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 Author Topic: The Theory of Stretchy Thingys  (Read 19216 times) Description: Ausdehnungslehre 1844 0 Members and 1 Guest are viewing this topic.
jehovajah
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 « Reply #180 on: April 01, 2014, 12:36:30 AM »

As misconceptions fall away I ask " What is the advantage of Grassmans method over traditional Trigonometry? "

Grassmann after all does not claim to remove trigonometry, only to attain its ends more quickly and earlier. In the meantime he also offers some insights into the nature of certain Practices and processes.
For example, what advantage is AB + BC  = AC?

I suppose that as one explores  Grassmans enterprise, this gradually becomes clearer. As it is, in this Vorrede , it offers tantalising glimpses of something!

For example: could the cosine and sine laws be derived using Grassmans " system?"

It is unlikely, because Grassmann relies on fundamentals like these to define his system. But it is tantalising that  AC ^2  as ( AB + BC)^2 is almost the cosine rule except for no trig values. But then, instead of keeping the anticommutative terms, they are cancelled! This means that the nearest possible identify is Pythagoras theorem.

For several reasons this is not easily obtained by this new parallelogram product. Instead some have used Grassmans trig line segments as primitives and used these to define the dot product. It is almost the same as the inner product but there is some confusion..

Whatever the result, it seems, the ideas have been yet again reworked by certain others to make their own consistent algebras.

This begs the question; what did Grassmann actually "teach", and was it understood by admiring students who passed it on faithfully, or was it just baudlerised for its inspiration and technique. But never fully presented?
Consider this video in which Norman uses the trig line elements and the inner product properties.

Normans nition of quadrance obscures the constructional issue for the parallelogram product. Also it introduces the dot product as a expression of entities called points which are rename trig line segments arranged in a grid.

While stating that we do not want to presuppose Pythagoras, while being motivated by his theorem, he nevertheless establishes a grid in which it is only possible to prove Pythagoras theorem by algebraic manipulation.

Again, this is the game algebraists play, when they hand wave and say: we do not need pictures, just clear algebraic rules!

The way I put it is geometry for the blind!

We sighted people know we need both.

Is this a demonstration of Grassmanns method? In fact it is not. It is however a presentation full of supposed Grassman terminology, but very little is solid Grassmann labelling. It is a reworking in his style.

Now I find Norman quite logical and systematic. I have pointed out how Grassmann appears whimsical and fanciful. This element of Grassmanns method is the one that seems to be rooted out by " serious" mathematicians. In 1861 his brother published his redacted version of Hermanns work.
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 « Reply #181 on: April 01, 2014, 07:02:37 AM »

Commentary
The use of gleich in German mathematical and philosophical texts is symptomatic. It reveals the context and the motivation of the author, for it is the authors choice to use gleich or gleich ung to denote some situation or expression.

Like "like" the word has many subtle meanings, because like is a fundamental comparison. In fact we often as students to compare( that is look for like things) and contrast( that is look for differences).

When considering analogous thinking, Simile , metaphor, likeness and analogy cover the same ground as does duality. Thus the humble= sign can be overworked.

Terms like polynomial, expression, equation, formulae, identity, constraint are often ill defined references to the same set of symbols.
As of yet I have not read the work of LaGrange, but I am aware that in his system, an expression of a constraint was written in some equation form. This tied together those symbols whose range and meaning were bounded by each other.

As an example, Möbius would constrain his subjects in the Barycentric calculus by writing or deriving an equation that expressed the interrelationship of his subjects parameters. He wrote AB +BA = 0 to define the concept of negative in his line segment descriptions. Thus his statement expresses is " line segment AB combined with line segment BA is set to 0"

Note it expresses nothing about length or orientation. Thus it conveys direction along the line segment can be opposed. The 0 is symbolic because it expresses Shunya, meaning everything. The constraint implies a sinle line segment universe!

We can regard some of Grassmanns symbolic expressions in this way, that is as constraints.
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 « Reply #182 on: April 01, 2014, 09:38:26 PM »

Commentary

Misconceptions fall by the wayside?

On every pair of parallel lines it is usual to put arrows indicating parallelism!

Suppose , now those arrows indicate direction. Then for each pair of parallel line segments we have an indicated direction. As we move these parallel pairs round cyclically we note , in the parallelogram, that the direction of poining( the principal direction) poits at some stage in the contra direction. Thus we have in a parallelogram the evidence for these pairs of parallel lines turning round and going the other way. This means, in the product of 2 adjacent line segments the value must switch sign as they cyclically interchange round the parallelogram!

The use of Grassmanns style pervades modern mathematics particularly Normans. This video shows a mixture of styles and how they are used to communicate mathematics and notions of space, especially trigonometric ones. We cannot step back from coordinate geometry or mechanics, but we can use the reference frame more intuitively and heuristically. The cyclical interchange  of points also connects the point notation to permutation and combinatorial theory.

I think it is important to note that the product of 2 line segments is a parallelogram. The introduction of the dot product was to formalise the trig line segment projection. It should not be made somehow " mysterious! " I note it takes Norman until lecture 28 in his wild Lin alg series to introduce the dot product! And that is not because he has not used it in his earlier lectures. The misconceptions around this method of analysis nd synthesis are many and confusing. In the end , Grassmann is all about trigonometric Analysis and solutions to issues.
The cyclical interchange of points?

Note also the clear definition of a negative area! This is down to Grassmann and is perhaps his major historical contribution! Until his book not even Bombelli had a clear definition of negative product in the plane. It is per force associated with rotation
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 « Reply #183 on: April 02, 2014, 03:13:33 AM »

Commentary

The obvious point of these translations, but perhaps one that escapes notice is what I call " observer as animator" .

To calculate, the procee involves animation at all levels. This the observer/ calculator must do. They must interact with space, in 3 d thus may involve building a model or sculpting a form. In 2d this may involve drawing a sketch consisting in line segments( Skesis or Schemata). But then the solution involves manipulating that form, either mentally or physically until it gives a known ratio to a given Metron. Mentally the form is rotated, translated reflected , topologically bent, covered, shrunk, expanded and otherwise distorted to give up its secrets. This the observer has to do. This the observer often does not realise or have a clue how to proceed doing this!

Grassmanns method provides a set of standard heuristic moves to bring closer the solution. The first is the parallelogram product, but in dynamic form. The second is the summation constraint that identifies the diagonals of the Parallegram. The third is the cyclic interchange of points which facilitates cancelling and elimination , the fift is the trig projections and their constraints. The swivelling line segments are a sixth .mthere are many other Grassmann only touches upon in the Vorrede. The Barycentric calculus, the tangential contact points, the absolute length labels etc etc. all tools that make for a faster heuristic approach to solving problems.

Returning to the product of a line segment with itself it is now apparent that AB^2 is a line segment of twice the length of AB , that is 2 AB . But what we now do, because it is a double line segment by parallel projection or translation of each parallel line( collinear line if you will allow) relative to each other, is treat it as a segmented line segment and bend or rotate one segment relative to the other!

Our language tells us that is what we do, for we say the square on the line! Because we cannot do it by translation it has to be constructed by rotation. So the finl step of the product

AC ^2 is to rotate one of the lengths to form perpendicular line segments AC and us this to product a square.

On the face of it we would perhaps do it in this situation to uphold Pythagoras theorem. But in fact, allowing this rotational construction from line segments is directly linked to the content of Book 2 of the Stoikeia.
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 « Reply #184 on: April 02, 2014, 09:05:24 AM »

Commentary
It occurs to me somewhat whimsically that we can now use Grassmanns swivelling operator or quotient Algebra  in the following way
Given (aBC)^2 => a2BC^2=>a1(BC + BC)--nb. The a now has to be distributed by the projection/ translation of BC by BC  usual in the construction of a parallelogram.
Since the second part is now free to swivel:

=> aBC  + aBC•e å => aBC  + aBC•e where i is $\sqr(-1)$ and å is a radian measure.
For the so called "complex numbers" we can write

=> aBC  + aBC•i

And we can say this refers to the diagonal of a square formed by rotating the second line segment by $\pi$/2 radians.
Whence
a$2$(BC +BC•i). Revealing the distribution of the factor over associativity rather than summation, and also describing square production as a rotation by an imaginary operator or quotient on a bisection of a line segment.

The whimsical development can be extended without hindrance to the general "complex number".
aBC + bCA•i .

What does this mean?

The interpretation of "complex numbers" in the Grassmann system is intuitive and geometrical and perhaps suffers by being pigeon holed as number!
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 « Reply #185 on: April 02, 2014, 10:51:21 AM »

The distributiyity over association occurs in this whimsical fashion .

aBC  + aBC•i is the diagonal of a square parallelogram.

Thus the parallelogram product of this square is
a2BC•BC•i

BC•i is a line segment rotated at the corner C of the square into a positin CD which is perpendicular to BC .

Thus I can rewrite the product as
a2BC•CD

Where BC•CD is a unit square.

Distributiyity over associativity is one of the features of the Grassmann parallelogram product and it has geometrical meaning. I discuss this in the earlier post about building with blocks!

Normally it is completely passed over or ignored in the notion of commutativity, but in older lists of arithmetic rules it used to appear in the section on distributive rules.
• here is used in both senses as arithmetic product and the now ubiquitous dot product! But it only takes the first product meaning in the quotient algebra. In the exterior algebra it just means the parallelogram product!
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 « Reply #186 on: April 02, 2014, 11:28:01 AM »

Again, I have to draw your attention to how whimsical this alternative way of thinking is! It is completely idiosyncratic and in one sense nonsensical!

However it is really quite radical because it is totally inclusive of the subjective and objective processing involved in the human mind.

Most scientists and mathematicians are taught to exclude the observer and to promote the observed! Here Grassmann by default includes all and attempts to clarify all! Every twist and turn, wart and blemish in our thinking s recorded and annotated in his method. Therefore it reveals precisely where we fudge things to get sn idealised form, here we reason by analogy, where we derive by ly logistic logic, where deductive and inductive reasons are used to support a claim, and where pure " fancy" has obscured a deeper intuitive significance or meaning.

It can be uncomfortable for some mathematicians to see that what they hold as true is only analogically so, or what they claim as " wrong" is actually a doorway to a deeper relationship, a metaphysical and or subjective processing one!

It is this which has made Grassmanns method ultimately triumphant!
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 « Reply #187 on: April 04, 2014, 07:10:19 AM »

Ausdehnungslehre p xii

This roughly is the content, which I have reserved to myself  for the second and last volume, at least as far as They have been processed by myself up till now, with it  the work will be closed !
The time when this second volume will appear I cannot utter, in relation to which,  by the diverse roles in which  my current office involves me,to me  it is impossible  to find those ( topics) the peace, which is necessary for  the processing of the same.
Thus  this first volume also builds  itself into a unique whole established for itself and complete in itself, and I keep it for a multi-purposed event  ,  in order to give permission to   publis:  together, this first volume  with the applications related to it;  as both volumes together, and  separated from the applications !

In practice it is according to the unavoidable, necessary presentation of a new expertise, therewith its rank and its significance becomes correctly understood,   to show like "bang!" its application and its relationship  to related content! The Introduction should also identically  serve this (purpose). This introduction is the nature of the matter, according to a more philosophical nature, and, even if I separated out this introduction out of the melding of the complete work, still this happens:  they recoil in horror through its philosophical form!
Around mathematicians   not like "bang!"; yet still, Specifically there is ruling  among mathmatician, and to a certain degree not with injustice, a conscious Dread of the philosophical characterisations of mathematical and physical content , and in practice  the most deep examinations of this Style, how it is promulgated  from Hegel  and his school , suffers a disclarity and an  arbitrariness which annihilated all the fruit of such deep examinations!
Pxiii
Regardless of which  things ( just mentioned) I believed it, the matter , to be  guilty,   by which  (in recompense impelled by a Muse) the new expertise   must be allocated its  place in the field of Wisdom, and therefore so as  to satisfy both promotions,   setting before you all an introduction , which without the fine grasp of the whole essentially causing loss( of interest), can be(come) thoroughly worked over, strenuously evaluated.
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 « Reply #188 on: April 07, 2014, 03:10:02 AM »

Commentary
Well that was fun!

For a digest of Hegel I recommend chadafrican YouTube site. Not only is it a good review but it is alternative to Grassmnn's assessment!

I also recommend searching for "analytical mechanics lagrange" on YouTube  to see how Newton Euler Lagrange and Hamilton shaped modern Mechnics while Grassmann was right on the outskirts of this group because of his isolation in the Prussian empire, and his low academic status.

We see a reason for his lack of advancement in his conviction that mathematicians in particular needed a good strong dose of Philosophy! He was not alone in thinking this in Prussia where Kant and Hegel Schiller and others were of the same mind. However Gauss was only slowly forming this opinion and only later encouraged Riemann to explore this issue.

Consequently Grassmann deliberately set his first volume the task of being the cat among the pigeons. Later he was to write: " that work was more for the reader of classical thought because of its more philosophical form".. He took on the mathematical establishment and lost. Consequently the 1861 version is a radicl redaction of the first volume, with it's overviews and applications
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 « Reply #189 on: April 07, 2014, 09:37:13 AM »

Commentary
Although I regard this as a Gross simplification, nevertheless it may be helpful to some readers.

Hegel's philosophy I characterise as mindfulness, in which the observer pays attention to both conscious and unconscious processes within their subjective processing. Consequently, it starts with tautology ( self referencing notions) and ends with tautology.

In this process, entirely subjective, the final tautology represents a " movement" without moving, that is a new understanding of the original tautology.

This process is called dialectic, and it is also called Platonic or Socratic dialectic or discourse.  It is a mark of the influence of Hegel that it is also called Hegelian Logic.

Because dialectic deals with notions in a process, it is noetic. This means it encompasses all aspects of the subjective experience. Therefore it has a place for intuitive thought and experience within its process.

It posits that all human experience is entirely subjective, and we only experience objectivity by denying part of that subjective experience. As a a consequence it can analyse all kinds of human experience as a result of this denial or suppression of " half" of the whole. The whole however cannot be determined because of an assumption of indeterminacy right at the start of Hegel's analysis! This essential tautological assumption  reveals " movement without moving" and " inversion" as fundamental notions of subjective processing.

While I fully acknowledge Grassmanns characterisation of the Mathematicians dread of philosophical discussion, I am still unsure of my translation of "nicht sogleich".

However for a mathematician Grassmann is attempting to show that if you fully label every aspect of your thought process you in fact make analogous thinking much easier and more creative. In addition you show the vanity of " abstract" thought and " algebraic" representation, because they are simply tautologies of analogous thinking.

The " strangeness" of Grassmanns logic is thus my experience of his thoroughly dialectic approach!
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 « Reply #190 on: April 10, 2014, 06:19:02 AM »

Commentary
This video explains the trig line segments algebraically, that is symbolically using some extra ideas like a function . While it may not seem related to the swivelling arm of  Grassmanns quotient operator, the generalisation is in fact a rational version of it.

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 « Reply #191 on: April 11, 2014, 09:58:43 AM »

AUSDEHNUNGSLEHRE pxiii1844continued

Also I note that likewise among the applications, those which are themselves related by contents related to Nature( physics, laws of Crystals), they can be thoroughly evaluated;  without that, the progress of the whole development would be disturbed thereby.
Through these applications to physics I especially believe the  importance , yes the indispensability of the new expertise and the analysis presented within it has been demonstrated.

That the same applications in their Concrete form , that means in  their transference into  Geometry, an admirable teaching content might provide,  of which an elementary treatment is definitely a thing  doable. Once again I am hopeful  to be able to provide supporting evidence in a part time way , in relation to which, to such a verification, according to its appropriateness in the work itself  I am able to bring into being as found not a single place!

Specifically it is an elementary  treatment of Statics, even if it should go forth in the same manifest and genera( also by construction presented!) results, unavoidably necessary, the labels of the Sum and the products of line segments to be taken on board, and the principal laws for these to be developed, and I am certain that no matter who has vehemently sought once again the taken on board bits of this labelling, it never again  will be given as extra insight!

Even if I have thusly completely adjudicated its Right in favour of the new expertise , (which at least partially its processing is published now), and in favour of it the  demands, which  in the field of wisdom it can make,  I will not cut it short in any way,therethrough i believe  to myself not to  draw close to the accusation of the pretense! Because then the truth calls out for its right; it is not the work of things which it brings to the conscious awareness and to  acceptance by you ; it has its nature and its therebeing is of itself; and to curtail its right out of  false modesty,it  is a betrayal of the Truth. But  I must demand the more forbearance of  hindsight  for all of everything which is what is due to my work on the expertise!
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 « Reply #192 on: April 18, 2014, 09:09:06 AM »

This paper located by Kram1032 makes for some useful practice in identifying the Grassmann forms or labels / handles. The presentation is elliptical, that is a lot of the detailed work is left out. The Clifford algebras are presented clearly and concisely and the fundamental forms or "objects" of the Algrbra are identified. These fundamental forms are the surfaces that most easily conform to the generalised products. Thus in 2d the fundamental forms are parallelograms and do conic curves . In 3d tetrahedrons and thus conic surfaces. The number of dimensional line segments Gives greater rotational freedom, but the " view" is selected by which products are set to -1,0,1.

It is hard to connect the dots! We start with points and line segments and then we get various products and finally a simple theory of Curves, here generalised to curved surfaces!

To find ot how you really have to read the rest of the Ausdehnungslehre!

I am still working on the above translations, but the point of this whole thread was to grasp Hermann's thinking. It turns out that it is very very radical for his time, but it's subject of meditation is just trigonometry and the geometric processes like extreme and mean etc that evaluate or proportion geometry. Thus it is useful and instructive to put aside the notions of vector and to pick up an old traditional course in trigonometry, especially one where the right angled triangle ratios are emphasised over the circular functions!  While the circular functions are useful, they are Eulers construction of the right angled triangle indexer or ratios. These ratios do not require the arc to be useful! They were extended to the arc measure by the Indian and Arabic scholars at a much later date. In many medieval images you will see that astronomers used a square sextant to measure the position of stars, not an arc. The square gave right angled triangle measurements directly!

Later use of the arc , minutes and seconds relied upon a accurate machining of a disc into equal degrees. 360 was chosen to accomodation Babylonian data, but in fact Babylonians still used the chord to diameter proportions!

The arc has one natural measure and that is the diameter of the circle to which it conforms. Having made a true disc b
Y mechanical means the rolling of the disc could only be measured by how far it turned to move through its diameter! Thus pi was always a symbol of a whole rotation of a disc. Euler threw a spanner in the works when he associated  pi to a hemi circle arc!

Thales geometry of the circle is the foundation of Trigonometry. Here the arc, the chord, the radius , the diameter all meet and they reveal themselves to be incommensurable! And yet this is the simplest of all curves, but the hardest to machine. It is special and makes straight lines special and planes special. Cones and cylinders also become special . Their specialness is in our human conscious perfecting of them. They do not occur naturally. They occur by a pragmatism whose inflexibility leads to ideal forms. They are thus totally subjective creations of our subjective processing.

Because they are subjective, the incommensurability is either to be exxpected( human error) or unrxpected( design specifications are fixed and simple). By them we learn that we interact with a changeable reality we cannot apprehend! We must snatch at ephemera and make them real to conceive of " reality"! Reality is what we make it, what we can make( fact) or what we can describe( myth) as processes by analogy. Both are vital to our sanity.

Panta Rhei.
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 « Reply #193 on: April 19, 2014, 09:42:32 AM »

I am currently researching Napiers Wonderful Logarithmic Tables.

We have seen how Grassmann introduces the logarithm in the quotient operator algebra, that is the quotient product for trig line segments in distinction to ordinary line segments. Tha analogy is exact.

Moreover, the life's work of Napier has the identical goal to that of Grassmann in as much as they both wanted to make mathematical processes easier on their fellows by carefully constructing notation or labels that encsulate a notion of process!

These articles make very clear how. Napier worked, although some historians are confused by his kinematic motivation. Do not trouble yourself over this, because his motivation was circular motion. In fact his logarithms of sines are a particular correspondence between the sines and the tangents of a angle measured. From the vertical direction. It seems confusing therefore that he should refer to the sine not the cosine, but in fact it is not. For a projected radius the measurement horizontally was often called the Haversine. In fact nautical tables preferred this layout to the one we get introduced to at school. The cosine table was quite a late introduction into the trigonometric toolkit, as most mathematicians actually used symmetry notions to determine the dine. Thus a notion that a projected radius splits a line into its cosine with the remaining part being the sine , although in fact inaccurate , was close enough to be a useful rule of thumb. The haver sine was in fact the correct notion.

Commentary

So both Grassmann and Napier recognised the logarithm as a correspondence notationlly. The indexial arithmetic is a contrived convention based on the geometric means in series with infinite terms, the fact that we say infinite terms shows how we have disconnected from the geometry. Napiers construction and velocity constraints initiate that the process woul never end, but would always be regular. Like the famous paradox by Zeno the faster runner never actually completes his race before the tortoise! However, pragmatically the faster runner passes the tortoise at a given time. Zeno's paradox only confuses when a notion of time is left out of consideration.

Later, Newton uses this same concept to frame his deductions in the Principia. Time, whatever it was( and it is only a comparison of motions) was crucial to avoid the argues using Zeno to refute infinite process deductions. This is not to say that such deductions should not be rigorously scrutinised, because many strange ideas have crept in under that particular line of reasoning!
Quote

This is a post i did on the proportioning based on my translation of the relevant actual text of Mirici Logarithmus.

In point of fact it is not at all difficult to conceive of a rotary machine ith just these properties!
A wheel has a straight arm extended radially beyond its perimeter to intersect with a scale set at right angles to the diameter. At the same time a rod drops from the point on the rim from where the armature extends, onto the diameter. The length of this rod is the sine of the angle measured anticlockwise from the diameter. This degree scale is set against the rotating wheel.

Because this rod is attached to the rim of the wheel the sine can be measured directly along the diameter from the rim, once the length is marked and the wheel turned to the diameter line , or alternatively the rod is swung to the radius of the wheel and marked off there.
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 « Reply #194 on: April 19, 2014, 11:36:30 AM »

Commentary,
I want to look at the concept of function in terms of 2 notions: purpose and process.

The concept of function has its origin in the Prussian Holy Foman Empire, where it's organisational role was paramount in social and militaristic settings. That the idea should be carried over into the structure of mathematics is uite natural . Determined and deterministic behaviours are the bedrock of mathematical consistency. Thus the purpose adoration of a function highlights it's labelling role. Such. A role encapsulates a notion of order, structure and relationship to a whole. It also acts as a characterisation and a definite distinguished or distinction.

The process role is precisely the duty and actions the function is tasked to fulfill. Consequently any expression of those duties is in fact a process or procedural statement. Mathematics is composed of procedural statements, formulae algorithms etc, and in order for these procedures to be about something numbers are defined in an objective sense. This distorts the underlying concept of numbers which are the Arithmoi. Indeed Arithmoi are spatial concepts, but entirely general and formal. They are the products of procedures acting on subjective interpretations, evaluations and distinctions. The du jective nature of mathematics has been obscured by misunderstanding of the role of an ideal, or form. It's function is to allow iterative interaction with space.

Because a function has a purpose it is labelled to identify that purpose. Many of the labels Grassmann creates have this function role also.
 « Last Edit: April 20, 2014, 12:31:42 AM by jehovajah » Logged

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