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Introduction
Warren McCulloch‘s seminal 1961 essay, “What is a Number, that a Man May Know It, and a Man that He May Know a Number?“, laid the groundwork for what would later be recognized as second-order cybernetics[1]. This profound question continues to resonate within the fields of cognitive science, cybernetics, and philosophy of mind. In this article, we’ll explore how the concept of number emerges from embodied cognition and how this perspective aligns with the principles of second-order cybernetics and scybernethics.
The Enactive Origins of Number: Processual Gestures
From a scybernethics perspective, numbers are not abstract entities existing independently of human cognition. Instead, they are enacted through a historical and phenomenological process, originating from the fundamental gesture of designation or indication[1]. This enactive view posits that our understanding of numbers is grounded in embodied experiences and sensorimotor activities.
Consider the etymology of “calculus,” which refers to small stones used for counting. This linguistic connection highlights how abstract mathematical concepts are rooted in concrete, physical actions. In ancient time (3400 BC), imagine Mesopotamian Sumerians merchants and geometers using pebbles to develop common references when sharing land or exchanging goods. They have most likely enacted calculation from the recurrence of repeated gestures of common designation or body measurement, like piles of goods of the same volume or steps/cubits for example, then symbolized and abstracted by small pebbles to transcend direct presence.
So abstract numbers are in this deep understanding the emerging result of an enkinaesthetic phenomenological sedimentations of gestures (i.e. habits metastabilized through homeostatic sensorimotricity and social coordinations), and not things-by-themselves as our inner and natural processual forgetting seems to imply (cf. Plato’s anamnesia). It exemplify how numerical cognition emerges very likely from practical, gestural and embodied interactions with the world.
Transduction and Double Cuts
The development of numerical systems illustrates a key concept in scybernethics: transduction. This process involves a process/form leap from sensorimotor gestures to abstract symbols used for intersubjective communication, agreement and coordination of action (cf. Maturana’s “languaging”). The invention of numbers represents a significant “double cut“:
- A separation between concrete experience and abstract representation
- A division between individual cognition and collective agreement
This transduction enabled the development of more sophisticated mathematical thinking and laid the foundation for scientific and technological advancements.
Second-Order Cybernetics and the Observer
McCulloch’s essay, though predating the formal articulation of second-order cybernetics, embodies its core principles[2]. Second-order cybernetics emphasizes the role of the observer in systems, recognizing that the act of observation influences the observed phenomenon[3]. This reflexive stance is crucial when considering how humans come to know numbers and how numbers shape human cognition.
In the context of numerical cognition, we must consider not only how humans understand numbers but also how the concept of number influences human thought and perception. This circular relationship exemplifies the self-referential nature of cognition that is central to second-order cybernetics.
The Cartesian Cut and Beyond
The Cartesian separation of mind and body, as well as subject and object, has profoundly influenced Western scientific thought, leading to the institutionalization of modern science. However, the enactive perspective on number challenges this dualism, emphasizing the embodied and situated nature of mathematical cognition.
As we move beyond the Cartesian paradigm, we can appreciate how numerical concepts arise from the interplay between embodied experience, cultural practices, and abstract reasoning. This holistic view aligns with the goals of second-order cybernetics in transcending simplistic subject-object distinctions.
Technological Implications
The development of computing machines represents another significant transduction in human history. Claude Shannon’s abstraction of communication as information, separated from embodied meaning, and the work of Alan Turing and John von Neumann in abstracting computation from matter, mirror the earlier abstraction of number from physical counting[5].
These technological developments highlight how our understanding of number and computation continues to evolve, influencing and being influenced by our cognitive models and technological capabilities.
Conclusion: Towards a Reflexive Understanding
McCulloch’s question about numbers and the humans who know them invites us to adopt a reflexive stance towards cognition and knowledge. By recognizing the enacted, embodied nature of numerical concepts, we gain insight into both the power and limitations of mathematical thinking.
The scybernethics perspective, with its emphasis on enaction and embodied cognition, offers a rich framework for understanding the circular relationship between numbers and human cognition. As we continue to explore this relationship, we may find new ways to bridge the gap between abstract mathematical concepts and lived experience (cf. the scyberspace), leading to more nuanced approaches in fields ranging from education to artificial intelligence.
In embracing the principles of second-order cybernetics, we acknowledge that our understanding of numbers is not a discovery of pre-existing truths, but a creative act of cognition shaped by our biological, cultural, and technological contexts. This reflexive awareness opens new avenues for research and invites us to reconsider the fundamental nature of mathematical knowledge and human understanding.
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Citations
[1] http://homepages.math.uic.edu/~kauffman/NUM.html
[2] https://en.wikipedia.org/wiki/Second-order_cybernetics[3] https://journal.emergentpublications.com/Article/ee387e54-c659-492c-86b4-26bae4bf69c5/github
[4] https://www.pangaro.com/glanville/Glanville-SECOND_ORDER_CYBERNETICS.pdf
[5] http://pespmc1.vub.ac.be/Papers/Cybernetics-EPST.pdf
[6] https://www.researchgate.net/publication/235251005_Second-order_Cybernetics_An_Historical_Introduction
[7] http://www.eolss.net/sample-chapters/c02/e6-46-03-03.pdf
[8] https://www.worldscientific.com/doi/pdf/10.1142/9789813226265_0001
Reference
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.