[ DimenTion and Imaginary Numbers ]

Me: What could be the relation between the scybernethics "dimention" (geometric quasi-dimension), related to the recursive spacialization (formalization) of a representation and the mathematical notion of imaginary numbers (Bombelli, Euler, Gauss and Schrodinger)?

NotebookLM:

The sources discuss the concept of "dimentionality" in scybernethics, emphasizing its quasi-bidimensional nature and its relation to the processual, but do not make a direct connection to imaginary numbers in mathematics. However, the sources offer several concepts that can be used to explore possible connections to imaginary numbers.

Here's an exploration of how scybernethics' "dimention" might relate to the mathematical notion of imaginary numbers:

• Quasi-Bidimensionality and the Dia-gram: Scybernethics uses the concept of quasi-bidimensionality to describe a space that is "not three, not four" [1-3]. The dia-gram is a formal tool that represents this concept [4]. It is a matrix with four quadrants, but the intersection of process and first-person perspective is self-reflexive, requiring the addition of a time dimension, making it not reducible to a formal, abstract spatialization [5]. This "necessary defect" is not easily represented by traditional spatial dimensions [5]. This relates to the idea of imaginary numbers by bringing in a dimension that is not easily reducible to the real number line.
• The Processual Dimension: The "dimention" in scybernethics is closely related to the "processual dimension", which is defined as what remains when form is subtracted [6-8]. Form is understood as anything that can be indicated or designated, while the processual dimension is dynamic, unobjectifiable, and vital for generating meaning [8, 9]. The processual dimension is not directly accessible through a third person perspective [5], but is felt as a dynamic, vital force that is linked to subjective, first-person experience [4, 7]. Like the imaginary component of a complex number, the processual dimension is not directly measurable but is essential to understanding the dynamic behavior of a system.
• Transduction: In scybernethics, transduction means the apparent transformation of a form into a process and vice versa [10]. This transformation is not a simple, one-to-one mapping, but rather a more complex dynamic, analogous to complex mathematical operations that bring together real and imaginary numbers [10]. For example, the mechanical gesture of calculation (a form) is transducted into an electronic equivalent (a process) through binary logic [11]. This could be related to the mathematical concept of imaginary numbers as numbers that allow for the mathematical transformation of otherwise impossible solutions to equations.
• The "Ambijective Gesture": The "ambijective gesture" is a cyclical process between objectification and subjectification, involving a co-determination of epistemological understanding and body-based meaning-making [6, 7, 12, 13]. It's a first-person process that is essential for making sense of ourselves [13]. It can be seen as a cycling gesture between objective and subjective perspectives, similar to the way imaginary numbers and their complex conjugates can be used to understand cyclical phenomenon in math. This cyclical movement between objectification and subjectification can be seen as being like an operation performed in the complex plane.
• Analogy and the Limits of Formalization: Scybernethics emphasizes analogical thinking over purely logical or formal approaches [9, 14]. Analogy in this context, is a relation of relations, a ratio² [15]. This highlights the limitation of formal systems to capture the entirety of experience [9]. The processual dimension, like the imaginary component of a complex number, challenges the limits of pure formalization and is necessary for understanding reality. This may suggest a relationship with the way imaginary numbers expand the scope of mathematical operations. This limitation is often expressed in scybernethics as the need to explore the "weaker" side of conceptual dipoles because it's the side that is harder to define [6, 7, 16]. This echoes the way imaginary numbers expand the realm of mathematical possibility.
• Recursive Self-Reflection: The concept of "dimentionality" in scybernethics also relates to recursive self-reflection, in which the process of observation and understanding includes the observer, leading to a "second-order" perspective [6, 7, 17]. The recursive nature of the process / first-person intersection in the dia-gram is self-reflexive [5]. The nature of imaginary numbers, with their relationship to the complex plane and their use in generating fractals and self-similar patterns, could be seen as analogous to the recursive aspect of self-reflection in scybernethics.
• Not Three, Not Four: The "Not three, not four" motto of scybernethics refers to the quasi-bidimensionality of the dia-gram and to the idea that a full understanding of reality requires more than the usual spatial dimensions [1, 2]. Similarly, imaginary numbers could be seen as a way to expand the space of numerical possibilities beyond the real number line and to address the limits of ordinary dimensional space. The idea of "not three, not four" also suggests an attempt to capture a dynamic complexity that traditional dimensions fail to address [5]. This links to the idea that complex numbers, including imaginary numbers, are needed to accurately describe complex, dynamic phenomena.

While the sources do not explicitly link scybernethics' "dimention" to imaginary numbers, the concepts of quasi-bidimensionality, the processual dimension, transduction, analogical thinking, recursive self-reflection, and the limitations of formalization in scybernethics offer an intriguing framework for understanding the necessity of imaginary numbers in mathematics to move beyond the real number line in order to capture more complex dynamic phenomena.