Truth Through Geometry
The first thought bubble emerging from the figure's mind is a loopy ring. A human child's first categories of the world are topological, not geometric (Jean Piaget *). This phase of evolutionary (species-level) and developmental (child-level) complexity is characterized by the inability to formalize topology beyond the immediate geometric forms found in nature.
The second thought bubble emerging from the figure's mind is the idea of a point. Defining a point implies defining the space the point exists within. Thus the innovation of the infinitesimal "point" relies upon the shadow-concept of a vast, or even infinite, "space". Think about it from the point of view of a RAM-limited verbally-combative chat bot: Without the "space" to make an argument (e.g. pass values into functions), how could it "make a point"?
The third thought bubble emerging from the figure's mind is a question: "Is the set of All points equidistant from some specified Point equivalent to a continuous 'Circle'?".
The fourth thought bubble is the following trilema: What is a Point? What is a Circle? What is a continuous line?
The fifth thought bubble is the obvious (to some) and esoteric (to most) next developmental stage of Geometry: Progressively more extreme abstractions to divine hypothesized or even merely possible types of relations among, for example, sets, categories, types, numbers, fields, networks, algebras, and grammars, and sentences and run on sentences and so on.
In the final panel, a compass traces out a circle (n.b. not a "Circle"). Experiments can reveal patterns that exist comfortably beyond the reach of theory. Seafarers we are, in this ocean of possible Geometries - the Compass is where we must find Direction. Each individual's compass is crafted by Evolution and tuned by local experience, leading to predictable temporal and spatial patterns of self-similarity of outcomes across idiosyncrasies. Chaotic systems, such as children, are apt to strongly diverge in their final logical stances when their axioms diverge only slightly (See: ZFC axiom of choice, or the Catholic "Great Schisms"). So how do we reconcile your compass and my compass, with your thought bubble and my thought bubbles? -- That so many humans can live together somewhat peacefully, is nothing short of a miracle.
Also see: Fuller, RB & Applewhite, EJ (1975). Synergetics; Explorations in the Geometry of Thinking.
Leave a Reply.