It just occurred to me the famous problem with Euclid’s fifth postulate can also be seen hidden in the first four postulates, which are said to be true because they are intuitively obvious. Let's look at them:
- To draw a straight line from any point to any point.
- To extend a finite straight line continuously in a straight line.
- To describe a circle with any center and radius.
- That all right angles are equal to one another.
The 5th postulate stood out because "it cannot be directly observed through construction." This is because all constructed lines are necessarily finite, whereas the fifth postulate assumes that lines go on infinitely:
- At most one line can be drawn through any point not on a given line parallel to the given line in a plane.
The way I see it, other postulates can be shown to have similar problems. For example, for the first: It is impossible to draw a perfectly straight line. No matter how straight the line you draw via construction, we can simply enlarge the magnification to show that it is not perfectly straight. The only way around this is to draw a line within your imagination, instead of using any form of pencil and paper. Once we move to a proof within imagination only, anything is possible, including the 5th postulate.
Hm, a quick Google search tells me I've just joined an ongoing debate with such thoughts. Here's the related wikipedia page: Existence theorem.
By the way, this thought occurred while I was reading this interesting essay: On the Claim that Non-Euclidean Geometry Is Needlessly Over-complicated. I do not yet accept its premise, since I sometimes think Non-Euclidean Geometries may be more coherent than Euclidean, but it has ideas worth thinking about.
This post is loosely related to my nascent study of neural nets, because while studying backpropagation, I discovered I needed to learn about derivatives, so while studying calculus for the first time, I discovered I needed to learn about infinity, so while studying infinity and time and relativity (again, as these are things I actually study often with my ponderously slow approach), I discovered the author linked in the "needlessly overcomplicated" post above. I have more to write in this area -- this current post is tip of a larger iceberg -- but we'll get to that eventually.
While I'm here, note there is a related, long-pondered thought where I propose that there is no such thing as true equality as we commonly understand equality: never are there two things which are perfectly identical, so therefore equality is always an approximation at best.
(Edit, some time later I discovered Korzybski's writing which extensively explores this insight on the impossibility of equality.)
To be continued some day...