 
the thought pattern, which through the knitting together of both comes to be presented, 
the out giving result of the knitting together.

Should both Limbs become distinguished, thusly we name  the one the fore limb( more forward one)  and the other the hind limb( the more afar one )

As the general sign of the knitting together we choose the sign "^"
Let now a, b be the limbs of the same, and indeed  a the fore limb and  b the hind limb, thusly we signify the out giving result of the knitting together with  (a^ b);

In which the brackets here should express: that the knitting together no more in separation of  their limbs should be manifested, rather as a monad ( unit) of labels.

The out given result of the knitting together can once again with other thought patterns be everyway  knotted, and thusly one reaches to a knitting together of more limbs  which therefore  at the immediately nearby  always appears as a knitting together of each 2 .

In order to make way for the ease of practice we enslave to ourselves  the usual "shortening off" bracket assigning, in which we specifically the together related  signs of a bracket  let go away, even if  the "of the discussed" opening sign[(]  either aside the beginning of the whole expression stands, or after an other opening  sign would  be following; to the by considering game,  in place of ((a^b)^c) we write  a^b^c

Commentary on §§1-2

Because I am reading ahead I gain insights about the purpose of the earlier paragraphs, which is useful when translating.

It is now apparent that the first paragraph on Like and Differring entities is about 4 identified ways of declaring 2 thought patterns like. Thus it is about equality, duality, congruency and identicallity , and by implication any other mode of expressing like entities. Moreover it is about when and how we declare entities to be like, and thereby the judgement we use to make such a declaration.

In a very real sense most of us have been trained to identify like entities, few have been trained to use their judgment to declare likeness or equality as a finding, especially so for mathematicins who have like entities given to them on a plate!

Comparing , contrasting and then concluding based on the results of those 2 prior processes  is essentially the 3 part structure or tool Hegel uses in his presentation. The conclusion allows a judgment to be expressed, and often recommendations for improvements . The conclusion can be as short or as long as the author wishes to expand on his or her insights. Hegel steps back to view the 3 part structure as essentially a historical or developmental process in history he called the Dialectic.

It is a clear movement to and fro between greater vision from obscurity and sometimes from grand apprehension to diminished comprehension of a new contingency, and then onwards, at each stag, in the moment,  being unaware of the necessities of the resolution that then emerges as a glimmer or a shower of sparks!

Marx and others felt that not only was this process historically inevitable, but that also acting in accord or mimicry of it was justifiable as invoking the inevitable solution/ resolution. The end thus justifying the means.

However if Hermann is taken as a careful student of Hegel we cannot justify that kind of behaviour. Rather what Hermann shows is that in analysing the current position in the dialectical process we may see more clearly where we were imperfect and hopefully synthesise a more perfect apprehension.

Thus by being rigorously analytical in the comparisons and contrasts of the concepts of like and differing we may better understand the use of the "=" label or sign, and not just in arithmetic or mathematic, but more widely and dynamically.

Certain conventions have to be set. But by setting them as simply and carefully as possible Hermann hopes to securely found the doctrine of thought patterns.

Thus he now moves onto binding and loosing . The reference here is the keys of Peter . "What is bound in heaven is bound here on earth and what is loosed here on earth is loosed also in heaven". By this he gives a certain gravitas to the rules he is "out giving" or evoking here. The evoked result or out giving result/ response, or even the responsive result are essential notions behind this knitting process..

Hermann here means to deal with fundamentally the notions of commutativity and associativity. Because we are introduced to these concepts in the arithmetical arts we do not apprehend where they derive from or what they ultimately mean, over the next few paragraphs Hermann will give concording demonstrations of these notions and their implications and in fact how knitting and unravelling , binding and loosing encode addition and subtraction at a fundamental level , but also multiplication and division at a barely less fundamental process level.

The implications for commutativity and associativity are however remarkably different for both levels of process.

After years of consideration I have concluded that it is more complete to start with a whole and then divide it. Thus division also known as subtraction at a more fundamental process level is narratively prior to multiplication and thus addition.

Thus I start with Analysis or repeated( ana) cutting( lysis) of the whole  eventually followed by Synthesis or binding( syn) creation( thesis from Zeus-made or brought about by Light/ Lightning). In this way addition and multiplication has a purpose: to reveal the whole, while division and subtraction have their purpose: to uncover the structure of the whole.

These re patently Pythgorean ideas but belong here in this kind of fundamental discussion of how we came to develop the qualification Mathematikos into the subject of. Mathematics.

Is Hermann saying this is the only synthesis possible? On the contrary, he is showing how carefully we can construct the fundamental skill sets required for an evoked resultant expertise. But it does not mean we can just choose whatever we like whenever we feel like it! The rigour and the consistency demanded for serious consideration and acceptance become the more onerous the more fanciful or complex the fundamental notions that are proposed! By always choosing the simplest notion Hermann not only lessens the burden of proof on himself, but also recommends its acceptance to us as a " trivial" matter.

Later we are surprised how sturdy an edifice he then builds on such "trivialities"!

However, some have used this freedom, and the general ignorance of the process to introduce desired trivia that in fact masquerade as trivia. As Norman points out: it is not a trivial matter to apprehend to ourselves infinite powers! The trivial nature that I speak of is not the trivia of words : that is" let us assume infinite powers", or "let us be god "; which indeed are trivia bordering on frippery, and in another age would have brought death upon those proposing it! Rather the trivial thing is that of action: can I or cannot I bind 2 limbs together? Can I bind such a bound thing to a third? Can I repeat such a process until I run out of things to bind or become exhausted or die?

Such trivia as these one can readily assent to or deny . But it is by assenting to such as these that we extend our consciousness of our ability to construct certain evoked results , and to rely on calculation as meaning something in our constructive ability.

For this reaon was at first repudiated and then by dialectical degrees bought into meaningful use after some proper adjustments in interpretation.

We shall see how Hermann sets out to do this at the beginning of his Grand enterprise.

The general Doctrine of the Thought Pattern (Ausdehnungslehre 1844)

§3 Now the special artwork of the knitting together therethrough becomes evoked, what becomes firmly held by considering the same knitting  as "advantageous result" , that brands: under  which  circumstances and in which "extensive magnitude"  the beneficial result is becoming set as remaining  like itself.

The singular every way varyings , which one , without the individually knitted together thought patterns themseves changing , can take in advance ( of specific details) is the varying of the brackets and the un-ordering of the limbs.

Let us take to mind first the knitting together thusly  that by considering 3 limbs the setting of the brackets no real distinction, that brands: no distinction of the advantageous result founds,

Therefore  that   a^(b^c) =a^b^c  exists.

Thus follows immediately nearby that one also in every multiple limbed knitting together of this artform  without its advantageous result varying the brackets can be let go away.

Because each bracket encloses immediately nearby a two limbed expression every way satisfying the there over firmly set appointed ( meaning)  , and this expression must once again as limb be evryway connected with an  other thought pattern, in short it steps an every way connecting from three thought patterns henceforward, for which we set out ahead ( of any thing else) that one could  let go away the brackets without the advantageous result of their knitting together varying.

Therefore  there, one is permitted to set in  place of each thought pattern the like entity to it , the total beneficial result, through the letting go away of each bracket also is becoming not varied.

Therefore


Or there, one is permitting in 2 expressions, which only through the setting of the brackets are distinguishing themselves continuous according to the plainly outwardly demonstrated Proposition the brackets let go away , thusly are both expressions also under  it alike, (there, they are alike to the same( brackets  losing) expression),
and one has the foregoing proposition in something of a more general thought pattern: