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’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!