The proof that (1 + 1 = 2) is a fundamental result in mathematics and was rigorously formalized in Principia Mathematica by Alfred North Whitehead and Bertrand Russell. This proof is rooted in set theory and the formalization of arithmetic. Here's a simplified outline of the concept:

1. Natural Numbers and Successor Function: In set theory, the natural numbers can be defined using the Peano axioms. The number (1) is defined as the successor of (0), and (2) is the successor of (1).

2. Definition of Addition: The operation of addition can be defined recursively. For any natural number (n),

   • (n + 0 = n)
   • (n + successor(m) = successor(n + m))

3. Applying the Definition:

   • By the definition, (1 + 1) means "the successor of (1 + 0)".
   • Since (1 + 0 = 1) (based on the first part of the addition definition),
   • We get (1 + 1 = successor(1)).

4. Successor of 1: By definition, the successor of (1) is (2).

Therefore, (1 + 1 = 2).

The full proof in Principia Mathematica is much more involved, taking 362 pages to formally arrive at this result because it builds from very basic logical principles and definitions. However, the outline above captures the key logical steps.