Date: September 14th, 2021 1:32 PM
so i actually decided to do geometry, and i have already completed eighty percent of the textbook Geometry: A High School Course, though without doing the exercises at the end of each chapter, although i have gone back now and am re-reading the textbook and taking notes and doing the exercises; it is a better way, but i noticed i learned about 85 percent without all that. however, i am being a completist and want to make sure i digested all of it properly. i stopped after vectors, and boy was that a super difficult section, i got stumped many a time. i only have the chapter on transformations, and then the last one on isometries left, and then i can say i actually read a textbook on geometry.
but that is, kind of, beside the point. oh and by the way, im super high on meth right now. anyways, moving on: the main thing is that i have in parallel (its a pun) been reading another book or two on geometry, one more classic in style, written in 1880s about Euclidean geometry, called Plane and Solid Geometry, where there is definitely no single mention of transformations, coordinates, or vector dot product. it is pure geometry, and it is fun in a different way, but it does not have the guided sense of building a spider's web of mathematical maturity as one goes up the ladder of pages. although there has been a moment in that book that is perhaps higher than any peaks i have experienced in the lang text. it has to do with finding the circumference of the circle as limit of an inscribed polygon perimeter, the beauty of it being that it relies on the idea of triangle inequality to show why the circumference is in fact the limit of this process. but overall, it is a lesser text. anyways, i also got another book going, called simply Geometry, and it is only concerned with geometrical constructions. it is a translation of a soviet mathematics club text, slightly idiosyncratic in its content choice and overall methodology. so long story short, i get to the section on parallel lines, and i read this part --
"Two lines that do not intersect, no matter how far and in what direction we extend them, are called parallel lines.
It is not possible to verify directly that two lines are parallel. How can one be sure that they will not intersect a mile from here? In the usual geometry, which is called Euclidean geometry, we assume that given a line and a point that is not on this line, there exists one and only one line passing through this point that is parallel to the given line."
well. i got to thinking, and here is my thought of the day... my deep meth induced psychosis laden commentary on it:
Euclidean geometry assumes tacitly that the local property of parallel lines extends into the global condition. It has no proof or verification that this is so. It is a mental construction, and therefore Euclidean geometry is itself a mental construction, though one that makes absolute sense locally, within the bounds of experience. in my view its validity in the global condition is likely to hold, but never to be proven strictly on a logically deductive basis. every local geometry is necessarily Euclidean, and likely so is global geometry; however, we have no way of strictly verifying this, either through direct experience or experiment, logistically, or within a logical-deductive system based on reason alone. It will forever have to be taken as fact by faith on a global scale, and it is the geometry of space on every local scale -- though once again, we only have experience to guide us, and strictly logically speaking, since we cannot show this to hold for the global case, the theorem brakes down for the idea of Euclidean geometry holding for EVERY local case. Though by the isotropy and homogeneity, etc of space, it can be deduced that Euclidean geometry, holding in every local space known to man so far, on every scale, should indicate that it is the global case as well. Our physical intuition is correct, the non-euclidean geometries, though logically self-consistent, and thus real on that level, are not in fact representations of real space. not to mention, our entire logical system breaks down as in the space of a "singularity" of a black hole inside the event horizon, or the theorized big bang singularity at the "beginning" of time -- in such notions of global scenario for entire space that exists containing all things we are running into a similar scenario, except we have no mysterious paradoxical singularity to stand for it, but I suspect it is another wall where reason cannot, maybe as yet, penetrate itself.