Seemeioon estin, ou meros outhen
The Grassmann concept of a Strecken is like an object oriented class definition. The class line has three properties: direction; length; points that fulfill some function. We might attribute some colour to these points to visually identify them.
However a point has no parts by Euclids definition.
We cannot write a list of observables for meaurables or orientations for a point. But we can and do write a list of subjective experiences and descriptions of a point. A point has the subjective parts ( properties): meaning; significance, hen we communicate about a particular point we communicate about its meaning and significance, that is wholly subjectively. Thus we give points a subjective reference frame which we carry about with us internally and use to subjectively identify experiences including experiences of topos or place.
Colouring a point is just that, giving a distinguishing experience of a topos with its meaning and ignificance hooked onto that experience like a coat on a coat hanger.. These are subjective structures, internal models and maps of external experiences.
Even though a Metron is deemphasised in a Grassmann Algrbra, it is still one of the properties of a line. To maintain tht property Grassmann uses the interior Algrbra of points to define a line as a product of points A,B. the usefulness of this is that these points mark off a Metron in the extensive Algrbra, by which coefficients are derived. Thus in this form of lineal algebra there is an implicit Metron, and this guides the use of any explicit Metron.
The notion of a vector has this metrical implication implicitly, and so is a good instance of a Strecken. Where a vector concept is sometimes confusing is where it is suggested to be somehow implicitly free of all these relationships implicit in a Strecken. The mixture of implicit and explicit use of properties is why the algebra is so subtle. . Very often, Grassmann draws on the intuitive implicit properties without explicitly stating the fact. This he inherited from his Fathers struggles with rigour.
Hermann corrected mistakes his father Judtus had made without sacrificing too much of the elegance in this way of thinking. Later researchers, for rigours sake, attempted to split subtle points into 2 ignoramus concepts only to find they lead to other ifficulties.
The blend of what you fudge and what you expose is demonstrable in any system, axiomatic or not. Axiomatic systems tend to set the fudges out at the beginning, but they still inhere in he system!
We have to live pragmatically, and that is why the pragmatic seemeia on is so important. It's a fudge, but it makes the whole ytem work usefully., we can hide all our fudges behind the seemeia on! That means subjectively we knowingly or unwittingly delude ourselves in order to get a pragmatic result.
The Schwerpunkt developed from the observation that a point exists in a topos, that is a place. This place is not explicitly referenced, it is subjectively referenced. However, the practice developed by Descartes, DeFermat and organised by Wallis to set up a reference frame called a fixed axial sytm.mthe Measuring line was used to model these axes which were set orthogonally to each other in a standardised format. What this meant was a point could be referenced by two " numbers". .
This is a misconception of the reference frame, and it has persisted to its detriment.
Those who wanted to break free from Cartesian coordinates could not put their finger on the problem. Grassmann did. The point has 2 properties in a reference frame: position and magnitude. We generally ignore the position and focus on the magnitude. Grassmann realised tht this type of point was different to a Euclidean point which has no parts, except subjective ones. The point had a position and a magnitude on the axes. This is then used to project onto a third point in space by parallel lines to the axes.mthis point does not have a magnitude in the reference plane it has only a position specified by coordinates.
However it could be given a magnitude using Pythagoras theorem, and so a Schwerpunkt could describe a conic section point!
If I switch to a polar coordinate frame thn every point in the plane has a position and a magnitude. Thevschwerpunkt deals with a major inconsistency in traditional reference frame theories.
Grassmann uses this understanding to define the inner and outer products of Strecken under "parallelogram multiplication".
What is a Strecke? The simplest and noblest notion is " a construction line". It is a subjective notion of our intention and application to construct. We conceive it before we even draw it, and its meaning grows as we construct. Once its job is done, it fades into the background
The angle between the Strecken becomes crucial. Up until this point it had not been considered, but his work on the ebb and flow of tides advanced his conception of the algebra. In the case of the lineal algebra the angle has to be included in the analysis, and that means the trig functions and surprisingly the exponential logarithmic functions.
His concept of parallelogram multiplication meant naturally that 2 Strecken in the same line and in the same direction would produce a zero parallelogram. Also two Strecken in the same line but directed contra would do the same( gleichgerichtet). He called this behaviour the "Aussere produkt". This seems to be because there is no projection line involved in this conception of the parallelogram. The Two Strecken form the outer perimeter of the parallelogram, and both flow out of each other ( auseinander tretenden)
However there was another case when the Strecken produced a zero parallelogram: if one Strecke was projected onto the other Strecke this designated a shortened Strecke. If two shortened Strecken lie against each other then their product will be zero. This perpendicular projection involves the cosine function( arithmeticsche produkt but now called the dot product) and as these Strecken fall within the given Strecken the parallelogram constructed by these Strecken is an inner product!
The Grassmann Outer product is about the strecken spreading out from one another as you step away the strecken like clock hands, the Inner Product is about the nearness( Annaherung) of the Strecken in this same process, but for the Inner Product it was important that the projection was perpendicular onto each other. In this way the strecken have a reciprocal value applicable .This actually makes the Grassmann inner product
AB* cos^2¢*sin¢ if ¢ is the angle between them.
The outer product therefore represented a construction based on parallel lines, while the inner product is based on perpendicular projection and then a parallel line construction. Although this is not the work up for covariant and contra variant vectors, it is the source of that technology. Grassmann specified a vertical projection( perpendicular ) for his inner product, but the Euclidean inner product works slightly differently in where it projects the Strecken to.
Now Grassmann was keen to put his results and discoveries in a second " Volume". Especially as he believed he had found out how to represent undulatory motion and angle in his algebra. He was so excited that he wrote this in his first Vorrede as an overview of good things to come, in case pressure ( of circumstance) delayed the publication of the second volume. How true that fear turned out to be, and then some. The uptake of his first volume was minimal! Yet it alerted Peano and Hamilton to a great genius. I have written what I have written about Gauss and Riemann, with some corrections I might add, but the plot is the same.
The inner and outer product are crucial to representing angle and undulation. The use of the exponential is also novel, but well founded.
The inner product never exceeds the pi angle! This is due to the insistence on drawing perpendiculars onto the other Strecke. As the angle between the 2 Strecken alters the outer product goes from an acute to an obtuse parallelogram in its outer product. The two Strecken must step out from each other, that is emerge from a common join or point. This product actually goes negative when the angle exceeds pi. However many ignore this formalism in geometrically constructing the product, something Grassmann warns against, we must observe all the conventions! Thus as Grassmann points out, you can represent every outer product by accounting for and interchanging the signed designation. To keep equality when designations for Strecken change you must change the sign of the whole system.
Grassmann has contrasted Strecken that were connected to each other by a join, whose directions or orientations were ticking apart like the hands of a clock with the vertically projected Strecken which got closer the more the projecting Strecken got further apart to each other , that is in a partition sense they were reciprocal to one another. What he meant by that I think is that the Strecken rotated apart the shadows they cast vertically on each other drew closer to each other, not in orientation but in "nearness". For the inner product this gave "geltenden Werthe", that is applicable values by parting in a reciprocal manner to the angle spread. This must be a reference to a table of values namely the Sine table. The dot product makes use of the cosine tables, but the Grassmnn inner product is a more complex combination.
Spend some time just appreciating how the different internal angle changes the sign of the product! This is for the exterior product.
The interior product is a bit more involved. Drop vertical /per pendicular lines onto each Strecke from the other Strecke.. That means for points ABC and Strecken AB, BC drop vertically onto BC from A and vertically onto AB from C. The two Strecken from B to the perpendiculars are used to form the inner product.. It can only form in an acute or obtuse angle so it never becomes negative. The standard or Gibbs vector inner product ( dot product or Euclidean product) does go negative unless a restriction is set on the angles.
Grassmann insist only on the construction. This means that as the Strecken pass the pi/2 boundaries they have to be prolonged backward to perform the construction.. The Strecken marked off are now in the contra directions of the outer product Strecken so produce a positive parallelogram product. Strictly speaking, these constructions produce strecken outside of the initial Strecken but in the same line with them ( gleichgerichtet ), so the inner product is distinct from the outer product.
The construction constrains us to use the angle between these projected Strecken to construct the parallelogram. This is not the same angle used by the outer product, so because Grassmanns construction of the parallelogram involves 2 same signed cosines( cos^2¢) and never uses the reflexive angles the result is always positive for the inner product. The construction of the outer product involves only the sine of the angle, so the resultant parallelogram switches sign when the angle becomes reflexive..
Grassmann noted that the inner product did not change sign when the Strecken designation was changed, so for the inner product he had commutativity
AB = BA
Grassmann utilised this fact in his Ebb and Flow of tides paper, to establish an identity between the angle , the trig and the exponential functions of the angle, he appears to have expressed it in degrees, but the point is it is an IDENTITY. This means that we do not evaluate the numbers we switch between the two to get a facility for visualising what is being modelled. We could say that it is a map onto the Grassmann product planes if that helps!
Grassmann then goes on to formally deduce the Eulerian form from his algebraic representation in inner and outer product form.
His point, briefly highlighted in this Vorrede was that his analytical method was as general if not more so than Eulers!
How come it produces an analogous identity? This is simply a consequence of Grassmann writing a linear combination of the outer and inner products and applying the combinatorial rules, it is to be observed that the outer and inner products coincide in the same manner as the i and the numeral products, but in zero rather than in 1. Using this as the angle measure and the exponential function he models a sine and cosine identity. Clearly an evaluation of the inner product is required akin to but not at all the same as radians, because they are an arc diameter ratio, the inner product is to all intents and purposes an area of a variable parallelogram.