Thursday, February 17, 2011

axioms of ideas

  A while back I thought of this idea that is somewhere between math and philosophy. The goal was to mimic geometry but apply the rules to ideas instead. So, in geometry they have points, I thought in my system I would have ideas like big, small, loud, inside, outside, etc as "points". In geometry the second thing they have is lines made up of points. In my system I have comparisons as "lines". An example line in geometry might be said to be "drawn" between two points. In mine, I "draw" a comparison between two ideas. Like "big is to loud". This compares big and loud. In geometry they have circles, in mine I have concepts. A concept encompasses several ideas, more general concepts include more ideas. The word encompasses is even similar to that in geometry of drawing a circle with a compass to include points on the inside. Also in geometry they have an axiom about parallel lines, remember in mine a line was a comparison like "big is to loud", so a pair of parallel comparisons becomes an analogy... "big is to loud as small is to quiet". If you took the SAT's this might look familiar. An analogy is a correlation between two comparisons... correlation in Latin literally means "running alongside" the way two parallel lines run alongside one another.
So to recap:

  1. Ideas = points
  2. Comparisons = lines
  3. Concepts = circles
  4. Analogies = pairs of parallel lines
Now for the more mathematical bit I need to set up the definitions formally the same way they do with geometry.

On Wikipedia it gives these as Euclid's axioms:
    Let the following be postulated:
    1. to draw a straight line from any point to any point.
    2.To produce [extend] a finite straight line continuously in a straight line.
    3.To describe a circle with any center and distance [radius].
    4.That all right angles are equal to one another.
    5.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.

And these are the ones I made with the words in bold above instead of the elements of geometry.
    let the following be postulated:
    1. to draw a comparison from any idea to any other idea
    2. to extend a comparison continuously through more and more ideas
    3. To describe a concept with any idea as it's center and to the increasingly more general (including more ideas)
    4. That all orthogonal comparisons are orthogonal in the same way    (More on this below)
    5. For every comparison there is an analogy, the second comparison not including elements of the first

Ok, so I've made a geometry with different words than they used in the original, right? 

An extended comparison is comparing even more ideas such as "big is to loud is to bright" (this could go on and on)

A pair of orthogonal comparisons is like "big is to loud" and "big is to wet" They're orthogonal in the sense that they both include the same point: {"big"}, and "loud" and "wet" are as different from one another as they can be without being closer to opposites.  This visual for this is "big is to loud" and "big is to wet" are like perpendicular lines that both start at the word "big".

The notion of parallel relates to analogous comparisons like this one "soft is to hard as inside is to outside" This is an analogy relating opposites, but "inside" might be on the extended comparison "inside is to core is to brain"(things within other things) and "soft" on the extended comparison "soft is to dim is to quiet"(matters of degree) and the two extended comparisons not share any ideas in common.


So anyway the idea is that having defined these axioms in the same way as the ones for geometry, any proof in geometry will apply to these axioms of thought as well. 

The cool thing is that I made a system for the words: ideas, comparisons, concepts, and analogies that is exactly the same as the system for geometry using the words: points, lines, circles, and parallel lines, everything that has been proven in geometry over the centuries is instantly known about this new topic as well. 

The best part was the funny way the words for the geometrical ideas were always very fitting to the way I had my system worded. Like a comparison is "drawn" between two ideas and a line is "drawn" between two points...every single word I used in mine was like that... Like people were thinking about ideas geometrically already but hadn't put it all together.  

1 comment:

  1. It sounds a bit like Descarte taken literally. I really like the idea however and need to think about it more before I comment in detail.

    Justin Giuffrida