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I have been delaying posting section 8 until I got some understnding of it, or some inspiration by its Muse. As usual, some other thing triggers an appreciation of a text I am working on, and the generalities of the Wolfram language are a case in point. But it is not these intricacies that contain the simple exposition. It was something else, fleeting which now escapes my words, my bonne mots!

This section should summarise Hermanns derivation because section 9 launches into a full exposition of the AusdehnungsLehre, and yet it does not quite do that. It seems to be a post ahead of what is coming as a closure of what has just been gone through.

What have we gone through in this dialectical progression? It seems like I have been taken through certain broad notions . Ideas and experiences that I have in common with others but not specified with others by debate or consultation and consensus or agreement.

A lot of these notions are not up for discussion generally, except if you have a philosopher as a friend! Commonly they are shied away from, replaced with some platitude or some workaround.

I am not decrying this pragmatic response. It has saved my life and sanity on many occasions. Keep It Simple Stupid or KISS was a mnemonic I was drilled in during sales training. As you can appreciate, I found that a hard rule to follow! Nevertheless it was and is a very serviceable rule of thumb. Literally my autism meant I could never understand why communication had to be so dumb!

That is not the same as saying my communication is so super intelligent that the rest of humanity clearly is beneath my understanding! No rather I found that I was autistic, hiding my ignorance behind grand sounding words I did not understand, but could enunciate fluently! My articulation was a skill of my Autism, not of my understanding!

I think of Stephen Fry a well known Wordsmith. Out of the sounds of words and the scripting of words he can create a beautiful entertainment. But like a song it advances no technology or makes no poor man warm in clothes and fed with food. Yet those words may emliven his spirit , set his heart and mind to achieve his needs despite the prevailing circumstance. Words do have a power if they are just so, the right word for the right time.

So Hermann has looked at these words that attempt to apprehend our most general experiences, which are right out at the edge of our intuition and of our experience of consciousness and being alive. And he does not say " now let's define them precisely!" for that is a vain agenda.

That is a vain agenda that troubles all Aristotelians. The taxonomy of apprehension that Aristotle above all established is the most Autistic Trait of a person with Aspbergers syndrome. Aristotle was a high functioning Autistic . His taxonomy dominates his logic. In accepting his logic one is forced to forever analyse, even when drawing conclusions.

Hegel recognised this, and contrasted it with the teaching methods of Plato and Socrates, teachers or mentors of Aristotle. Aristotle found his balance with Plato. If he could have stayed he would never have left him, but geopolitics played him a hand he could not avoid or refuse. He had to leave Greece , but then was drawn back into the very centre of the Greco Macedonian Empire. His unique view shaped Alexanders view of the world as a system . Helemism flows from that view.

But Plato was a Pythagorean student. We have no record that he attained the qualification of Mathematikos, but he clearly sought that qualification. Eudoxus and other Pythagoreans like Euclid taught at his Acadrmies, and established sister academies. Both Euclid and Eudoxus were Mathematikoi. They achieved that qualification. Aristotle did not because he had differing opinions. Plato may simply have loved being a perpetual student!

The core concept of Pythagorean philosophy is synthesis. The core symbol of that is the Mosaic. And the core method of achieving this synthesis was by revelatIon from the Musai or Muses.

This seems perhaps different to the Socratic Philodophy as some teach, but in fact it is very much in accord with it. The Musai are the curators of all knowledge, skill, art and aesthetics, rhetoric, poetry , dance and Musical Harmony. Phusis, the sister of Harmony is a representation of the oppositional , contradictory nature that has to be harmonised. Thus Physics deals directly with these physical contradictions or opposites, + and – , positive and negative, pros and cons , evaporation nd condensation. ,,, etc. with a view to harmonising them.

Socrates suggested that the human had a soul that simply knew everything, but a body that frustrated that knowledge , in a sense. By guiding that person through skilled questions that person could remember what his soul already knee.

This is a pure Tautology. Pure in that it is the purest form of apprehension the human mind or consciousness can achieve. Everything is already known, either by the Soul or the Musai. Humans simply have to ask the right questions to recall this knowledge.

But then there were infinite regression! Human consciousness knows about endless loops of thought. That is a fundmental limit of our ability to know. A loop is the simplest tautology. We think and " advance" our thinking by tautology. But we simply rename the loop as we go round it again. Thus it is the sme but different!

It is this Same but different, or same and different tautological structure that Socrates sought to build, and Hegel managed to build by his logical categories.

Unlike Aristotles taxonomy and Ontology ( a taxonomy of being or existence notions), Hegel sets out all the possible categories based on the distinctions in language. Thus rather than studying nature and creating a new category for each discovery, Heglian listed all the possible categories language itself can distinguish. It is a very large number, but it is not infinite.

Given or realising these categories it is possible to divide our experience into real and Fomal. In such a way that the fomal always imperfectly mirrors or captures the real.

Realising this, Hegel also realised that we have trapped ourselves by our own processes of understnding. He then realised that the way to absolute freedom is by synthesis not analysis. The synthesising of our notions actually makes a finite set of constraints, onditions or categories infinite in application, because each category can have an infinite number of Statuses or conditions!

Synthesis means we can model the whole universe with a brick or we can build from bricks a more detailed model of the universe or the universal. The single or the complex structure is the same subject, but the detail we can go to is Fractal.

Clearly it is unsatisfactory to go straight for the single brick model! But starting with the Hegelian categories and working through the thousands of them, but still a finite and doable number , we could build an absolute model, the best we can hope to achieve. Along the way we attain to what Hegel called the absolute rationality or Reason of our conscious experience, whether we categorise it as a god or Philosophical reason our aesthetics determines our individual style.

So Hermann applied this kind of systems or category logic to Mathrmatics. The first thing that wrnt out the window into the rubbish bin was the term Mathematics! He and Robert coined the term doctrine of thought patterns. Yes they allowed Mathematics to be retained for a while( because we love it so!) but ultimately because of its contradictions the name will be consigned to the dustbin of history!

Preceding with thought patterns then how do we get at the subject content of Mathematics? We do this by specifying generalities. That is we specify general notions.

There are many general notions, love, peace , space... So which ones do we choose to specify?

The sections up to 5 identify those general notions.

Now I m using the word notion here but the german in Hegel is Begriff! One of my favourite mentalist words! It is literally the grip by which we handle, think and manipulate things real or imagined..

I have purposely consistently translated that as handle/ label.. As a mathematician and familiar with Algebra and programming that is the kind of terminology I would most appreciate.. So to hear the word use to denote a notion is very interesting and satisfying.

So Hermann identifies or labels 2 general contrasting generlities : Same and Differring, but only after initiating them in the very act of reification. Hermann needs these 2 generalities in order to specify the subject area of the doctrine of thought patterns. They bring embedded within them the labels or handles for Discrete and Continuous. They also embed a dynamic activity of reification as thought patterns or thought forms, entirely set by conscious or unconscious thought processes and commands.

Hermann establishes these general notions but shapes them in Mathematical language markers! Thus he sets up a system of leading questions! The shaping readily leads the educated mind to the direction Hermann wants to go. To anyone else it undoubtedly leads to confusion!

Now, I criticised Hamilton for using legerdemain or sleight of hand when he skipped over from ordinal numbers to Cardinal numbers. I could do so because he sought to slip that distinction into a syllogistic format. Here however Hermnn is using a dialectic format, and so almost anything goes that leads to a resolution! In addition because he wants to get at Mathematics from a more general platform it is required for him to show how he does that. He does it by shaping the labels as if they were leading questions. At this general level that shaping gives you or the reader sparks of insight and connections, which a mathematician would never see!

That kind of insight justifies tha approach . Or does it? You will have to decide that yourself.

Having specified his generlities by various labels and handles he then specifically discusses the  whole lot in 4 categories and labels or attaches handles to each and within each.

The labels I hope I have identified or picked up on, but the moste important are unmissable. Intensive and extensive, step rise, step extension, climax, extending out of, algebraic, symbolic arithmetic, tally marks, combinatorial, bundle, tied together, ordering: next to or under, row, column, rank,file, array, space, constraint condition status , setting, layout. Etc. and any others I may have not mentioned in that list, such as list itself!

So we have reduced by stages and shaping from the most general notions to a set of more specific notions which will service the general Mathematical discussions Hermann wants to pursue.

So finally as a heads up in section 8 Hermann states that there is a form that is posted ahead from this discussion of the subject in 4 . It is a kind of general array found everywhere in the doctrine of the extending and extensive magnitudes. The only proviso is that its use should not require writing out the whole of it again ineach branch or an offshoot ! In that case it would be useless. However where it applies it facilitates the common development of many branches and offshoots at one go!

To Hermann that was a prize worth reaching for. He seems to label it the capstone or final array.  In true Hegelian dialectic, this new summit of realisation now becomes the foundation  of the Doctrine of Extensive or extending magnitudes.

My earlier mistake was to assume the hewing into 4 had already been done! In fact the Ausdegnungslehre that follows takes us up to and through that hewing process. The general capstone array is despatched ahead to that upcoming event.

Norman on the fundamentals

<a href="http://www.youtube.com/v/Ek0URXLCZCE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Ek0URXLCZCE&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/Ek0URXLCZCE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Ek0URXLCZCE&rel=1&fs=1&hd=1</a>

We can see how Grassmann avoids these issues by using generalities in an Hegelian dialectic rather than an Arustotelian logic.

I have just reworked section 6 and hopefully clarified some of the structural sentencing.

It seems reasonable to consider algebraic as a synonym for symbolic arithmetic. The tally mark stands for a crowd of things , and is a symbol for that crowd used in arithmetic. However now Hermann has laid out briefly another or an extended symbol typology to clearly denote or stand for a crowd of things when viewed as a combination. The closest to this conceptually is the set theory of Cantor et al.

 As usual we must resist the desire to restate Hermann in words defined and derived by someone else. His signs distinguish between crowds of things as Tally mark and/ or as Combination. The concept Menge is the a priori of the set notion, but Hermann here leaves Menge undefined. He merely distinguishes two aspects of the experience of Menge!

I have completed retranslating section1 with my newly acquired adjectival Skills 

I must say they do help to better express Hermanns thoughts rather than my own. At the same time I am surprised how little of his ideas I mangled in translation! Lol?

So I think I have finished correcting section 3 in terms of a more literal translation to Hemanns words..
The main clarification is the manifesting of space to the thought process in it proportional arrangement is vital to developing a proper kinematical expertise.

Thus the following video shows how the kind of likenings and or manifestations used to denote something affect our modelling of it and our apprehension and ultimately how true it is to empirical or primitive experience of the senses.

Do not be blinded by advanced mathematics. If you can't experience it  " It ain't necessarily so!"
<a href="http://www.youtube.com/v/XkAPv5s92z0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/XkAPv5s92z0&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/XkAPv5s92z0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/XkAPv5s92z0&rel=1&fs=1&hd=1</a>

Just to mark this ground breaking event and to note the use of the word " Expertise". This is an example of real expertise, and geometry, Phorometry and Mechanics, the kinematical skills Hermann alludes to in section 3

<a href="http://www.youtube.com/v/R3WwbVY3YbY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/R3WwbVY3YbY&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/R3WwbVY3YbY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/R3WwbVY3YbY&rel=1&fs=1&hd=1</a>

The everyway standing of §7 requires some background reading .
https://www.marxists.org/reference/archive/hegel/works/hl/hl217.htm
Hegel's approach or cognisance of the topic is later and based on his categorisation of linguistic or label distinguishers. In so doing he critiques and adapts the ideas as presented by Leibniz and  Kant.
http://dataspace.princeton.edu/jspui/bitstream/88435/dsp01ww72bb53m/1/Diehl_princeton_0181D_10142.pdf

I have yet to find an originating reference to Gauss, so if he did contribute to the concept it must be in response to Kant's and Leibniz philosophical examination of extensive and Intensive properties of space and spatial objects. Gauss was as a huge force in establishing a scientific standard or convention for fundamental weights and Measures that would be internationally acceptable and applicable. The word "dimensions" and dimensional analysis derive from these kinds of concerns and movements towards a common system of units. The first and easiest to establish were the extensive magnitudes. Attempts to establish intensive magnitudes as a system of units took a lot longer and required innovative measuring techniques and comparisons. The quantity of light falling on a retina for example is an intensive measure, but it is couched in an extensive format.
http://www.arcaneknowledge.org/science/magnitude/magnitude.html

http://books.google.co.uk/books?id=IUFTcOsMTysC&pg=PA236&lpg=PA236&dq=intensive+extensive+magnitude+gauss&source=bl&ots=0uXnhRc1Uc&sig=Ghp3vDEiEaUCuRF_AQ_xQixz0Rk&hl=en&sa=X&ei=S4FkVJ6jA4_g7QaRy4G4Bw&ved=0CB8Q6AEwAA#v=onepage&q=intensive%20extensive%20magnitude%20gauss&f=false

A critical simmary of Grassmanns work . Read from the heading Grassmann to the heading Riemann.

http://books.google.co.uk/books?id=ldzseiuZbsIC&pg=PA90&lpg=PA90&dq=Gauss+intensive+magnitude&source=bl&ots=UmNFqA7C-0&sig=pbPf_uAd4Ak48ZMMhSxa8s0IpMM&hl=en&sa=X&ei=U4ZkVL2pBIKwadyjgJgL&ved=0CCIQ6AEwAQ#v=onepage&q=Gauss%20intensive%20magnitude&f=false