Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

Commentary

Something is special, specific or particular, not of itself but because we have a category of specifics, items which vary from one anther so no 2 are the same. But we can also have a specific item as a unique element of a category of likes . Thus this one element differs from every other, but every other element or item is identical to each of the others.

Thus the concept of special requires a group of differing items and or a group of uniformly identical items in order to support the perception of special or specific.

But these groups are structured ino lists and sublists , categories and subcategories by this same process of identifying likes and unlikes which resides in the process of forming a group of differing entities. Necessarily this process means we gather like things together in order to distinguish the differing things, and thereby we construct sub groups of like things which subgroups differ from one another precisely by what defines the elements within a subgroup.

More deeply we only have like elements if those elements differ in some way! Thus if there is no difference we cannot distinguish like elements from just one element, the same element! Unless we make or allow this distinction like things would not exist as a concept: there would only be one unique item in a sea of unique items!

However differences are also perceivable as like properties, especially if some practice or job requires both elements to be employed together to perform some task. In that case the differing elements can be seen as a combined entity necessary for the performance of some Activity

The label of the algebraic form requires a Metron. This Metron is a measure of how much like and unlike elements we might need to combine to create or synthesis some combinatorial entity. The use of a Metron encourages counting discretely, but we must not forget the continuous nature of the Metron.
Thus the algebraic form encompasses discrete continuous and contiguous entities which themselves will have a combined form or format.

But having said that this contradictory statement about likes and differing groups is only apparent. The distinctions merge but not so completely that one requires some measuring tool to assign ratios of each to every presentation! Thus they can be continuous or contiguous or discrete it does not matter, as long as it is recognised that in the moment of gazing on the apparition of them they appear as a unit with the likes being the foundational presuppositions for the unlikes and vice versa.

The underpinning idea is the structural category subcategory division of apprehension. While Aristotle is one of the great Taxonomists, it is not apparent that he used his taxonomic predilection as a tool of synthesis not just a way of recording Analysis. Hegel insists on both analysis and synthesis in his philosophy. The knower can only be realised as a self referencing entity that also references other self referencing entities is a Hegelian dialectical stage. Hermann is using these ideas and approaches to induct us into a freeer more flexible mindset.

[jehovajah]

Commentary
No.5 is a crucial bridging step to the derivation of the labels for extending magnitudes. It is worth pondering for a while.

The theory of Types or Categories that underpins it is a contribution to logic which is original to Hegel. In one great schema he sets out a basis fr Cantor set theory and Gödels incompleteness theorem. Both are compounded within his categorical system of thought. Within his synthetical system one does not come to a halt or stalemate. As in the Aristotelian system, rather one is inspired to resolve the situation dialectic ally and to move forward to a new category of representation.

If you learn nothing else from this induction learn to embrace contradiction as a creative state of affairs by which one may move forward to a new synthesis in a categorical way that frees your " Geist" from imperfect limitations.

This is no trite or simple process or procedure. Rather to resolve such contradictory situations may require huge efforts in Analydis and dsynthesis to ensure that foundations for the new resolving category are secure. Thus Hermanns one man effort in deriving the Lineal Algebra will serve as a detailed example of how this can be done.

You will note that in doing do many familiar labels and names take on an altered meaning or apprehension, and logos and analogos as conceptual thought processes are given equal weight. Validity is not just the gift of a "logical" or Aristotelian argument.it is also the gift of an argument by analogy . In this way a dialectical process can be sustained on one or both types of argumentation or demonstration.

We have been encultured into thinking" logic" is merely Aristotelian logic, and only that logic is " sound". However those who study logic know this is not the case. The so called ontological argument for the existence of God is a famous example, but the categories of argument even ibAristotelianlogic run into the thousands! Most of us only know a handful of the Arisyotelian arguments, let alone anyone else's!

In addition the Platonic and Pythagorean forms of logic are often obscured by or viewed through the Aristotelian. By introducing his system of categories and subcategories Hegel liberated the subject of logic from an academic backwater into the mainstream of geopolitical real politic and religious dialectical progression. In addition he provided a strong critical analysis of where these subjects were dodging the issues, and avoiding the inevitable progression to human absolute realisation of Geist.

Hegel is difficult because he is unrelenting. Similarly Hermann is difficult because he is unrelenting . Nothing must escape the scrutiny of analysis or the practice of synthesis.

[jehovajah]

Lord Xenus YouTube site provides a great reading on the Classical philosophers by Charlton Heston. In my opinion the section on Hegel, in 2 parts is very clear and informative .

https://www.marxists.org/reference/subject/philosophy/works/ge/gadamer.htm

This link to Gadamers essay of the Science of Logic by Hegel is a bit more technical but also helpful.

The philosophical background that Kant inspired in the German / Prussian Renaisance especially impacted on the Humboldt educational reforms through Humboldts own philosophical conceptualisation of Self actualisation. Thus the content of the German philosophical idealism was directly applicable to the process of reorganising the Prussian education system to produce home grown intellectual and technological talent.

Philosophers like Fichte provided a philosophical basis for reorganisingScirnce and mathmatics on clear logical lines. It was this kind of Philodophy that heavily directed and influenced Justus Grassmann snd Robert Grassmann. They were motivated to implement its implied programmes into the primary education of the young especially in Stettin(Sczeczin).

In this regard, Hermann introduced the more up to date critique of Fichte as advanced by Hegel into the work of the Grassmann household. However this was not an uncritical or slavish acceptance of Hegelian philosophy by Hermann. It is clear that in the Grassmann household these philosophical concepts were discussed and put through the ringer to squeeze out every last practical application. However in the main Hermann seems to have bern heavily influenced by Hegel's contemporary style and lectures.

It has been mooted that Schliermacher played an important role in formulating Hermann and Roberts views, but it is clear that the philosophically rich environment and time in which they grew up lacked no champion for modern Prussian Idealistic Philodophy! They literally could take their pick.

So why Hegel?

I think because Hermann like myself , wanted a universal all encompassing apprehension, and yet still appreciable by simple rules and patterns. Hegel's patterns, despite the complexity of the content they deal with are very easily graspable and very fractally  simple. The pattern of category and sub categories  is fundamentally a pattern of 3!

AB + BC = AC

fundamentally and resonantly models, is analogous  to, is a metaphor and simile of every structural aspect of the Hegelian system . Thus Hegels categorical system of logic, and his fundmental dialectic operation or process is all captured or alluded to in that simple childish, childlike observation Hermann made when he was a young child.

[jehovajah]

Hegel and Mathrmatics a Marxist Perspective

https://www.marxists.org/reference/subject/philosophy/works/ru/kolman.htm

A resource on Hegel.
https://www.marxists.org/reference/archive/hegel/index.htm

It is important to realise that Hegel's system was immensely influential but heavily criticised. Hermann Grassmann was one of its critics! So was Marx. Nevertheless we find mentioned a corpus of Mathemtical essays and research by Hegel which may be source material for Hermanns utilisation of the Hegelian system or method.

The attractiveness of the Hegelian approach is that you bring what you have to it, start from where you are and engage in a dialectical process a la Hegel to create your own imperfect understanding that nevertheless is a new foundation for further dialectical" purification". Ultimately you arrive at a stage which is a status of your intellectual or reasoning capability at that precise time or moment which has resolved many contradictory statements or apprehensions in a positive rather than negative direction giving you greater freedom and flexibility to progress to further resolutions and solutions.

While Hegel assumed this was a process uncovering an absolute "truth" , one is never in a position to determine that absolutely. Instead one gives or is capable of giving many processes that arrive at the same result, thus establishing a basis for moving forward in the dialectical process.in a positive fashion.

The principle of Acceptance is the most fundamental principle that I have ever formulated. It is a principle that is akin to the axiom of choice, but more fundamental. It is an empirical principle by which all empirical experience is founded. Without the principle of acceptance I assert that no thought or consciousness is distinguishable as self aware and worse still is distinguishable by any thing referred to by the label "language". Thus experience and thought of experience and self awareness of experience , being and Nothing etc are empty in precisely the sense Hegel describes and even in the sense of a holographic projection.

The limit of my labelling and there referential ability, or if you like the content of a conceptual label, is the limit of my acceptance, or the limit of acceptance. In this way acceptance is how everything that may be thought or languaged is reeified .

It does not make sense to say that without it nothing exists, because "I" saying that , do so having accepted , consciously or worse yet unconsciously that which roots and rises up in that moment of acceptance, or which "manifests" out of "somewhere", both of which labels again do so in a "moment", which itself is apprehended in that same moment.  I cannot get beyond that languaged description either by language or by thought or by experience. The moment I do go beyond,  I justify that acceptance has operated and been operating at every level . Essentially I am an entity in a fractal consciousness engaging by acceptance with a fractal distribution of that consciousness.

Moment therefore is a fundamental concept , primitive idea which the dialectic process synthesises into time, a more complex and dialectically advanced notion. Indeed the notion of a process, any process whatsoever draws upon a primitive sequencing and ordering in experience which itself is perceived in a primitive moment.

The more analysis that is attempted the more attribution and generality  has to be appended to my primitive moment. Thus "a priori " is simply Italian for acknowledging this impassable foundation of consciousness.
Kant's Transcendental schema is simply a more elaborate version, or an elaboration of the same experience.

Hegel thus starts his synthesis and dialectical process of synthesis at this general, universally so, moment.Hamilon in his Theory of couples subtitles it as a science of Pure Time, because he starts in the primitive moments of an assumed progressive sequence of moments, as the reader apprehends them by recollection.

Thus Hegel does not shrink from grasping the thorny "turbulent" or fractal nature of a priori " knowing" or knowledge. The principle of acceptance includes the principle of assumption, the analogy of rooting and rising up, the process of setting down arbitrarily, anywhere and building fom there.

The principle of acceptance is therefore just stating and identifying what every philosopher and philosophy does at its outset. It includes the concept of postulates, aitema in Greek, as well as axioms ( from the Greek for axle and axis) as predicate concepts of the subjective action of assumption or accepting. This language model or analogy may help some to apprehend what is active in the distinction being made by the statement of the principle that is acceptance, but it does not describe an active act, but rather a completed or instantaneously past action. As you may feel, we are at the ultimate limit of our ability to describe this in the language model.

A meditative experience of these things is recommended, but the subjective experince of meditation is a private profundity. It cannot describe to another what has been experienced. The best individuals can do is accept that others have had similar or identical experiences and actively share time and discourse with those they are " intuitively" drawn to. For what purpose? Well dialectically let's say it is good for a few beers!  

[jehovajah]

In the previous post I highlighted the limitation of languaging. Thus at a stroke logic is limited.
However what presents itself then unfettered by language is the Sensory mesh and that Hegelian unit or category which is everything to do with the senses prior to languaging. The enity or entities I refer yo as sensory mesh or meshes and their Essential quality or qualities. The idea of Essnce here is in contrast inaction to their being or entities/ ontological status. Again the principle of acceptance underpins these a priori experiences.

The signal, or state change in a sensory mesh is of fundamental importance, mainly because it is a precognitive and for the major major part unconscious interaction between the a priori entities and structures.

I shall say that a computational machine of the autonomous sort we label as a computer would be a sufficient analogy in this discussion. The various inputs have a correspondence to senses and to empowerment. The various outputs have a correspondence with consciousness, determined action and the like. For example we may usefully correspond the output screen to the visual and imagination functions within our experiential continuum.

Between the inputs and the outputs is a technological world of a priori interaction, which, in the Hegelian Analydis must include prior and posterior action and interaction! Thus feedback and feed forward interactions must be in the processes that give conscious outputs that " seem" to require a posterior I knowledge nd knowing, or to require us to " learn" before we can act wisely or safely. However in many animal cases this " instinct" is endemic in the unfolding behaviour of a new born. Thus giving credence to information communication which is non languaged!

For want of a better description we call this "symbolic coding"! The history of symbolic coding is long, but for computing Alan Turing cannot be ignored. Because of his work in encoding and breaking codes he was able to conceive of a communication mode that was symbolic and evanescent of a natural language nd yet not a recogniseble language at all.

In this regard symbols become important. What symbols refer to becomes important, and how they refer zooms into view. Reference by simple " attachment" or tying together or association is crucial. Such an attachment could be visual , kinaesthetic or auditory, wht ties them together is more often thn not position in a sequence in a sequence of moments. To make the idea more complex it ids the time position in a time sequence that links or ties symbols to their referents.

It is tempting, like the Greek seemeia or seemeioon to think that indication links 2 things together, that some indicator or indication is sufficient to identify what things are linked, but this is not the case, evn practically. For example if one points in the direction of n object evn if it is the only object in that direction it is still not clear what is being referred to. By physically touching or tying the object to the observer it is implicitly clear what the reference is.

Thus a spatial link between symbol and reference is necessary nd this is conducting the attention along a time dependnt sequence where at each moment the symbol the linking strategy and the reference are percirved as a unit or a Heglian category.

In more direct analogy I refer you to the power of symbol, or logo or symbolic parphenelia in religious or secular contexts, pictograms, or syllabaries or syntax manuals as examples of effective communication of specific referrents whether quantitative or fomal, or qualitative and experiential.

Labels and symbols provide a means of communicating about something in a generally none specific way capable of many interpretations..

How we label,and refer to extensive and extending objects snd quantities and qualities is crucial to how these labels may be organised into notations that help or hinder our communicatin in these Language restricted domains. Principally because so much language when referring to these experiences is functionally " empty" of any additional content beyond what is being and can be labeled by a symbol .

[jehovajah]

I am continuing the transition of the Induction by Hermann Grassmann in the 1844 version of the Ausdehnungslehre, but I can already grasp the purpose of the mixed product which is the subject of this thread and indeed the subject of Grassmanns referred paper.

The Grassmann use of the Hegelian method and dialectic is sufficiently distinct to focus on it as a particular form of a more general Hegelian conception.

It is necessary to analyse anything into at least 2 related but contradicting conceptions, or cognisances. Grassmann chooses a complex contradicting pair each of which is itself a contradicting pair! The fundamental characteristic of these primitive contradicting pair is that they are continuous and discrete, and further thst distinction relies on a further or deeper but more general contradiction between sameness and differing( active sense). There is Also a completed sense ( same and difference). The consideration of these tense or time differenced experiences is also considered in the full treatment.

So we already feel the complexity, and with it the dialectical pull of a dialectical process of solution or resolution of the contradictions. We might in the vernacular call this simplification but we would be misled if we then related that to easier or more fundamental! So it is called a dialectic resolution.

The closing in product and the spreading out product are two differing processes of describing the same thing.

If we restrict the discussion to the parallelogram we might conventionally say they both refer to the area of a parallelogram. Yet in doing so we pose more questions than we answer. What is area ? Why is area calculated as it is?
The dialectic system requires 2 processes in contradictory statement form. Hermann provides this by using a label called the closing in product and a label called the spreading out product to refer to the product of a parallelogram.

These products are perceptible in the geometric expertise, not in the Arithmetic one, where area is the only conceptual relative!

To sharpen the contradiction Hermann chose labels | and ^ for the closing in product and the spreading out product for constituent elements which are entities in geometry, say points, limes, planes, volumes etc.

For these elements he chose labels usually alphabetic letters. Also, to give himself more flexibility he used tally marks( numerals) to both economise on the use of alphabetic symbols/ labels but also to subtly and efficiently indicate sameness and differing, or same and different.

The binding symbols were chosen as+*–/. The subtlety of these symbols is key to his notational representation. Consequently it is vital to read his contextual exposition of the meaning of the symbols in context. It is also vitally crucial to read and understand his Vorrede and his Induction. Without these the tendency is to follow Aristotelian logic rather then Hegelian! Plus, the dialectic process is not evident in the Aristotelian process.

Induction is an Aristotelian principle , exposited by Aristotle from older Pythagorean and Platonic teachings. Intuition and induction refer to the interaction of the Musai on the sensitive soul or spirit of a human or animate consciousness. Hegel, starting with Aristotle goes beyond, particularly to synthesis of all that is attained or apprehended by analysis as set out in a book misnamed the Metsphysics.

So Grassmanns induction sets us up for the Heglian type process. The two products are contradictory in some way. One relies only on the geometric entities the other relies on a projection of the entities into each other.

Using the projection product , the closing in product, we can derive the arithmetical formula for area of a parallelogram. The arithmetical product is a correspondence to the geometrical product which is merely an entity projected into another entity using entities of construction. What results can be labelled, and the labels themselves now reveal a common Arithmetic form! Thus b|h is the same as bh or bxh. We are here identifying a product of a geometrical process of construction with a conventional arithmetic formula!

Now what about a^b? This geometrical product which is a construction using a parallel projection has no arithmetical formulary! Thus Grassmann reveals that arithmetic is a restricted model of what is geometrically possible! Resorting to arithmetic is like poking your own eye out! And being thus self blinded do you, oh mighty pedagogue, deign to teach the young who see more perfectly than you?

What we have is 2 contradicting statements about the parallelogram for example if we restrict the entity to line segments. What is the interpretation of the spreading out product in arithmetic? It has no interpretation. Thus we must resolve to a more general concept.

What is the interpretation in Geometry? The answer to both products in this case is the parallelogram.

The succeeding question is to what use can these differing products be put in Geometry?

Further analysis was required before this could be answered and the analysis revealed several other differing products that were arguably more primitive. These Grassmann organised into 3 types. Commutative, anticommutative and imaginary or roots of unity!

I say " organised" but in fact they are mutually antithetical. They contradict in the required way to have a Hegelian category of their own. These more primitive products are not arithmetical, but Geometrical. They were universally named as Higher arithmetic, but this was and is a mistake that Hermann addresses at the outset, the misfounding of Geometry on a formal axiomatic definitional basis rather than on a real, empirical experimental one. Setting this " right" allowed him to avoid the mental contusions of those trying to make geometrical objects, entities and processes into formal arithmetical ones!. It also avoided the profound confusion!

What did he discover by this result of further analysis? The products represent different projections, vertical, parallel and ROTATIONAL. Immediately a fourth projection suggests itself: none vertical or parallel! This is precisely the principle of projective geometry pioneered by DesArgues, but of course of ancient origins.

Suffice it to say that Grassmann was now free to construct mixed geometrical products! In this case the | product was defined as a vertical projection product ( although this does not have to be the case as it could be a projective geometry projection) and the ^ product was defined as a rotational product.

[jehovajah]

How can AB be both the rotational projection and the parallel projection? This lovely contradictory statement inspired further analytical research which happily was inspired by Lagrange's Analytucal Mechanics and work done on solving the Ebb and Tide problem. Grassmann was able to identify the rhombus as the parallelogram that contained the resolution. In addition it linked not only the trigonometric projections but also the hypertrigonometric proections to the rhombus parallelogram. From this he was able to easily draw on Eukers wrk with the hyperbolic functions and establish the link to the imaginary  calculus functions!

So Grassmanns Hegelian approach was doing what Hegel said it should, bringing together contradictions into a new dialectic resolution that promoted higher and more productive reasoning!

From Euler certain factors and eigen values are transated into the mixed form for the geometric product that is an archetype for the Quaternions, that is a Geometrucal archetype from which Hamiltons Quaternioms may be " derived" by analogical equation.

Because it is an analogical equation it cannot be described in arithmetical terms. However the rise of set and Group Theory with ring theory allowed analogues to be described as operators. Thus the operation in an analogous group is cross applied to an arithmetic situation to generate a result or value which frankly would not be arrived at by ordinary arithmetic means.

Now the term Algebraic came to be used to describe these general group theoretic sets , classes, categories etc. why this is so is perhaps what Grassmann is about to explain in the next few sections of the Einleitung.

[jehovajah]

http://books.google.co.uk/books?id=lpAmDtAh-nQC&pg=PA4&dq=justus+gunther+Grassmann+verbindungslehre&hl=en&sa=X&ei=hytCVOyhKcbd7Qa1rIGACw&ved=0CCUQuwUwAA#v=onepage&q=justus%20gunther%20Grassmann%20verbindungslehre&f=false

There is a whole deep underpinning of ideas that Justus Guther Grassmann set out for school children between 7 to 9 years old and for teachers with little real understanding of geometry. These ideas were published for his school district prior to 1827, (1817) using the title " Verbindungslehre".

Prior to this study I had taken the term to mean " group Theory" at least in an early nascent sense. But as I looked into the German useage it became clear that Connection and connecting was the paramount activity. Thus it is a much broader conception than the label " group Theory" .  And yet it is here perceived as an elementary subject!

Perceptions of what school kids should know and be capable of doing change constantly, so it is of interest to read this material at some stage in the study of Hermanns background education.

It is of interest that Justus proposed these ideas prior to Dedekind and to Riemann as part of a small international group of researchers looking for a natural foundation for the sciences, the philosophies and mathematics. The idea was clearly that of a theosophical group who looked for natural analogies of spiritual or Reason based conceptions. In that regard Hegel's work and philosophy starts with the evident assumption that the processes of the natural world are the processes of the absolute, or the free Reason to which he sees everything eventually perfecting by this process itself. He called the process Dialectic, and it is a synthesis process.

The study of how all these things are connected to each other is thus important, but Justus and his group are interested in a very specific subset of things, namely the 2 dimensional geometric entities, lines ,rays and points. The connections between these things and various combinations of these thins in connection with the tally markers is what he called the study of connections.

I also recognised that Lehre is more pedagogical than the word study, and perhaps teachings or doctrine is a better emphatic translation, but that is not as reader or student friendly as study, so I will keep an eye out for any dogmatic tones I maybe overlooking.

[jehovajah]

Ausdehnungslehre   1844
Induction

6. Out of the criss crossing of these  both contradictory statements , from which the former relates itself to the artform of creating whole and the latter relates itself  to the elements of creating, go forth   the 4  categories of the thought forms/ patterns; and out of which the  to them inter communicant branches of the study of thought forms/ Patterns go forth. And indeed the discrete thoughtForm/ pattern separates initially  there according to that  in Tally mark and Combination (  bundle).

Tally mark is the algebraically discrete thought form/ pattern , that is it is the summary grabbing together of the entity  as like "set" things ; the Combination is the combinatorially discrete thought pattern/ form that is it is the summary grabbing together of the entity  as differing "set" things. The expertise of the Discrete things is thus the study of Tally markers and the study of Combinations (the study of things  connecting to things)

That herethrough the label of the Tally mark is completely  exhausted ( prescribed/assigned) and exactly circumscribed, and plainlly so the label of the Combination, is wholly hardly needy of a further after thought( proof/evidence) . And there the contradictory statements through which these Definitions have  gone forth from here , the simplest definitions, in the label of the mathematical thought form/ pattern, are without mediation with Givens .

Thus  herethrough the above derivation, in sufficiently far reaching manner is well finalised correctly.(QED!)

I make still only a remark:

How that this  contradictory statement between both thought forms is expressed, through the different signing related to their elements, onto a very pure cognisance, in which the Tally mark  linked entity with  one and the same symbol will be signified (1); the Combination linked entity  will be signified with differing things; in the remainder completely  arbitrary  signs( the printers block!)wil be utilised.

 Now hereafter every crowd of things( specified system)How that  can be so well apprehended as Tally marker ,
how that  can become  so well apprehended as Combination:  each of those according  to the differing manner of expressing tracking, well hardly permits a passing comment!


Footnote to this section page xxvi
The label of the Tally mark and the Combination has already for  17 years in " Treatises" penned of my Father , running over the label of the pure doctrine of the Tally Marks, which in the Curriculum of the Stettin High Schools from 1827 has been  distributed in print, on a completely ancestrally related manner has developed , but without having reached the Attention of a bigger Audience.

[jehovajah]

Commentary
In section 6 Hermann claims to have finalised his goal of deriving the labels of the Tally mark and the Combination. It is doubtful if the reader even knows until that point that it was a goal of his. However, in the dialectic process the usual Goal / objective and operations and testing to achieve that goal, the TOTE method is not as simple as in the syllogistic approach. This is because Analysis is specific in orientation while synthesis is general , seemingly arbitrary , in orientation.

To be sure one can have a goal in building a house, but there is so much variety possible that tying down a specific design is more of a guideline than a specific outcome! What results is what results! It may be specific to the design, but everyone knows that to achieve that level of accord requires a particularly strict and literal, even robotic diligence, and a willingness to tear down what is constructed if it deviates in any way! Only one in 10 million would ever be that demanding, and they would certainly be classed as Autistic, or worse still a perfectionist.

Hermann may be autistic, but he is certainly not a perfectionist! His rigour varies in these descriptions, because he is human and not a robot. On many occasions he asserts the conclusion as self evident, or not requiring further proof! This is always an appeal to common sense, but it is not logically " sound" in the Aristotelian sense. It is an appeal to pragmatic sensibility, and to the principle of exhaustion. What exhausts one mind or person may not exhaust another. Thus these appeals constitute a " logical" weakness in that further analysis is possible at these junctures.

The " point" is when do we stop analysis? The Pythagoreans state through Euclid that we stop when we get the signal to stop, the seemeioon! The seemeioon is often translated as point, but it literally means indicator . The indicator has no parts! This is Euclids declaration at the beginning of his philosophical synthesis the Stoikeia. Analydis stops when there are no more parts to disintegrate! But long before that analysis stops due to exhaustion of the analyser!

Hermann thinks he has gone deep enough, or reache far enough to justify his definitions of the Tally mark and the Combination. But we may go further today, especially with the fabulous tools we have for measuring and perceiving.

Nevertheless few will undertake the task of digging deeper. In fact even Hamilton was astonished at how far reaching Hermanns analysis was and is! There may be some who claim to have gone deeper or further! And I may be one of those, but I doubt it as of now. The metaphors of deeper or wider or further are empirical, and based on extending magnitudes in a real expertise. However Hermann is not concerned with the real expertises here. His sole purpose is to construct the formal expertise of Formdnlehre, or thought pattern doctrine. Until now I had not even realised this level of distinction consciously, although intuiti Ely I recognise every intention Hermann has as an experience I had when I was classing myself as seriously depressive! This was before I reclassified myself as Autistic.

Unlike Hermann I cannot exhaust a thing! I recognise constantly that what I write, know or understand is already incorrect and out of date! Thus all my writings are thought dumps, just like core dumps in the old days of computing. I do not defend a position as Aristotle may imply is possible, because I think it is impossible in a changing environment to have anything but approximate knowledge. New information leads to new insights and the overturning of old constructs .

Where Scirnce goes wrong, but they are human after all, is to try to prevent that dialectical process! A professor would rather sit on or throw out of his class a person who threatens his cherished dreams and ideas. Science would rather denounce inventors as fallacious than lose access to thrif cash cow. Etc.

We all crave certainty, but there is none in my opinion. What saved me from insanity a la Cantor was the fractal zoom! Once I could accept infinite regression, something Kant refused to do, I could leave my interminable analysis alone! I could park it anywhere and go for a walk On the beach! I could start building and synthesing anywhere just to see what resulted. I was free at last.

Hegel has given me a philosophical mentoring in a way that does not make him the Don, but rather makes the process the dialectical one a useful pattern tool to utilise. Hermann in using this tool, however modified provides me with an example of what can be constructed , if one wishes and is dedicated enough. Unfortunately, Hermann believed or acce
Ted what hewasconstructing was " the Truth", and for this he suffered great pains and
Erdonal privations . The key is not the concept" the Truth" but the concept of total acceptance and dedication. I have accepted and dedicated myself to a handful of things inmy life, and that has built my experiential continuum as is. I recognise that once I stop accepting change I will become cut off from life, thus the fewer things I have to change the easier it is to continue..

The only constant in my universe is Change! Panta Rhei!

[jehovajah]

I place this video here because the issue is dealt with in Hermanns preamble to the derivation of the 4 categories in section 6. In particular we can see the Hegelian conceptualisation of nature and natures reason and the Geist or the Absolute of reason have a realisation in this discussion.

Why do physicists struggle? Because physicists and philosophers have not applied Hegelian logic or Philosophy. Is the situation only describable by quantum mechanics? The answer is no. Prior to quantum mechanics was Hegelian logic and his Phenomenology of Geist. Bohr was wrong in accusing all western philosophers because he did not know Hegelian Philosophy. In fact he directs us to mystical mentors Buddha and pragmatists like Lao Tse who nevertheless was based in traditional Chinese philosophy, and thus Indian Philodophy, that everything is connected in a complementarity that is contradictory until resolved, precisely the doctrine of Hegel.

Do I negate mystics by Hegel? On the contrary, Hegel embraces them all, and they Hegel. What I negate is the science that is so local that Bohr has to go to china to find a philosophy that matches his thinking!

<a href="http://www.youtube.com/v/BFvJOZ51tmc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/BFvJOZ51tmc&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/BFvJOZ51tmc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/BFvJOZ51tmc&rel=1&fs=1&hd=1</a>

[jehovajah]

If we seek a definition of Ausdehnung and Ausdehnungslehre section 7 gives us one that will blow your socks off!

As far as I am aware extensive and intensive magnitudes were a creation of Gauss. Hermann defines the intensive magnitudes as being Created through algebraically continuous likes. Whatever way you cut that you get a continuous uniformity, the kind of entity that looks the same no matter how much it is scaled, added to or subtracted. That is to say you can look into the space and not know where you are relative to anywhere else in the Spacbut the Ausdehnung or extensive magnitude , or if you like the extending or extensive magnitude is defined as a creation through combinatorially continuous differings! We could say contiguous differings form the thought pattern of the Ausdehnung, and the Ausdehnungs Lehre is the study of those contiguous differings.

But the foundation or the prior art for apprehending these Ausdehnungen is function theory and the differential and integral Calculus! Actually it is not quite that bad. All we need is the foundational concepts of these subject areas.

Ausdehnungslehre is thus the doctrin of extensive and extending magnitudes.

Here is an accessible lecture about the Issues contemporary with the work of Grasmanns Justus and Hermann. What is not clear, until you read the Astronomical Principles in particular, Leibniz as an instance Lagrange as a modification and Euler as a root and branch refunding is the role of extensive and extending magnitudes in the calculus. These are none other than the infinite sums and series with which Newton established the dynamic nature of His Fluxions, and which Leibniz eventually arrives at as a general method for integration, and Lagrange explores in terms of coefficient multipliers( a Newtonian method overlooked by others) and Euler sets out as foundational principles of his exposition of  Calculus.

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

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

[jehovajah]

Ausdehnungslehre 1844

induction

7. Plainly, in a  similar manner the Continuous thought  pattern/ form (or the magnitude there concording ) separates itself into  the Algebraically continuous thought  pattern/ form (or the intensive  magnitude) and into the Combinatorially continuous thought pattern/ form( or the extensive magnitude). Thus The intensive magnitude is that entity which is reeified through the creating whole of the Like entity. The extensive magnitude , or the extending magnitude, is that entity which is reeified  through the creating whole of the Differing entity . That yonder entity builds The foundation of the function doctrine , of the differential and the integral Calculus   as everyway varying magnitude, this hereby entity builds The foundation of the extending/ extensive magnitude doctrine

There, from both these  branches : the former, (the study of Tally marks), is nursed as a higher branch  (to become subordinate),  but yet the latter appears as an until now unknown branch; so it is necessary, without farther delay this difficult tracking  through the label of the continuous fluid and fleeting entity to declare closely: how in the Tally mark the singularising/ unit defining steps  hereforward , in the Combination the separating/specifying of the " together!" thought steps  hereforward. (Thus also in the intensive magnitude the singularising/unit defining  of the Elements, which   in accord with  their label indeed have still separated/specified the intensive magnitude, but which Only in their "essential nature", to be like themselves,  are  making a representation of  the intensive magnitude)

On the other hand in the extensive magnitude the separating of the elements which indeed as far as they make a representation of a single/ unit  magnitude are united , but which plainly only in their separating apart from one another are "constituting" ( specifying the constituents of) the magnitude.

Thus there exists likewise the intensive magnitude -the fluidly reeified  Tally mark; the extensive magnitude - the fluidly reeified  Combination. The latter one is essentially a stepping out of one another of the elements, and  a firm attachment of the same elements as out of one another entities.

The creating whole element,  by the latter it, appears as a self varying entity, that means through a differing system of status/ condition markers it appears as a  "from afar through travelling" entity , and the Gathered System of these differing status/ condition markers plainly makes a representation of  the field of study of the extending / extensive magnitude.

On the other hand, by the Intensive magnitude,  the creating whole of the same itself delivers a continuous Rank Array,  of self like status/ condition markers,  Quantity of which is plainly the intensive magnitude .

As an exemplar for the extensive magnitude we can at the best the bounded Line( line Segment) choose!  elements of which essentially are stepping  out from one another , and plainly therethrough the lins are constituting   as extending .

On the other hand as an exemplar  of the intensive  magnitudes some kinda  "with assigned strength" endowed  Point, in which here the elements do not themselves pour their innards outwards., rather present themselves only in the climax , thus  making a perfect stair step a representation of  a climax.


Also here shows itself  the difference that is placed onto a beautiful cognisance of the denotation: specifically in terms of the intensive magnitude, which  the doctrine of functions is making content out of, one does not distinguish the elements through specific symbols  , rather where specific symbols are stepping here forward , there is the complete everyway varying magnitude therethrough signified.
On the other hand in terms of the extending magnitude,  or by the  concrete representation of which , by the line,   the differing elements also become signified With  differing symbols( from the printers blocks ) , directly how in the combination doctrine.

Also it is clear how every real magnitude can become manifested in a dual cognisance. As intensive and extensive; specifically the line also becomes manifested as an intensive magnitude ; even if one from the art form looks away  , how their elements are out one another, and simply the quantity apprehends; and plainly so can the with power endowed Point as extensive magnitude become thought, in which one the power itself sets forth in form of a line.

Historically the Discrete thing  has before developed itself under the 4 branches of Mathematics as the Continuous  thing ( there, that to which "dissected  limb Perceptions" lies nearer than this), the Algebraical thing before as Combinatorial thing( there, the like thing more easily summarily grabbed together becomes than the differing thing). There to here is the Tally marks doctrine the earliest thing to root and rise up, doctrine of combination and differential calculus are same timeish, and the doctrine of extending magnitude, from them all in their abstract thought forms/ patterns must be the latest, while on the other side to it more concrete ( althoughl of a restricted nature) development, the doctrine of space, already relates to the most earliest time.

[jehovajah]

It urns out section 7 is an important stage in the induction. There is quite a bit of it yet to translate, but I have updated the last posted translation because it revealed to me how narrow my focus had been in my initial exploration of the ideas of Hermann.

Many of you readers, like me would not have heard of Hegel, except perhaps in a Marxist lderogatory! Certainly I would not have placed him amongst any mathematical greats, and in any case mathematics is not about Philodophy?  I could not disagree more. It was my study of the foundations of mathematics, that is reading David Hilberts book at university that made an otherwise inglorious university career fundamentally worthwhile!

While I did not grasp the richness of this enquiry, but rather stumbled on it during day's of aimless mind numbing questing for the " spark" , the djin that would lay Mathematics so called, out before me like sn open book, I now realise my quest was answered in that stumbling.  Yet Hegel was not a name I recall, nor Grassmann from the first reading of that Book.

And certainly, in the thread Fractal Foundations where I track down many of the great " Mathematicl" philosophers Hegel does not appear.

The obscure nature of the Ausdehnungslehre is due almost entirely  to Hegelian logic! Once you have an experience of Hegel's logic, categories and rhetorical style you find Hermann as remarkably clear! The density of thought in Hegel's works is palpable. He is not nor was not a light read. Similarly neither was Aristotle. I am beginning to glimpse that perhaps Aristotle has been much maligned by modern critics who do not read the Greek, but rather comment on someone's translation! No doubt I shall find out in time.


Nevertheless Hegel, unlike Newton and Hamilton both admirers of Aristotle, used Aristotle as a jumping off point , following Kant's lead toward a more contemporary redaction of Aristotles ideas. Whereas Kant sought to transcend philosophical reasoning by his analytical sagacity, Hegel takes a more immersive approach.

By immersing himself in the act of philosophising Hegel draws out the distinctions of all philosophers. But then he goes further. He synthesises all these distinctions back into a whole. It is that process of analysis and re synthesi that is the dialectic process. Ultimately it is a learning process during which many seeming contradictions are resolved and the instigator comes away with a new appreciation and apprehension.

In undergoing this experience Hegel developed a pattern that progressed his study. The pattern is often referred to as thesis, antithesis and synthesis. Hegel referred to it as the dialectic . Based on the Platonic dialectic it nevertheless is new questions are not just asked and answers pursued to yet more questions etc, rather contradictory statements are pitted against one another and resolutions sought actively. There is always a resolution. Hegel was positive about this, and he calls this approach the positive dialectic.

Of course there is a negative dialectic, and in that case no resolutions are sought, rather greater distinctions and separations occur. Hermann clearly apprehended the positive dialectic as a way forward .

The second and intriguing fact about Hegel is that he wrote mathematical papers applying his method . Apparently these papers are archived somewhere but only Marx could apprehend them. It is a moot point as to how much Hegel's actual Mathematical ideas were known or available to Hermann.

In any case Hermann in his induction demonstrates what comes about by applying the Hegelian dialectic. Many contradictions or contradictory statements now make sense in this Hegelian approach. Many curious effete statements of what he has achieved that I noticed in the Vorrede now make sense. The generation of new labels now makes sense. And as you will shortly read, the duality or rather the dual affinity of the point and the line now make perfect sense.

Norman Wildberger explores this duality between point and line extensively in his Universal Hyperbolic Geometry. It is therefore a pleasant surprise to realise that it is a feature of Hermnns induction that arises naturally out of the application of the Hegelian dialectic process.

[jehovajah]

This page link to my Bombelli thread indicates my early apprehension of Grassmann, but despite its intuitive astuteness I can safely say I did not have a clue what Grassmann had done or how he arrived at his exposition. Only in this last year since translating the Vorrede and  the Einleitung have I really developed an understanding that is transforming into a skill at applying the Hegelian-Grassmann dialectic! At least in understanding his writings!

http://www.fractalforums.com/complex-numbers/bombelli-operator/msg47192/#msg47192