Whitehead's Geometry: Mathematical and Experiential
Extensive connection, extensive abstraction, straight lines, flat loci, and strain feelings
In Whitehead’s Process and Reality, space, time, and the interconnectedness of actual entities are central to his metaphysical framework.
Extensive connection, extensive abstraction, straight lines, flat loci, and strain feelings are all key concepts that reflect different aspects of how entities are related spatially, temporally, and experientially in the universe.
Extensive Connection
At the foundation of Whitehead’s metaphysics is the concept of extensive connection, which describes how actual entities (the fundamental building blocks of reality, understood as events or processes) are interconnected within the spatio-temporal continuum. These entities are not isolated, static points, but rather dynamic events whose relationships to other events define their positions in space and time. Each actual entity is embedded in a web of relations, and these relations constitute the structure of the universe.
This interconnectedness is extensive because it spans both spatial and temporal dimensions, making the universe a continuous, relational whole rather than a collection of discrete, self-contained parts.
Extensive Abstraction
While extensive connection describes the real, relational structure of the world, extensive abstraction refers to the process by which we conceptually simplify and analyze these relations. Extensive abstraction involves isolating certain features of the interconnected web of events, such as points, lines, or regions in space. It allows us to focus on particular aspects of space and time, such as specific locations or directions, without needing to account for the full complexity of the universe’s relational structure.
For example, points, straight lines, and flat loci are not fundamental realities in Whitehead’s system but are abstractions—conceptual tools that help us interpret the spatial relations within the extensive continuum. Points are abstract loci derived from the interconnected field of events, while straight lines and flat loci are abstracted representations of spatial continuity and direction.
Straight Lines and Flat Loci
Straight lines and flat loci emerge from the process of extensive abstraction. A straight line is a geometrical abstraction that simplifies the relational structure of the extensive continuum into a linear path between two points, while a flat locus is an abstract, two-dimensional surface within the continuum. These abstractions help us navigate and understand space by providing simplified models of the more complex reality of spatial relations.
However, in Whitehead’s metaphysics, these geometrical abstractions do not capture the full, lived experience of space and time, which includes the forces and tensions that arise in the actual connections between entities. They serve as useful but incomplete representations of the deeper, dynamic relationships that actual entities experience.
Strain Feelings
Where geometrical abstractions like points, straight lines, and flat loci help us conceptualize spatial relations, strain feelings reflect the experiential side of these relations. Strain feelings are the tensions, forces, and resistances that actual entities experience as they interact spatially and temporally. These feelings arise from the physical extension of entities and their relational positioning within the continuum.
For Whitehead, actual entities "prehend" or feel their relationships with other entities, and strain feelings are one way they experience the spatial and temporal tension inherent in these relationships. As entities interact within the extensive continuum, they feel the push and pull of their spatial and temporal connections, generating feelings of strain. This experiential aspect is an important part of how entities are situated within the continuum and reflects the lived reality of spatial relations, which goes beyond mere abstraction.
The Role of the "Point"
In classical geometry, a point is defined as an infinitesimally small, non-extended location in space. However, in Whitehead’s metaphysical system, a point is not an independent, fundamental entity but an abstraction derived from the extensive connections of actual entities. It is a conceptual simplification that helps us focus on specific aspects of spatial relations. Points are abstracted loci within the continuum, useful for describing spatial relations but ultimately derived from the interconnected web of actual events.
In this sense, points, like straight lines and flat loci, represent the results of extensive abstraction. They help us understand spatial relations in simplified, geometrical terms but do not capture the full reality of the extensive connections between actual entities, nor the strain feelings that entities experience within those connections.
How They All Relate
- Extensive connection describes how actual entities are interconnected within the spatio-temporal continuum, forming a relational web.
- Extensive abstraction is the process of simplifying and conceptualizing these connections, leading to abstract entities like points, straight lines, and flat loci.
- Straight lines and flat loci are geometrical abstractions that represent simplified, idealized versions of spatial relations.
- Strain feelings are the lived, felt experiences of tension and force that actual entities experience in relation to one another within the extensive continuum.
- Points are abstractions from the extensive connections of actual entities, helping us conceptualize specific locations in space but ultimately derived from the broader relational structure of the universe.
While geometrical abstractions like straight lines and points offer useful ways to simplify and model space, strain feelings highlight the experiential side of these spatial relations, revealing how entities actually feel and interact within the spatio-temporal continuum. Both aspects—abstraction and experience—are crucial for understanding Whitehead’s philosophy of space, time, and the interconnectedness of actual entities.