The Ausdehnungslehre of Hermann Grassmann 1844 reprinted in 1877

[jehovajah]

http://m.youtube.com/Zq8JOi2b11k
<a href="http://www.youtube.com/v/Zq8JOi2b11k&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Zq8JOi2b11k&rel=1&fs=1&hd=1</a>

Napoleon defends the revolutionary spirit.

[hermann]

Hello Jehovajah,

Die Ausdehnungslehre is now available by Amazone.
http://www.amazon.de/Hermann-G%C3%BCnther-Grassmann-Die-Ausdehnungslehre/dp/3836402270/ref=sr_1_1?s=books&ie=UTF8&qid=1437067183&sr=1-1
I found now information which Issue has been reprinted.

The other material is not available at the momennt.
http://www.amazon.de/mathematische-physikalische-Veranlassung-Mathematisch-Physikalischen-Wissenschaften/dp/B00GI4DN88/ref=sr_1_2?s=books&ie=UTF8&qid=1437067786&sr=1-2

May be some one is interrested in the reprint.

Hermann

[jehovajah]

Thanks Hermann for your continuing support of this thread.

As you may know I am concentrating on the root ideas Justus Grassmann intentionally sowed in a generation of young children in Stettin during the time of the French revolutionary expansion. At the time Revolution in everything was very much on people's minds .

Thus the work Justus attempted to and failed to get off the ground in a big way in his day nevertheless steadily grew its support and fan base.

This work that Hermann did, lives and breathes that earlier work and carries it forward to a new dimension of applicability. Without that background I find I am always sensing a deeper generality to the 1844 version. With that work I now know precisely what Hermann was thinking about and what shortcomings he was addressing.

[jehovajah]

This eloquent exposition of the analogy of being and becoming gives a modern take on what Hermann spends so much time in the Vorrede establishing.


https://m.youtube.com/watch?v=g6W9fATLQdk
The rooting and rising up analogy or metaphor as a structural definition of becoming is foundational to his deriving of the rules of Like, and likening and likeness.
Like and differing are fundamental to our perceptions of being and becoming and the differentiation and distinguishing into magnitudinal experiences .

[jehovajah]

Every now and then we re invent the wheel.

This video by Norman extends his Realm concept to some innovative Tropical Algebras. I point out that Hermann Grassmann has a big following in the educational system in the tropics. Consequently this algebra will be familiar to those who are his students! It is found in the early part of the Ausdehnungdlehre 1844

http://m.youtube.com/watch?v=1_ZfvQ3o1Ac

[jehovajah]

Storming' Norman!
This continues his careful construction of an Algebra.

http://m.youtube.com/watch?v=oWJIQdo1vpQ

[jehovajah]

http://m.youtube.com/watch?v=3lKOeyHzyrI
Suddenly Norman reveals that he has been preparing the ground for an assault on the area now called combinatorics.

It is important to note that combinatorics is a newish trend, but in Hermnns time it was not even considered as a mathematical topic!

In any case Norman by way of algebra guides the reader viewer into a Begriff ! That is a handle/ lable system that enables the participants to chat naturally but precisely about thought patterns.

The essentially of Order and repetition and distinction are covered by a simplistic convention which quickly reveals complexity levels in ones apprehension . It is these simple conventions which turn off so many students by their inanity and triviality on the one hand, but subtle and surprising utility on the other !

It takes a good teacher to emphasise the need to pay attention to the simple things more than the complex.

Hereby is revealed the combinatorial foundations of all Algebras, and though really symbolic arithmetics they do at the same time discover a deeper combinatorial process in our thought patterning.

It is worth re reading the first 3 sections of the Ausdehnungslehre 1844 in the light of this revisionist Begriff by Norman. Nevertheless the very general style adopted by Hermann is still way ahead of our time!!

[hermann]

Hallo Jehovajah,

Thanks for the link Normans Video on Higher Data Structure.

During my seasons holiday I have worked on Data Structures from a software developers point of view by developing some packages for abstract data structures for example I have a generic list package, a package for manipulation of bit arrays and digits. So I have started implementing the basics of mathematics from a software developers point of view.

Instead of a mathematical prove I have to convince my Ada-Compile, that my code is correct Ada-Code and I have to watch my programm if it behaves as expected.
May be I find some time to present the results on my page on Ada-Data-Structures.

As usal the holidays are over and I have to concentrate on my daily bussiness.

Hermann
P.S Here is the actual specification of my linked list:
(The generic parameter may also be a linked list.)

generic
   type Value_Type is private;
package Primitive_List_Generic is

   type Value_Array_Type is array (Integer range <>) of Value_Type;

   type List_Type is limited private;

   function Is_Empty (List : List_Type) return Boolean;
   function Length (List : List_Type) return Integer;

   procedure Append (List : in out List_Type; Value : Value_Type);
   procedure Append (List : in out List_Type; Value_Array : Value_Array_Type);

   Empty_List : exception;

   function First (List : List_Type) return Integer;
   function Last  (List : List_Type) return Integer;

   function Convert_To_Array (List : List_Type) return Value_Array_Type;

   Out_Of_Range : exception;
   Undefined_Element : exception;
   function Get_Slice (List : List_Type; Start, Stop : Integer) return Value_Array_Type;

   type Element_Type is private;

private

   type Element_Access_Type is access Element_Type;

   type List_Type is record
      Number_Of_Elements : Integer := 0;
      First_Element      : Element_Access_Type;
      Last_Element       : Element_Access_Type;
   end record;

   type Element_Type is record
      Value : Value_Type;
      Next  : Element_Access_Type;
   end record;

end Primitive_List_Generic;

[jehovajah]

 http://m.youtube.com/watch?v=pD1ZZ127b_g
The role of Sanskrit in the intellectual climate of Hermanns time and beyond to our time .
We must remember Eulers great knowledge of Indian philosophical thought.
Hermann became a key professor of the Indan influence on western language through Sanskrit ...

[jehovajah]

Thankyou Hermann. You have helped me decide what I am going to do with some old papers and studies I have to sort through from the loft!

I am currently, as you are, in a bind. I would like to progress my work on the Grassmanns translations but I have to prepare for my family so things are ticking over very slowly. However I am heartened by your communications and interests. These constructivist processes are crucial to understanding the Grassmanns state of mind.

The Formenlehre or doctrine of thought patterns is so broad that it is easily missed and segmented into differing subject boundaries. The rise of combinatorial studies has been a welcome but mysterious adjunct to Mathematical doctrine! The historical provenance of this collection of studies is often obscured in topological, mathematical and computational subjects. I believe the Grassmanns indeed pioneered a Key or set of keys to understanding how we can think and model patterns in our sensory experience.

The connection to language and particularly Sanskrit is so fundamental that mathematicians are blinded to it. This blindness arises out of Aristotelian subject boundary wars! This is why Hegel, the Grassmanns and the Prussian Renaisance was so powerful. Aristotle was debunked by this time! Sir William Rowan Hamilton to Newton were greatly enamoured of Aristotle, sir William Gilbert was not! Constructivism arises by overthrowing Aristotelian Platonism and sourcing back to Pythagorean schools of thought. Through these sources comes the treasures of all past great civilisations and language cultures!

So for me computational science is tha natural home for so called Mathematics. In fact it may well be the modern conceptualisation of Hermanns tentative suggestion: " Formenlehre" .

The stumbling block has been the shifting mis identification of Algebra! 
Historically it really arises as the Indian method of generalisation. There it has a long and perfected tradition based on constructivist ideology. It was not called Al Jibr which was an epithet given to it by Al Khwarzimi! It has many Sanskrit names for each of its departments and applications but let us say Gita and Sutra are Sanskrit indicators that we are involved with this system of thought patterning and expression.

So Al Jibr has a vernacular meaning: " mind fluff!" . This is a classical joke! Scholars from Islam found this one of the most difficult processes to apprehend! This was because you had to become  devotee to an Indian guru to be properly trained in all these things. Outsider: the Greeks, the Arabs, Christians did not want to give up their belief systems to learn this stuff! And yet it was clearly a higher learning and facility.

Bombelli is perhaps the most influential but least famous westerner who demonstrated the engineering, constructivist power of this thought system, but there are many artists and engineers who guided by the Pythagoreans filtered this system into Western thinking including Wallis, and by default Newyon, De Moivre and Cotes.

The Greek model was the Arithmoi, or do we are told. But in fact this was the Pythagorean school of thought redacted by Plato! Little is truly known about Pythagoras in the west, but of course there are Indian Buddhist traditions about all our great heroes including Jesus!

I mention this only to encourage you Hermann in a great tradition of language study and implementation derived from Sanskrit pioneers. Your computational explorations are language process explorations and this is what drives you, me The Grassmanns and NJWildberger!

http://m.youtube.com/watch?v=a8Ufs4lowc4

[jehovajah]

http://m.youtube.com/watch?v=BzMT1m8b-QM
The Ansicht continues. Those familiar with Mathematica will recognise these structural or syntactical markers. What you will gain from this is the generality of view with specificity of adapting the general to the particular.

These are Grassmannian concerns, thoroughly but not exhaustively set out in the Ausdehnungslehre 1844

[jehovajah]

http://m.youtube.com/watch?v=ALq8Rrr2mfg
What the Ausdehnungslehre is used for by theoretical physicists! But they do not publicise that they are using Grassmanns inductive methodology .

[jehovajah]

http://m.youtube.com/watch?v=sW_IkMQEAw
Norman reveals the heart of the inductive method of defining an Arithmetic.
For those not familiar with Justus Grassmann this was the key stumbling block in his Logicsl Doctrine of Mathematics, explored in his work for Nature Clients and Physicists! .
At the moment I am not clear if this is how Hermann solved his Fathers difficulty by utilising an inductive methodology to define everything as Norman illustrates, but if it is it certainly is a testament to sophisticated understanding of thought processing or thought patterning everywhere in evidence in the Ausdehnungslehre of 1844.

We can clearly see that Algebra is a symbolic Arithmetic, and that Geometry is an allied explanatory system to this exploration of inductive thought patterning. The Begriff or labelling/ handle designing becomes a separate but important co dependent expertise and practice to put these Notions into a useful and efficient aid to combinatorial process that underpins calculus: that is direct counting using calculii or calculation.

The geometrical interpretation serves as a reference guide only. Soon the thought patterning itself sparks off remarkable analogous thinking that unifies widely separated subject areas that rely on counting expertise!

Is Geometric Algebra the Grassmann Formenlehre?

As wonderful as GA and its derivatives are  I do not yet think so! The Methodology and Systemology of the AusdehnungsLehre is probably better evolving into the area now called Homotopy.

The lineal Symbol was chosen by Grassmann not as a line, but a representation of both intenive and Extensive magnitude. Thus his methodology can be applied to spaciometric topologies or to spaciometric metaphors of intensive experiences . That is : the topologies we choose to locate in our experiential continuum both inside and outside the boundary of our skin or proprioceptive sensors.

For induction or recursion read fractal iteration .
Z= Z+C is Mandelbrots famous definition of " almost similarity" . This inductive thought pattern underpins all Fractal geometry so called and speaks to the dynamical nature of space-time or the aether.

To go from there to Spinors( Twistors) and n-dimenional Fourier reference frames is perhaps a surprising connection to some, but reflects the triumph of the Grassmanns labours and research into putting human philosophy onto a "scientific " or rather Hegelian basis!
I do not think Justus subscribed to Hegelian philosophy, but Hermann certainly embraced it and probablynRobert did too to a lesser extent since he had his own agenda to promote.

Do not get too worried bout the technical detail! As many researchers will tell,you after years of labour" Hermann has already sweated the hard stuff!"

[jehovajah]

One of the main readons I am diving in to the technical detail and background of the Ausdehnungslehre is because historically Hermanns plans to produce a second Volume that I have dubbed the Schwenkenlwhre were thwarted by events. Yet much of the material survives in a reworked form in the 1862 redaction produced in collaboration with a sympathetic editor who nevertheless imposed his own slant on the finished product : his brother Rovert!

I have written before about this collaboration but suffice it to say Robert saved Hetmann and the family nme fom obscurity ! Hemanns reprint of the 1844 version in 1877 with intense Anotation additional appendices and back references is an attempt to correct this after the material became critically significant. Fom these notes and the sketch of the Schwrnknlehre I hope to come to understand Hermanns original thought patterns with regard to arc segments as lineal symbols .

My suspicion is that the general theory especially the Elementar presentation extends simply to this arc segment description of rotationlal extensive magnitudes but not so intuitively to rotational intensive magnitudes( whatever they may be in our experiential continuum) ! I suspect that spiral forms fulfill this rotational intensive magnitude concpt, thus requiring both the arc and the lineal segments to describe.

In my opinion thebarcvnd lineal segments together form the fundmental basis for representing spacetime / the aether, both extensively and intensively . It is possible thereby to model an inherent computational consciousness in spaciometric dynamics.

[jehovajah]

http://m.youtube.com/watch?v=_AaOFCl2ihc
This is such a bad history of what happened followed by a nice explanation of a Clifford Algebraic form that I posted it to call it out.
The most general algebra is the lineal Algebra of Hermann Grassmnn . Hamilton acknoledged this while Gibbs admitted he had no idea what Grassmann was communicating generally! Bill Cliftord became an avid Grassmann student all his short life! Hamilton,,though not openly abgrassmann student nevertheless recognised Hermann as his master.
Lord Kelvin, Llewis Carroll and many others took up a jingoistic campaignn aginst the Irish mathematician and used Gibbs as a pawn in his game . Heaviside merely reworked Maxwells equations developing his own vector Analysis from Hamiltons and Maxwels as he went along.
Kelvin also browbeat Maxwell into publicly reversing his high opinion of the Quaternions, forcing him academically to rewrite his equations in the kelvin Gibbs vector notation !