What happens when the obvious is not so obvious? How do we proceed if what we take for granted actually requires deeper exploration? Mathematics is full of those situations, of those conflicts so to speak. The search for a mathematical proof can become an exhausting... or directly impossible mission. However, the Principia Mathematica of Alfred North Whitehead and Bertrand Russell accepted the challenge at the beginning of the 20th century, and includes a notable example: The difficulty of proving that 1+1=2
How do we learn mathematics? How are we first exposed to that world? Personally, I can say it was a combination of drawings, objects, even songs. In first grade we were taught to use a basic abacus, which became a gateway to understanding large numbers. More advanced operations gave rise to more complex phases of study... and to our elementary hatred of numbers and formulas (!)
During that whole long process, we also learned to accept certain things in mathematics. And I say 'accept', because obtaining proofs is something completely different. With that in mind we arrive at the YouTube channel Half as Interesting, which shared an excellent video focused on Principia Mathematica, a three-volume work on the foundations of mathematics written by two titans, Alfred North Whitehead and Bertrand Russell (yes, the same one as the paradox). However, that work is usually cited for a very particular reason: Its proof that 1+1=2
1+1=2, and other mathematical proofs
The video explains that at the beginning of the 20th century there was a kind of mathematical crisis: In each of its branches, when you need to prove something, you are depending on proofs that already exist. And those proofs depend on other proofs, which in turn depend on others... and I imagine you can already see the problem. Sooner or later you reach the point of assumptions, things that as the video says "feel right" or seem like common sense. Those assumptions are different for each branch of mathematics, and there was no package of "golden rules" to prove their truth.
The objective of Principia Mathematica was to correct that, or rather, to shape a system of mathematics based on pure logic, trying to minimize the number of axioms, primitive notions, and rules of inference. Unfortunately, the work did not fulfill its goal, but it left some very interesting snacks along the way, including the proof that 1+1 equals 2
At first glance, the proof seems written in an alien language. In fact, it includes several references to other sections of the volumes, so an adventurer will have to explore more than 300 pages to understand what it is trying to communicate: This appears on page 378 of the first edition, on page 362 of the second edition, and on page 360 of the "reduced version". The complete proof appears on page 86 of the second volume (1st edition), with the comment that the proposition is "occasionally useful". Principia Mathematica may be a book "of mathematicians, for mathematicians", but we are glad to know that at least they had a good sense of humor...
