Grassmann's Vision

Grassmann's Vision and Semantic Unfolding in Lineal Geometric Analysis: Ascending to the Exterior Algebra and Descending Back to its Commutative Shadow via Temporal Orders

The genesis of the notion of an exterior algebra originated in Grassmann's "Die Lineale Ausdehnungslehre'', where he set the purpose to develop a universal apparatus for research in geometry based on the notion of extension. Leibniz had already put forward this vision for the development of a genuine type of "geometric calculus'' not based on artificial choices of coordinates and other subjective conventions, but pertaining to the geometric objects themselves as an articulation of the process of semantic unfolding from the infinitesimal to the global. In this sense, Grassmann's theory of extension should be understood as a precise manifestation of Leibniz's goal in relation to geometry, i.e. referring to geometric magnitudes being amenable to procedures of measurement.

Composition and Unfolding into Layers

At the primordial stage, Grassmann's notion of extension is based on the conception of a lineal process of joining and separating at different nested, hierarchically organized layers, in which traces of prior layers in the process are replicated and adjoined to a present layer. The important thing that Grassmann realized and tried to explicate algebraically is that lineal extension requires necessarily the operation of multiplication, i.e. the operation specifying the type of product formation, or more generally, the composition type of geometric magnitudes that effect the unfolding of analysis to higher layers.

The semantic unfolding of geometric analysis according to Grassmann's lineal extension theory is based on the notion of a geometric type of product, i.e. a product pertaining to multiplication of geometric magnitudes. In Grassmann's conception the process of extension at the first layer initiates in terms of directed line segments, whereas implicitly also exists a "zero layer'' identified with scalars. The operation of multiplication is thought of indirectly in relation to the operation of addition, i.e. if there is an operation identified as addition that operates within a single layer, then any operation satisfying the distributive laws in terms of this addition is called multiplication.

In this manner, Grassmann introduced the exterior product as a means of composition or, "connection'' in his words, according to the satisfaction of the distributive law with respect to the addition of directed line segments, i.e. vectors in current terminology. What is characteristic of a distributive law is the property of invariance under transition from one side to the other in the equational expressional of this law.

The Exterior Product

The exterior product is an associative multiplicative operation that, with reference to the first layer, takes two directed line segments A and B at the first layer and produces a directed area or parallelogram at the second layer. This can be accomplished in two ways that they are not symmetric. Either the emerging parallelogram is the product of replicating B a total of A times along the linear extension of A and then concatenating or adding these based replications, or it is the product of replicating A a total of B times along the linear extension of B and then adding these based replications again.

In this way, they are conceived as opposite extension processes from the linear layer to the bilinear or area-bounding layer. In other words, they are mutually cancelling each other in the precise manner that their product is being oriented oppositely. The fact that the multiplication product area of the linear extensions A and B is directed or signed, according to the above, means that this geometric product is non-commutative.

We conclude that the multiplicative exterior product of two linear directed segments is non-commutative, i.e. it depends on the order of their composition in the two possible ways. In this way the orientation of the produced signed area is dependent on the order of composition, and thus it is signed.

If we identify the layer that the independent linear extensions A and B are located as the first exterior power space, then the layer that the product signed area is located is called the second exterior power space. In a totally analogous fashion, the lineal extension process proceeds to higher exterior power spaces using the property of associativity of the exterior product and the property of distributivity with addition.

Temporal Order and Geometric Replication

Analogously to the directed area element, the lineal extension process makes meaningful in a higher layer the directed volume element for three corresponding independent linear extensions. Due to the associativity property, this can also be conceived in a temporally ordered way by replicating the directed area of two of them along the linear extension of the third and then concatenating to obtain the directed volume element.

Due to the antisymmetric property of the exterior product if two elements at a layer have a common element of a lower layer, then their product is zero. This provides a conceptual understanding of the lineal extension process driven by the application of this multiplicative product. More precisely, it is named exterior because the non-nullity of the product of two extensive geometric magnitudes requires that each one of them is located geometrically to the exterior of the other making them independent.

The non-commutativity of the exterior product operation applied to two independent directed linear extensions is due to the anti-symmetry in the order of their composition, which gives rise to a specific orientation (clockwise or anticlockwise) of the completed parallelogram area.

The crucial thing in this case is that each of these directed linear extensions serves as a temporal order for the transportation or flow of the other along its extension. In this sense, there exist two distinct temporal orders with respect to which the parallelogram area can be potentially completed, but they are not equivalent, i.e. they differ in the way they induce an orientation on this area. Therefore, the existence of these two potentialities due to the choice between two independent temporal orders as modes of flow determination, represented by the non-commutativity of the exterior product, pertains to the apeiron aspect of reality.

Ontophainetic Platform and Spectral Shadow

The temporal interpretation of the ordered non-commutative exterior product of two independent directed linear extensions pertains to the apeiron aspect of reality. As such, it can only be reflected on an ontophainetic platform in statu-nascendi, manifesting the time-space of the pertinent present of its instantiation as a locus of synchronization, that is, in spectral commutative algebraic terms.

This is what we may call a commutative spectral shadow of the non-commutative apeiron that is expressed by the existence of two distinct temporal orders for the potential completion of the parallelogram area. The basic characteristic of this commutative shadow is that it is locally of an infinitesimal nature, or more precisely, it is of a cohomological nature.

This realization of the "actual taking place of reality'' bears two consequences: First, due to the locally infinitesimal nature, the commutative shadow pertains locally to some potential linear connection differential form that can be integrated along the boundary of the parallelogram. Second, the commutative shadow has meaning only in parameterizing locally a whole equivalence class of symmetric shadows, more concretely, only in the context of the cohomology class it specifies.

Descending and Ascending

In the same way that the non-commutative exterior algebra is an algebraic representation of Grassmann's lineal extension process from layer to layer starting from directed linear extensions, where the scalars or points are only implicitly assumed, the commutative shadow offers an inversion of this process, in terms of commuting infinitesimal units, where each one of them corresponds to a directed extension, culminating to the determination of points according to distinct linear extensions becoming dependent at that point.