Proving 1+1=2

This is my first philosophy of math post, and I got the ideas from reading Scott Aaronson.

I saw some wiseacre on an internet forum make a reference to this famous page in Russell and Whitehead's Principia Mathematica, and challenging people to prove that 1+1=2. So I thought,

What does it mean to prove that 1+1=2?

Generally when you prove something in formal math, you start by defining axioms, then use logical inference rules to derive the result. If you've learned Geometry, you are probably most familiar with Euclid's axioms for plane geometry.

[Interesting geometry fact: Do you remember learning that when two parallel lines are intersected by a transversal line, pairs of interior angles are congruent? There's no derivation of why that's true -- it's just accepted as an axiom of Euclidean geometry! But it's like the axiomatic black sheep -- much more complicated than Euclid's other four axioms. And it's also what makes the system Euclidean geometry, instead of a different geometry.]

Today, the most common axiom system mathematicians use as a foundation for proofs is called Zermelo-Fraenkel Set Theory. And in ZF set theory, 1+1=2 has a relatively simple proof (although I personally haven't seen it yet)! 1+1 is also easy to prove in a simpler axiom system called Peano Arithmetic.

In the last few centuries, there's been a huge quest by mathematicians to "justify" basic math statements like 1+1=2, in order to "place mathematics on a more secure foundation". That's partly what all this axiom system stuff is about.

Axiom systems are great; in fact, they are an area of concentration for me. But we can make axioms be whatever we want. If you get a kick out of seeing 1+1=3, you can just define a new axiom system where that's the case. You still have to avoid logical contradictions, though, so I hope you weren't too attached to the old semantic meaning of "+" and "=".

So, what is the ultimate secure foundation for 1+1=2? Well, I'm afraid 1+1=2 is part of the foundation itself. Yes, the axioms of set theory prove it, but that's not why anyone is actually inclined to believe that 1+1=2. On the contrary, mathematicians surely listened to their powerful intuition that 1+1=2 when they chose which axioms to accept as part of the "foundation of math".

So where does the 1+1=2 intuition come from? I think you can break it down a little more by seeing 1+1 as the definition of our intuitive notion of two-ness. That intuition, I'm guessing, is a connection we make when the fundamental perceptual notions of repetition or similarity are invoked. Perhaps you are holding a Jolly Rancher, and also a Sweet Tart. They're not identical, but they're both foods, and they're certainly both physical objects with pretty well-defined spatial boundaries. So two is afoot.