Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

I recall a vision I had when looking at Eulers equation for polyhedrons

F +V – E–2 =0

At the time I was working in the thread Fractal Foundations of Mathrmatics and I was researching polynomials as a conceptual notion. I could see the direct connection to a spatial object in degree 3 polynomials based on Eulers equation

S + F – E + V – 2 = 0
Where S is the number of spatial objects.

I put S in because I clearly had a 1-1 polynomial equation as a cubic

0.X³ + k.X² –m.X¹ + n.X⁰ – 2 = 0

I understood that as a journey through spatial ordering starting with points, sequencing them into vertices, then connecting the vertices with points to form line segments, and at that moment both faces and the whole object was reified!

Or more mysteriously points are formed sequentially into line segments and then as each line segment extends into another a face is formed and the line segments take a turn into another plane, gradually building face by face until the whole solid is formed.

But then the proces did not stop! It continued to build a line of solod forms and then a plane of solid forms and then face by face a solid fom of solid forms!

But then it reiterated the same synthesis plan at that level over and over!

It clearly was a Fractal ! The solid form did not have to be solid throughout it could be a mesh ,mand each mesh was at a different "level" of complexity, but completely captured or encoded by a polynomial of degree n!
In which case the alternating foresigns had a meaning of direction. We had missed that simple synthetical requirement .

On the way through I realised as I had suspected that n dimensional space is where we live already, we just had to clear away the fanciful notions of needing more than three mutually orthogonal axes!

I also saw the interplay of how at each level the solid form becomes the 0 of the next or the point of the next , I could not decide because what role did the numerals play in physical space? Especially as X⁰ was already accounted for.

It has taken time but I gradually realised one thing was missing: me the observer, calculator and manipulator! The numerals represent my conscious involvement in the whole polynomial expression, in formulating the equation and in utilising the spatial orientations of the structure to unravel a commensurable Arithmos as a count tally.

I did not understand it back then but it is becoming clear now as I work through Grassmanns exposition.

Whatever we humans record and commit to myth , memory and legend, mystery teachings, curricula and public eduction systems,writings in particular and general , despite being all screwed up, messed up , chopped about etc, will still holographically reappear in someone's consciousness whole and complete!

The genes will reassemble to construct an exact replica! What cannot be guaranteed is that it will be utilised by that individual/ animate  to whom it first reappears, but eventually it will reappear to someone who can and will utilise it to human advantage.

Without elaborating the Hegelan dalectic corresponds to an evolutionary developmental schema and particularly that of a virus or junk DNA scenario. If life evanesces anywhere it will eventually attain to levels of consciousness at which we are and beyond. if the universe will allow.

This , as measured by the geometrical structuring concepts of crystals is as inevitable as the day is long. We would not accept it out of deference to God, but space is a fractal that emerges out of an infinite regress/ progress synthesis.

[jehovajah]

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§6 The new setting out ahead( of everything else) , which we are from afar to here fusing, is the setting out ahead, that the output result of the analytical knitting together  is clear (one assigned), or with other words, that,

even if the one limb of the synthetical knitting remains unvaried in everyway, the other limb therefore varying itself , then also the output result every time varies itself.

Hereout outputs itself  immediately nearby , that
                          
                              a ^ b v b =( is like) a
                                
exists, because  a ^ b v b = a denotes the thought pattern, which with  b synthetically knitted  a ^ b  gives!

Now   a is such a thought pattern ( one assigned) and is everyway pleasing of  the  unambiguously clear quality of the result, therefore the singleton thought pattern  is outwardly demonstrating the validity of the above likening as .

Hereout once again henceforward goes that

           a ^ (b v c) =( is like) a ^ b v c
exists.

In order to specifically bring the second expression around onto the first , one can replace b in it with ((b v c)^c) and hold out

a ^ b v c =( is like) a ^ ((b v c)^c) v c;

This is, according to §4

         is like(=)     a^(b v c)^c v c,
and this once again concords to the thusly plainly outwardly demonstrating Proposition
                       =(is like)a ^ (b v c)

Therefore is the first expression also like to the last : (QED)

There one can these conclusions completely soundly once again reiterate, even if multiple limbs in the bracket come to the fore, thusly one has the proposition:

" Even if the synthetical knitting is a simple one, and the inter communicant analytical knitting is an absolutely clear ( one assigned) one,  thusly can one concording to a synthetical sign the Bracket whimsically set or let go way.

We denote then (  if that absolutely clear ( one assigned) quality onto a general cognisance is finding place):
 the synthetical Knitting Addition,;
and the inter communicant analytical Knitting Subtraction."

What the ordering of the limbs troubles, thusly follows that

a ^ b v c =( is like) a v c^ b exists;

 because

a ^ b v c =( is like)b^ a v c = b^(a v c) =a v c^b

Thusly that :

also, therefore, we   have demonstrated as concording, the everyway toutable quality of 2 limbs, of which one limb has a synthetical foresign the other an analytical foresign,

Thusdirectly the completely clear (one assigned) quality of the analytical output result is set out before ( anything else)

And now under this setting out ahead( of everything else) the propositions of this  paragraph gain power, while the propositions of the previous also  then still gain power, even if the output result of the analytical Knitting many assigned is!

Footnotes
• By considering Game entity  of such a many assigned quality delivers nothing direct, how itself later comes to be in order to show, the extensive / extending magnitude Doctrine in abundant crowds, rather then also Arithmetic displays  itself, and it is therehere the firmly set differentiating also  important for it(the Doctrine). Specifically Addition and Multiplication as simple knittings together displaying  themselves  ; and while the subtraction always is clear ( one assigned), thusly is it only the Division, as long as the Null not as Divisor appears: the propositions of the previous § generally are of the half empowering for the division only  ,  while the propositions of this § only with the restriction empowers, that the Null does not appear as divisor. Out of the not- tracking of these circumstances must the most urgent  Objections  and everyway whirrings ( fireworks!) henceforward arise , how it also toward the division has happened!

•• a far later placed besides everyway search, upon which to base   the rules for the knitting together of multiply assigned Magnitudes  has guided me to the conviction That above all one must transform the multiply-assigned magnitudes Initially into one assigned magnitude,  importantly before one  can apply on them  a random set- knitting  . I have   of this conviction  in my  Ausdehnungslehre from 1862, in the added remarks to No,348 and to No,477 expression everyway distributed, and i have together at like moment  besides former position demonstrated, how one can transform the multiply assigned magnitudes into one assigned magnitude. Also my Arithmetic( Stettin 1860, printed and everyway laid out by R. Grassmann ) lays this conviction to ground.(1877)!

[jehovajah]

Commentary §6

The overriding result of this paragraph is to demonstrate that all the threads for the use of bracketing and ordering of any synthetical and analytical process have been woven to completion.

The implications are astonishingly far reaching!

Because of his approach Hermann sits right on the borderline between 0- dimensional systems and n- dimensional systems. He can easily pass into all of them very generally, as these sets of propositions and demonstrations show.

But he concludes by mentioning the one assigne and multiply assigned magnitudes. These are new concepts that arise out of his deep thought patterning of space. Later we will get a clearer representation of these terms, but for now just think of the process of solving a rank array matrix by Gauss's elimination method. That is we reduce a system of likes row by row until we end up with a singleton limb with a single unique valuation . From this, as we shall see we can then go back up the analytical chain and evaluate every row( or alternatively we could do it by columns).  Each row is therefore the multiply assigned magnitude, and thus the sense of multiple asignment is multiple variables/ line segments/ spaces etc, etc.

It is hard to believe that we have placed ourselves in such a powerful commanding position simply by adopting Hermanns dialectical approach!

I have one dissent and that is over Multiplication, but I have explained that already.

[jehovajah]

One of my major stumbling locks with understanding Hermanns reasoning was how he came to the conclusion AB = –BA for products.

The reason for my difficulty, and perhaps that of many mathematicians in his time: quit apart from being sloppy, and quite apart from recoiling at a bit of Philosophy was simply the ingrained weak case thought patterns!
The Arithmetical thought patterns and number bonds etc are the first we are forced to memorise without question! Consequently if ever anyone dared to ask if the thought patterns were "true" the response would range from seemingly sensible but reinforcing examples to downright vilification in front of the whole school!

One soon learned to please the teacher, but then later to be pleased by ones interaction with the material. But one never quite has the time and finally not the inclination to start over! This is where the Grassmanns were different. Each for his own reasons and predilections seemingly engendered by Justus Grassmann the father of Robert and Hermann, to start from the very beginning and build up.

Hermann was self activated by his father as was Robert, they both received a free school, Pestalozzi  like education. Robert showed early progress, Hermann seemed to be much slower. He surprised his family by winning the prestigious mechanics award for original if confusing work on the Ebb and Flow of Tides, and also because he was nearly the only entrant into the competition.. Despite this he was held back while Robert soared up to university, doctorate, and district regulator for the teaching of mathematics. Hermann had to plod on working to help pay his way and only in brief moments of liberty able to work on his pet project the Ausdehnungslehre.

Finally he printed off what he had achieved gambling on a lot of interest, but the ventUre failed to make any money whatsoever and represented a financial loss as well as a slap in the face.

We have this flawed masterpiece because Robert rescued it from oblivion. We have lost the purported second volume the Kinematics of Swinging which would have built on his paper the Ebb and flow of Tides, and instead hit a rehashed Ausdehnungslehre Robert style. Anyway, it was a modest success and Hermann eventually was happy to keep his name on it. To his joy he was now able to reprint his original 1844 one with annotations. He had moved on a long way since then but still regarded the 1844 work as the fundamental ground of his later works, articles, essays and arguments.

Without reading it one cannot appreciate how fundamentally different it is to any other philosophical treatment of Mathematical principles.

So here we see that starting with 3 elements and 2 processes, synthetic and analytical one derives results that apply to knitting together cats and dogs equally as well as any production line process and of course to addition and subtraction.

One had every reasonable expectation that these rules carry over into the product case. Dropping down from 3 elements to 2 in the context of 3 brought a surprising and troubling result! If his reasoning was correct , and it was Hegelian dialectic so it should be rigorously sound, then A general principle had been missed by all mathematicians, but especially Mechanics and Geometers: anti commutativity in any any 2 limbed combination in the context of a third limb!

Or rather we recognise it in the synthesis mode, but ignore it in the multiplication or product mode of combination,.

Hermann has yet to demonstrate this but it follows from these propositions because of their general nature and applicability to multi-limbed synthetic and analytic knittings, within a n-step/ stage system governed by rigorous rules and constraints.

Hermann in the footnotes states that we should kick up a fuss over the slack or even non tracking of these important differings and differentiation.

I can recommend Norman Wildbergers WildlinAlg series as the closest to Hermanns systems as written that I have seen as a regular university level course.  Of course some specialist have embraced Hetmanns ideas directly or indirectly through Hestenes geometric Algebra, but no one puts it like Hermann, not Robert, not Justus, not Norman not David. It is a worthwhile and educative read .

I now feel that the Ebb and Tide paper also needs to be read , but that Hermans treatment of Hamiltonian quaternions represents a maturation if those ideas even beyond swinging arms into rotating arms and arcs.

We will get thereby to the Fourier and Laplace Transforms by and by as descriptions of general kinematics in any pace including fluid Dynamics.and wave deformations in any medium including magnetic plasma currents.

[jehovajah]

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§7.  Through the analytical everyway journeying one reaches to the Indifferent and the Analytical thought pattern. The former one holds out through the analytical Knitting of two like thought patterns, therefore a v a represents the Indifferent thought Pattern, and indeed is the same independently from the Value of a .

In practice a v a =( is like ) b v b;  because b v b represents the thought pattern which with b synthetically knitted b gives as a result, such a  thought pattern is a v a, there b ^ (a v a)  = b^ a v a = b exists.

Now In the surrounding catchment in which at the like moment the output result of the analytical knitting one assigned is , therehere also  a v a becomes set like b v b. There thuswith the Indifferent thought pattern under the performed setting out ahead( of everything else) always only One  value represents, thusly the needy quality outputs itself thereout  , it  to fix( the concept) through a central sign.

We choose thereto for the blink of an eye the sign &,

and besign the thought pattern (& v a) with (v a) ,

and name (v a)  "the pure analytical" thought pattern; And indeed,  (if the synthetical Knitting the Addition was), "the negative" thought pattern!

That (a v &) and (a ^  &) are like a,
that further ^(v a) is like v a and v (v a) is like ^a directly outputs itself,
in which one only has  to substitute the plainly represented fully standing expression of this thought pattern,
  thuslike the correct quality of these likenings to overview looking all around it.

The analytical thought pattern directed toward the Addition we were naming in the special case  "the negative" thought pattern and the Indifferent, in relating onto the Addition and Subtraction, we are naming Null.

Footnotes
There is an everyway farmed out  undertaking, even if one , (to "the by considering  Game") , by considering the addition and subtraction in Arithmetic, according to which  the herehere related rules  for positive tally Markers one has the concording demonstrated, it afarhere still especially wants to establish ( as demonstrated)  for  negative tally Markers. In which one namelike the negative Tally Mark as such defines, which to a added Null gives as a result , thusly one means here with the Adding( in which the label of the same adding process is  placed upon immediately nearby only for Positive Tally Marks )
either the same Knitting together cognisances , for which the Fundamental rules empower, which invoke the general label of Addition ,

Or one other knitting together.
In the former case  the concording demonstration  is unnecessary,  the wider rules then for the negative Tally markers, are already therewith demonstrating; in the latter case it is impossiblle,  if the label of the Addition of such Tallymarks not of something still  should become concording! Thusly plainly by the "fractions" in the comparative statement leaning against  the entire Tally Markers.

[jehovajah]

Commentary on §7

The indifferent and the analytical thought pattern are two of the most general ideas regarding dynamical situations that I have ever come across.

Earlier I precluded the power of these general notions by jumping, once again, too soon to a conclusion! These ideas are not about zero or negative numbers or evaluation but about unchanging results independent of anything else and about the analytical thought pattern itself!

The analytical thought pattern is precisely the tool needed to break down a whole into its constituent parts for further examination. In arithmetic it is naturally associated with negative tally marks, but the footnote here says that for the sake of rigour that needs to be demonstrated, and suggests 2 lines of demonstration. In the first it is within the definition of summation, and so needs no proof( this is the course we have been following in these discussions) but in the second case it is a bit more tricky. We have to analyse the comparative statement comparing it against the whole of the Tallymarks concept to find those parts that give rise to the negative tally marks. It is clearly a tricky task otherwise Hermann would have demonstrated it in the 17 or so years between the 2 books. He gives no reference to where he has done this particular demonstration, but he is convinced it can be done.

In this regard Norman Wildbergers construction of the Integers is a very relevant demonstration concording to Grassmanns style..
<a href="http://www.youtube.com/v/YDBLXCFrihc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YDBLXCFrihc&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/YDBLXCFrihc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/YDBLXCFrihc&rel=1&fs=1&hd=1</a>

We shall see next where all this careful demonstrating can now take us, and it ain't to establishing Arithmetic folks! Numbers are of passing importance as you have just seen?

[jehovajah]

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§8.we have until here the label of the addition purely  formally held onto, in which (position) we appointed it through the value of certain set knittings.

This formal label remains also always of the uniquely general  knitting.  Yet this is not The Artform,  how we reach toward this label artform in the individual branches of the Mathematic  . Much more outputs itself in them out of the creation whole of the magnitudes themselves a centrally acting "knitting" cognisance, which  because therethrough , that those formal rules upon it are applicable ,  presents itself  as addition in the plainly given to us general Sense.

We track specifically two magnitudes( thought patterns), 

which through advance setting of the same created whole cognisance go henceforward , and which we  name " in like sense created whole" , thusly is clear, how one can array both so: besides one another ; 

that both a complete entity produce, in which  "of  both sides" content comes to be thought related to it, that brands: the parts which both hold within, come to be in a "together!"-Thought entity ,and this whole then with both those magnitudes  "in  like sense created whole"  comes to be thought, in a like manner.

Now it is easy to show that this knitting together is an Addition, that brands: that it is a simple one , and its analysis is a one-assigned one. Initially I can whimsically  adhere together and whimsically everyway tout ( for exchange), because the parts , which " together!"-thought become, thereby-considering   remain the same, and their succession not of other entity may (be), there they are all " like" ( as through like created whole was rooting and rising up)

Therefore it also exists  its analysis is one-assigned; because were this not the case, thusly must by considering the synthetical knitting, while the one limb entity and the output  result  the same remain, the other limb differing Values can take besides; then from these Values must be of which  one greater than the other, therefore then must to the latter yet still parts from afar to come; then therefore  also would arrive to the output result the same parts from afar, the output result also an other Output result  to become , contradicting the setting out ahead ( of everything else).  Therefore there also the inter-communicant  analytical  knitting one-assigned is, thusly is the synthetical knitting as Addition to be apprehended, the inter-communicant Analytical knitting as Subtraction, and it  to empower according to the  set down rules.related for these knittings all in §§3–7 .

It was outputting itself there in that place , that the rules of these Knittings, then also remain  unvarying in everyway to endure, even  if the limbs become negative . We everyway liken the negative magnitudes with positive ones so we can say  they be in"  running into 'against set' sense created", and thuslywell we can adhere together the magnitudes in like ( ' against set) sense created as the magnitudes in " running into' against set' sense" created  under the Name" like - artform  magnitudes" , and therefore,  the real label of the addition and subtraction for like-artform magnitudes is appointed mainly  upon this cognisance.

[jehovajah]

Commentary §8

In this paragraph Hermann demonstrates the application of his method. The previous set up has created, rooted and risen up a set of rules . These are formal rules tHey do not apply in this form to spatial magnitudes. Nor do they conform to the usual mathematical formalities of addition and subtraction, which are partly based on real magnitudes.

So now he sets out to how the formal ideas apply and then how that subjects the. Entity to the formal rules in & 3–7. However he does not apply these rules to so called numbers, but rather to distinct magnitudes as quantity magnitudes.

We start with their creation whole by some process. Well by the same process we can  adhere them together to make a combined form. This demonstrates simple synthetical knitting,

Then he demonstrates that the analytical knitting process requires that the result is one- valued. That idea is explained by example in that demonstration,

Now in demonstrating that magnitudes satisfy these 2 characteristics he relies on the real experience of magnitudes of bound quantity. He simply has not done  any formal set up to justify his assertion that there must be one magnitude bigger than another! The veracity of this assertion comes from real spatial experience of bound quantities whimsically collected.

 He then goes on to bind positive and negative magnitudes together. There is no other explanation of these negative magnitudes other than they must run into against the positive magnitudes , but be like the positive magnitudes in every other respect.  Thus yhis is a spatial identification of negative magnitudes:,they simply run in the opposing direction right into and through the positive ones.

The only way to thought bind them is to rebrand them as like artform  magnitudes, except of course we have to run in the opposing direction to the positive magnitudes.

Compare yhis construction of magnitudes to Normans formal algebraic one.. Is this less rigorous?

I do not think so because we have set out the rigorous general rules, here Hermann is simply identifying how we have to arrange real objects in real space to comply.

The rules remain enduring!

However, please note Norman has introduced an abstraction via the tally marks and the Arabic numerals, a set of labels he calls Natural numbers. He therefore has to rigorously define these labels and how they behave. Hermann has not done anything about number so far because he is focused on real spatial objects. He cannot define how real objects behave, he just has to have descriptions of what they actually do.mthus addition and subtraction are not mathematical, they are spatial interactions between spatial objects. That is where Hermann starts.

[jehovajah]

Most people will react to so called symbolic Algebra as if they were given a dry biscuit or a sour grapefruit! It is a deadly dry dire experience resulting in tasteless uninspiring forms! But it neede not be.

Firstly the forms are not the object of the algebra. Newton did not see the artistic merit of Algebra, preferring to call it analysis, tools by which he analysed the more beautiful and wonderful dynamics in the world God had created. The most beautiful expression of these analytical results was in the formal astrological and geometrical designs drawn in ancient treatises.

Wallis on the other hand felt he needed to capture the thought patterns of geniuses like Newton, Viete, Harriot, and other pioneers in symbolic algebra especially Bombelli.  But you can't see thought patterns, without training in the ways and transformations of algebraic symbols.

We can also draw hope from the role of symbols in the written arts. Not only does the typeface have a pleasing aspect , but also the referrents that the words invoke they are the wondrous  things.

So in translating hermanns writings , piece by piece one is always bordering on the valley of dry bones ! One instinctively hopes that these bones can live beyond the typeface or printers art, and read like a racy novel, or an absorbing adventure or some wondrous tale!

Ones aesthetic tastes are dashed if the baleful and unrelenting referrent is but an endless sequence of parading numbers, numerals even of the Hindu Arabic arts. Gloom descends, the will to proceed fails, rigor mortis sets in and one like mathematics feels as cold as Death!

So here we learn that analogy design, art, simple construction creativity and engaging interaction with real processes in the real world are on offer. We are no longer in the Maul of Mathematics but in the open plains and uplifting hills of the doctrine of thought patterns and cognisances of actual experirnces. The formal script can now be read as a description of dynamic life nd growth, planting, rooting and rising up, self assembling and constructing into a beautiful fractal whole.

I was pissed as hell because Hermann sidestepped the problem with the multiplication, but it was the way he leapt over it that brought me to see that I was stuck in a gestalt bind by the way I was taught to do arithmetic! No explanation, insistence on rote learning, constant urging to move along and not loiter over these "trivialities" !

So I really thought I had come upon a gold mine hen I read book seven of the Stoikeia regarding the Arithmoi. There I found the source of factorisation and multiplication as Division. But I had not read books 5 and 6 the Logos Analogos methods that underpin book 7 . I know Hemann has and what is in these books s artifice, bespoke design, construction, skilled combinings, all dynamically linked to a segmented line by parallel projection, rotational projection and perspective projection. This production of other forms by these projections are the real dynamic and organic source of Multiplication, the multiplication Hermann defines in §9

This "multiplication " or projection also encompasses ith in it " division" , again by projections, but perspective projection is the one which gives us fractions and rational numbers so called. This kind of multiplication is found naturally in the organic growth of a contiguous crystal, in the organic cellular growth and division and rearrangement found in mitosis nd meiosis.

This is the track Hemann is on, not the number/ numeral symbolic tally mark one. Even the highest accountant wants to see the figures realised in some real world object, real estate, means of production and groups of people and organisms working to produce!

[jehovajah]

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§9. We have until here only tracked one  synthetical  knitting artform, for itself and in its everyway holding out ( in space) attribute toward the inter communicant analytical knitting. It now arrives thereupon, to layout the relationship of two differing synthetical knitting artforms. At the End goal the one knitting artform must  be appointed  concording to its label  artform,  through the other knitting artform.

This label concording depends on the artform, how an expression, which both knitting cognisances holds within, without varying of the total output result, can become bespoke designed .

The  simplest artform how "both  in an expression"  knittings could come to the fore is the artform that the result of one knitting becomes "cast onto" the second knitting(, therefore if "^" and "₩" the signs of both knittings are, thusly the everyway holding out in space attribute of both depends on the bespoke designed  knittings, which are permitted with the expression a ^  b ₩ c  to become taken forward. 

Even if the second knitting should  like- measuredly relate  itself onto both limbs  of the first knitting , thusly the second knitting presents  itself as the simplest bespoke designing, that one could   "cast onto" each limb of the first knitting the  second knitting , and then these individual outputs as limbs of the first knittings cognisance one could set. 

If this bespoke designing can  come to be taken forward without varying of the total output  result , that therefore brands:  a ^  b ₩ c = (a ₩ c) ^ (b ₩ c) exists , thusly   the second knitting together, the   inter communicant  "to that  first" knittiting together,   we name "  next higher step/ stage/ rank"

In particular by considering this second knitting: both limbs are depending in like cognisance ,   on the first knitting,  
thusly, therefore, 
that concording  empowers thuslywell for the hind limb of the new everyway connecting,  how for the everyway connecting of the new forelimb, 

and  further the first knitting is a simple one, and its inter communicant analytical  knitting a one- assigned one, 

thusly name we the last one " Multiplication", 

while we for the first knitting already above  the name Addition had firmly set. 

The artform, it is this  mainly : how from forward going entities herein, that brands: even if a knitting together artform is not yet given, such a knitting artform  can come to be appointed together with the " therebesides  to affiliate itself"  higher step/ stage/ rank .

Therehere we track also the addition as the knitting of the first step/ stage/ rank,  therefore the multiplication as the knitting of the second step/ stage/ rank.

From now we choose   the usual signs appointed for this knitting together artform  at the places of   the general knitting signs, and indeed we choose for the multiplication the direct besides one another writing style.

Footnoteg
As a third step/stage/rank one would present the same principle of the Exponentiating,  but what we are over going  here is the short way.The remainder of the label-concording  for these knittings here are only a formal concording , and can come to be everyway bodied, first in the individual expertises through Real definitions lying in the Nature of the Thing.

[jehovajah]

Commentary on§9

Unterwerfen is a direct German transliteration go the Latin subjugate. Thus the design idea is that the first knitting should be su jug ate to the second knitting .  But for this to happen simply the first knitting must be within the system of the second. Hermann had already covered this requirement in the induction, so this was a simple notational reference to that system.

However, in kerping with the knitting metaphor I used a knitting term, casting on, by which knitters join together different sections of knitting making each section subject to the whole design as a part..

The German umgestalten means something like to design around a something or someone . Thus the unspecified design is closely dependent on the subject of the design. In this case the design pacification seems a bit arch, that is it has to end up looking like the distribution law diesn't it?  However the principle of the art is not to choose a specific format, but to choose The format that can be taken ever forward!

Thus perhaps for the first time , I don't know this law was identified as one of the most general and necessary laws which applies to all spaciometric knittings!. But that it is crafted and designed is not disputed here but blatant.  The Grassmanns as a family believed Mathrmatics was a constructed art.

In the previous paragraph Hemann began o fit the generalities to a more specific interpretation. Here he becomes even more specific about how these knittings are going to be distributed in space..

The everyway distributed in space brhaviour of the knittings is particularly highlighted by the analytical knitting, because it is one valued, thus a subtraction in one direction may have no effect in every other direction while decreasing the whole.. Consequently addition increases the whole , but now we know that increase may be one directional only. Analysis has revealed this possibility..

In the footnote Hermann mentions that the principle of exponents is the formall description of the design model. Consequently polynomials represent a many staged system of knittings. Laying these out in space, the everyway holding out in space of the systems gives rise to Hemanns concept of an n-stage or n- dimensional system. At every stage higher than the first the knitting process is a product, but that product by design has the structure of a sum . Thus multiplication is a sum of more primitive multiplications ad infinitum .

Why is the first knitting different? It is different by design. It is designed to be the level at which we start.  In space we will find we are free to choose that level! Today we set that level at the Planck length. However supposedly the Planck length has no preferred direction! As you can see this is crazy talk. The designer sets the system at all times, not the space!

Because of this we can never know space formally. The real experience of space is always beyond our formal designs, but we can still shape our systems around space.

One other point, the design of the product process has 3 simplest formats, parallel projection, perspective projection and circular projection. Of the circular projection we would perhaps interpret the hybrid polar coordinate system as an example ,mbut the simple true circular projection is more like a chain link, and the more circular dimensions it has the further out it can systematically reach. However it cannot reach a point by a direct route . It is a system of arc segments of a circle.

[jehovajah]

Reading ahead, there is only one more general point of significance to make, which comes about from considering division as a fraction or ratio product , that is as a multiplication product as designed, division designed in this way is not one valued, it is multi-valued, and so some supplementary rules have to be applied to the real situation, to choose the appropriate result. 

Now because of this consistent design brief of subjugating addition  and subtraction processes to multiplication ones commutativity or  exchangeability of factors is not a property of multiplication!  Hermann discusses this result in a general way and points to the uniting ability in formal arithmetic as its source. Thus for Hermann the real application of the designed system must dominate over the formal system and attribute to the foundations only what can be demonstrated at every stage in a real system according to the nature of the magnitudes.

[jehovajah]

However, at this stage Hermann considered the commutativity of the first knitting induces that the designed second knitting would be empowered to operate in both directions. What this means precisely he does not specify until later. But at this stage what is good enough for the forelimb of the new knitting process is good enough for the hindlimb.

This induction is based solely on the detailed behaviour of the first knitting, and all the possible
 Exchanges in that situation. There one can interchange the limbs in a pair binding as long as the output result is not changed.

Similarly the limbs of the new binding can be exchanged round if the output result does not change.

Later we will see this written out explicitly, but the constraint on the output is not demonstrated. Consequently Hermann realised that he could not design a process that remained unchanged at the second stage/ rank/ step. The nature of the first step  being limb like and simple constrains the output, and the everyway holding out into space attribute. Going up a stage gives extra directions , in fact manifoldly many , by which the output resultant limbs may be knitted together , and the resultant output is not invariant!

What can be carried forward then is the design of multiplication by the forelimb, and multiplication by the hindlimb . Commutativity, therefore must come from the special or specific nature of the real magnitudes.

Formal numbers are not so constrained and based upon their unreal nature , there inherent imaginary nature, commutativity is defined by the factorisation table for any given number.

The factorisation table arises by a division algorithm, thus it does not come from the multiplication process just designed by Hermann . This Euclidesn division or divisor algorithm which identifies all the whole number factors of a number , and relates indeed to the factoring of a larger object in space by a smaller object in space, became the basis of the taught methods of number bonds and factor bonds confusingly renamed multiplication.

At the same time the amalgamating of the division product with the multiplication knitting process is also confusing. Nevertheless it is consistent with the knitting process that multiplication should increase a product while decreasing it should rightly be called division .

The 2 systems are clearly of use but also should be clearly differentiated.. And commutativity, like there are only 3 dimensions of space should be let go to its special and rare place in the system of things, because it is rare that any production process is commutative.

Again the interplay of 3 limbs even if the foresign of one is completely different to all the other foresigns  reveals much about our design processes under Analysis.

[jehovajah]

Designing the second and third stage knittings represents an interesting challenge for the foundations of The doctrine of thought patterns.

We see here that Hermann elected to use simple subjugation of the first knitting to the second stage knitting. However the format he used concentrated on only 2 limbs , which is the formal knitting thought pattern admittedly, but Hermann first applied it in the context of 3 limbs in which any 2 are combined to output a resultant limb in that context .

Because the forelimb is overtly or explicitly the first knitting format, he has 3 limbs, but of differing step/ stage/ rank. Ideally one would like to see the symmetry a ₩ b ₩ c, identifying with the first development of knitting, played out according to those already set out rules.

This clearly is the associative format and it would be of interest to see if under the given constraints associativity carries forward.

In Addityion, the design Herman chose is in fact a binomial design, because the forelimb is a combination of 2 whimsically chosen limbs . Norman calls these bilinear forms but they could also be called bilimbal ones. In any case it should become apparent that the Binomial expansion will play a fundamental role in the development of the knitting processes of each stage as they trickle down to the first stage.

However again the forelimb could be chosen as a trinomial, in keeping with he initial design, and the rules for knitting together adduced from this situation.

As you can see the increase in complexity of the thought patterning soon becomes overwhelming . However I am sure today's modern symbolic systems could del ith the math, making design a lot simpler.

The impact of all this on this thread is that Hermann is designing a product rule that is mixed in order to capture the tspatial Verhältniss of the quaternions, that is the everyway holding out in space attribute of these product arrangements called quaternions. It may be that his bilimbal Analysis of products is leading to or missing a more efficient product based on these design musings.

In any case the one important concept that came to Hamilton in a flash on broom bridge was that a fourth axis was required to do the calculation evaluations with! This fourth axis was ignored simply because mathematicians were not considering the ghost in the machine, their own conscious contribution to calculation. They were assuming the bus drove itself, all the while pushing it along by their own calculation effort. From the outset they attempted to keep the human involvement out of it, foolishly because it is clear but ignored that every calculation proceeds by some agency. In there time it was the human conscious, in our time we can call upon the combination of the electronic programmed conscious, and the programmed operating system, each an electronic model of human lcalculating behaviours.

The fourth axis in quaternions represents the measuring tool or carpenters measure, by which the carpenter accounts for everything else in the proportioned way of the Pythagorean school of thought. Without it we as calculators are at a loss as to how to proceed. Ith it we are trained to run through proportioning with some alacrity, but no where as fast as a modern ompuing system.

The point is no product can ever be functional if the calculator has no clue how o proceed with it. To proceed ith a calculation one has to lay it out in space either symbolically or as actual arrangment transformations of magnitudes.

For example, a fixed plane in space hardly exists, but without the notion we would be unable to lay out calculations to determine height , distances reas etc., all of which are purely formal but also relate able to spatial objects and aspects .

[jehovajah]

When designing a product one needs to keep in mind the tessalating Fractl it produces in space .

Hermanns main design product was the retinal parallelogram , the parallelepiped , the crystall facet structure of higher dimensions / stage / rank products.

Theolar coordinate product produces Shunyasutras, that is curved annulus portions , spherical shell portions

We have seen cylindrical, conical and toroidal as well as tetrahedral product designs as fractals on the forum and in the gallery. All of these product designs are differing in complexity of expression, especially when to display the result they have to be evaluated by the standard Cartesian product , but essentially we have a general solution in the form of the exponential product.

However, the simplest design pattern Hermann chose was one in which the general product of a stage was the sum of other products from the same stage. Thus a polynomial  product

AC is the same as AB + AD , where C is B + D and is a knitting in the stage that C is at  now A can not  be at the same stage as C but it must be a differing system which subjugated C. We may be used to regarding axes as independent , that means not influenced by anything else, but in fact the whole concept of constructing systems within systems, especially as it applies to products links them together in a causal dependency .

In fact we have to realise that this is a cognisance, a way of apprehending 2 independent rules as cooperating in expressing certain forms. The cooperation or dependent - like relationship we call a function or a polynomial , a rule expressing the causl link we perceive .

Once again the cusal link is emergent in our conscious perception, not in the product. Design we are placing at the foundation of our systems. Thus it is important that the product rules we design are the simplest possible ones for the fractal division of space.


Unfortunately that design constraint only gives us a simplistic view of the attributes of space. We can design more complex systems but we need computers and motion capture and fast Fourier transforms and laPlace transforms to even approach expressing these products.

Apart from the aesthetic beauty potential in these more complex products , why go there? Astrologers and presumably Mathematicians have no real interest in such complexity, but Natural philosophers do , in order to express the behaviour of chemical and organic growth nd interaction patterns , and to model statistical and stochastic processes, especially where no causal function expression seems to exist..

Again why do we need to go there when we can video the behaviour at high speed? We seem to need to express proportional relationships in order to feel that we understand or rather everyway stand a behaviour, but real expertise is ofte based on experiences where the proportions are felt not symbolised. Nevertheless some do encode this expertise symbolically and for those these proportions make and give sense.

We need both types of expertise, clearly .