Parallel thoughts meet at infinity, similar to how parallel lines meet at infinity in Euclidean geometry. There are other kinds of Non-Euclidean geometry and other kinds of thinking patterns, but a theory of thinking can be based on these various geometries which would be derived from basic axioms from which theorems could be in turn be derived. Surely this has been done, and probably done by the Classic Greeks. It is a kind of thinking based on a rigid consistency to which I am ill adapted.
This formal thinking is quite different from my feedback form, which I like to think of as based on experience, postulated cause and effects, and feedback to help form new actions which are more appropriate to the presenting situations. It seems to me that all thinking is wild and speculative—even the formal types seem to have an abundance of non-sense—and it is the feedback from reality that is most helpful for predictable future actions. That has been my trend in the past, but for this exploration I will set up some standard Euclidean axioms and begin a formal type of thinking about thoughts and the conduct that may be derived from these thoughts. This will rarely lead to worthwhile actions, but it might lead to some humorous thoughts.
“Let the following be postulated“:
- “To draw a straight line from any point to any point.” This becomes – “It is possible to draw a straight line of thoughts from any thought to any other thought.”
- “To produce [extend] a finite straight line continuously in a straight line.” becomes, “A straight line of thoughts can be drawn between any two thoughts and extended indefinitely.”
- “To describe a circle with any centre and distance [radius].” becomes – “A circle of thoughts can be drawn equidistant from any particular thought.”
- “That all right angles are equal to one another.” becomes – “Thoughts can be opposite to other thoughts and others can be equidistant from both of those.”
- The parallel postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” becomes – “A line of thoughts intersecting parallel lines of thoughts will produce the same outcomes of thoughts in each of the parallel lines of thoughts.”
The Elements also include the following five “common notions”:
- “Things that equal the same thing also equal one another.” becomes – Ideas that are equal to the same thing also equal one another.
- “If equals are added to equals, then the wholes are equal.” becomes – If equal thoughts are added to equal thoughts, then the wholes produced are equal.
- “If equals are subtracted from equals, then the remainders are equal.” becomes – If equal thoughts are subtracted from equal thoughts then what remains is equal.
- “Things that coincide with one another equal one another.” becomes, Thoughts that coincide with one another equal one another.”
- ”The whole is greater than the part.” becomes – The whole of ideas is greater than the parts of the idea.