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Commentary

Even though I have not completed transacting §4 I can now see where Hermann is going with all this bracketing not changing the advantageous result.
Firstly the word Ergebeniss itself :" out giving" is a simple enough activity but what does it signify? Eventually I have come to recognise it as the "output result". Although the action is different the similarity of giving or putting something out there wherever that may be is essentially the notion.

Thus the output is what is actually completed and knitted together.

If we consider a machine process the in put is what is placed into the process. The process then "gives out" the finished result. Thus we have the "in putting" of the ingredients and this is followed by the " out giving" of the result.

The process of inputting brackets does not change the out putted / out given result of the knitting.

Section 4 is about how the brackets change the ordering of the knitting Process but not the result of the process, the out giving/ out put result.

I glimpsed this briefly while working through Hamiltons step derivations., but could not express it clearly. The output result is unaffected by the bracketing of the elements . The bracketing of the elements are whimsical, thus for a firmly fixed sequencing( this is the Fundamental ordering) the whimsical bracketing reorients commutative and associative  arrangements of the elemental processing.

Associativity and commutativity require a fundamentally fixed ground on top of which an order of combination is formally and externally imposed without varying the order of elemental sequencing. The real elemenys remain unchanged the formal combinations of them may vary without affecting the output/ out giving result..

Now onto this we can construct a definition of Tally marks, and notation for accounting praxis and sequence of notational symbols called Tallies.

The general Doctrine of the Thought Pattern (Ausdehnungslehre 1844)

§4 were( onto the other side for a "knitting together") only the everyway  toutable quality ( with a view to exchange) of the both limbs firmly set, thusly would there be drawn thereout no other following result able to come to be.

This appointed meaning yet coming  from afar to  the "in the previous § completed" meaning , thusly follows: that also by multiple limbed expressions the ordering of the limbs for the total  output result is equally valid, in which one can specifically easily show , that every  two, themselves following on top one other, are allowing  limb everyway touting ( for exchange).

In practice, according to the immediate last outwardly demonstrated  proposition (§3) two such limbs,( to which entity one  will concording demonstration provide of everyway toutable quality ) ,  can one enclose in brackets without varying the total output result, further  these limbs under themselves are everyway touting for exchange .

Without varying the  output results which out of them has been represented "knitting together" ( how we do plainly set out ahead of everything else), therefore, without the  output result of the whole "knitting together" varying, there one places every thought pattern which to its like entity  can be set

and at last now can the brackets, again so set become, how they were at the beginning.

Thuswith is the everyway toutable quality( for exchange) of two one other following limbs outwardly demonstrated. (QED)

Now therefore one may bring therethrough setting forth of this process  of every limb onto every whimsical position ; thusly is the ordering of the limbs mainly equally valid.

Therefore this Result firmly fixed together with the result of the previous paragraph

We reify such a" knitting together"  making a way by the Short route, for which the output  beside us Appointed meanings empower,
we reify naming it  a Simple entity .

Now a still further-going appointed meaning is  no more possible for the artform of the knitting, even  if one does not return onto the Nature of the knitted together thought patterns, and we therehere the achieved knitting  pace to the "ontop  loosing" , or  to the analytical process.

In hermanns treatment you can't have commutativity without associativity, but you can have associativity without commutativity.

There is one more output result from the far field.

The significance is in the fact that Hermann starts his thought patterns doctrine with the relationships between 3 points and thus 3 Strecken or line segments. From this he notices the law of 2 Strecken in the plane

AB + BC = AC

One of the fundamental "equations" of thought patterns.

This I have called the law of 3 points, the law of 2 Strecken; and the product law or the law of 3 Strecken or the parallel product I have distinguished with those labels. Of course one can reduce them to line segments connecting 3 points in space!

Notice that Last flippant remark!

Suddenly I went from the plane to 3d ( or rather n-d!) space!.

Because our knowledge of Astrology is screwed up, and by now quite paltry few will have heard of much past Pythagoras and trigonometry. But I was introduced to Herons formula as a way of finding areas of triangles. I had no clue of its significance or its history, nor indeed the significance of the Pythagorean school of thought. Without knowing it my educational background was modelled on the Aristotelian school of platonic thought, not the Euclidean! And yet Euclids name was bandied about like a religious Icon.

Wallis accused Descartes of Huge Plaguarism, taking the work of Harriot as his own! Harriot was dead and not likely to object, so why not? Descartes had to survive in Europe and that meant patronage. He was never going to work as a serf, especially with his habit of staying in bed thinking most mornings!

Thought patterns, that was his Fortè and setting up a system of algebraic or symbolic geometry was his joy.. The truth about his coordinate system, as usual is a little twisted.

I read La Geometrie briefly expecting to find the x and y axes. Instead I found reference to Ptolemaic and circle theorems based on Euclids Stoikeia apparently, but in fact drawn from the mechanical philosophical texts of Omar Khayyam and other Islamic scholars who favoured Aristotle.

Thales and Apollonius were also studied as part of the advanced curriculum in the Arabic Academies of Europe( Spain)  and Baghdad . Undoubtedly copies of the Stoikeia by the Neopythagoreans came into the possession of the Islamic scholars, but by that time they were all fully enamoured of Aristotle. Thus came centuries of wrongly interpreting the Stoikeia as a course in Geometry for " babes" who first want to learn or need to learn the elements of mechanics!

Later wallis came upon a Greek version of the Stoikeia among some papers returned by English Merchants, and Isaac Barrow returned from a Harrowing trip to the far east and Europe with papers and knowledge by studying in the great libraries and seminaries around the Arabic empire. His principal interest was in Apollonius and his Conics, of which he could only "obtain" an incomplete version.

Nevertheless for his efforts and peril he was given a chair of geometry in Cambrudgeshire ( Gresham College) to work on English translations . Somehow Wallis got hold of the documents pertaining to Euclid in Greek and began a lifelong translation of those at his seats and positions in Oxford. It is possible that Bartow sold him the material for economic reasons, and because he took on the additional burden of the seat of Mathrmatics, newly established at Cambridge university which offered a higher Stipend but more duties!

Both Wallis and directly Bartow were responsible for directing Sir Isaac Newtons studies, and Newton, having been baffled by Astrology he found in a penny pamphlet in the market fair went on to learn from barrow the fundamentals of Euclidean geometry and the fundamentals of Apollonian Conics.

Of course Wallis's books on the Stoikeia were directed reading for Newton, who surpassed both his course tutors and the authors of books he was directed to study.

So it was that Mathematics came to hold both geometrical reasonings( thought patterns) and formal Aristotelian thought patterns, Euclid versus Aristotle, in a seemingly good mix.

At least that was the case in England and the British empire. In France the mechanical philosophy was compared against documents that survived the purge of the rabid clerics who during the plague and the crusades demonised all Arabic learning, documents that were preserved in the Roman catholic libraries, and inconsistencies were found.

Consequently , since Aristotle was right, Euclid had to be wrong, and the Stoikeia was rewritten several times to make it fit this view. The most famous of these reworkings was that of LeGendre a French mechanical Engineer. His text book on the Geometry of Euclid contains so many mistaken views that it is no wonder mathmatics took a wrong and embarrassing turn!

The Fifth postulate problem arose among the Islamic scholars who in studying the Stoikeia assumed that every proposition was derived or derivable from preceding ones.. This was a very Aristotelian syllogism, based on his taxonomic predilrctions( OCD of the highest kind, probably Ausbergers Syndrome), and no one, owing from the Tekne or mechanical philosophies recognised the Stoikeia as a course in Philosophy, Pythagorean philosophy as opposed to Aristotelian Academics and Platonic style learnings.

By the time LeGendre got his hands on it the Kantian notion of Axioms was just being formulated, and the Axioms of Euclid now took on a new meaning. No longer were they just axles or axes on which the wheel of learning turned, they became self evident truths. Self evident only if you were an artisan!

Because classicl scholars could not bring themselves to sully pure, holy reason with the material mundanities of mechanics, what is self evident to a mechanic is not so to a classical scholar. What is supposedly self evident to a scholar is logic, reason, spirit. By this means mythology , opinion, convention all brcome the starting point for classical scholars and Mechanics only in so far as some mythological hero employed such artisanship!

Needless to say mathematics was in a mess by this time, and Gauss in particular was looking for a way out of it. He thought Lobochevsky might have hit upon it, completely undermining Bolyai's work in favour of his own. He certainly was not going to let Bolyai take the credit of saving Mathematics!

Later a curious manuscript came into his hands, while he was busy surveying. It was Grassmanns document under the cover of a letter by Möbius. Perusing it quickly I am sure he found it ver confusing as did Möbius, but at the same time intriguing. Because of the hypnotic quality of the writing I am sure Hermanns book set off a train of thought in Gauss mind.

Gauss fairly critiqued the style and presentation of what is an was an incomplete imperfect masterpiece. It failed on the most basi Acadrmic standard of clarity of Aristotelian logic! It failed because indeed it was not Aristotelian, it was Hegelian!

What does this have to do with Herons formula and Pythagorean philosophy? Everything. Aristotle only lead human thought down one path, while Pythagorean scholars explored the n-dimensional space in which we live. In book 2 the segmented line is introduce as a fundamental of proportion. From it the Pythagoreans set out their understanding of all proportions in all dimensions. The link was the circle or rotation of the elements of the proportions. To enter into space from the segmented line you must have at least 3 points.

If those points remain interconnected by collinearity as the line segments rotate then you have or define what is a plane! Alternatively, if a circle passes through all three points while constrained to be firmly fixed at a displace et from a point in that plane, its centre , then the circle is entirely in that plane..

These are very hard constraints to achieve mechanically in thought, but in nature rigid axial rotations and rigid cutting tools are our best approximations to all of these. The Lathe delivers all these surfaces depending on rigidity. This is " axiomatic" that based on the axial or axis , to any skilled Artisan.

Hermann thus started his method where the earlier aPythagorean rtisans said one should, on the complex combination of the circle, 3 points and 3 straight line segments, on an axially lathed surface, called a plane.

<a href="http://www.youtube.com/v/8rjxOFAzBa4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/8rjxOFAzBa4&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/8rjxOFAzBa4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/8rjxOFAzBa4&rel=1&fs=1&hd=1</a>

What is directly relevant here is that the combinations of the 3 limbs are kept firmly fixed. Thus the everyway ordering of the limbs , according to the proposition should not change the out put result. The result for the straight line should be the same for the result for any and every triangle including Pythagoras theorem, Thales theorems etc. and this should be the same for every plane in das Raum!

Oh yes Descartes started off algebraic geometry by fixing at least 2 line segment directions in what later became called generalised coordinate style and firmly associated to LaGrange!. It was Wallis who fixed the cross axes we now so easily call Cartesian.

And Gauss backing Lobachevsky and Riemann over Grassmann and Möbius eventually proved not as fruitful as he hoped.

It will become apparent, but not commented upon, that the "=" sign has made an appearance in the discussion. The "=" sign is fundamental to the labelling scheme and appears in the context of Like entities.

To say the sign is not commented upon is unfair, but pertinent. Since the Induction Hermann has been setting up the prior intuitions for Like things culminating in his 4 statements or propositions of "like" declaration. Thus Hermann has been commenting on the "=" sign all along .

It is pertinent, because the trained mathematician is too easily drawn into a narrow cootation of that sign. So narrow in fact that other signs have to be employed to denote all that Hermann means to expound as Like entities.

Therefore, as soon becomes obvious the "=" sign is not the same as the arithmetical or even mathematical counterpart. Herman uses it to mean like entities. We then have to pay attention to the rhetoric to determine which of the 4 likes he is employing and why.

In particular many " algebraic" manipulations are not the same in Hermans method . Many long strings of like things may indeed appear to be algebraic manipulations, but in fact are not. They are chains of like things governed by some simple constraint.

What then are so called algebraic manipulations?

They are a mnamonic chain of permutations of a sequence of entities. Mnemonic because they are a memory tool of a much deeper analysis, which Hermann gives, and indeed Hamilton does the same in his theory of Couples. However Hamilton derives his via a model of displacements in Time. One is amused by how clever his model is but not lead to think this is anything more thn an intellectual game.

Hermann on the other hand derives his explanations from a philosophical exposing of the functioning of the human mind. Thus, as we shall see, the effect is quite different because now we must examine ourselves and how we think and form logical and arithmetical conclusions!

Now exposed is how we cover and uncover certain thought patterns to output a conclusion or result. Now is clearly set out how we ignore certain patterns, promote parts of patterns, rely on prior propositions and insights, bend or extend notations , overlook certain details while focussing on others in order to present an output result

And to what end?

For Grassmann the grand visions of LaGrange, Euler and LaPlace speak of a higher " truth" a clearer and more able Mechanics, a powerful and more flexible dynamics that could display kinematics and Ballistics and Astrological phenomena so much more accurately.

Truly he felt what he was doing, and encouraging the Prusdian people's to excel in was nothing more than the Capstone method of all mathematics, which would reveal to us sublime truths never before grasped about the very Nature of Nature itself!

Bearing that in mind you may grasp why he was so animated to look again at the foundations of  mathematics, because he saw that he could rescue it from its profound difficulties, into which it had fallen, and liberate the minds of men as the French revolution had " liberated" the French people , starting with the Prussian intelligentsia abd the rest of the world to follow.

This was highly renaissance and Romantic thinking, and it was extant throughout a wide European demographic. It empowered a whole movement to reform the institutions of Europe, and Hermmann's works and writings reflect that background .

I have mentioned before the wide ranging impact of Hermanns seminal work, and the redaction by his Brother Robert. In context hermanns work is a masterly summary of the status of advanced mathematical thinking in Europe during his time. But even more it provided an entrance into a higher state of learning for a poorly equipped Prussian Scientific community. By it he hoped to give Prusdian science and Technology an edge to compete with the other industrialising nations.,

But , as the saying goes, A prophet is not recognised in his own countr! His work in fact inspired innovations in Italy, Britain and America sooner than it did in his own struggling country.

As far as Prusdian philosophy is concerned it is a short work, and it is incomplete, but it is very accessible to philosophers. Mathematicians on the other hand recoiled at its philosophical style, even as they recoil today at Newtons Astrological principles, and it is from these negative minds that the myth arose concerning the obscure nature of the work. It did not make it easier using the renowned Philodophy of Hegel, because the establishment in Prussia was very negative toward Hegel . But the work itself is not obscure, rather it is groundbreaking and troubling.

It forces the reader to think very rigorously and in a definite pattern!

The general Doctrine of the Thought Patterns (Ausdehnungslehre 1844)

§5  The analytical everyway journeying retains therin: that one seeks the other entity  to:  the output result of the Knitting together and the one limbed entity of the same (knitting) .
Therehere there are attentive to a knitting together two analytical everyway journeying artforms, each is becoming sought according to the specifically from the knitting forelimb or hindlimb ;

And both everyway journeying artfoms  deliver then only one equal/ like output result, even if both limbs of the originating knitting are everyway toutable(for exchange) . There also  the analytical everyway journeying can become apprehended as knitting together, so we differentiate the originating  or Synthetical knitting together and the "on top " loosing or analytical knitting.

In what follows we now become immediately nearby to the synthetical knitting in the sense of the previous paragraphs as one simple knitting , in order  to set out ahead ( of everything else) and as sign for the same the sign"^ " is kept held in place,
 the far against view: for the inter communicant knitting together the around turned sign "v "  is chosen,  there here both artforms fall together from the same knitting.

and indeed thusly , that we construct here,
By considering the analytical knitting
  the output result of the synthetical knitting relative to the forelimb
-What is given( relative)to the forelimb  ?

Thusly concording to here  a v b this that thought pattern signifies, which  with b synthetically knitted gives ( the result) a, so that at all times a v b ^ b=a  therefore exists.

Herein lies "thus"- like locked in, that a v b v c denotes the indicated thought pattern  , which with c and then with c  synthetically knitted gives a as a result, that is branding therefore also according to §4 the indicated thought pattern  which with the same "Values" in the around turned succession, or with b ^c synthetically knitted gives a as a result
That brands

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

And there the same concluding succession for whimsically many  limbs is empowering, thusly follows, that also the ordering  arrangement of the limbs, which have analytical foresigns, like valid is, and one is permitted to close these limbs in a bracket,only if one the in bracket toward the back foresign turns around

Now Hereout follows further, that let it be

             a v (b v c) =(is like) a v b ^ c
In practice one has out of the definition of analytical knitting together

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

This expression is once again everyway pleasing of the so plainly outwardly demonstrating rule
      = (is like)   a v ((b v c)^c)^c
        =(is like)   a v (b v c ^c) ^c

And this last one at last is everyway pleasing of the definition of analytical knitting together

         (is like)= a v b ^ c,

Therefore also the first expression is like to the last.

We express this result in words and we combine it with the the beforehere achieved results, thus we hold out the proposition:


This is the most general results, to which by considering the taken asside settings out ahead of everything else we can reach.

Afar against that, out of the same things does not go henceforward, that one can let go away a bracket, which an analytical sign enclosed, and a synthetical entity before itself has. Much more a new setting out ahead( of everything else) must first become constructed thereto .

It is clear that while matrices were not conceived clearly by Hermann, the determinant was. That is not to say Hermann invented it. Cauchy and Gauss it seems were hot on its trail through attempting to understand when systems of equations were solveable., but even they were following up on clues found by Chinese Mathematicians and Indian Astrologers,

The rank array that LaGrange and Euler and LaPlace made central to mechanics is Hermanns grand framework into which he hopes to transform all thought patterns.

Thus his presuppositional work is about dealing with n-limbed knittings or combinations, where the combinations are permutable. This is much more complex and richer than we can express, but fortunately it is fractal. What Hermann had uncovered is the Fractal power of 3 . As he frequently states if it holds for 3 then it holds for as many limbs as you like!

Thus he did not have to really work with n limbs, like a computer has to, he only had to work with 3, the rest conformed by the same rule.

<a href="http://www.youtube.com/v/67pMZbTQxP8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/67pMZbTQxP8&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/67pMZbTQxP8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/67pMZbTQxP8&rel=1&fs=1&hd=1</a>

The second part

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

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

Normans introduction to the determinant is very clear , and I particularly like the det() lable that inputs a matrux( rank array) and outputs a number!

Commentary

I have referred to Normans work, not only because he respects and innovates on Hermanns ideas, but because his videos relate to some aspect of the translation of Hermanns work.

Here however, the algebraic manipulation Norman refers to is purely relating to the BODFAS rules. But in the far view is the mysterious Book 2 of the Stoikeia.

It is glibly passed off as Euclidean Algebra, but what it is really is proportioning a line and then applying that proportion to quadratic or 2 dimensional figure.

Here Norman shows how simple staged or successive application of the fundmental line segments can relate shapes and areas together.
http://youtu.be/_nAgDNz6ETQ
<a href="http://www.youtube.com/v/_nAgDNz6ETQ&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/_nAgDNz6ETQ&rel=1&fs=1&hd=1</a>

Many lines were segmented and explored and found to relate areas like the golden ratio, or or some other pattern of areas. The Pythagoreans reveled in these affine summations, and many mythological stories contain references to divine proportions marked off on a line.