this post was submitted on 20 Oct 2024
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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.
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?
1.xxx and 1.xxy are still 2 numbers, so 1+1=2.
Gottem.
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