Learning how to read. I had great difficulty learning to read. At least part of that problem was a reflection of moving to many different schools and a chaotic home life. Some of it was poor teaching. Part of it was moving to another country for almost a year, which meant repeating the last half of the fourth and part of the fifth grade. This is a critical period because it was at the point when reading practice is switched into reading in subjects where a certain skill level is assumed---which I didn't have.
What saved me was reading science fiction in the early 8th grade. For some reason, I learned how to project my imagination into the stream of words and see a world that didn't exist. At some point I just learned how to read, without knowing it. Somekind of switch turned on. Another feature was saying the words out loud. If I could say the words quietly they eventually became a silent voice in my mind. Another part of this learning was the improved ability to concentrate for longer periods so I could get through longer and longer passages.
About the same time, this home practice for enjoyment transfered over to the world history series taught in middle and high school. I especially enjoyed studying maps and imagining places in the world along with photographs.
At some point probably in the 1990s I realized that reading and writing formed a world that could never be mastered. There was no greatest upper bound to the skill level and concepts. This realization came up when I was remembering my struggles in studying math. Math was another subject I went over and over and developed a near phobia---except when I finally understood something. Then I got a big thrill and went around for weeks on somekind of intellectual high.
The central understanding for me in math was that there were many different representations of a general concept. The easiest one for me to gain entry was through a geometric or visual representation. This approach was highly poo-pooed in early college math series. Nevermind. I filled pages and pages of drawings working my way through concepts. A mechanical draughting machine with Clearprint graph paper helped, tremendously. Here is a beautiful illustration of the relationship between an indefinite and definite integral:
http://commons.bcit.ca/math/software/viscalc.html
There are a lot of subtle concepts of space embedded in this relatively simple illustration. Start with idea that the drawing itself is a series of orthogonal projections (O) and a drafting machine makes these projections. When you draw this illustration you use 90 degree perpendicular projections with the machine head vertical arm at 90 degrees, horizontal arm at 0 degrees. That means you are doing an orthogonal projection. This is the core of what's called a Euclidean space. A Euclidean space is not exhausted by orthogonal projections (EO), since E allows for angle projections as long as line length and angle remain unchanged. Here is the wiki:
One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (that is, subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections. (See Euclidean group.)
http://en.wikipedia.org/wiki/Euclidean_space
I think the equivalence relation between logic and space can be illustrated:
(algebraic-set-logic) <-> (geometric-analytic-space).
Meditating on Felix Klein's Erlangen Program was a huge intellectual adventure---something of a mystic experience. Wow did I wish I could command that kind of imaginary force. The whole of the program, in its most abstract sense was the foundation of the Renaissance and artistic prospective of illustionary space---which on the intellectual plane pre-figured modernity. So somehow the human mind had already discovered the geometry of this projection, but never understood its profound philosophical-social-physical science implications. To pick an arbitrary date, say 1514 and Durer's Meloncolia the point in his life where reason threatened to leave him, along with his fat kid inspiration, love, the intellect, and mastery of the world in geometry... to Felix Klein's paper in 1872 was 358 years later. And Durer was a later generation if you pick Brunellschi instead sometime before he died in 1446.
The collective we have worked on these ideas for a very long time. Do they get us closer to grasping the nature of the universe? I don't think so, but I would hate to abandon the project. I mean what else is there to live for? Just to eat and die? Fuck you!
CG