the point isn't to prove that the triangle is a triangle it's to prove that the system of mathematics you made up actually works
this post was submitted on 17 Jul 2024
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hey! get out of here with your math!!
Help! He's logicing all over us!
Until you prove that you can't prove that the system you made up works.
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.
I think the statement "this system is consistent" is a practical statement that is unprovable in a sufficiently powerful consistent system.
Can you help me understand the tone of your text? To me it sounds kinda hostile as if what you said is some kind of gotcha.
Just explaining that the limitations of Gödel's theorems are mostly formal in nature. If they are applicable, the more likely case of incompleteness (as opposed to inconsistency) is not really a problem.
Oh, what if the Riemann hypothesis is such a statement then? Or any other mathematical statement. We may not have any use for them now, but as with all things math, they are sometimes useful somewhere unexpected.
It's extremely unlikely given the pathological nature of all known unprovable statements. And those are useless, even to mathematicians.
Math is also used to make a statement/model our universe. And we are still trying to find the theory to unify quantum mechanics and gravity. What if our math is simply inconsistent hence the theory of everything is not possible within the current mathematical framework?
Sure when you are solving the problems it is useless to ponder about it, but it serves as a reminder to also search for other ideas and not outright dismiss any strange new concept for a mathematical system. Or more generally, any logical system that follows a set of axioms. Just look at the history of mathematics itself. How many years before people start to accept that yes imaginary numbers are a thing.
Dunno what you're trying to say. Yes, if ZFC is inconsistent it would be an issue, but in the unlikely event this is discovered, it would be overwhelmingly probable that a similar set of axioms could be used in a way which is transparent to the vast majority of mathematics. Incompleteness is more likely and less of an issue.
Then it doesn't work
No, see Gödels Incompleteness theorem
It's very counter intuitive. As the other commenter suggested I was referring to Gödel and his incompleteness theorem.
Actually if the system you made up doesn't work it would be possible to prove that it does inside that system as you can prove anything inside a system that doesn't work.
That is why my comment is not entirely accurate it should actually be: Until you prove that if the system works you can't prove that the system works.
Can you spot the difference in the logic here?
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.
You can tell it's a triangle because of the way it is.
How neat is that??
Yay nostalgia!
You have no idea how much I need it friend. Now...can I pet that dawg??
I depend on it.
I feel that brother. Stay strong
That's pretty neat!
Philomena Cunk proof right there
that won't work for yoda, though. for him, there is no triangle. there's just doangle or donotangle.
How have I not heard this before 😆
I nearly failed geometry because I didn't understand what my instructor wanted from me.
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.
Non-Euclidian horrors beyond comprehension
Commonly known as reality
But isn't a curve just many angles next to one another?
Yes, but infinitely many.
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.
Even philosophy 101 can give you a ton of reasons why looking at it just isn't enough
Especially Philosophy 101
How Can Triangles Be Real If Our Eyes Aren't Real?
"If we can't prove these two triangles are similar while not being congruent, the world is doomed."
"Oh my god. We need a 10th grader with at least a B-average, stat!"
LoL, actual.
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...
In one of my last CS classes, we did proofs and would use "by observation" for this kind of thing.
Equilateral triangle. All three sides are equal length and it has three interior angles that add up to exactly 180 degrees.
Are you telling me that "you can see this is a triangle" ??! You can see ?? How dear you say that!