Could someone explain to me Bertrand Russell's proof that 1+1=2?

Why do people have to get some complicated; if you have one apple and someone else gave you another apple you have two apples.

Seriously… have you read PM?

http://books.google.com/books?id=sbTVAAAAMAAJ&dq=principia%20mathematica&pg=PA357#v=onepage&q=&f=false

It’s obvious! (Okay, seriously… PM makes no sense to me, but that’s because it’s formal logical proofs proving from NOTHING which is way out there).

Apparently I linked to Volume II which doesn’t cover your question, and google books hasn’t scanned a public domain version of Volume 1…

Take one apple. Place it on the tabe. Thats ONE. Take a second apple place it on the table next to the first one. Thats TWO. lol. Im just messing. I have no idea.

It’s gibberish unless you work through the preceding ~300 pages. And I haven’t. ;)

What R&W (as philosophers and logicians) were aiming at with this work as a whole was to show that the underpinnings of arithmetic could be built up from logical principles. Gottlob Frege had tried this with Die Grundlagen der Arithmetik in 1884. Russell, perhaps the only academic in the world at the time who was a fan of Frege, himself pointed out a fatal error in Frege’s system in a letter to him.

Principia Mathematica ultimately failed in this objective. In order to dodge all the paradoxes, R&W created a very complicated system of types (ramified type theory) which did not fold out naturally from basic logical principles. Well into their exhausting work (IIRC Russell said he never fully recovered from the mental strain of its preparation) they saw that they had erected a series of walls that made practicing actual mathematics using their system nearly impossible and they introduced an ad hoc rule that a stroke allowed the walls to be breached. Additionally, future logicians (Alfred Tarski most prominently I think) were to point out the mistakes they had made in not making a distinction between the formal language they were building and the language they used to discuss the system (meta-language).

At the same general time period you also had a number of people attempting to remove paradoxes from Georg Cantor’s set theory. Eventually it was sufficiently patched up by Frankel, Zermelo and Skolem to become ZF set theory. ZF set theory (or ZFC with the addition of the Axiom of Choice) and its relations are generally regarded as satisfactory theories for mathematical foundations. The community of mathematicians using these systems to create formal proofs like R&W did is very small. While mathematicians may have occasional recourse to the axioms of set theory, they overwhelmingly prefer to work in natural language and with much higher level concepts to make proofs shorter and more elegant. Also, since Gödel, there is a recognition that even having such foundational theories in hand, we can never be absolutely certain that they are consistent and will never lead to an error.

An interesting footnote to the whole story is the more recent discovery that a tweak saves Frege’s original system from the known paradoxes (see this article).

I remember vividly reading a short mention of PM and the 1+1=2 proof in some story when I was about 10 years old and thinking at the time it was indeed about the silliest thing in the world.

Martin Davis wrote a wonderful book for the lay reader, which while not super easy, sets up much the events I’ve mentioned in earlier posts in a clear historical context. All of this activity helped to kick start computer science.