“As the radius of the circle of what is known expands, we become aware of the expanding circumference of our own ignorance. ... We will increase our chances of success if we have the wisdom and humility ….”
This reminds me of a metaphor I heard at a lecture in December 2011, but a sphere of knowledge was used there, not a circle. The lecturer suggested that our ignorance comprises questions that we know how to ask but to which we do not yet know the answer (the “known unknown”). As our sphere of knowledge increases, so does our ignorance. Our knowledge is measured by the volume of the sphere, and our ignorance is measured by the surface area of the sphere. Our wisdom, suggested the lecturer, is the ratio between our knowledge and our ignorance. Because the volume of a sphere increases faster than the surface area and the radius grows, we are getting wiser. You can see this lecturer presenting this metaphor in a more recent YouTube video (https://youtu.be/hgQS3ZBLStE?t=1h26m27s).
Walking away afterward, I asked my (then) fourteen-year-old daughter what she thought about that metaphor. She thought it was pretty reasonable until I asked her why she thought our knowledge could be represented as a three-dimensional sphere in a Euclidean space. Then she started thinking. Maybe more dimensions? The greater the number of dimensions, the slower wisdom would increase in this metaphor (see https://en.wikipedia.org/wiki/N-sphere#/media/File:N_SpheresVolumeAndSurfaceArea.png).
Then I asked her why she thought there could not be any “bubbles of ignorance” inside our sphere of knowledge and why is it a sphere rather than some complicated “fractal-like” volume for which the surface area might increase faster than the volume increases. She said that probably she was a little tired by the end of the last of several long lectures she had listened to that afternoon and so had not really been thinking when the metaphor was presented.
Before we can speak about a geometry of knowledge, we need a topology of knowledge. Epistemologists specialize in the nature of knowledge. I sought an epistemologist who had published something about the topology of knowledge as a starting point for perhaps turning the metaphor into something useful. I found a fellow in Germany. Corresponding with him, I learned that he was using the term in a slightly different sense. He was indeed exploring interesting questions, but not the question that interested me. Further, he did not know of anymore who had approached my question. Consequently, I cannot speak intelligently about any topology of knowledge, let alone a geometry of knowledge.
Turning to wisdom, I am not convinced that the ratio of knowledge to ignorance would be a good way to measure wisdom (although I can see how such a definition of wisdom might be attractive to people who think they have gained a lot of knowledge in their lives). Rather, wisdom might be related to the ability to avoid making decisions and taking actions that result in great harm. If so, then increases in logos without equal or greater increases in pathos and ethos could mean decreasing wisdom.
I try to find the wisdom in Leonard Cohen’s “Story of Isaac”: “You who build these altars now to sacrifice these children, you must not do it anymore.”