Friday, December 5, 2008

2+2=4: a proof

In case you had any doubts, here is a proof that 2+2=4.

Of course, if by "2" we mean "apple" and by "4" we mean "orange", then the statement is false. It should be clear that "2+2=4" has a specific meaning, and if we change any of its meaning, we've changed the statement. Natural numbers, such as 2 or 4, have specific meanings. They are things which obey the Peano axioms. If they don't obey the Peano axioms, they are not really natural numbers, and we might as well be talking about apples and oranges.

The Peano axioms thoroughly logical and simple to state. But I'm not going to cover it in detail, since you can just peruse the Wikipedia article for more.

For every natural number n, the Peano axioms define the "successor of n", or S(n). Every natural number, except zero, is the successor of another natural number. All natural numbers can be expressed this way:

0
S(0)
S(S(0))
S(S(S(0)))
S(S(S(S(0))))
...

We have names for each of these numbers: 0, 1, 2, 3, 4, ...

And so, by "2+2=4", we really mean this:

S(S(0)) + S(S(0)) = S(S(S(S(0))))

Not only do natural numbers have a specific meaning, but the symbol "+" has a specific meaning. It is defined with the following two axioms:

n + 0 = n
n + S(m) = S(n + m)

So here's the rest of the proof:

S(S(0)) + S(S(0)) = S( S(S(0)) + S(0) )
= S( S( S(S(0)) + 0 ) )
= S(S(S(S(0))))

Fairly simple, eh? But, hey, maybe if you find a way to tap into the power of the other 90% of your brain, you will prove the impossible. Either that or your dreams will be crushed and the resulting cynicism will negatively affect the rest of your life.

A harder problem would be to prove that n + m = m + n. I think you might even have to use the axiom of induction for that one.

In other news, I'm taking the Putnam tomorrow! Also, I'm sure this will come as shocking news: I'm going to minor in math! Yay!

2 comments:

intrinsicallyknotted said...

Muahahaha! We've got our claws into you now--soon you'll be changing your major to math!

Good luck on the Putnam, and have fun! I'll have to look at the problems after it's all over (or perhaps you could post a few of them here).

miller said...

I call Peano's rot the Spaghetti axioms.

See link at:

https://groups.google.com/forum/#!topic/sci.math/hBqGwOwoRaY



Also, visit my New Calculus site for the first and only rigorous formulation in the history of humans: http://thenewcalculus.weebly.com