That assumes that 1 and 1 are the same thing. That they’re units which can be added/aggregated. And when they are that they always equal a singular value. And that value is 2.
It’s obvious but the proof isn’t about stating the obvious. It’s about making clear what are concrete rules in the symbolism/language of math I believe.
This is what happens when the mathematicians spend too much time thinking without any practical applications. Madness!
The idea that something not practical is also not important is very sad to me. I think the least practical thing that humans do is by far the most important: trying to figure out what the fuck all this really means. We do it through art, religion, science, and… you guessed it, pure math. and I should include philosophy, I guess.
I sure wouldn’t want to live in a world without those! Except maybe religion.
Just like they did with that stupid calculus that… checks notes… made possible all of the complex electronics used in technology today. Not having any practical applications currently does not mean it never will
I’d love to see the practical applications of someone taking 360 pages to justify that 1+1=2
Isn’t 1 and +1 well defined by the Peano Axioms by using the intersection of all infinite successor functions and starting at the empty set?
It depends on what you mean by well defined. At a fundamental level, we need to agree on basic definitions in order to communicate. Principia Mathematica aimed to set a formal logical foundation for all of mathematics, so it needed to be as rigid and unambiguous as possible. The proof that 1+1=2 is just slightly more verbose when using their language.