The dissemination of Grassmann’s ideas to the larger mathematical public in Germany intensified with the interest in this scholar’s achievements shown by Alfred Clebsch (1833–1872) in the early 1870s. The premature death of Clebsch prevented him from deepening his adaptation, but the friends and disciples in the Clebsch school continued the reception of Grassmann’s work. I intend to show the important role of Clebsch’s school, and in particular that of Felix Klein (1849–1925) in making Grassmann’s work ac
Basis and dimension are two elementary notions in the theory of vector spaces. The origin of the term ‘basis’ comes from the possibility of expressing any element of a given set as a linear combination of the basis elements. Therefore, the origin lies in a question of generation; on the other hand the condition of unicity brings out the question of independence. The connection between generation and dependence is certainly one of the most interesting characteristics of the concept of basis: any maximal set of independent vectors or any minimal set of generators, is a set of independent generators and vice versa and such a set is a basis. Moreover, the dimension, beyond its “natural meaning”, is the merging point from which the question of invariance is to be drawn out. Indeed, the fact that all bases have the same number of elements entails two results: there cannot be more than a certain number of independent vectors, and fewer than the same number of generators. With a suitable starting point in the presentation of definitions and first properties on dependence and generation, these different aspects seem quite logically connected and easily explainable, but historically, the development of these two concepts was less straight-forward. For various reasons, in the approach to the concept of basis, the connections between dependence and generation were not always exhibited. Therefore the concept of dimension could only partially be drawn out, and some of its aspects were smothered, or even considered as obvious and assumed to be true without proof. On the other hand, the relation between the dimension of a subspace and the rank of any system of linear equations by which it can be represented, played a role in the history of the concepts of basis and dimension.
Michael Crowe has shown in his History of Vector Analysis (Crowe 1985) that Grassmann’s earliest mathematical work, the Theory of Tides, contains almost all of the key vectorial notions that appeared four years later in the Ausdehnungslehre. On the other hand, Grassmann never published the Theory of Tides—it first appeared in 1911 in Grassmann’s collected works (Grassmann 1840). Many of the physical applications, however, did appear in 1877, the last year of Grassmann’s life, as “Die Mechanik nach den Prinzipien der Ausdehnungslehre” in the Mathematische Annalen. The only essential difference between this later version and the 1840 appearance appears to me to be ostensibly minor changes of notation. I believe, however, that it is just this difference that points to the contribution that the Theory of Tides can make to our understanding of how the Ausdehnungslehre came to be.
So I start wit an undefined scatter of points. I distinguish two points A,B. they are completely arbitrary, except the tool I use for a synthesis process imposes some limitations.
The first construction action I define is to use a pair of dividers and fix them on A and B. this gives me an instrumental copy of something I will call displacement §
§AB is an algorithm or method. It does not affect the points per se, it affects the observer and the tool used.
Now does it make sense to use a divider? Only in the plane or in contact with an in elastic surface, which nevertheless is markable. Already you can see how this method of analysis/ synthesis sets certain constraints to be achievable or pragmatic.
§AB makes sense only in a certain set of circumstances.
Confining the observer and actioner to those circumstances enables me to write
§AB = §BA.
But in fact this says nothing about points. It says that the measuring instrument ends up in the same fixed configuration. Thus we immediately fall upon the notion of an exterior algebra!
Leaving the points to one side, and concentrating on the dividers I can compare gape and set up an additive Alebra of gape. This is an exterior age ra, a prior one necessary before I can develop a concept of displacement as gape, and the practice of measuring using gape.
So dosplacent is an exterior algebra associated to the scatter points. It does not synthesise anything from points entirely of points. This observation is at the heart of the notion of an interior algebra and it is the notion of closure. This idea is that if we are talking points then everything should be about points? Later we will see that the reatriction is even stricter.
So a second synthesis I could do is construct aset of points from the scatter set which are centred around A with the same displacement §AB. This forms a spherical surface of points. However,nab spherical surface is not a point, so now I have constructed a new object that is not a point. It is an exterior topology even though it is embedded in the scatter points.
There is an algebra constructive on this surface, but we tend to call it a hyper geometry. I just call it a Spaciometry. However it cannot be constructed without some prior Spaciometry of these spherical shells,
¢AB is not the same as ¢BA.
However the two do intersect and form a new collection of points called a circle which is the identical form for both
So €AB = €BA.
We can construct an exterior algebra on this circle between the points, which is again embedded within the scatter points but does not include A and/or B .
The first interior algebra, one which is closed for all it's elements, which is set talk for it includes A and B is the set of points on any curve that goes through both A and B.
However, since we call this a curve or a line it is not a point, and so it is called an exterior construction, even though it is really interior!
So what is an interior construction for A and B ? Currently it is defined as the midpoint , a single point on the exterior product straight line between A and B.
This confusion is not uncommon in a developing subject. At the level of a point algebra it seems a moot point about which construction is exterior and which is interior, but the notion of multiple and extensive use of a Metron underpins it. A line is clearly this kind of extensive multiple form made of points..
To distinguish the two the + sign is used
So p(A+B)=(A+B)/2 = p(B+A)
This construction in the plane is the standard line bisection, but in 3d space it is a lot more complex, but still true..
The upshot is that most construction actions in Grassmann analysis produce or invoke exterior algebras. A few with stringent conditions create interior algebras.
Now I have to discuss the use of the contra notion and the – signal.