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.
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
The practical application isn’t the proof that 1+1=2. That’s just a side-effect. The application was building a framework for proving mathematical statements. At the time the principia were written, Maths wasn’t nearly as grounded in demonstrable facts and reason as it is today. Even though the principia failed (for reasons to be developed some 30 years later), the idea that every proposition should be proven from as few and as simple axioms as possible prevailed.
Now if you’re asking: Why should we prove math? Then the answer is: All of physics.
The answer to the last question is even simpler and broader than that. Math should be proven because all of science should be proven. That is what separates modern science from delusion and self-deception
It lays the foundations for automated proof verification, which is going to help with the development of new theorems as well as automated reasoning about computer programs.
But like… what does the proof even entail? I feel if you asked a child (or maybe me) what the proof was they’d say “well the definition of those two numbers, and the definition of plus means that 1+1=2”. What else is there?
Proving it from the definition is quite easy. The hard part is to build up all the concepts that you need to define the numbers and the operation in the first place. What exactly that entails depends on what axiom system and system of logic you are using. For example, here is the Metamath proof of 1 + 1 = 2, where you can click to see all the axioms, definitions and theorems involved.
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.
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.
We all know that math is just a weirdly specific branch of philosophy.
Physics is just a weirdly specific branch of math
Only recently
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
The practical application isn’t the proof that 1+1=2. That’s just a side-effect. The application was building a framework for proving mathematical statements. At the time the principia were written, Maths wasn’t nearly as grounded in demonstrable facts and reason as it is today. Even though the principia failed (for reasons to be developed some 30 years later), the idea that every proposition should be proven from as few and as simple axioms as possible prevailed.
Now if you’re asking: Why should we prove math? Then the answer is: All of physics.
The answer to the last question is even simpler and broader than that. Math should be proven because all of science should be proven. That is what separates modern science from delusion and self-deception
It lays the foundations for automated proof verification, which is going to help with the development of new theorems as well as automated reasoning about computer programs.
But like… what does the proof even entail? I feel if you asked a child (or maybe me) what the proof was they’d say “well the definition of those two numbers, and the definition of plus means that 1+1=2”. What else is there?
Proving it from the definition is quite easy. The hard part is to build up all the concepts that you need to define the numbers and the operation in the first place. What exactly that entails depends on what axiom system and system of logic you are using. For example, here is the Metamath proof of 1 + 1 = 2, where you can click to see all the axioms, definitions and theorems involved.
I don’t even know where to start with that page. I feel like the curtain has really been pulled back today!
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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.