The Algebra that Hermann Grassmann developed over an extended period started with some initial impulses from his Father Justus.
Justus was reacting Legendrian geometry to make it more logically consistent with Arithmetic. The reason was that he was on the side of the philosophical debate that said Mathematics was synthetic, man made, and therefore wholly investable in some improved form. The other side was adopted by Kant who believed mathematics to be of divine origin, revealed to humans through conversations with divine beings , and passed around by discourse.
In part, the full reply to Kant was given by Ficht who developed all of "modern" mathematics from a few " axioms" as they were called. In fact they are a late Greek concept most notably used by Newton in his Principia, and favoured by Kant. Kant hoped to show that the few axioms discernible in mathematics could not account for all of Mathematics. Unfortunately, Galois Theory and some modern philosophers were convinced otherwise. They had access not only toa mechanical texts, but Islamic texts showing how few axioms were in fact necessary!
Justus was among the first contributors to what we now call Ringtone and Group theory, he worked with some crystallographers who believed in the dynamism in nature. They were impressed by the Newyonian Method of Fluxions, more than the Differential Geometry of Leibniz, and they were thoroughly persuaded by Descartes fixed coordinate methods. They made progress in classifying crystals and there reflective properties using an odd mixture of properties which gave consistent results in determining the type of a crystal. It was a kind of group Algebra that fascinated Justus who co sidered it an extension to arithmetic. This is where Ausdehnung Groesse came from. It was. Theory he imbued his boys with, and the primary school children he was responsible for educating in the Stetin( Sczeczin) district.
The first lesson he taught was on points and how they determined Strecken, length and direction of travel. This was Hermanns first introduction to the negative sign, but also the algebraic notation of geometry.
Briefly, he learned AB = –BA
Then he learned how that helped to describe processes in a system of 3 points, A,B,C.
He learned AB + BC = AC .
There were any number ( 6 to be precise) of combination of symbols which made sense of that expression in regard to a joining of 2 Strecken in a straight line. But he noted it still made sense if they joined at a relative Nile providing you did not go beyond the Strecken given, and AC was never given.
This I have called the Law of 2 Strecken AB and BC .
Then, when he was learning about multiplication, he was introduced to the rectangle. Justus had a problem with the logicality of arithmetical multiplication. It just was not logical. However it seemed intuitively obvious from the point of view of the geometry of the rectangle. Thus his theoretical and philosophical position was somehow geometry imparted a logic of its own, different to Ariyhmetical logic and just as valid. He could mot articulate this intuition consistently. Time and again he was forced to say, look at the diagram of a rectangle when it came to multiplication.
The problem is deep, and I have discussed it in previous posts. What Hermann learned was that the Strecken were juxtaposed and that was then used to construct a rectangle by parallel lines. But he also noticed that the same construction gave s parallelogram, and that was of the same " area", the area was not important at this stage, the notation was . By using parallel lines with different intercepts many equal parallelograms could be constructed.
It dawned on Hermann to try out his law of 2 Strecken on the multiplication. His father had taught the combinatorial rules of arithmetic, including, associativity, distributiyity and commutativity. In England we just get taught BODMAS!
The result was as expected until he decided to use a third Strecke as the replacement for the sides BA and BC which had parallels CD and AD.
Thus BC + CD :== third Streck BD.
This was all set up as required for multiplication, the Strecken diverging.
The shocking consequence was that BA.BC had to be equal to – BC.BA
He was in trouble. His father had just explained the commutativity law, and this was not it! It actually took him several months of figuring it out to finally accept it, and then he was hooked!
Could his notation be extended to include division? In his day Dedekind had just floated the idea of real numbers, that is continuous magnitudes, but the only theoretical division notion was rational polynomials. Although they were called fractions, those who were responsible for higher arithmetics knew them as a derivation from rational polynomials. These rational polynomials were actually devised from the notion of Arithmoi, handed down through the Arabic scholarship, and the writings of the Neoplatonists. They were sometimes thought of as Gematria, a corruption of Greek Geometresse, and taught in the higher Academies and schools of learning in Baghdad, Spanish universities, and in northern Italian universities.
The trivium and the quadrivium formed the basis of a liberal arts Education in the west and what came to be called Number theory and then Arithmetic derives from these beginnings. The topics of rational and irrational numbers derived from the solution of rational polynomials.
Rational polynomials have a fundamental geometrical origin, and it is these forms that most resemble the Greek Arithmoi. What also was swept along with these number thoughts was the underpinning work by Eudoxus on Logos Analogos.. This work showed how lines were symbols of different magnitudes, but as lines on a Mosaic, they could be quantified, that is metric used or segmented into units , monads, that were used as Metrons in Katametresee.
The work of Eudoxus was often obscurely presented. It is mixed in with numerology and Gematria nd often in Arabic Jewish circles this was called Kaballah, secret esoteric knowledge! In fact it is the logos Analogos method that harmonises all measurement, all astrological lore and establishes laws for dynamic behaviours. Hooke , Barrow and others who were deep into geometry drew much from it, but it was Newton who mastered it's deep significance. All his profound works are based on its deep algebra, and he did not care to share his Algebra with every person. He felt it was a private communing with god, and his masters, the ancient Greeks in particular presented it in the symbols of geometry.
Wallis, besides many innovations was also keen to establish an anglicised version of Algebra. Drawing on the works of Diophantus, Bombelli, Herriot, Viete, he formed the strong opinion that Descartes was a plagiarist of Herriots work. He certainly was watchful of the deceitful practices in European intellectual society. Thus he it was that went to the defence of Newtons primacy in the method of Fluxions.
Wallis also fixed Descartes original coordinate Method to fixed orthogonal axes! This form persists as the standard representation of Cartesian coordinates. However, the original text by Descartes shows axes at any fixed angle. These notions were thoroughly understood by Lagrange and Laplace who provide notions called generalised coordinates, for use in mechanics and astronomy.
This is the source that Hermann tapped into, as he developed his method of analysis and notation or terminology.. While studying Lagrange to enter a mathematical competition, he noted how his notation and terminology simplified the celestial mechanics. It also introduced him to the power of vertical projection!. Suddenly he understood that ratios for trig functions were just that , functions for the curve of the circle. Similar functions could be set up for any curve. The role of these functions was to tabulate the ratios involved in the curves. From these ratios the points of the curve could be displayed on the appropriate set of axes. The trig ratios used on an arbitrary curve meant that the sine ratio could be used to describe any curve. The right triangle could be used to model any curved or straight line in the plane, and by spherical trigonometry it could extend to 3d space.
This was vital for Hermann, as it introduced him to his concept of the inner product as a ratio of Strecken, and this provided him with his division or rational algebra! It also linked him powerfully to extant tables of sine and hyperbolic sines. Thus his manipultions by his algebra could always be evaluated by these tables!
It was as simple as that! Correctly do the manipultions, and then look up the solution in the tables! The amount of calculation he avoided by his notation was immense. His method not only simplified complex pages of calculation, it also clarified exactly what relationships were being used, and what concepts were invariant. It provided general classes of solutions, not just one specific solution.
He had shown that there was an extension to Arithmetic that was algebraic, symmetric and powerful, and could handle any dimension or axes in space.
He had shown the powerful network of segmented lines that used parallel line projection to enmesh the very fabric of space in a quantised way yhat was algebraic and arithmetic and much more. His Ausdehnungsgroesse were lines of extension that signified any orientation and magnitudes in those orientations, and they formed a mesh of compass directions, a multicompass vector network for every reference point in space.
http://youtu.be/yAb12PWrhV0 here Norman sets this up in the planar case.
The inner product is a projective algebra based on the ratios of lines. One application is the modelling of curves by tabulation of these ratios. Another application is the modelling of dynamic motions by the differential calculus based on these ratios. The continuous nature of the inner product is its essential difference to the outer product. While orientation changes in the outer product, the quantity of the vector does not. Thus for a dynamic system a characteristic dimension or set of dimension exists , and within that product they have controlling determination of the variation in quantity of the inner product, that is its unit size. The ratio of the inner product to the outer product is a characteristic of the divergence or convergence of the characteristic line segments.
Grassmann saw all of this and more by 1844. Hamilton published his research on Quaternions in 1843. In 1853 Hamilton published his first book on Quaternions for the scientific community. Then he set about refining it in an attempt to catch up ith Grassmnns work, apparently unaware at how Hemann's Ausdehnungslehre had been very poorly received, and also not aware that Grasmann was not hs academic peer!
In 1861 Robert, Hermnns brother, in forced collaboration with Hermann redacted the work and presented a more mathematical and algebraic treatment. . This was better received nd alerted the Prussian intellectuals to the fact that the solution to Riemanns challenge about the grounds of geometry had been provided by this work.
Much reworking of the method took place, and not all f it correct, but all of it well intentioned. It was the year after Grassmann died that Bill Clifford came upon this work, saw Grassmns challenge to apply it to undulatory theory and the pendulum, and attempted to deal with this in generality using he quaternion algebra of Hamilton he was studying as a model of rotation.
Before he died, Grassmnn had also studied Quaternions , and concluded that his method described rotations even more generally than Quaternions!. His method could model Quaternions and appeared to simplify the algebra. The solution Hamilton achieved was by brute force! He had mixed the conjugate with the direct imaginary forms without realising. I jk = –1 was in fact a twisted basis, the K, under homologous examination actually maps to the conjugate! The non commutativity stems from this mapping. A commutative group of Quaternions is actually possible, but whether it is also a division ring I have not checked.
Grassmans insight all those years ago was that the law of 3 strecken revealed that dealing with components of a summation Strecken is commutative, but when they are equated to a third Strecken rotation becomes a hidden parameter. The solution was provided by the Cotes Euler identity and the anti commutativity property. This allowed the general Strecke to be represented by a exponential identity. With the trigonometric ratios , this could be extended to model for any conic and eventually for any curve.
Clifford died before he could take this further, the multivector ( MULTI compass vector network) at any point in space was clearly described by this exponential form, but rarely used or understood until much later, and then mostly in complex number form.
There is a crucial quatenionic form to this exponential form, which fulfills Grassmnns wish to deal with the spheroidal surfaces of space naturally. But from the extension of these segmented magnitudes into any direction the Clifford algebras provide an n dimensional version for which there is currently only one application that uses Clifford algebras to display crystall spaces!
The Clifford algebras I find get too technical too quickly. This is a reflection of how algebraic the understanding of it really is. No one can give intuitive examples of it except as crystals?!
The answer is to realise the fractal nature of space , and how these MULTI compass vector networks mesh dynamically to define fractal regions of space.
The law of addition of precisely 2 Strecken.
The Strecken must be given, that is a relative reference frame must pin point 3 points. The points must be identified in sequence. The pairs of points in sequence mark the beginning and end of the segments and the common point of join, which must also be the beginning of the second Strecken.
This is dometines referred to as "tip to toe".
This addition of the Strecken is a construction not a calculation. It is a useful way of recording the process of locating a point relative to axes in the directions and thus parallel to each Strecken. In this sense, the law of 2 is about the components of a " vector" or the coordinates of a. Point.. For this reason the law of 2 Strecken is useful for parametrising the points on a line or curve.. Eigen vector functions are a case in point..
The law requires 3 points because that defines a plane for straight line segments so in general the Law defines a plane.min that regard the Strecken are usually the same kind of magnitude. Just oriented differently. Every Strecken is then regarded as symbolic of a whole family of parallel lines with that orientation illing the plane. This sets up a mesh of points in the plane which are the fundamental objects of position and reference in the plane.
The segments can be used to represent differing magnitudes. In this case the points represent these combined magnitudes, not positions in space.. But if the magnitudes are measured relative to the segments symbolic of displacement in the plane, then a coefficient of proportion can map the combined magnitudes directly onto the reference plane. This provides an overlay onto the reference plane that fundamentally enables measurements in a plane to be located.
This may sound like graphing on a plane, which it can be compared to. But this is more generalised than just Cartesian Coordinates based on Wallis's fixed axes. It is more like Descartes original presentation. The concept of invariance now becomes clear, because different Strecken sums may be transformable from one grid network into the other by transformation rules.
The law of 2 Strecken underpins the component wise description of a third Strecken which is found in the law for precisely 3 Strecken.
This Law arises in the context of multiplication of precisely 2 Strecken.
There must be 3 points, and the Strecken must diverge from a single point to the othe 2 points. There is a sequence to how this divergence is constructed given by the order in which the 2 Strecken are given or set up. . Because rotation about an initial point is involved, the direction of rotation is specified, and conventionally it is anticlockwise from the first Strecke to the second Strecke.
We realise that rotation is involved in the law of 2 Strecken, but the sequence of constructing actually guides the constructor unambiguously to the correct relative orientations. Here in the law of 3 Strecken the significance of Strecken orientation, Strecken divergence and the direction of displacement along Strecken is instructively highlighted.
Having set up the Strecken , ab represents the product of a construction process using the family of parallel lines of each Strecke. The product is clearly a parallelogram., and since the Strecken are given, it seems that we can commute the symbols and get the same parallelogram. In fact , following the formal set up instruction leads to a parallelogram that is flipped relative to the initial one.
There are 3 notions of commutativity: swap the symbols; swap the process order; carry out the process order with the labels( symbols) interchanged so that geometrical objects are relabelled . The third means that the fixed construction does not chage, just how one describes it or refers to it.. The first method of just swapping symbols is almost like this 3rd kind,,but it differed in that the observer twists the symbols in their minds not on the construction! It is a habit formed by learning multiplication tables which focus only on the resultant not the process.
It is the third Strecke that reveals how that fact results in a sign change which indictes rotation. One might have guessed at pi radians of rotation, but it is in fact more complex than that . It is not a rotating in the plane, it is one out of the plane for any whole object of points, but it is a reflection in a line in the plane for a figure sheared along the parallel lines.
Thus, the negative sign means More than just rotation, it means reflection and shear and any combination of those 3 transformation.
One of the striking things about parallel line constructions is the conservation of magnitude under shear. This sgnitude is not actually visually experience! The shape seems to stretch so finely under shear that one would not guess that nothing was invariant under such a trauma , but we have always been able to demonstrate that provided the projection is strictly parallel then something of the form resins invariant.
This invariant thing has been called Area, but by that is meant the count of Arithmoi within this stretched form remains constant, because the inner product increases inversely proportionally to the outer product( logos analogos).what we see is space twisting and stretching, but the inner measure responding inversely. This is the proper conservation law for space; constancy in proportioning.
All our conservation laws are based on this proportion relationship between inner and outer products.
So now if we set c as the combination of a + b, then c has to be the same kind of magnitudes as a and b.
We can then write
ab = (c—b)(c—a)= cc — ca — bc + BA = 0 + ba — 2 ab( because ab = cb = ac and allowing commutativity)
==> 2ab = ba — ab
So set ab = —ba to give
2ab = 2ba.
But that is a contradiction! To preserve commutativity we have to introduce anti commutativity .
The other alternative is to set BA = 3ab !
When we allow anti commutativity
AB =–BC = –CA
-ab = ba drops out naturally from the manipultions.
The alternatives are instructive. The transformations of the parallelogram introduce signs that signal these departures from the formal rules. It was presumably unacceptable to scale up as a solution to maintain commutativity, because if you look yo can see that there is no reason then why ba cannot flip which gives you ab = ba =0!
So the rule is the Strecken must diverge and the sequencing must be clockwise. Commuting sequencing is signalled by a negative sign. Anti commutativity is established . This is a direct result of identifying the law of 2 Strecken as components of a third vector or Strecke. This third Strecke introduced rotation and stretch into the formalism,,something that only circular projection can eliminate
Anti commutativity arises from the closed loop nature in space, we are mentally ambiguous about clockwise and anticlockwise. When we set up a firm convention we introduce anti commutativity for certain process paths to achieve the same outcome or point of reference.. We are surrounded by these anti commutative process paths and have learned to ignore them by focussing on the resultant. However, for an Algrbra of space we have to pay close attention to our process paths if we do not want to end up in the wrong place!
It turns out that the sign conventions only deal with part of the problem, and consequently they need to be precisely specified + and minus – , no matter how handy are inadequate to the task.
This video is the best introduction to the Grassmann initial Algebra any where on the web!
You will see the application of the law of 2 Strecken and the law of 3 Strecken masterfully explained and clearly defined .
The only rule missing is the inner product rule which is explained else where.
The whole of Norman's Wildlinear Algebra series, plus the underpinning Wildtrig series constitutes the basics of the 1844 Grassman Initial Algebra. The dot product as defined by Norman is the Grassman inner product , but missing the ratio with the outer product. This product is a ratio derived from vertical projection and the scalar product rules for the Grassmann product, the law of 3 Strecken.
The issue of anti commutativity comes up in many guises, and the differences are often pushed to one side or discounted. Many logical or systematic compromises are made to develop a smooth notation and a soothing rhetoric. There is no doubt that Hermann revised his initial ideas and impulses as difficulties presented themselves. This was a was a work in progress and revisions were inevitable.msome very subtle inconsistencies were glossed over, and anti commutativity was a casualty in the collaboration with others.
Grassmann as the developer may have hls subtle distinctions which rightly or wrongly informed his choices. As these surfaced as major problems he would redact to ensure greater consistency. Thus his method was honed toward perfection but never perfect.
The issue of 3 points and the three connected segments between them was a subtlety that many sweep away. The notational choices were dictated by his father initially , and the introduction of the cotta or negative sign was not well thought through in the days of its early introduction. The sign rules of Bombelli from Brahmagupta, and popularised in the west by several geometers like Herriot, Viete, Descartes , De Fermat and Bombelli, were made rigorous or codified by the Prussian mathematical institutes and the French Ecole. Together, they attempted to establish a consistent mathematical operator signage or notation. The negative or contr sign was one that everyone agreed was clear! However it turns out that it is not at all clear, and both Hamilton and Grassmann explore this fundamentally. The square root of –1 is a case in point.
Grassmans original law of 2 Strecken, the basis for the component version of vectors, reveals the difficulty only in the case of the law of 3 Strecken, where it has universally been adopted that a + b = b + a. This is not justified, because of the inconsistencies it covers, it is merely set out by fiat.
Such an approach is justifiable as a axiomatic game, in which the rules are set and the aim is to find any inconsistencies. The trouble is that the inconsistencies are in the premise! The greatest premise at this time was the perfection of Arithmetic. Consequently the sacredness of the operators /,*,+, –. Along with these sacred operations were the sequence rules BODMAS, and by these and other accommodations, arithmetic was deemed always to give the correct answer. It was customary to be " corrected" by mathematics, arithmetic in particular and to be judged accordingly.
Many inconsistencies exist in arithmetic, but no one was allowed to point these out! In the meantime the hole of mathematics was redefined in terms ofArithmetic. There is a historical reason for that, but it is a synthetically one. The early Greek geometers and the Pythagoreans in particular, nalysed the world around them and came to the notion of the seemeioon. This notion is mistranslated as point.
The seemeioon is the indicator, it indicates the end of analysis and the beginning of synthesis.
The seemeioon is not insular it is uncountable and it is dynamic. From it our real experiential continuum cn be synthesised, in coordination with every other seemeioon. The seemeia represent the interface between our conscious experience and that which we are experiencing, nd so they are mysterious, imaginary, subjective, mythical and mystical.
A dynamic seemeioon can be experienced as drawing a line, by distinguishing every other seeimeioon it passes " through". We could say it lights them up as it passes through them, or we could say they light up in sequence. There is a lot we could say, but we instead redact to the new concept of line!
This process of focussing on a distinguished experience as a new object is related to definition of that new object. These new objects can similarly be used to define a new object called a surface.
Now I am using the word object because I want to enforce an observer object paradigm. But in experience, each of these defined things is a subject! Thus my subjective experience I am portraying in objective definitional language. In short I am reacting my subjective experience to the identified subject, nd if it is agreed by mother person, then we may both agree that it is an object.,that is to say it is a shared subject that we both subjectively experience, but we do not want to elaborate on our individual subjective experiences!
Some symbolises love this brevity. , and mistake it for clarity. Thus a potentially auditory experience is replaced by a visual one, and a truism is spoken thusly "a picture is worth a thousand words!"
The benefits of this kind of conciseness must be weighed against the deficits of it. It turns out that as time progresses many iterations of former reductions evolve, because they are not fit for purpose, in some instance and must be adapted. In those case accretion may make the evolved form completely untenable and thus it needs be replaced by a newer insightful form.
Grassmann's law of 2 Strecken is in the context of 3 points. 3 points presents a cyclical dimension not evident but still present in 2 points. In 2 points we experience the cycle in terms of sides of a Strecken. Thus going from A to B can be represented as moving along one side of the Strecke while going from B to A as moving along the other side. Getting from one side to the other involves a full half turn rotation at a point that marks the end of the segment. When there are 3 points , the lines close the figure and can be thought of as enclosing or excluding the side notion!
The journey around the closed figure is different also, and we distinguish clock and anticlockwise cycles.
We have a clear distinction between lines and figures nd points in these descriptions, but all are connected by the dynamic seemeioon. The difficulties arise in symbolic notation. For where a line is the symbol that captures graphically all the freedoms we have discussed A is a label, and is severely restricted. AB is formed from 2 labels and gain restricts so that we have to write BA to convey the full senses in a line symbol.
The introduction of the contra symbol then complicates it because it is not usually defined. Usually we are left to surmise that this is some contra experience, but not told precisely what.
Grassmann in restricting the 3 points to 2 Strecken was making a notation about points , not lines. Thus we cn agree that whatever rout we take, if we end up at the same point, then the sameness is in the Point! Even when expressed in lines, to equate all journeys to the same point is to make a definition of duality or equality of point.
At the level of lines we cannot make that duality because they are clearly distinct.
In the context of figure construction, however, we can adduce new constructive relations, and indeed we have to adduce a new point and some new lines. The lines we adduce are parallel lines, and in order to adduce them we must identify or adduce a new point.
For points A,B,C we can find a fourth point D through which parallel lines can be drawn to make the figure ABCD.
Now this opens up a broader freedom of construction. How do we find this point D ? Do we find the point and then construct lines and then define them as parallel? Or do we allow parallel lines to exist and so through that construct the point D?
These are clearly not the same processes, and yet we end up with he same constructed figure. So again, to equate these 2 processes is to say that the sameness is in the final destination, the constructed figure.
However,mthe notation in imposing an arithmetical model reveals an inconsistency AB has to be made equal to —ba.
What is being revealed? The tautology of notation and process, redaction and lack of careful definition throws up a clear marker that something is not precisely being accounted for, nor is relativity being accounted for.,
The quick fix was to accept it in the notion of multiplication , but not in addition nd to move on. As a consequence we get a pattern of sign interchange as we go from summation to construction.
We can remove this sign alternation, but not if we want to do arithmetic. Over the years this facility has been honed to create powerful links for calculation. But inevitable complexity and tautology requiring flexibility of thought and analogy to gain " mastery"