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Author Topic: The Ausdehnungslehre of Hermann Grassmann 1844 reprinted in 1877  (Read 3867 times)
Description: Hopefully a steady translation of the whole 1844 version over time.
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jehovajah
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« Reply #60 on: April 18, 2016, 03:01:14 AM »

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

This series of videos is interesting because it reflects the application of the Mechanical principles as derived in classical times particularly by LaGrange and D'Alembert . Thus it is directly the kind of systems Hermann was studying while applying his Frderung!
The matrix presentation is conceptually derived fom Cayley and others, but that contribution would not have been so well received had not Justus Grassmann tackled the combinatorial problem of symbolic arithmetic!

The combinatorial problem is how to organise the large number of equations that describe any mechanical system on the page! These considerations apply to any arranging of symbols on the page. We see that the summation sigma becomes useful to organise these procedural combinations. Basically these summations are arrays of staus/condition markers.

These arrays as combinations appear at all levels of organisation, and Justus particularly derives it from studying the presentation of mineralogy taxonomies and classifications.

So the arrays are fundamental to the description. Hermann chose to use summation notation, Cayley decided the arrays themselves were more accessible.

For presentation the matrix notation is a great boon  but the content of the presentation was worked out before Justus and Hermann and Cayley. However it is Justus and Hermann and all Ring and Group theorists(Lee, Abel) who highlight the combinatorial layout of the symbols on the page

Hermann took it to be alongside a geometrical derivation in association , but later realised it was a combinatorial methodology for mathematical thinking about Foms! The line segmnt then became symbolic for a directed magnitude of any form or intensity.
« Last Edit: May 28, 2016, 11:48:47 AM by jehovajah » Logged

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« Reply #61 on: April 18, 2016, 07:12:24 AM »

One puzzle in the translation of the Vorrede immediately becomes clear. Newton when using a line as representing a force in general resolved it into its sine and cosine components! These components were orthogonal lines which are related to the original line by a vertical projection . This Senkrecht line is responsible for the colliding product concept in which the resultant of any two general lines is an interleaving sum of the two pairs of the projections.
Thus two lines a,b, sum to a + b which is acos + bcos\theta + asin + bsin\theta
And the product of the two general line segment the area of the parallelogram derived using the orthogonal components in an interleaving fashion !
(acos + bcos\theta)(asin + bsin\theta)-(acosasin + bcos\thetabsin\theta)
Which can be rearranged and simplified
« Last Edit: May 28, 2016, 11:58:34 AM by jehovajah » Logged

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« Reply #62 on: May 16, 2016, 10:49:40 PM »

Thanks for mentioning this. I took the freedom to download the original, so I can read it in the original German.
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« Reply #63 on: May 28, 2016, 12:07:38 PM »

Thanks for mentioning this. I took the freedom to download the original, so I can read it in the original German.
Your Welcome.
As you can see I am stuck at the moment by real workd problems but am itching to progress the transation/ exploration.  This thread is for all who want to translate the originals into English and to discuss or comment on Hermanns thinking process, so please contribute your translation or interpretation, if you can .
As you may notice my German is amateur but good enough for me . I wrote Kannenberg on one forum but he has not replied to me. His translation is the currently accepted standard, but panders too much to modern thought for my liking xxx
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« Reply #64 on: August 24, 2016, 09:09:03 AM »

My hope is to translate Justus pamphlet " To the Mathematics and Nature Patrons"  first so that I can understand how Hermann organises his combinatorial thinking or thought forms ,
However I would encourage anyone who has a passion for Grassmanns 1844 Ausdehnungslehre or questions to post them here.
Any translations also are welcome , in whole or in part.
While Hegels philosophy is crucial to Hermanns larger objectives for shaping his work, discussions on Hegel himself should be posted in the philosophy section. Hegels dialectic process as it applies to mathmatics in general would be appropriate here also. To my knowledge only Karl Marx specifically wrote on the subject, although Bertrand Russel and AN Whitehead might well be said to comprehensively tackle that subject.
  cheesy
Sometimes I forget to point out the relevance to Fractalers!
At a fundamental level programming and function / procedural forms derive from the Ausdehnungslehre. Turing ,Dirac, Russel,Mach, Peano, Levi,Ricci were all directly influenced by the ideas in this book and those developments from it.

So anyone who is writing an app might be able to implement the Grasmann 'arithmetic" directly.
I know one person has done or attempted this in Mathematica, but no one has stated they have designed a fractal generator based on these principles . They probably exist but an explicit list of them would be handy. .
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« Reply #65 on: September 22, 2016, 02:18:37 AM »

When bishop Berkely asked what a Fluxion was he did not intend to elucidate the answer but rather to defend the Faith. Amongst the faithful he caused a clerical split: those who supported ate British view and therefore Newton, and those who supported the European view and therefore DesCartes school . That school was revolutionised by the French and so a godless mechanical philosophy based on rational logic and logical propositional mechanics / mathematics of the astrologers was championed in Europe . The clergy was thus under pressure to defend the traditional view of the faith.
Newtonnwas seen as the head of this particular snake and he was ruthlessly attacked in his own time.  Despite the French, and the British and Kant defending and promoting Newtons concepts it still took 600 years for him to be recognised as the father of modern Astronomy.

So Fluxions were confused with European infinitesimals by some Leibnizians , whereas the Newtonian infinitesimal was not a number at all but a moment in time.
A Fluxion was a dynamic quantity which had a temporal property, and It was represented by a line in dynamic flux. .

We might well regard Herakleitos as the progenitor of this dynamic conception, but it is perhaps the Pythagorean school that codified the dynamics of the line segment as part of a rotational system. However the western redacters of their work imposed a static sense to their philosophical discourse in the Stokeia. Newtonnwho read the original derived the pure dynamics of the Greek thought and carried this through into his own works.

It is in his papers on motion that he introduces the infinitesimal time stamp so characteristic of his notion of Fluxions.

It is certain that Newton intuitively worked in a lineal,algebraic way, but he despised revealing his surmisings in algebraic notation, something Wallis pleaded with him to do.

So it should be no surprise that the Ausdehnungs gre are Fluxions. Certainly by the time Justus had presented his combinatorial findings to the Nature and Mathematics lovers societies , the dynmic in nature as symbolised on the page  were being set as a true foundation .

Hermann Grassmann was inculcated with these ideas and viewed the Strecke as a jostling entity. These dynmic entities arevNewtonian fluxions
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« Reply #66 on: September 22, 2016, 10:47:24 AM »

Fluents are Newtons intensive magnitude or quantity associated with a Fluxion. So for example displacement is a Fluxion in a dynamic system . The fluent is the velocity or speed of the displacement.
Or mass change isbanfluxion in a pressure dynamic, then density variation becomes a fluent in fluid dynamics alongside rotationnshearingband general stressing of a fluid volume.

If a Fluxion is combined with a Fluxion the result is a Fluxion . Hermanns  line segment process and notation highlighted this lineal algebra

The product of fluxions was dealt with by Newton as resulting in a Fluxion of a different magnitude . Thus a linealnfluxion becomes an areal Fluxion when in product with another Fluxion. However it was Hermann that dotted the I 's and crossed the t!s here, and really exponded on the role of product design .

Several fundamental products were systematised in his process or method .

Newton dealt with the fluent of a product of Fluxions. This often causes debate amongst those who use the differential geometry the dynamic differential is different . The Fluxions modified by its temporal property not its spatial one..
Thus if a,b are fluents of X,Y then bX + aY is the fluent of the Fluxion product XY .

The the product of fluents is discussed in higher order fluents , but suffice it to say that the product of fluents is a vanishingly small quantity. Whereas we effectively ignore it, in point of fact we fractalise it by applying the same product rule. The extended expressions so obtained of course are an infinite process of smaller ans smaller fractions. As such a process the principle of exhaustion may be justifiably applied and that is why we truncate it at the first 2 terms of the fluent.

We are not determining the area of the supposed parallelogram in determining the fluent .,we are determining the intensity of the fluxionnvariation or change and that is mostly in the gnomon to the parallelogram not in the corner or verticial form . This proportional relationship is why we can use the fractalnexpressionnover and over with certainty , something the proponents of infinitesimals in Europe failed to grasp.
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« Reply #67 on: May 21, 2017, 10:52:36 AM »

http://m.youtube.com/watch?v=-wokVaaGFgA

In this remarkable series Norman is at last able to present the general beauty of his rational trigonometry in the context of lineal algebra.
It has profound implications for teaching so called mathematics.
There is one misconception he holds on to and that is angles. He misconceives what an angle is, or rather he deals with the general nd pervasive misconception of the term " angle" .

The arc length is dealt with both by the Pythagorean school, Sir Robert Coates, Euler and indeed Newton and DeMoivre. It is a corespondence between the diameter( radius) of a disc and how far its centre moves along the diameter as it rolls on a flat surface. It is a ratio between non homogenous magnitudes where the eternal curve is compared irrevocably to a Rectilineal line by a process of point to point matching. The point on the curve is called the tangent point the Rectilineal line through this point and only through this singleton point is called the tangent line.

In terms of infinite processes this is an unimaginable definition! In pragmatic terms we accept it from a diagram as existing as drawn. There is no logical connection between the drawing nd the statement of definition. The drawing is always perisos( approximate) the definition artios( perfectly fitting)

So the arc as a magnitude is quantified by sub arcs, but the usefulness of these sub arcs depends on a correspondence or ratio( logos) that can be developed in an analogos( proportional) way.

Thus what Norman makes explicit is that ratio and proportion theory Eudoxus sets out in books 5 and 6 of Euclids Stoikeia.

While Norman rightly points out the infinite nature of certain ratios when reduced to a single standard form, it is the attitude that asserts we can ignore pragmatics that is questionable, and indeed misleading.

Eulers and Cotes theorem involve the cosine and sine Ratios, not infinite expressions of the same distinguished not as polynomials, or even power series , but as complete functions!

But here Norman demonstrates how the doctrine of extensive magnitudes leads to algebraic simplicity,uniformity and applicability to multiple dimensions AS directions.

It is the notion of orientations as dimensions that makes lineal algebra so powerful . In addition the direction of travel in a given orientation is fundamental to a specific conception of Dimendion in space. Dimension in general relies upon metrons by which quantification processes are established. So we have dimensions of space, mass and time, and dimensionless quantities like angle which are ratios. When I draw an arc length, I identify a dimensionless quantity called annngle or a corner. It is dimensionless precisely because it is encoded by a ratio, and recorded in ratio tables of approximate "Values" or results of a division process.

The tables have always been exact(artios) or approximate( perisos) results of a difference process called division. The power series expressions of this difference process or division provide a systematic way to find these differing values, and should not be used to replace the underlying ratios.

In the modern world of driving nd the McIntyre world of gears arc length is important, but for construction the sine tables have been specifically calculated to help engineers pragmatically. It is only through the discoveries of men like Ōrsted, Ampre Faraday and Boscovich that the long held belief in the vorticular motion of natural powers has established the applicability of the sine ratios to these invisible yet powerfully manifested phenomena.

As part of this approach to mathematical description it is imperative to understand that circular arcs are legitimate extensible magnitudes that describe rotational dynamics in an algebraic way set out in the Ausdehnungslehre
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« Reply #68 on: May 24, 2017, 06:16:58 PM »

Hallo Jehovajah,

thanks for the link to Norman Wildbergers lectors on trigonometrie.
It comes very close to what I have worked out on Clifford Algebra on my geometric algebra page.
http://www.wackerart.de/mathematik/geometric_algebra.html#clifford
Which is inspired by the actual discussion in the geometric algebra section here in fractal forums.

Hermann
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« Reply #69 on: October 14, 2017, 10:47:59 AM »

I have not abandoned this topic,
Failing eyesight and personal happenstance have caused me to concentrate my reserves on what is manageable and effective.

Now I know that Justus Grassmanns work is crucial to apprehending both Hermanns and Roberts works, which indeed were carrying his torch forward to a greater audience. .

I hope modern technology will help me compensate for my disabilities and provide the topic with a useful contribution of translations and insights .

I invite those interested to contribute in a similar manner so that we might all learn together the Presentation of these ancient topics in a modern way.
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« Reply #70 on: October 15, 2017, 08:00:59 AM »

<a href="http://www.youtube.com/v/f81crFAj5UE&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/f81crFAj5UE&rel=1&fs=1&hd=1</a>

http://m.youtube.com/watch?v=f81crFAj5UE

Here Norman clearly sets out the inductive step that underpins the Ausdehnungslehre .
The first point is historically the bits and pieces lay scattered throughout geometrical experience. What Justus Grassmann did was to align the whole onto a consistent combinatorial basis. It is this combinatorial presentation that allows induction to flourish in defining more complex stages( stfen) .

It is these stages that are inductive which allow iteration or recursion to be inherent in the construction of notions, notations and processes and procedures.

Thus we embed, by the combinatorial method identified by Justus, but not invented by him, the fundamental concept of Fractal Geometry : almost self similarity at every stage!

When Benoit pulled together his intuition based on visual forms he repeated the same practices of the ancients but more boldly and bravely than his teachers, who had become trapped by convention and dread of getting the answer wrong!

The ancients appreciated that which could be exactly described and that which was forever approximate. They did not fear the approximate, but of course they venerated the exact. But the real dynamic mechanical world relied on applying exact concepts approximately, intuitively and iteratively. The principle of exhaustion deals with that issue, and thus with fractal geometry as a pragmatic expertise, obscured by exact aesthetic rhetoric, and defensible logics( systems of debating or argumentation to defend a presentation).

What is truth? One hopes that logic would capture it, but must realise that logic itself is not truth!
One would hope logic would be true, but realise that it is as bent as the foundations it is presented upon!

So, as impressive as Aristotles linguistic and grammatical rules approach is, it is as flawed as the persons who utilise it in isolation from their ensemble of senses. The Pythagoreans worshipped the Musai, they venerated all the gifts and arts and senses they identified in any gifted individual willing to develop their gift into an expertise.

The method of training was not by examination but by Koan, by overcoming a task identified as truly given to you by the Musai. You were welcomed into the seniority of the Pythagoreans not by some written exam pass but by being clearly capable, knowledgeable and expert in the gift given you by the Musai. Then as recognised Mathematikos you could discourse with the other gifted Mathematikoi and mutually enhance one another's expertise.

Fractal geometry offers that opportunity, in a way the old Academic subject boundaries doesnot. It clings on to life and dynamism, not dry dead bones however beautifully arranged.
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