Der Ort der Hamilton schen Quaternionen in der Ausdehnungslehre

[jehovajah]

The dot product is defined by Grassmann in his definition of the closing in product, which is derived from his earlier definition of the shadow casting product and the projection product. The regressive or rather special pleadings product we will retire as unfruitful

The dot product is named by Bill Cliffird several years after Grassmanns death.

The dot product is a product formed from the summation of other products , which products are composed from rows or sequences or ordered sets of elements!

Now I am specifying a very general idea or process. Thus I am being specific about something very general! This is dialectic process in action! Soon I will get very general about something specific , and then the dialectic moves forward into a new synthesis of these ideas.

What are these elements? In general they cn be anything at all.  Sometimes I may experience these elements as a complete collection or structure or flow. Then I can utilise them as is in the specification of the dot product. At other times I may have to engage in a process of collecting nd ordering these elements.

What is the product composed from these elements ? This is as simply as possible to pair each element in one set or sequence or row with just one element in a second row. This is an association, a mapping or a linking of two elements . This pairing may be physicl, or by mapping . The product thus may be apparent and local but it ay alo be invisible and mentally apprehended.

When each elemental product is composed then the dot product is formed by gathering them all together. This gathering may be physical, so creating a region of these products, or mental creating a list on paper or in memory, or a transformation by counting, resulting in a label or name with more or less contextual meaning.

This is the dot product process and it's result, but now I am going to specify it for one of a number of very special cases.  The case I choose is if the elements are numerals.
I have 2 sets of ordered numerals. The sets are specified to be of the same number of elements. Each element in both sets has an ordinal numeral subscript, and the elements are paired by these subscripts. The resulting pairs are a product, but I now specify these products as factor multiplications.

The factor multiplications are performed and all the results added together to give a result which will be called the dot product of the 2 ordered sets of numerals. The result is a single number coded in numerals.

Thus there are several important processes in the dot product calculation.

Now I am going to specify the dot product for a set of numerals and a set of line segments. Both sets are ordered, with a numeral subscript. The dot product proceeds giving numeral line segment pairs. These pairs are products but they are geometrical products. All these products are summed and the result is a geometrical product a numeral line segment pair.

The summation of numeral line segment pairs is the parallelogram law of 2 line segments established by Grassmann.

[jehovajah]

Let a and b be the labels for the sets of ordered numerals. Then a•b is the label for the dot product of those 2 sets of ordered numerals. That represents a_ib_i
Where a_i,b_i are the individual numerals with their subscripts.

Let A be the label for a set of line segments and similarly B. Then a•A is the label for the dot product of one of the set of numerals with one of the set of line segments in order.
That represents a_iA_i where a_iA_i is each individual,numeral line segment pair.

Similarly for b•B

Now [a#b] is the Cayley table composition from a and b. It is an ordered  table of products a_ib_j where I,j run independently through the subscripts of the numerals.

Similarly [A#B] gives the composition table of product A_iB_j

But which of Hermanns products is similar to this of the line segments? In this case it is the closing in product [A_i|B_j].

The Cayley composition table simply needs a distinguishing mark between the line segment products to become the line segment products needed for the closing in product.

There is no summation at this stage. So I propose to extend the dot product to cover this case. In this instance the one to one mapping is Onto pairs with the same subscripts.

The closing in product is thus

[a#b] • [A#B]  which represents [a•A | b•B] and this is

a_ib_j[A_i|B_j].

I could denote this as the dot product of Square matrices.

[jehovajah]

I can define the cross producer as a numerical line segmnt  product.  This means that the line segment set of ordered line segments is dotted with a set of  numerical elements ordered by subscript. The difference in this numerical set is the process by which it is established. This process involves " cross" products of more primitive numerals. But no summation of the compound result.

Although I have called this " dotted" this is clearly not dotting because there is no summation. I perhaps should just call it pairing or dotpairing to maintain the link to the dot product . Cayley pairing is also a possibility to maintain link to the Cayley composition table. Whatever these are composition processes for products prior to summation to give an overall product type. Products thus involve composition and summation of primitive products.

The cross product is thus the beginning of many other possible product processes for numerals. These resultant ordered sets are then dotted with the line segmnt set to give a numerical line segment product.

I can define Cayley table composition using the ordered sets and arrays within the Cayley tables themselves.

[a#b] gives us a matrix A [c#d] gives us a matrix C

Collect the rows of A from the ordered array in A. Label them r,s,t.. Similarly for C label the columns m,n,o...

The composition of the Cayley tables now becomes

| r•m r•n r•o....|
|s•m s•n s•o....|
|t•m t•n t•o.....|
|........             |

This composition is clearly the same as or analogous to the fundamental Cayley composition from the ordered sets of elements. By reducing the array to rows or columns the matrix becomes an ordered set of rows or an ordered set of columns. Thus the composition is the same process at a " higher" stage of complexity.

[jehovajah]

Ausdehnungslehre 1844. Induction

3. The pure mathematic therehere is the expertise of the "Special Experiential Continuum", as of one  generated through the Thought process. The special experiential continuum apprehended in this sense We denote as a thought Form/thought  Pattern, or at worst a Form/Pattern. Therehere pure mathematic is a study of Form/ thought Form, that is the study of Pattern/ thought Pattern.

The Name "Magnitude doctrine" is not central to the fully gathered Mathematic, in which the same Name on an essential  branch of the same mathematic, on  combination theory finds no application  of  it, and on the arithmetic only in an  uncentrally related sense. There against, the expression "form " again seems to be too wide and the name thought form/ thought pattern a better fit.
Alone, "form" in its purest consideration, devoid of all real content is plainly not a different entity, as a thought form, and thus with the expression is inter communicant.

Before we move over to the partitioning  of the study of thought pattetn we must cut out a branch, which one untilhere has made judgement towards it with Injustice (an injustice to which one has maintained until now), specifically Geometry. Already out of the above set out label light shines in, that Geometry, so plainly like Mechanics, onto a real experiential continuum has to be restored. Specifically this must be Space/ 3d Space. And it is clear how the label of space in no way can be created whole  from thought Process, rather the same continues confronting, as a Given entity. Whoever is wanting to make the opposing division more important, is expected by the Muse to undertake  to do the work  the necessary derivation of 3 dimensions of space out of the  purely thought established ones, an exercise whose solution itself immediately presents as impossible!


Foot note
*the logic serves a pure mathematical side, which one can as a formal logic designate and its content jointly reprocessed according to my Bother Robert and me , and from the former in his 2nd Tome the Study of Form/ Pattern Stettin 1872 is represented in a centrally related form  
**The labels of magnitude come to be introduced into  Arithmetic through the act of Tallying or counting, the language therefore differentiates very well extending more about  or extending less  about ,(what the Number relates to ) from extending magnitude about  or diminishing magnitude about (what the magnitude relates to.)

[jehovajah]

Commentary

There is a lot of perspective changing going on in this section. Having narrowed the discussion to formal expertises Hermann then divides the formal expertises into 2 antithetical subjects: dialectic process and pure mathematic process. These are all thought processes, thought acts in general and in particular. The general thought process was examined for universal or general laws of thought. . The monad of thought was sought, so thought could be formed and measured in a Pythagorean sense. This exploration was pursued by a dialectic process, and not the traditional Socratic Platonic Pythagorean dialectic. Hermann was enamoured of the Hegelian dialectic process.

Pure mathematic was a thought process or thought activity that explored special cases of thought, established regimes of thought. Pure mathematic was not mathematics, it is more broader than that, but the term mathematic is so suggestive that this is easily overlooked. Pure Mathematik here refers to the reasonings performed habitually by a qualified Pythagorean expert, an Astrological reasoning that incorporates Geometrical representation.

This comparative set up of the division of the formal expertises conditions how we proceed with the analysis.. The general qualitative assessment of thought now needed to be transformed into a quantitative assessment. This was achieved by Hermann by an anthropomorphic or rather entititic representation of the experiential continuum. General thought processes becomes  general thought of a special entity.. In particular this works better for the special cases dealt with by pure Mathematic. Pure mathematic can be viewed as a special entity  that thinks through special cases, but only comes into existence by this thought act.

This was viewing the qualitative and expansive experiential continuum by a quantitative form developed through thought. This form or pattern is a step back from the activity of thinking to view it as a whole , and thus as a form or a pattern.

This form or pattern is usually approached from the Greek classical idea of Eideon, which is a visible form or an imagined form. It is the subject of the Platonic Socratic philosophy of Idea/ Form.

In choosing form over Idea Hermann departed from the classical concepts of surreal entititees to a promotion of real palpable entities. For example Dedekind started his concoction of numbers using just Ideas. Hermann is signalling that he is on the side of the constructive nature of mathematics and reason.

So now thought patterns are the entity that Hermann wants to go on to specify. That thought pattern is precisely the thought patterns treated philosophically in the Euclidean Stoikeia. This was an undergraduate level philosophy course defining thought patterns .. It contrasts strikingly with Aristotelian Logic , based on Aristotles grammatical analysis of language.

The special entity which comes into being as the thinking by it is being done, or it is a "by thought realised " special entity; this is the entity that applies the formal expertise of pure Mathematic. It is the special experiential continuum thought of in this particular way, and when the special experiential continuum is viewed in this way then it can be termed as a thought pattern.

When one concentrates on a triangle and specifies all its parts and relations that in and of itself is a special entity, a thought pattern specific to the triangle. This is similar for all spaciometric forms. Once that is grasped one can realise that thought patterns dominate all ones thinking. This is the pure mathematic. The dialectic therefore searches for the monad of all these special entities. This monad is the Metron from which all special entities may be constructed just like a basic unit of a mosaic.

Is there a basic unit of thought? This is the general quest of Hegelian dialectic in particular.

[jehovajah]

Commentary

In the formal expertise we define space as 3 dimensional. But we can also define it as 1,2,3....,n dimensional. However in Geometry we cannot define space as 3 dimensions because we cannot logically limit the dimensions to 3! Space is space by undeniable experience. If we label space we come across undeniable experiences: we are not limited to 3 ways of labelling space. There is no way we can logically prove that assertion that space is 3 dimensional. This makes space and geometry not part of the formal expertises, where everything has to be established by definition and thought acts that concord with definitions and other defined realities or definitional systems.

Mechanics and Geometry are part of the real expertises. We apprehend them as self evident and undeniable..their veracity confronts us every second of every minute of every day of our experiential continuum.

Since we cannot define space as 3 dimensional we cannot define it as n dimensional either. Space is as we experience it. However we can establish foal systems in which these dimensions are defined. These fomal systems must be consistent with each other. The question then is are they of any use to the real expertises? And if so how?

The most fundamental geometrical pattern is the closed loop. This is discovered by rigigid and non rigid rotations, strings and fluids to be so, undeniably in our experience. Through it we  percieve other more ubiquitous patterns: spheres vortices etc. But it is the formal constrained defined circle that we begin our study with!

This pattern is formal and abstract and not found in space. Yet everything we start to know about geometrical space depends on it! Hermann is saying this is a formal model which is not geometry or mechanics, but a formal model we apply to space. It literally tells us nothing about space. It tells us everything about how we perceive space! It tells me that we have a restricted blinkered view of space!

Nevertheless, this view has proved technologically useful, mechanics refining it to fit as is. But the success of this view has blinded us to alternatives and to freedom of thought pattern .

Their is fractal regression underpinning all that we claim to know. This has been held back from view by scared philosophers. Kant for example hid it behind his transcendent schemes. Others called it a monster! As a consequence we do not know what it is we don't know! Hermann sought to clarify this at the outset. Placing geometry among the real expertises has a profound effect on the formal expertises!

The Axiomatising of Geometry was a big injustice, it was Also a big mistake. The parallel postulate shenanigans did much to denigrate the reputation of real expertise based Grometry and Mechanics.Gauss was reputedly ashamed at this fiasco in geometry and sought a new geometry, but did not know how to get there. Hermann did, by restoring Geometry to a real experiential continuum. Space.

[jehovajah]

Ausdehnunglehre  1844
Induction §3 continued

Now  someone would (Imagine if it was like  he has to deliver this thought process), the entity still , the name of Mathematic still stretch it onto  also Geometry  to  Darling it! ; so could we indeed ourselves let this happen, even if  also it will allow ourselves   to  stand on the other side to our name The study of thought form or some random equivalent; yet still we must  in advance thereof advise of  it afar, that then that Name, because it conceals the "most differing" entity  within itself, also necessarily with time must become tossed aside as superfluous. The position of Geometry toward the theory of thought form depends on  the commensurate arrangement, in which the likening/manifesting  of space stands to  pure thought process

Now, Even If, like we were saying,  it  confronts by the thought process  that likening/ manifesting as self evident Given entity, so  therewith is still not asserted  that the likening/manifesting of Space was coming to us first out of the tracking of the space like thing , rather it is a primitive likening/ manifesting, with which the opened state of our senses for the sensible world is inherently given to us, and which is plainly adhered into us from the outset, like the body to the Soul.

Onto like cognisance (a like way of thought): it proportionally arranges itself with Time, and with the kinematics founded on the likenings/ manifestings  of Time and of Space , wherefore also the pure study of kinematics ( Phorometry)  with the same right as Geometry  one   has counted also as a mathematical expertise .

Out of the likening/ manifesting of kinematics ,  out of the manifesting of the conflicting statement from originating cause and Effecting  flows mediated the labels of the kinematical Skill;  so that thus Geometry, Phorometry and Mechanics shine out onto the primitive likenings/ manifestings of the sensible world as applications of the study of thought form/ thought Pattern.

[jehovajah]

Commentary
Hermanns use and reliance on Hegelian logic shines through this difficult section.. Here the 3 propositional sections are combined to set up the dialectic progression into the next section.

The original division represented a contradicting state of affairs or Verhältniss, the second statement narrowed it down into a contradiction between pure mathematics and the Dialectic process and the third one establishes that the study of form or thought forms by a special entity is the best way forward. These special forms I have highlighted as patterns

Then he uses Geometry to advance his case, by arguing geometry has bern unjustly treated and wrongly set up as an axiomatic discipline. In his day in Prussia that statement fell on fertile ground. It was obvious to Gauss something was wrong with geometry which had singularly failed to solve the fifth postulate problem after nearly a thousand years of trying! What was wrong Gauss did not know, and he directed Riemann to put out the call to physicists to rescue mathematics and geometry

However the Grassmanns had been working on the solution starting with Justus, who corresponded with a small international group of Ring and Group theorists and crystallographers. They felt the dynamics of crystals held the key to a real geometry one in accord with nature, not human thought.

However Justus struggled with Aristotelian logic, and foundered on the multiplication problem. He could not logically justify multiplication, all he could do was point to a geometrical figure!

Hermann on the other hand studied and read Hegel's writings and his dialectic or logic. Some have seen the similarity between Hegel and Plato in their dialectics. What no one has pointed out is how this logic differs to Aristotelian logic, upon which islamic and most European Academics relied.

Actually Socratic and Platonic dialectic derives from the Pythagorean school in which koans were given to students rather than linguistic analyses. The koans were intended to exercise every aspect of the disciples being until an epiphany occurred whereby the Musai or a Muse might reveal a profound " insight"  or a truth or ability or gift.

In this way the student was assessed to have progressed until it was agreed by the senior scholars that the student had become one of them , a Mathematikos.

This " qualification" was a discipleship based recognition, not a piece of paper. You literally became a
Lifelong " monk" or inhabitant of a "Monas" tery. Such a building was also called a Mousaion or the house of the Muses. It was decorated with " Mosaics" and it was a place of culture , learning and arts and customs.

Thus the student was immersed in a collegiate of thought, discourse debate and empirical observation and Muse led insights. Aristotle disagreed with this approach. Nevertheless while he could he remained with Plato. However he was forced to leave for geopolitical reasons, the same reasons that thrust him to the very top of the burgeoning Greek empire as tutor to the Emperor Phillips sons Alexander and Phillip.

Thus for a very brief period Aristotle outshined all Platonists. And in that position he ditched the Pythagorean system and created his own, while still claiming to be a Platonist. His school quickly passed away and but for the Islamic scholars may well have been forgotten under Plato. However he was once again raised to high prominence in the Islamic empire and thus obscured the alternative Pythagorean traditions of learning to this day.

Thus when Hegel began his philosophical work, it was not Aristotle he turned to, but to Plato, and from him eventually to his own unique style of logic , without perhaps knowing it he had recovered the original Pythagorean system in whole or in part. It was this kind of dialectic Hermann used to unravel the whole of western European thought for neatly 1000 years..

The Hegelian dialectic moves in circles, it moves in levels; it is very fractal and very thorough,and it always moves forward to a new synthesis based on the thesis and antithesis in a contradictory statement.

[jehovajah]

Normans Complexions!

Note how " vectors" easily replaces the mumbo jumbo of " number". These are special entities, special thought patterns or foms. And Norman is studying these forms in a geometrical setting.

You have to have a special state of mind or thought to apply these patterns expertly! This was Hermanns Förderung, his promoted way of analysing and analogising, apprehending(interpreting) and applying.

<a href="http://www.youtube.com/v/MTddX4Qo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/MTddX4Qo&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/MTddX4Qo-io&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/MTddX4Qo-io&rel=1&fs=1&hd=1</a>

[jehovajah]

Anschaung  is someone making something visible to others , making something apparent to others, likening something to another individual , manifesting something to another person or persons thought process or perception.

It is a complex reflexive in that the people ultimately perceiving do so through someone else drawing attention to Some analogy which they feel is apt and hope the perceiver will draw a perception from, conveying to their minds something of what is in his mind.

[jehovajah]

Pure mathematics is an expertise of a special entity a specific state of mind a special experiential continuum. If the special experiential continuum is understood in this way then we cn call it a thought form a pattern of thought or in a poor sense a form.
The idea that pure mathematics is the study of magnitudes, that is the experience of extensivity, is misleading! There are several branches of mathematics which have nothing to do with extensivity! On the other hand form seems to be too wide a concept, which is why the terminology thought Form or even a form of thought seems more appropriate, not so bad or poor a fit.

Granted if one restricts form to just the abstract idea without any real content, just the shell of a form then the use of the term form does correspond, but in Amy case to be clear it is best to centre on the term thought form.

Because of this possibility to introduce bad thinking into the development I suggest the word pattern or thought pattern as a more general term for the idea Hermann is trying to express. This does mean that the clear connection to form in space is less clear! It requires one to see all forms as a more general pattern of extensive experiences, but it prima facia includes all patterns made principally by drawing : thus drawings, sketches, scripts , plans, schematic diagrams, flow charts etc.

The Greek word Gramme which derives from the verb which covers all drawing actions is a good correspondence. This word is usually translated " line", but that is only part of its conceptual meaning.


Pure mathematics is thus the study of the patterns of thought of an expert in a special or specific state of mind! It is thus entirel formal, not real!

However we still want to use the word mathematics, because we love it so! For that reason Hermann extends the word across the real formal divide , temporarily. Eventually , he foresees, the word mathematics will be discarded because of its internal inconsistencies! Then the study of thought patterns will come into its own!
For now (1844) he was willing to put up with the new terminology being pushed to one side, because he also loved the word mathematics like a little darling!

Having established his terminological root he wants to go on to show how that overflows into the work and terminology of the Ausdehnungs Lehre. However to do that he has to restructure the contemporary perception of pure mathematics by first removing geometry from is deadly grip, establishing geometry as a real expertise like Mechanics, based on the real experiential continuum not the special state of mind of a few experts!

The real experiential continuum on which geometry is based , derived from and verified by is Space! "Space, the final frontier" is a well used phrase that captures the extensivenesses of the real space of Hermanns conception.

In line with his scheme or set up he now draws attention to a third set of skills called Phorometry.. This is the study of pure motion in space! Today the word has been sidelined to cover the specialist work of an optician diagnosing vision problems in the eye. This is a direct result of the dialectic tussle between pure mathematics and the study of thought patterns!

Rather than recognise the difficult conceptual problem, pure mathematicians have reacted defensively. They divided their subject into pure and applied, and left the applied mathematicians to their fate! Gradually applied mathematics has bern absorbed into Mechanics, physics, chemistry , and most importantly Computational science.

Pure mathematicians are thus a dying breed, as Hermann foresaw, and their relevance has gradually slipped into the netherworlds, backstreets and the boondocks of societal perceptions. Only in Academia have they been able to maintain some kind of mythical and mystical evanescence! However computational science in my view has made it more likely that they will soon disappear as an active cadre of intellectual expertise.

Today Mathematica is the new face of mathematics. Without realising it, and under the cover of the darling name Mathematics, Hermanns "study of thought patterns" has been realised in the computational sciences. In my opinion we need to make the break and revise the concept of mathematics along similar lines to the proposals that Hermann now proceeds to work through.

Phorometry is often confused with stereometry, and even photometry. But in Hermanns experince it was a broad field of expertise delivering direct measurement procedures, tools and equipment for making displacement measurements in space. It is a fundamental application of trigonometric concepts, and thus is allied to land survey, astronomical survey, triangulation as well as optical survey and opthamolocical survey and diagnosis!

Hermann sets out geometry, Mechanics and Phorometry as the real " mathematics"  and tHese are the thought patterns he wants to study further using inter communicant formal systems!

It is worth noting that Hermann was not wedded to the exact words Formenlehre. He was equally happy with other terminology expressing the same conceptual base.

[jehovajah]

We cannot use Grassmanns analytical and synthetical method to distort the real expertise of measurements of space. But we can use pure Mathrmatics by formal set up, or math magic to do just thst!!

<a href="http://www.youtube.com/v/nXF098w48fo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/nXF098w48fo&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/nXF098w48fo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/nXF098w48fo&rel=1&fs=1&hd=1</a>

You have to beware of experts and their expertise, especially if they are based on a formal or theoretical expertise
<a href="http://www.youtube.com/v/XwhtHUiAOjM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/XwhtHUiAOjM&rel=1&fs=1&hd=1</a>
<a href="http://www.youtube.com/v/XwhtHUiAOjM&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/XwhtHUiAOjM&rel=1&fs=1&hd=1</a>

Hermann distinguishes between the expertise of pure mathematics and the expertise of real "mathematics" which he denotes as the study of patterns applied to real empirical measurement skills or crafts like Geometry, Phorometry and Mechanics.

The manifestations of kinematics leads by mitigation of the contradictory statement of cause and effect or originating change and its outworking to the labels of the kinematical skill!

[jehovajah]

The Astrological Principles of Newton and the System of worlds represents a setting out of a philosophy of quantity and it's detailed application. Because Group and Ring theiry had not been identified or isolated as subject domains, newtons presentation stands out as a landmark innovative approach to Astrological mathematics.

In the same way, Hermann Ausdehnungslehre with his Putative Schwenkungslehre stands out as a landmark revision of Newtons Astrological principles in terms of group and ring theoretic terminology.

The material and work covered by both philosophies is identical, but the presentation is almost completely antithetical. Newton was secretive and dismissive of Algebraic gibberish, while Hermann was open and enamoured of Algebra. However , without the Hegelian dialectical approach he would not have been able to encompass the breadth and depth of Newtonian thought and empirical exploration.

The underpinning lineal algebra is the same, but expressed terminologically by Hermann where Newton expressed it Rhetorically.

But why stop at Newton? We may safely return to the Euclidean Stoikeia as a philosophical treatise that encapsulates all and more of what both Hermann and Newton treat of. Leaving aside Aristotles digression from the main Pythagorean principles , it is possible to see a clear path from the Pythagorean school of thought through Plato and Euclid and Apollonius et al. to Newton and then to the Grassmanns and other group and ring theorist.

Today's use of group theory and ring theory in physics is thus not a predilection intellectual snobs( only!) but a fundamental clarification of the formal system which is applied to a real expertise system.

Where Hermann stands out, by Hegelian logical process is that he fully recognises this from the outset, where most who follow his work but not his philosophy do not.

[jehovajah]

Since we are leaving the world view of Aristotelian logic and have embarked on the journey of Hegelan logic it is probably a good idea to review chadafricans videos on YouTube regarding Hegel, and also Gregory B. Sadlers.
I can recommend those 2 channels to give you a clear introduction to an unfamiliar but important philosophical and phenomenological treatment of human experience.

<a href="http://www.youtube.com/v/7Hh8OGEsmqA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/7Hh8OGEsmqA&rel=1&fs=1&hd=1</a>

<a href="http://www.youtube.com/v/6L6zndZs8Js&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/6L6zndZs8Js&rel=1&fs=1&hd=1</a>

Many contradictions apparent to the Aristotelian world view are now acceptable in the dialectical process of Hegel. Thus many issues that bothered me in my early translation attempts of the Vorrede are actually not harbingers of doom to the Grassmann method. In fact they drive a synthesis at a new level.

Is Hegelian logic sound?

The question itself is interesting. Do the results of the dialectic ring like a bell? This test of well foundedness or good foundry process giving a perfect result is an auditory not a visual or kinaesthetic test, leaving aside the important vibration sense through the proprioceptive sensors. Thus a grammatical language analysis, the basis of  Aristotelian logic is an inappropriate vehicle for making this judgement call. The soundness is verified by empirical concordance( übereinstimmung!) and is a phenomenological assessment. Thus from the outset of this induction Hermann has promoted the soundness of the Hegelian dialectic and how it is assessed.

[jehovajah]

Lord Xenu is another recommended YouTube channel for a more traditional overview of the philosophers in their context.

I also draw your attention to Grassmanns own criticism of the Hegelian school. Thus Hermann does not slavishly conform to any academic or scholastic consensus on Hegel, neither in his own time nor in the time and contexts of those critics and admirers who followed. In any case it is hard to say whether any such putative consensus exists!

The student Karl Marx in critiquing Hegel drew his own conclusions on inadequate research portfolios. And in many instances critics have failed to grasp or grapple with the Heglian dialectical method, systematic approach or exemplars or even Hegel's own extensive analysis and synthesis of his contemporary context.

The Hegelian logic is first proposed by some as Aristotelian, because of the kindred spirit of deep unflagging analysis and total exploration of totality. However, no Aristotelian scholar seems able to finally say Hegel is a student of Aristotle, because he appears to unfailingly contradict himself in his writings where Aristotle, par excellence taxonomised every experience into a systematic taxonomy which isolates contradictions . In that difference lies the resolution that Hegel is not a slave student of Aristotle but a master of his own philosophy, which encompasses as far as is possible all philosophy and philosophers to his time.

Philosophers after Hegel have had to deal with Hegels philosophy, and most have consigned it to impenetrable obscurity as an academic trick to remove it from the preeminent position it holds. As the saying goes "they prefer the old wine because it is sweeter!". And again" new wine must go in new wine skins, lest the skin bursts and all is lost!"

Hermann stated that he was going to use the mindset of the Hegelian school to join with his own mindset to produce a remarkable independent system of analysis and synthesis . Thus he is fully ready to tackle the contradictory statements of specific analyses to resolve them , by mediation or mitigation into a new broader whole. We see this first in his trepidation, and disconcerting horror at stumbling onto a non commutative product process! Until then his experience, training, expectation and understanding led him to see multiplication in only one way. To his credit he held his nerve, did not throw out the result as nonsense, which an Aristotelian point of view actually demands, and found a higher level of distinction( or a deeper level if you so wish) that resolved the issue.

The problem for mathematics as an axiomatic system is that many demonstrations rely on contradiction to support the validity of an assertion!. Consequently Hegelian logic, if accepted, makes pure mathematic unsound wherever it relies on a contradiction to establish a truth!

What to do? Either bury your head in the sand or move on through the Hegelian dialectic process to a new resolution.! This is why Hermann predicted the " death" of pure mathematics, the obsolescence even of the name Mathematics, as time progressed and the internal contradictions became intolerable.

Yes, for now we still treat mathematics like some darling, to be indulged, to be wreathed in the some of nostalgia; but that time is fast fading away , and the time of the Fomenlehre is upon us, the time when the patterns of thought are the paramount subject of study and application.

So now we move on by a seeming contradiction to the idea of a study of thought foms. Hermann stated that the term "the study of magnitudes" was an inadequate descriptor of pure Mathematics as a formal system, even more so when Geometry is removed from its banal clutches. And yet he calls the next section the derivation of the terms or labels for extensive magnitude theory! - or the study of extensive magnitudes!

The resolution lies in the label Ausdehnung. Rather than a dead static object Ausdehnung is a dynmic, living subjective experience of changeable and changing form/ pattern. Formenlehre begins with a primitive dynamic that manifests itself in primitive, a priori manifestations. These are real not formal experiences, so to speak . From these primitive distinctions the dialectic process begins to build a greater outcome. Ultimately the result of the dialectic process is the Monad of the Neoplatonists or the Neo Pythagoreans . The logos analogos sumbola sunthemata process that restores the summetria of the Monad from the disparate pieces into which it apparently, to the mind at least , has fragmented. This is Shunya of the Sanskrit Philsophical schools of thought , before it becomes divided by the various philosophical and Yogic traditions.

Hegel goes beyond to one more level: that of absolute mind. This is the oneness of  Buddha, the Nirvana of self awareness that references only itself. It is absolute in that sense requiring no other to validate itself. Like Aristotles unmoveable mover it is self motivating , everything else it moves relative to itself. Thus the only thing that can move it is itself . Thus it is the Absolute Mover.

In this analytical result Hegel, Newton and Aristote concur, but their synthetical paths to this description differ widely. For Hegel this absolute is real, empirical and other to the individuals subjective progression of development. To Aristotle it ultimately was a categorical taxonomy. Once he had come to this conclusion he then goes on to taxonomise the inverse, and then the reverse and so on. Newton resolves his system into n all powerful God who establishes and maintains a mechanical universe.

Regardless of these ultimate drivers Hermann pursues the liberating outcome of his analytical and synthetical method as a young enthusiast encumbered by having to make his way in the world. Thus he never attains to the final conclusion of his ideas, calling on others to carry his torch onward and upward to some presumed glorious apex of the whole redesigned and more functional Mathematics layered into and manifesting in the kinematical dynamics of Space, and Nature in and of that space.