Grassmann began his insight by compounding two Euclidean notions , the semeion and the Monas. This combination i have called the pragmatic semeion.
This combination provides a kind of elastic material which is extensible, the marking substance or an elastic void which is extendable. This is the origin of the wave particle duality of space. For example the marking substance may be a fluid, but it is extensed over a surface and eventually runs out. Thus it isa deformable substance that has a limit. The marking matrial may be a solid like chalk or graphite, in which case it is spread over a suface by friction rubbing off and removing small parts or particles of the material and attaching it upon the surface. This too runs out and has a limit, but it lasts longer, a time comparison as well as a surface and length comparison can be made.
The 2 marking materials are analogous in application, and we can use one to under-stand , that is to found the explanantion of the other. this is a com-prehension, a holding together of two experiences in order to explain what is the same and what is different about them. This is a formal tautology, and tautologies underpin all our schema for exposition and explanation.
Thus at the analysis of the seeion we have a triple" void, particle and fluid. They all have wave properties, but it is mor pronounced in the fluid experience. They all have intensities and densities and thus are perceptible, and they all have continuity contiguity and discontinuity attributes. These exist at the level of the pragmatic semeion, and of course, as the stoikeioon was extended into smaller and smaller scales of space this triple extends into the quantum description of reality.
However, Grassmann was concerned at this stage with establishing the properties of his pragmatic semeion, which he denoted as punkt, or point.
It is necessary to distinguish between the point in Grassmanns Audehnungslehre and the general notion of poin in euclidean philosophy. The general "point" in Euclidean philosophy is the semeion, and this is a subjective notion that really is not a point at all! It is a supernatural mark. The modern notion of a point is in fact a combination of the mark, the limit process of disaggregation, and the monas.
Sometimes i use the word monas where you might feel it is more appropriate to use the word monad, but the 2 words are related but distinct. A monad is an instance of monas, and monas is a much larger idea than en or one or unit. Monas is that subjective experience of an instance of an idea or object schema which makes us cry out "one!". One way of describing it is using set theoretic language very precisely. This in part is what A. N. Whitehead and Bertrand Russel Attempted to do, inspired i might add by Grassmann's conception
For my purposes, i find it simpler to note the combination of the notions of Monas and Semeion into the notion of punkt as being sufficient denotation at this stage.The overall context is understandable: artisans and engineers and manufacturers and artists require a pragmatic semeiom in order to construct and form and shape.
To construct we therefore need the continuous nature of the pramatic semeion and at the very least the contiguous nature. There are 2 ideas of continuity but really it is a tautalogy between continuity and contiguity, a combination used to comprehend the ideas of continuity, to understand this experience. Again it relates t the fluid and the particulate and the void. On the back of the fluid we can analogise the particulate as amalgamated into a continuum, with the very last possible status of continuum between 2 particles being contiguity. Thus amalgamation structures are combinations of monads that overlap, while aggregation structures are combinations of monads that are contiguous.
These are 2 different apprehensions of space. Monads could be just contiguous, allowing then a uniformity of density to exist in a monadic description of space, given a unit of density or intensity. However monads could overlap. and this allows for a description of space in which the density is non uniform, and variable. This amalgamation form of monadic combination passed on naturally to the employment of Euclidean geometry to chemistry. In a very real way the chemical notation calculus draws on the same set of ideas that Grassmann develops more fully in his Ausdehnungslehre.
When we extend the pragmatic semeion, we therefore need a notion of continuity. As an arisan, there are many continuous sructures to utilise, from plants to rocks to flowing rivers etc. For Grassman a fortunate set of arguments influenced his understanding of the set of numbers: Dedekind was arguing strongly for a set called the real numbers, and these were a continuous set of numbers different to fractions and thus rational numbers. With the help of Cantor he was able to dupr most mathematicians into distinguishing between the rationals and the reals. There is no real distinction despite what you may be told.
With the "number" concept queering the pitch no one could see the connection to thte continuous nature of the monas, nor could he comprehend the Eudoxian scheme of comparison that underlay the monas and the arithmoi. Thus they all went along with the idea that they were discovering something "new". As it turns out they were correcting old and inadequqte notions of the arithmoi, but still not "getting it."
Grassmann to be fair did not fully get it, but he saw the connection between the semeion and the monas and how the continuity in the reals enabled him to describe an elastic extension, and this could be done by multiplication.
We have to cut back to Hamilton to fully comprehend the algebraic nature of multiplication. In his paper on couples Hamilton fully exposits the algebra of the arithmoi on the basis of an ordinal monas. Hamilton derives his arithmetic in all the stages upt to and including the reals in the most groundbreaking and beautiful way. He achieves in sketch what Russel and Whitehead laboured over for years to establish the link between logical relations and arithmetic. However it was an arithmetic of vectors, a vector algebra, something which mathematicians were struggling with, especially to comprehend the imainaries. Hamilton succeeds in comprehending the imaginaries in an alogous way to how he comprehends the arithmetic of the reals, but he fudges one step: how the imaginaries are numbers. What he in fact shows is that imaginaries are analogous vector using a different monas that canot be amalgamated with the reals, but are an amalgum of two real arithmetics with different vectors. It is the vectors that are joined and treated s a new monas, and the way they are joined is specified by a set of relations.
It is this specifying of how monas's are to be related or amalgamated which appears arch, because we have been fed the lie that "numbers" are natural, not constructed. The euclidean view, following Eudoxus is that the Arithmoi are constructed. What is natural or phusis is the monas, which by the way is actually supernatural like the semeion! The definition of monas naturalises it. It has to be a real object drawn from a number of like objects as a typical example. Thus when Grassmann combines the semeion and the monas he supernaturalises the monas and naturalises the semeion! The point then is a force to be reckoned with!
Using this etendable continuous nature of the practicle semeion but islolating the extension to the reals and the unit to the pramatic semeion Grassman had a multiplicative form of extension, or "production", but more importantly he had an orientation for that production. His notio combined both extension and orientation he realised. He does not explore orientation as i will in a later post, but accepts it straight from Euclid as an important aspect of the pragmatic semeion, which he never lets go.
The next greeek idea was the gnomon. This actually joined to geher 2 of his punkt extensions into a new monad which was in fact the edges of a parallelogram. Euclid in his construction passes from a formless extent to a form using the gnomon. The epiphaneia form he explores is the next level of monas, which subsumes within it the punkt monas in an easily apprehensible way.
The algebra of this form thus has sub algebras amd combinatorial algebras, which Grassmann fully realisd nd began to explore. His choice of notation was entirely new and thus obscure. It reequires attentive reading to understand the connection he is making. If you have read Hamilton you will understand why. The ground covered seems so familiar it is an effort to see where th difference that makes the difference is. I hope that by distinguishing in the greek i might excite the attention to the differences. In Grassmann's day however, those he would be writing for would be classical scholars, and the use of differences in language to excite the attention would not be as successful an approach as coming up with new appealing notation.
Fortunately over time, both these things have happened and Grassmann's notions may be more accessible as a result. However, what i am saying is that Grassmann is in fact drawing out a much older conception held within Euclid's Stoikeioon.