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 Author Topic: The Ausdehnungslehre of Hermann Grassmann 1844 reprinted in 1877  (Read 7851 times) Description: Hopefully a steady translation of the whole 1844 version over time. 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « Reply #15 on: March 26, 2015, 10:22:32 PM »

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction  ofthe simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§14  The extending magnitude representation then, only comes to seem as a simple entity  , if the varyings, which varyings the creating whole element outwardly suffers, are continuously becoming set like one another ,

So that  therefore,
if
through  one varying, out of an element a another element b henceforward is going , both which relate to that simple extending magnitude representation,
then next ,
through a like varying, out of b an element  c of the same extending magnitude representation becomes created whole,

then also In addition  this like quality  indeed comes to be having a place to find!

If a and b  come to be apprehended as continuously alongside one another bordering elements,  this like quality by considering the continuous creating whole should by passages there  find a place.

We can name such a varying a fundamental varying, through which out of one element of a continuous thought pattern a nearby alongside bordering entity come to be created whole,  and then we can come to be declaring :

Quote
"the simple extending magnitude representation let be such a representation which through continuous continuing forth of the same fundamental varying goes henceforward"

Now In the same sense , in which the varyings become like  oneanother set, we also become able to like set the "therethrough created whole" representation  , and in this Sense, specifically that: the "through like varyings" entity comes to be self like set upon the same "cognisance created whole" entity , we name the simple extending magnitude  representation of the former step "an extending magnitude" or "an  extending (magnitude) of the former step" or "a line segment".

Therefore the simple extending magnitude representation comes to be related to  the "extending magnitude"   , if we look away from the elements, which elements the former enholds, and hold fast only the artform of the creating whole Activity;

and then, while    two extending magnitude representations can become  oneanother like set only if they the same elements enhold,

Then thusly  two "extending magnitudes"  are already created whole on like manner ( that brands, through the same varyings) , even if also 'they are created whole without the same elements to enhold.

At the last The totality  of all line segments,  which through  continuing forth of the same And  of the"running  into against set"  fundamental varying of the same are createable whole, we name a System. ( or a field of study) of former step.

All The line segments  related to the same system of former step therefore become  created whole through continuing forth, either of the same fundamental varying, or of the "running into against set" fundamental varying.

Before we go over the knitting of the line segments, we want to everyway alongside show the set  down labels in the previous § through the application to Geometry.

The like quality of the varying cognisance comes to be here through like quality of the direction everyway representing;

therehere, the unending direct line represents itself here as system of former step  , as simple "extending of former step", the bounded direct line. What there  would be named as a like artform, appears here as parallel, and  the parallelism  features like case way its two parallel sides as parallelism in the same and in the" rinning into against set" sense.  We can holdfast The name of the line segment  in intercommunicant sense for the geometry ,

and therefore here under " like line segments",  such bounded lines we can everyway stand, which line segmnts have like direction and length.

Footnotes
• the abstract assigning of this originating concrete Nomenclature requires  wholly no legal finishing permission, there the names of the Abstract originating all have concrete assigning .
•• I present  now the expression "field of study " before the expression " System",  which many times in other peoples sensibilities  Is like used
••• Thusly This distinguishing is important for the Geometry  , that it not least would be by consideration carried to the associate teaming of the geometrical statements and demonstrations , if one this distinction through simple Nomenclature fixes, where to I would be pleased to thump forward some type of expression around " like running"  and "against running" .
 « Last Edit: April 13, 2015, 05:20:46 AM by jehovajah » Logged

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 « Reply #16 on: March 26, 2015, 11:09:26 PM »

Commentary §14

This was very difficult to piece out, and I am still working on the footnotes,

The if, then then also structure took some finding, but it was important to show how Hermann considers every case.

The important statement about overlooking fundamental elements is important to understand. While Hermann synthesises from the ground up he also synthesises from the top down. So when a constructive or logical difficulty occurs by an Aristotelian analysis, he can look beyond to the higher goal and impose conditions from the top down. This is the detail of the Hegelian Dialectic process.

You do have to think about this carefully. Is he constructing in a manner that is basically unsound? Or is it pragmatically a brilliant solution to move forward?

The two § 13 and 14 are a detailed setting out of his method, using labels. The interpretation in a geometrical sense is but one interpretation! Therefore one has to attempt to grasp the generalities he uses as labels. Here he concludes that his definition or work through of the line segment label is safe enough to use for geometry providing certain distinctions are fixed and held firm, and these seem to be parallelism and direct lines. However the generality of his set up allows me to establish arc segmnt , circular arc segmnts , both concentric and around different centres but with same radius as geometrical candidates.

Again note how direction, that is orientation and translation are his general varying models / labels.

When he states his aim to make the extensive representation simple by setting all varyings equal or better like, he ie establishing an equally segmented geometry in every orientation. . But the building of such systems have to find a place in reality. So a crystal may grow along to axes but if it is to grow in thethirdaxis it has to be the planar crystalline form that is extending in this 3rd direction. No one has seen a single row of molecules forming a third axis!

This is the physical meaning of subjugation: a plane is subjugate to a three dimensional art form or its development, so it is practical  not to get hooked up on the axes and to focus on the " higher" step form.
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 « Reply #17 on: March 27, 2015, 10:17:41 AM »

Commentary §§13-14

Well if you are feeling spaced out by these 2 paragraphs, join the club!

I think this morning I have a better insight on them than I did last night, when my brain refused even to read the translations!

The labels are really where we start, but by way of introduction Hermann starts with the " point " label. But almost immediately he replaces that with the representational extending form! This is the most general idea or form he can express.

This form actually has 3 interchangeable expressions. : the extending magnitude is the mst general metrical one, but this relates to the previous or prior representation as being an extending( magnitude) of the first step! . Thus an n- dimensional extending magnitude is an extending of the first step from an n-1 dimensional extending magnitude..

If you recognise this as a recursive definition you will be on the right track.

The recursive definition starts with the third expression of the representational extending magnitude  that is the line segment. Hermanns definition starts with the line segment and builds from there.

And we know how the line segment is generated from the creating element the point.

Because of this general structure to his recursive definition he has to ensure that the next level variation " finds a place" in a real system. How do we check on that?

His answer is to look away from the elements that are varyin and look at the developing form. From this one can determine the next step variation, if it exists.

Thus for a cube, many of us would think: that can't happen. But mathematicians try to explain the hyper cube as casting a shadow that is 3 dimensional. While this is an ingenious attempt it is not happening, it is not real. What is real is a cube moving in space! Thus we can trace out the form such a shape makes, and it is this form that is hermanns guide to the next level.

There are several other alternatives: a cube rotating in space, or a cube expanding in space. The requirement for orthogonality would perhaps restrict these suggestions, but as you can read Hermann applies constraints to fit the representational extending magnitude into a real situation, not to limit the choices and freedom we have to imagine or design next level extending magnitudes.. Orthogonality is not the same as independent!

So before going on to the general notion of addition and subtraction, Hermann provides a demonstration of how to fit the general definition to a specific interpretation: in this case geometry. He shows how his general definition is applicable to geometry by use of parallelism in any and all oientations( directions) . Length is an afterthought:?it arises from the bound nature of the line segment. The fundamental system is undefined as to length, merely being remarked as " endless", and it is the line.

Notice how the line is built up from countless line segments, not the other way round.

As I pointed out earlier parallelism is one property that fulfills the definitions, the circular arc is another.

In passing, the simple extending magnitude representation is intercommunicant with the notion of a " uniform" representation. Indeed the word uniform means formed by the same measure in every orientation, and that measure is defined as unit,unity or Monas. All philosophers start with this uniform space or tablet and develop from there. I do not think one starts from the Fractal, non uniform space and attempts to build from there!

In the Fractal Foundations thread I attempted to do just that , and perhaps that is why I perceive the circle as the fundamental unit  not the square. With the circle, sphere or circular arc you cannot hide away the fundamental discontinuities in real space , or rather the need for many scales of circles to cover space! It is fractal all the way down!
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jehovajah
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 « Reply #18 on: March 27, 2015, 03:36:22 PM »

I have reemphasised both translations in the light of the last comment.
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 « Reply #19 on: March 27, 2015, 04:50:53 PM »

Hermanns way of checking the geometrical application  of his labels suggests the following :

Geometry----------------------------------------------------------------Fractal Geometry
Line segment : totality line.                                           Arc segment and radius ( circular arc) : totality circle
Parallrlogram: plane.                                                   Spherical : Trochoidal surface

Parallelepiped: space.                                                      Spherev: fractal space

Moving cuboid traces a surface.                                       Hypo trochoid, epi trochoidal surface of moving
sphere

And the first chapter now becomes about n-dimensional space models not just the straight line.

Norman does a great job using his affine  2-point line and a projective 1-point or a sloped line , but Hermann is not setting this up in these two §. He is setting up the general n- dimensional system , and illustrating it in the 3 dimensional case.
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 « Reply #20 on: March 28, 2015, 02:29:12 AM »

This video should sound like gobbledegook. However I would like you to retun to it from time to time to see if it becomes more sensible.
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 « Reply #21 on: March 28, 2015, 02:50:03 AM »

This video should sound like gobblrde gook. However I would like you to retun to it from time to time to see if it becomes more sensible.

Actually that made a lot of sense.  Tensors are a more generic form of vector transformations.
Like scalar and dot product are just Tensors is actually very enlightening.

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 « Reply #22 on: March 28, 2015, 07:05:36 AM »

This is advanced in that it is dealing with some topics in chapter 2 of the Ausdehnungslehre, but it is worth beating in mind that these weird discussions ultimately derive from Peano's use of Hermanns ideas ,

Compare with §§13-14 to see if you can pick out any like things.

When you have done that several times then allow yourself to think how much more general Hermanns treatment is than these!

Which presentation is the easier to follow?
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 « Reply #23 on: March 28, 2015, 07:57:07 AM »

Again this is advanced in the sense it refers to work in the 1862 complete version of the Ausdehnungslehre, but this lecturer has captured something of the Förderung that Hermann uses to develop his outputs from. The expressions on the board are general rules, whimsical and seemingly arbitrary but it shows how these expressions arise out of our psychology, our interaction with space, the system which naturally fits the topic under discussion!

The calculus argument is simply an approximation argument , what Newton called the fluent of the Fluxions in space. These fluents all derive from the binomial series  expansion and are all approximations.

The idea that these things "occur naturally" is Maths speak. Hermann demonstrates how much philosophical background preparation or a priori preparation is required to get this to happen. That is what Ausdehnungslehre 1844 is principally about! What mindset or Förderung makes all these expressions easy and simple to derive in a lecture, or teaching system? F

inding these cognisances were Hermanns life's work. His aim was singlehandedly to do all the hardworking, sweat the details, wander in the dark, take all the risks to find the best way to put the reader in the stadium position! As the observer the reader has greater freedom greater overview, and can feel like the lord and master of what he surveys! But that is only(!) because Hermann has worked to put him in that position..

The reason is historical , and relates to the Humboldt educational reforms. Hermann wanted to help his Prussian nation and people to become self actualised individuals, so they could take their place on the worlds platforms equal to the French and hopefully surpassing them.

What does that mean for us today?

We can benefit in the stadium position, but someone or some group needs to continue to do the hardwork, to use the method outlined in the induction to further apply his method and insights and heuristic style and label and product design goals and constraints.
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 « Reply #24 on: April 10, 2015, 10:29:07 AM »

I am impressed with this lecturer and recommend his course here, and his style.

It is a bridge between Hermanns style and the modern style now ubiquitous. However I am researching Hermanns thought patterns warts and all, and the more general approach Hermann adopts .

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 « Reply #25 on: April 10, 2015, 10:55:44 AM »

@flexiverse.

Thanks for the contribution and the insight.

At the moment I am looking at Justus Work, but the thread on the place of Quaternions revealed to me that Tensor is from the Italian Tensoré not directly from the Latin Tensorus. This means that Peano translated Ausdehnung directly into Italian as Tensoré, but Hamilton was discussing the stretching of the hypotenuse in a right angle triangle principally. Thus Tensorus relates directly to the tension in this stretching length in both cases but Hamilton focuses on the right angled triangle as defining the length, whereas Hermann and Peano are labelling general spaces that extend in whimsically chosen directions in whimsically chosen oientations.

The vector label was adopted From Hamiltons work and obscured the familiar line segment label. Because of convention vector has taken on an algebraic notational meaning, which is more natural for Algebra and number lists, and a bit limiting for geometry, or space intuition.

However Pavell has tackled these issues in an Impressive style reflecting Hermanns own style helpfully.
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 « Reply #26 on: April 10, 2015, 11:16:55 AM »

The totality of a creating element for a system of a certain step/ stage / rank is now conventionally called a span! While this may help some to recognise what Hermann is referring to, my point is that Hermann has a reputation of being obscure, which I deny. It is because he is crystal clear that his contemporaries felt he was not subtle enough!  But by being this clear one can see the analogy wherever it crops up.

Pavell is very good at drawing this out, even if the examples are in set language and described in general set language.
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 « Reply #27 on: April 16, 2015, 03:32:32 PM »

Ausdehnungslehre 1844
Section 1
The Extending Magnitude

Chapter 1
Addition and Subtraction  of the simple extensive/ extending Magnitudes of the former Step/ rank/ Stage or the Line Segments

§15. If the continuous creating whole of the line segment comes to be thought interrupted within, broken underfoot in its waygoing  , then around hereconcording, continuing forward again to come to Be, thusly appears the complete line segment as knitting of 2 line segments, which  continuously  close themselves up besides one another , and from which the one appears as continuing  forth of the other

Both the line segments, which  the limbs of this knitting build a represention of are " in the same sense" created whole(§8), and the output result is the knitting from the beginning element of the former to the end element of the latter, if both continuously  in one another are " laid" ,that brands, are so presented that the end element of the first at the same moment is the beginning element for the second.

Forward running we besigned the line segment from the beginning element $\alpha$ ( compare with fig.2) to the end element $\beta$ with [$\alpha$$\beta$],  [$\alpha$$\beta$] and  [$\beta$$\gamma$] are in the same sense created,
Thus is  [$\alpha$$\gamma$] the output result of the above indicated  to one side knitting, if  [$\alpha$$\beta$] and  [$\beta$$\gamma$]  are the limbs.

We have already (§8) concordingly demonstrated above,  that this knitting, there it represents the unioning of The" in like sense created whole" magnitudes, as Addition, and their intercommunicant analytical unioning as subtraction apprehended must become,

And therehere all rules of these knitting artforms for it empowers.

We have here still only the centrally acting assignings concordingly  to demonstrate, which the negative magnitude upon our field of study achieves. Specifically , around the initial , the assigning of the subtraction Side by side like showing to make to us , thusly  we can thereout that

[$\alpha$$\beta$]  +  [$\beta$$\gamma$] =  [$\alpha$$\gamma$]  exists,
Thus directly   [$\alpha$$\beta$]  and   [$\beta$$\gamma$] in like sense created are,  we can the conclusion draw, that plainly thusly  generally

[$\alpha$$\beta$] = [$\alpha$$\gamma$] – [$\beta$$\gamma$] exists( compare fig.2)
Therefore, that brands, if we now to us the In the subtraction usual nomenclature serve up,

Quote
" the remainder is, if  minuend and subtrahend  with their end element one lays onto one another, the line segment is from the beginning element of the Minuend to the element of the Subtrahend"

One sets in the latter format  $\alpha$ and $\beta$ identical, thusly one outwardly holds
[$\alpha$$\alpha$] = [$\alpha$$\gamma$] – [$\alpha$$\gamma$]

That brands, like Null!

Further is everyway pleasing of the label of the negative line segment.

(–[$\alpha$$\beta$]) = 0–[$\alpha$$\beta$] = [$\beta$$\beta$]–[$\alpha$$\beta$]= [$\beta$$\alpha$]

That brands, the line segment [$\beta$$\alpha$], which to an other [$\alpha$$\beta$] its label concording to (§13) running into against set is, appears also in its relating to the Addition and subtraction as the running into against set magnitude to that one.

There at last now. a + (–b) =  a  –b exists.,

Thusly one has , if [$\alpha$$\gamma$] and [$\gamma$$\beta$] in the running into against set sense created whole are

[$\alpha$$\gamma$]  +  [$\gamma$$\beta$] = [$\alpha$$\gamma$]  +(–[$\beta$$\gamma$]) = [$\alpha$$\gamma$] –  [$\beta$$\gamma$] = [$\alpha$$\beta$]

That brands, if also both line segments are created whole in the running into against set sense, their Sum is the line segment from the beginning element of the first to the end element of the second continuously laid besides them.

And we can this result with the above grabbing together summary, declare

Quote
" if one  knits together 2 like-artformed line segments continuously, that brands so knitted that  the end element of the former becomes the  beginning element of the latter, thusly the Sum of both is the line segment from the beginning element of the former to the end element of latter.";

And in which entity it as Sum is besigned, thusly  it should lay therein expressed, that all rules of addition and subtraction for this knitting style empower.

Still I want to herebesides have a Following to close, which for the wider development,   abundantly fruitful  is, specifically that , if the bounding elements of a line segment in the same System themselves both around a like line segment vary , then the lying between the 2 new bounding elements line segment to the former is like.

In practice let the originating line segment be [$\alpha\beta$] ( compare fig3) and [$\alpha\alpha$']  = [$\beta\beta$ ']

Thusly is to show, that if all named elements are related to the same system
[$\alpha'\beta'$]  = [$\alpha\beta$]  be.

But it is [$\alpha'\beta'$] = [$\alpha'\alpha$] + [$\alpha\beta$]  + [$\beta\beta'$]
concording to the definition of the Sum,

And there
[$\alpha'\alpha$] = –[$\alpha\alpha'$] = –[$\beta\beta'$] exists

Thusly heaves up themselves [$\alpha'\alpha$] and  [$\beta\beta'$] by considering the Addition, and it is really by working through
[$\alpha'\beta'$]  = [$\alpha\beta$]

Footnotes
•This besigning of the line segment is Only for a forward runnining line segment. The enduring besigning of the same through its boundong elements can firstly become everyway standing, if we the knitting of the elements  have learned to Recognise  to be( see the second section § 99)
The besigning  [$\alpha\beta$] is chosen in the Ausdehnungslehre from 1862 for the product of both elements $\alpha$ and $\beta$ , which  if [$\alpha$ and $\beta$ are points, represents the line part between $\alpha$ and $\beta$, whererom the line segment therethrough distinguishes itself, that in these ones only Length and direction, but in those ones at the same time, the position and unending-like, direct line comes to be held fixed, to which the line part is relating.

Therefore it is here around, thusly more therebesides, to firmly hold, that the besigning of the line segment through $\alpha\beta$ only a forward running Aide is, the material measured besigning $\beta-\alpha$ can concording to the principle of representing first in §99 com to be given. (1877)
•• above all, compare here  §7
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 « Reply #28 on: April 18, 2015, 08:46:03 AM »

Commentar on §15

Once again Hermann surprises by the elegance and sparseness of his rhetoric. I had anticipated a lot of case by case examples. Instead he confirms that what he is putting aside is confirmed by the prior work in the induction, and that his focus on the analytical is the necessary and sufficient demonstration..

This calculus of labels is hardly ever taught, but it is a crucial Förderung to master. Later on it will be the background or foundation on which he builds some new tautological meanings  for the Greek symbols.
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 « Reply #29 on: April 18, 2015, 12:21:29 PM »

The figures referred to can be found here

http://www.fractalforums.com/complex-numbers/the-theory-of-stretchy-thingys/45/

I will attach them in this thread when I find the file on my computer.
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