Nobody is practically concerned with the "incompleteness" aspect of Gödel's theorems. The unprovable statements are so pathological/contrived that it doesn't appear to suggest any practical statement might be unprovable. Consistency is obviously more important. Sufficiently weak systems may also not be limited by the incompleteness theorems, i.e. they can be proved both complete and consistent.
You just reminded me of having to prove that math signs work and do what they do from basic axioms to integers and rational numbers using logical proofs... Damn that was interesting but SO tedious...
Well, at that level I think it's more to show you know how to prove it. You're working under the assumption the axioms of the system you've been told work.
"Just look at it". Visual proofs are not proofs. They're demonstrations. For a species that is so easily fooled by things we see, visualization cannot be taken as rigorous proof.
Yes but what if one side is so slightly curved that it's invisible to the naked eye? Then your total angles would be 179.99 degrees and it's not a triangle.
From a calculus perspective, you may be able to define it as infinitely many angles infinitely close to one another, but I don't think that'd be a particularly good definition.
I remember using a compass or protractor a few times in geometry class and feeling I could never ever possibly need this because I have no interest in designing things.
A decade later I found myself working for an aircraft manufacturer and needing to verify angles and dimensions for fabricated parts was understandably a critical part of the job.
I loved geometry. It made algebra make sense. Plus I had a really awesome geometry teacher. He looked like Shel Silverstein and was super pumped every day to teach math.
I loved geometry. It's the class where I first got experience programming. I just sat in class programming stuff on my calculator not really paying attention. I did fine in the class luckily.
Totally unrelated, but I (30 yo) recently realized I'm almost certainly ADHD. There definitely weren't any identifiable signs before that people should have noticed...