One other point I will put here (Ha, it took a page, it it is more of a hologram than a point). Your visualization of points causing lines, lines causing circles, and that leading to spheres put me in mind of my visualization of the three dimensions.
I was trying to tie pure math to real world referents. Pure math, by definition, has no connection to reality, since practical math is defined as "math with a real world application".
To demonstrate, the square root of negative one is necessary for pure math, but is logically impossible as it would be both positive and negative at once. Thus, it was once taken to be the archetypical proof that pure math DID NOT DESCRIBE the "real World".
Then it was discovered that electronics requires this square root to describe something in practical engineering without which electronics cannot be explained, and from being "proof" of the "non-reality" of "pure math", it became or biggest piece of evidence that the only dif between pure math and practical is, we have found a practical use for the latter.
On the other hand, the world is flat.
The Euclidean "pure math" in which parallel lines never meet is practical in nearly every application on Earth. We live on Flatland, a sphere so large it has been "flattened" from our perspective to such a degree that the horizon appears as a straight line unless we physically turn around 360 degrees, and see that straight line nonetheless bends in a circle.
Then we notice ships sinking below the horizon, and after observation, begin to deduce the movement of the stars as we learned to navigate. Eventually, Astrologers and navigators found that, in practical terms, if one sails far enough north or south, parallel lines do meet, while if one travels east or west, latitudes are not parallel by def because they difer in length. We go thru Ptolemy, Copernicus. Lobachevski, and Einstein, and realize that, on a large enough scale, parallel lines behave accord to Lobachevsky and Einstein's geometry, and space is bent...but we still speak of sunrise and sunset!
When I taught math, I had some success in explaining the three dimensions in terms of real world referents.
You have given me a new, simpler insight into my insight, by pointing out that nothing or Zero is the reference point, from which somethingness emerges, and nothing is defined by its position on a positive/negative line
To my students. I started by pointing out that the line is light and/or electromagnetism. That is the real world referent for the pure math concept of “the line.” (“As Above, so below” a Babylonian insight that describes fractals. Note that even the straight lines of light have “wavelength and their own three dimensionality.)
The second dimension on a graph is left or right. On a graph, this is shown as a straight line perpendicular to the pos/neg line. But in the real world, the first, linear, dimension is north south, while the second is east west, not left right, and is parallel to the line through circularity, being bent.
And as circles create spheres, the third dimension is down and up, or more accurately, down and out. This is the bending of space itself, gravity.
That much we can grok
The next question is, what, if anything do spheres make?
Oh. the world is only flat on a small enough scale. As for intersecting spheres, I suspect they create holograms, and intersecting holograms create language, but that is the barest suspiscion.Statistics: Posted by Twain Shakespeare — 11 Jun 2012, 06:40
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