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 Author Topic: The Theory of Stretchy Thingys  (Read 19217 times) Description: Ausdehnungslehre 1844 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « on: November 09, 2013, 09:24:20 PM »

So I want to discuss Grassmanns Vorrede and possibly go on from there into his Einleitung to the 1844 Ausdehnungslehre which I feel is about dynamic extendable magnitudes and the analysis and synthesis of these spatial dynamisms in an arithmetical symbolic way. That is, Grassmann defines, quantitatively what his subject is and then analyses and synthesises them.

http://www.drhertz.org/properties.pdf

This is a " complete list" of the axioms and definitions in the real numbers. This is what I was introduced to on my first analytical course at university, without any explanatory motivation. Of course I thought it was Algebra. Nobody told me it was philosophy in algebraic notation.

Now I know this too to be false. This was and is symbolic arithmetic. Algebra, for what it is worth is the Arabic name for " mind twisting rhetoric!" of course you would not put yourself through it if it did not have some use. The use is general and analogous thinking, dealing not with one specific calculation but the method or algorithm for all arithmetical calculations of that type.

However, this was a group theoretical presentation of the real numbers. It was not an investigation into the foundations of arithmetic. In the time of Grassmann, Justus his father and other pioneering group and ring theorists were attempting to rebuild mathematics from the natural and meta mathematical world in which they existed. Justus group of collaborators were early crystallographers, seeing in the dynamic world a geometrical order that produced crystals dynamically. They hoped by studying these crystal formations to understand how to construct a more applicable and relevant Arithmetic, or rather to extend the arithmetic of commerce into an arithmetic of natural processes.

Justus therefore analysed arithmetic and geometry and logic down to its nuts, bolts and washers and attempted to construct a mechano set of principles that built a sound mathematics.

He ran into difficulty with Legrndres redaction of Euclid, which everyone mistakenly assumes is about geometry, and with the concept of multiplication.

The multiplicative axioms are the problem. They unravel the whole of arithmetic because they do not logically come from considering geometrical forms!
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jehovajah
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 « Reply #1 on: November 10, 2013, 04:20:49 AM »

Most of us have played with bricks.

When you set up an array of bricks say 2 by 4  we know we can set that array in any orientation on the ground.  I also know that you would immediately sy that there are 2 ways to construct this array.

Well then , imagine the bricks scattered across the room. To construct the array in the centre of the room you could collect each brick individually and return it to the centre and carefully place it in contact with the previous brick if there was one. Once you got to 4 you could then suddenly place the next brick on the top or quarter turn side of the first brick. The sequence is extended,by carefully returning and placing the bricks next to the previous one and in contact with the " row" of bricks below one brick to one brick. You woul stop at the end of the bottom or first row with your last brick in the second row on top of the last brick in the first row.

In so doing you have created and defined a row of 4 bricks and an array of 8 bricks consisting in 2 rows of 4

The combinatorial product of your effort can be defined as a 4 by 2 array, where the 4 signifies the count of bricks in a row and the 2 signifies how many rows., and  the word array means they are patterned not scattered, and indeed they are as contiguous as possible.

Well that was 2 products of a possible 3 using that particular method. The third variation is to start the second row below the first in contact with the ground. The second was to stack the second row on top of the first row.

Well of course there are more methods of forming this array. Now we know what it looks like we could save time by collecting 4 bricks in sequence before returning to the centre to place them carefully.. We save on the repeated foraging, but now at the centre we have to decide which sequence to place the bricks in, and of course we have 4! Choices now.

There are many other scenarios that result in the same product but involve different choices in layout of the array. The "other" choice you may have considered traight away( why?) is where the trip involves collecting 2 bricks at a time and constructing the array 2 bricks at a time.

We still have several choices if we lay the bricks in pairs. We could for example lay the bricks in pairs in a row that totals four. But the one you are probably screaming out now is the laying of bricks in pairs, but carefully placing the pairs on top or to the topside on the ground or below the first pair on the ground so they are contiguous. The next pair is placed on top, above or below the previous pair until 4 pairs have been laid.

These are the pragmatic choices one is faced with when constructing or synthesising a mosaic array .

Now ,after all of this we are expected to note that the 4 by 2 array has become a 2 by 4 attay.. It should be clear that this is a rather meagre reduction of the combinatorial choices we have just touched upon. In addition we are supposed to assent to the term commutate as if it means nothing spaciometric ally. The fact is it is a symbolic , literal device referring only to marks or symbols on the page.

Giving our assent to it is all that is sought!  The methods and processes are distinct and distinguishable and do not say ab = ba!

To get to this statement we have to agree that row counts will precede column counts, thus 4 x2 means 4 in a row,2 rows.
2 x4 means 2 in a row 4 rows.

If I now apply this to the measurements of a single block say 3 cm laid out horizontally to me , left to right on the ground and 2 cm vertically on the ground, forward of me then I get

4(3) x2 for one method  and 2(3) x4 for the other arrangement

Equating these 2 formulae is equating the lengths of one side of the bricks! I have ignored the depths of the bricks.

Similarly I can write
4 x 2(2) and 2 x 4(2) to equate the other dimension.

However, now my notation has to be defined. The brackets are to signify the measurement of the fundamental brick in the direction of the row or column.

All this is usually finessed away. It is confusing to write it down in longhand and the student is not shown this instead they are expected to just pick this up from the demonstration and the exercises.

It is very tempting to ignore this detail, but it in fact is crucial to understanding multiple forms in symbolic notation. It was discovered or noticed by Hermann Grassmann, and it is the reason why logicians have fundamental difficulties with deriving multiplication

Let us use the full notation of this form now

4(3)x2(2)  can be symbolically equated to 4x2(3:2)

This format can now be defined to represent the foundation of multiplication of scalars.or ordinary arithmetical multiplication. The bracketed  pair are the dimensions of the brick and the notation relates them in a proportion or ratio format. The ratio format can now be defined as information about the single brick and representative of a single brick. The scalars now multiply to 8 the known numbers of bricks, and we have now captured the relevant information  for counting and identification purposes that should enable anyone to construct this mosaic array in any orientation in space.

2x4(3:2), and 4x2(2:3) and 2x4(2:3) are all different constructions, but the scalar commute in this precise sense. Not in the undefined sense hitherto taught.

Of course this is still a meagre representation of all the possible construction algorithms, and this is where Grassmann gradually realised that the stretch of these dimensions by an extensive method was very important to describing position in space. Of course he was not to know the Pythagoreans had already developed this idea.

There are 2 important things to note here: by removing Arabic numerals the discussion becomes more geometric and a direct link to the Arithmoi is almost obvious. The arrays of 1 are geometric Arithmoi.

Secondly, because Norman has defined the arrays to be. Row arrays he misses the Law of transitive distributiyity in the associative law. He in fact illustrates it with the circles, but fails to name it the wedge product.

It's real significance is in crossing the operator imposed boundary. The so called operations are combinatorial arrangements so
2.(3.6) = (2.3).6 = 3.(2.6) but
In vector group algebra  the wedge (2+3) can be scaled  6(2+3) and this is the same wedge formation as (6,2 +3) or (2 +6.3) where the combinatorial+ joins the Strecken of different " lengths" into wedge forms which are rotated relative to each other.

The wedge form is a modern notation for the transitive distribution just pointed out. Because it is a proper multiplication it is confusing why modern notation writes it as a +

This is because of abelian group notation.

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jehovajah
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 « Reply #2 on: November 10, 2013, 08:15:45 AM »

The young Abel was active at the same time as Justus Grassmann was formulating his ideas.
http://en.wikipedia.org/wiki/Niels_Henrik_Abel,
There is every opportunity that Justus and Abel were involved with the same or overlapping groups of mathematical researchers looking into extending and rationalising mathematics. Certainly Justus referred to his work in mathematics as Verbindungslehre, and looked at a combination of Zahlenlehre and Kombinationlehre. At the time, Galois and others were looking into the solution to polynomial equations and the combinatorial relationships  of roots figured heavily in that line of theory .

A year before Abel died, In 1827 Justus published a paper on his ideas that caught Hermanns imagination. It was a paper setting out how numbers and algebraic notation could be combined to extend notation for Arithmetic. It is likely that it drew upon the same vein of ideas initiated by Abel.

http://en.wikipedia.org/wiki/Abelian_group

Certainly much of what Abel laid out appears in Grassmanns Ausdehnungslehre. What distinguishes Hermanns work is clear: while others were looking at the general polynomial group, Hermann was looking at Geometry. He realised that die Begränzte Linie could stand as a symbol for many magnitudes measured in geometry. Only later did he realise it could symbolise motion and displacements in der Raume or 3d space.

He knew that he needed to have combinatorial actions that represented addition and multiplicatio, from his Fathers paper, but he does not seem to have known about closure in a group. This was fortunate. It meant his ideas were unbounded by convention, and eventually he realised that he could subsume nearly all of mathematics into his methodology of analysis and synthesis.
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 « Reply #3 on: November 10, 2013, 09:44:53 AM »

The Wikipedia article has improved somewhat since I started my research . It could be I understand it more, who knows, but I can recommend it as an overview.
http://en.wikipedia.org/wiki/Hermann_Grassmann

I still Have issues with some concepts  said to be Herrmnns, particularly the "linear " or vector references. The purest concept of Hermanns thinking is given in his Vorrede, as we shall see, but lineal means just that, a line! Ausdehnung is to reach or stretch out especially an arm as in pointing or exercising. This is therefore a theory based on stretching out lines.

Mathematicians often like to talk about extending a method in a certain way to make it more general or applicable, and I am positive Hermann was aware of this meaning, which is why he specifically puts lineale as the adjective.

Another way to think of it is as a tailor stretches out his tape measure to get the dimensions of a suit , or an architect does the same to measure out the positions of his footings , they perform a lineal Ausdehnung. They measure Ausdehnungs Größe and record them in a way that enables them to construct or synthesise some product in space. Hermann simply recognised this everyday procedure as a mathematical method of analysis and synthesis. The next thing he did which was down to his fathers interest was define the underlying group and ring structure for this method. He therefore intuitively recognised points and lines as fundamental elements. By trial and error he isolated the begränzte Linie as his fundamental mathematical object, which allowed him to see lines and pairs of points as duals of each other.

The third thing he always kept in mind was the dynamic nature of these entities. None of his terminology was ever meant to be spatially fixed, everything was meant to be variable and dynamic.

When Hermann rejected 3d space, he did not mean he was aware of some other higher alien esoteric spaces. He was saying that in der Raume the 3 dimensions have never been significant for proper construction. Every person knows that measurements have to be taken in any number of orientations to construct a form . Thus a form determines the dimensions most useful.

At one time he considered calling his method Formenlehre, because of the importance of form. The spanning of a space by a basis actually goes against his general idea. The basis was to be revealed as the kernel or the least structure that accounted for every point and which every vector could be reduced to, but this often involved using coefficients that were not convenient. If one insisted on unit coefficients or integer coefficients then there were a lot more bases to play with!

The diophantine constraints meant that certain bases could be used to solve or represent those types of mathematics, and other constraints led to other types of "spaces". The space where the coefficients are continuous or smooth is usually what is meant by a "vector" space, and this space reduces to a basis of 3 dimensions, but these dimensions are orthogonal by convention. It is possible to use more generalised reference frames.

It became possible to generalise this method outside of geometry into node or graph theory and category theory. The methodology has proved to be so robust that many computer programmers have been able to use it to structure their programming languages and concepts
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 « Reply #4 on: November 10, 2013, 08:37:06 PM »

There are several crucial passages in the Vorrede where my poor German meant I was confused by the language construction. But my heart and mind immediately apprehended what Hermann was saying. These crucially were the Law of 3 points and the Law of 4 points.

The law of 3 points involves only 2 Strecken. It was hard to rid my mind of some inept training in vectors. I did not know then that Strecken were run of the mill constructs, just line segments. I knew they were lines on the page, I knew they were constructed or construction lines, I knew they were symbolic, yet I still could not grasp  what they were!

Later I managed to gradually isolate their significance, and to appreciate their utter utter simplicity.
Then I found the definition Grassmann gave in the Einleitung, after an intense dreamlike passage about how things may be outwardly different but inwardly they were alike, that distinctness could be used to collect things into groups of likeness, and that likeness can be used to distinguish groups of entities, and this duality between likeness and difference was a powerful human experience that broke down barriers. He said that entities that had these properties could be represented by die Begränzte Linie( Strecke).,the line which was constrained between points is a Strecke.  It's not just a line segment, it is a symbol of some discrete magnitude,,and every discrete magnitude could be symbolised by a line. The lines themselves could be distinguished by orientation in space relative to each other.

If I have 2 points A,B they constrain a line AB that starts at A and ends at B . The points if fixed are distinct from the line. The line is that stretchy thingy that stretches betwween the two points from A to B. Needless to say the stretch from B to A is a related but entirely different stretch. The negative sign signified this at a time when the negative sign was not understood as a rotation operator. In fact more than rotation is occurring. It is an entirely different stretch. The notion of contra, coined by Hamilton is most useful in marking this distinction.

However Grassmann was constrained by the use at the time and he was learning about the negative quantities at the time. What he noted was that commuting the points on the page represented a different stretch in space. Commuting the points did not switch the points in space, it switched the mental focus of the observer. The negative sign does not quite capture that distinction Hamilton in his essay on conjugate functions or couples describes how the notatin conducts the mind of the observer on mental journeys, switches of points of view, rotations of orientation. This he called contra as in the contra step. It was as if a mirror reflected the stretch from a point behind to the poin in front, as the stretch from the point in front reaches out to the point behind the mirror.

So Hermann noticed AB could be an identity with –BA, but unfortunately  he used the = sign. This creates a stronger affinity with arithmetic than Grassmann eventually realised he needed. In part it was this notation that held him back, and also made communication with others slippery.  His rhetoric is clear, but for mathematicians trained in symbolic interpretation the redefinition of symbols often went unnoticed. More attention was given to the + and= sign than the rhetoric which mitigated their meaning.

The 2 Strecken in the law of 3 points A,B,C are AB and BC. AB combined at B with BC gives a stretch AC. Grassmnn noted this notation was" true"  in 2 situations.: when C was on the straight line extended from AB beyond B and when C was on the segment between A and B.,

He wished it was always true that the segment AC was always given by the combination of the other 2 segments . But then he relished if he omitted the segment AC then the 2 Strecken always represented a stretch from A to C via B. this was the simplest way that it could be true.

You may struggle to grasp this realisation. AC is not the line segment AC it is the stretch from A to B and then from B to C..

But then he could identify the line segmentn AC wih this unique pair of stretches thus he could once again use an equal sign, but of course it would have been clearer to use and identity sign.

Again CA was the  contra of AC  and the negative switched the tems in the addition and commuted the letters within the terms!

It was the realisation of the law of 4 points that convinced Hermann he was onto something he could devote his life to.
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 « Reply #5 on: November 11, 2013, 03:21:52 AM »

The law of 4 points occurred to Hermann when he was studying some ideas of his dad about multiplication. As far as I know, Justus had difficulty placing multiplication in his logical and constructive " map"'of mathematics. In the end he resorted to a geometrical  definition of multiplication. Basically he said if you stare at the rectangle you will intuit that multiplication is soundly based!

I have explained the difficulty he had and why in the second post.  The difficulty arises from thinking numbers are real objects.  They are our response to a dynamic world and they are imaginary. The process of that response is called Logos, and it is a significant analytical concept discoverd by the Greeks , in particular the Pythagoreans

The Pythagoreans went further and compared logos with analogos, the basis of Eudoxus teachings in books 5 and 6 of the Stoikeia. However Justus had Legendres redaction of Euclid's Stoikeia and he felt therefore that Euclid was about geometry and his geometry was faulty logically .

Euclid's Stoikeia is not about geometry, but about philosophy, and it is an introductory course to Pythagorean philosophy. The real source of geometry was another subject called mechanics. Newton expressed this in his opening statements to the Principia.
Mthe artisan develops or discovers mechanical relations, tools that produce these mechanical objects. The skilled artisan refines these mechanical relations and objects in a meditative reflection called geometry .  This reflection abstracts and perfects these relations and records these ideas in symbolic notation.

The Pythagoreans went beyond this and looked at all ideas and forms in this abstract reflective way. This was and is the essence of philosophy. They uncovered invariant relationships at this level of abstraction and it is this discovery that intrigued Socrates. He playfully introduced the concept through the questions and discourses he had with his students amongs whom was Plato. Which one is reality? The sensory experiences or the meditative reflective experiences that reveal these invariant forms?

There is no answer. The listener must decide for themselves, but their decision will affect the whole course of their lives!

So Justus and many of his contemporaries who studied geometry through Legendres redaction found many logical inconsistencies not found in the Stoikeia, but introduced by Legendre 's redaction, which combined mechanics with the philosophical abstractions.

The problem with modern multiplication is that it was decided to base it on number bonds at some stage in the late 17th early 18th century. The renaissance men also failed to grasp multiplication from the quadrivium or Trivium in the seats of higher learning. The meaning of the Arithmoi was lost in multiple translations, and the consequence was the make do make piece called number.

So it was as if n a dream that Grassmann realised that the dynamic construction of a rectangle and more importantly the parallelogram  was the foundation of multiplication. He imagined the two required sides to be active and dynamic and they pushed forward past each other to create a parallelogram .,they also rotated relative to each other to generate different parallelograms. The crucial role of parallel lines in creating these multiple forms was clearly apparent.

By constructing a parallelogram Hermann saw he could construct multiple copies that tessellated space!  That was a vision of an extensive measure that was fractal . The pattern of multiplication spreads out into space a one side by the other side.  It also resides in the single constructed parallelogram.

From a point A the two segments fan out to B and C. Through B a parallel line to AC is constructed, the through C a parallel line to AB is constructed, they meet at D and a form is made ABDC

AB x AC  = ABCD

Not very revealing until a third Strecke is introduced into the plane . For simplicity sake let us call it AD

Now the behaviour that makes this work is the proposition that says triangles on the same base and between the same parallel lines are DUAL( isos) . It was also generally known that parallelograms with the same constraints were dual. Hermann realised that if he made the combinatorial process of multiplication represent the construction of parallelograms then he could use this dual property with different looking parallelograms on the same Strecke !

So now considering say AB as dynamically extending along the same parallel line as he first constructed, he knew that knew that if AC was fixed then all these different forms were dual! And similarly for AC  . The point D might move further or closer so that the parallelogram at a went from acute to rectangular to obtuse, but ABCD would still be the same or dual.

It is important not to jump to the conclusion that we are talking about area. The area is not defined. This is the shape we will utilise as a Metron to count space. It is the notion of counting space that we have lost by losing contact with the Arithmoi.  The word area is derived from the notion of counting monads in an Arithmos.mthe original Monas is this constructed shape.

Now we are restricted if we think Arithmoi were just rectangles. They could be any pattern that tessellated, and even those that did not tessellated perfectly were used. The best concept I have found is to think of mosaics.

The third Strecke allowed Grassmann further insight he decided to use this to define the rules of addition and multiplication. In the parallelogram the Strecken were symbolic of every Strecken in the constructed tessellated space. Therefore a Strecke which was located at a point H say could be considered as brought into or projected into the same parallel Strecke in the defining parallelogram. Similarly what was demonstrated in one parallelogram was applicable to all .

A parallelogram construction must have two Strecken fanning out from one point. They do not converge to one point. That signifies construction so that AB. AC means construct a parallelogram whereas AB . CA is undefined.

The introduction of the contra concept allows this second one to be defined as well as BA.CA the converging case.

If a Strecken follows another through a common point then this is the law of three points situation, so the parallelogram contains two different law of three points that result in AD as the identity.

AC + CD does not = AB + BD, but they both have the same identifying Strecke . Grassmann overlooked this and thought he had commutativity. He clearly does not but at this stage I cannot quite explain what it is. It is certainly some kind of projective or reflective duality. I even think that Klein named it a glide reflection but I not certain

Emboldened Grassmann used the third Strecke to demonstrate distributivity of the new product and this is when he received the greatest shock?

AB = –BA
AC = AB + BC and CA = – AC = –( AB + BC)* = –(BC + AB) = –BC + –AB = CB + BA

Note the * on the bracket. Something else besides normal negation is going on and it seems to involve this anti commutativity through and through.

Finally AB can identify AD + DB and AC can identify AD + DC

Assuming commutativity naively

These should all cancel but they do not!

The first pair are legal constructions so they cancel, but the second 2 pairs are not legal constructions.
Converting them to legal constructions we get
0 = –DA.DB – DB.–DA + DC.–DA– –DA.DC

0 = –DA.DB + DB.DA – DC.DA +DA.DC
Carrying the signs through transitively creates the problem the right hand side reduces to 0 , but the constructions were changed to achieve this so
The switching of symbols is confusing and misleading when mixed with the rules for signs. But he was determined to make it work so he eventually concluded that

This ensures that everything coomutes to give zero but obscures what is really happening, and it is Cayley who pins it down in his matrix algebra and multiplication. Transposition is key to the difficulty, but by grasping the nettle and defining ab= –ba Grassmann captured an important part of the solution. The anti commutativity and the sign change crops up again and again.

Grassmanns product was therefore confused and that allowed him to make connections he might otherwise have steered away from.

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 « Reply #6 on: November 11, 2013, 04:40:31 PM »

The complexities of creating a intuitive and useful notation or terminology are hard to express, but it should be clear by now that it is a very messy and arcane business. Much is assumed, much is neglected, and much needs to be refined by applying the notation and ideas.  Grassmann took years to cobble together a utilitarian notation. It was not as smooth as it is portrayed now. There are many logical flaws and holes, but he had astrong framework in Arithmetic to guide him.

His notation can lead to inconsistencies. However he responded to this by setting bounds for applicability. Gradually what emerges is many methods specific to certain spatial problems , but all methods initiated in a common group and ring theoretic framework.

Thus he did not create a mathematics,,but rather he created a methodology of analysis and synthesis.. Certain tools were more useful and evident than others , some were strictly limited to a class of problems. Working in the subject of geometry he single handed lay created by his methods and by trial and error a set of handling labels and mathematical entities thst were very useful. This was the lineal stretchy things! The Strecken

He had a law of 2 and 3 Strecken which encapsulated summation and multiplication in an extensive sense. He had a difficulty which he resolved by defining Strecken multiplication as construction of a parallelogram, but it was hazy with regard to commutativity. He defined a property or commutativity to cope with most of the issues in that part of multiple firms of Strecken . He intuitively felt that direction of the constructed forms changed in commuting the factors, but he did not know why. It just seemed to be a recurring theme.

,but he went on to demonstrate some quite remarkable proofs and to simplify many presentation. His method was not flawless, but it had very real consequences in terms of computation.

This set of affairs evpxisted with the roots of unity, where the foundation was flimsy but a lot of great results came out of using what little was known certainly, even if it was" false".

The anti commutative rule hides many years of work trying to figure out the flaw in commutativity. Their are many accommodations to the simple rules initially laid down before it achieved a secure role in Grassmanns methodology.

Grassmann recognised he needed fellow researchers, and critics, because the task was too great for him.
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 « Reply #7 on: November 13, 2013, 04:48:52 AM »

It occurred to me, while trying to step out auseinander treten and ineinander treten with my feet, grasp what  Grassmann was saying  about the Senkrecht Strecke in the next important stage of his arithmetic, that the stossender Strecken may not just be fanning out from one corner as I imagined. These parallel lines would project out of every point of the defining Strecken , principally the three points that define the two Strecken A,B,C.

From A would project AB and AC . From C would project CD, and from B would project BD. these Strecken would thus be stossender, crashing into one another at the points of intersection, the 4 points A,B,C,D.

AB//CD,AB=CD
AC//BD, AC = BD
And we could write AB.AC = CD.BD
And  AB.BD = CD.AC

Thus in the parallelogram direction and quantity of magnitude are inhered into the formal labels.
Going back to the law of 3 points I note Grassmann makes a point of apprehending the quantity of magnitude and the direction of each Strecke.

The equality sign is the misleading sign. The multiplication concept is not being extended it is being nalysed by Grassmann prior to extension.

If the forms above identify construction of a parallelogram, then the = should be replaced by —> to make a construction method.
AB.AC —>CD.BD

would then indicate that the sides are projected onto the following sides in that order by the parallel line projection.

Similarly
AB.BD —>CD.AC

Is a construction by parallel line projection. It is clear that the direction of the second projection has been reversed relative to the first construction method.

Commuting the 2 sides  should produce the same construction. We are not changing the orientation of the Strecke or their relationship in space. We are simply changing the order of notation

AC.AB —>BD.CD

It is clear that this projection maps divergent Strecken onto convergent Strecken

And

BD.AB —>AC.CD
This is glide deflection  mapped onto another glide deflection although the first structure is fractured by the commutation. But the non commuted structure mapped a glde deflection onto a fractured structural operation.

There is a sense in which the commutation has reversed some aspects of the construction process, but to define this Grassmann placed a third Strecke in the plane of the parallelogram set as structurally defined by these multiplication definitions..

Then he introduced the summation definition in the law of 3 points using the sides of the parallelograms to identify Strecken relationships.
To keep it simple  let AD be the third Strecke.

Now instead of construction forms being disallowed we can interpret them.

The constructions give a pair of parallel lines, a parallelogram ith in the original and 4 half completed constructions, whichever commutation of the Strecken is chosen

This shows that in the parallelogram the combinatorial form of multiplication is commutative as before, but when we relate to the triangle we get a anticommutative result!

AD.AD = (AB + BC)(AB + BC) = AB^2 + BC^2 + AB.BC  + BC.AB

Here we get 3 sets of parallel lines squared and 2 parallelograms.

Grassmann deliberately chose to make the parallelograms cancel! This was the beautiful result he was after, Pythagoras theorem. To get it he had to define anti commutativity,

I do not think he did it lightly , but knowingly and with trepidation. But as I have found it does not affect the commutativity of the parallelogram! By making this definition he had both the Pythagoras theorem, and the commutativity in the parallelogram where he needed it for calculation.

We can now distinguish the " fan" product from the " wedge" product. The fan product produces a closed construction of a parallelogram, the wedge product an open construction of a parallelogram , and these extend outside the original parallelogram. By adopting Grassmans definition we can swing the open ones inside the frame of the original so they are completed.
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 « Reply #8 on: November 14, 2013, 01:15:00 AM »

The fan product and the wedge product are isolated from the combinatorial product of constructing a parallelogram ABCD

Fan product
AB.AC

Wedge product
AB.BD

The fan product constructs the parallelogram by converging the parallel lines to point D in the projection by parallel lines. The fan product is defined only in parallelograms.

The wedge product constructs the parallelogram only by one of the projections being in the negative sense of the projecting parallel line this creates anti commutativity, but it has to be defined in the triangle form, where it creates the triangle as a Pythagorean form and cancels the parallelogram forms, by definition, not by overlapping. The wedge product can be defined for the triangle and for the parallelogram. In the parallelogram it does not affect commutativity, but it does introduce sign into the form.

This was not obvious to Grassmann all at once. By trial and error he refined his concept of summation and combinatorial product, and extended it to apply to 1 dimensional constructions and higher dimensional constructions. The wedge product suddenly defined a whole set of arithmetics of dimensional forms in a way that the fan pricy did not.

However the fan product came to have a role in Clifford algebras as we may get to later.

The next major development for his arithmetic was division.

He was working on tide calculations using trigonometric ratios, when he came across the hyperbolic functions. The revelation was that these functions were tabulated using the right triangle.

The relation of many functions to the right triangle is overlooked in the general introduction to graphing. In fact Descartes in his geometry gave no special position to the right triangle in analysing graphs, developing a generalised coordinate system. However the Greeks used the right triangle as a fundmental building block for relating curved figures to the circle, and rectilineal forms to the circle.

Bombelli is perhaps the first Algebraists to use the carpenters set square extensively to solve binomial and trinomial problems and then to extend to the quartile and quintic solutions, and sketching out how to find the roots of polynomials of degree six!

It was Wallis who simply fixed the axis orthogonally to gain immediate and permanent access to the Pythagorean formulary in deriving polynomial equations. He thereby showed the common form of the Parabola in its simplest expression, losing a lot of geometrical information but gaining the ability to formulate the conic equations.

The hyperbola therefore was defined as a function of the ratios in the right triangle, this meant that the exponential function could be mapped by right angle data points , and in so doing the link between the exponential and the hyperbolic functions was revealed. The ratios were similar to those of the circle and thus the terms Cosinus and Sinus were initially used . Eventually cosh and sinh was preferred

Grassmann realised the power of a vertical projection as opposed to a parallel projection.this gave him axes to pre calculated tables and meant he could always evaluate his algebraic foms. It also gave him access to stepwise rotation as the Strecken diverged by dynmic rotation. This he utilised in his Ebb and flow calculations, giving him access to swinging pendulums and other periodic motions. Finally he realised that by the Strecken projecting vertically onto each other or the Strecke extended he could define a new Strecke and the Strecken together could construct a smaller inner parallelogram, he had a n method of division and a set of tables to evaluate that division or fraction or ratio.

So he defined division of Strecken, particularly in an example to do with  hyperbolic  ratios. From this definition he was able to derive the square root of –1 as a division product of Strecken.

Grassmann now had a combinatorial ring based on these lineal Strecken and he used this confidence in the symbolic arithmetic to develop the lineal stretchy thingies as a powerful new geometrical language.

He needed help to further his research and development. He pleaded for help, but for -7 years non came. His book the Lineale Ausdehnungs Lehrer was eventually burned to provide warmth or damaged in storage in the ware house.

It was only rescued later by his brother Roberts driven intervention.

The Grassmann inner product is different to the usual inner product as we shall see.
Nomans establishment of division clearly shows the role of ratios in its description. However the real history of division is not as notmn has described it. This is a modern approach which does not recognise the fundamental role division played in the definition of the Arithmoi.
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 « Reply #9 on: November 14, 2013, 07:57:42 AM »

Flicking through an online copy I actually found some figures for the first time!
https://archive.org/stream/dielinealeausde00grasgoog#page/n315/mode/2up
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 « Reply #10 on: November 16, 2013, 12:58:20 PM »

So far, this is the best introduction to the fan and wedge products as I call them. The transitive distribution of the scalar over the wedge product was first brought to my attention here. Norman imply defines it and it looks different because of the wedge.. In fact it is a normal consequences of commutative  switching and associative bracketing.
2(AB)=A(2B) =(2A)B

It is the spaciometric significance of this switching which is fundamental, and the fact tha we have to define scalars and other entities to fully apprehend it,

Thus AB=0 is not a number product!

When the Grassmanns extended arithmetic to geometry symbolically they kept the arithmetical signage for the different entities. So the fan product and the wedge product look the same, but the geometry is different. Defining them to both produce the construction of a parallelogram is not to say that Grassmann ignored the differences. In fact the anti commutativity shows that he did not.

The fan product applied to three points and their 2 Strecken actually produces a parallelogram, but Strecken addition is defined by the wedge product. The wedge product produces a parallelogram only if the Strecken anti commute. This means that I can define the third Strecke by the wedge and then square the third Strecke .mthe combinatorial table produces combinations that have to be defined by the geometry and that's what gives anti commutativity. This does not happen with the fan product because no summation is defined for that configuration.

However, the fan structure has a use in Barycentric coordinates. A combinatorial form is defined by taking fractions of the fan Strecken such that they sum to 1. Effectively the fan structure is replaced by the wedge structure but only in the case where the vector scalars are these eigen values, hat is fractions that add to one (eigen meaning one single thing) this version of combinatorial Strecke picks out individual points on the line  given by the wedge sum –a + b.

By adhering to these strict formalisms Grassmann demonstrated a method of notation that was safe and meaningful. The many pitfalls of indiscriminate use of these symbols were well explored by Grassmann.

Most people feel that numbers are natural. They do not realise that numbers are a strict formalism which must be adhered to. Thus some procedures give the correct answer even when he formalism is completely shot to hell! It's a tough call when you are told your answer is wrong evn though numerically correct. The reason is the unsafe formalism used.

However I learned that Grassmann carefully distinguished his elements and how they sequence and combine, and then extended them to lineal combinations of products and summations!

Note the term lineal combinations has nothing to do ith linear polynomials. Linear polynomials have at most 2 terms. Libel combinations are combinations of lines( Strecken ) and can have any number. Count.

Because the fan product is so point specific, the Barycentric combination of the Strecken is not identified with another Strecke even though the word line is used.,strictly the Barycentric or fan structure focuses on points. The wedge product identifies or is identified by other Strecke.
Several definitions of signed area follow on from the wedge product. In particular the surprising last definition . The cyclical nature of the vectors , and the hidden fan definition make it work by negating each of the 3 vectors in each definition of the fan. This is again due to the restrictions in how we may define a vector component wise and the anticommutative rule in a triangle wedge product.
This is also a powerful link to rotation in terms of the cyclic direction.

All these formalisms Justus Grassmann implemented in his Maths course throughout Sczeczin.in 1820's.
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 « Reply #11 on: November 16, 2013, 07:08:25 PM »

Now the word vector actually was coined by Hamilton in relation to his Quaternions. Grassmann kept the reality of what he was dealing with clear in his mind:Segmented lines in parallel formation. But the common meaning of stretch also he retained, and the dynamism of the stretch, however it is done, the difference between a point and a stretch from one point to another has to be kept clear.

The broader context of affine goemetry and projective geometry really need to be explained clearly, and Norman does that .
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 « Reply #12 on: November 18, 2013, 11:23:19 AM »

At last the definition of  Strecke in symbolic arithmetic. It is quite hard to explain how difficult it is to get a fundamental definition of the Begräntzte Linie!

The system Norman establishes in his Maths foundations series is highly recommended by me. It is not perfect, but is the most accurate system devised on arithmetic symbols. The fact that Notman defines a vector as a Strecke shows his intuitive understanding of Grassmanns thinking.

Because he thinks like Grassmann he makes some mistakes in his thinking, like Grassmann, but don't we all!

The fundamental difference between Norman and Eudoxus is Norman constructs the Arithmoi using a number metaphor, while Eudoxus starts with magnitudes and isolates them by ratios or logos. Then dealing with fundamental processes of division he analyses the structures and relations of the ratios. When this is done he can construct syntheses of these ratios to define multiple , subtraction and addition as combinatorial processes.

The Arithmoi come from these logos analogos relationships as being metaphors for forms.

Standardising form on the triangle and the sector, Eudoxus can build or synthesise complex mosaics and count space! Thus the Arithmoi are these various grid or graph paper patterns that tessellated space perfectly OR approximately. The overriding process was to count space.

Grassmanns Strecken allow him to follow Eudoxus synthesis of the Arithmoi. He did not recognise this all at once, but his tutor was Laplace and Lagrange, as well as versions of LeGendres Geometrucal redaction. His insights enabled him to develop a combinatorial method of analysis and synthesis with wide application. The method was under constant revision and refinement, and do Normans system is the latest in a long line of developments.

The commutative property , so called, of the vector sum is problematic

Suppose AC = AB + BC= AD + DC

Then 2AC = 2AB + 2BC =( AB+AD) + (BC+DC) under the rules for adding components. This clearly does not match up, so the second= sign is invalid.

We have to do a bit of jiggery pokery to get it to work, and that is why anti commutativity arises in the triangle product rules.

I do not see it as a major issue, but it is one to be aware of interns of how we make this method work best for us, especially if we want to use n dimensional bases.

The fundamental issue here is tackled in the imaginary "number " history. Bombelli recognised certain combinations as adjugate and others as conjugate. The adjugate combinations I have called wedge products and or summations, thus the fan products or summations are conjugate, according to Bombelli.

This subtle distinction accounts for much of the misuse of the – sign in elementary treatments of analysis of Spatz relationships. The drive to keep it simple runs rough shod over subtle complexities that show up in the analysis of rotation and reflection. These explain why we have to use 1/2 angles to describe rotations in 3 dimensions and n dimensions.

Norman here defines the current formalism. It is based on only allowing the wedge sum in the fundamental definition of the addition concept. This you will recognise as you must go along before you can go up. This restriction hides the fundamental problem discussed above and seems to give a consistent Algrbra. However, things come unstuck when multiplication products are defined, and nobody tells you why?
The above is why!

Another way of covering over the issue is to make a distinction in vectors. So we define axial or basis vectors. We may even call them primitive or intuitive vectors. I just called them fundamental vectors. The point is tha we do not define are arithmetic in terms of these vectors but in terms of combinations of these vectors, called affine combinations..

This is analogous to Eudoxus not defining Arithmoi in terms of the magnitudes, but rather in terms of the ratio of magnitudes, or the logos. This is how we conceptualise real experiences as abstract entities: we compare. Them and it is the comparison we use to refer to them.
The philosophical issue is deep, but basically the experiences we have are subjective, the definiions on those experiences can never be communicated, and they are tautological, which is not a bad thing. However, it is better if two experiences are compared and contrasted, this means each party in a communication can hold his own experience in comparison to a common experience, the common experience is often considered to be objective, but that is merely a definitional convenience.

Now the common experience becomes the means of communication.

Simply if you have a bed and then I try to describe it to someone, you will be amazed at how difficult it is yo communicate your experience to someone who also has a bed, but not your bed! His comparison between the lingual reference and his experiential reference only partially helps. However if you both stand by the bed then the subjective experiences are different, but the comparison is made on the same object, even if from different perspectives( and that is another issue!).  The two can now agree the comparisons are the same, identical, congruent etc, even when they are subtly different!

However, the tautology now has 2 bases which allows for the referential process to gradually change and develop with experience while constraining how it develops.

Few consider the role of tautology in developing our logics, but  Vasil Penchev, a philosopher in Bulgaria drew my attention to this .
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 « Reply #13 on: November 18, 2013, 04:23:53 PM »

In this context Norman asks what is a circle. Turns out to be a proportion of 6 terms. This is twice the number of terms in a similar triangle proportion, because 2 specify the centre and the third the radius  in terms of quadrance. The first three terms are quadratic, that is quadrance forms.

To understand the full context you need to get book 11 of the Stoikeia.,then the definition of the meet and the join of points and then finally the quadrature of the circle..
http://youtu.be/B-5EVtEOSAU
http://youtu.be/huR1yKwx-DM
The discussion about the radians is a side issue. The quadrance and spread have  major applications in all geometries and that is its significance.

Contrasting the proportion for a line with that of a curve is highlighted by this video
http://youtu.be/eMt09NN1WOE
In which it is noted that circularity simplifies the descriptions, and emerges as the fundamental property of convex polygons.
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 « Reply #14 on: November 19, 2013, 04:27:39 AM »

Pythagoras theorem and Thales theorem are intimately connected.. The sphere is the goal of the introductory course called the Stoikeia of Euclid. Points and lines , planes and circles all derive from the interactions and relations of spherical surfaces.
Norman went forward to numbers to then go back to redact historical development of Maths. I went back to the Pythagoreans and found the mosaic that underlies all their modelling of space. From this they synthesised their Komodo , which was spherical. It was not just one sphere it was an infinite fractal of spheres in a regression of spheres. Apollonius was the greatest philosopher of the Pythagorean construction of the Kosmos.

It was Apollonius who redacted Euclids Alexandrian text to place greater emphasis on the circle and circle constructions in the early parts of the course. The seemeioon was the central concept of everything. It was where the compass placed , magically drew circles of every size in concentric splendour. It was also where the divider, placed marked off segments of a line which became centres for other circular constructions. And by these constructions circular lines of intersecting circles could be drawn and points of intersecting curves could be identified. These points of intersection defined linear relations some of which were good or straight. These were dual points from mutual circles  of radii that fanned out from two points or centres.

The equipment made the marks, but the marks were used to improve the equipment. So there was a symbiotic relationship between tools and constructions.

A true rule was constructed against the dual points, but for greater distances taut light string or tape is still used. Line of sight was used under the assumption of direct communication with the senses. Optiks and Data apprehended the sensory experience and drawings were made, but measurement was left as an individual choice.

The Arithmoi of the Pythagoreans were rich an varied.. But eventually the preparation of clay tiles standardised the pieces used in mosaic constructions. In space nets were constructed by intersecting string in a knots. These served as points for further nort constructions. It was clear that points of intersection and lines of taut cord were crucial to performing any gematrial or combinatorial reasoning.
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