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.
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.
In base 2 binary for example the digits are 0 and 1. Counting from 0 up would look like 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, etc
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.
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.
Not a math wizard here: wouldn’t either of the 1s stop being 1s if they were anything but exactly 1.0? And instead become 1.xxx or whatever?
In base 2 binary for example the digits are 0 and 1. Counting from 0 up would look like 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, etc
In that case 1 + 1 = 10
1.xxx and 1.xxy are still 2 numbers, so 1+1=2.
Gottem.