Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

Hermann is going to go on to describe the role of Analogy( analogous thinking) Ancestry, and Criticism in developing an art oskillset of wider and further developments of each development. This is in short keys to developing creativity or creative thought patterns or skills in the doctrine of thought patterns. This is also called Heuristics.

For Mathematicians I can recommend Polyas how to solve it books, as accessible introductions to the field of Heuristics( which from the Greek peripatetic teachers means hero worship, or copying your hers principles of action).

I have not kept up to date with the field since I was a student, particularly since the advanced book I bought after the Polya was off putting! It presented the topic in a set of series of state variables, and thus long abstract summands. Had I known then precisely what I know now I would have recognised this as an attempt to carry ot Hermanns rigorous method using rank arrays, almost verbatim!

The next time I glanced over such seres was in a very brief survey of Turins state logic analyses for his Turin machine, so I have no doubt that over the past 40 years some headway has been made in heuristic algorithms based on hermanns insights here.

Machine learning probably owes its development to hermanns rigorous analysis and insistence on a summary sketch overviewe of the constraints in the field of study.

As far as I know this was how Lagrange and Euler operated so successfully and innovatively for so long .

[jehovajah]

This is a nice bedtime story for the kids! Lol!

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

[jehovajah]

This video is showing the use of LaGrangian principles in modern Physics.  As students of LaGrange, both Hermann and Hamilton incorporate his methodology into their thought patterns. As a result Hamilton went on to recast physics in a new systematic way, a kind of matrix of Status markers , or constraint makers. This came to be called the Hamiltonian. The idea was to get the Hamiltonian into a minimum state.

Hermann did not develop into that specificity. He rather delineated the most general principles and the most general way to set up a system.

Using his approach he went on to develop the whole lineal Algebra of n-stages or n-rank algebra. Hamilton was very impressed, and rushed to catch up. In his work " The Elements" he makes the throw away remark that his discussion, of course can be generalised to n-tuples! However the readers may now realise how hard that would prove to be, if it were not for Hermanns groundbreaking lineal Algebra.

I think Hamilton would have been able to achieve it, eventually, but as it was he had Hermanns work as a guide and a common expertise and quick sketch overview generated by LaGrange and LaPlace. However he died before fully completing his work, and the tide had turned against his quaternions, so few, except his students bothered to carry on his work in his style.

Instead Grassmanns style was picked up as an " orphan" idea, few even bothering to check if he was still alive! St Vainant found out that he was very much alive! Peano on the other hand translating Hermanns ideas into Italian was able to develop and implement them in his Italian school in a way that lead to Levi and Ricci developing Tensor analysis..

Gibbs took Grassmanns and Hamiltons work and presented it as his own development. It is known that he first presented it as a set of cribb notes on useful ideas and methods for physicists. As the popularity of these notes grew Gibbs was developing into an acolyte of Kelvin, and his Fourier analytical approach. This set the ground for a statistical mechanics which seemed the way forward to deal with the increasing systems complexity. Kelvin and Maxwell had pioneered this statistical and probabilistic approach of Gauss and Boltzmann.

Gibbs had an incentive to develop his notes , but he could not understand Grassmann and he secretly loathed Hamiltons quaternions, and he found that most of the established mathematicians also hated the doctrine of the imaginaries. Kelvin in particular was very scathing, so Gibbs had to include some nonsense to cover over Hermanns use of the imaginaries.

Gibbs presented a version which came to be called vector analysis. It was incomplete and underwent further development by many including Heaviside and Dirac.

However it was the usefulness of the statistical and later probability approach that returned favour to Hamilton. And his mechanics.

The mixture of Grassmann and Hamilton occurred in this fashion but underlying this was the short lived work of Bill Clifford. And Bill Clifford was a died in the wool Grassmanian

Is the Grassmann method the way forward?

<a href="http://www.youtube.com/v/hCSLqS0jKpk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/hCSLqS0jKpk&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/hCSLqS0jKpk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/hCSLqS0jKpk&rel=1&fs=1&hd=1</a>

[jehovajah]

Commentary

The systems within systems can be developed in 2 ways: the straight line segment way and the curved arc segment way. They may seem different but they are not.

The straight line development hides the inherent rotations within the systems. It is thus a diminished descriptor of our real experience. How we came to elevate it to a divine position akin to " the way things are" beats me.

The curved arc segment development is a much richer , more complex model of our experience. We are forced to set in place constraints. This is the LaGrangian approach to describing a system or modelling behaviours. As a consequence one develops a fuller richer realisation of a mathematical model in which explicitly the constraints set are apparent and all assumptions are explicitly laid out.

The issue then becomes about defining the system strictly enough to be able to identify measurable parameters . These then become the subject of empirical data collection.

These 2 systematic ways of describing a system that is changing caught the imagination of the brightest minds, but ultimately it was dumbed down .. Here however Hermann does not allow this, and ultimately this is why quantum and Classical mechsnics differ. However they describe the same experience from different perspectives..

That statistical and probabilistic mechanics naturally attaches to the LaGrangian point of view becomes clear. Probability is a development based on the unit circle. And statistics requires a bell shaped curve to speak generally about huge quantities. Both require an apprehension of curved variation not merely summative direct variations.

Thus Bohr and Einstein created a false dichotomy as the mathematics used was really the only difference, not the empirical behaviours. But the blinkers were on and have remained on until today!

Let us begin to see clearly, as Grassmann wished and hoped for.

[jehovajah]

Commentary
The LaGrange methodology reworked Grassmann style

<a href="http://www.youtube.com/v/zhk9xLjrmi4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/zhk9xLjrmi4&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/zhk9xLjrmi4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/zhk9xLjrmi4&rel=1&fs=1&hd=1</a>

http://youtu.be/-QVENB3aEvY

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

This was one of the key methodologies Hermann worked through and reworked according to his way of out ordering . Today it is still recogniseable as Lagrange's methodology, but the notationlly style has moved on through linear Algebra. It probably has a Clifford Algebra notation now, but the spark for all of this presentational style of the substance was Hermann and Hamiltons reworking of LaGranges method. Lineal Algebra rigorous and synoptically overviewed briefly , and Quaternions dissected the details presentationally so the reader and student could grasp geometrically what was being done methodically.

You still have to drop down into the detail, and calculate the elements after you have worked out how to measure them, but essentially it is all mapped out before you, and you can sure footedly progress toward a calculated solution.

[jehovajah]

Ausdehnungslehre 1844
Induction

15.  On the entity (every time (reified) points of The development) is the art form of the wider development essentially evoked through a leading idea,  which either is not the variant entity, as an every way meditated Analogy to related and already known branches of the wisdoms, or which is, and this is the best case, a  direct ancestral linking to the" first available  "for" seeking" Truth 


 The analogy is only a necessary aid/ catalyst, there it comes in from afar to play in related subject fields ; even if it plainly does not thereupon come to sit besides,  henceforward,  through this way to heave up the relationship to a related branch, and thus to pull up a forward running analogy with this branch .


The ancestry linking seems to be strange to the study field of pure expertise and to the mostly everywhere entity, the mathematical one.  It is impossible, Alone without it,  a "random any" new truth to discover;

 through blind combination one does not arrive at the achieved results thereto , rather what on has to combine and onto which manner must be evoked by the leading idea, and this idea  further around  before it  has  everyway effected/ manifested itself   through the expertise itself  , can only in the thought pattern of the ancestral relating  appear. Therehere it is some glorifying thing this ancestry linking on the  expertise-like study field. It is specifically, even if it is from  right art form, the "in one entity together showing whole developing" rank array, which the art form drives to the new truth , but with still not "out of one the other" embedded moments of development, and therehere also, in starting, only first as dark presentiment, the "out of one the other embedding "of that moment Inheres at the same instant, the discovering of the truth and the critique of that driven everyway entity .


Footnote on page xxxii
This case steps in by considering the "here to process "expertise in active relating onto the Geometry, by the viewpoint of whose "half"  I nave mostly brought into relationship the way of analogy  before you

[jehovajah]

Commentary

Ancestral linking is a rather specific rendering. One might make it wooly and in distinct, but I feel that the direct concept clarifies the otherwise befuddling statements about a really indispensable tool for experts and mathematicians.

The use of the array concept serves a similar purpose. While it is arguable that Hermann did not think of matruces when he formulated these guidelines, it is clear that LaPlace, LaGrange and Euler did think in terms of arrays of terms organised across the page. Some were series some were tables for constructing series etc . The word used was probably systems .  It is the systematic nature of these methods that drills through, no matter how long or complex the sequence or array.

The use of Analogy is very important, but do not use the obvious well used one, it won't get you to new developments . Use ones that come out of the Farfield, or the left field. But the best way forward is if you have a leading idea that calls to ancestral relations! Such an idea admittedly is dark and blind, more of a feeling that there is a progenitor or forebear out there somewhere. Using ones expertise and this idea And the array , a snapshot of everything in its place and under its constraints, the expert can push the system back towards its origins, to at the sat suddenly discover its progenitors , new concepts of a more general applicability..

It should be clear that Hermann is very advanced in his thinking at this stage, even equal to Euler , LaGrange and LaPlace, who he was studying very closely. But he had the edge which was his Hegelian analytical and synthetical model and his notation of line segments to represent and label both intensive and extensive magnitudes. Because of this label scheme he could not only follow the leading mathematicians and engineers and Astrologers of his day, he could also out think them!

[jehovajah]

http://scienceworld.wolfram.com/biography/Laplace.html

The unified physical view is the notion that Hermann calls the Überblick, or the Übersicht. This unified view which both LaGrange and LaPace subscribed to, was expressible in terms of an array of parameters and constraints with degrees of freedom systematising what could be expected from the whole System.

The transform idea is consistent with this approach and powerfully coordinates relationships between any 2 formats, and in particular between the lineal and the curvilinear, which at the time was thought to be only approachable via the Calculus.

Hermann had a different view, as did Hamilton, and the use of quaternions and line segments considerably simplified the concepts, which otherwise remained buried in differential equations.

Helmholtz was reported to opine that differential equations were the only safe way to describe the laws of nature. Many of his time were thus convinced that mathematics was the only way to understnd the natural laws, and the higher the mathematics the more profound the truths!

My opinion is, for hat it is worth, this is patent nonsense.

In researching LaPlace I was reminded that we cannot solve the n body problem for gravitational systems, and yet we can send rockets to mars and beyond. The contradictory statement is worthy of the Hegelian dialectical treatment!

[jehovajah]

I have completed a brief survey of LaGrange and engineer, LaPlace an Astronomer and Euler a polymath, to see what contribution they made to hermanns understanding of formal expertises. It seems clear that LaGrange made the biggest impact on him while LaPlace and Euler taught and refined the differential and integral calculus .

<a href="http://www.youtube.com/v/kV3oj207pUk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/kV3oj207pUk&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/kV3oj207pUk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/kV3oj207pUk&rel=1&fs=1&hd=1</a>

The sophistication of the calculus employed by LaPlace and Euler opened up the application of formal systems to physical empirical behaviours. In particular tools in calculus became apparent and eventually identified as grad,div, curl. The integral calculus helped to reveal the connections between recognised formulae of Newton and differential forms that seem to describe physical mechanics.

Here Euler shows that a differential equation form for a fluid point in a pressure field is still a formula of proportions captured by Newtons insight.

This was important, because LaGrange with his multivatiable approach had determined a potential differential form which he felt had to be the principle of mechanics! This was his one consistent view of mechanics and engineering, the result of his deep thinking about mechanical systems and engineering. Euler corresponded with LaGrange on this point over a number of years, and his opinion was that Newtons principles were necessary and sufficient for the whole of Mechanics  . Eventually LaGrange changed his opinion . In other words differential equations do not form the necessary and sufficient formulation to set up formal systems for mechanics! Newtons direct and simple proportional measures and ratios is the recognition that as humans we can only measure and observe directly not by differential equations or any calculus whatever. Direct experience of intensive and extensive magnitude is the only place where we can begin.

The French drive to take Newton and Leibniz calculus to its fullest extent resulted in some great work and formula building, but it did not advance basic and fundamntal research and innovation. Advances come by intuitive feelings and realisations of possible ways forward, or deeper apprehension of guiding principles, or striking analogies that break up the consensus view to turn up a new relationship between old developments and new ones.

At this level it is probably hard to think that Hermans work could be important compared to these 3, but we know how important his ideas actually were, since just about everything is now notated in vector form. In addition, much of the great 3's work would be inaccessible to the majority of those who now access it, and that is a painfully small number of people in the world.


Why have so many found Hetmanns work so powerful? It is not because it is so general, but because it uses human scale thought patterns and intuitive measures . He starts at the most mysterious and mystical and brings it right down to the pragmatic and systematic and human.

[jehovajah]

The Lagrangian principle of the minimum, or Hamiltons principle of least action.

http://youtu.be/08vJyA-XD3Q
http://www.youtube.com/watch?v=08vJyA-XD3Q

These mathematical principles appear by observer thought pattern, not really as rules or laws of nature. Hermann really finds this kind of none rigorous thinking very worrying. It is sophistry like most mathematical conservation laws. This was however what LaGrange for along time felt was a fundamental principle of Nature as did Hamilton. Euler was not convinced that Mathmagicins could deserve such a promotion to universal principle. It did not rely on any empirical behaviour, rather it relies on the mathematician fooling himself and others.

[jehovajah]

Just thought of a few new symbols for operators:-

<+,+>,<*,*>

They mean add in the direction of <,> so the result is not a heap agglomerate but rather an ordered string.

Similarly the multiplication means set out the sub products as sequences added according to <,>. So 3*>3 means 3+>3+>3.

Of course I could stack the 3's and I would use a *^ symbol  or some latex generated equivalent.

Does this help or just give us more to do in being rigorous ?

It certainly makes clear that 4*>3 is not the same as 3*>4, but it lacks the extensive magnitude data which is the really useful part of hermanns method.

Do the factors need to be distinguished in this way if they are just tally marks or counters/ scalars?

What really lies behind this notion, (ancestral linking alert!) is the orientation and translation of extensive magnitudes, and that for me is more valuable than tally mark operators.

They may serve as a bridge to the underlying extensive and intensive magnitudes.

[jehovajah]

The notion of a product in the geometry of real space is intimately connected with projection. We might call these projections constructions but I prefer to call them parallel and circular and scaling projections.

Viewed in this way the sequence of projection and other constraints become important.

The notion of a system in a system means one system is " coupled" to the other by design. Thus one projection is always followed by the other and this is ensured by the continuous/ contiguous constraint.

If the projections were not constrained to be continuous/ contiguous then in fact the projections could occur without sequence or order. Hermann carefully avoided this in §10 when he specified rule governed extensions.

Discrete extensions and the combinations do not produce geometrical products in the same way . A lot more rules and constraints are required. The potential for " chaotic" forms is greater in the discrete section.

System combination has to be individually specified and so does any potential product. Without coupling of the projection process random unpredictable forms are possible.

[jehovajah]

The notion of coupling and grounding the systems distinguishes Hermann and engineers from pure mathematicians today.

Despite the general language Hermann has one very clear goal imposed by Lagrange, the systems must conform to real engineering systems ! Thus Hermann starts with 3 points in real pace. These 3 points specify a plane in real space( really they specify a formal plane which we embed in real space) within that formal plane 3 line segments are constructible by those points , and any continuous or contiguous pair can define the rule governed elemental directions. Once chosen, they set the fundamental reference of the attributed system, and they deliver a coupled system of one element inside another element.

Contrast that with an axiomatic thought pattern. A point is specified real or formal ( already we lose track with reality!) from that point 2 orientations are whimsically chosen.( so how is that specifiable?) we draw infinite lines I each orientation through the point.( where do we fo this ? We have no plane specified to do this! Ah, of course! We draw this all on the blackboard! But I do not live in the blackboard!) then we mark of a second point on each line of orientation and define an oriented line segment thereby. Oh yes, that also defines the plane( er.. What if the paper is actually lying on a curved cylinder? Oh, we just straighten it ou? Well that's alright then!).

Now we constrain the student/ operator to only draw lines parallel to these 2 infinite lines.. Next we constrain the student to always place the tip of the line segment in one direction of the 2 possible. Then we constrain the student to only extend a line segment from its tip., in accordance with the 2 orientations.

We fo not specify a process order to these extensions. You can do them in any order you like.( whimsically). This is commutativity of extensive addition. There is no idea of a system in a system or system coupling. The coupling is imposed by the student/ operators whim, and is therefore not predictable. In any engineering system a process that can behave in this way is uncoupled and cannot be relied on, even though it never goes outside the bounds of a similar but coupled system!

Hermann called this incoherent system behaviour, there are many such systems in reality , and they are a valid field of study, but they are not conforming to the doctrine of the extending / extensive magnitude.

In such decoupled systems there is no obvious cause and effect relationship. Thus we are left with a statistical and probabilistic approach. Since Classical Mechanics is based on these coupled systems it is not hard to understand why Quantum mechanics seems untenable. The fact is, as Hermann states, we impose these rules on real dynamical systems because we can't make coherent sense otherwise.

However, what statistics and probability studies have shown us is that there is a " periodicity" to certain measurable phenomena, which very roughly approximate the conic section curves. Now Statistics was in its infancy in Hetmnns day, not even a branch of mathematics, but probability had a long history by then, nd was related to games of chance. As Einstein said, no serious philosopher would place the laws of nature as described by Newton as the handiwork of God, on the basis of a crap shoot! " God does not play dice!"

However LaPlace and LaGrange we're not of the opinion god was even a hypothetical entity in their systems ! Therefore you see the first attempts to apply statistical analysis to their observations of dynamical behaviours. In fact Gauss was rather celebrated for using statistical Anlysis of errors to correctly calculate the position of a rarely sighted comet. And Boltzman, well he was something else! His visionary thinking was far ahead of his contemporaries.

Suffice it to say that when Hetmann defines a product as an extending out one, it's constraints only make sense in a coupled system. Thus imposing ab = –ba is axiomatically constraining the system to be extensively coupled. Because the axiomatic approach has not got a clue as to the whys and wherefores these constraints are whimsical and hive no insight. Worse than that they engender a profound confusion. Our mathematician has no real geometrical intuition and cannot convey to an engineer what it is he is requiring in a real system. An engineer has to go figure that out himself!

If an engineer realised he was being taught a system originally designed by an engineer he would go to LaGrsnge or LaPlace and gain better insight. However since these 2 were highly symbolic in their presentation he would perhaps miss the lementary points on which their systems are based. This is where Hermann comes in. He exposits the elementary concepts in this induction. However, since he was attempting to get a professorship out of this insight, he starts out at a too philosophicl level for just about everyone!

So there you have it: the poor ngineers is left at the mercy of some brmused mathmatician who has his own agenda to push.

Earlier I wrote that quantum mechanics is probably based on the arc segment rather than on the direct line segment. Here I have elucidated that it is based on the statistics of uncoupled systems. The connection is the circle. As a closed system the circle naturally models uncoupled systems each circle has its own uncoupled centre. When Galileo observed the Jovian system, it was seen by him to be " absolute" which simply means independent of nothing else. The concept of uncoupled is in that precise sense absolute.

However, being absolute or uncoupled does not mean it cannot be fractal! Thus it is part of a larger system to which it's parts are coupled only through a principal centre. So what am I saying? Is an uncoupled system coupled or not? Remarkably and statistically and probabilistic ally that may depend on the scale we use to observe and measure! Thus the conundrum between quantum and classical mechanics may just be a fractal scale artefact!

The doctrine of the extending magnitude, by design, deals only with coupled systems, as Lagrange and LaPlace originally conceived mechanics, but today we have by accident and innovation combined moved on to a broader range of dynamical system.

[jehovajah]

Commentary §§14, 15

This sets straight the curved and warped opinions of Hegel!

http://youtu.be/TvFu6ak_SGk
http://www.youtube.com/watch?v=TvFu6ak_SGk

Gegensatz I have translated as "contradictory statement", so this falls into the fallacy of thesis, antithesis, and synthesis.

Gegen means against or going against. It is very general . In some cases it means "opposingly" against in others it means "at the side of" against. Thus the 3 part structure of his work does not neatly fall into the commonly stated pattern.

As I translated Hermann I noticed that the Gegensätze were sometimes alternatives or advancements or analogous ways of stating the same thing!

It seems to be important that the movement to a 3rd position is a natural occurrence of thinking about the issues! The ancestral linking description is also indicative of Hegels lecture performance. The use of contemporary analogies is also a trait of Hegel. Finally the stumbling in the dark until discovering the analogy of the right phraseology is descriptive of Hegls lecture performance

It would seem that Hermann was present at at least one of his lecture performances, even before he started to write the Ausdehnungslehre

[jehovajah]

As a working translation I would think of Gegensätze as comparing and contrasting statements. Thus Hegels method is to compare and contrast , and then draw his own conclusions or resolutions or figuring out of the issues.

The fact that he write in a structured way, a fractal pattern of 3 relates to his deep religious and mystical sensibility, and representative of the Trinity: the Father,Son, and Heilige Geist!  In this way he uses the trinity as an all pervading Analogy, and a structure of the shape or form of Gest as spirit or mind in our experiential continuum.

Ancestral linking is involved in this example of revelatory truth, the Father is intimated by the Son, and the full revelation comes through the Holy Spirit! Similarly we may have a form before us evoking in us some deeper connection. By employing Geist ( mind or spirit) that deeper connection is suddenly revealed to us!